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[
"The GHP scaling limit of uniform spanning trees in high dimensions",
"The GHP scaling limit of uniform spanning trees in high dimensions"
] | [
"Eleanor Archer ",
"Asaf Nachmias ",
"Matan Shalev "
] | [] | [] | We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d-dimensional torus Z d n with d > 4, the hypercube {0, 1} n , and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree. c ) converges in distribution to P ∈ M 1 (X GHP c ) if for any bounded continuous function f : (X c , d GHP ) → R we have lim n E n f = Ef , where E n and E are the expectation operators corresponding to P n and P. As usual, if {X n } and X are random variables taking values in (X c , d GHP ), we say that X n (d) −→ X if the corresponding pushforward measures of X n converge in distribution to that of X.(2)of Claim 4.2. Hence by Lemma 2.5,≤ r ℓ /4 by (2) of Claim 4.2. Hence by Lemma 2.5,Since t + J ≤ λ r ℓ /3 (X) by construction, and W = Γ n , it also follows that (Γ n ∪ Γ x ) \ Γ 5r ℓ /6x ⊂ W ∪ X[t + J + r ℓ /24, M ℓ √ n]. | null | [
"https://arxiv.org/pdf/2112.01203v2.pdf"
] | 244,798,610 | 2112.01203 | fdf45263f7615b592f269eb7a87f9c1e880bcd57 |
The GHP scaling limit of uniform spanning trees in high dimensions
12 Apr 2022 April 14, 2022
Eleanor Archer
Asaf Nachmias
Matan Shalev
The GHP scaling limit of uniform spanning trees in high dimensions
12 Apr 2022 April 14, 2022arXiv:2112.01203v2 [math.PR]
We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d-dimensional torus Z d n with d > 4, the hypercube {0, 1} n , and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree. c ) converges in distribution to P ∈ M 1 (X GHP c ) if for any bounded continuous function f : (X c , d GHP ) → R we have lim n E n f = Ef , where E n and E are the expectation operators corresponding to P n and P. As usual, if {X n } and X are random variables taking values in (X c , d GHP ), we say that X n (d) −→ X if the corresponding pushforward measures of X n converge in distribution to that of X.(2)of Claim 4.2. Hence by Lemma 2.5,≤ r ℓ /4 by (2) of Claim 4.2. Hence by Lemma 2.5,Since t + J ≤ λ r ℓ /3 (X) by construction, and W = Γ n , it also follows that (Γ n ∪ Γ x ) \ Γ 5r ℓ /6x ⊂ W ∪ X[t + J + r ℓ /24, M ℓ √ n].
Introduction
Consider the uniform spanning tree (UST) of the d-dimensional torus Z d n with d > 4 or another transitive high-dimensional graph such as the hypercube {0, 1} n or a transitive expander graph. In this paper we show that the Brownian continuum random tree (CRT), introduced by Aldous [1,2], is the Gromov-Hausdorff-Prohorov (GHP) scaling limit of such USTs.
Convergence of such USTs to the CRT in the sense of finite dimensional distributions has been established in the work of Peres and Revelle [28]. The novelty of the current paper is proving that this convergence holds in the stronger GHP topology. This implies the convergence in distribution of some natural geometric quantities of the USTs (which were not known to converge prior to this work) and allows us to express their limiting distribution explicitly. For example, it follows from our work that the diameter and the height seen from a random vertex of these USTs, properly rescaled, converge to certain functionals of the Brownian excursion, as predicted by Aldous (see [2,Section 4]). Additionally, it implies that the simple random walk on these USTs converges to Brownian motion on the CRT. We discuss these implications in Section 1.3.
Our main result is as follows.
Theorem 1.1. Let T n be a uniformly drawn spanning tree of the d-dimensional torus Z d n with d > 4. Denote by d Tn the corresponding graph-distance on T n and by µ n the uniform probability measure on the vertices of T n . Then there exists a constant β(d) > 0 such that
T n , d Tn β(d)n d/2 , µ n (d) −→ (T , d T , µ)(1)
where (T , d T , µ) is the CRT equipped with its canonical mass measure µ and
(d)
−→ means convergence in distribution with respect to the GHP distance between metric measure spaces. Remark 1.2. We take the convention of Aldous [2, Section 2] that the CRT is coded by two times standard Brownian excursion, although different normalizations are sometimes used elsewhere in the literature.
Our result shows that high-dimensional USTs exhibit a strong form of universality, a common phenomenon in statistical physics whereby above an upper critical dimension, the macroscopic behaviour of a system does not depend on the finer properties of the underlying network. For USTs the upper critical dimension is well-known to be four as for the closely related model of loop-erased random walk (LERW). Above dimension four LERW rescales to Brownian motion, see [19]. In lower dimensions the scaling limits are markedly different. On Z 2 it was shown by Lawler, Schramm and Werner [20] that LERW rescales to SLE 2 , and Barlow, Croydon and Kumagai [7] later established subsequential GHP scaling limits for the UST. This was later extended to full convergence in a result of Holden and Sun [13]. On Z 3 , much less is known, however the breakthrough works of Kozma [16] and Li and Shiraishi [23] on subsequential scaling limits of LERW enabled Angel, Croydon, Hernandez-Torres and Shiraishi [4] to show GHP convergence of the rescaled UST along a dyadic subsequence. Their scaling factors are given in terms of the LERW growth exponent in three dimensions, which was shown to exist by Shiraishi [31]. Finally, in four dimensions, a classical result of Lawler [18] computes the logarithmic correction to scaling under which the LERW on Z 4 converges to Brownian motion. Schweinsberg [30] showed that with these logarithmic corrections to scaling, the finite-dimensional distributions of the UST on the four dimensional torus converge to those of the CRT, analogously to [28]. Various exponents governing the shape of the UST in Z 4 are given in the recent work of Hutchcroft and Sousi [15]. Our proof of GHP convergence does not encompass the four dimensional torus (see Problem 7.3).
In the rest of this section we first present the standard notation and definitions required to parse Theorem 1.1. We then state the most general version of our result, Theorem 1.5, handling other high-dimensional underlying graphs such as expanders and the hypercube. We close this section with a discussion of the various corollaries mentioned above and the organization of the paper.
Standard notation and definitions
A spanning tree of a connected finite graph G is a connected subset of edges touching every vertex and containing no cycles. The uniform spanning tree (UST) is a uniformly drawn sample from this finite set. Given a tree T we denote by d T the graph distance metric on the vertices of T , i.e., d T (u, v) is the number of edges in the unique path between u and v in T .
We follow the setup of [26,Sections 1.3 and 6] and work in the space X c of equivalence classes of (deterministic) metric measure spaces (mm-spaces) (X, d, µ) such that (X, d) is a compact metric space and µ is a Borel probability measure on (X, d), where we treat (X, d, µ) and (X ′ , d ′ , µ ′ ) as equivalent if there exists a bijective isometry φ : X → X ′ such that φ * µ = µ ′ where φ * µ is the pushforward measure of µ under φ. As is standard in the field, we will abuse notation and represent an equivalence class in X c by a single element of that equivalence class.
We will now define the GHP metric on X c . First recall that if (X, d) is a metric space, the Hausdorff distance d H between two sets A, A ′ ⊂ X is defined as Furthermore, for ε > 0 and A ⊂ X we let A ε = {x ∈ X : d(x, A) < ε} be the ε-fattening of A in X. If µ and ν are two measures on X, the Prohorov distance between µ and ν is given by d P (µ, ν) = inf{ε > 0 : µ(A) ≤ ν(A ε ) + ε and ν(A) ≤ µ(A ε ) + ε for any closed set A ⊂ X}. Definition 1.3. Let (X, d, µ) and (X ′ , d ′ , µ ′ ) be elements of X c . The Gromov-Hausdorff-Prohorov (GHP) distance between (X, d, µ) and (X ′ , d ′ , µ ′ ) is defined as
d GHP ((X, d, µ), (X ′ , d ′ , µ ′ )) = inf{d H (φ(X), φ ′ (X ′ )) ∨ d P (φ * µ, φ ′ * µ ′ )},
where the infimum is taken over all isometric embeddings φ : X → F , φ ′ : X ′ → F into some common metric space F .
It is shown in [26,Theorem 6 and Proposition 8] that (X c , d GHP ) is a Polish metric space. Denote by M 1 (X GHP c ) the set of probability measures on (X c , d GHP ) with the Borel σ-algebra. We say that a sequence of probability measures {P n } ∞ n=1 ⊂ M 1 (X GHP The CRT is a typical example of a random fractal tree and can be thought of as the scaling limit of critical (finite variance) Galton-Watson trees. As we shall explain in Section 3, we do not directly approach the CRT in this paper; therefore we have opted to omit the definition of the CRT and refer the reader to Le Gall's comprehensive survey [21] for its construction (see also [2]) as a random element in X c . Except for this, by now we have stated all the necessary definitions required for Theorem 1.1.
The general theorem
We now present the general version of Theorem 1.1 which will imply the GHP convergence of the UST on graphs like the hypercube {0, 1} m or transitive expanders. Our assumptions on the underlying graph are stated in terms of random walk behavior but should be thought of as geometric assumptions. For a graph G, two vertices x, y and a non-negative integer t we write p t (x, y) for the probability that the lazy random walk starting at x will be at y at time t. When G is a finite connected regular graph on n vertices we define the uniform mixing time of G as t mix (G) = min t ≥ 0 : max
x,y∈G |np t (x, y) − 1| ≤ 1 2 ,(2)
We will assume the following throughout the paper. This is the same assumption under which Peres and Revelle establish finite-dimensional convergence in [28].
Assumption 1.4. Let {G n } be a sequence of finite connected vertex transitive graphs with |G n | = n.
1. There exists θ < ∞ such that sup
n sup x∈Gn √ n t=0 (t + 1)p t (x, x) ≤ θ; 2. There exists α > 0 such that t mix (G n ) = o(n 1 2 −α ) as n → ∞.
Both items in Assumption 1.4 imply that the graph sequence is in some sense of dimension greater than four. The first item is a finite analogue of the condition that the expected number of intersections of two independent random walks is finite; in Z d this happens if and only if d > 4. The second item (which clearly holds on the torus on n vertices once d > 4, since this has mixing time of order n 2/d ) heuristically ensures that different parts of the UST that are distance √ n apart behave asymptotically independently. We do not claim that these conditions are optimal (see the discussion in [25,Section 1.4]), but they are enough to yield convergence to the CRT in the most interesting cases. Theorem 1.5. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let T n be a sample of UST(G n ). Denote by d Tn the graph distance on T n and by µ n the uniform probability measure on the vertices of T n . Then there exists a sequence {β n } satisfying 0 < inf n β n ≤ sup n β n < ∞ such that
T n , d Tn β n √ n , µ n (d) −→ (T , d T , µ)
where (T , d T , µ) is the CRT equipped with its canonical mass measure µ and
(d)
−→ means convergence in distribution with respect to the GHP distance.
The sequence {β n } is inherited from the main result of Peres and Revelle, see [28, Theorem 1.2] (we restate this as Theorem 3.1 in this paper). Note that Theorem 1.1 is not a special case of Theorem 1.5 since the latter does not guarantee a single scaling factor β, rather a sequence β n (which is the best one can hope for in the context of Theorem 1.5 since one can alternate between different graph sequences).
Proof of Theorem 1.1 given Theorem 1.5. For the torus Z d n with d ≥ 5, Peres and Revelle proved that there exists β(d) ∈ (0, ∞) such that [28,Theorem 1.2] holds with β n = β(d); see the choice of β n at the end of Section 3 of [28] as well as Lemma 8.1 and (17) in that paper. Hence, this and Theorem 1.5 readily imply Theorem 1.1.
Furthermore, see Lemma 1.3 and Section 9 of [28], in graphs where additionally two independent simple random walks typically avoid one another for long enough (see the precise condition in [28,Equation 6]), we can take β n ≡ 1. This family of graphs includes the hypercube and transitive expanders with degrees tending to infinity. In the same spirit, for the d-dimensional torus, β(d) → 1 as d → ∞. Moreover, it is also immediate to see that Assumption 1.4 holds for a sequence of bounded degree transitive expanders (see for instance [28,Section 9]) and hence Theorem 1.5 holds for them as well.
Corollaries
Pointed convergence
In order to establish some of the corollaries alluded to above, it will be useful to rephrase Theorem 1.5 in terms of pointed convergence. Roughly speaking, this means that we consider our spaces to be rooted, and we add a term corresponding to the distance between the roots in the embedding in Definition 1.3. We refer to [10,Section 2.2] for the precise definition. We start with the following observation, which is a trivial consequence of a coupling characterization of the Prohorov distance (see [26, Proof of Proposition 6]). Lemma 1.6. Suppose that (X n , d n , µ n ) n≥1 , (X, d, µ) are in X c with (X n , d n , µ n ) → (X, d, µ) deterministically in the GHP topology. Let U n be a random element of X n sampled according to the measure µ n , and U be a random element of X sampled according to the measure µ. Then
(X n , d n , µ n , U n ) (d) → (X, d, µ, U )
with respect to the pointed GHP topology, as defined in [10, Section 2.2].
Due to transitivity, in our setting the root can be an arbitrary vertex O n rather than uniformly chosen. Combining Theorem 1.5 with Lemma 1.6 and Skorohod representation theorem we deduce the following. Theorem 1.7 (Pointed convergence). Let {G n } be a sequence of graphs satisfying Assumption 1.4, let T n be a sample of UST(G n ) and let O n be an arbitrary vertex of G n . Denote by d Tn the graph distance on T n and by µ n the uniform probability measure on the vertices of T n . Then there exists a sequence {β n } satisfying 0 < inf n β n ≤ sup n β n < ∞ such that
T n , d Tn β n √ n , µ n , O n (d) −→ (T , d, µ, O)
where (T , d T , µ, O) is the CRT equipped with its canonical mass measure µ and root O, and
(d)
−→ means convergence in distribution with respect to the pointed GHP distance defined in [10, Section 2.2].
Diameter distribution
The diameter of a metric space (X, d) is sup x,y∈X d(x, y) and denoted by Diam(X). When X is a tree, it is just the length of the longest path. The study of the diameter of random trees has an interesting history. Szekeres [33] proved in 1982 that the diameter D n of a uniformly drawn labeled tree on n vertices normalized by n −1/2 converges in distribution to a random variable D with the rather unpleasant density
f D (y) = √ 2π
3 n≥1 e −bn,y 64 y 2 (4b 4 n,y − 36b 3 n,y + 75b 2 n,y − 30b n,y ) + 16
y 2 (2b 3 n,y − 5b 2 n,y ) ,(3)
where b n,y = 8(πn/y) 2 and y ∈ (0, ∞). Aldous [1,2] showed that this tree, viewed as a random metric space, converges to the CRT and deduced that D is distributed as
2 · sup 0≤t1<t2≤1 e t1 + e t2 − 2 inf t1≤t≤t2 e t ,(4)
where {e t } t∈[0,1] is standard Brownian excursion. Curiously enough, up until 2015 the only known way to show that (4) has density (3) was to go via random trees and combine the Aldous and Szekeres results. Wang [34], prompted by a question of Aldous, gave a direct proof of this fact in 2015.
A uniformly drawn labeled tree on n vertices is just UST(K n ) where K n is the complete graph on n vertices. Applying Theorem 1.5 we are able to extend Szekeres' 1983 result to USTs of any sequence of graphs satisfying Assumption 1.4. Corollary 1.8. Let {G n } be a sequence of graphs satisfying Assumption 1.4, let T n be a sample of UST(G n ) and let {β n } be the sequence guaranteed to exist by Theorem 1.5. Then
Diam(T n ) β n n 1/2 (d) −→ D ,
where D is the diameter of the CRT, i.e., a random variable defined by either (3) or (4).
Proof. Let D n = Diam(T n ) and let g : [0, ∞) → R be bounded and continuous. The function h : X c → R defined by h((X, d, µ)) = Diam(X) is continuous with respect to the GHP topology; indeed, for any two metric spaces X 1 and X 2 we have | Diam(X 1 ) − Diam(X 2 )| ≤ 2d GHP (X 1 , X 2 ). Thus the composition g • h : X c → R is bounded and continuous. By Theorem 1.
5 we conclude E g•h((T n , dT βn √ n , µ n ) → E[g•h((T , d, µ))] where (T , d, µ) is the CRT. Therefore, E[g(D n )] → E[g(D)
] as required.
Height distribution
Given a rooted tree (T, v), the height of (T, v) is sup x∈T d(v, x), i.e. the length of the longest simple path in T starting from v, and denoted by Height(T, v). The study of the height of random trees predates the study of the diameter. In 1967, Rényi and Szekeres [29] found the limiting distribution of the height of a uniformly drawn labeled rooted tree on n vertices normalized by n −1/2 ; we omit the precise formula this time (it is also unpleasant). Aldous [1,2] realized that the limiting distribution is that of the maximum of the Brownian excursion.
The following corollary is an immediate consequence of Theorem 1.7. The proof goes along the same lines as the proof of Corollary 1.8; we omit the details. Corollary 1.9. Let {G n } be a sequence of graphs satisfying Assumption 1.4, let T n be a sample of UST(G n ) and let β n be the sequence guaranteed to exist by Theorem 1.5. Let v n be an arbitrary vertex of G n . Then
Height(T n , v n ) β n n 1/2 (d) −→ 2 sup t∈[0,1] e t ,
where {e t } t∈[0,1] is standard Brownian excursion.
SRW on the UST converges to BM on the CRT
A particularly nice application of Theorem 1.5 together with [10, Theorem 1.2] allows us to deduce that the simple random walk (SRW) on UST(G n ) rescales to Brownian motion on the CRT. The latter object was first defined by Aldous in [2, Section 5.2] and formally constructed by Krebs [17]. In what follows, we let P (On) n (·) denote the (random) law of a discrete-time SRW on UST(G n ), started from O n , and let P (O) (·) denote the law of Brownian motion on the CRT as constructed by Krebs, started from O. Theorem 1.10. Let {G n } be a sequence of graphs satisfying Assumption 1.4, let T n be a sample of UST(G n ), and let (X n (m)) m≥0 be a simple random walk on T n . Then there exists a probability space Ω on which the convergence of Theorem 1.7 holds almost surely, and furthermore, on this probability space, for almost every ω ∈ Ω the spaces ((T n , d n , µ n , O n )) n≥1 and (T , d, µ, O) can be embedded into a common metric space (X ′ , d ′ )(ω) so that
P (On) n 1 β n √ n X n (2β n n 3 2 t) t≥0 ∈ · → P (O) ((B t ) t≥0 ∈ ·)(5)
weakly as probability measures on the space D(R ≥0 , X ′ (ω)) of càdlàg functions equipped with the uniform topology.
Proof. The existence of such a probability space Ω follows from the Skorohod representation theorem since the space of pointed compact mm-spaces endowed with a finite measure is separable by [
Firstly, if ν n (x) = deg x, then d GHP (T n , dT n βn √ n , µ n ), (T n , dT n βn √ n , 1 2n ν n ) ≤ 1 βn √ n , so that T n , d Tn β n √ n , 1 2n ν n (d) −→ (T , d, µ)
with respect to the GHP distance as a consequence of Theorem 1.5 and the triangle inequality. It therefore follows from [10, Theorem 1.2] that if (Y n (t)) t≥0 is a continuous time SRW on G n with an exp(1) holding time at each vertex, then
1 β n √ n Y n (2β n n 3 2 t) t≥0 (d) −→ (B t ) t≥0(6)
as n → ∞, almost surely on Ω. This result then transfers to the SRW sequence (X n (·)) n≥1 in place of (Y n (·)) n≥1 by standard arguments using the strong law of large numbers and continuity of the limit process. We refer to [5,Section 4.2] for an example of such an argument.
Organization
We begin with some preliminaries in Section 2 where we introduce the standard definitions of loop-erased random walk, mixing time and capacity which are central to the proof. We also record some stochastic domination properties of USTs, and prove there a general result regarding negative correlations of certain expected volumes in the UST (see Claim 2.12).
Next in Section 3 we present the main argument of the proof, while delegating two useful estimates, Theorem 3.3 and Theorem 3.6, to Section 4, and a third useful estimate, Lemma 3.7, to Section 5. In Section 6 we present a necessary though rather straightforward abstract argument combining the result of Section 3 with the results of [28] to yield Theorem 1.5. Lastly, in Section 7 we present some concluding remarks and open questions.
Acknowledgments
We thank Christina Goldschmidt for many useful discussions. This research is supported by ERC starting grant 676970 RANDGEOM, consolidator grant 101001124 UniversalMap, and by ISF grant 1294/19.
Preliminaries
In this section we provide an overview of the tools used to prove Theorem 1.5. Throughout the section, we assume that G = (V, E) is a finite connected graph with n vertices. We will use the following conventions:
• For an integer m ≥ 1 we write [m] = {1, . . . , m}.
• For two positive sequences t(n), r(n) we write t ∼ r when t(n)/r(n) → 1.
• For two positive sequences t(n), r(n) we write t ≫ r when t(n)/r(n) → ∞.
• We omit floor and ceiling signs when they are necessary.
• Through the rest of this paper, the random walk on a graph equipped with positive edge weights is the random walk that stays put with with probability 1/2 and otherwise jumps to a random neighbor with probability proportional to the weight of the corresponding edge. If no edge weights are specified, then they are all unit weights.
Loop-erased random walk and Wilson's algorithm
Wilson's algorithm [35], which we now describe, is a widely used algorithm for sampling USTs. A walk X = (X 0 , . . . X L ) of length L ∈ N is a sequence of vertices where (X i , X i+1 ) ∈ E(G) for every 0 ≤ i ≤ L − 1.
For
an interval J = [a, b] ⊂ [0, L] where a, b are integers, we write X[J] for {X i } b i=a .
Given a walk, we define its loop erasure Y = LE(X) = LE(X[0, L]) inductively as follows. We set Y 0 = X 0 and let λ 0 = 0. Then, for every i ≥ 1, we set λ i = 1 + max{t | X t = Y λi−1 } and if λ i ≤ L we set Y i = X λi . We halt this process once we have λ i > L. The times λ k (X)
| LE(X)|−1 k=0
are the times contributing to the loop-erasure of the walk X. When X is a random walk starting at some vertex v ∈ G and terminated when hitting another vertex u (L is now random), we say that LE(X) is the loop erased random walk (LERW) from v to u.
To sample a UST of a finite connected graph G we begin by fixing an ordering of the vertices of V = (v 1 , . . . , v n ). At the first step, let T 1 be the tree containing v 1 and no edges. At each step i > 1, sample a LERW from v i to T i−1 and set T i to be the union of T i−1 and the LERW that has just been sampled. We terminate this algorithm with T n . Wilson [35] proved that T n is distributed as UST(G). An immediate consequence is that the path between any two vertices in UST(G) is distributed as a LERW between those two vertices. This was first shown by Pemantle [27].
To understand the lengths of loops erased in LERW we will need the notion of the bubble sum. Let G be a graph and let W be a non empty subset of vertices of G. For every two vertices u, w ∈ V (G), define
p t W (u, w) = P u (X t = w, X[0, t] ∩ W = ∅) ,
where X is a random walk on G. We define the W -bubble sum by
B W (G) := ∞ t=0 (t + 1) sup v∈V p t W (v, v).
Note that since the random walk on G is an irreducible Markov chain on a finite state space, we have that P(X[0, t] ∩ W = ∅) decays exponentially in t and hence this sum is always finite. Another bubble-sum we will consider is when the random walk is killed at a geometric time (rather than when hitting a set W ). Let T ζ be an independent geometric random variable with mean ζ > 1. We define
p t ζ (u, w) = P u (X t = w, T ζ > t), B ζ (G) := ∞ t=0 (t + 1) sup v∈V p t ζ (v, v).
Definition 2.1. We say that a random walk X on a finite connected graph G starting from an arbitrary vertex is bubble-terminated with bubble-sum bounded by ψ if it is killed upon hitting some set W and B W (G) ≤ ψ, or alternatively, if it is killed at time T ζ − 1 and B ζ (G) ≤ ψ.
Both bubble-sums allow us to bound the size of the loops erased in the loop-erasure process. As in [14] and [25, Claim 3.2] we have the following.
Claim 2.2. Let G be a finite connected graph and X be a bubble-terminated random walk on G with bubblesum bounded by ψ. For any finite simple path γ such that P(LE(X) = γ) > 0 of length L we have that the random variables
{λ i+1 (X) − λ i (X)} L−1 i=0
are independent conditionally on {LE(X) = γ} and furthermore
E[λ i+1 (X) − λ i (X)| LE(X) = γ] ≤ ψ ,for all 0 ≤ i ≤ L − 1.
Proof. In the case that X is killed upon hitting W , see [ [25], thus the expected difference between two consecutive times has the same bound, and the proof is the same.)
When X is killed at T ζ − 1 where T ζ is an independent geometric random variable with mean ζ > 1, the proof can be deduced from the previous claim. Indeed, we add a new vertex ρ to G and edges (ρ, u) for every u ∈ G with weights on them so that the probability to visit ρ from every vertex in a single step is equal to 1/ζ for any u ∈ G. Call the resulting network G * . A random walk on G * started from v ∈ G and terminated when hitting ρ has the same distribution as a random walk on G with geometric killing time.
Mixing times
Recall the definition of the uniform mixing time above Assumption 1.4. It follows that for every t ≥ t mix we have that 1 2n
≤ P u (X t = v) ≤ 2 n ,(7)
where X t is the random walk. Even though in this paper we mainly use the uniform mixing time as defined in (2) we also use a more classical version of distance between probability measures on finite sets. Recall that the total variation distance between two probability measures on µ and ν on a finite set X is defined by
d TV (µ, ν) = max A⊂X |µ(A) − ν(A)| .
It is a standard fact (see [22,Section 4.5]) that if t ≥ kt mix , then for any vertex x
d TV (p t (x, ·), π(·)) ≤ 2 −k .(8)
Capacity
The capacity of a set of vertices quantifies how difficult it is for a random walk to hit the set. It is a crucial notion when one wishes to analyze the behavior of Wilson's algorithm. Let {Y i } i≥0 be a random walk on G and for
U ⊂ V (G), let τ U = inf{i ≥ 0 : Y i ∈ U }. Given k ≥ 0 we define the k-capacity of U by Cap k (U ) = P π (τ U ≤ k). If W ⊂ V (G)
is another subset of vertices we define the relative k-capacity Cap k (W, U ) = P π (τ W ≤ k, τ W ≤ τ U ). Note that the relative capacity is not symmetric in W, U .
We will see later that the capacities of certain subsets determine the expected volumes of balls in UST(G). Here we collect some useful facts about the capacity. By the union bound, when G is regular we always have the upper bound
Cap k (V ) ≤ kπ(V ) = k|V | n .(9)
The capacity is defined for the lazy simple random started at stationarity. When k is significantly larger than the mixing time, the starting vertex does not make much difference as the following claim shows.
Claim 2.3. Let G be a connected regular graph. Let u ∈ V , let U ⊆ V be nonempty, let r = r(n) ≫ log(n) · t mix (G)
and assume that t = t(n) is a sequence so that t(n) ∼ r(n). Then, for large enough n,
P u (τ U < t) ≥ 1 3 Cap r (U ).
Proof. See [25,Claim 1.4].
We will also use the following lemma.
Lemma 2.4. Let G be a connected regular graph. Let W ⊂ U be subsets of vertices and k, s, m ≥ 0. Assume that
Cap k (W, U \ W ) ≥ s .
Then we can find at least L = ⌊s
/(m + k/n)⌋ disjoint subsets A 1 , . . . , A L of W such that m ≤ Cap k (A j , U \ A j ) ≤ m + k n ,
for all j = 1, . . . , L.
Proof. We first observe that if A is a subset of W and v ∈ W \ A then by (9) we have that
Cap k (A ∪ {v}, U \ A ∪ {v}) ≤ Cap k (A, U \ A) + k n .
Secondly, we observe that if
A 1 , . . . , A L ′ are disjoint sets so that ∪ L ′ j=1 A j = W , then Cap k (W, U \ W ) = L ′ j=1 Cap k (A j , U \ A j ) .
With these two observations in place, we now perform an iterative construction of the subsets. We add vertices from W to A 1 until the first time that In order to obtain useful lower bounds on the capacity, we state a well-known relationship between the capacity of a set A and the Green kernel summed over A. Given a set A ⊂ G and k ∈ N we define
Cap k (A 1 , U \ A 1 ) ≥ m. By the first observation we have that Cap k (A 1 , U \ A 1 ) ≤ m + k/n. Then we add vertices from W \ A 1 to A 2 until the first time that Cap k (A 2 , U \ A 2 ) ≥ mM (k) (A) = x,y∈A G (k) (x, y),(10)
where
G (k) (x, y) = E x k i=0 ½{X i = y}
. This is useful due to the following characterization of capacity.
Lemma 2.5. Let G be a connected regular graph. For all A ⊂ G,
Cap k (A) ≥ k|A| 2 2nM (k) (A) .
Proof. The proof is the same as that of [9, Theorem 2.2], but instead considering a stationary starting point distributed according to π, noting that G (k) (π, x) = k n for all x ∈ G n by transitivity, and specifically using the measure µ(x) = ½{x∈A} |A| .
The following bound on E M (k) (P ) where P is a random walk path will be useful.
Lemma 2.6. Let G be a connected regular graph with n vertices. Let m and k be two positive integers and let P be a random walk path of length m started at v ∈ V (G). Then
E M (k) (P ) ≤ 2m m+k t=0 (t + 1)p t (v, v) .
Proof. The proof goes by the same argument as in [14,Lemma 5.6].
Furthermore, in order to lower bound the relative capacity, we define the k-closeness of two sets U and W by Close k (U, W ) = P π (τ U < k, τ W < k) .
It follows from [28, Lemma 5.2] together with (9) that on any finite connected regular graph G, if W = X[0, T ] where X is a random walk on G started at stationarity, and T is a stopping time, then for any set U ⊂ G,
E[Close k (U, W )] ≤ 4E[T ]kCap k (U ) n ≤ 4k 2 |U |ET n 2 .(12)
Lastly, recall the two bubble sums defined in Section 2.1. One of the uses of the capacity is to bound such bubble sums.
Claim 2.7. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let W ⊂ G n be a set of vertices such that Cap √ n (W ) ≥ c, then
B W (G) ≤ θ + 4 c 2 .
Proof. This follows by exactly the same proof as in [25, Claim 3.14].
Claim 2.8. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let ζ > 0 be given. Then
B ζ −1 n 1/2 (G n ) ≤ θ + 2ζ −2 .
Proof. Take any v ∈ G n . Then, similarly to [25, Claim 3.14], since ∞ t=0 (t + 1)(1 − x) t = x −2 and using (7),
B ζ −1 n 1/2 (G n ) = ∞ t=0 (t + 1)p t (v, v) 1 − ζn −1/2 t ≤ θ + 2 n ∞ t= √ n (t + 1) 1 − ζn −1/2 t ≤ θ + 2ζ −2 .
Stochastic domination properties
The UST enjoys the negative correlation property, i.e., the probability that an edge e is such that e ∈ UST(G) conditioned on f ∈ UST(G) for some other edge f is no more than the unconditional probability. Moreover, Feder and Mihail showed that for every increasing event A that ignores f , the probability of A given f ∈ UST(G) is no more than the unconditional probability. This led to the following result. The same proof leads to a slightly more generalized version.
Lemma 2.10. Let (G, w) be a weighted network and suppose that (H, w ′ ) is a network such that V (G) ⊆ V (H) and that for every edge
(v, u) with w((v, u)) = 0 we have w((v, u)) = w ′ ((v, u)). Then, UST(G) stochastically dominates UST(H) ∩ E(G).
Later in the paper, we will apply Lemma 2.10 in the following context. To study UST(G) using Wilson's algorithm, it will sometimes be convenient to add an extra vertex to G called the sun, and for every vertex v ∈ G add an extra edge from v to the sun. We give well-chosen weights to these new edges and call the new graph the sunny graph. Lemma 2.10 tells us that the UST of the sunny graph, intersected with E(G), is stochastically dominated by UST(G). This idea was previously used in [35] and [28].
We will also make use of the following well-known lemma. Here G/A denotes the graph obtained from G by identifying all vertices in A with a single vertex. Lastly, let W be a set of vertices, and let A 1 and A 2 be disjoint subsets of W . In what follows we consider UST(G/W ). Given an integer k and j ∈ {1, 2}, let I j (k) denote the vertices of G that are connected to W in UST(G/W ) by a path of length k such that the last edge on the path to W is an edge that one of its original endpoints belonged to A j (including A j itself). Also, let X j = X j (k) = |I j (k)|. Claim 2.12. Let G be a finite connected graph, take any k ≥ 1 and let W, A 1 , A 2 be as above. Then, for UST(G/W ) and for every M > 0,
E[X 2 |X 1 ≤ M ] ≥ E[X 2 ].
Proof. We will first show that for every v ∈ G, the events {X 1 > M } and {v ∈ I 2 } are negatively correlated. Fix some v ∈ G such that v ∈ I 2 has positive probability. Condition on v ∈ I 2 and on γ 2 , the path from v to A 2 . The UST conditioned on W and γ 2 has the distribution of UST(G/(W ∪ γ 2 )). Hence, by Lemma 2.11 we have that UST(G/(W ∪ γ 2 )) is dominated by UST(G/W ). Therefore, by Strassen's theorem [32], there exists a coupling of the two measures such that UST(G/(W ∪ γ 2 )) ⊆ UST(G/W ). This means that every vertex connected to W through A 1 by a path of length at most k in UST(G/(W ∪ γ 2 )) will also be connected by the same path to A 1 in UST(G/W ). Therefore, we have that
P(X 1 > M | v ∈ I 2 (k), γ 2 ) ≤ P(X 1 > M ).
Then by averaging over γ 2 and taking complements we obtain
P(X 1 ≤ M | v ∈ I 2 (k)) ≥ P(X 1 ≤ M ).
Therefore, inverting using Bayes' rule, we have for every v with P(v ∈ I 2 (k)) > 0 that
P(v ∈ I 2 (k) | X 1 ≤ M ) ≥ P(v ∈ I 2 (k)).
Summing over v yields the result.
The lower mass bound
The starting point of the proof of Theorem 1.5 is the work of Peres and Revelle [28].
d Tn (x i , x j ) β n √ n
converge jointly in distribution to the k 2 distances in T between k i.i.d. points drawn according to µ.
For the proof of Theorem 1.5 we take the same sequence β n guaranteed to exist by the theorem above. As we shall see in Section 6, the convergence of Theorem 3.1 is equivalent to what is known as Gromov-weak convergence, which does not imply GHP convergence. In order to close this gap in this abstract theory, Athreya, Löhr and Winter [6, Theorem 6.1] introduced the lower mass bound condition and proved that this condition together with Gromov-weak convergence is in fact equivalent to GHP convergence; we discuss this further in Section 6. The main effort in this paper is proving that the lower mass bound holds under Assumption 1.4; this is the content of the following theorem.
Theorem 3.2. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let T n be UST(G n ). For a vertex v ∈ T n and some r ≥ 0 we write B Tn (v, r) = {u : d Tn (v, u) ≤ r} where d Tn is the intrinsic graph distance metric on T n . Then for any c > 0 and any δ > 0 there exists ε > 0 such that for all n ≥ 1,
P ∃v ∈ T n : |B Tn (v, c √ n)| ≤ εn ≤ δ.
In other words, the random variables
max v {n|B Tn (v, c √ n)| −1 } n are tight.
In the rest of this section we prove Theorem 3.2, delegating parts of the proof to Section 4 and Section 5. For the rest of this section as well as Section 4, {G n } is a sequence of graphs satisfying Assumption 1.4 and T n is UST(G n ).
Bootstrap argument
The main difficulty in Theorem 3.2 is that it is global; that is, it requires a lower tail bound on the volumes of the balls around all vertices simultaneously. Our approach is to prove a strong enough local version of this bound, that is, a bound for a single vertex, and use a bootstrap argument to obtain a weaker (yet sufficient) global bound. The idea is to use the observation that if there is one vertex x ∈ T n such that |B(x, r)| is small, then either |B(x, r 2 )| is also small, or otherwise there are many vertices v ∈ B(x, r 2 ) such that |B(v, r 2 )| is small. Provided that these two latter events are sufficiently less likely than the former, this allows us to define a sequence of events on the balls of dyadic radii |B(x, r 2 ℓ )|, each with strictly stronger tail decay than the previous one. We will iterate this observation enough times until the probability of the final event is o 1 n , at which point we will apply the union bound and conclude the proof. Thus, our goal will be to iteratively improve the bounds on
P B x, c √ n 2 ℓ ≤ ε 4 ℓ c √ n 2 ℓ 2 ,(13)
where x is a fixed vertex (our graphs are transitive so the choice of x does not matter) and ℓ = 0, . . . , N n where N n , the number of iterations, will be chosen suitably as we now explain.
Since we will use Wilson's algorithm to sample branches in UST(G n ), it will be important in our arguments in Section 4 that the radius c √ n 2 l we consider at each step is significantly longer than the mixing time of a random walk on G n . Therefore, we require that c √ n 2 Nn ≫ n 1 2 −α (recall the constant α from Assumption 1.4), so the number of iterations N n can be at most of order log n. We will see in the proof of Theorem 3.2 that for this bootstrap argument to work with only log n steps, it will be convenient to obtain bounds on (13) that are sub-polynomial in ε.
A natural strategy to bound the probability in (13) is to first sample a single branch joining x to a predefined root of UST(G n ), consider the volumes of balls in subtrees attached to this branch close to x, and show that the sum of these volumes is very unlikely to be too small. This strategy almost gives sufficiently strong tail decay, but there is one step at which the tail decay is not sub-polynomial. This problem arises in the first step since there is a probability of order ε that the path joining x to a root vertex is of length less than √ εn.
This is not a fundamental problem since if this path is short, then it means we just picked a short branch when longer branches to different roots were available. However, it is not convenient to condition on picking a long branch to a well-chosen root since this conditioning reveals too much information about UST(G n ), which makes it difficult to control other properties of the branch, primarily its capacity and the capacity of its subsets. It is also inconvenient (though probably possible) to continue choosing a few more branches until we reach a certain length.
The simplest way we found to circumvent this issue is to first sample a branch Γ n between two uniformly chosen vertices of G n and perform the bootstrap argument discussed above conditioned on Γ n and the event that it is a "nice" path, a property we will define later that will include, amongst others, the event that Γ n is not too short. Then, using Wilson's algorithm we may sample other branches of UST(G n ) by considering loop-erased random walks terminated at Γ n ; thus Γ n can be thought of as the backbone of UST(G n ), and provided Γ n is sufficiently long we can sample the branch from x to Γ n and consider its extension into Γ n to make it longer if necessary. With this modified definition of a branch, it is then possible to prove a conditional sub-polynomial tail bound in ε for (13), and then to prove Theorem 3.2 by decomposing according to whether Γ n is "nice" or not.
Throughout the rest of this paper, and in accordance with Theorem 3.2, we fix c > 0 to be a small enough parameter and ε > 0 which can also be chosen to be small enough depending on c and set
N n = α 10 log 2 n r = c √ n,(14)
and for any scale ℓ ∈ {0, . . . , N n }
r ℓ = r 2 ℓ ε ℓ = ε 4 ℓ k ℓ = ε 1/2 ℓ r ℓ .(15)
Theorem 3.3. Let {G n } be a sequence of graphs satisfying Assumption 1.4, let T n be UST(G n ) and denote by Γ n the unique path between two independent uniformly chosen vertices. Then for any δ > 0 there exist c ′ , ε ′ > 0 such that for all c ∈ (0, c ′ ) and all ε ∈ (0, ε ′ ) there exists N = N (δ, c, ε) such that for any n ≥ N we have that, with probability at least 1 − δ,
(I) Cap √ n (Γ n ) ≥ 2c,
(II) For any scale ℓ ∈ {0, . . . , N n } and subsegment I ⊆ Γ n with |I| = r ℓ /3 we have that
Cap k ℓ (I, Γ n \ I) ≥ ε 1/6 ℓ k ℓ r ℓ n , (III) |Γ n | ≤ ε − 1 10 √ n.
Definition 3.4. For the rest of this paper, given c and ε as above we denote by E n,c,ε the intersection of the events in (I), (II), (III) of the above theorem.
Remark 3.5. The reader may notice that although c is fixed in Theorem 3.2 and (14), it is now treated as a variable parameter in Theorem 3.3. This is intentional since we need |Γ n | ≥ c √ n to overcome the problem of branch length mentioned above, and we cannot ensure this with high probability when c is fixed; only when it is small. To prove Theorem 3.2, we start with a fixed c, but our first step is to reduce it if necessary so that the statement of Theorem 3.3 holds as well. We will then prove the theorem with this smaller value of c. This poses no problem since the assertion of Theorem 3.2 with smaller c is stronger; this is also discussed in the proof of Theorem 3.2.
Next we assume that E n,c,ε holds for some positive c and ε, and let x be a vertex of G n . Let Γ x denote the loop-erasure of the random walk path starting x and stopped when it hits Γ n (if x ∈ Γ n then Γ x is empty). For an integer s ∈ (0, c √ n) we denote by Γ s x the prefix of Γ x of length s as long as |Γ x | ≥ s; otherwise, i.e. if |Γ x | < s, we denote by Γ s x the prefix of the path in UST(G n ) from x to one of the two endpoints of Γ n such that this path has length at least s. This is possible since by part (I) of Theorem 3.3 and (9), if E n,c,ε holds, then |Γ n | ≥ 2c √ n. If the two endpoints of Γ n can be used, we choose one in some arbitrary predefined manner. In Section 4.3 we will prove the following.
Theorem 3.6. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let T n be UST(G n ). Denote by Γ n the unique path between two independent uniformly chosen vertices and for a vertex x ∈ G n and s > 0 let Γ s x be as described above. Then for any c > 0 there exist ε ′ > 0 and a constant a > 0 such that for every ε ∈ (0, ε ′ ) there exists N = N (c, ε) such that for any n ≥ N and any ℓ ∈ {0, . . . , N n } we have
P Cap k ℓ Γ 5r ℓ /6 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≤ ε 1 6 ℓ k ℓ r ℓ n and E n,c,ε ≤ e −a(log ε −1 ℓ ) 2 .
Given these two estimates we are now ready to proceed with the proof of Theorem 3.2. Our strategy is as follows. On the event
Cap k ℓ Γ 5r ℓ /6 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≥ ε 1 6 ℓ k ℓ r ℓ n ,(16)
we can condition on Γ n and Γ x and then apply Lemma 2.4 with m = Our aim will be to test each of the intervals {A i } in turn to see if the trees hanging on A i contribute at least r 2 ℓ ε ℓ to B(x, r ℓ ). We will test these intervals conditionally on Γ x ∪ Γ n and on the outcome of the previous tests. Here we encounter a significant difficulty since the failure of some past tests introduces a complicated conditioning which we cannot access directly by contracting some edges.
To overcome this difficulty we proceed as follows. Conditioned on Γ n ∪ Γ x ⊂ UST(G n ), we contract Γ n ∪ Γ x to a single vertex (still remembering the original edge-set) to form the graph G n /(Γ n ∪ Γ x ). By the UST spatial Markov property [8,Proposition 4.2], we have that UST(G n ) is distributed as the union of Γ n ∪ Γ x and the UST of this new graph. Before proceeding, we then add a new vertex called the sun, denoted by ⊙, to the graph G n /(Γ n ∪ Γ x ), and add an edge from every vertex to the sun with weight chosen so that a lazy random walk on G n ∪ {⊙}/(Γ n ∪ Γ x ) will always jump to the sun at the next step with probability 1 k ℓ . Then, we identify the sun with Γ n ∪ Γ x , remembering the edges emanating from the sun. This ensures that when we run Wilson's algorithm on the remaining graph, rooted at the contracted vertex, random walks will always be killed when they hit the sun, so typically they only run for time of order k ℓ .
On the graph G n /({⊙} ∪ Γ n ∪ Γ x ) we will often say "hit A" when A is a subset of {⊙} ∪ Γ n ∪ Γ x . The meaning of hitting A in the graph G n /({⊙} ∪ Γ n ∪ Γ x ) is to hit {⊙} ∪ Γ n ∪ Γ x by traversing an edge whose original endpoint belonged in A. In some cases it will be convenient to start a random walk at a uniform vertex U in the original graph G n , and project the start point onto G n /({⊙} ∪ Γ n ∪ Γ x ); in this case "hit A" also includes the event U ∈ A.
By Lemma 2.10, conditionally on Γ n ∪ Γ x , we have that UST(G n /(Γ n ∪ Γ x )) stochastically dominates
UST(G n /({⊙} ∪ (Γ n ∪ Γ x )) ∩ E(G n /(Γ n ∪ Γ x )
). Therefore, we can couple the two USTs together such that every edge e not adjacent to the sun in UST(G n /({⊙} ∪ (Γ n ∪ Γ x )) also appears in UST(G n /(Γ n ∪ Γ x )). When we expand {⊙} ∪ Γ n ∪ Γ x in UST(G n /({⊙} ∪ (Γ n ∪ Γ x )) and then remove ⊙ and its incident edges, we obtain several connected components, one of which contains x. By stochastic domination, the component containing x is a subset of UST(G n ). Therefore, let B ⊙ (x, r ℓ ) denote the set of vertices connected to x by a path of length at most r ℓ that does not intersect the sun after expanding {⊙} ∪ Γ n ∪ Γ x in the sunny graph. By stochastic domination, if we can prove a lower tail bound for B ⊙ (x, r ℓ ) on the sunny graph, it automatically transfers to a lower tail bound for B Tn (x, r ℓ ) on the original graph.
Recall that, given Γ n ∪ Γ x , each of the A i 's defined above is a subset of Γ n ∪ Γ x . When working on the graph G n /({⊙} ∪ Γ n ∪ Γ x ), we let I i (k ℓ ) be the set of vertices connected to the contracted vertex in UST(G n /({⊙} ∪Γ n ∪Γ x ) by a path of length at most k ℓ , such that the last edge on this path has an endpoint in A i . Note that this is equivalent to being connected to A i by a path of length at most k ℓ not touching Γ n ∪ Γ x after expanding the path and separating ⊙ to obtain a subset of UST(G n ). We also include A i in I i (k ℓ ) and set X i = X i (k ℓ ) = |I i (k ℓ )|.
Let B ⊙ j = { j i=1 X i ≤ 16ε ℓ r 2 ℓ }
and (for notational convenience) interpret B ⊙ 0 as an almost sure event. In Section 5 we will prove the following lemma.
Lemma 3.7. Conditionally on Γ x ∪ Γ n , let B ⊙ j be as defined above on the graph G n /({⊙} ∪ Γ n ∪ Γ x ). Then for each j ≤ (2 13 e) −1 ε
− 1 3 ℓ P B ⊙ j B ⊙ j−1 , (Γ n ∪ Γ x ), Cap k ℓ (Γ 5r ℓ /6 x , Γ n ∪ Γ x \ Γ 5r ℓ /6 x ) ≥ r ℓ k ℓ ε 1 6 ℓ n ≤ 1 − 1 160e ε 1/6 ℓ .
This has the following immediate corollary.
Corollary 3.8. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let T n , Γ n and Γ x be as in the previous theorem. Then for any c > 0, any ε > 0, all n large enough and any ℓ ∈ {0, . . . , N n }, we have
P |B Tn (x, r ℓ )| ≤ 16r 2 ℓ ε ℓ , Cap k ℓ (Γ 5r ℓ /6 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ) ≥ ε 1 6 ℓ k ℓ r ℓ n ≤ exp −bε −1/6 ℓ , where b = (5e 2 2 18 ) −1 . Proof. Given that Cap k ℓ (Γ 5r ℓ /6 x , (Γ n ∪Γ x )\Γ 5r ℓ /6 x ) ≥ ε 1 6
ℓ k ℓ r ℓ n , we can condition on Γ n ∪Γ x and obtain intervals
(A j ) (2 13 e) −1 ε −1/3 ℓ j=1
on the graph G n /({⊙} ∪ Γ n ∪ Γ x ) as described above. Applying Lemma 3.7, we then deduce that
P |B ⊙ (x, r ℓ )| ≤ 16r 2 ℓ ε ℓ Cap k ℓ (Γ 5r ℓ /6 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ) ≥ ε 1 6 ℓ k ℓ r ℓ n , Γ n ∪ Γ x ≤ (2 13 e) −1 ε −1/3 ℓ j=1 P B ⊙ j B ⊙ j−1 , (Γ n ∪ Γ x ), Cap k ℓ (Γ 5r ℓ /6 x , Γ n ∪ Γ x \ Γ 5r ℓ /6 x ) ≥ r ℓ k ℓ ε 1 6 ℓ n ≤ 1 − 1 160e ε 1/6 ℓ (2 13 e) −1 ε −1/3 ℓ ≤ exp −bε −1/6 ℓ .
To conclude, we average over Γ n ∪ Γ x , then transfer this result from B ⊙ (x, r ℓ ) in UST(G n /({⊙} ∪ Γ n ∪ Γ x ) to B Tn (x, r ℓ ) in UST(G n ) using the stochastic domination result of Lemma 2.10, as explained above.
We now have all the tools to prove Theorem 3.2.
Proof of Theorem 3.2. Let δ > 0. We define the events
A ℓ = ∃x ∈ T n : |B(x, r ℓ )| ≤ ε ℓ r 2 ℓ and |B(x, r ℓ+1 )| ≥ ε ℓ+1 r 2 ℓ+1 , B ℓ = ∃x ∈ T n : |B(x, r ℓ )| ≤ ε ℓ r 2 ℓ ,
for ℓ ∈ {0, . . . , N n }. We decompose by writing
P ∃x ∈ T n : |B(x, c √ n)| ≤ εc 2 n ≤ P(¬E n,c,ε ) + Nn−1 ℓ=0 P(E n,c,ε ∩ A ℓ ) + P(E n,c,ε ∩ B Nn ).(17)
In what follows we will show that given δ > 0 we can find ε and c small enough and N large enough so that the sum above is at most 3δ. This yields the required assertion of the theorem since the quantity P(∃x ∈ T n : |B(x, c √ n)| ≤ εn) is non-decreasing as c decreases.
We first apply Theorem 3.3 and find ε and c small enough and N large enough (depending on δ) that the first term is at most δ for all n ≥ N . To control the second term in (17) we note that if A ℓ occurs, then |B(v, r ℓ+1 )| ≤ ε ℓ r 2 ℓ = 16ε ℓ+1 r 2 ℓ+1 for all v ∈ B(x, r ℓ+1 ), and the number of such v is at least ε ℓ+1 r 2 ℓ+1 . Therefore using Theorem 3.6, Corollary 3.8 and Markov's inequality, we have for all n large enough that
Nn−1 l=0 P(E n,c,ε ∩ A ℓ ) ≤ Nn−1 l=0 P E n,c,ε and v ∈ T n : |B(v, r ℓ+1 )| ≤ 16ε ℓ+1 r 2 ℓ+1 ≥ ε ℓ+1 r 2 ℓ+1 ≤ n Nn−1 l=0 ε −1 ℓ+1 r −2 ℓ+1 e −a(log ε −1 ℓ ) 2 + e −bε − 1 6 ℓ ,(18)
By making ε smaller and N larger if necessary we can guarantee that the term in the parenthesis on the right hand side is at most ε 10 ℓ for all n ≥ N . This shows that the sum can be smaller than δ as long as ε is small enough and N is large enough.
Finally, for the third term we recall that r Nn = cn 1 2 − α 10 and ε Nn = εn − α 5 , and use Theorem 3.6, Corollary 3.8 and the union bound to bound P(E n,c,ε ∩ B Nn ) ≤ n e −a log 2 (ε −1 n α/5 ) + e −bε − 1 6 n α/30 , which tends to 0 as n → ∞, so it is smaller than δ as long as n is large enough. Provided n is sufficiently large, we have therefore bounded (17) by 3δ, concluding the proof. (We can then reduce ε if necessary so that the bound holds for all n ≥ 1).
Proofs of Theorems 3.3 and 3.6
In this section we prove Theorem 3.3 and Theorem 3.6. Due to the results of [25], this essentially boils down to proving only capacity estimates. In both cases, we will bound capacity using Lemma 2.5. In Section 4.1 we prove two claims that we later use in the proofs of Theorem 3.3 and Theorem 3.6 in Section 4.2 and Section 4.3 respectively.
Two claims
In what follows we take z = 1/20 and assume that {G n } is a sequence of graphs satisfying Assumption 1.4. Our first claim shows that with very high probability any loop-erased trajectory (that has bounded bubblesum) has a rather long subinterval which is derived from a (relatively) short segment of a random walk trajectory (which in turn will have a long subinterval with good M (k) and closeness values by the subsequent claim).
Claim 4.1. Fix ψ > 0 and c > 0. There exists ε ′ > 0 such that for every ε ∈ (0, ε ′ ) there exists N such that for all n ≥ N and for all scales ℓ ∈ {0, . . . , N n } the following holds. Let X be a random walk on G n which is bubble-terminated (see Definition 2.1) with bubble-sum bounded by ψ and let Γ be its loop erasure. Also fix j ∈ N and χ = min{ z 3ψ , 1 24 }. Then with probability at least
1 − exp − ε − z 3 ℓ log(1/ε ℓ ) either |Γ| < jr ℓ 24 , or there exists t ∈ [(j − 1)r ℓ /24, jr ℓ /24] such that for all integers 1 ≤ m ≤ χε − 2z 3 ℓ log ε −1 ℓ , λ t+mε 5z 3 ℓ r ℓ (X) − λ t+(m−1)ε 5z 3 ℓ r ℓ (X) ≤ ε z ℓ r ℓ .
Proof. We set M ℓ = ε −5z/3 ℓ . On the event that |Γ| ≥ jr ℓ /24, we divide Γ[(j − 1)r ℓ /24, jr ℓ /24] into M ℓ /24 consecutive disjoint subintervals of length r ℓ /M ℓ . For m ∈ {1, . . . , M ℓ /24} we say that the m-th interval is good if
λ (m+1)r ℓ M ℓ (X) − λ mr ℓ M ℓ (X) ≤ r ℓ M 3/5 ℓ = ε z ℓ r ℓ .
As we assumed that the bubble sum is bounded by ψ, it follows from Claim 2.2 that conditioned on Γ and the event {|Γ| ≥ jr ℓ /24}, the collection of events that the m-th interval is good are independent. Furthermore, by Claim 2.2, Claim 2.7 and Markov's inequality the probability of each such event is at least
1 − ψ M 2/5 ℓ .
Hence, the probability that a sequence of χε
(1 − ψM −2/5 ℓ ) χε − 2z 3 ℓ log ε −1 ℓ ≥ ε 2χψ ℓ ,
where we used the inequality 1 − x ≥ e −2x valid for x > 0 small enough. Since there are M ℓ 24 intervals in total, we can form M ℓ 24 (χε
− 2z 3 ℓ log ε −1 ℓ ) −1 disjoint runs of χε − 2z 3 ℓ log ε −1 ℓ
consecutive intervals. Since the events are independent conditionally on Γ, we deduce that the probability that none of these runs contain only good intervals is at most
1 − ε 2χψ ℓ M ℓ 24 χε − 2z 3 ℓ log ε −1 ℓ −1 ≤ exp − ε 2χψ−5z/3 ℓ 24χε −2z/3 ℓ log ε −1 ℓ ≤ exp − ε 2χψ−z ℓ 24χ log ε −1 ℓ ≤ exp −ε − z 3 ℓ log ε −1 ℓ ,
where in the last inequality we used the fact that χ = min z 3ψ , 1 24 by assumption.
For the next claim recall the definitions of M (k) in (10) and of Close k (U, V ) in (11). We show that with very high probability, any random walk interval of length of order ε z ℓ log ε −1 ℓ r ℓ has a slightly shorter subinterval of length of order ε z ℓ r ℓ , such that its value of M (k) and its closeness to the rest of the path are very close to their expected values given by Lemma 2.6 and (12). This is done by finding many well separated intervals and employing the fast mixing of the graph to obtain independence. Claim 4.2. Fix some χ, c > 0. There exists ε ′ > 0 such that for every ε ∈ (0, ε ′ ) there exists N such that for all n ≥ N and for all scales ℓ ∈ {0, . . . , N n } the following holds. Let X be a random walk on G n , started from stationarity. Let M > 0 and fix some interval I ⊂ [0, M √ n] with |I| = 1 2 χε z ℓ log ε −1 ℓ r ℓ . Also let W ⊂ G n be fixed. Then with probability at least 1 − 2e − χz
6 (log ε −1 ℓ ) 2 there exists a subinterval J = [t − J , t + J ] ⊂ I such that (1) |J| = 2ε z ℓ r ℓ , (2) M (k ℓ ) (X[J]) ≤ r ℓ 4 (3) Close k ℓ X[J], X t + J + r ℓ 24 , M √ n ∪ X[0, t − J − r ℓ 24 ] ≤ r ℓ k 2 ℓ M n 1.5 . (4) Close k ℓ (X[J], W ) ≤ r ℓ k 2 ℓ |W | n 2 .
Proof. We write I = [t − I , t + I ] and then further subdivide I into χ 5 log ε −1 ℓ segments of length 2ε z ℓ r ℓ separated by buffers of length 1 2 ε z ℓ r ℓ , that is, we set
I j = [t − j , t + j ) := t − I + 5 2 jε z ℓ r ℓ + 1 4 ε z ℓ r ℓ , t − I + 5 2 (j + 1)ε z ℓ r ℓ − 1 4 ε z ℓ r ℓ(19)
for each non-negative integer j ≤ χ 5 log ε −1 ℓ . It will be important soon that the length of the buffers satisfy 1 4 ε z ℓ r ℓ ≥ 1 4 ε z Nn r Nn ≫ n 2α 3 t mix for all n large enough by Assumption 1.4, (14) and (15). We also set
X avoid I = X 0, t − I − r ℓ 36 ∪ X t + I + r ℓ 36 , M √ n .
We condition on X avoid I and define for each j the event
E j = M (k ℓ ) (X[I j ]) ≤ r ℓ /4 and Close k ℓ X[I j ], X avoid I ≤ r ℓ k 2 ℓ M n 3/2 and Close k ℓ (X[I j ], W ) ≤ r ℓ k 2 ℓ |W | n 2 .
Note that since |I| = 1 2 χε z ℓ log ε −1 ℓ r ℓ , we have that t − j − t − I ≤ r ℓ 72 and t + I − t + j ≤ r ℓ 72 for all j and all ℓ provided ε is small enough. Thus,
Close k ℓ (X[I j ], X avoid I ) ≥ Close k ℓ X[I j ], X 0, t − j − r ℓ 24 ∪ X t + j + r ℓ 24 , M √ n .
Hence E j implies that the interval I j satisfies the conditions (1) − (4). Note that the events {E j } j are not independent, but that was why we introduced the buffers. Let {Y j } j≤ χ 5 log ε −1 ℓ be independent random walks started from stationarity and run for time 2ε z ℓ r ℓ , set
E ind j = M (k ℓ ) (Y j ) ≤ r ℓ /4 and Close k ℓ Y j , X avoid I ≤ r ℓ k 2 ℓ M n 3/2 and Close k ℓ (Y j , W ) ≤ r ℓ k 2 ℓ |W | n 2 ,
and note that conditioned on X avoid I the events {E ind j } j are independent. Now by Lemma 2.6 and Assumption 1.4 we have (provided ε < 1 and c < 1/6, for example) that
E M (k ℓ ) (Y j ) ≤ 2θε z ℓ r ℓ .
Since |X avoid I | ≤ M √ n, it also follows from (12) that
E Close k ℓ Y j , X avoid I X avoid I ≤ 8ε z ℓ r ℓ k 2 ℓ M n 3/2 ,
and that
E[Close k ℓ (Y j , W )] ≤ 8ε z ℓ r ℓ k 2 ℓ |W | n 2 .
Consequently, by Markov's inequality and independence we get that
P none of {E ind j } occur X avoid I ≤ ((16 + 8θ)ε z ℓ ) χ 5 (log ε −1 ℓ ) ≤ e − χz 6 (log ε −1 ℓ ) 2 ,(20)
as long as ε is small enough depending on θ. To conclude, note that as long as n is large enough, we can couple the independent walks {Y j } and {X[I j ]} so that
P(∃j : X[I j ] = Y j ) ≤ χ log(1/ε ℓ ) 5 2 −n 2α 3 ≤ e − χz 6 (log ε −1 ℓ ) 2 .(21)
Indeed, assume we coupled the first j − 1 pairs and condition on all these pairs. By the Markov property and since the buffers between distinct I j 's are longer than n 2α 3 t mix , the starting point of X[I j ] is 2 −n 2α 3 close in total variation distance to the stationary distribution by (8). Therefore, we may couple it to the first vertex of Y j so that they are equal with probability at least 1 − 2 −n 2α 3 (see for instance [22,Proposition 4.7]). Moreover, once their starting points are coupled, we can run the walks together so that they remain coupled for the remaining 2ε z ℓ r ℓ steps. Hence (21) holds for large enough n and we combine with (20) in a union bound to conclude that
P none of {E j } occur X avoid I ≤ 2e − χz 6 (log ε −1 ℓ ) 2 .
Proof of Theorem 3.3
As mentioned in Section 2.4, to sample Γ n we will use a coupling with the sunny graph G * n = G * n (ζ) introduced in [28], obtained from G n by adding an extra vertex ρ n known as the sun, and connecting it to every vertex in v ∈ G n with an edge of weight (deg v)ζ √ n−ζ (so that the probability of jumping to ρ n at any step is ζn −1/2 ). It follows from Lemma 2.10 that the graph UST(G * n ) \ {ρ n } obtained from the UST of G * n by removing ρ n and its incident edges is stochastically dominated by the UST of G n . Therefore, there is a coupling between UST(G n ) and UST(G * n ) such that UST(G * n ) \ {ρ n } ⊂ UST(G n ); moreover if Γ * n denotes the path between u and v in UST(G * n ), then Γ n = Γ * n in this coupling provided that ρ n / ∈ Γ * n . Note that this sunny graph is different to the sunny graph used in the statements of Lemma 3.7 and Corollary 3.8. As outlined in Section 3.1, Lemma 3.7 and Corollary 3.8 refer to later stages of the overall proof strategy.
Consequently, it will be convenient to work with a path Γ * n = Γ * n (ζ) sampled as follows. Let T and T ′ be two independent geometric random variables with mean ζ −1 n 1/2 . Given T , let X be a random walk run for T − 1 steps started from u ∈ G n and let LE(X) be its loop erasure. Given T ′ and X, run X ′ , a random walk started from v ∈ G n and terminate X ′ after T ′ steps. Write T X for the minimum between T ′ and the first hitting time of X. Let Γ * n be the path between (u, v) in LE(X) ∪ LE(X ′ [0, T X ]), if such a path exists. Otherwise, let Γ * n = ∅. Lemma 4.3. For every δ > 0 there exists ζ > 0 such that for all large enough n there exists a coupling of Γ n and Γ * n (ζ) such that Γ n = Γ * n (ζ) and is non-empty with probability at least 1 − δ.
Proof. For every path Γ from u to v let H(Γ) = Γ be equal to Γ if ρ n / ∈ Γ and H(Γ) = ∅ if ρ n ∈ Γ. Run Wilson's algorithm on the graph G * n (ζ) initiated at the points ρ n , u and then v, and note that the hitting time of ρ n is a geometric random variable, and moreover that given τ ρn , the walk until time τ ρn is distributed as a random walk on G n . Consequently, H( Γ * n ) has the distribution of Γ * n . By the discussion above, we can find a coupling of (Γ n , Γ * n ) where these paths are equal whenever ρ n / ∈ Γ * n . Under this coupling, we have that (Γ n , Γ * n , H( Γ * n )) are all equal with probability 1 − P(ρ n / ∈ Γ * n ). As H( Γ * n ) has the law of Γ * n , this is in fact a coupling of Γ n and Γ * n where the paths are equal with probability 1 − P(ρ n / ∈ Γ * n ). By [25,Claim 2.9], this probability tends to 1 as ζ → 0. For the final part of the claim, note that Γ * n is clearly non-empty on this good event.
Proof of Theorem 3.3. Our main effort is to show that part (II) holds with high probability. Indeed, that part (I) and (III) occur with probability at least 1 − δ as long as ε > 0 and c > 0 are small enough is a consequence of [25, Theorem 1.1 and Theorem 2.1].
Let δ > 0. We appeal to Lemma 4.3 and obtain ζ > 0 so that P(Γ n = Γ * n (ζ)) ≤ δ/4. Denote by B the event of part (II) of Theorem 3.3. For the rest of the proof, we think of δ and ζ as fixed, we set ψ = θ + 2ζ −2 and χ = min{ z 3ψ , 1 24 }, and decrease both c and ε until we eventually obtain that P(B c ) ≤ δ. Recall that Γ * n (ζ) is generated using two independent random walks with geometric killing time which we denote X and X ′ . Setting M = 8 ζδ , we can thus write
P(B c ) ≤ P(Γ n = Γ * n (ζ)) + P |X| + |X ′ | ≥ M √ n + P |X| + |X ′ | ≤ M √ n and B c .
The first event has probability at most δ 4 by the above. Since M = 8 ζδ , the probability of the second event is also bounded by δ/4 by Markov's inequality. For the third term, first decrease c if necessary so it is less than 1 2M (this will be useful at the end of the proof), then let ℓ ≤ N n be a fixed scale and let I ⊂ Γ * n (ζ) be some segment with |I| = r ℓ /3. It therefore has at least r ℓ /6 vertices either on LE(X) or on LE(X ′ ) and hence contains at least one interval of the form LE(X)[(j −2)r ℓ /24, (j +1)r ℓ /24] or LE(X ′ )[(j −2)r ℓ /24, (j +1)r ℓ /24] (that is, an interval of the form [(j − 1)r ℓ /24, jr ℓ /24] plus two buffers of length r ℓ 24 both before and after the interval) for some j ≤ 24M √ n r ℓ . Since Assumption 1.4 holds, we deduce from Claim 2.8 that X and X ′ are bubble-terminated random walks with bubble sum bounded by ψ. Hence we may apply Claim 4.1 and the union bound to learn that the probability that there exists a scale ℓ and j as above such that the event of Claim 4.1 does not hold for X or X ′ is at most
∞ ℓ=0 M √ n r ℓ /24 exp − ε − z 3 ℓ log(1/ε ℓ ) = 24M c ∞ ℓ=0 2 ℓ · exp − ε − z 3 ℓ log(1/ε ℓ ) = 24M c ∞ ℓ=0 2 ℓ · exp − ε − z 3 4 ℓz 3 log(4 ℓ ε −1 ) ,
which can be made to be smaller than δ/4 by decreasing ε appropriately. Thus we may assume without loss of generality that I contains an interval of the form LE(X)[(j − 2)r ℓ /24, (j + 1)r ℓ /24] for some j that we fix henceforth, and that there exists a time t ∈ [(j − 1)r ℓ /24, jr ℓ /24] such that for all integers m satisfying
1 ≤ m ≤ χε − 2z 3 ℓ (log ε −1 ℓ ) we have λ t+mε 5z 3 ℓ r ℓ (X) − λ t+(m−1)ε 5z 3 ℓ r ℓ (X) ≤ ε z ℓ r ℓ .(22)
We write X[t 1 , t 2 ) for the corresponding part of X, that is, we set t 1 = λ t (X) and t 2 = λ t+χε z ℓ log(ε −1 ℓ )r ℓ (X). It holds by construction that
t 2 − t 1 ≥ χε z ℓ log ε −1 ℓ r ℓ and t 2 ≤ M √ n .(23)
We now apply the union bound using Claim 4.2 with W = X ′ [0, M √ n] to get that the probability that there exists a scale ℓ and i ≤
2M
√ n χε z ℓ log ε −1 ℓ r ℓ such that (1) − (4) of Claim 4.2 do not hold for the interval
(i − 1) 1 2 χε z ℓ log ε −1 ℓ r ℓ , i 1 2 χε z ℓ log ε −1 ℓ r ℓ is at most ∞ ℓ=0 2M √ n χε z ℓ log ε −1 ℓ r ℓ · exp − χz 6 log ε −1 ℓ 2 ≤ ∞ ℓ=0 2M (2 · 4 z ) ℓ χε z log ε −1 c ε 4 ℓ log 4 ℓ ε χz 6 ,
which can be made smaller than δ/4 by decreasing ε appropriately. Therefore we henceforth assume that all such intervals contain a good subinterval satisfying (1) − (4) of Claim 4.2.
Since [t 1 , t 2 ] must contain an interval of the form (i − 1) 1 2 χε z ℓ log ε −1 ℓ r ℓ , i 1 2 χε z ℓ log ε −1 ℓ r ℓ by (23), it now follows that [t 1 , t 2 ] contains a subinterval J = [t − J , t + J ] satisfying conditions (1) − (4) of Claim 4.2 with
W = X ′ [0, M √ n]. Since |J| = 2ε z ℓ r ℓ ≥ λ t+mε 5z 3 ℓ r ℓ (X) − λ t+(m−2)ε 5z 3 ℓ r ℓ (X) for each 2 ≤ m ≤ χε − 2z 3 ℓ (log ε −1 ℓ ) by (22), there must exist some m * ≤ χε − 2z 3 ℓ (log ε −1 ℓ ) such that t − J ≤ λ t+(m * −1)ε 5z 3 ℓ r ℓ (X) < λ t+m * ε 5z 3 ℓ r ℓ (X) ≤ t + J .
We set A = (LE(X))
[t+(m * −1)ε 5z 3 ℓ r ℓ ,t+m * ε 5z 3 ℓ r ℓ )
, so that A ⊂ I ⊂ Γ n , so that |A| = ε 5z 3 so that the tail bound of Theorem 3.6 holds. We therefore also assume that {|Γ x | > r ℓ 3 }. Under E n,c,ε we have that Cap √ n (Γ n ) ≥ 2c and hence by Claim 2.7 we have that B Γn (G n ) ≤ θ + 1 c 2 = ψ, so X is bubble-terminated random walk with bubble sum bounded by ψ. Now divide X([0, M ℓ √ n]) into 2M ℓ √ n(χε z ℓ log ε −1 ℓ r ℓ ) −1 disjoint consecutive intervals of length 1 2 χε z ℓ log ε −1 ℓ r ℓ . Also note that 2M ℓ √ n ≤ χε z ℓ log ε −1 ℓ e χz 12 (log ε −1 ℓ ) 2 r ℓ for all ℓ ≤ N n provided that ε is small enough as a function of χ and c (i.e., depending on θ and c). By the union bound and Claim 4.2, provided n exceeds some N (c, ε) the probability that all of these consecutive intervals contain a subinterval satisfying points (1) − (4) of Claim 4.2 is therefore at least
1 − 2M ℓ √ n(χε z ℓ log ε −1 ℓ r ℓ ) −1 e − χz 6 (log ε −1 ℓ ) 2 ≥ 1 − e χz 12 (log ε −1 ℓ ) 2 e − χz 6 (log ε −1 ℓ ) 2 = 1 − e − χz 12 (log ε −1 ℓ ) 2 .
In particular, since any interval I ⊂ [0, M ℓ √ n] of length χε z ℓ log ε −1 ℓ r ℓ must contain an entire consecutive interval of the form above, we deduce that, provided n ≥ N (c, ε),
P ∀I ⊂ [0, M ℓ √ n], |I| = χε z ℓ log ε −1 ℓ r ℓ : ∃J ⊂ I satisfying (1) − (4) of Claim 4.2 ≥ 1 − e − χz 12 (log ε −1 ℓ ) 2 .(25)
We next apply Claim 4.1 with j = 1 to obtain that, provided n ≥ N (c, ε), with probability at least
1 − exp − ε − z 3 ℓ log(1/ε ℓ )(26)there exists t ≤ r ℓ 3 such that for all 1 ≤ m ≤ χε − 2z 3 ℓ (log ε −1 ℓ ), λ t+mε 5z 3 ℓ r ℓ (X) − λ t+(m−1)ε 5z 3 ℓ r ℓ (X) ≤ ε z ℓ r ℓ .(27)
We write X[t 1 , t 2 ) for the corresponding part of X, so that t 1 = λ t (X) and t 2 = λ t+χε z ℓ log(ε −1 ℓ )r ℓ (X). It holds by construction that
χε z ℓ log ε −1 ℓ r ℓ ≤ t 2 − t 1 ,(28)
and moreover since we assumed that {|Γ x | > r ℓ 3 } and {τ Γn ≤ M ℓ √ n}, we clearly have that t 2 ≤ M ℓ √ n. On the event E n,c,ε , it therefore follows from (25) and (28) For the rest of the proof we assume that such a J exists and that E n,c,ε holds. By part (1) of Claim 4.2 and (27),
|J| = 2ε z ℓ r ℓ ≥ λ t+mε 5z 3 ℓ r ℓ (X) − λ t+(m−2)ε 5z 3 ℓ r ℓ (X) for each 2 ≤ m ≤ χε − 2z 3 ℓ (log ε −1 ℓ ). Therefore there must exist some m * ≤ χε − 2z 3 ℓ (log ε −1 ℓ ) such that t − J ≤ λ t+(m * −1)ε 5z 3 ℓ r ℓ (X) < λ t+m * ε 5z 3 ℓ r ℓ (X) ≤ t + J . Now set A = (Γ x ) [t+(m * −1)ε 5z 3 ℓ r ℓ ,t+m * ε 5z 3 ℓ r ℓ )
.
Note that, by construction, it holds that A ⊂ Γ r ℓ 3
Therefore, since Close k ℓ (·, ·) is monotone and subadditive in each argument (by definition and the union bound), applying (3) − (4) of Claim 4.2 we deduce that
Close k ℓ A, (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≤ Close k ℓ X[t − J , t + J ], X t + J + r ℓ /24, M ℓ √ n + Close k ℓ X[t − J , t + J ], W ≤ r ℓ k 2 ℓ (M ℓ √ n + |W |) n 2 ≤ 2ε 10z 3 ℓ r ℓ k ℓ n · M ℓ ε 1 2 − 10z 3 ℓ r ℓ √ n ,
(where we used k ℓ = ε 1/2 ℓ r ℓ and |W | ≤ ε −1/10 ℓ √ n = M ℓ √ n on the event E n,c,ε ). Consequently, since z = 1/20, recalling that M ℓ = ε −1/10 ℓ and assuming without loss of generality that c, ε < 1/2, we obtain that
Cap k ℓ Γ r ℓ /3 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≥ Cap k ℓ A, (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≥ Cap k ℓ (A) − Close k ℓ A, (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x ≥ 2k ℓ r ℓ ε 10z 3 ℓ n 1 − ε 7 30 ℓ r ℓ √ n ≥ k ℓ r ℓ ε 1 6 ℓ n .
To summarize, we showed that Cap k ℓ Γ
r ℓ /3 x , (Γ n ∪ Γ x ) \ Γ 5r ℓ /6 x
is large enough on the event E n,c,ε whenever
{τ Γn ≤ M ℓ √ n} and the relevant events in Claim 4.1 and Claim 4.2 occur so that we can find A as above. Theorem 3.6 therefore follows on taking a union bound over (24), (25) and (26), choosing ε ′ small enough as a function of c and requiring that n is large enough as a function of ε and c (since χ and ψ were themselves functions of c).
Proof of Lemma 3.7
In this section we prove Lemma 3.7. Throughout we assume that the index n, the scale ℓ and the paths Γ n ∪ Γ x are fixed. We also take the setup of Section 3.1, as outlined above Lemma 3.7. This means that we condition on Γ n ∪ Γ x and add a sun ⊙ to the graph G n /(Γ n ∪ Γ x ) with weights chosen so that a lazy random walk will jump to the sun at the next step with probability 1 k ℓ . We also assume that the intervals A j ⊂ Γ x for j = 1, . . . , (2 13 e) −1 ε − 1 3 ℓ are predefined as described in Section 3.1. For the rest of this section we work on the graph G n /({⊙} ∪ Γ n ∪ Γ x ). Recall that
r ℓ = r 2 ℓ , ε ℓ = ε 4 ℓ , k ℓ = ε 1/2 ℓ r ℓ , |A j | ≤ 2 13 eε 1/3 ℓ r ℓ , Cap k ℓ (A j , (Γ n ∪ Γ x ) \ A j ) ≥ 2 11 eε ℓ r 2 ℓ n .(29)
When we talk about capacity and relative capacity in this section, we are always referring to these quantities on the original graph G n .
Recall also that, for each j ≤ (2 13 e) −1 ε − 1 3 ℓ , we let I j (k ℓ ) be the set of vertices connected to the contracted vertex in UST(G n /({⊙} ∪ Γ n ∪ Γ x ) by a path of length at most k ℓ , such that the last edge on this path has an endpoint in A j . This also includes vertices originally in A j before the contraction. Since ℓ is fixed for this section, we also set X j = |I j (k ℓ )|.
Claim 5.1. Assume that Γ x and Γ n satisfy (16) (and therefore (29)). Fix a scale ℓ and consider the graph G n /(Γ n ∪ Γ x ∪ {⊙}) as described above. Then, for every j ∈ {1, . . . , (2 13
e) −1 ε − 1 3 ℓ } we have E[X j ] ≥ n · Cap k ℓ (A j , (Γ n ∪ Γ x ) \ A j ) 2e ≥ 2 10 ε ℓ r 2 ℓ .
Proof. By Wilson's algorithm, for every v ∈ G n , we have that v ∈ I j (k ℓ ) if a random walk starting at v hits Γ n ∪ Γ x ∪ {⊙} at A j and its loop erasure is of length at most k ℓ . Therefore,
P(v ∈ I j ) ≥ P v (τ ⊙ > k ℓ ) · P v (τ Aj < k ℓ and τ Aj < τ (Γn∪Γx)\Aj | τ ⊙ > k ℓ ),
where all hitting times refer to hitting times of the lazy random walk. First note that P(τ ⊙ > k ℓ ) =
1 − 1 k ℓ k ℓ ≥ 1 2e
. Then, given τ ⊙ > k ℓ , the lazy random walk until time k ℓ is distributed as a lazy random walk on G n /(Γ n ∪ Γ x ). Since all degrees in G n are equal we get
E[X j ] = v∈Gn P(v ∈ I j ) ≥ v∈Gn P v (τ Aj < k ℓ and τ Aj < τ (Γn∪Γx)\Aj in G n /(Γ n ∪ Γ x )) 2e = n · Cap k ℓ (A j , (Γ n ∪ Γ x ) \ A j ) 2e ,
and we conclude the proof using (29).
Recall that our goal is to find a lower bound for the probability that j+1 i=1 X i is large given that j i=1 X i is small. To this end, let Φ j be the (random) edge-set consisting of all simple paths of length at most k ℓ in UST(G n /({⊙} ∪ Γ n ∪ Γ x )) that end in the contracted vertex through
A 1 ∪ . . . ∪ A j . Note that Φ j determines { j i=1 X i ≤ 16ε ℓ r 2 ℓ }
and that conditioning on Φ j = ϕ j for some set of edges ϕ j means precisely that the edges of ϕ j are in the UST (open edges) and all other edges touching a vertex v of ϕ j , such that the path in ϕ j from v to A 1 ∪ . . . ∪ A j is of length at most k ℓ − 1, must not belong to the UST (closed edges). These open and closed edges determine Φ j . Thus, to condition on Φ j = ϕ j , we erase the closed edges and contract all the open edges to a single vertex which coincides with Γ n ∪ Γ x ∪ {⊙}, and call the remaining graph G n (ϕ j ). By the spatial Markov property of the UST [8,Proposition 4.2] we have that UST(G n (ϕ j )) together with ϕ j is distributed precisely as UST(
G n /({⊙} ∪ Γ n ∪ Γ x )) conditioned on Φ j = ϕ j . Note that the event { j i=1 X i ≤ 16ε ℓ r 2 ℓ } occurs if and only if |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ where V (ϕ j ) are the vertices touched by ϕ j . Claim 5.2. Let ϕ j ⊂ E(G n ) be such that P(Φ j = ϕ j ) > 0 and |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ .
Let γ be a simple path in G n (ϕ j ) that ends at the contracted vertex. Let (Y t ) t≥0 denote a lazy random walk on G n (ϕ j ) started from a uniform vertex U of the original graph G n and killed upon hitting the contracted vertex of G n (ϕ j )/γ, that is, the upon hitting the vertex corresponding to the contracted edges {⊙} ∪ Γ n ∪ Γ x ∪ ϕ j ∪ γ. Denote by V (Γ n ∪ Γ x ∪ ϕ j ∪ γ) the set of vertices of G n touched by the edges in Γ n ∪ Γ x ∪ ϕ j ∪ γ and let M ⊂ V (Γ n ∪ Γ x ∪ ϕ j ∪ γ) be a fixed subset of vertices of G n . Then
P(Y hits M ) ≤ 64ε ℓ r 2 ℓ n + 4|M |k ℓ n .
(Recall here that to "hit M " means to hit the contracted vertex via an edge that originally led to M ).
Proof. Let ∆ = deg(G n ), i.e. the degree of vertices in the original graph G n (recall that by Assumption 1.4 all vertex degrees are equal), and let
V bad = v ∈ V (G n ) \ V (Γ n ∪ Γ x ∪ ϕ j ) : deg Gn(ϕj) (v) ≤ ∆ 2 .
In other words, V bad is the set of all vertices of G n that are not in the contracted vertex of G n (ϕ j ) that are adjacent to at least ∆/2 closed edges. Since |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ , the number of closed edges is no more than 16∆ε ℓ r 2 ℓ . Hence the number of vertices touching a closed edge is at most 32∆ε ℓ r 2 ℓ and each vertex in V bad contributes at least ∆/2 to this count, so |V bad | ≤ 64ε ℓ r 2 ℓ . Recall that, when we originally added the sun to G n /(Γ n ∪Γ x ), we chose the weights so that the probability that a lazy random walk on G n /(Γ n ∪ Γ x ) would jump to the sun at the next step is always 1 k ℓ . In the graph G n (ϕ j ), we have now contracted some edges and closed some other edges. For any x ∈ G n (ϕ j ), these operations can only increase the probability that Y will jump directly to the sun from the vertex x. Therefore, by coupling, we can separate the sun and its incident edges, and obtain an upper bound for P(Y hits M ) by instead bounding the same probability for a lazy random walk on (G n (ϕ j )/γ) \ {⊙} with an independent Geo( 1 k ℓ ) killing time. We denote this second lazy random walk by Y ′ . To control capacity on (G n (ϕ j )/γ)\{⊙} we will need to work with the stationary measure on (G n (ϕ j )/γ)\ {⊙}, which we denote by π ′ . (The bound on |V bad | above will then help us to compare π ′ with the uniform measure). We define π ′ on all of G n by remembering the edges from before the contraction. In particular, this means that for u ∈ G n , we have
π ′ (u) = ∆ − N cl (u) v∈Gn (∆ − N cl (v)) ,
where N cl (v) denotes the number of closed edges incident to v in G n (ϕ j ).
We now observe the following. If u ∈ G n \ V bad , then
π ′ (u) ≥ ∆/2 n∆ ≥ 1 2n .
Also, for every u ∈ G n , provided that c < 1/32 and ε < 1, we have that
π ′ (u) ≤ ∆ n∆ − 32∆ε ℓ r 2 ℓ ≤ 2 n .
In what follows, these two observations mean that we will be able to switch between π ′ and U and vice versa provided we multiply by 2. In particular, we can write
P U (Y ′ hits M ) ≤ P U ∈ V bad + 2P π ′ (Y ′ hits M ) ≤ 64ε ℓ r 2 ℓ n + 2 ∞ t=0 P π ′ (Y ′ t ∈ M ) = 64ε ℓ r 2 ℓ n + 2 ∞ t=0 π ′ (M )P Geo 1 k ℓ ≥ t ≤ 64ε ℓ r 2 ℓ n + 4|M | n ∞ t=0 1 − 1 k ℓ t = 64ε ℓ r 2 ℓ n + 4|M |k ℓ n .
We will use Claim 5.2 to prove the following upper bounds.
Lemma 5.3. Let ϕ j ⊂ E(G n ) be such that P(Φ j = ϕ j ) > 0 and |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ . Then
(i) E[X j+1 | Φ j = ϕ j ] ≤ 5 · 2 13 · e · ε 5/6 ℓ r 2 ℓ . (ii) Var(X j+1 | Φ j = ϕ j ) ≤ 68ε ℓ r 2 ℓ E[X j+1 | Φ j = ϕ j ],
Proof. We condition on Φ j = ϕ j throughout this proof so our probability space is that of UST(G n (ϕ j )). To prove (i) we condition on Φ j = ϕ j and take any v ∈ G n (ϕ j ) \ {⊙}. By Wilson's algorithm on the graph G n (ϕ j ), we have that P(v ∈ A j+1 | Φ j = ϕ j ) is upper bounded by the probability that a lazy random walk started at v hits A j+1 before it hits the sun. If (Y t ) t≥0 is such a random walk starting from a uniform vertex of G n , by Claim 5.2 and (29) we have that
E[X j+1 | Φ j = ϕ j ] ≤ nP(Y t hits A j+1 ) ≤ 64ε ℓ r 2 ℓ + 4|A j+1 |k ℓ ≤ 5|A j+1 |k ℓ ≤ 5 · 2 13 · e · ε 5/6 ℓ r 2 ℓ ,
where we also used the upper bound on |A j+1 | in (29).
To ease notation in the proof of (ii) we write P(·), E[·] and Var(·) for P(· | Φ j = ϕ j ) and the corresponding expectation and variance. We have
Var(X j+1 ) = u,v∈Gn P(u, v ∈ I j+1 ) − P(u ∈ I j+1 )P(v ∈ I j+1 ).(30)
= v u P(u ∈ I j+1 | v ∈ I j+1 ) − P(u ∈ I j+1 ) P(v ∈ I j+1 ).
Fix some v, and rewrite the inner sum as
n P(U ∈ I j+1 | v ∈ I j+1 ) − P(U ∈ I j+1 ) ,
where U is a vertex chosen uniformly from G n . We decompose the event v ∈ I j+1 according to γ v , the path from v to A j+1 in G n (ϕ j ) which is of length at most k ℓ and obtain that
P(U ∈ I j+1 | v ∈ I j+1 ) − P(U ∈ I j+1 ) = γv P(γ v ⊆ UST(G n (ϕ j )) | v ∈ I j+1 )[P(U ∈ I j+1 | γ v ⊆ UST(G n (ϕ j ))) − P(U ∈ I j+1 )].
To compare P(U ∈ I j+1 | γ v ⊆ UST(G n (ϕ j ))) and P(U ∈ I j+1 ) we note again from the spatial Markov property [8,Proposition 4.2] that the rest of UST(G n (ϕ j )) given γ v ⊆ UST(G n (ϕ j )) is the UST on the graph obtained from G n (ϕ j ) by contracting γ v . By coupling Wilson's Algorithm running on each of the two graphs (G n (ϕ j ) and G n (ϕ j )/γ v ), the difference between the two quantities can be upper bounded by the probability that a random walk starting from a uniform vertex of G n hits γ v before it hits the new sun ⊙. By Claim 5.2, this is bounded by
64ε ℓ r 2 ℓ n + 4k 2 ℓ
n uniformly for all γ v with |γ v | ≤ k ℓ . As γv P(γ v ⊆ UST(G n (ϕ j )) | v ∈ I j+1 ) sums to 1 we obtain that
n(P(U ∈ I j+1 | v ∈ I j+1 ) − P(U ∈ I j+1 )) ≤ 64ε ℓ r 2 ℓ + 4k 2 ℓ .
Plugging this into (30) and using (29) we obtain
Var(X j+1 ) ≤ v (64ε ℓ r 2 ℓ + 4k 2 ℓ )P(v ∈ I j+1 ) ≤ (64ε ℓ r 2 ℓ + 4k 2 ℓ )E[X j+1 ] = 68ε ℓ r 2 ℓ E[X j+1 ]. Recall that B ⊙ j = { j i=1 X i ≤ 16ε ℓ r 2 ℓ }, and Φ j is the random edge-set induced by ∪ j i=1 I i (k ℓ ). Under B ⊙ j ,
we have no information about the structure of Φ j , other than that |Φ j | ≤ 16ε ℓ r 2 ℓ (and this was important for the factorization in the proof of Corollary 3.8). However, in order to prove Lemma 3.7, we will need the following lower bound.
Lemma 5.4. It holds that
P Φ j ∈ {ϕ j : E[X j+1 |Φ j = ϕ j ] ≥ 2 9 ε ℓ r 2 ℓ } | B ⊙ j ≥ ε 1/6 ℓ 80e .
Proof. Recall that we are working on the graph G n /(Γ n ∪ Γ x ∪ {⊙}). Suppose that j i=1 X i ≤ 16ε ℓ r 2 ℓ , and note that this event can be written as the disjoint union of all possible ϕ j such that P(Φ j = ϕ j ) > 0 and |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ . When conditioning on Φ j = ϕ j for some ϕ j we work on the graph G n (ϕ j ), as defined above Claim 5.2. Note that by Lemma 5.3, we have that for every ϕ j with |V (ϕ j )| ≤ 16ε ℓ r 2 ℓ and P(Φ j = ϕ j ) > 0 that E[X j+1 |Φ j = ϕ j ] ≤ 5 · 2 13 eε 5/6 ℓ r 2 ℓ . Furthermore, by Claim 2.12 and Claim 5.1 we have that
E X j+1 j i=1 X i ≤ 16ε ℓ r 2 ℓ ≥ E[X j+1 ] ≥ 2 10 ε ℓ r 2 ℓ .
Write E ′ and P ′ for the expectation and probability operators P(· | B ⊙ j ) and E · | B ⊙ j on G n /({⊙}∪Γ n ∪Γ x ). We have that
E ′ [X j+1 |Φ j ] ≤ 5 · 2 13 · e · ε 5/6 ℓ r 2 ℓ a.s., E ′ [E ′ [X j+1 | Φ j ]] = E ′ [X j+1 ] ≥ 2 10 ε ℓ r 2 ℓ .
Therefore
2 10 ε ℓ r 2 ℓ ≤ E ′ [X j+1 ] ≤ 2 9 ε ℓ r 2 ℓ + P ′ [E ′ [X j+1 | Φ j ] ≥ 2 9 ε ℓ r 2 ℓ ] · 5 · 2 13 · eε 5/6
ℓ r 2 ℓ . Rearranging, we deduce that
P ′ E ′ [X j+1 | Φ j ] ≥ 2 9 ε ℓ r 2 ℓ ≥ ε 1/6 ℓ 80e ,
as required.
Lemma 5.5. Suppose ϕ j is such that P(Φ j = ϕ j ) > 0 and E[X j+1 | Φ j = ϕ j ] ≥ 2 9 ε ℓ r 2 ℓ . Then
P X j+1 ≤ 16ε ℓ r 2 ℓ | Φ j = ϕ j ≤ 1 2 .
Proof. The result is a straightforward application of Chebyshev's inequality, similarly to [14,Lemma 6.13]. First note that it follows from Lemma 5.3(ii) that
Var(X j+1 | Φ j = ϕ j ) ≤ 68ε ℓ r 2 ℓ E[X j+1 | Φ j = ϕ j ] ≤ 68 2 9 E[X j+1 | Φ j = ϕ j ] 2 ,
where in the last inequality we used that E[X j+1 |Φ j = ϕ j ] ≥ 2 9 ε ℓ r 2 ℓ by assumption. Using this again we therefore deduce that
P X j+1 ≤ 16ε ℓ r 2 ℓ Φ j = ϕ j ≤ P X j+1 ≤ 1 2 5 E[X j+1 |Φ j = ϕ j ] Φ j = ϕ j ≤ 2Var(X j+1 |Φ j = ϕ j ) E[X j+1 |Φ j = ϕ j ] 2 ≤ 1 2 .
Proof of Lemma 3.7. By Lemma 5.4, given that j i=1 X i ≤ 16ε ℓ r 2 ℓ , we get with probability at least ε
1/6 ℓ /80e that ϕ j satisfies E[X j+1 | Φ j = ϕ j ] ≥ 2 9 ε ℓ r 2 ℓ .
For every such ϕ j , by Lemma 5.5, we get that given Φ j = ϕ j , we have that X j+1 ≥ 16ε ℓ r 2 ℓ with probability at least 1/2. We conclude that
P B ⊙ j B ⊙ j−1 , (Γ n ∪ Γ x ), Cap k ℓ (Γ 5r ℓ /6 x , Γ n ∪ Γ x \ Γ 5r ℓ /6 x ) ≥ r ℓ k ℓ ε 1 6 ℓ n ≤ 1 − ε 1/6 ℓ 160e ,
as required.
6 A criterion for GHP convergence 6
.1 GP convergence
We first aim to address the convergence provided in Theorem 3.1. Recall our definitions and notation from Section 1.1 (in fact this section can be seen as a direct continuation of Section 1.1).
Definition 6.1. Let (X, d, µ) and (X ′ , d ′ , µ ′ ) be elements of X c . The Gromov-Prohorov (GP) pseudodistance between (X, d, µ) and (X ′ , d ′ , µ ′ ) is defined as
d GP ((X, d, µ), (X ′ , d ′ , µ ′ )) = inf{d P (φ * µ, φ ′ * µ ′ )},
where the infimum is taken over all isometric embeddings φ :
X → F , φ ′ : X ′ → F into some common metric space F .
Thus d GP is a metric on X GP c which is the space X c where we identify all mm-space with GP distance 0. There is a useful equivalent definition of convergence of mm-spaces with respect to the GP distance. Given an mm-space (X, d, µ) and a fixed m ∈ N we define a measure ν m ((X, d, µ)) on R ( m 2 ) to be the law of the m 2 pairwise distances between m i.i.d. points drawn according to µ. Theorem 6.2 (Theorem 5 in [12]). Let (X n , d n , µ n ) and (X, d, µ) be elements of X GP c . Then d GP ((X n , d n , µ n ), (X, d, µ)) −→ 0 , if and only if for any m ∈ N ν m ((X n , d n , µ n )) ⇒ ν m ((X, d, µ)) ,
where ⇒ denotes standard weak convergence of measures on R ( m 2 ) . This is still not quite the setting of this paper since USTs are random mm-spaces. Thus let M 1 (X GP c ) denote the space of probability measures on X GP c . Each element P ∈ M 1 (X GP c ) therefore defines random measures (ν m ) m≥2 and we additionally have annealed measures on R ( m 2 ) , given bỹ ν m (P) := X GP c ν m ((X, d, µ))dP for each integer m ≥ 2. It is often more straightforward to prove deterministic weak convergence of the measuresν m for each m ≥ 2, rather than distributional weak convergence of the random measures ν m for each m. For example, the conclusion of Theorem 3.1 can be restated as ν m UST(G n ), d n /(β n √ n), µ n ⇒ν m (CRT) ,
for any fixed m ≥ 2. However, this does not immediately imply that the USTs converge to the CRT in distribution with respect to the topology of (X GP c , d GP ), e.g. see [12,Example 2.12 (ii)]. Indeed, the random mm-spaces need not be tight. It is not hard to show that this is not the case in our setup. Lemma 6.3. Suppose that (G n ) n≥1 is a sequence of graphs satisfying Assumption 1.4. Let d n denote the graph distance on UST(G n ) and µ n the uniform probability measure on its vertices. Then there exists a sequence (β n ) n satisfying 0 < inf n β n ≤ sup n β n < ∞ such that (UST(G n ), 1 βn √ n d n , µ n ) converges in distribution to the CRT with respect the topology of (X GP c , d GP ).
Proof. We appeal to [12, Corollary 3.1] and verify conditions (i) and (ii) there. Condition (ii) is precisely (31). To verify condition (i) we use [12,Theorem 3] (and recall that by Prohorov's Theorem the relative compactness of the measures is equivalent to their tightness) and verify conditions (i) and (ii) there (see also Proposition 8.1 in [12]). Condition (i) is just saying thatν 2 is a tight sequence of measures on R, which follows from (31). Lastly, Theorem 3.2 directly implies condition (ii) [12,Theorem 3].
We remark that the use of Theorem 3.2 in the last line of the proof above is an overkill and it is not too difficult to verify condition (ii) of [12, Theorem 3] directly.
GHP convergence and the lower mass bound
The key to strengthening the GP convergence of [28], as stated in Lemma 6.3, to GHP convergence is the lower mass bound criterion of [6]. In [6, Theorem 6.1] it is shown that GP convergence of deterministic mmspaces together with this criterion is equivalent to GHP convergence. In this paper we require an extension to the setting of random mm-spaces (i.e., measures on mm-spaces); it is not hard to obtain this using the ideas of [6] and we provide it here (Theorem 6.5).
As in [6, Section 3], given c > 0 and an mm-space (X, d, µ) we define m c ((X, d, µ)) = inf x∈X {µ(B(x, c))} .
We begin with a short claim about deterministic mm-spaces. Claim 6.4. Let (X n , d n , µ n ) be a sequence of mm-spaces that is GP-convergent to (X, d, µ), i.e., d GP ((X n , d n , µ n ), (X, d, µ)) → 0 .
Suppose further that for any c > 0 we have inf n m c ((X n , d n , µ n )) > 0 .
Then, for every ε > 0 inf x∈supp(µ) µ(B(x, ε)) ≥ lim inf n→∞ inf x∈Xn µ n (B(x, ε/2)) > 0.
Proof. Fix some x ∈ supp(µ) and ε > 0. Then µ(B(x, ε/4)) ≥ b for some b = b(x, ε) > 0. Put δ = min{b/2, ε/12}. By the GP convergence there exists N ∈ N such that for every n ≥ N there are isometric embeddings taking X n and X to a common metric space (E, d ′ n ) such that the Prohorov distance between the pushforwards of their measures is smaller than δ. Therefore we may assume that X n and X are both subsets of some common metric space. We abuse notation and write µ and µ n in place of their respective pushforward measures. Since the GP distance is at most δ we get that b ≤ µ(B(x, ε/4)) ≤ µ n (B(x, ε/4 + δ)) + δ.
for n ≥ N . Hence by our choice of δ we get µ n (B(x, ε/3)) > 0. Therefore, we can find some y n ∈ X n such that d ′ n (x, y n ) < ε/3. Also, for any δ ′ ∈ (0, ε/6), we can find N 2 ∈ N such that for n ≥ N 2 we have inf y∈Xn µ n (B(y, ε/2)) ≤ µ n (B(y n , ε/2)) ≤ µ(B(y n , ε/2 + δ ′ )) + δ ′ ≤ µ(B(x, ε)) + δ ′ .
Hence, taking the lim inf on the left hand side and then taking δ ′ → 0 we obtain that for all x ∈ X lim inf n→∞ inf y∈Xn µ n (B(y, ε/2)) ≤ µ(B(x, ε)), and the claim follows by taking the infimum over x ∈ X.
We now state and prove the main goal of this section; as we state immediately afterwards, it readily shows that Theorem 3.2 implies Theorem 1.5. Theorem 6.5. Let ((X n , d n , µ n )) n≥1 , (X, d, µ) be random mm-spaces and suppose that (i) (X n , d n , µ n ) (d) −→ (X, d, µ) with respect to the GP topology.
(ii) For any c > 0, the sequence m c ((X n , d n , µ n )) −1 n≥1 is tight.
Then (X n , d n , µ n ) (d)
→ (supp(µ), d, µ) with respect to the GHP topology.
Proof. The metric space (X c , d GP ) is separable (see [6, Figure 1]), hence by the Skorohod Representation theorem, there exists a probability space on which the convergence in (i) holds almost surely. We will henceforth work on this probability space, and may therefore assume that (X n ) n≥1 and X are embedded in a common metric space where d P (X, X n ) → 0 almost surely. We will show that on this probability space, we have that (X n , d n , µ n ) −→ (supp(µ), d, µ) in probability with respect to the GHP topology, giving the required assertion.
Let ε, ε 2 > 0. By (ii), we have that there exists some c 1 > 0 and N 1 ∈ N such that for every n ≥ N 1 we have P inf x∈Xn µ n (B dn (x, ε/2)) ≤ c 1 ≤ ε 2 .
Hence by Fatou's lemma P lim sup n inf x∈Xn µ n (B dn (x, ε/2)) > c 1 ≥ 1 − ε 2 .
Meaning, with probability larger than 1 − ε 2 , we can find a (random) subsequence n k such that for every k ∈ N we have that m ε/2 ((X n k , d n k , µ n k )) = inf x∈Xn k µ n k (B dn k (x, ε/2)) > c 1 . Problem 7.3. Let T n be a uniformly drawn spanning tree of the 4-dimensional torus Z 4 n . Denote by d Tn the corresponding graph-distance in T n and by µ n the uniform probability measure on the vertices of T n . Let γ n be the sequence appearing in [30, Theorem 1.1], uniformly bounded away from 0 and infinity. Does the lower mass bound of Theorem 6.5(ii) hold for the sequence T n , dT n γnn 2 (log n) 1/6 , µ n n≥1 ?
Finally, one may also ask whether USTs rescale to the CRT under the weaker assumptions of [25], under which the authors prove that the sequence of rescaled UST diameters is tight. In particular, they do not assume transitivity but instead require that the graph is balanced ; that is, there exists a constant D < ∞ such that max v∈Gn deg v min v∈Gn deg v ≤ D for all n. It is straightforward to extend the proof of Theorem 3.2 to this setting by carrying the constant D through all our computations, but since we are still restricted by the assumption of transitivity of [28] for Theorem 1.5 we have chosen to keep the notation simple and have not pursued this here.
Department of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel Emails: [email protected], [email protected], [email protected]
d H (A, A ′ ) = max{sup a∈A d(a, A ′ ), sup a ′ ∈A ′ d(a ′ , A)}.
and so forth. By the second observation we deduce that we can continue this way until at least L =
Lemma 2. 9 .
9[24, Lemma 10.3]. Let G be a connected subgraph of a finite connected graph H. Then, UST(G) stochastically dominates UST(H) ∩ E(G).
Lemma 2 .
211. [24, Exercise 10.8]. Let (G, w) be a finite network. Let A ⊆ B be two sets of vertices. Then, UST(G/A) stochastically dominates UST(G/B).
Theorem 3. 1 .
1[28, Theorem 1.2]. Let {G n } be a sequence of graphs satisfying Assumption 1.4 and let T n be UST(G n ). Denote by d Tn the graph distance on T n and by (T , d, µ) the CRT. Then there exists a sequence {β n } satisfying 0 < inf n β n ≤ sup n β n < ∞ such that the following holds. For fixed k ≥ 1, if {x 1 , . . . , x k } are uniformly chosen independent vertices of G n , then the distances
e disjoint subintervals A 1 , . . . , A L of Γ k ℓ (A j , (Γ n ∪ Γ x ) \ A j )) ≥ 2 13 er ℓ k ℓ ε eε ℓ r 2 ℓ n for all j = 1, . . . , L. Moreover, since the cardinality of ∪ L j=1 A j is at most 5r ℓ 6 , the number of j's such that |A j | ≥ 2 13 eε1 3 ℓ r ℓ is at most 5 6·2 13 e ε − 1 3 ℓ . Hence the number of j's for which |A j | ≤ (2 13 e)ε we relabel the sets so that A i for i = 1, . . . , (2 13 e) upper bound on their size and forget about the other sets.
intervals are all good is at least
that the probability that [t 1 , t 2 ] does not contain a subinterval J = [t − J , t + J ] satisfying conditions (1) − (4) of Claim 4.2 is bounded by e − χz 12 (log ε −1 ℓ ) 2 .
Due to the buffers of length r ℓ /24 present in the beginning and ending of I it follows that (Γ * n \ I) ⊆ X[0, t − J − r ℓ /24] ∪ X t + J + r ℓ /24, M √ n ∪ W . Hence by (3) − (4) of Claim 4.2 we getwhere we used k ℓ = ε 1/2 ℓ r ℓ and |X| + |X ′ | ≤ M √ n. Consequently, since we chose c < 1 2M and z = 1/20, we can reduce ε if necessary so thatas required. Finally, to cover the case where I is primarily contained in LE(X ′ ) rather than LE(X), note that we can reverse the roles of X and X ′ above to obtain a fourth contribution to the probability of δ 4 . This concludes the proof.Proof of Theorem 3.6We assume that E n,c,ε holds and let ℓ ≤ N n be a fixed scale throughout the proof. The proof will involve applications of Claim 4.1 and Claim 4.2; for these we will take ψ = θ + 1 c 2 , take χ = min{ z 3ψ , 1 24 }, take M ℓ = ε −1/10 ℓ and take W = Γ n . These four variables will assume these values throughout the proof.Proof. Let x be some vertex of G n and let X be a random walk started from x and let τ Γn denote the time at which X hits Γ n , so that Γ x = LE(X[0, τ Γn ]). We start by upper bounding the time until X hits Γ n . On the event E n,c,ε we have from Theorem 3.3 (I) that Cap √ n (Γ n ) ≥ 2c. It therefore follows from Claim 2.3 that for each i ≥ 1,Consequently, taking a product over i ≤ M ℓ it follows thatProvided that ε is small enough as a function of c, this is much smaller than the required bound on the probability of Theorem 3.6; hence we work on the event {τ Γn ≤ M ℓ √ n} for the rest of the proof. Furthermore, if {|Γ x | ≤ r ℓ 3 }, then Γ 5r ℓ /6 x contains a segment I ⊂ Γ n with |I| = r ℓ 3 (see the definitions above Theorem 3.6). Hence on the event E n,c,ε it follows from Theorem 3.3 (II) (which we just proved in the previous subsection) thatℓ k ℓ r ℓ n , convergence implies convergence in probability, we get by assumption (i) that lim n→∞ P d P ((X n , d n , µ n ), (X, d, µ)) > ε ∧ c 1 2 = 0.Hence we can find N 2 ∈ N such that for every n ≥ max{N 1 , N 2 } with probability at least 1 − 3ε 2 the three eventsoccur. Let x ∈ supp(µ). Since the Prohorov distance between µ and µ n is smaller than ε ∧ c1 2 , we have thatHence µ n (B dn (x, 2ε)) > 0.Thus, supp(µ) ⊆ X 2ε n . We use the same argument to obtain that under this event, X n ⊆ supp(µ) 2ε and conclude that P(d GHP ((X n , d n , µ n ), (supp(µ), d, µ)) > 2ε) ≤ 3ε 2 .We therefore get that (X n , d n , µ n ) converges in probability (hence, in distribution) to (supp(µ), d, µ) in the Gromov-Hausdorff-Prohorov topology, as required.Proof of Theorem 1.5. Lemma 6.3 shows that the UST sequence converges in distribution with respect to d GP to the CRT (X, d, µ) so that condition (i) of Theorem 6.5 holds. Theorem 3.2 verifies that condition (ii) holds, and lastly, it is well known (see[1,Theorem 3]) that supp(µ) = X. The conclusion of Theorem 6.5 thus verifies Theorem 1.5.Comments and open questionsCombining with self-similarity of the CRT, Theorem 1.1 can also be used to recover the UST scaling limit in other settings. For instance, Theorem 2 of[3]entails that the branch point between three uniformly chosen points in the CRT splits the CRT into three smaller copies of itself, with masses distributed according to the Dirichlet( 1 2 , 1 2 , 1 2 ) distribution, and where each copy is independent of the others after rescaling. This together with Theorem 1.1 shows the following., the torus on (approximately) n vertices and d > 4. Sample a Dirichlet( 1 2 , 1 2 , 1 2 ) random variable, that is, a uniform triplet (∆ 1 , ∆ 2 , ∆ 3 ) on the 2-simplex. Conditioned on this, let G ⌊∆1n⌋ , G ⌊∆2n⌋ , and G ⌊∆3n⌋ be disjoint and attach each to an outer vertex of a 3-star. Let T n be the UST on the resulting graph and µ n the uniform measure on its vertices. Then (T n , 1Next, building on the corollaries in Section 1.3, one can also ask finer questions about the structure of the UST in the mean-field regime. One in particular is the convergence of the height profile.Problem 7.2. Take the setup of Theorem 1.5, and set H n (r) = #{v ∈ G n : d Tn (O, v) = r}. Does the process (H n (rβ n √ n)/ √ n) r>0 converge to its continuum analogue on the CRT? (That is, the Brownian local time process (ℓ(r)) r≥0 defined in[11,Theorem 1.1]).This does not follow straightforwardly from the GHP convergence of Theorem 1.5 since that only captures the convergence of full balls of diameter √ n with volumes of order n. (On the other hand, it is straightforward prove convergence of the rescaled volume profile V n (r) = s≤r H n (s) from GHP convergence).Next, our paper addresses the general mean-field case but leaves the upper critical dimension case of Z 4open. Here the mixing time is really of order n 1/2 , but it was shown by Schweinsberg[30]that Gromov-weak convergence to the CRT still holds with an additional scaling factor of (log n) 1/6 . Our proof of the lower mass bound does not immediately transfer to the 4-dimensional setting. However, it is possible that it is attainable to do so using the recent results of Hutchcroft and Sousi[15].
Hence by Claim 6.4, on this event we have that inf x∈supp(µ) µ(B d (x, ε)) ≥ c 1 . Next, since almost sure References. Hence by Claim 6.4, on this event we have that inf x∈supp(µ) µ(B d (x, ε)) ≥ c 1 . Next, since almost sure References
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| [] |
[
"MNRAS Letters; accepted A RECORD OF THE FINAL PHASE OF GIANT PLANET MIGRATION FOSSILIZED IN THE ASTEROID BELT'S ORBITAL STRUCTURE",
"MNRAS Letters; accepted A RECORD OF THE FINAL PHASE OF GIANT PLANET MIGRATION FOSSILIZED IN THE ASTEROID BELT'S ORBITAL STRUCTURE"
] | [
"Matthew S Clement ",
"Alessandro Morbidelli ",
"Sean N Raymond ",
"Nathan A Kaib "
] | [] | [] | The asteroid belt is characterized by an extreme low total mass of material on dynamically excited orbits. The Nice Model explains many peculiar qualities of the solar system, including the belt's excited state, by invoking an orbital instability between the outer planets. However, previous studies of the Nice Model's effect on the belt's structure struggle to reproduce the innermost asteroids' orbital inclination distribution. Here, we show how the final phase of giant planet migration sculpts the asteroid belt, in particular its inclination distribution. As interactions with leftover planetesimals cause Saturn to move away from Jupiter, its rate of orbital precession slows as the two planets' mutual interactions weaken. When the planets approach their modern separation, where Jupiter completes just short of five orbits for every two of Saturn's, Jupiter's eccentric forcing on Saturn strengthens. We use numerical simulations to show that the absence of asteroids with orbits that precess between 24-28 arcsec yr −1 is related to the inclination problem. As Saturn's precession speeds back up, high inclination asteroids are excited on to planet crossing orbits and removed from the inner main belt. Through this process, the asteroid belt's orbital structure is reshaped, leading to markedly improved simulation outcomes. | 10.1093/mnrasl/slz184 | [
"https://arxiv.org/pdf/1912.02833v1.pdf"
] | 208,857,430 | 1912.02833 | 5571e22d115c1ebd0aa877c5da0b8973809c7a81 |
MNRAS Letters; accepted A RECORD OF THE FINAL PHASE OF GIANT PLANET MIGRATION FOSSILIZED IN THE ASTEROID BELT'S ORBITAL STRUCTURE
Matthew S Clement
Alessandro Morbidelli
Sean N Raymond
Nathan A Kaib
MNRAS Letters; accepted A RECORD OF THE FINAL PHASE OF GIANT PLANET MIGRATION FOSSILIZED IN THE ASTEROID BELT'S ORBITAL STRUCTURE
Preprint typeset using L A T E X style emulateapj v. 12/16/11MNRAS Letters; accepted Keywords: minor planetsasteroids: generalplanets and satellites: formationplanets and satellites: dynamical evolution and stability
The asteroid belt is characterized by an extreme low total mass of material on dynamically excited orbits. The Nice Model explains many peculiar qualities of the solar system, including the belt's excited state, by invoking an orbital instability between the outer planets. However, previous studies of the Nice Model's effect on the belt's structure struggle to reproduce the innermost asteroids' orbital inclination distribution. Here, we show how the final phase of giant planet migration sculpts the asteroid belt, in particular its inclination distribution. As interactions with leftover planetesimals cause Saturn to move away from Jupiter, its rate of orbital precession slows as the two planets' mutual interactions weaken. When the planets approach their modern separation, where Jupiter completes just short of five orbits for every two of Saturn's, Jupiter's eccentric forcing on Saturn strengthens. We use numerical simulations to show that the absence of asteroids with orbits that precess between 24-28 arcsec yr −1 is related to the inclination problem. As Saturn's precession speeds back up, high inclination asteroids are excited on to planet crossing orbits and removed from the inner main belt. Through this process, the asteroid belt's orbital structure is reshaped, leading to markedly improved simulation outcomes.
INTRODUCTION
As the giant planets grow within the primordial gas disk, the combination of the Sun's radial force and gravitational torques from the disk and other planets rapidly shepherd them into a mutual resonant configuration (Masset & Snellgrove 2001;Morbidelli & Crida 2007). This scenario is consistent with the number of resonant giant exoplanets discovered (eg: GJ 876 and HR 8799, among others, Mills et al. 2016). After the nebular gas dissipates, the giant planets' orbits continue to evolve via interactions with leftover planetesimals in the primordial Kuiper Belt. Scattering events between these objects and the outermost gas giants preferentially displace material inward, while Jupiter tends to scatter objects out of the system entirely (Fernandez & Ip 1984). These small exchanges of angular momentum cause the giant planets' orbits to diverge, eventually destroying the resonant chain. The Nice Model describes how the global instability induced by this escape from resonance sculpts the primordial solar system into its modern form Morbidelli et al. 2005;Gomes et al. 2005;Nesvorný & Morbidelli 2012).
The precise timing of the instability has been the subject of a number of recent studies. A delayed instability would imply a correlation with the late heavy bombardment (a perceived spike in lunar cratering ∼400 Myr after gas disk dispersal), the existence of which is now in doubt (Zellner 2017;Morbidelli et al. 2018;Quarles & Kaib 2019). Furthermore, simulations of the Nice Model's effects on the fully formed inner planets routinely over-excite the fragile terrestrial worlds on to orbits where they collide with one another or are lost from the system (Brasser et al. 2013;Kaib & Chambers 2016). The "Jumping Jupiter" (Brasser et al. 2009;Roig et al. 2016) model attempts to resolve this issue by requiring that Jupiter and Saturn's semi-major axes diverge in a step-wise manner towards their modern locations as the result of a close encounter with one of the ice giants. However, studies of the scenario argue for weaker instabilities (Deienno et al. 2018) and large period ratio jumps (Toliou et al. 2016) that are low probability outcomes of statistical studies of the instability (Nesvorný & Morbidelli 2012;Deienno et al. 2017). Furthermore, authors must post-process simulation results by modifying the asteroid belt's initial inclination distribution in order to provide good matches to the modern orbital structure (Roig & Nesvorný 2015;Deienno et al. 2016Deienno et al. , 2018. Recent work argued that an early (just a few Myr after gas dissipation) instability (Nesvorný et al. 2018) might fix the terrestrial destabilization problem (Clement et al. 2018(Clement et al. , 2019a) without requiring such a specific jump. Such an evolutionary model has the advantage of providing a natural explanation for the disparity between the inferred geological accretion time-scales of Earth ( 50 Myr; Kleine et al. 2009) and Mars ( 5 Myr; Dauphas & Pourmand 2011). However, Xenon measurements from Comet 67P (Marty et al. 2017) are at odds with an instability occurring before the end of Earth's magma ocean phase. Other successful models (eg: Levison et al. 2015;Izidoro et al. 2015) for terrestrial evolution resolve problems related to Mars' size and formation time by invoking non-uniform disk conditions. In arXiv:1912.02833v1 [astro-ph.EP] 5 Dec 2019 particular, the "Grand Tack" model Walsh & Levison 2016) resolves the small Mars problem by arguing that Jupiter migrated into the terrestrial region during the nebular gas phase; thereby truncating the distribution of embryos and planetesmials near Mars' modern orbit. However, all schemes require a giant planet instability at some time to explain the outer solar system (see recent review in Raymond et al. 2018).
While the Nice Model is successful at explaining most aspects of the solar system's dynamical state, studies of its consequences in the asteroid belt are all plagued by a common pitfall (O'Brien et al. 2007;Deienno et al. 2016Deienno et al. , 2018Clement et al. 2019b). Specifically, numerical simulations overpopulate the high inclination parameter space in the inner main belt (we define the inner belt as the region of asteroids with semi-major axes less than 2.5 au). In this letter, we examine the dynamical processes responsible for this shortcoming. Additionally, we propose a mechanism through which these high-inclination asteroids are naturally removed that is compatible with any of the various terrestrial evolutionary models.
THE ASTEROID BELT INCLINATION PROBLEM
The spatial orientation of orbits in the solar system precess circularly, or rotate, on time-scales much longer than their actual orbital periods. The perturbative effects of these variations within the Keplerian problem (particularly those of the Jupiter-Saturn system) have long been known to drive dynamics in the asteroid belt (Poincare 1892;Morbidelli & Henrard 1991a). In the secular theory of solar system evolution (eg: Milani & Knezevic 1990;Morbidelli & Henrard 1991a;Murray & Dermott 1999), the long-term behavior of the eight planets' orbital eccentricities (e i ) and longitudes of perihelia ( i ) are described by the solutions to the secular equations of motion:
e i cos i = 8 j M ij cos (g j t + β j ) e i sin i = 8 j M ij sin (g j t + β j )(1)
The same analysis can be applied to the precession of the planets' inclination nodes; specifically the behavior of the orbital inclination (sin i/2) and longitude of ascending node (Ω). Secular resonances occur when an object precesses at a rate equal to one of the solar system's dominant eigenfrequencies. These eigenfrequencies are denoted g 1 -g 8 for the planets' eccentricity vector precessions, and s 1 -s 8 for the inclination node precessions. The ν 6 resonance is comprised of the orbital semi-major axes and inclinations with precession rates that match the g 6 rate of 28.22 arcsec yr −1 , and cuts across the modern asteroid belt (see Fig. 4). There is a clear deficiency of inner belt asteroids with inclinations above the ν 6 resonance and precessions slower than g 6 relative to Nice Model predictions (Morbidelli et al. 2010). In the modern asteroid belt, the ratio of large (D > 50 km) asteroids with inclinations above the ν 6 resonance, to those below is ∼0.08 (referred to in the subsequent text as the ν 6 ratio).
In the current version of the Nice Model, the orbital eccentricities and inclinations of asteroids are excited by secular resonances rapidly moving across the belt region (O'Brien et al. 2007;Deienno et al. 2018). Because simulations of the instability begin with the giant planets in a more compact configuration, their orbits precess at different rates than they currently do. The locations of their respective secular resonances are displaced as the planets' orbits change during, and after the instability. Of most importance for the inner asteroid belt, the ν 6 resonance must traverse from ∼4.5 au to its modern location at ∼2.05 au, and the ν 16 (inclination nodes precessing equal to s 6 ) resonance must sweep from ∼2.8 au to ∼1.9 au . The ν 16 resonance moves inward, and encounters the inner belt first. This process excites inclinations, but leaves the eccentricities of asteroids unaffected. Because of the characteristic shape of ν 6 (Fig 4), its movement only excites the eccentricities of low-inclination asteroids in the inner belt (often on to planet crossing orbits). Through these processes, many asteroids are stranded on relatively stable, highinclination, low-eccentricity orbits in the inner main belt (Morbidelli et al. 2010;Clement et al. 2019b).
The fraction of inner main belt asteroids isolated above ν 6 is tied to the smoothness of the giant planets' migration. Studies of smooth migration utilizing artificial forces substantially deplete the a/i parameter space below ν 6 , and simultaneously overpopulate the high-i region of phase space (Morbidelli et al. 2010;. However, lower ν 6 ratios can be achieved with a "Jumping Jupiter" style instability (Brasser et al. 2009;Roig & Nesvorný 2015;Deienno et al. 2016Deienno et al. , 2018, or when the full chaos of the event is considered (Clement et al. 2018(Clement et al. , 2019b. While many inner belt asteroids' inclinations are still over-excited in these scenarios, a substantial number survive giant planet migration with inclinations below ν 6 because the important secular resonances do not linger at any particular location (Clement et al. 2018).
DEPLETION MECHANISM
Dynamical instabilities are inherently stochastic, and each follows a unique path. When studying the Nice Model, authors typically select systems in which the giant planets' final orbits are closest to those of the modern solar system (Clement et al. 2018;Deienno et al. 2018). However, this does not guarantee that the simulated planets followed the same evolutionary path as the real ones. As Saturn's orbit moves away from the Sun following the instability, its precession rate continues to decrease, thereby lowering g 6 to its modern value. Thus the crux of the Nice Model's ν 6 problem has been in finding a mechanism to deplete asteroids that precess slower than g 6 after the instability strands them above ν 6 . However, these previous studies have neglected the precise effects induced by Jupiter and Saturn's specific modern configuration. Presently, the solar system's two most massive planets lie just inwards of a mutual 5:2 mean motion resonance (MMR), with Jupiter completing ∼4.97 orbits for every two of Saturn's. Secular precessions are known to speed-up near MMRs (Milani & Knezevic 1990;Morbidelli & Henrard 1991a). Fig. 1 demonstrates the behavior of the g 6 rate as Jupiter and Saturn approach their modern configuration. Perturbative derivations of the three-body secular Hamiltonian typically expand the problem in Taylor series, and subsequently neglect mass terms of order two or higher (for a full discussion of secular resonances in the asteroid belt see Milani rard 1991a,b). This simplification holds when the bodies' mean longitudes (λ i in Delaunay variables) are nonresonant. When the two objects approach a MMR, the quadratic mass term is no longer negligible, and the precession rate g i increases asymptotically. Our first set of simulations are designed to measure the solar system's g 6 eigenmode as Jupiter and Saturn approach the 5:2 MMR. We perform 3,200 integrations of the modern solar system with the M ercury6 hybrid integrator (Chambers 1999). In each run, Saturn's semi-major axis is decreased by 0.005 au, and all other orbital elements are left unchanged. Each system is integrated for 10 Myr, and the secular amplitudes and frequencies are calculated via Fourier analysis of the simulation time outputs (Šidlichovský & Nesvorný 1996). Through this process, we generate the curve presented in Fig. 1.
As the value of g 6 lowers and rises, the ν 6 resonance shape sweeps from right to left and back in a/i space. We argue that this reversal in sweeping of the ν 6 resonance explains the depletion of asteroids with precession rates less than the current value of g 6 (Figs. 2 and 3) and inclinations above ν 6 in the inner belt region (Fig. 4, top panel).
NUMERICAL SIMULATIONS
Next, we use numerical simulations to demonstrate how the reversal of g 6 by just ∼2.5 arcsec yr −1 can affect a uniform population of main belt asteroids about the ν 6 secular resonance (migrating the Jupiter-Saturn period ratio from ∼2.45-2.49). We use the GEN GA (Grimm & Stadel 2014) integration package for this phase of our study. We first test different migration time-scales by performing three separate simulations of the solar system and 10,000 massless test particles. Asteroid orbital elements are selected randomly from uniform distributions of non-planet-crossing orbits (2.0 < a < 4.0 au, 0.0 < e < 0.5, 0.0 < i < 40.0 • and 0-360 • for angular orbital elements). Simply put, the migration is achieved by minor alterations to Saturn's semi-major axis (by cubic interpolation utilizing GEN GA s built in Set Elements function) such that the Jupiter-Saturn period ratio evolution follows an exponential function of time un- Simulation minimum of g 6
Perihelion precession rates in the modern asteroid belt D <50 km D >50 km Figure 2. Distribution of orbital precession rates as a function of semi-major axis for all known asteroids with constrained orbits (Knežević & Milani 2003). The horizontal lines represent the current value of Saturn's g 6 eigenfrequency, and the minimum value obtained from Fig. 1. Red points correspond to large asteroids for comparison with the bottom panel of Fig. 4 (note that the total number of points is less here as this figure is zoomed in on the range of 15 < g < 35). The asteroids in between the red lines near ∼3.1 au are members of the high-inclination collisional family (31) Euphrosyne (Novaković et al. 2011). After the break-up of Euphrosyne below the bottom red line, the family members filled the gap that was presumably emptied by primordial migration as the result of semi-major axis spreading due to the Yarkovsky Effect (eg: Bottke et al. 2001).
til reaching the modern value. Each system is integrated for an additional 100 Myr to remove quasi-stable asteroids. Our selected migration speeds (τ mig ) are loosely based on studies of Saturn's smooth migration's effect on the asteroid belt's structure (as we seek to study uniform migration after the Nice Model instability; Minton & Malhotra 2011). Specifically, our fastest migration (τ mig =3 Myr) is selected to equal the slowest sweeping of ν 6 (through the bulk of it's migration from 2.8-2.1 au) that permits the asteroid belt's survival. Because we are only interested in the final phase of migration and clearing in the young solar system, our selected migration speeds are quite slow. The results of these simulations are summarized in Table 1 and Fig. 3. Through the full migration process, the ν 6 ratio consistently drops from 1.98 to less than unity. Asteroids with high inclinations that would have been unaffected if g 6 had never dipped below ∼28 arcsec yr −1 are quickly swept up and elevated in eccentricity via resonant perturbations (Fig. 3). Once excited, they are eventually removed from the belt as the result of encounters with the terrestrial planets (largely Earth and Mars). Through this process, the inner belt's overall ν 6 ratio is substantially reduced. Furthermore, these results are independent of the migration time-scale selected (3.0, 10.0 and 30.0 Myr). Since Saturn's final migration has a strong effect on the inner main belt's population above the ν 6 resonance regardless of migration speed, we limit τ mig to 5 Myr for the remainder of our study (Morbidelli et al. 2010;Toliou et al. 2016).
As evidenced by the reduction of the ν 6 ratio by a factor of three or so, this mechanism is successful at removing asteroids near ν 6 , but cannot be solely responsible for Table 1 Initial conditions and results for simulations of a uniform distribution of asteroids: the columns are as follows: (1) the simulation number, (2) the total simulation time, (3) Saturn's average migration speed for the first 0.04 au, and (4-5) the initial and final ratios of inner main belt (a < 2.5 au, i < 40 • ) asteroids above to those below the ν 6 resonance.
Run τ migṙSat ν 6 ratio i ν 6 ratio f Nesvorný 1996;Knežević & Milani 2003). The colour of each point corresponds to the object's inclination. Note that, as with Fig. 2, the total number of points here is not 10,000 since many asteroids precess faster or slower than the range of values plotted (particularly in the outer main belt).
generating the modern ratio. Therefore, the ratio must also be limited in Jupiter's jump phase (Roig & Nesvorný 2015;Deienno et al. 2016;Toliou et al. 2016;Clement et al. 2018;Deienno et al. 2018;Clement et al. 2019b). In our next set of simulations, we investigate asteroid belts formed via terrestrial accretion models (Clement et al. 2018) that experienced a range of jumps. We begin by selecting all surviving asteroids from successful simulations of the Early Instability scenario in Clement et al. (2019b) that finished with P Sat /P Jup <2.8 (using the nomenclature of that work these are runs 1, 3, 6, 1b and 2b). Each system in Clement et al. (2019b) was evolved for 200 Myr, through the Nice Model instability and giant impact phase of terrestrial planet formation. Because a giant planet instability of arbitrary timing is invoked to explain the outer solar system in all terrestrial planet formation models, our initial conditions can be considered roughly independent of evolutionary scheme . To improve statistics, we generate 10 separate, 1,000-particle belts by randomly choosing asteroids from these completed simulations, and slightly altering their semi-major axes, eccentricities and inclinations. These small positive and negative deviations are made via random sampling of Rayleigh distributions (σ a = 0.025 au, σ e = 0.025 and σ i = 1.0 • ). All 8 planets, and the three largest modern asteroids are included for (Clement et al. 2019b): the columns are as follows: (1) the simulation number, and (2-3) the initial and final ratios of inner main belt (a < 2.5 au, i < 40 • ) asteroids above to those below the ν 6 resonance. these simulations. We provide the initial and final ν 6 ratios for these simulations in Table 2. The median initial ν 6 ratio for our simulations is ∼1.4 (as compared to the modern solar system value of ∼0.08). After 100 Myr of evolution, the overall result of Fig. 3 and Table 1 holds. In the two outlier simulation (runs 3 and 8), the sweeping of ν 6 destabilized many low-i, as well as high-i asteroids. Therefore, while the inclination parameter space above ν 6 was well depleted, the final ratio was still poor.
The inclination structure of a successful simulation (run 5) is plotted in Fig. 4. While the top panel (the post-planet formation belt) broadly matches the concentrations of modern asteroids in different radial bins, the inner main belt is significantly over-populated above the ν 6 resonance. Contrarily, the middle panel (results of this study) is in better overall agreement with the observed belt (final ν 6 ratio of 0.33). From this figure, it is clear that the sweeping most efficiently removes highinclination objects in the inner main belt. Since all objects with a 2.3 au have high inclinations initially (the result of the location and movement of ν 16 , as discussed above), the region's final a/i structure is altered more dramatically than the 2.3 a 2.5 au region.
CONCLUSIONS
The Nice Model Morbidelli et al. 2005;Gomes et al. 2005;Nesvorný & Morbidelli 2012;Deienno et al. 2017) offers the most consistent explanation for the solar system's precise dynamical state. Numerous authors have investigated the model's effect on the asteroid belt's orbital distribution (Morbidelli et al. 2010;Roig & Nesvorný 2015;Deienno et al. 2016;Clement et al. 2018;Deienno et al. 2018;Clement et al. 2019b). However, previous studies have consistently struggled to match the asteroid belt's inclination population about the ν 6 resonance. We have shown that this is likely resolved when Jupiter and Saturn's precise approach to their 5:2 MMR is considered along with the aforementioned works. Our current work, coupled with the well developed Nice Model, thus represents a comprehensive picture of the young solar system's formation and early evolution. NNA13AA93A. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications.
Figure 1 .
1Evolution of the solar system's g 6 eigenfrequency during Saturn's final phase of migration. The modern value of g 6 is denoted by a bold point. The figure's minimum is at P Sat /P Jup = 2.46 and g 6 = 25.89.
Figure 3 .
3Final asteroid precession rates from our τ mig = 30 Myr simulation of a uniform distribution of asteroids (table 1, run 3). Precession rates are computed by frequency modulated Fourier Transform (see:Šidlichovský &
Figure 4 .
4Inclination distribution of our simulated asteroid belt compared with the observed structure. The vertical dashed lines correspond to the semi-major axes of several important mean motion resonances with Jupiter. The bold dashed lines represent the approximate, eccentricity-averaged orientation of the ν 6 secular resonance in a/i space. The top panel depicts the initial conditions for a successful simulation. The middle panel illustrates the same belt's inclination structure following 5 Myr of Saturn's final migration inducing a reversal in the sweeping of ν 6 , and 100 Myr of subsequent evolution in the presence of a steady-state solar system. The bottom panel shows the present day asteroid belt's a/i distribution (only bright objects with absolute magnitude H < 9.7, approximately corresponding to D > 50 km, are plotted). United States. M.S.C. and N.A.K. thank the National Science Foundation for support under award AST-1615975. S.N.R. acknowledges NASA Astrobiology Institutes Virtual Planetary Laboratory Lead Team, funded via the NASA Astrobiology Institute under solicitation NNH12ZDA002C and cooperative agreement no.
& Knezevic 1990; Morbidelli & Hen-2.40
2.42
2.44
2.46
2.48
2.50
P Sat /P Jup
Table 2
2Results for simulations of asteroid belts generated via 200 Myr planet formation simulations
ACKNOWLEDGEMENTSThis material is based upon research supported by the Chateaubriand Fellowship of the Office for Science and Technology of the Embassy of France in the
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| [] |
[
"Consistent LDA ′ +DMFT -an unambiguous way to avoid double counting problem: NiO test",
"Consistent LDA ′ +DMFT -an unambiguous way to avoid double counting problem: NiO test"
] | [
"Pis'ma V Zhetf \nInstitute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia\n",
"I A Nekrasov \nInstitute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia\n",
"N S Pavlov \nInstitute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia\n\nInstitute for Metal Physics\nRussian Academy of Sciences\nUral Branch\nS.Kovalevskoi str. 18620990EkaterinburgRussia\n",
"M V Sadovskii "
] | [
"Institute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia",
"Institute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia",
"Institute for Electrophysics\nRussian Academy of Sciences\nUral Branch\nAmundsen str. 106620016EkaterinburgRussia",
"Institute for Metal Physics\nRussian Academy of Sciences\nUral Branch\nS.Kovalevskoi str. 18620990EkaterinburgRussia"
] | [] | We present a consistent way of treating a double counting problem unavoidably arising within the LDA+DMFT combined approach to realistic calculations of electronic structure of strongly correlated systems. The main obstacle here is the absence of systematic (e.g. diagrammatic) way to express LDA (local density approximation) contribution to exchange correlation energy appearing in the density functional theory. It is not clear then, which part of interaction entering DMFT (dynamical mean-field theory) is already taken into account through LDA calculations. Because of that, up to now there is no accepted unique expression for the double counting correction in LDA+DMFT. To avoid this problem we propose here the consistent LDA ′ +DMFT approach, where LDA exchange correlation contribution is explicitly excluded for correlated states (bands) during self-consistent band structure calculations. What is left out of Coulomb interaction for those strongly correlated states (bands) is its non-local part, which is not included in DMFT, and the local Hartree like contribution. Then the double counting correction is uniquely reduced to the local Hartree contribution. Correlations for strongly correlated states are then directly accounted for via the standard DMFT. We further test the consistent LDA ′ +DMFT scheme and compare it with conventional LDA+DMFT calculating the electronic structure of NiO. Opposite to the conventional LDA+DMFT our consistent LDA ′ +DMFT approach unambiguously produces the insulating band structure in agreement with experiments. 71.27.+a, 71.28.+d, 74.25.Jb, | 10.1134/s0021364012110070 | [
"https://arxiv.org/pdf/1204.2361v1.pdf"
] | 119,276,019 | 1204.2361 | 663886498f7e5f97de09062b423f78e87c67083c |
Consistent LDA ′ +DMFT -an unambiguous way to avoid double counting problem: NiO test
11 Apr 2012
Pis'ma V Zhetf
Institute for Electrophysics
Russian Academy of Sciences
Ural Branch
Amundsen str. 106620016EkaterinburgRussia
I A Nekrasov
Institute for Electrophysics
Russian Academy of Sciences
Ural Branch
Amundsen str. 106620016EkaterinburgRussia
N S Pavlov
Institute for Electrophysics
Russian Academy of Sciences
Ural Branch
Amundsen str. 106620016EkaterinburgRussia
Institute for Metal Physics
Russian Academy of Sciences
Ural Branch
S.Kovalevskoi str. 18620990EkaterinburgRussia
M V Sadovskii
Consistent LDA ′ +DMFT -an unambiguous way to avoid double counting problem: NiO test
11 Apr 2012Submitted March 2012arXiv:1204.2361v1 [cond-mat.str-el]7120-b7127+a7128+d7425Jb
We present a consistent way of treating a double counting problem unavoidably arising within the LDA+DMFT combined approach to realistic calculations of electronic structure of strongly correlated systems. The main obstacle here is the absence of systematic (e.g. diagrammatic) way to express LDA (local density approximation) contribution to exchange correlation energy appearing in the density functional theory. It is not clear then, which part of interaction entering DMFT (dynamical mean-field theory) is already taken into account through LDA calculations. Because of that, up to now there is no accepted unique expression for the double counting correction in LDA+DMFT. To avoid this problem we propose here the consistent LDA ′ +DMFT approach, where LDA exchange correlation contribution is explicitly excluded for correlated states (bands) during self-consistent band structure calculations. What is left out of Coulomb interaction for those strongly correlated states (bands) is its non-local part, which is not included in DMFT, and the local Hartree like contribution. Then the double counting correction is uniquely reduced to the local Hartree contribution. Correlations for strongly correlated states are then directly accounted for via the standard DMFT. We further test the consistent LDA ′ +DMFT scheme and compare it with conventional LDA+DMFT calculating the electronic structure of NiO. Opposite to the conventional LDA+DMFT our consistent LDA ′ +DMFT approach unambiguously produces the insulating band structure in agreement with experiments. 71.27.+a, 71.28.+d, 74.25.Jb,
INTRODUCTION
During the last 15 years the so called LDA+DMFT approach (local density approximation + dynamical mean-field theory) became a common tool to describe band structure of real strongly correlated materials [1,2,3,4,5,6]. In this approach the results of LDA band structure calculations are supplemented with local Coulomb (Hubbard) interaction term for those states which are counted as strongly correlated. Formally the LDA+DMFT Hamiltonian can be written aŝ
H =Ĥ LDA −Ĥ DC + 1 2 i=i d ,l=l d mσ,m ′ σ ′ ′ U σσ ′ mm ′nilmσnilm′ σ ′ − 1 2 i=i d ,l=l d ′ mσ,m ′σ J mm ′ĉ † ilmσĉ † ilm ′σĉ ilm ′ σĉilmσ .(1)
Here U σσ ′ mm ′ are the most important matrix elements of Coulomb matrix (Coulomb repulsion and z-component of Hund's rule coupling) and J mm ′ are spin-flip terms of Hund's rule couplings between the strongly correlated electrons (assumed here to be d-states, enumerated by i = i d and l = l d ). The prime on the sum indicates 3) E-mail: [email protected] 3) E-mail: [email protected] 3) E-mail: [email protected] that at least two of the indices of operators have to be different, andσ =↓ (↑) for σ =↑ (↓).
The LDA part of the Hamiltonian (1) is given by:
H LDA = − 2 2m e ∆ + V ion (r) + d 3 r ′ ρ(r ′ )V ee (r−r ′ ) + δE LDA xc (ρ) δρ(r) ,(2)
where ∆ is the Laplace operator, m e the electron mass, e the electron charge, and
V ion (r) = −e 2 i Z i |r − R i | , V ee (r−r ′ ) = e 2 2 r =r ′ 1 |r − r ′ |(3)
denote the one-particle potential due to all ions i with charge eZ i at given positions R i , and the electronelectron interaction, respectively.
The E LDA xc (ρ(r)) in (2) is a function of local charge density which approximates true exchange correlation functional E xc [ρ] of density functional theory in the framework of local density approximation [7]. The form of the function E LDA xc (ρ(r)) is usually calculated from perturbation theory [8] or numerical simulations [9] of the "jellium" model with V ion (r) = const. Once we choose some basis set of one-particle wave functions ϕ i (e.g. to do practical calculations and explicitly express matrix elements of the Hamiltonian (2)), we can obtain ρ as:
ρ(r) = N i=1 |ϕ i (r)| 2 .(4)
Finally a termĤ DC is subtracted in Eq. (1) to avoid double-counting of those contributions of the local Coulomb interaction already contained inĤ LDA via Hartree term and E LDA xc (ρ(r)). Since there does not exist a direct microscopic or diagrammatic link between the model (Hubbard like) Hamiltonian approach and LDA it is not possible to expressĤ DC rigorously in terms of U , J and ρ. Thus there is no unique and accepted expression forĤ DC (see e.g. Ref. [10]).
One popular expression forĤ DC is the Hartree like (fully localized limit) expression [11]:
H DC = 1 2 U n d (n d − 1) − 1 2 J σ n dσ (n dσ − 1). (5)
Here, n dσ = m n il d mσ = m n il d mσ is total number of electrons on interacting orbitals per spin, n d = σ n dσ , U is Coulomb (Hubbard) repulsion and J is the exchange or Hund's rule coupling obtained usually from constrained LDA procedure [12]. The n d value can be obtained either from LDA calculations or can be recalculated during the DMFT loop. Practically, the values obtained are pretty close to each other.
Below we introduce the consistent LDA ′ +DMFT approach, which allows one to avoid the double counting problem unambiguously. To illustrate the advantages of this new approach we shall apply it to calculations of the band structure of the well known prototype of charge transfer insulating system NiO.
CONSISTENT LDA ′ +DMFT APPROACH
One of the possible ways to solve the double counting problem is to perform Hartree+DMFT or Hartree-Fock+DMFT calculations (see for the overview of the concept Ref. [13]). This approach uses the advantage of knowledge of diagrammatic expression for Hartree or Hartree-Fock terms. Thus, performing Hartree-Fock band structure calculations for real materials we do know exactly what portion of interaction is, in fact, explicitly included. Then obviously, the double counting term should be chosen in the form of Eq. (5). However, up to now we are unaware of any Hartree+DMFT or Hartree-Fock+DMFT calculations for real materials.
In fact, Hartree-Fock band structure calculations are in some sense a large step backwards from DFT/LDA approach, which was so successful in description of many real materials. Even in the case of strongly correlated systems DFT/LDA is recognized as a best starting point for further model Hamiltonian treatments, such as e.g. LDA+DMFT method.
In view of this we suggest a kind of compromise between Hartree-Fock and DFT/LDA starting points to be followed by DMFT calculations. As described above main obstacle to express double counting term exactly is exchange correlation E LDA xc (ρ(r)) portion of interaction within LDA. It seems somehow inconsistent to use it to describe correlation effects in narrow (strongly correlated) bands from the very beginning, as these should be treated via more elaborate schemes like DMFT. To overcome this difficulty for these states, we propose to redefine charge density (4) in E LDA xc as follows: This approach to describe realistic strongly correlated systems we shall call the consistent LDA ′ +DMFT.
ρ ′ (r) = i =i d |ϕ i (r)| 2 (6)
It is in precise correspondence with the standard definition of correlations, as interaction corrections "above" Hartree-Fock. We explicitly exclude contributions to E LDA xc from (strongly) correlated bands, where correlations are treated via DMFT, while we take all electrons into account in LDA calculations for all other (non correlated) bands.
RESULTS
Following many recent works [10,14,15] (and references therein) we choose as a testing system the prototype charge transfer insulator NiO. LDA band structure calculations for NiO were performed within the linearized muffin-tin orbitals (LMTO) basis set [16]. In the corresponding program package TB-LMTO v.47 the E LDA xc was taken in von Barth-Hedin form [8]. In the Fig. 1 we present LDA densities of states (left panel) and band dispersions (right panel) of NiO. Band dispersions consist of two separate sets of bands: the O-2p bands (from -3 to -9 eV) and Ni-3d bands, crossing the Fermi level (from 1.5 to -3 eV). Dashed lines in Fig. 1 for metallic (LDA produces metallic state for NiO) electron densities r s =2-6 are known to be of the order of 1 eV [9]. Further we perform DMFT calculations using LDA and LDA ′ Hamiltonians, which include all states (without any projecting). DMFT impurity solver used was Hirsh-Fye quantum Monte-Carlo algorithm [17]. Inverse temperature was taken β = 5eV −1 (2321 K) and 80 time slices were used, with 10 6 Monte Carlo sweeps. The use of very high temperature does not lead to any qualitative effects in the results, allowing us to avoid unnecessary computational efforts. Parameters of Coulomb interaction were chosen as typical for NiO [10,15]: U =8 eV and J=1 eV. To obtain DMFT(QMC) densities of states at real energies, we employed the maximum entropy method [18].
In the Fig. 2 Within conventional LDA+DMFT we obtain the metallic solution, which contradicts experiments. This fact can be explained as follows. We already mentioned that LDA and LDA ′ calculations results differ mainly by the values of charge transfer energy ∆ = |ε d − ε p |. In fact, we observed [19] that double counting correction essentially affects ∆, or the other way around, the different values of ∆ require the different values of double counting corrections to obtain the same results. In its turn, the different values of double counting correction can lead either to metallic or insulating solutions for the same set of other parameters [10,19].
Once we employ the consistent LDA ′ +DMFT approach, we obtain the charge transfer insulating solution for NiO, which agrees well with other LDA+DMFT calculations for NiO [10,15] and experiment [20], confirm- ing the effectiveness of our approach. Namely, the peak at -2 eV which consists almost in equal parts from Ni-3d and O-2p states is nothing else but Zhang-Rice bound state (in agreement with Ref. [15]). Lower Hubbard band formed mainly from Ni-3d states is located lower in energy than Zhang-Rice band. Conducting band is just the upper Hubbard band dominated by Ni-3d states.
As an additional check of consistency of our approach we also performed LDA ′ +DMFT calculations for SrVO 3 . The results obtained are in good agreement with those obtained in Ref. [21], further validating our proposed LDA ′ +DMFT approach as an effective and unambiguous method of band structure calculations for strongly correlated systems.
CONCLUSION
In this work we proposed the consistent LDA ′ +DMFT approach, which solves the problem of non-uniqueness of the double counting correction. By excluding LDA exchange correlation contribution for correlated states within the self-consistent LDA calculations (e.g. for Ni-3d states) we end up with just Hartree like portion of interaction for (strongly) correlated states. Then we know exactly, what should be subtracted as a double counting correction term, while merging LDA ′ and DMFT. We tested our consistent LDA ′ +DMFT approach, calculating the band structure of NiO. We obtained the insulating solution without any additional fitting parameters and in general agreement with experimental data [20], while in other LDA+DMFT works for NiO the double counting correction was either treated as an adjustable parameter [10], or the special form of double counting term was introduced [15] to achieve agreement with experiment.
We thank A. Poteryaev for providing us QMC code and many helpful discussions. This work is partly supported by RFBR grant 11-02-00147 and was performed within the framework of programs of fundamental research of the Russian Academy of Sciences (RAS) "Quantum mesoscopic and disordered structures" (12-Π-2-1002) and of the Physics Division of RAS "Strongly correlated electrons in solids and structures" (012-T-2-1001).
excluding the contribution of the density of strongly correlated electrons. Then this redefined ρ ′ (r) is used to obtain E LDA xc and perform the self-consistent LDA band structure calculations for correlated bands at the initial stage of LDA+DMFT, while correlations of d-electrons are left to be treated via DMFT. This means that what is left for correlated states out of interaction on the LDA stage would be just the Hartree contribution of Eq. (2). At the same time all other states (not counted as strongly correlated) are to be treated with the full power of DFT/LDA and full ρ in E LDA xc . Now, the problem of double counting correction is uniquely definedit should be taken in the form of the Hartree like term, given by Eq. (5).
Fig. 1 .
1LDA and LDA ′ calculated band dispersions and densities of states of NiO. The Fermi level EF is at zero energy.
LDA ′ +DMFT (lower panel) results for NiO. Different lines represent partial Ni-3d(t 2g ) (solid line), Ni-3d(e g ) and oxygen O-2p (dash-dot line) contributions to density of states. To obtain O-2p states DMFT(QMC) self-energy was analytically continued to real frequencies by Pade approximant method. For both conventional LDA+DMFT and consistent LDA ′ +DMFT calculations we used H DC of Eq. (5) with n d recalculated on each DMFT iteration step. Corresponding values of H DC are 62 eV (n d =8.7) and 58.13 eV (n d =8.2) for conventional LDA+DMFT and consistent LDA ′ +DMFT respectively. The total occupancies of Ni-3d states within LDA and LDA ′ calculations were 8.5 and 8.3.
Fig. 2 .
2Consistent LDA ′ +DMFT (lower panel) and LDA+DMFT (upper panel) partial densities of states for NiO. The Fermi level is at zero energy.
show conventional LDA results. Full lines correspond to LDA ′ calculations without E LDA Overall changes can be characterized as an almost rigid shift of oxygen states down in energy by about 1 eV for LDA ′ calculations, while Ni-3d states are only slightly modified due to change of Ni-O hybridization. In other words LDA ′ calculations lead to the change of charge transfer energy ∆ = |ε d − ε p | by about 1 eV. Rather small influence of E LDA xc on Ni-3d states is not surprising, since E LDA xcxc
on Ni-
3d states, namely, with redefined charge density (6) in
E LDA
xc .
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| [] |
[
"FINITE EDGE-TRANSITIVE ORIENTED GRAPHS OF VALENCY FOUR WITH CYCLIC NORMAL QUOTIENTS",
"FINITE EDGE-TRANSITIVE ORIENTED GRAPHS OF VALENCY FOUR WITH CYCLIC NORMAL QUOTIENTS"
] | [
"Jehan A Al-Bar ",
"Ahmad N Al-Kenani ",
"ANDNajat Mohammad Muthana ",
"Cheryl E Praeger "
] | [] | [] | We study finite four-valent graphs Γ admitting an edge-transitive group G of automorphisms such that G determines and preserves an edgeorientation on Γ, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Γ and we classify all examples. We show there are five infinite families of such pairs (Γ, G), and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the subclass of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.Normal quotients and basic graphs of cycle type: Let OG(4) denote the family of all pairs (Γ, G), where Γ is a finite simple connected undirected graph of valency 4, and G ≤ Aut(Γ) is a vertex-transitive and edge-transitive group of | 10.1007/s10801-017-0749-3 | [
"https://arxiv.org/pdf/1612.06024v1.pdf"
] | 119,149,562 | 1612.06024 | 5900ed984a2da968e9bafdfa9fd1936a3881ea43 |
FINITE EDGE-TRANSITIVE ORIENTED GRAPHS OF VALENCY FOUR WITH CYCLIC NORMAL QUOTIENTS
19 Dec 2016
Jehan A Al-Bar
Ahmad N Al-Kenani
ANDNajat Mohammad Muthana
Cheryl E Praeger
FINITE EDGE-TRANSITIVE ORIENTED GRAPHS OF VALENCY FOUR WITH CYCLIC NORMAL QUOTIENTS
19 Dec 2016arXiv:1612.06024v1 [math.CO]
We study finite four-valent graphs Γ admitting an edge-transitive group G of automorphisms such that G determines and preserves an edgeorientation on Γ, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Γ and we classify all examples. We show there are five infinite families of such pairs (Γ, G), and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the subclass of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.Normal quotients and basic graphs of cycle type: Let OG(4) denote the family of all pairs (Γ, G), where Γ is a finite simple connected undirected graph of valency 4, and G ≤ Aut(Γ) is a vertex-transitive and edge-transitive group of
Introduction
Finite edge-transitive oriented graphs of valency four have been studied intensively because of their links to maps on Riemann surfaces [6,8]. They are simple undirected graphs of valency 4 which admit an orientation of their edges determined and preserved by the action of a vertex-transitive and edge-transitive automorphism group. The work of Marušič (summarised in [6], but see also [7,9,11]) demonstrated the importance of a certain family of cycles, occurring as subgraphs, for understanding the internal structure of these graphs. This approach has been exploited recently by Marušič andŠparl [10] for the sub-family of weak metacirculants, to give a classification scheme for this sub-family. On the other hand, pursuing a different approach, recent work of the authors in [1] suggests that cycles occurring as normal quotients play a special role. In this paper we study these graphs which have a cycle (oriented or unoriented) as a normal quotient, and classify the graphs with at least two 'independent' cyclic normal quotients (as defined in Definition 1.1). We compare this classification with the analysis in [10] for weak metacirculants in Subsection 1.1.
automorphisms with a specified G-orbit ∆ on ordered vertex-pairs consisting of one ordered pair for each edge. In the literature a G-action with these properties is said to be 1 2 -transitive. An edge orientation is defined as follows: orient each edge {x, y} of Γ from x to y if (x, y) ∈ ∆. Then Γ is said to be G-oriented. Our notation suppresses the orbit ∆, and we note that the edge-orientation is determined by G up to possibly replacing ∆ by {(x, y) | (y, x) ∈ ∆} (which corresponds to reversing the orientation of each edge).
For (Γ, G) ∈ OG(4) with vertex set X, and a normal subgroup N of G, the normal quotient Γ N of (Γ, G) has as vertices the N -orbits in X, and a pair {B, C} of distinct N -orbits forms an edge of Γ N if and only if there is at least one edge {x, y} of Γ with x ∈ B and y ∈ C. The quotient Γ N is proper if N = 1 (so that Γ N is strictly smaller than Γ). There is a constant ℓ, independent of the adjacent pair B, C, such that each vertex of B is joined by an edge in Γ to exactly ℓ vertices of C, [1, Proposition 3.1]. If ℓ = 1 then the kernel of the G-action on Γ N is semiregular and hence equal to N , and we say that (Γ, G) is a normal cover of (Γ N , G/N ).
If all edges from B to C have the same G-orientation, then Γ N inherits a Ginvariant orientation from Γ. For any normal subgroup N , by [1, Theorem 1.1], either (Γ, G) is a normal cover of (Γ N , G/N ) and (Γ N , G/N ) ∈ OG (4), or the quotient Γ N is degenerate: consisting of a single vertex (if N is transitive), or a single edge (if the N -orbits form the bipartition of a bipartite graph Γ), or Γ N is a cycle possibly, but not necessarily, inheriting a G-orientation of its edges. Thus, apart from these degeneracies, the family OG(4) is closed under the normal quotient operation, and much can be learned from studying its 'basic members', namely those pairs (Γ, G) ∈ OG (4) for which all proper normal quotients are degenerate. A study of these basic pairs was initiated in [1].
In this paper we focus on pairs (Γ, G) ∈ OG(4) with at least one cyclic normal quotient. The alternative approach in [10] mentioned above also uses certain graph quotients to subdivide the weak metacirculants in OG(4) into four broad classes. Our discussion in Subsection 1.1 shows that the graph quotients corresponding to two of these four classes can never arise as normal quotients in our sense, and only those for 'Class I' of [10] can be cyclic normal quotients of pairs in OG (4). Moreover, because of the major focus in [10] on 1 2 -arc-transitive graphs Γ (namely those for which (Γ, Aut(Γ)) ∈ OG(4)), fewer examples are found than in our situation. We make more detailed remarks in Subsection 1.1 (in particular giving definitions of these concepts).
Questions and results: There are many natural questions that arise: how many different cyclic normal quotients can a pair (Γ, G) ∈ OG(4) possess? In particular, how many if (Γ, G) is basic? Can a (basic) pair (Γ, G) have cyclic normal quotients, some of which inherit an edge orientation while others do not, and if so are there restrictions on the numbers of them? We begin by gathering a few observations concerning these questions in Theorem 1. We denote by C r a cycle of length r; and we say that a cyclic normal quotient is G-oriented or G-unoriented if it does, or does not, inherit a G-invariant edge orientation, respectively.
(c) a pair (Γ, G) ∈ OG(4) with at least 2n normal quotients Γ Ni ∼ = C ri and Γ Mi ∼ = C si , for 1 ≤ i ≤ n, such that gcd(r i , r j ) = gcd(s i , s j ) = 1 for i = j, and such that each Γ Ni is G-oriented while each Γ Mi is G-unoriented.
The pairs (Γ, G) used in the proof of Theorem 1 (c) for n > 1 are not basic and we do not know of any basic pairs with these properties. Problem 1. Decide if the number of unoriented cyclic normal quotients of a basic pair (Γ, G) ∈ OG(4) can be unboundedly large.
The multitude of cyclic normal quotients in the examples examined to prove Theorem 1 are all quotients of one or two particular cyclic normal quotients. For example, in the proof of Theorem 1 (c) using (Γ, G) from Construction 2.6, the quotients Γ Ni are all quotients of a single normal quotient Γ N ∼ = C r with r divisible by all the r i , and all the Γ Mj are quotients of a single normal quotient Γ M ∼ = C s with s divisible by all the s j . Definition 1.1. Two cyclic normal quotients Γ M , Γ N of (Γ, G) ∈ OG(4) are independent if the normal quotient Γ K is not a cycle, where K =Ñ ∩M , withÑ ,M the subgroups consisting of all the elements of G which fix setwise each N -orbit, or each M -orbit, respectively.
If Γ M , Γ N are cyclic normal quotients of (Γ, G) ∈ OG(4) and the normal quotient Γ K is a cycle, where K =Ñ ∩M , then both Γ N and Γ M are isomorphic to (G/K)normal quotients of Γ K , and hence both or neither of them are G-oriented, according as Γ K is G-oriented or not. Thus if one of Γ N , Γ M is G-oriented and the other is G-unoriented, then Γ N , Γ M must be independent. The graphs used to prove Theorem 1(a), (b) do not have independent cyclic normal quotients, but those used to prove Theorem 1(c) do have such quotients. Our main result classifies all pairs (Γ, G) ∈ OG(4) with independent cyclic normal quotients of given orders. (4), let x be a vertex, and suppose that (Γ, G) has independent cyclic normal quotients Γ N ∼ = C r and Γ M ∼ = C s , where r ≥ 3, s ≥ 3.
Theorem 2. Let (Γ, G) ∈ OG
Then G x ∼ = Z 2 , and the following hold:
(a) at least one of Γ N , Γ M is G-unoriented, say Γ N is G-unoriented; (b) (Γ, G) ∈ OG(4)
is a normal cover of (Γ, G) ∈ OG(4), which has independent cyclic normal quotients Γ N ∼ = C r and Γ M ∼ = C s such that N ∩ M = 1;
(c) Γ, G are as in one of the lines of Table 1, and Γ M (and Γ M ) are G-oriented if and only if the entry in column 3 is 'yes'.
We give several constructions of families in OG(4) in Section 2, and prove Theorem 1. Then we prove Theorem 2 in Section 3.
Remark (a) Some of the pairs in Theorem 2(c) have larger cyclic normal quotients, for example, if (Γ, G) = (Γ(r, s), G(r, s)) is as in line 1 of Table 1 with s even, then Γ N = C r extends to Γ N2 = C 2r (see Lemma 2.7). In this case Γ N 2 , Γ M are independent for (Γ, G), and N 2 ∩ M = 1. Applying Theorem 2 to (Γ, G) with these cyclic normal quotients of orders 2r and s, we find that the pair (Γ, G) is given by line 2 of Table 1. Thus (Γ(r, s), G(r, s)) ∼ = (Γ + (2r, s), G + (2r, s)) when r is odd and s is even. (b) For each of the pairs (Γ, G) in Theorem 2, Γ is arc-transitive (see Definition 2.1 and Construction 2.10). However there may still be 1 2 -arc-transitive graphs Γ with automorphism groups G such that (Γ, G) ∈ OG(4) has independent cyclic normal quotients Γ M , Γ N , since the normal quotient Γ = Γ M∩N may admit certain automorphisms making it arc-transitive which do not lift to automorphisms of Γ. In the next subsection we discuss this possibility further.
Problem 2. Describe the maximal cyclic normal quotients of all the pairs (Γ, G) in Table 1, and in particular decide whether there are any further relations between these graph families (beyond the isomorphisms given in the remark above.
1.1. Comparison with the approach of Marušič andŠparl [10] for weak metacirculants. The first infinite family of 1 2 -arc transitive graphs, constructed in [3], consisted of 'metacirculant graphs' which had been introduced by Alspach and Parsons [2] in 1982. Several other constructions of 1 2 -arc transitive graphs also turned out to be metacirculants. Recently Marušič andŠparl [10] embarked on a thorough analysis of a (proper) sub-family of OG(4), called weak metacirculants, with a special focus on those which are 1 2 -arc transitive. Marušič andŠparl [10] in [10] define a graph Γ to be a weak (m, n)-metacirculant relative to an ordered pair (ρ, λ) of its automorphisms, if Γ has mn vertices, the automorphism ρ has m cycles of length n on vertices, the cyclic subgroup λ permutes the ρ-cycles transitively, and ρ λ = ρ r , for some r such that gcd(r, n) = 1. The subgroup ρ, λ is thus a metacyclic group which acts transitively on the vertices of Γ. If, in addition, λ m fixes a vertex (in which case λ m fixes a vertex in each ρcycle), then Γ is called a metacirculant. A graph may be a weak metacirculant relative to more than one pair (ρ, λ) (see Construction 4.2), and according to [10, p. 368], it is an open question whether or not all weak metacirculants are in fact metacirculants (relative perhaps to some other pair of automorphisms).
We classified in Theorem 2 the graph-group pairs (Γ, G) ∈ OG(4) which have independent cyclic normal quotients. We wish to understand which of them are weak metacirculants relative to pairs of automorphisms lying in G. We hope thereby to gain a better understanding of how our normal quotient analysis compares with the analysis in [10] which focuses on alternating cycles. As a bsais for this discussion we make the following assumptions. (4), and suppose that Γ is a weak (m, n)-metacirculant relative to (ρ, λ), for some ρ, λ ∈ G. Let H := ρ, λ and R := ρ , so H ≤ G, the subgroup R is normal in H, and H is transitive on the vertices of Γ.
Hypothesis 1.1. Let (Γ, G) ∈ OG
Since the focus in [10] is on 1 2 -arc transitive graphs, and since no Cayley graph of an abelian group is 1 2 -arc transitive (see [10, beginning of §3]), the authors there assume that H is non-abelian. From this they deduce that the number m of R-orbits is at least 3 ([10, Proposition 3.2]). We lose little generality if we also assume that m n Γ G ρ λ Conditions on r, s Name for Γ in [10] s r Γ(r, s) G(r, s) µ ν at least one odd X o (s, r; 1) s r/2 Γ + (r, s) G + (r, s) µ 2 µν both even X e (s, r/2; 1, 0) Table 2. (4), and studied further in [4,13,14,15,16], is defined according to the nature of a certain quotient graph defined modulo the R-orbits X i (1 ≤ i ≤ m) on vertices (that is to say, the ρ-cycles). This quotient is slightly different from our graph quotients in that it also encodes, for a vertex x in X i , the number of edges from x to vertices in X j , for each j. Four different kinds of quotients are identified in [10], and these are used to subdivide the weak metacirculants in OG(4) into four classes, denoted I, II, III, IV. The reason that four different behaviours are observed in [10] is that some of the quotient graphs in [10] do not correspond to a normal quotient of (Γ, G) ∈ OG(4) for any G. In fact, it is only the quotients arising for graphs in Class I of [10] which can possibly occur as cyclic normal quotients in our sense (see Lemma 4.1). Moreover, for (Γ, G) ∈ OG(4), it is possible to have different choices of ρ, λ ∈ G leading to different quotients Γ R of Γ which may or may not be normal quotients of (Γ, G), and if they are normal quotients, then they may or may not be G-oriented. In Construction 4.2 we give explicit examples of metacirculants (Γ, G) ∈ OG(4) with different pairs (ρ, λ) of elements in G illustrating each of these possibilities.
Now we turn to the property of having independent cyclic normal quotients, which is characterised in Theorem 2. In that result we do not assume a priori that the edge-transitive group contains a weak metacirculant subgroup. However, all of the pairs (Γ, G) in the outcome of Theorem 2 turn out to be metacirculants. We obtain more Class I weak metacirculants in Theorem 2 than those obtained in [10] because our assumptions are valid for some arc-transitive graphs as well as 1 2 -arc transitive graphs. For example, if Γ is 1 2 -arc transitive, then it follows from the proofs of [10, Lemmas 4.2 and 4.3] that the quotient Γ R is G-oriented, whereas the graphs in lines 3-6 of Table 1 correspond to examples in Theorem 2(c) having a G-unoriented normal quotient which can occur as Γ R for suitable choices of ρ, λ.
Our approach can bring additional insights to the work in [10]. In the light of our discussion it makes sense to restrict to the Class I weak metacirculants for which Γ R is a cyclic G-oriented normal quotient of (Γ, G). Using Theorem 2 we find all such graphs with independent cyclic normal quotients. Corollary 1.2. Suppose that Hypothesis 1.1 holds with Γ R a cyclic, G-oriented, normal quotient of (Γ, G) of length at least 3. Suppose also that (Γ, G) has independent cyclic normal quotients Γ N ∼ = C r , Γ M ∼ = C s , as in Theorem 2. Then (Γ, G) is a normal cover of (Γ, G) ∈ OG(4) such that Γ is a weak (m, n)-metacirculant relative to (ρ, λ) and one of the lines of Table 2 holds.
This corollary is proved in Section 4. The Class I metacirculant graphs which are 1 2 -arc transitive were proved in [10, Theorem 4.1] to be precisely the connected 4valent tightly attached 1 2 -arc transitive graphs (see also [14]). To assist in comparing Corollary 1.2 with that classification we give in Table 2 also the names of the graphs Γ used in [10, Examples 2.1 and 2.2].
Constructions and proof the Theorem 1
In this section we examine several infinite families of graph-group pairs in OG(4), and describe their cyclic normal quotients. In Subsection 2.6 we prove Theorem 1.
2.1. Notation. For fundamental graph theoretic concepts please refer to the book [5]. For a subset Y of the vertex set of a graph Γ, the induced subgraph [Y ] is the graph with vertex set Y and the edges of [Y ] are those edges of Γ with both vertices in Y . An induced subgraph will inherit an edge-orientation from an edge-orientation of Γ.
A permutation group N on a set V is semiregular if the only element of N fixing a point of V is the identity; also N is regular if it is both transitive and semiregular.
For (Γ, G) ∈ OG(4), if an edge {x, y} is oriented from x to y, then we call y an out-neighbour of x, and we call x an in-neighbour of y. By a neighbour of x we mean an in-neighbour or an out-neighbour.
We say that graph-group pairs (Γ, G) and (Γ ′ , G ′ ) are isomorphic if there exist a graph isomorphism f from Γ to Γ ′ and a group isomorphism ϕ :
G → G ′ such that, for all x ∈ V Γ and g ∈ G, (x g )f = (xf ) gϕ .
For a group K with an inverse-closed generating set S such that 1 K ∈ S (that is S −1 = {s −1 |s ∈ S} is equal to S), the Cayley graph Cay(K, S) is the graph with vertex set K and edges {k, sk} for k ∈ K, s ∈ S. The facts that S generates K and S is inverse-closed imply that Cay(K, S) is connected and undirected, respectively. The definition of adjacency implies that K acts faithfully by right multiplication as a vertex-regular group of automorphisms of Cay(K, S). Also, S is the set of neighbours of the vertex 1 K , and the subgroup of Aut(K) leaving S invariant acts naturally as a subgroup of automorphisms stabilising 1 K .
2.2.
Preliminaries on cyclic normal quotients. We first show that oriented and unoriented cyclic normal quotients of (Γ, G) ∈ OG(4) can be distinguished by the action on vertices of the normal subgroup.
Lemma 2.1. Let (Γ, G) ∈ OG(4) have a cyclic normal quotient Γ N , and letÑ be the subgroup consisting of all elements of G fixing each N -orbit setwise.
(a) If Γ N is G-unoriented, thenÑ = N and is semiregular on V Γ. (b) If Γ N is G-oriented, thenÑ contains G x , for each vertex x.
Proof. Let x ∈ V Γ, let y, y ′ be the out-neighbours of x, and let z, z ′ be the inneighbours of x. Let B, B ′ denote the N -orbits containing x, y respectively. Suppose that Γ N = C r , for some r ≥ 3.
(a) Suppose that Γ N is G-unoriented. Then B ′ contains also one of the inneighbours of x, say z, and the other neighbours y ′ , z ′ lie in a third N -orbit distinct from B, B ′ . Consider the stabiliserÑ x of x inÑ . By the definition ofÑ , the subgroupÑ x must fix the N -orbit B ′ setwise, and hence must fix the unique outneighbour y of x it contains. ThusÑ x fixes each of y and y ′ . SimilarlyÑ x fixes each of z, z ′ . It follows from the connectivity of Γ thatÑ x = 1. ThusÑ is semiregular on V Γ, and in particularÑ = N .
(b) Now suppose that Γ N is G-oriented. Then B ′ contains both out-neighbours y, y ′ of x. In this case G induces a cyclic group Z r on Γ N , and the setwise stabiliser G B of B fixes each N -orbit setwise, that is to say,
G B =Ñ . Since G x < G B , we have G x <Ñ .
Using this lemma we prove that two oriented cyclic normal quotients cannot be independent.
Lemma 2.2. Let (Γ, G) ∈ OG(4), and let Γ N = C r , Γ M = C s , for some r, s ≥ 3,
where N, M consist of all elements of G fixing setwise each N -orbit, or each Morbit, respectively. If both Γ N and Γ M are G-oriented, then Γ N ∩M = C t is Goriented, for some multiple t of lcm{r, s}. In particular Γ N , Γ M are not independent.
Proof. By Lemma 2.1(b), the subgroup K := N ∩ M contains G x , for a vertex x, and in particular K = 1. This implies, by [1, Theorem 1.1], that Γ K is degenerate, and since it has order at least lcm{r, s} ≥ 3, it must be a cycle of length a multiple t of lcm{r, s}. Since Γ N is isomorphic to a quotient of Γ K , it follows that Γ K is G-oriented.
2.3.
Examples with many cyclic normal quotients. The first family of graphs we consider consists of the lexicographic products C r [2.K 1 ] with a natural orientation on their edges.
Construction 2.3. Let r ≥ 3 and let Γ be the graph with vertex set
X = Z r × Z 2 and edges {(i, j), (i + 1, j ′ )} for all i ∈ Z r , j, j ′ ∈ Z 2 , that is, Γ = C r [2.K 1 ]
, the lexicographic product of C r and 2.K 1 . We orient the edges so that
(i, j) → (i+1, j ′ ), for all i, j, j ′ . Let G = Z 2 ≀ Z r = {(σ 1 , . . . , σ r )τ ℓ | 0 ≤ ℓ < r, each σ k ∈ Z 2 } where (σ 1 , . . . , σ r ) : (i, j) → (i, j + σ i ), and τ : (i, j) → (i + 1, j), and let B = {(σ 1 , . . . , σ r )| each σ k ∈ Z 2 } = Z r 2 , the 'base group' of G.
By [1, Lemma 3.6], G preserves the edge orientation, (Γ, G) ∈ OG(4), and (Γ, G) is basic of cycle type. We parametrise all the cyclic normal quotients of (Γ, G) with the set of divisors of r, and show that (Γ, G) does not have independent cyclic normal quotients. A cyclic normal quotient Γ N is maximal if there is no normal subgroup K of G, contained in N , and such that Γ K is cyclic of order larger than Γ N . Lemma 2.4. Let r, Γ, G, X be as in Construction 2.3, and for a divisor c of r let
N (c) = B · τ c . Then N (c) is normal in G and Γ N (c) = C c and is G-oriented, if c ≥ 3, or is K 2 or K 1 if c = 2 or 1, respectively. Moreover each proper normal quotient of (Γ, G) is equal to Γ N (c) for some c, and Γ B = Γ N (r) is the unique maximal cyclic normal quotient. Proof. It was shown in [1, Lemma 3.6] that Γ B ∼ = C r and is G-oriented. By definition N (c) contains B and, since G/B ∼ = Z r , the group N (c) is normal in G and G/N (c) ∼ = Z c . Also the graph Γ N (c) is isomorphic to the quotient of (Γ B , G/B) relative to the normal subgroup N (c)/B ∼ = Z r/c of G/B, so Γ N (c) ∼ = C c and is G- oriented, if c ≥ 3, and is K 2 , K 1 if c = 2, 1 respectively.
Let N be a nontrivial normal subgroup of G, and consider Γ N . It was shown in the proof of [1, Lemma 3.6] that each N -orbit on vertices is a union of some B-orbits. Thus B fixes each vertex of Γ N and so Γ N = Γ BN . Since G/B ∼ = Z r it follows that BN = N (c) for some divisor c of r, and hence Γ N = Γ N (c) . If c = r then N (r) = B, and we conclude that Γ B = Γ N (r) is the unique maximal cyclic normal quotient of (Γ, G).
Γ G Generators for G Conditions on r, s Γ(r, s) G(r, s) µ, ν, σ - Γ(r, s)
H(r, s) µ, σν, τ s even Γ + (r, s) G + (r, s) µ 2 , µν, σ both even Γ + (r, s) H + (r, s) µ 2 , σµν, τ both even Table 3. Table for Definition 2.1 2.4. Graphs with both independent oriented and unoriented cyclic normal quotients. To prove the second part of Theorem 1, we use a family of graphs given in Definition 2.1. We define an edge-orientation and a group of automorphisms preserving it. We also describe a certain subgraph in some cases. These graphs were considered in [7, Section 3] from the point of view of their alternating cycle structure (see also [9,Example 2.4]).
Recall the concept induced subgraph from Subsection 2.1. Note that, if k is even, then the integers representing a given element of Z k are either all even, or all odd, and hence in this case elements of Z k have a well-defined parity. First we define the undirected graphs and subgroups of their automorphism groups.
Definition 2.1. Let r, s be integers, each at least 3. Define the undirected graph Γ(r, s) to have vertex set X := Z r × Z s , such that a vertex (i, j) ∈ X is joined by an edge to each of the four vertices (i ± 1, j ± 1). Also, if r, s are both even define
X + := {(i, j) ∈ X | i, j of the same parity}
and let Γ + (r, s) = [X + ], the induced subgraph. Finally, define the following permutations of X, for (i, j) ∈ X,
µ : (i, j) → (i + 1, j), ν : (i, j) → (i, j + 1), σ : (i, j) → (−i, j), τ : (i, j) → (−i, −j),
and define the groups as in Table 3, where in lines 3 and 4 (r, s both even), we identify µ, ν, σ, τ with their restrictions to X + , and consider the subgroups G + (r, s) and H + (r, s) acting on X + . Also letM = µ, σ = D 2r , M = µ = Z r , N = ν ∼ = Z s , and N ′ = ν 2 , τ σν , and note that G(r, s) =M × N and H(r, s) = (M × N ′ ). τ (with s even; note that (σν) 2 = ν 2 ).
Let M t = µ t for t | r, and N t = ν t for t | s. If r and s are both even, we also consider the following subgroups restricted to their actions on X + :
M + = µ 2 , σ = D r , M + = M 2 = Z r/2 , and N + = N 2 ∼ = Z s/2 .
Recall that, for a graph Γ, an action of a group H is 1 2 -transitive on Γ if H ≤ Aut(Γ) and H is transitive on the vertices and the edges of Γ, but is not transitive on arcs.
-transitive on Γ + (r, s). Proof. The N -orbits in X are the sets B i := {(i, j) | j ∈ Z s }, for i ∈ Z r .
If s is odd, then by the definition of Γ, for each i, each vertex of B i is connected by a path in Γ to each vertex of B i+1 , and hence Γ is connected. Suppose then that s is even and define B even
i = {(i, 2j) | j ∈ Z s } and B odd i = B i \ B even i , for i ∈ Z r .
Then the definition of Γ implies, for each i, that each vertex of B odd i (or B even i ) is connected by a path in Γ to each vertex of B even i+1 (or B odd i+1 , respectively). Therefore, if r is odd, Γ is connected. On the other hand if r and s are both even, then Γ is disconnected with connected components [X + ] and [X \ X + ]. In this case the induced subgraph Γ + is connected. This proves (a).
If r is even, then it follows from the discussion above that ∪ i even B i and ∪ i odd B i form the parts of a bipartition of Γ, and similarly Γ is bipartite if s is even (the parts of the bipartition being unions of M -orbits). On the other hand if r, s are both odd then |V Γ| is odd so Γ is not bipartite. If both r, s are even then Γ + is bipartite with bipartition {(i, j)|i, j even}, {(i, j)|i, j odd}. This proves (b).
It is straightforward to check that each of µ, ν, σ, τ preserves the edge set of Γ = Γ(r, s), so that G := G(r, s) and H := H(r, s) lie in Aut(Γ). Also, if both r, s are even, then µ 2 , µν, σ, τ all leave X + invariant, so G + := G + (r, s) and H + := H + (r, s) are contained in Aut(Γ + ), where Γ + = Γ + (r, s). The subgroup µ, ν of G acts regularly on X and so also, if s is even, does the subgroup µ, ν 2 , τ σν of H. Hence So H also is edge-transitive but not arc-transitive on Γ. Similar arguments show firstly that (i) the subgroup µ 2 , µν of G + acts regularly on X + , that G + x = σ , and that G + is 1 2 -transitive on Γ + ; and secondly that (ii) the subgroup µ 2 , σµν of H + acts regularly on X + , that H + x = τ , and that H + is 1 2 -transitive on Γ + . This proves part (c). First we define an edge-orientation on the graphs in Definition 2.1 which leads to independent cyclic normal quotients, one oriented and the other not. We note that this family of oriented graphs was studied in [7, p.50-51. Props
Γ + N + is G-unoriented while Γ + M + is G-oriented.
Proof. By Lemma 2.5, G, G + are 1 2 -transitive on Γ, Γ + respectively, and it is straightforward to check that the edge-orientation of Construction 2.6 is preserved in each case. If at least one of r, s is odd, then Γ is connected, by Lemma 2.1 and hence (Γ, G) ∈ OG (4). Similarly if both r, s are even, then Γ + is connected, by Lemma 2.1 and hence (Γ + , G + ) ∈ OG (4).
Each of N, M,M is an intransitive normal subgroup of G. The M -orbits in X are C j = {(i, j) | i ∈ Z r }, for j ∈ Z s , and are the same as the orbits ofM , and the N -orbits in X are the subsets B i = {(i, j) | j ∈ Z s }, for i ∈ Z r . Also each of N + , M + ,M + is a normal subgroup of G + and is intransitive on X + . The M +orbits in X + are C + j := C j ∩ X + , for j ∈ Z s , and are the same as theM + -orbits in X + , and the N + -orbits in X + are the subsets B + i := B i ∩ X + for i ∈ Z r . By the definition of Γ, for each x = (i, j) ∈ B i , each of B i−1 and B i+1 contains one out-neighbour and one in-neighbour of x, and it follows (as in [1, Proposition 3.1(b)]) that Γ N is a connected G-arc transitive graph of valency 2 and order rs/|B i | = r, so Γ N = C r and is G-unoriented. Similarly, for each x = (i, j) ∈ C j , both out-neighbours of x lie in C j+1 and both in-neighbours of x lie in C j−1 , so (as in [1, Proposition 3.1(a)]) Γ M = ΓM is a connected, G-oriented, G-edge transitive graph of valency 2 and order rs/|C j | = s, that is, Γ M = C s and is G-oriented. If in addition s is even (so r is odd) then each N -orbit B i is the union of two N 2 -orbits, namely B even i = {(i, j)|j even} and B odd i = {(i, j)|j odd}. For j even, the vertex x = (i, j) has one out-neighbour in each of B odd i−1 and B odd i+1 , and for j odd, (i, j) has one out-neighbour in each of B even i−1 and B even i+1 . Since r is odd, it follows that Γ N2 is the cycle C 2r and is G-unoriented. Part (a) now follows from Definition 1.1, since N ∩M = 1. Part (b) also follows since the B + i = B i ∩ X + , C + j = C j ∩ X + , M + =M ∩ G + , M + = M ∩ G + , N + = N ∩ G + andM + ∩ N + = 1.
2.5.
Graphs with independent unoriented cyclic normal quotients. We now give two constructions of oriented graphs with independent unoriented cyclic normal quotients. The graphs in the first construction are the graphs Γ(r, s) and Γ + (r, s) of Definition 2.1 with a different edge-orientation from that in Construction 2.6. Construction 2.8. Let r, s be positive integers, with r ≥ 3, and s even, s ≥ 4, and recall the graph Γ(r, s) of Definition 2.1 with vertex set X = Z r × Z s . Define an edge-orientation as follows.
if j is even then (i, j) → (i + 1, j + 1) and
(i, j) → (i − 1, j − 1), if j is odd then (i, j) → (i + 1, j − 1) and (i, j) → (i − 1, j + 1).
In other words, (i, j) → (i + 1, j + (−1) j ) and (i, j) → (i − 1, j − (−1) j ). Note that this orientation is well-defined since elements of Z s have a well-defined parity as s is even. If r and s are both even, then this edge-orientation restricts to an edge-orientation of Γ + (r, s) = [X + ]. Proof. It is easy to check that each of µ, ν 2 , τ preserves the edge-orientation of Γ, and with a little care, that σν does also. Thus H preserves the edge-orientation of Γ, and H + preserves the edge-orientation of Γ + .
(a) Suppose that r is odd. Then by Lemma 2.5, Γ is connected and H acts (4). Now M + , N + are normal subgroups of H + . The N + -orbits in X + (with X + as in Definition 2.1) are B i ∩X + , for i ∈ Z r . Let x ′ = (i, j) ∈ B i ∩ X + . Then each of B i−1 ∩ X + and B i+1 ∩ X + contains one out-neighbour and one in-neighbour of x ′ . Hence, as in [1, Proposition 3.1(b)], Γ + N + is a connected H + -arc transitive graph of valency 2 and order |X + |/|B i ∩ X + | = r, so Γ + N + ∼ = C r and is H + -unoriented. Similarly the M + -orbits are C j ∩ X + , for j ∈ Z s , and for each x ′ = (i, j) ∈ C j ∩ X + , each of C j−1 ∩ X + and C j+1 ∩ X + contains one out-neighbour and one in-neighbour of x ′ , and so Γ + M + ∼ = C s and is H + -unoriented. By Lemma 2.1, N + , M + are the kernels of the H + -actions on Γ + N + , Γ + M + , respectively. Since N + ∩ M + = 1, the quotients Γ + N + , Γ + M + are independent.
The graphs in the final construction are standard double covers of the graphs Γ(r, s) of Definition 2.1.
Definition 2.2. The standard double cover of a graph Γ with vertex set X is the graph Γ 2 with vertex set X 2 = {x δ |x ∈ X, δ ∈ Z 2 } such that {x δ , y δ ′ } is an edge if and only if δ = δ ′ and {x, y} is an edge of Γ.
Note that Γ 2 has the same valency as Γ and twice the number of vertices. Construction 2.10. Let r, s be positive integers, with r, s ≥ 3, and let Γ 2 (r, s) be the standard double cover of the graph Γ(r, s) of Definition 2.1, so Γ 2 (r, s) has vertex set X 2 as in Definition 2.2. Define an orientation on the edges of Γ 2 (r, s) as follows.
(i, j) 0 → (i + 1, j + 1) 1 and
(i, j) 0 → (i − 1, j − 1) 1 (i, j) 1 → (i + 1, j − 1) 0 and (i, j) 1 → (i − 1, j + 1) 0 .
In other words, (i, j) δ → (i + 1, j + (−1) δ ) δ+1 and (i,
j) δ → (i − 1, j − (−1) δ ) δ+1 .
We extend the automorphisms defined in Definition 2.1 to maps on X 2 as follows.
For (i, j) δ ∈ X 2 , µ : (i, j) δ → (i + 1, j) δ , ν : (i, j) δ → (i, j + 1) δ , σ : (i, j) δ → (i, −j) δ+1 , τ : (i, j) δ → (−i, −j) δ and let G 2 (r, s) = µ, ν, σ, τ = M × N , where M = µ, στ ∼ = D 2r , and N = ν, σ ∼ = D 2s .
Lemma 2.11. Let r, s, Γ = Γ 2 (r, s), G = G 2 (r, s) be as in Construction 2.10. Then G preserves the edge-orientation, Γ is connected if and only if r, s are both odd, and in this case (Γ, G) ∈ OG(4), and Γ N = C r , Γ M = C s are independent cyclic normal quotients, and each is G-unoriented.
Proof. Careful but straightforward checking shows that each of the generators of G preserves edges, and that each preserves the edge orientation of Γ. It is well known and easily proved that a standard double cover of a graph Σ is connected if and only if Σ is connected and not bipartite. It follows therefore from Lemma 2.5 that Γ is connected if and only if r, s are both odd. Suppose this is the case. The subgroup µ, ν, σ is normal in G of index 2, and is regular on vertices. The stabiliser G x of the vertex x = (0, 0) 0 is τ , which interchanges the out-neighbours (1, 1) 1 and (−1, −1) 1 , and the in-neighbours (1, −1) 1 and (1, −1) 1 of x. Thus G is 1 2 -transitive on Γ and preserves the edge-orientation so (Γ, G) ∈ OG (4).
The N -orbits in X 2 are the subsets B i = {(i, j) δ |j ∈ Z s , δ ∈ Z 2 }, for i ∈ Z r . Each vertex in B i has one out-neighbour and one in-neighbour in B i+ε , for ε = ±1. It follows that Γ N = C r and is G-unoriented. Similarly the M -orbits in X 2 are the subsets C j = {(i, j) δ |i ∈ Z r , δ ∈ Z 2 }, for j ∈ Z s . Each vertex in C j has one out-neighbour and one in-neighbour in C j+ε , for ε = ±1, and hence Γ M = C s and is G-unoriented. (b) Again let r, Γ, G be as in Construction 2.3, and this time choose n distinct odd primes p 1 < p 2 < · · · < p n and take r = i p i . Again (Γ, G) is basic in OG(4) of cycle type, by [1,Lemma 3.6], and by Lemma 2.4, we have G-oriented normal quotients Γ N (pi) = C pi relative to N (p i ), for i = 1, . . . , n.
(c) Choose 2n odd primes p 1 < p 2 < · · · < p n and q 1 < q 2 < · · · < q n (where some p i , q j may be equal), and take r = i p i and s = i q i in Construction 2.6. By Lemma 2.7, (Γ, G) ∈ OG(4), and (Γ, G) has a cyclic G-oriented normal quotient Γ N = C r , and a cyclic G-unoriented normal quotient Γ M = C s . For i = 1, . . . , n, consider N pi = ν pi and M qi = µ qi , as in Definition 2.1, and note that N pi × M and N × M qi are both normal in G and intransitive on vertices. The normal quotients Γ Np i ×M and Γ N ×Mq i are isomorphic to quotients of Γ M and Γ N , and are in fact isomorphic to C pi and C qi , respectively, with the former G-oriented and the latter G-unoriented. This completes the proof.
Proof of Theorem 2
In this section we analyse the structure of pairs (Γ, G) ∈ OG(4) with independent cyclic normal quotients. The first lemma yields a proof of parts (a) and (b) of Theorem 2. Parts (a) and (b) of Theorem 2 follow from Lemma 3.1. Also, from part (b), (Γ, G) is a normal cover of (Γ, G), and so the order of the vertex stabiliser G x is equal to the order of a stabiliser in G of a vertex of Γ. Thus (even to prove that stabilisers have order 2) it is sufficient to consider the case where N ∩ M = 1. We therefore make this assumption from now on. We use the following notation.
Notation 3.1. We assume that (Γ, G) ∈ OG(4), and that Γ N = C r (G-unoriented), Γ M = C s , and A = Aut(Γ M ), ϕ, π 1 , π 2 are as in Lemma 3.1, with N ∩ M = 1. We use the notation introduced in the proof of Lemma 2.1, so x ∈ V Γ has outneighbours y, y ′ and in-neighbours z, z ′ . We let B = x N , B + = y N , B − = (y ′ ) N , the pairwise distinct N -orbits containing x, y, y ′ , and C = x M , C + = y M , C − = (y ′ ) M , the M -orbits containing x, y, y ′ (so
C + = C − if Γ M is G-oriented). Write Aut(Γ N ) = a, c | a r = c 2 = 1, a c = a −1 ∼ = D 2r
where a maps B to B + and c fixes B. We also write
Aut(Γ M ) = A = b, d | b s = d 2 = 1, b d = b −1 ∼ = D 2s or b | b s = 1 ∼ = Z s
according as Γ M is G-unoriented or G-oriented, respectively, and b maps C to C + and (if Γ M is G-unoriented) d fixes C.
3.1.
Case Γ M is G-oriented. First we derive information about generators of Gϕ.
Lemma 3.2. Using Notation 3.1, and assuming that Γ M is G-oriented,
(a) G x = h ∼ = Z 2 is such that hϕ = (c, 1). (b)
The following all hold, where either t = 1, or t = 2 divides gcd(r, s): , 1), (c, 1) ,
|G| = 2rs t , |V Γ| = rs t , N ϕ = (1, b t ) , M ϕ = (a tand Gϕ = (a i , b), M ϕ , where either i = 0 or i = t − 1 = 1. Proof. Since G is edge-transitive on Γ, G x = 1. Let h ∈ G x \ {1}.
Then h fixes both B and C setwise and so h ΓN ∈ c and h ΓM = 1. Since h = 1 it follows that hϕ = (c, 1) and G x = h ∼ = Z 2 , proving part (a).
Note that M contains h, by Lemma 2.1. Since M ϕ ≤ a, c × 1, it follows that M ϕ = (a t , 1), (c, 1) , for some t | r. Hence |M | = 2r t , |G| = 2rs t , and |V Γ| = rs t . Also N ϕ ≤ 1 × b so N ϕ = (1, b ℓ ) for some ℓ | s, and we have |N | = s ℓ , |G| = 2rs ℓ . Therefore ℓ = t divides gcd(r, s). Since Gϕπ 2 = b , G contains an element g such that gϕπ 2 = b. All such elements satisfy gϕ = (a i c δ , b) for some i, δ. We may replace g by gm for some m ∈ M , and assume that δ = 0 and that 0 ≤ i < t. It remains to prove that t ≤ 2. To see this, note that Gϕ contains (
a i , b).(a i , b) (c,1) = (a i , b).(a −i , b) = (1, b 2 ), which lies in N ϕ.
We consider the cases t = 1 and t = 2 separately. Recall the concepts of regular, Cayley graph, and isomorphism of graph-group pairs from Subsection 2.1. Proof. Suppose that t = 1 and identify G with Gϕ. Then |V Γ| = rs and G = a, c × b . Moreover the group K = (a, 1) × (1, b) , is normal in G, acts regularly on V Γ, and G is the semidirect product K.G x . By [1, Remark 4.1 and Lemma 4.2], we may assume that Γ = Cay(K, S 0 ∪S −1 0 ) for a 2-element generating set S 0 for K such that S 0 ∩ S −1 0 = ∅, and we may identify x = 1 K , S 0 = {y, y ′ }, and S −1 0 = {z, z ′ }. The group K acts by right multiplication and G x = (c, 1) ≤ Aut(K) acts naturally on V Γ = K. Now y = (a i , b j ) for some i ∈ Z r , j ∈ Z s , and so y ′ = y (c,1) = (a −i , b j ). In particular i = 0 since y ′ = y.
Since N ≤ K, the N -orbits are the cosets N (a k , 1) for k ∈ Z r , and as in Notation 3.1, (a, 1) maps B = 1 N = N to B + = y N = N (a i , 1). However (a, 1) maps B to N (a, 1) and hence i = 1. Also M ∩ K = (a, 1) is transitive on each M -orbit, and the M -orbits are the cosets M (1, b k ) for k ∈ Z s . As in Notation 3.1, M (1, b) and hence j = 1. Thus S 0 = {(a, b), (a −1 , b)}, and since K = S 0 , at least one of r, s must be odd.
(1, b) maps C = 1 M = M to C + = y M = M (1, b j ). However (1, b) maps C to
It follows that each vertex (a k , b ℓ ) has out-neighbours (a k±1 , b ℓ+1 ), and so the map f : (a k , b ℓ ) → (k, ℓ) defines a graph isomorphism from Γ to the graph Γ(r, s) of Construction 2.6. Also the map (a, 1) → µ, (1, b) → ν, (c, 1) → σ extends to an isomorphism ϕ ′ from G to the group G(r, s) of Construction 2.6, and f, ϕ ′ define an isomorphism from (Γ, G) to (Γ(r, s), G(r, s)).
Lemma 3.4.
Under the assumptions of Lemma 3.2, if t = 2 then both r and s are even, and (Γ, G) is isomorphic to the graph-group pair (Γ + (r, s), G + (r, s)) in Construction 2.6.
Proof.
Suppose that t = 2 and identify G with Gϕ. Then r, s are both even, |V Γ| = rs 2 , and G = (a i , b), (a 2 , 1), (c, 1) of order rs, where i = 0 or 1, by Lemma 3.2. Moreover since Gπ 1 = a, c it follows that i = 1. Then the group K := (a 2 , 1), (a, b) is a subgroup of G of index 2, and as G x = (c, 1) ∼ = Z 2 , we have K x = 1 and K acts regularly on V Γ. As in Lemma 3.3, we may assume that Γ = Cay(K, S 0 ∪S −1 0 ) for a 2-element generating set S 0 for K such that S 0 ∩S −1 0 = ∅, and we may identify x = 1 K , S 0 = {y, y ′ }, and S −1 0 = {z, z ′ }. The group K acts by right multiplication and G x = (c, 1) ≤ Aut(K) acts naturally on V Γ. Thus y = (a j , b k ) for some j ∈ Z r , k ∈ Z s of the same parity, and y ′ = y (c,1) = (a −j , b k ).
Since N ≤ K, the N -orbits are the cosets N (a ℓ , 1) for even ℓ ∈ Z r , and N (a ℓ , b) for odd ℓ ∈ Z r . As in Notation 3.1, (a, b) ∈ K and maps B = 1 N = N to B + = y N , and we have y N = N (a j , 1) if j is even and N (a j , b) if j is odd. However (a, b) maps B to N (a, b) and hence j = 1, and so k is odd (since j, k have the same parity). Also M ∩ K = (a 2 , 1) is transitive on each M -orbit, and the M -orbits are therefore the cosets M (1, b ℓ ) for even ℓ ∈ Z s , and M (a, b ℓ ) for odd ℓ ∈ Z s . As in Notation 3.1, (a, b)
maps C = 1 M = M to C + = y M = M (a, b k ). However (a, b) maps C to M (a, b) and hence k = 1. Thus S 0 = {(a, b), (a −1 , b)}.
It follows that each vertex (a k , b ℓ ) has out-neighbours (a k±1 , b ℓ+1 ), and so the map f : (a k , b ℓ ) → (k, ℓ) defines a graph isomorphism from Γ to the graph Γ + (r, s) of Construction 2.6. Also the map (a, b) → µν, (a 2 , 1) → µ 2 , (c, 1) → σ extends to an isomorphism ϕ ′ from G to the group G + (r, s) of Construction 2.6, and f, ϕ ′ define an isomorphism from (Γ ′ , G) to (Γ + (r, s), G + (r, s)).
3.2.
Case Γ M is G-unoriented. We identify G with Gϕ, and we first derive a short explicit list of possibilities for G. Suppose that (1, d) ∈ N C and let x ′ := x (1,d) . Note that x ′ = x (since N is semiregular), and that so N is either (1, b 2 ), (1, d) with s even, or (1, b), (1, d) . Thus |G| = 2rs or 4rs, and |V Γ| = rs or 2rs, and hence |M | = r or 2r, respectively. Since (c, 1) ∈ M , a similar argument identifies the possibilities for M . If |G| = 4rs then (i) holds. If |G| = 2rs then r, s, |V Γ|, M, N are as in (ii) and M × N has index 2 in G. Since a ∈ G ΓN , G contains an element of the form g = (a, g ′ ) for some g ′ and, adjusting g by an element of N , we may assume that g ′ = 1 or b. Since M does not contain (a, 1) it follows that g = (a, b) and (ii) holds.
x ′ = x (c,d)(1,d) = x (c,1) . Thus x, x ′ ∈ x N ∩ x M = B ∩ C. Since N ΓM is a normal subgroup of G ΓM = b, d containing d, it follows that N ΓM contains dd b = db −1 db = b 2 ,
Suppose now that N C = 1, so also M B = 1. Then N ΓM is a semiregular normal subgroup of G ΓM = b, d = D 2r , and N ΓM does not contain d, so we have the following possibilities for N : either N = (1, b t ) for some divisor t of s, or s is even and N = (1, b 2 ), (1, bd) and in this latter case we set t = 1. Then |N | = s t , |G| = |G ΓN |.|N | = 2rs t , |V Γ| = rs t , and so |M | = r t . Hence t divides gcd(r, s) and a similar argument shows that either M = (a t , 1) , or r is even, t = 1, and M = (a 2 , 1), (ac, 1) . In all cases K := M × N is semiregular on V Γ (as it does not contain (c, d)) and has t orbits of size rs t 2 . Now each K-orbit is a disjoint union of r t orbits of N , and it follows that an N -orbit and an M -orbit meet in at most one vertex. In particular B + ∩ C + = {y}. This implies that z ∈ C + (since we are assuming that z ∈ B + ) and hence z ∈ C − and z ′ ∈ C + . Now y K contains C + = y M and hence contains z ′ , which implies that B − = (y ′ ) N = (z ′ ) N ⊂ (z ′ ) K = y K . Hence y K contains {y, y ′ , z, z ′ }, and it follows, since K ✂ G, that all edges from vertices in x K go to vertices in y K , and since Γ is connected this implies that K has at most two orbits in V Γ. Thus t = 1 or 2. If t = 1, then K is regular so G = K.G x as in (iv). Thus we may assume that t = 2. Then r, s, M, N, |V Γ| are as in (iii) and K has index 4 in G. Since G ΓN contains a, G contains an element of the form g = (a, g ′ ) with g ′ ∈ b, d . Now N contains (1, b 2 ) so we may assume that g ′ = d δ b δ ′ where each of δ, δ ′ is 0 or 1. The subgroup M does not contain (a, 1), and so the element g ′ = 1. Also if g ′ = d then M would contain g(c, d) = (ac, 1), which is not the case. Hence g = (a, b) or (a, db), so G is as in (iii). Now we analyse cases Lemma 3.5(i)-(iv) separately. Proof. Suppose that Lemma 3.5(i) holds. Then K := (a, 1), N is a regular normal (index 2) subgroup of G, and as in Lemma 3.4, we may assume that Γ = Cay(K, S 0 ∪ S −1 0 ) for a 2-element subset S 0 ⊆ K \ {1}, with S 0 ∩ S −1 0 = ∅ such that K = S 0 , x = 1 K , S 0 = {y, y ′ }. Now K acts by right multiplication, G x = (c, d) ≤ Aut(K) acts naturally on V Γ, and S 0 is a G x -orbit. Thus y = (a i , d δ b j ) for some i ∈ Z r , j ∈ Z s , δ = 0 or 1. This means that y ′ = y (c,d) = (a −i , d δ b −j ) and since S 0 = {y, y ′ } generates K, it follows that δ = 1, i and j are nonzero, and gcd(i, r) = 1.
Since N ≤ K, the N -orbits are the cosets N (a k , 1) for k ∈ Z r , and as in Notation 3.1, (a, 1) maps B = 1 N = N to B + = y N = N (a i , 1). However (a, 1) maps B to N (a, 1) and hence i = 1. Thus y = (a, db j ). To determine the orbits of M , we note that (c, 1) = (c, d)(1, d) ∈ M maps u ∈ K to u (c,d) (1, d) and so the M -orbit containing (1, b k ) is the set product (a, 1) 1) {(1, b), (1, db)} and hence j = 1. Thus S 0 = {(a, db), (a −1 , db −1 )} and since S 0 generates K we see that both r and s must be odd.
{(1, b k ), (1, db k )}. By Notation 3.1, (1, b) maps C = 1 M to C + = y M = (a, 1) {(1, b j ), (1, db j )}. However (1, b) maps C to (a,
Finally, defining f : 1 , and the group isomorphism ϕ : G → G 2 (r, s) extending (a, 1) → µ, (1, b) → ν, (c, 1) → στ, (1, d) → τ , we obtain an isomorphism (f, ϕ) from the pair (Γ, G) to the pair (Γ 2 (r, s), G 2 (r, s)) in Construction 2.10. Proof. Here r, s are both even. The approach is similar to that of the previous lemma. If Lemma 3.5(ii) holds, let K := (a, b), N , and if Lemma 3.5(iii) holds, let K := (a, d δ b), N (where δ = 0 or 1). In either case K is a normal subgroup of G of index 2, and K is regular on vertices (since K ∩ G x = 1, implying that G = KG x ). Thus we may assume that Γ = Cay(K, S 0 ∪ S −1 0 ) for a 2-element generating set S 0 for K such that S 0 ∩ S −1 0 = ∅, and we may identify
V Γ → X 2 by (a i , b j ) → (i, j) 0 , (a i , db j ) → (i, j)x = 1 K , S 0 = {y, y ′ }. Suppose first that Lemma 3.5(ii) holds. Then y = (a i , b j d δ ′ ), for some i ∈ Z r , j ∈ Z s , δ ′ ∈ Z 2 , with i, j of the same parity. Now S 0 is a G x -orbit so y ′ = y (c,d) = (a −i , b −j d δ ′ ). Since S 0 ∩ S −1 0
= ∅, we have y ′ = y −1 , and hence δ ′ = 1. However S 0 is then contained in the proper subgroup (a 2 , 1), (
1, b 2 ), (a i , b j d) of index 2 in K, contradicting the fact that S 0 = {y, y ′ } generates K.
Thus Lemma 3.5(iii) holds. Suppose first that δ = 0. Then as y ∈ K, we have y = (a i , b j ) for some i ∈ Z r , j ∈ Z s of the same parity. This implies, as in the previous paragraph, that y ′ = y (c,d) = (a −i , b −j ) is equal to y −1 , which is a contradiction. Thus δ = 1. If y = (a i , b j ) with both i, j even, then we again find y ′ = y −1 , a contradiction. Hence y = (a i , db j ) with both i, j odd. Since N ≤ K, the N -orbits are the cosets N (a 2k , 1) and N (a 2k+1 , db), for 0 ≤ k < r/2, and as in Notation 3.1, the element (a, db) ∈ G maps B = 1 N = N to B + = y N = N (a i , db j ) = N (a i , db) (since (1, b 2 ) ∈ N ). However (a, db) ∈ K acts by right multiplication, and maps B to N (a, db). Hence i = 1 and y = (a, db j ). Also M lies in K, and the M -orbits are the cosets M (1, b 2k ) and M (a, db 2k+1 ), for 0 ≤ k < s/2. As in Notation 3.1, the element (a, db) ∈ G maps C = 1 M = M to C + = y M = M (a, db j ). However (a, db) ∈ K acts by right multiplication, and maps C to M (a, db). Hence j = 1 and y = (a, db), y ′ = y (c,d) = (a −1 , db −1 ).
Each vertex w of Γ = Cay(K, S 0 ∪ S −1 0 ) has out-neighbours yw, y ′ w, so for i, j even, w = (a i , b j ) has out-neighours (a i+1 , db j+1 ) and (a i−1 , db j−1 ), while for i, j odd, w = (a i , db j ) has out-neighours (a i+1 , b j−1 ) and (a i−1 , b j+1 ). Thus the map f : V Γ → X + defined by (a i , b j ) → (i, j) if i, j are both even, and (a i , db j ) → (i, j) if i, j are both odd, determines a graph isomorphism from Γ to Γ + (r, s), which maps each oriented edge w → yw, w → y ′ w of Γ to an oriented edge of Γ + (r, s) with the edgeorientation of Construction 2.8. Also the map (a 2 , 1) → µ 2 , (1, b 2 ) → ν 2 , (c, d) → τ, (a, db) → τ σµν, extends to a group isomorphism ϕ : G → H + (r, s), and we obtain an isomorphism (f, ϕ) from the pair (Γ, G) to the pair (Γ + (r, s), H + (r, s)) in Construction 2.8. Proof. Suppose that Lemma 3.5(iv) holds, so |M | = r, |N | = s, |V Γ| = rs. There are two possibilities for M and two for N . For any of these M, N , the group K := M × N has index 2 in G and acts regularly on the vertices of Γ. Hence we may assume that Γ = Cay(K, S 0 ∪ S −1 0 ) for a 2-element generating set S 0 for K such that S 0 ∩ S −1 0 = ∅, and we may identify x = 1 K , S 0 = {y, y ′ }. Now y = mn for some m ∈ M, n ∈ N , and y ′ is the image of y under conjugation by (c, d) ∈ G x . Since y ′ = y −1 , it is not possible for both m (c,d) = m −1 and n (c,d) = n −1 to hold. Therefore M, N are not both cyclic. Interchanging r and s if necessary, we may assume that n (c,d) = n −1 , and it follows from Lemma 3.5(iv) that s is even and N = (1, b 2 ), (1, bd) . We will prove that (a) holds. (In the case where m (c,d) = m −1 it will then follow, on interchanging r and s in our arguments, that (b) holds.)
Since n (c,d) = n −1 , the element n satisfies n = (1, db j ) for some j ∈ Z s , and we have y = m(1, db j ), y ′ = m (c,d) 1), (ac, 1) (see Notation 3.1, since these elements lie in G and induce a on Γ N ). In the second case, the element (a, d) maps B to (N (ac, 1)) (c,d) = N (a −1 c, 1) = N (ca, 1), and since this set must contain y = m(1, db) we have m = (ca, 1). However this implies that y = (ca, db) which has order 2 and hence y −1 = y ∈ S 0 , a contradiction. Therefore M = (a, 1) . The element (a, 1) maps B to B + = N (a, 1) and since this set contains y it follows that m = (a, 1) and y = (a, db), y ′ = (a −1 , db −1 ). Now the fact that S 0 = K implies that r must be odd.
It remains to identify the graph-group pair. The oriented edges of Γ are the pairs u → yu and u → y ′ u for u ∈ K. An easy computation shows that, for each i ∈ Z r and for 0 ≤ j < s 2 ,
(a i , b 2j ) → (a i+1 , db 2j+1 ) and (a i , b 2j ) → (a i−1 , db 2j−1 ), (a i , db 2j+1 ) → (a i+1 , b 2j ) and (a i , db 2j+1 ) → (a i−1 , b 2j+2 ).
It follows that the bijection f : K → Z r × Z s given by f : (a i , b 2j ) → (i, 2j) and f : (a i , db 2j+1 ) → (i, 2j + 1) defines a graph isomorphism from Γ to the graph Γ(r, s) such that the oriented edges u → yu and u → y ′ u of Γ are mapped to oriented edges according to the edge-orientation defined in Construction 2.8. Also the map ϕ given by ϕ : (a, 1) → µ, ϕ : (1, b 2 ) → ν 2 , ϕ : (1, db) → τ σν, ϕ : (c, d) → τ
We give examples to show that, for a given (Γ, G) ∈ OG(4), different pairs (ρ, λ) can lead to different behaviours for the quotient Γ R , even with the same subgroup H = ρ, λ .
Theorem 1 .
1Given a positive integer n, there exists (a) a basic pair (Γ, G) ∈ OG(4) with exactly n pairwise non-isomorphic normal quotients, all G-oriented cycles; (b) a basic pair (Γ, G) ∈ OG(4) with at least n normal quotients, all Goriented cycles of pairwise coprime lengths;
Lemma 2 . 5 .
25Let r, s, Γ(r, s), G(r, s), H(r, s), M, N , Γ + (r, s), G + (r, s), H + (r, s) be as in Definition 2.1. Then (a) Γ(r, s) is connected if at least one of r, s is odd; while if both r, s are even then Γ(r, s) has connected components [X + ] and [X \ X + ], and Γ + (r, s) is connected. (b) Γ(r, s) is bipartite if and only if at least one of r, s is even; while if both r, s are even, then Γ + (r, s) is bipartite.
and (if s is even) H(r, s) are 1 2 -transitive on Γ(r, s); and if r, s are both even, then G + (r, s) and H + (r, s) are1 2
G and H are vertex-transitive. The stabiliser in G, or in H, of the vertex x = (0, 0) is σ , or τ , respectively. The element σ acts on the four neighours of x by interchanging (1, 1) and (−1, 1) and interchanging (−1, −1) and (1, −1), and the element σµν −1 ∈ G maps the edge {x, (1, 1)} to the edge {x, (1, −1)}. Thus G is edge-transitive but not arc-transitive on Γ. If s is even, then τ interchanges (1, 1) and (−1, −1) and interchanges (1, −1) and (−1, 1), and the element τ µσν ∈ H maps the edge {x, (1, 1)} to the edge {x, (−1, 1)}.
p.159, Proposition 3.3], where they were characterised by properties of their alternating cycles.
Construction 2. 6 .
6Let r, s, Γ(r, s), Γ + (r, s) be as in Definition 2.1. Define an edgeorientation of Γ(r, s) such that (i, j) → (i ± 1, j + 1) for each (i, j) ∈ X. If r, s are both even this restricts to an edge-orientation of Γ + (r, s) = [X + ].
Lemma 2. 7 .
7Let Γ = Γ(r, s), G = G(r, s), N, N 2 , M,M , Γ + = Γ + (r, s), G + = G + (r, s), N + , M + ,M + be as in Definition 2.1, with edge-orientations of Γ, Γ + as in Construction 2.6. (a) If at least one of r, s is odd, then (Γ, G) ∈ OG(4) and Γ N = C r , ΓM = Γ M = C s are independent cyclic normal quotients; Γ N is G-unoriented while Γ M is G-oriented. Moreover if s is even then Γ N2 = C 2r and is G-unoriented.
(b )
)If both r, s are even, then (Γ + , G + ) ∈ OG(4) and Γ + N + = C r , Γ + M + = Γ + M + = C s are independent cyclic normal quotients;
Lemma 2. 9 .
9Let s be even, s ≥ 4, and let r ≥ 3. Let Γ = Γ(r, s), H = H(r, s), N ′ , M , and, if r is even, also Γ + = Γ + (r, s), H + = H + (r, s), N + , M + , be as in Definition 2.1, with edge-orientations of Γ, Γ + as in Construction 2.8.(a) If r is odd, then H preserves the edge-orientation of Γ, (Γ, H) ∈ OG(4), and Γ N ′ = C r , Γ M ∼ = C s are independent, H-unoriented, cyclic normal quotients. (b) If r is even, then H + preserves the edge-orientation of Γ + , (Γ + , H + ) ∈ OG(4), and Γ + N + = C r , Γ + M + ∼ = C s are independent, H + -unoriented, cyclic normal quotients.
1 2
1-transitively, so (Γ, H) ∈ OG(4). Now M , N ′ are normal subgroups of H. TheN ′ -orbits are B i = {(i, j) | j ∈ Z s }, for i ∈ Z r . Let x ′ = (i, j) ∈ B i .Then each of B i−1 and B i+1 contains one out-neighbour and one in-neighbour of x ′ . Hence, as in [1, Proposition 3.1(b)], Γ N ′ is a connected H-arc transitive graph of valency 2 and order rs/|B i | = r, so Γ N ′ ∼ = C r and is H-unoriented. Similarly the M -orbits are C j = {(i, j) | i ∈ Z r }, for j ∈ Z s , and for each x ′ = (i, j) ∈ C j , each of C j−1 and C j+1 contains one out-neighbour and one in-neighbour of x ′ , and so Γ M ∼ = C s and is H-unoriented. By Lemma 2.1, N ′ , M are the kernels of the H-actions on Γ N ′ , Γ M , respectively. Since N ′ ∩ M = 1, the quotients Γ N ′ , Γ M are independent. This proves part(a).(b) Suppose now that r is even, r ≥ 4. Then by Lemma 2.1, Γ + is connected and H + acts 1 2 -transitively, so (Γ + , H + ) ∈ OG
By Lemma 2.1, N, M are the kernels of the actions of G on Γ N , Γ M respectively, and since M ∩ N = 1 it follows that Γ N , Γ M are independent. 2.6. Proof of Theorem 1. Let n be a positive integer. (a) Let r, Γ, G be as in Construction 2.3. Choose an odd prime p and set r = p n . By [1, Lemma 3.6], (Γ, G) is basic in OG(4), and by Lemma 2.4, its normal quotients relative to intransitive, nontrivial normal subgroups of G are precisely the G-oriented quotients Γ N (c) = C c , for c | r and c > 1. There are precisely n possibilities, namely for c = p i with i = 1, . . . , n.
Lemma 3 . 1 .
31Let (Γ, G) ∈ OG(4), and suppose that Γ N , Γ M are independent cyclic normal quotients, where N, M consists of all elements of G fixing setwise each N -orbit, or each M -orbit, respectively. Let Γ = Γ N ∩M , G = G/(N ∩ M ), N = N/(N ∩ M ) and M = M/(N ∩ M ). (a) Then one of Γ N , Γ M is G-unoriented, say Γ N = C r for some r ≥ 3, and the other Γ M = C s , for some s ≥ 3, may be G-oriented or G-unoriented. (b) The quotient (Γ, G) ∈ OG(4) has independent cyclic normal quotients Γ N ∼ = C r and Γ M ∼ = Γ M such that N ∩ M = 1, and (Γ, G) is a normal cover of (Γ, G). (c) If N ∩ M = 1, then the map ϕ : g → (g ΓN , g ΓM ) defines a group monomorphism from G to D 2r × A, such that Gϕπ 1 = D 2r , Gϕπ 2 = A, where π 1 , π 2 are the natural projection maps of D 2r × A on D 2r , A respectively, and A = D 2s or Z s according as Γ M is G-unoriented or G-oriented, respectively. Proof. Part (a) follows from Lemma 2.2. By Definition 1.1, Γ = Γ N ∩M is not a cycle, and hence, by [1, Theorem 1.1], (Γ, G) is a normal cover of (Γ, G) and (Γ, G) ∈ OG(4). Now N and M are normal subgroups of G, and by construction the corresponding G-normal quotients satisfy Γ N ∼ = Γ N and Γ M ∼ = Γ M . These quotients are independent since N ∩ M = 1. Thus part (b) is proved. If N ∩ M = 1 then the map ϕ is a monomorphism from G to Aut(Γ N ) × Aut(Γ M ) = D 2r × A, and Gϕπ 1 = G ΓN = D 2r , Gϕπ 2 = G ΓM = A.
Lemma 3 . 3 .
33Under the assumptions of Lemma 3.2, if t = 1 then at least one of r, s is odd and (Γ, G) is isomorphic to the graph-group pair (Γ(r, s), G(r, s)) in Construction 2.6.
Lemma 3 . 5 .
35Using Notation 3.1, identify G with Gϕ, and assume that Γ M = C s is G-unoriented. Then G x = (c, d) ∼ = Z 2 , and one of the following holds.(i) |V Γ| = 2rs, M = (a, 1), (c, 1) , N = (1, b), (1, d) , and G = M × N ; (ii) r, s are both even, |V Γ| = rs, M = (a 2 , 1), (c, 1) , N = (1, b 2 ), (1, d) , and G = M, N, (a, b) ; (iii) r, s are both even, |V Γ| = rs 2 , M = (a 2 , 1) , N = (1, b 2 ) , and G = M, N, (a, d δ b), (c, d) , where δ = 0 or 1; (iv) |V Γ| = rs, G = (M × N ).G x with M × N regular on V Γ, where either M = (a, 1) or r is even and M = (a 2 , 1), (ac, 1) , and either N = (1, b) or s is even and N = (1, b 2 ), (1, bd) . Proof. By Lemma 2.1, the subgroup of all elements of G fixing setwise each Norbit, or each M -orbit, is equal to N , or M , respectively, and both N and M are semiregular on V Γ. Since G is edge-transitive on Γ, G x contains an element h that interchanges y and y ′ , and hence h interchanges the N -orbits B + and B − . Thus h ΓN = c and G x ∩ N = N x has index 2 in G x . Since N is semiregular it follows that N x = 1 and so G x = h ∼ = Z 2 . Similarly h interchanges C + and C − so that h ΓM = d. This implies that h = (c, d), proving the first assertion. Next we study the setwise stabiliser N C of C = x M in N . Since N ∩ M = 1 we have N C ∼ = N ΓM C ≤ G ΓM C = d , and it follows that N C ≤ (1, d) . Similarly M B ≤ (c, 1) . Note that (1, d) ∈ N C if and only if (c, 1) ∈ M B (since (c, d) ∈ G x ); in particular N C = 1 if and only if M B = 1. Also we may assume, without loss of generality, that z, z ′ lie in B + , B − respectively (see Notation 3.1).
Lemma 3. 6 .
6If Lemma 3.5(i) holds then r and s are both odd and (Γ, G) ∼ = (Γ 2 (r, s), G 2 (r, s)), as in Construction 2.10.
Lemma 3 . 7 .
37The case Lemma 3.5(ii) leads to no examples, while if Lemma 3.5(iii) holds, then δ = 1 and (Γ, G) ∼ = (Γ + (r, s), H + (r, s)), as in Construction 2.8 .
Lemma 3 . 8 .
38If Lemma 3.5(iv) holds then either(a) r is odd and s is even, M = (a, 1) , N = (1, b 2 ), (1, bd) , and (Γ, G) ∼ = (Γ(r, s), H(r, s)) of Construction 2.8; or (b) r is even and s is odd, M = (a 2 , 1), (ac, 1) , N = (1, b) , and (Γ, G) ∼ = (Γ(s, r), H(s, r)) of Construction 2.8.
( 1 ,
1db −j ). Now K acts by right multiplication and M ≤ K, so the M -orbits are C 2k := M (1, b 2k ) and C 2k+1 := M (1, db 2k+1 ) for 0 ≤ k < s 2 . The vertex x lies in C = C 0 , and y lies in the image C + of C under the action of the element (c, b) = (1, bd)(c, d) ∈ G (see Notation 3.1, since this element lies in G and induces b on Γ M ). However (c, b) maps C to (M (1, bd)) (c,d) = M (1, b −1 d) = M (1, db), and since this set must contain y = m(1, db j ), we have j = 1. Thus y = m(1, db). The N -orbits are the subsets N u for u ∈ M . The vertex x lies in B = N , and y lies in the image B + of B under the action of the element (a, 1) if M = (a, 1) , or (a, d) = (ac, 1)(c, d) if M = (a 2 ,
Table for Corollary 1.2 m ≥ 3. The subdivision introduced in [10] of the family of weak metacirculants in OG
We may assume that N, M consists of all elements of G which fix setwise each N -orbit, or each M -orbit, respectively. Part (a) of Theorem 2 follows from Lemma 3.1(a), and we may assume that Γ N is G-unoriented. Part (b) of Theorem 2 follows from Lemma 3.1(b), and. Γ , G) ∈ Og, ) have independent cyclic normal quotients Γ N , Γ M of orders r, s respectively. 2as explained just before Notation 3.1) for the rest of the proof we may assume that N ∩ M = 13. Proof of Theorem 2. Let (Γ, G) ∈ OG(4) have independent cyclic normal quotients Γ N , Γ M of orders r, s respectively. We may assume that N, M consists of all elements of G which fix setwise each N -orbit, or each M -orbit, respectively. Part (a) of Theorem 2 follows from Lemma 3.1(a), and we may assume that Γ N is G-unoriented. Part (b) of Theorem 2 follows from Lemma 3.1(b), and (as explained just before Notation 3.1) for the rest of the proof we may assume that N ∩ M = 1.
G-Oriented If Γ M Is, Then, |G x | = 2 and line 1 or 2 of Table 1 holds. If Γ M is G-unoriented then. by Lemmas 3.5, 3.6, 3.7, and 3.8, |G x | = 2 and one of the lines 3, 4, 5 or 6 of Table 1 holds. This completes the proof. 4. Weak metacirculants with independent cyclic normal quotients First we see that under Hypothesis 1.1, it is only the graphs in Class I of [10If Γ M is G-oriented then, by Lemmas 3.2, 3.3, and 3.4, |G x | = 2 and line 1 or 2 of Table 1 holds. If Γ M is G-unoriented then, by Lemmas 3.5, 3.6, 3.7, and 3.8, |G x | = 2 and one of the lines 3, 4, 5 or 6 of Table 1 holds. This completes the proof. 4. Weak metacirculants with independent cyclic normal quotients First we see that under Hypothesis 1.1, it is only the graphs in Class I of [10]
If Γ R is a normal quotient of (Γ, G), then Γ lies in Class I or IV of [10] relative to (ρ, λ), and if Γ R is cyclic then Γ lies in Class I. If Γ R is a normal quotient of (Γ, G), then Γ lies in Class I or IV of [10] relative to (ρ, λ), and if Γ R is cyclic then Γ lies in Class I.
Conversely, for Γ in Class I or IV relative to (ρ, λ), either H is regular on vertices, or Γ lies in Class I and (Γ, H) ∈ OG(4) with Γ R an H-oriented cyclic normal quotient. Conversely, for Γ in Class I or IV relative to (ρ, λ), either H is regular on vertices, or Γ lies in Class I and (Γ, H) ∈ OG(4) with Γ R an H-oriented cyclic normal quotient.
By [1, Thm 1.1], there are no edges joining vertices in the same R-orbit, and moreover there is a constant k such that, if there is an edge between some vertex in the R-orbit X i and some vertex in X j , then each vertex of X i is adjacent to exactly k vertices in X j . These conditions are not satisfied for Classes II and III of [10, Section 3], and hence only Classes I and IV can correspond to normal quotients of (Γ, G). Moreover, since the quotients for Class IV have valency 4, it is only the quotients for Class I which can arise as cyclic normal quotients. (b) Suppose that Γ is in Class I or IV of [10] relative to (ρ, λ), and that H is not regular on vertices. Then H x = 1 for x in the R-orbit X 1 , and since the action induced by H on the R-orbits is cyclic (induced by λ), H x fixes each R-orbit setwise. If H x fixed each vertex adjacent to x then. Suppose that Γ R is equal to Γ N for some N ✂ G (that is to say the sets of R-orbits and N -orbits on vertices are the same). since Γ is connected, it would follow that H x would fix all vertices, contradicting our assumption that H x = 1Proof. (a) Suppose that Γ R is equal to Γ N for some N ✂ G (that is to say the sets of R-orbits and N -orbits on vertices are the same). By [1, Thm 1.1], there are no edges joining vertices in the same R-orbit, and moreover there is a constant k such that, if there is an edge between some vertex in the R-orbit X i and some vertex in X j , then each vertex of X i is adjacent to exactly k vertices in X j . These conditions are not satisfied for Classes II and III of [10, Section 3], and hence only Classes I and IV can correspond to normal quotients of (Γ, G). Moreover, since the quotients for Class IV have valency 4, it is only the quotients for Class I which can arise as cyclic normal quotients. (b) Suppose that Γ is in Class I or IV of [10] relative to (ρ, λ), and that H is not regular on vertices. Then H x = 1 for x in the R-orbit X 1 , and since the action induced by H on the R-orbits is cyclic (induced by λ), H x fixes each R-orbit setwise. If H x fixed each vertex adjacent to x then, since Γ is connected, it would follow that H x would fix all vertices, contradicting our assumption that H x = 1.
Thus H x moves some vertex y adjacent to x, say y ∈ X 2 , and hence X 2 contains at least two vertices adjacent to x. For Γ in Class IV, the four vertices adjacent to x lie in four distinct R-orbits and hence we conclude that Γ lies in Class I of [10], and that two R-orbits distinct from X 1 , namely X 2 and, say, X 3 , each contain two vertices adjacent to x. Thus H x interchanges the two vertices in Γ(x) ∩ X 2 , and since H preserves the edge-orientation, it follows that H is edge-transitive, so (Γ, H) ∈ OG(4), with Γ R a cyclic normal quotient. Finally the fact that H x interchanges the two vertices in Γ(x) ∩ X 2 implies that Γ R is an H-oriented cycle. This last fact can also be deduced from the proofs of [10, Lemmas 4.2 and 4Thus H x moves some vertex y adjacent to x, say y ∈ X 2 , and hence X 2 contains at least two vertices adjacent to x. For Γ in Class IV, the four vertices adjacent to x lie in four distinct R-orbits and hence we conclude that Γ lies in Class I of [10], and that two R-orbits distinct from X 1 , namely X 2 and, say, X 3 , each contain two vertices adjacent to x. Thus H x interchanges the two vertices in Γ(x) ∩ X 2 , and since H preserves the edge-orientation, it follows that H is edge-transitive, so (Γ, H) ∈ OG(4), with Γ R a cyclic normal quotient. Finally the fact that H x interchanges the two vertices in Γ(x) ∩ X 2 implies that Γ R is an H-oriented cycle. (This last fact can also be deduced from the proofs of [10, Lemmas 4.2 and 4
as in line 1 of Table 3. Then, using Lemma 2.7 we see that (Γ, G) ∈ OG(4), and that, for ρ, λ as in (a), (b) or (c) below, Γ is an (r, r)-metacirculant relative to (ρ, λ), H := ρ, λ = µ, ν is regular, and the quotient Γ R , for R := ρ , is a cycle C r . (a) Let ρ = µν and λ = µ. Then Γ R is G-oriented but is not a normal quotient of (Γ, G) since N G (R). G Let Γ = Γ(r, = G(r, R) ; = H, Construction 4.2. Let r be an odd integer, r ≥ 3,. The R-orbits are {(i + ℓ, ℓ) | ℓ ∈ Z r }, for i ∈ Z r .Construction 4.2. Let r be an odd integer, r ≥ 3, and let Γ = Γ(r, r) and G = G(r, r), as in line 1 of Table 3. Then, using Lemma 2.7 we see that (Γ, G) ∈ OG(4), and that, for ρ, λ as in (a), (b) or (c) below, Γ is an (r, r)-metacirculant relative to (ρ, λ), H := ρ, λ = µ, ν is regular, and the quotient Γ R , for R := ρ , is a cycle C r . (a) Let ρ = µν and λ = µ. Then Γ R is G-oriented but is not a normal quotient of (Γ, G) since N G (R) = H. (The R-orbits are {(i + ℓ, ℓ) | ℓ ∈ Z r }, for i ∈ Z r .)
Let ρ = ν and λ = µ. Then Γ R is a normal quotient of (Γ, G) that is not G-oriented. Let ρ = ν and λ = µ. Then Γ R is a normal quotient of (Γ, G) that is not G-oriented.
Let ρ = µ and λ = ν. Then Γ R is a G-oriented normal quotient of. Let ρ = µ and λ = ν. Then Γ R is a G-oriented normal quotient of (Γ, G).
Now we study the case where (Γ, G) has independent cyclic normal quotients and where Γ R is a G-oriented normal quotient, so Γ is of Class I of. by Lemma 4.1Now we study the case where (Γ, G) has independent cyclic normal quotients and where Γ R is a G-oriented normal quotient, so Γ is of Class I of [10], by Lemma 4.1.
with Γ R ∼ = C m a G-oriented normal quotient of (Γ, G), and that (Γ, G) has independent cyclic normal quotients. Then for a possibly different pair (ρ, λ) of elements from H, the same conditions all hold and (Γ, G) has a possibly different pair of independent cyclic normal quotients. Lemma 4.3. Suppose that Hypothesis 1.1 holds. one of which is Γ RLemma 4.3. Suppose that Hypothesis 1.1 holds with Γ R ∼ = C m a G-oriented normal quotient of (Γ, G), and that (Γ, G) has independent cyclic normal quotients. Then for a possibly different pair (ρ, λ) of elements from H, the same conditions all hold and (Γ, G) has a possibly different pair of independent cyclic normal quotients, one of which is Γ R .
OG(4). Moreover, by Theorem 2, the stabiliser G x of a vertex x has order 2. We may assume that M (respectively N ) is equal to the kernel of the G-action on Γ M (respectively Γ N ). Let Γ = Γ M∩N. etc., as in Lemma 3.1. Since Γ M , Γ N are independent. Thus G = HG x and |G : H| ≤ 2, since H is vertex-transitiveProof. Let Γ M , Γ N be independent cyclic normal quotients of (Γ, G), for normal subgroups M, N of G. We may assume that M (respectively N ) is equal to the kernel of the G-action on Γ M (respectively Γ N ). Let Γ = Γ M∩N , G = G/(M ∩ N ), etc., as in Lemma 3.1. Since Γ M , Γ N are independent, (Γ, G) ∈ OG(4). Moreover, by Theorem 2, the stabiliser G x of a vertex x has order 2. Thus G = HG x and |G : H| ≤ 2, since H is vertex-transitive.
Assume then that Γ T is cyclic. Since Γ M is a quotient of Γ T , the pair Γ N , Γ T are independent cyclic normal quotients, and as T ⊆ R, each R-orbit is a union of T -orbits. Thus the arguments of the previous paragraph may be used to replace (ρ, λ) by a new pair (ρ ′ , λ ′ ) from H, and replace R by T , and Γ R by Γ T , so that all assertions hold for the independent cyclic normal quotients Γ N , Γ T . Finally we prove Corollary 1.2. Proof of Corollary 1.2. Suppose that Hypothesis 1.1 holds with Γ R ∼ = C m a Goriented cyclic normal quotient of (Γ, G) ∈ OG(4), where m ≥ 3. Suppose also that, as in Theorem 2, (Γ, G) has independent cyclic normal quotients. ; Γ R , So G = R , Λ . ; Γ N ∼ = C R , Γ M ∼ = C S, Γ , G , Γ M ∼ = Γ M ∼ = C , Thus we may assume thatR does not contain M . Suppose next that M properly containsR, that is, Γ M is a quotient of Γ R . Then Γ N , Γ R are independent cyclic normal quotients and all the assertions hold without changing (ρ, λ). So now we may assume also that M does not containR. Let T := R ∩ M and note that T ≤ H since R ≤ H. If Γ T is not cyclic then Γ M. = M ∩ N , G := G/T and Γ := Γ T ,the pair (Γ, G) is a normal cover of (Γ, G) ∈ OG(4). m are as 1, respectively. Thus, so far we have proved that m, n, Γ, G, and the conditions on r, s are as in the appropriate line of Table 2. Finally we prove that Γ is a weak (m, n)-metacirculant. Consider first line 1 of Table 2. Then G = µ, ν, σ where µ, ν, σ, are as in Definition 2.1. It is easy to check that Γ is a weak (s, r)-metacirculant relative to (µ, ν), so the assertions for line 1 of Table 2 all hold. The graph Γ is the graph X o (s, r; 1) defined in [10, Example 2.1] (although in [10] both of r, s are assumed to be odd. If indeed both of r, s are odd then Γ, and hence also Γ, is 1 2 -arc transitive (see [10, Theorem 4.1] and its proof). Now consider line 2 of Table 2. Then G = µ 2 , µν, σ where again µ, ν, σ, are as in Definition 2.1. This time Γ is a weak (s, r/2)-metacirculant relative to (µ 2 , µν), so the assertions for line 2 of Table 2 all hold. The graph Γ is the graph X e (s, r/2; 1, 0) defined in [10, Example 2.2] (although in [10], r/2 is assumed to be even and atSince Γ R is a normal quotient of (Γ, G), there is a normal subgroupR of G, containing R, with the same vertex-orbits as R, and such thatR is equal to the kernel of the G-action on Γ R . Moreover, since Γ R = ΓR is G-oriented, the group G/R induced on Γ R is cyclic of order m, and is therefore equal to the cyclic group induced by λ on Γ R , so G = R , λ . Suppose first that some R-orbit is a union of j of the M -orbits in V Γ, where j ≥ 1. Then each R-orbit is a union of j of the M -orbits, and Γ R is isomorphic to a quotient of Γ M . Since m ≥ 3 and Γ R is G-oriented, it follows that Γ M is also G-oriented. Thus G induces a regular cyclic group on Γ M , and as H is vertex- transitive, H induces the same group, so G = M, λ ′ for some element λ ′ ∈ H. Moreover we may choose λ ′ such that λ, λ ′ induce the same action on Γ R , and hence λ ′ = λρ i for some i. Also M ∩ H is the subgroup ρ ′ , where ρ ′ = ρ j , and Hypothesis 1.1 holds with (ρ ′ , λ ′ ) in place of (ρ, λ), and M ∩ H in place of R (noting that ρ λ = ρ r implies that (ρ ′ ) λ ′ = (ρ ′ ) r , and gcd(r, n) = 1 implies that gcd(r, o(ρ ′ )) = 1). Since Γ M is G-oriented, M contains G x by Lemma 2.1(b). Hence M = (M ∩ H)G x , Γ M∩H = Γ M (the 'new Γ R '), and all the assertions hold relative to (ρ ′ , λ ′ ), and the normal quotients Γ N , Γ M∩H . Thus we may assume thatR does not contain M . Suppose next that M properly containsR, that is, Γ M is a quotient of Γ R . Then Γ N , Γ R are independent cyclic normal quotients and all the assertions hold without changing (ρ, λ). So now we may assume also that M does not containR. Let T := R ∩ M and note that T ≤ H since R ≤ H. If Γ T is not cyclic then Γ M , Γ R are independent cyclic normal quotients and all the assertions hold without changing (ρ, λ). Assume then that Γ T is cyclic. Since Γ M is a quotient of Γ T , the pair Γ N , Γ T are independent cyclic normal quotients, and as T ⊆ R, each R-orbit is a union of T -orbits. Thus the arguments of the previous paragraph may be used to replace (ρ, λ) by a new pair (ρ ′ , λ ′ ) from H, and replace R by T , and Γ R by Γ T , so that all assertions hold for the independent cyclic normal quotients Γ N , Γ T . Finally we prove Corollary 1.2. Proof of Corollary 1.2. Suppose that Hypothesis 1.1 holds with Γ R ∼ = C m a G- oriented cyclic normal quotient of (Γ, G) ∈ OG(4), where m ≥ 3. Suppose also that, as in Theorem 2, (Γ, G) has independent cyclic normal quotients Γ N ∼ = C r , Γ M ∼ = C s , where N, M are the kernels of the G-actions on Γ N , Γ M respectively. By Theo- rem 2, we may assume that Γ N is G-unoriented, and by Lemma 4.3, we may assume that one of the independent cyclic normal quotients is Γ R . (Note that, from the proof of Lemma 4.3, Γ R may be replaced by a cyclic G-oriented normal quotient of order a proper multiple of the original m in Hypothesis 1.1, but in the exposition we can, and will, continue to use m as the order of Γ R .) Since Γ R is G-oriented, we have Γ R = Γ M ∼ = C m , so m = s. By Theorem 2 again, setting T := M ∩ N , G := G/T and Γ := Γ T , the pair (Γ, G) is a normal cover of (Γ, G) ∈ OG(4), and Γ, G and Γ M ∼ = Γ M ∼ = C m are as 1, respectively. Thus, so far we have proved that m, n, Γ, G, and the conditions on r, s are as in the appropriate line of Table 2. Finally we prove that Γ is a weak (m, n)-metacirculant. Consider first line 1 of Table 2. Then G = µ, ν, σ where µ, ν, σ, are as in Definition 2.1. It is easy to check that Γ is a weak (s, r)-metacirculant relative to (µ, ν), so the assertions for line 1 of Table 2 all hold. The graph Γ is the graph X o (s, r; 1) defined in [10, Example 2.1] (although in [10] both of r, s are assumed to be odd). If indeed both of r, s are odd then Γ, and hence also Γ, is 1 2 -arc transitive (see [10, Theorem 4.1] and its proof). Now consider line 2 of Table 2. Then G = µ 2 , µν, σ where again µ, ν, σ, are as in Definition 2.1. This time Γ is a weak (s, r/2)-metacirculant relative to (µ 2 , µν), so the assertions for line 2 of Table 2 all hold. The graph Γ is the graph X e (s, r/2; 1, 0) defined in [10, Example 2.2] (although in [10], r/2 is assumed to be even and at
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The University of Western Australia. Praeger) Also affiliated with: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics M019. Jeddah, Saudi Arabia; Stirling Highway, Crawley, WA 6009, Australia E-mail address35King Abdulaziz UniversityAll authors. Jehan A. Al-bar: [email protected]; jaal[underscore][email protected] E-mail address, Ahmad N. Al-kenani: [email protected]; [email protected] E-mail address, Najat M. Muthana: [email protected](All authors) King Abdulaziz University, Jeddah, Saudi Arabia (Cheryl E. Praeger) Also affiliated with: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics M019, The University of West- ern Australia, 35 Stirling Highway, Crawley, WA 6009, Australia E-mail address, Jehan A. Al-bar: [email protected]; jaal[underscore][email protected] E-mail address, Ahmad N. Al-kenani: [email protected]; [email protected] E-mail address, Najat M. Muthana: [email protected];
. E Najat, M Muthana, second email): najat[underscore][email protected] E-mail address, Cheryl E. Praeger: [email protected] address, Najat M. Muthana (second email): najat[underscore][email protected] E-mail address, Cheryl E. Praeger: [email protected]
| [] |
[
"Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds",
"Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds"
] | [
"Takayuki Moriyama ",
"Takashi Nitta "
] | [] | [] | A complex contact structure γ is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Ker γ of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k-vectors on a complex contact manifold splits into the sum of O( k Ker γ) and O(L ⊗ k−1 Ker γ) as sheaves of C-module. The theorem induces the short exact sequence of cohomology of holomorphic k-vectors, and we obtain vanishing theorems for the cohomology of O( k Ker γ). | 10.1142/s0129167x1850091x | [
"https://arxiv.org/pdf/1710.05127v1.pdf"
] | 54,621,589 | 1710.05127 | 2d272ded9d59dcac73bfd9d07c6b0d2d06f19eb4 |
Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds
14 Oct 2017 September 19, 2018
Takayuki Moriyama
Takashi Nitta
Splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds
14 Oct 2017 September 19, 2018
A complex contact structure γ is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Ker γ of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k-vectors on a complex contact manifold splits into the sum of O( k Ker γ) and O(L ⊗ k−1 Ker γ) as sheaves of C-module. The theorem induces the short exact sequence of cohomology of holomorphic k-vectors, and we obtain vanishing theorems for the cohomology of O( k Ker γ).
Introduction
Originating in physics, contact geometry is a mathematical formulation of classical mechanics. Contact geometry describes a geometric structure which appears in any constant energy hypersurface in the even-dimensional phase space of a mechanical system. In mathematics, the concept of contact structure appears explicitly in the work of Sophus Lie, and implicitly perhaps much earlier. By using a sheaf coefficient cohomology theory, Gray developed the idea and introduced a concept of almost contact structure [5]. He considered the deformation of a global contact structure in the terminology of homological algebra. Boothby and Wang studied the homogeneous manifolds associated with the contact transformation group [2]. Furthermore, Kobayashi introduced the complex contact structure and developed several results of complex contact geometry [6]. The complex contact structure is associated with a quaternionic structure with respect to the twistor correspondence [4] [10] [8]. From the beginning of the study of contact structures, it has been known that contact structures have a deep relationship with sheaf cohomology, for example, Gray'work.
Let M be a complex manifold of dimension 2n + 1. We consider a system {(U i , γ i )} of an open covering {U i } of M and holomorphic 1-forms γ i on U i such that γ i is a contact 1-form, that is, (dγ i ) n ∧ γ i = 0 on U i , and γ i = f ij γ j for a holomorphic function f ij on U i ∩ U j . We say that such systems {(U i , γ i )} and {(U ′ i ′ , γ ′ i ′ )} are equivalent if there exists a holomorphic function g ii ′ on each intersection U i ∩ U i ′ so that γ i = g ii ′ γ ′ i ′ on U i ∩ U i ′ , and call an equivalent class of {(U i , γ i )} a complex contact structure on M . We denote by γ = {(U i , γ i )} the contact structure on M . A pair (M, γ) is called a complex contact manifold. The contact structure γ induces a line bundle L on M by the transition function {f ij }. The canonical bundle K M is equal to L −n−1 since L n+1 ⊗ 2n+1 T * ∼ = C by the global section (dγ i ) n+1 ∧ γ i . The contact structure γ is an L-valued 1-form on M . We regard γ as a bundle map from the holomorphic tangent bundle T of M to L and denote by Ker γ the kernel of the map γ :
Ker γ = {v ∈ T | γ(v) = 0}.
There exists a short exact sequence of sheaves :
0 → O(Ker γ) → O(T) γ → O(L) → 0.
Let X be a holomorphic vector field on M . The Lie derivative L X γ i of γ i with respect to X is a 1-form on U i . The set {L X γ i } is not a global form on M . However, the restriction L X γ i | Ker γ to Ker γ defines a global section {L X γ i | Ker γ } of the tensor L ⊗ Ker γ * of L and the dual of Ker γ. We call a vector field X a contact vector field of γ if L X γ i | Ker γ = 0 for each i. The system of equations {L X γ i | Ker γ = 0} is a global and holomorphic equation for X. Such a vector field generates a contact automorphism. We define aut(M, γ) to be the set of contact vector fields of γ. As an analogy of the real contact structures [7], Nitta and Takeuchi showed that for any element s of O(L), there exists a unique contact vector field X of γ such that γ(X) = s. Moreover, the correspondence O(L) → aut(M, γ) is isomorphic [11]. It means that the holomorphic tangent sheaf O(T) splits into O(Ker γ) and O(L) as sheaves of C-module. LeBrun showed that O(T) does not split into the sum of O(Ker γ) and O(L) as sheaves of O-module on Fano manifolds [8]. Therefore, the splitting of O(T) does not directly induce that of the sheaf O( k T) of k-vectors.
In this paper, we show the splitting of O( k T) into the sum of O( k Ker γ) and O(L ⊗ k−1 Ker γ) as sheaves of C-module. We extend the map γ to a bundle map from
k T to L ⊗ k−1 Ker γ by v 1 ∧ · · · ∧ v k → k j=1 (−1) j−1 γ(v j ) ⊗ v 1 ∧ · · · ∧ v j−1 ∧ v j+1 ∧ · · · ∧ v k
and denote the map also by γ for simplicity. Since the kernel of the map γ is just the space k Ker γ, we obtain the short exact sequence of the sheaves :
0 → O( k Ker γ) → O( k T) γ → O(L ⊗ k−1 Ker γ) → 0.
We shall extend to the equation L X γ i | Ker γ = 0 for a 1-vector X to a global and holomorphic equation for k-vectors. Let ∇ be a connection of the line bundle L such that ∇ 0,1 =∂. For a 1-vector X, the local equation L X γ i | Ker γ = 0 is given by the global
equation (d ∇ γ)(X) + d ∇ (γ(X))| Ker γ = 0 which is holomorphic. The direct extension (⊗ k d ∇ γ)(X) + d ∇ (⊗ k−1 d ∇ γ(γ(X)
))| Ker γ = 0 for a k-vector X is global but not holomorphic whenever ∇ is holomorphic. In order to find a global and holomorphic equation for holomorphic k-vector fields, we decompose the space k Ker γ as a sum of primitive parts with respect to the symplectic structure on Ker γ. Then we obtain such a equation which induces the following splitting theorem for O( k T) :
Theorem 1.1. Let k be an integer from 1 to 2n + 1. The sequence
0 → O( k Ker γ) → O( k T) → O(L ⊗ k−1
Ker γ) → 0 splits as sheaves of C-module. In particular, the sequence
0 → H i ( k Ker γ) → H i ( k T) → H i (L ⊗ k−1 Ker γ) → 0
is exact for each i = 0, . . . , 2n + 1.
We generalize the theorem to the splitting of O(L m ⊗ k T) and the exact sequence of H i (L m ⊗ k T) under a condition for m and k (see Theorem 3.8). As an application, we obtain the following vanishing theorem for H i (L m ⊗ k Ker γ) on compact Kähler manifolds by Kodaira-Akizuki-Nakano vanishing theorem :
Theorem 1.2. If M is a compact Kähler complex contact manifold with c 1 (M ) > 0, then H i (M, L m k Ker γ) = {0} for k and m satisfying one of following three conditions i ≤ 2n − k, m ≤ −[ k+1 2 ] − n − 1, 1 ≤ k ≤ 2n + 1, ∀i, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1, i ≥ k + 1, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.
We also have a similar result for vanishing of cohomology in the case of c 1 (M ) < 0 (see Theorem 4.1). Moreover, on CP 2n+1 , we show the vanishing theorem for H i (O(m) k Ker γ) by Bott vanishing theorem (see Theorem 4.3).
This paper is organized as follows. In Section 2, we prepare some propositions for complex symplectic vector spaces. The space of k-vectors is decomposed into a sum of primitive parts with respect to the symplectic structure. In Section 3, the equation L X γ i | Ker γ = 0 for a 1-vector X is extended to a global and holomorphic equation for k-vectors. By using the equation, we prove Theorem 1.1 (see Theorem 3.4). The theorem is generalized to the splitting theorem for O(L m ⊗ k T) (Theorem 3.8). On CP 2n+1 , L is given by O(2) and we also obtain the splitting of O(m) ⊗ k Ker γ. In the last section, we show two kinds of vanishing theorems (Theorem 4.1 and Theorem 4.3).
Complex symplectic vector spaces
In this section, we prepare some propositions for complex symplectic vector spaces in order to show our main theorems in Section 3.
Symplectic structures
Let V be a complex vector space of dimension 2n. A complex symplectic vector space is a pair (V, ω) of V and a non-degenerate skew-symmetric bilinear form ω on V . If we regard the symplectic structure ω as the isomorphism from V to the dual space V * , then the 2-tensor ⊗ 2 ω of ω is the isomorphism
⊗ 2 ω : ⊗ 2 V → ⊗ 2 V * .
It induces the isomorphism ⊗ 2 ω : ∧ 2 V → ∧ 2 V * . A 2-vector w 0 is defined by
⊗ 2 ω(w 0 ) = ω.
If we take a basis e 1 , . . . , e 2n of V such that ω = e * 1 ∧ e * 2 + · · · + e * 2n−1 ∧ e * 2n , then w 0 is represented as w 0 = e 1 ∧ e 2 + · · · + e 2n−1 ∧ e 2n and ω(w 0 ) = n.
Decomposition of the space of k-vectors
Let k be an integer from 1 to 2n. We define an operator L :
∧ k V → ∧ k+2 V by L(X) = X ∧ w 0 for X ∈ ∧ k V . We consider a 1-form θ as a map from ∧ k V to ∧ k−1 V by θ(v 1 ∧ · · · ∧ v k ) = k j=1 (−1) j−1 θ(v j )v 1 ∧ · · · ∧ v j−1 ∧ v j+1 ∧ · · · ∧ v k
for v 1 , . . . , v k ∈ V and denote the map also by θ for simplicity. Let l be an integer with 0 ≤ l ≤ k. We regard an l-form θ 1 ∧ · · · ∧ θ l as a map θ 1 ∧ · · · ∧ θ l :
∧ k V → ∧ k−l V by θ 1 ∧ · · · ∧ θ l (X) = θ l • · · · • θ 2 • θ 1 (X) for X ∈ ∧ k V . Let Λ be an operator Λ : ∧ k V → ∧ k−2 V defined by Λ(X) = ω(X) for X ∈ ∧ k V .
Then we obtain the formula (ΛL − LΛ)(X) = (n − k)X for X ∈ ∧ k V , and inductively,
(ΛL r − L r Λ)(X) = r(n − k − r + 1)L r−1 X (1) where we define L 0 = id. A k-vector X is called primitive if Λ(X) = 0. If follows from the formula (1) that Λ s L r X = r! (r−s)! (n − k − r + 1)(n − k − r + 2) · · · (n − k − r + s)L r−s X, r ≥ s, 0,
r < s (2) for a primitive k-vector X. Let ∧ k e V denote the space of primitive k-vectors :
∧ k e V = {X ∈ ∧ k V | Λ(X) = 0}.
Then we have the following decomposition of the space ∧ k V of k-vectors :
Proposition 2.1. If k ≤ n, then ∧ k V = ∧ k e V + L ∧ k−2 e V + · · · + L [ k 2 ] ∧ k−2[ k 2 ] e V.
If k > n, then
∧ k V = L k−n ∧ 2n−k e V + L k−n+1 ∧ 2n−k−2 e V + · · · + L [ k 2 ] ∧ k−2[ k 2 ] e V
where [m] means the Gauss symbol of m.
Transformation associated with the decomposition
We define a linear transformation T on ∧ k V by
T (X) = c 0 X + c 1 LΛX + c 2 L 2 Λ 2 X + · · · + c [ k 2 ] L [ k 2 ] Λ [ k 2 ] X = [ k 2 ] i=0 c i L i Λ i X for constants c 0 , c 1 , c 2 , . . . , c [ k 2 ]
, then we obtain the following :
Proposition 2.2. The transformation T is isomorphic if and only if the constants c 0 , c 1 , . . . , c [ k 2 ] satisfy r s=0 c s r! (r−s)! (n−k+r+s)! (n−k+r)! = 0 for any r = 0, . . . , [ k 2 ].
Proof. In the case k ≤ n and 0 ≤ r ≤ [ k 2 ], the equation (2) implies that
L s Λ s L r X = r! (r−s)! (n−k+r+s)! (n−k+r)! L r X, s ≤ r, 0, s > r for X ∈ ∧ k−2r e
. It yields that
T (L r X) = [ k 2 ] s=0 c s L s Λ s L r X = r s=0 c s r! (r − s)! (n − k + r + s)! (n − k + r)! L r X (3) for X ∈ ∧ k−2r e .
In the case k > n and k − n ≤ r ≤ [ k 2 ], we also have the same equation
(3) for X ∈ ∧ k−2r e . Hence Proposition 2.1 implies that T is an isomorphism from ∧ k V to itself if and only if r s=0 c s r! (r−s)! (n−k+r+s)! (n−k+r)! = 0 for each r = 0, . . . , [ k 2 ].
Splitting of sheaves on complex contact manifolds
Let (M, γ) be a complex contact manifold of dimension 2n + 1 and L the line bundle associated with the contact structure γ. We denote by D the subbundle Ker
γ of T. Let ∇ be a connection of L with ∇ 0,1 =∂. The covariant exterior differentiation d ∇ γ of γ is a smooth section of L ⊗ 2 T * .
The isomorphisms associated with d ∇ γ
Let d ∇ γ| D denote the restriction of d ∇ γ to D. Then, d ∇ γ| D is a holomorphic section of L ⊗ 2 D * which is independent of the choice of a connection ∇. We identify d ∇ γ| D with the holomorphic bundle map from D to L ⊗ D * . Then the map d ∇ γ| D :
D → L ⊗ D * is isomorphic since dγ i is non-degenerate on D. In general, the k-th tensor ⊗ k d ∇ γ of d ∇ γ is a smooth section of L k ⊗ k 2 T * which is regarded as the smooth bundle map ⊗ k d ∇ γ : k T → L k ⊗ k T * .
Let e be a local frame of L and A a connection form of ∇ with respect to e. The contact structure γ is given by γ = e ⊗ γ 0 for a holomorphic 1-form γ 0 . Then
d ∇ γ = e ⊗ (dγ 0 + A ∧ γ 0 ). For 1-vectors v 1 , . . . , v k , the L k -valued k-form ⊗ k d ∇ γ(v 1 ∧ · · · ∧ v k ) is written by e k ⊗ {(dγ 0 + A ∧ γ 0 )(v 1 ) ∧ · · · ∧ (dγ 0 + A ∧ γ 0 )(v k )} = e k ⊗ {⊗ k dγ 0 (v 1 ∧ · · · ∧ v k ) + γ 0 ∧ (⊗ k−1 dγ 0 )(A(v 1 ∧ · · · ∧ v k )) −A ∧ (⊗ k−1 dγ 0 )(γ 0 (v 1 ∧ · · · ∧ v k ))}.
It implies
⊗ k d ∇ γ(X) = e k ⊗ {⊗ k dγ 0 (X) + γ 0 ∧ (⊗ k−1 dγ 0 )(A(X)) − A ∧ (⊗ k−1 dγ 0 )(γ 0 (X))} (4) for any k-vector X. We remark that the map ⊗ k d ∇ γ is not holomorphic whenever ∇ is holomorphic. However, the restriction ⊗ k d ∇ γ| D is the holomorphic map from k D to L k ⊗ k D * . The induced map ⊗ k d ∇ γ| D : O(⊗ k D) → O(L k ⊗ k D * )
is isomorphic on sheaves, and it is extended to an isomorphism
⊗ k d ∇ γ| D : O(L m ⊗ k D) → O(L k+m ⊗ k D * )
for each m ∈ Z and k = 1, . . . , 2n + 1.
In the case m = −1 and k = 2, we have the isomorphism
⊗ 2 d ∇ γ| D : O(L −1 ⊗ 2 D) → O(L ⊗ 2 D * ). By considering d ∇ γ| D as a holomorphic section of L ⊗ 2 D * , there exists a holomorphic section w of L −1 ⊗ ∧ 2 D such that ⊗ 2 d ∇ γ| D (w) = d ∇ γ| D
as w 0 in Section 2.1. The section w is independent of the choice of a connection ∇. (4) and (5) imply that
The map
F : k T → L k ⊗ k T * We fix an integer k such that 1 ≤ k ≤ 2n + 1 and define a map F : k T → L k ⊗ k T * by F (X) = [ k 2 ] i=0 (k − i)! k!i! ⊗ k−2i d ∇ γ((d ∇ γ) i (X)) ∧ (d ∇ γ) i for X ∈ k T, where (d ∇ γ) 0 is the identity map id and (d ∇ γ) i is the i-th wedge d ∇ γ ∧ · · · ∧ d ∇ γ of d ∇ γ. We have (d ∇ γ) i (X) = e i ⊗ {(dγ 0 ) i (X) − i(dγ 0 ) i−1 (A(γ 0 (X)))} (5) for each i = 0, . . . , [ k 2 ] since (d ∇ γ) i = e i ⊗ {(dγ 0 ) i + iA ∧ γ 0 ∧ (dγ 0 ) i−1 }. The equations⊗ k−2i d ∇ γ((d ∇ γ) i (X)) = e k−i ⊗ {⊗ k−2i dγ 0 ((dγ 0 ) i (X)) − i ⊗ k−2i dγ 0 ((dγ 0 ) i−1 (A(γ 0 (X)))) +γ 0 ∧ (⊗ k−1−2i dγ 0 )((dγ 0 ) i (A(X))) − A ∧ (⊗ k−1−2i dγ 0 )((dγ 0 ) i (γ 0 (X)))}(6)
where we use A((dγ 0 ) i−1 (A(γ 0 (X)))) = γ 0 ((dγ 0 ) i−1 (A(γ 0 (X))) = 0 in the first line. It yields that
⊗ k−2i d ∇ γ((d ∇ γ) i (X))| D = e k−i ⊗ {⊗ k−2i dγ 0 ((dγ 0 ) i (X)) − i ⊗ k−2i dγ 0 ((dγ 0 ) i−1 (A(γ 0 (X)))) −A ∧ (⊗ k−1−2i dγ 0 )((dγ 0 ) i (γ 0 (X)))}.(7)
The second term in the right hand side of the equation (7) is 0 except
for 1 ≤ i ≤ [ k 2 ] and the third term is 0 except for 0 ≤ i ≤ [ k−1 2 ]. Hence F (X)| D = [ k 2 ] i=0 (k − i)! k!i! {⊗ k−2i d ∇ γ((d ∇ γ) i (X)) ∧ (d ∇ γ) i }| D = e k ⊗ [ k 2 ] i=0 (k − i)! k!i! {⊗ k−2i dγ 0 ((dγ 0 ) i (X)) ∧ (dγ 0 ) i − [ k 2 ] i=1 (k − i)! k!(i − 1)! ⊗ k−2i dγ 0 ((dγ 0 ) i−1 (A(γ 0 (X)))) ∧ (dγ 0 ) i − [ k−1 2 ] i=0 (k − i)! k!i! A ∧ (⊗ k−1−2i dγ 0 )((dγ 0 ) i (γ 0 (X))) ∧ (dγ 0 ) i D .(8)
The
restriction F | D : k D → L k k D * satisfies F | D (X) = F (X)| D for X ∈ k D. If X is a holomorphic section of k D, then it follows from γ 0 (X) = 0 that F | D (X) = F (X)| D = e k ⊗ [ k 2 ] i=0 (k − i)! k!i! ⊗ k−2i dγ 0 ((dγ 0 ) i (X)) ∧ (dγ 0 ) i D is a holomorphic section of L k ⊗ k D * .
Hence we obtain the following :
Lemma 3.1. An L k -valued k-form F | D (X) restricted to D is holomorphic for any holo- morphic section X of k D. The lemma implies that F | D is regarded as a map from O( k D) to O(L k ⊗ k D * ). The map F | D is written by F | D (X) = ⊗ k d ∇ γ| D [ k 2 ] i=0 (k − i)! k!i! (d ∇ γ| D ) i (X) ∧ (w) i for X ∈ O( k D) since (d ∇ γ| D ) i = ⊗ 2i d ∇ γ| D (w) i . We define the transformation f : O( k D) → O( k D) by f (X) = [ k 2 ] i=0 (k − i)! k!i! (d ∇ γ| D ) i (X) ∧ (w) i for X ∈ O( k D)
. Then the map F | D is the composition of ⊗ k d ∇ γ| D and f . Proposition 2.2 implies that the following :
Proposition 3.2. The map F | D : O( k D) → O(L k ⊗ k D * ) is isomorphic. Proof. It suffices to show that f is isomorphic since ⊗ k d ∇ γ| D is isomorphic. At each point x ∈ M , the linear map f x : k D x → k D x is written by f x = [ k 2 ] i=0 (k − i)! k!i! L i Λ i
where L and Λ are operators as in the previous section associated with the symplectic
structure (d ∇ γ| D ) x on D x . The map f x is isomorphic by Proposition 2.2 since each coefficient c i = (k−i)! k!i! is positive. Hence F | D is also isomorphic. 3.3 The map G : Γ(L ⊗ k−1 D) → Γ(L k ⊗ k T * ) We define a map G : Γ(L ⊗ k−1 D) → Γ(L k ⊗ k T * ) by G(s) = [ k−1 2 ] i=0 (k − 1 − i)! k!i! d ∇ ⊗ k−1−2i d ∇ γ((d ∇ γ) i (s)) ∧ (d ∇ γ) i
for s ∈ Γ(L ⊗ k−1 D). Let s be a section of L ⊗ k−1 D. The section s is locally written as s = e ⊗ s 0 for a section s 0 ∈ k−1 D. It follows from the equations (6) and
(d ∇ γ) i (s) = e i+1 ⊗ (dγ 0 ) i (s 0 ) that ⊗ k−1−2i d ∇ γ((d ∇ γ) i (s)) = e k−i ⊗{⊗ k−1−2i dγ 0 ((dγ 0 ) i (s 0 )+γ 0 ∧(⊗ k−2−2i dγ 0 )((dγ 0 ) i (A(s 0 )))}.(9)
We remark that the second term in the right hand side of the equation (9) is 0 except for
0 ≤ i ≤ [ k 2 ] − 1. It yields that d ∇ ⊗ k−1−2i d ∇ γ((d ∇ γ) i (s))| D = e k−i ⊗ {d ⊗ k−1−2i dγ 0 ((dγ 0 ) i (s 0 ) +dγ 0 ∧ (⊗ k−2−2i dγ 0 )((dγ 0 ) i (A(s 0 ))) +(k − i)A ∧ ⊗ k−1−2i dγ 0 ((dγ 0 ) i (s 0 ))}| D .
Hence
G(s)| D = [ k−1 2 ] i=0 (k − 1 − i)! k!i! d ∇ (⊗ k−1−2i d ∇ γ((d ∇ γ) i (s))| D ) ∧ (d ∇ γ) i | D = e k ⊗ [ k−1 2 ] i=0 (k − 1 − i)! k!i! d(⊗ k−1−2i dγ 0 ((dγ 0 ) i (s 0 ))) ∧ (dγ 0 ) i + [ k 2 ] i=1 (k − i)! k!(i − 1)! (⊗ k−2i dγ 0 )((dγ 0 ) i−1 (A(s 0 ))) ∧ (dγ 0 ) i + [ k−1 2 ] i=0 (k − i)! k!i! A ∧ ⊗ k−1−2i dγ 0 ((dγ 0 ) i (s 0 )) ∧ (dγ 0 ) i D .(10)
Then we have Proposition 3.
3. An L k -valued k-form {F (X) + G(γ(X))}| D restricted to D is holomorphic for any holomorphic k-vector X.
Proof. If we take s as the image γ(X) of a k-vector X, then s 0 = γ 0 (X), and it follows from the equations (8) and (10) that
{F (X) + G(s)}| D = e k ⊗ [ k 2 ] i=0 (k − i)! k!i! {⊗ k−2i dγ 0 ((dγ 0 ) i (X)) ∧ (dγ 0 ) i + [ k−1 2 ] i=0 (k − 1 − i)! k!i! d(⊗ k−1−2i dγ 0 ((dγ 0 ) i (γ 0 (X)))) ∧ (dγ 0 ) i D .
Hence {F (X) + G(γ(X))}| D is holomorphic for a holomorphic k-vector X.
The splitting of sheaves O( k T) as C-module
We have the following theorem:
Theorem 3.4. Let k be an integer from 1 to 2n + 1. The sequence
0 → O( k D) → O( k T) → O(L ⊗ k−1 D) → 0
splits as sheaves of C-module. In particular, the sequence
0 → H i ( k D) → H i ( k T) → H i (L ⊗ k−1 D) → 0
is exact for each i = 0, . . . , 2n + 1.
Proof. Let s be a holomorphic section of L ⊗ k−1 D.
We take an open set U of M where the bundles L and D are trivial. Then we can take a holomorphic k-vector Y on U such that γ(Y ) = s as follows. We fix a local frame e of L on U . The contact form γ is given by γ = e ⊗ γ 0 for a holomorphic 1-form γ 0 on U . We can choose a local frame {e 1 , . . . , e 2n } of D and a local section e 2n+1 of T on U such that γ(e 2n+1 ) = e. If s is written by s = s i 1 ...i k e ⊗ e i 1 ∧ · · · ∧ e i k−1 on U , then we take a section Y by Y = s i 1 ...i k e 2n+1 ∧ e i 1 ∧ · · · ∧ e i k−1 . Now we consider the smooth section F (Y ) + G(s) of L k ⊗ k T * . Proposition 3.3 implies that {F (Y ) + G(s)}| D restricted to D is a holomorphic section of L k ⊗ k D * . Hence, there exists a holomorphic section h of k D such that
{F (Y ) + G(s)}| D = F | D (h)
by the isomorphism F | D in Proposition 3.2. We define X by X = Y − h. Then we obtain the holomorphic k-vector X on U satisfying the following equations
(i) γ(X) = s, (ii) {F (X) + G(s)}| D = 0.
We take such local sections X 1 and X 2 on open sets U 1 and U 2 , respectively. The first condition (i) implies that the difference
X 1 − X 2 is in k D on U 1 ∩ U 2 . We also have the equation F | D (X 1 − X 2 ) = F (X 1 − X 2 )| D =L m ⊗ k T → L m+k ⊗ k T * by F m (X) = [ k 2 ] i=0 c m,i ⊗ k−2i d ∇ γ((d ∇ γ) i (X)) ∧ (d ∇ γ) i for X ∈ L m ⊗ k T. By the same argument in Lemma 3.1, the restriction F m | D induces the map from O(L m ⊗ k D) to O(L m+k ⊗ k D * ). m ≤ −n − [ k 2 ] − 2, 1 ≤ k ≤ 2n + 1, −n − 1 ≤ m ≤ −k − 1, 1 ≤ k ≤ n, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1. Proof. The map F m | D is the composition of ⊗ k d ∇ γ| D and the transformation f m of O(L m ⊗ k D) defined by f m (X) = [ k 2 ] i=0 c m,i (d ∇ γ| D ) i (X) ∧ w i for X ∈ O(L m ⊗ k D). At each point x ∈ M , the linear map (f m ) x : L m x ⊗ k D x → L m x ⊗ k D x is writ- ten by (f m ) x = [ k 2 ] i=0 c m,i L i Λ i . Proposition 2.2 implies that (f m ) x is isomorphic if r s=0 c m,s r! (r−s)! (n−k+r+s)! (n−k+r)! is not zero for each r. If k ≥ 2, then r s=0 c m,s r! (r − s)! (n − k + r + s)! (n − k + r)! = (m + n + r + 1) · · · (m + n + 2) (m + k) · · · (m + k − r + 1) for any r = 1, . . . , [ k 2 ]. It yields that r s=0 c m,s r! (r−s)! (n−k+r+s)! (n−k+r)! = 0 for each r if m ≤ −n − [ k 2 ] − 2 or m ≥ −n − 1. By the assumption m ≤ −k − 1 or m ≥ −[ k 2 ], the linear map (f m ) x is isomorphic if m ≤ −n − [ k 2 ] − 2, ∀k, −n − 1 ≤ m ≤ −k − 1, 1 ≤ k ≤ n, m ≥ −[ k 2 ], ∀k.
Hence, we finish the proof.
We define a constant c ′ m,i by
c ′ m,i = 1 (k + m)(k + m − 1) · · · (k + m − i) i! for each i = 0, 1, . . . , [ k−1 2 ]
. We remark that these constants are well-defined since m+k < 0
and m + k − [ k−1 2 ] > 0 in the cases of m ≤ −k − 1 and m ≥ −[ k 2 ], respectively. We define a map G m : Γ(L m+1 ⊗ k−1 D) → Γ(L m+k ⊗ k T * ) by G m (s) = [ k−1 2 ] i=0 c ′ m,i d ∇ ⊗ k−1−2i d ∇ γ((d ∇ γ) i (s)) ∧ (d ∇ γ) i for s ∈ Γ(L m+1 ⊗ k−1 D)
. Similarly to Proposition 3.3, we have Proposition 3.7. An L k+m -valued k-form {F m (X) + G m (γ(X))}| D restricted to D is holomorphic for any holomorphic L m -valued k-vector X.
By repeating the proof of Theorem 3.4 with F m and G m instead of F and G, then we obtain
Theorem 3.8. The sequence 0 → O(L m ⊗ k D) → O(L m ⊗ k T) → O(L m+1 ⊗ k−1 D) → 0
splits as sheaves of C-module, and the sequence
0 → H i (L m ⊗ k D) → H i (L m ⊗ k T) → H i (L m+1 ⊗ k−1 D) → 0
is exact for each i = 0, . . . , 2n + 1 if k and m satisfy one of following three conditions
m ≤ −n − [ k 2 ] − 2, 1 ≤ k ≤ 2n + 1, −n − 1 ≤ m ≤ −k − 1, 1 ≤ k ≤ n, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.
The splitting of the sheaves O(l m ⊗ k T) on CP 2n+1
On the odd dimensional projective space CP 2n+1 , there exists a standard contact structure γ written by γ = z 0 dz 1 −z 1 dz 0 +· · ·+z 2n dz 2n+1 −z 2n+1 dz 2n in the homogeneous coordinate. Let l denote the hyperplane bundle O(1) on CP 2n+1 . The associated bundle L is given by l 2 = O(2) and the contact structure γ is regarded as a section of l 2 ⊗ T * . We consider the short exact sequence
0 → l m ⊗ k D → l m ⊗ k T γ → l m+2 ⊗ k−1 D → 0
for m ∈ Z and 1 ≤ k ≤ 2n + 1. If m is even, then the splitting of O(l m ⊗ k T) is induced by Theorem 3.8 since L = l 2 . From now on, we assume that m is odd. We define a constantc m,i byc m,0 = 1 and
c m,i = 1 (k + m 2 )(k + m 2 − 1) · · · (k + m 2 − i + 1) i! for each i = 1, . . . , [ k 2 ]
. These constants are well-defined since k + m 2 − i + 1 = 0 for any i. We fix a connection ∇ of l and define a map F m :
l m ⊗ k T → l m+2k ⊗ k T * by F m (X) = [ k 2 ] i=0c m,i ⊗ k−2i d ∇ γ((d ∇ γ) i (X)) ∧ (d ∇ γ) i for X ∈ l m ⊗ k T. The restriction F m | D induces the map from O(l m ⊗ k D) to O(l m+2k ⊗ k D * ). Proposition 3.9. The map F m | D : O(l m ⊗ k D) → O(l m+2k ⊗ k D * ) is isomorphic if m satisfies m ≤ −2n − 2[ k 2 ] − 3 or −2n − 3 ≤ m. Proof. If k ≥ 2, then r s=0c m,s r! (r − s)! (n − k + r + s)! (n − k + r)! = ( m 2 + n + r + 1) · · · ( m 2 + n + 2) ( m 2 + k) · · · ( m 2 + k − r + 1) for any r = 1, . . . , [ k 2 ]. By the same argument in Proposition 3.6, the map F m | D is isomorphic if m ≤ −2n − 2[ k 2 ] − 3 or −2n − 3 ≤ m.
We define a constantc ′ m,i bỹ Proof. We fix a frameẽ of l on U . Then γ is given by γ =ẽ 2 ⊗ γ 0 for a holomorphic 1-form γ 0 on U . Let X be a holomorphic l m -valued k-vector X on M . By the same argument in the proof of Proposition 3.3, we have
c ′ m,i = 1 (k + m 2 )(k + m 2 − 1) · · · (k + m 2 − i) i! for each i = 0, 1, . . . , [ k−1 2 ]. These constants are well-defined since m 2 + k − [ k−1 2 ] = 0. We define a map G m : Γ(l m+1 ⊗ k−1 D) → Γ(l m+2k ⊗ k T * ) by G m (s) = [ k−1 2 ] i=0c ′ m,i d ∇ ⊗ k−1−2i d ∇ γ((d ∇ γ) i (s)) ∧ (d ∇ γ) i for s ∈ Γ(l m+2 ⊗ k−1 D). Then we have{ F m (X) + G m (s)}| D =ẽ 2k+m ⊗ [ k 2 ] i=0c m,i {⊗ k−2i dγ 0 ((dγ 0 ) i (X)) ∧ (dγ 0 ) i + [ k−1 2 ] i=0c ′ m,i d(⊗ k−1−2i dγ 0 ((dγ 0 ) i (γ 0 (X)))) ∧ (dγ 0 ) i D .
Hence { F m (X) + G m (γ(X))}| D is holomorphic.
By repeating the proof of Theorem 3.4 with F m and G m instead of F and G, then we obtain Theorem 3.11. The sequence
0 → O(l m ⊗ k D) → O(l m ⊗ k T) → O(l m+2 ⊗ k−1 D) → 0
splits as sheaves of C-module, and the sequence
0 → H i (l m ⊗ k D) → H i (l m ⊗ k T) → H i (l m+2 ⊗ k−1 D) → 0
is exact for each i = 0, . . . , 2n + 1 if k and m satisfy one of following conditions
m ≤ −2n − 2[ k 2 ] − 4, 1 ≤ k ≤ 2n + 1, m : even, −2n − 2 ≤ m ≤ −2k − 2, 1 ≤ k ≤ n, m : even, m ≥ −2[ k 2 ], 1 ≤ k ≤ 2n + 1, m : even, m ≤ −2n − 2[ k 2 ] − 3, 1 ≤ k ≤ 2n + 1, m : odd, m ≥ −2n − 3, 1 ≤ k ≤ 2n + 1, m : odd.
Vanishing theorems for cohomology of k D
In this section, we apply the splitting theorems to the cohomology of k D and obtain the vanishing theorems. From now on, we denote by E k D the tensor E ⊗ k D of vector bundles E and k D for simplicity.
Vanishing of the cohomology on compact Kähler complex contact manifolds
We have the following vanishing theorem of the cohomology on compact Kähler manifolds :
Theorem 4.1. If M is a compact Kähler complex contact manifold with c 1 (M ) > 0, then H i (M, L m k D) = {0} i ≤ 2n − k, m ≤ −[ k+1 2 ] − n − 1, 1 ≤ k ≤ 2n + 1, ∀i, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1, i ≥ k + 1, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1. If c 1 (M ) < 0, then H i (M, L m k D) = {0} i ≥ k + 1, m ≤ −n − [ k 2 ] − 2, 1 ≤ k ≤ 2n + 1, i ≥ k + 2, m = −n − 1, 1 ≤ k ≤ n, i ≤ 2n − k − 1, m = −k, 1 ≤ k ≤ n, i ≤ 2n − k, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.
Proof. By Theorem 3.8, the sequence
0 → H i (M, L m k D) → H i (M, L m k T) → H i (M, L m+1 k−1 D) → 0(11)
is exact for each i if m and k satisfy one of following three conditions
m ≤ −n − [ k 2 ] − 2, 1 ≤ k ≤ 2n + 1, −n − 1 ≤ m ≤ −k − 1, 1 ≤ k ≤ n, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.(12)
It follows from Serre's duality that
H i (M, L m k T) * ∼ = H 2n+1−i (M, Ω k L −m K M ) ∼ = H 2n+1−i (M, Ω k L −m−n−1 ).(13)H i (M, L m+1 k−1 D) = {0}(14)for k + 1 ≤ i ≤ 2n + 1, m > −n − 1 and 1 ≤ k ≤ 2n + 1. The condition (15) is written by H i (M, L m ′ k ′ D) = {0} for k ′ + 2 ≤ i ≤ 2n + 1, m ′ > −n and 0 ≤ k ′ ≤ 2n.(15)
We only consider the vanishing for k ′ ≥ 1 since the case k ′ = 0 that H i (M, L m ′ ) = {0} for 2 ≤ i and m ′ > −n is induced by the Kodaira-Akizuki-Nakano vanishing theorem. The first condition (14) induces the second one (15) for k ≥ 1. Hence, (12) and (14) imply that
H i (M, L m k D) = {0} if m satisfies one of two conditions i ≥ k + 1, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1, i ≥ k + 1, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.(16)
The Serre's duality implies that
H i (M, L m k D) * ∼ = H 2n+1−i (M, K M L −m k D * ) ∼ = H 2n+1−i (M, L −k−m−i ≤ 2n − k, m ≤ −[ k+1 2 ] − n − 1, 1 ≤ k ≤ 2n + 1, i ≤ 2n − k, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1.(18)
We remark that the condition k ≤ −n − 1 implies that 2n − k ≥ k + 1. It follows from (16) and (18)
that H i (M, L m k D) = {0} for i ≤ 2n − k, m ≤ −[ k+1 2 ] − n − 1, 1 ≤ k ≤ 2n + 1, ∀i, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1, i ≥ k + 1, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1.
In
H i (M, L m k D) = {0} i ≥ k + 1, m ≤ −n − [ k 2 ] − 2, 1 ≤ k ≤ 2n + 1, i ≥ k + 2, m = −n − 1, 1 ≤ k ≤ n.(19)
Serre's duality implies
H i (M, L m k D) = {0} i ≤ 2n − k, m ≥ −[ k 2 ], 1 ≤ k ≤ 2n + 1, i ≤ 2n − k − 1, m = −k, 1 ≤ k ≤ n.(20)
Hence it completes the proof.
Remark 4.2. Salamon proved that any (p, q)-cohomology H p,q (M ) vanishes for p = q if M is the twistor space of a quaternion manifold with a positive scalar curvature [10].
In the proof, he also obtained the vanishing of the cohomology of k D on the twistor space (Equation (6.4) in [10]). He used the notation of L and E as L 1 2 and L − 1 2 D in our notation, respectively. These results are improved to the case of compact Kähler complex contact manifolds with c 1 (M ) > 0 by the same argument, and the vanishing is translated into
H i (M, L m k D) = {0} ∀i, −n ≤ m ≤ −k − 1, 1 ≤ k ≤ n − 1, i = k, m = −k, 1 ≤ k ≤ n.(21)
The first condition in (21) is equal to the second condition in Theorem 4.1. The second one in (21) is independent of our theorem. However, we remark that the first and third conditions in Theorem 4.1 are not induced by the vanishing (21).
The vanishing of the cohomology on CP 2n+1
In this section, we show the vanishing theorem for H i (l m ∧ k D) = H i (O(m) ∧ k D) on CP 2n+1 by using Bott's vanishing formula [3]. We have the short exact sequence
0 → D → T γ → l 2 → 0.
It induces the exact sequence
0 → l m k D → l m k T γ → l m+2 k−1 D → 0(22)
for m ∈ Z. Then we have a vanishing of the cohomology as follows
Theorem 4.3. H i (l m k D) = {0} if, in the case m is even i = 2n + 1, m ≤ −2n − 2 − 2[ k+1 2 ], 1 ≤ k ≤ 2n + 1, i = 2n + 1 − k, m = −2n − 2, 1 ≤ k ≤ n, ∀i, −2n ≤ m ≤ −2k − 2, 1 ≤ k ≤ n, i = k, m = −2k, 1 ≤ k ≤ n, i = 0, m ≥ −2[ k 2 ], 1 ≤ k ≤ 2n + 1
and, in the case m is odd
i = 2n + 1, m ≤ −2n − 3 − k, 1 ≤ k ≤ 2n + 1, ∀i, −2n − 2 − k ≤ m ≤ −2k + 1, 1 ≤ k ≤ 2n + 1, ∀i, −2n − 3 ≤ m ≤ −k, 1 ≤ k ≤ 2n + 1, i = 0, m ≤ −k + 1, 1 ≤ k ≤ 2n + 1.
Proof. By applying Theorem 3.11 to the sequence (22), we obtain the short exact sequence
0 → H i (l m k D) → H i (l m k T) → H i (l m+2 k−1 D) → 0 (23) if m ≤ −2n − 2[ k 2 ] − 4, 1 ≤ k ≤ 2n + 1, m : even, −2n − 2 ≤ m ≤ −2k − 2, 1 ≤ k ≤ n, m : even, m ≥ −2[ k 2 ], 1 ≤ k ≤ 2n + 1, m : even, m ≤ −2n − 2[ k 2 ] − 3, 1 ≤ k ≤ 2n + 1, m : odd, m ≥ −2n − 3, 1 ≤ k ≤ 2n + 1, m : odd.(24)
By applying Bott's vanishing formula to Serre's duality H i (l m k T) * ∼ = H 2n+1−i (Ω k (l −m−2n−2 )), H i (l m k T) = {0} holds except for the following cases
i = 2n + 1 − k, m = −2n − 2, i = 2n + 1, m < −2n − 2 − k, i = 0, m > −k − 1.(25)
It follows from (23) and (25) that
H i (l m k D) = H i (l m+2 k−1 D) = {0} if i = 2n + 1 − k, m = −2n − 2, i = 2n + 1, m ≤ −2n − 3 − k, ∀i, −2n − 2 − k ≤ m ≤ −k − 1, (m = −2n − 2), i = 0, m ≥ −k.(26)
In the case that m is even, (24) and (26)
i = 2n − k, m = −2n, 0 ≤ k ≤ n − 1, i = 2n + 1, m ≤ −2n − 2 − 2[ k+1 2 ], 0 ≤ k ≤ 2n, ∀i, −2n + 1 ≤ m ≤ −2k − 2, 0 ≤ k ≤ n − 1, i = 0, m ≥ −2[ k+1 2 ] + 2, 0 ≤ k ≤ 2n.(28)
We remark that the case k = 0 of (28) is contained in Bott's vanishing formula. We only consider the vanishing of H i (l m k D) for k ≥ 1. We summarize (27) and (28)
i = 2n + 1, m ≤ −2n − 2 − 2[ k+1 2 ], 1 ≤ k ≤ 2n + 1, i = 2n + 1 − k, m = −2n − 2, 1 ≤ k ≤ n, ∀i, −2n ≤ m ≤ −2k − 2, 1 ≤ k ≤ n, i = 0, m ≥ −2[ k 2 ], 1 ≤ k ≤ 2n + 1.(29)
In the case that m is odd, by repeating the above argument the conditions (24) and (26) imply that H i (l m k D) = {0} for i = 2n + 1, m ≤ −2n − 3 − k, 1 ≤ k ≤ 2n + 1, ∀i, m = −2n − 2 − k, 1 ≤ k ≤ 2n + 1, ∀i, m = −2n − 1 − k, 1 ≤ k ≤ 2n, ∀i, −2n − 3 ≤ m ≤ −k − 1, 1 ≤ k ≤ 2n + 1, ∀i, m = −k, 1 ≤ k ≤ 2n − 1, i = 0, m = −k, k = 2n + 1, i = 0, m ≥ −k + 1, 1 ≤ k ≤ 2n + 1.
(30)
Applying (29)
i = 0, m ≥ −2[ k 2 ], 1 ≤ k ≤ 2n + 1, i = k, m = −2k, 1 ≤ k ≤ n, ∀i, −2n ≤ m ≤ −2k − 2, 1 ≤ k ≤ n, i = 2n + 1, m ≤ −2n − 2 − 2[ k+1 2 ], 1 ≤ k ≤ 2n + 1(31)
and, in the case m is odd i = 0, m ≥ −k + 1, 1 ≤ k ≤ 2n + 1, ∀i, m = −k, 1 ≤ k ≤ 2n + 1, ∀i, m = −k − 1, 1 ≤ k ≤ 2n, ∀i, −2n − 1 − k ≤ m ≤ −2k + 1, 1 ≤ k ≤ 2n + 1, ∀i, m = −2n − 2 − k, 1 ≤ k ≤ 2n − 1, i = 2n + 1, m = −2n − 2 − k, k = 2n + 1, i = 2n + 1, m ≤ −2n − 3 − k, 1 ≤ k ≤ 2n + 1.
The conditions (29) A contact structure on CP 2n+1 is unique, up to automorphisms [11]. Hence Theorem 3.11 and 4.3 hold for any contact structure on CP 2n+1 . 2) which is given by a = a ij z i dz j in the homogeneous coordinate. If a ij = −a ji and (a ij ) is non-degenerate, then A induces a contact structure on CP 2n+1 . By applying Theorem 4.3 to D = N (1), we obtain the vanishing formula for the cohomology H i (∧ k N (m + k)).
0 by the second condition (ii). ThenX 1 = X 2 = 0 on U 1 ∩ U 2 since F | D isisomorphic. Hence the correspondence of s to X provides a right inverse of the map γ : O( k T) → O(L ⊗ k−1 D) as a C-module map. It induces the splitting of O( k T) and the exactness of H i ( k T) for each i. It completes the proof.
Remark 3. 5 .
5In the proof, h is independent of the connection ∇ since {F (Y ) + G(s)}| D and F | D do not depend on ∇ by Proposition 3.2 and 3.3. The k-vector X = Y − h is also independent of ∇. Hence, the splitting of the sequence in Theorem 3.4 is independent of the choice of the connection.3.5 The splitting of the sheaves O(L m ⊗ k T) as C-module In this section, we generalize Theorem 3.4 to the splitting of O(L m ⊗ k T) under a condition for m and k. Let m be an integer such that m ≤ −k − 1 or m ≥ −[ k 2 ]. We define a constant c m,i by c m,0 = 1 andc m,i = 1 (k + m)(k + m − 1) · · · (k + m − i + 1) i! for each i = 1, . . . , [ k 2 ]. These constants are well-defined since m + k < 0 and m + k − [ k 2 ] + 1 > 0 in the cases of m ≤ −k − 1 and m ≥ −[ k 2 ], respectively. We define a map F m :
Proposition 3. 6 .
6The map F m | D : O(L m ⊗ k D) → O(L m+k ⊗ k D * )is isomorphic if k and m satisfy one of following three conditions
Proposition 3 . 10 .
310An l 2k+m -valued k-form { F m (X) + G m (γ(X))}| D restricted to D is holomorphic for any holomorphic l m -valued k-vector X.
In the case of c 1 (M ) > 0, the first Chern class c 1 (L −m−n−1 ) of the line bundle L −m−n−1 is negative if m > −n − 1 since c 1 (L) = − 1 n+1 c 1 (K M ) > 0. By applying the Kodaira-Akizuki-Nakano vanishing theorem [1] to the last cohomology in (13), we have H i (M, L m k T) = {0} for k + 1 ≤ i if m > −n − 1. Hence, the sequence (11) implies H i (M, L m k D) = {0},
* = L −1 D. We apply the vanishing in (16) to the last cohomology in (17), and obtain H i (M, L m k D) = {0} if m satisfies one of two conditions
the case of c 1 (M ) < 0, the first Chern class c 1 (L −m−n−1 ) is negative if m < −n − 1. Then the Kodaira-Akizuki-Nakano vanishing theorem implies H i (M, L m k T) = {0} for k + 1 ≤ i if m < −n − 1. It follows from (11) that
imply that H i (l m k D) = {0} and H i (l the vanishing H i (l m+2 k−1 D) = {0} for (27) by H i (l m k D)
as H i (l m k D) = {0} for even m and
and (30) to Serre's duality H i (l m k D) * ∼ = H 2n+1−i (l −m−2n−2−2k k D), we obtain H i (l m k D) = {0} if, in the case m is even
and (31) in the case of even m, and (30) and (32) in the case of odd m induce the conditions in theorem. Hence it completes the proof.
Remark 4. 5 .
5In algebraic geometry, the null correlation bundle N on CP 2n+1 is defined by the short exact sequence0 → N → T(−1) → O(1) → 0 where T(−1) = O(−1) ⊗ T ([9]). It induces to the following :0 → N (1) → T → O(2) → 0. The bundle N (1) = O(1) ⊗ N is the kernel of a transformation A : T → O(
Acknowledgements. The authors would like to thank Professor S. Nayatani for his useful comments and advice. The first named author is supported by Grant-in-Aid for Young Scientists (B) ♯17K14187 from JSPS.
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Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. S Kobayashi, Springer-Verlag70New York-HeidelbergKobayashi, S. Transformation groups in differential geometry, Ergebnisse der Math- ematik und ihrer Grenzgebiete, Band 70. Springer-Verlag, New York-Heidelberg, (1972).
Fano manifolds, contact structures, and quaternionic geometry, Internat. C Lebrun, J. Math. 63LeBrun, C. Fano manifolds, contact structures, and quaternionic geometry, Inter- nat. J. Math. 6 (1995), no. 3, 419-437.
Vector bundles on complex projective spaces. C Okonek, M Schneider, H Spindler, Progress in Math. 3 Birkhäuser. Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Progress in Math. 3 Birkhäuser, Boston, Mass. (1980).
Quaternionic Kähler manifolds. S Salamon, Invent. Math. 671Salamon, S. Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143- 171.
Contact structures on twistor spaces. T Nitta, M Takeuchi, J. Math. Soc. Japan. 391Nitta, T.; Takeuchi, M. Contact structures on twistor spaces, J. Math. Soc. Japan 39 (1987), no. 1, 139-162.
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| [] |
[
"The radial distribution of Blue stragglers in Galactic Globular Cluster NGC 6656 − Clues on the dynamical status",
"The radial distribution of Blue stragglers in Galactic Globular Cluster NGC 6656 − Clues on the dynamical status"
] | [
"Gaurav Singh ",
"R K S Yadav ",
"\nDepartment of Physics and Astrophysics\nAryabhatta Research Institute of Observational Sciences\nManora Peak\n263002NainitalIndia\n",
"\nUniversity of Delhi\n110007Delhi\n"
] | [
"Department of Physics and Astrophysics\nAryabhatta Research Institute of Observational Sciences\nManora Peak\n263002NainitalIndia",
"University of Delhi\n110007Delhi"
] | [
"MNRAS"
] | We present dynamical status of the Galactic globular cluster NGC 6656 using spatial distribution of Blue Straggler Stars (BSSs). A combination of multi-wavelength highresolution space and ground-based data are used to cover a large cluster region. We determine the centre of gravity (C grav ) and construct the projected density profile of the cluster using the probable cluster members selected from HST and Gaia DR2 proper motion data sets. The projected density profile in the investigated region is nicely reproduced by a single mass King model, with core (r c ) and tidal (r t ) radius as 75 ′′ .2 ± 3 ′′ .1 and 35 ′ .6 ± 1 ′ .1 respectively. In total, 90 BSSs are identified on the basis of proper motion data in the region of radius 625 ′′ . An average mass of the BSSs is determined as 1.06 ± 0.09 M ⊙ and with an age range of 0.5 to 7 Gyrs. The BSS radial distribution shows a bimodal trend, with a peak in the centre, a minimum at r ∼ r c and a rising tendency in the outer region. The BSS radial distribution shows a flat behaviour in the outermost region of the cluster. We also estimate A + rh parameter as an alternative indicator of the dynamical status of the cluster and is found to be 0.038 ± 0.016. Based on the radial distribution and A + rh parameter, we conclude that NGC 6656 is an intermediate dynamical age cluster. | 10.1093/mnras/sty2961 | [
"https://arxiv.org/pdf/1810.13139v2.pdf"
] | 119,524,285 | 1810.13139 | a5ac4a2038d470561467512241891fdcf78617a9 |
The radial distribution of Blue stragglers in Galactic Globular Cluster NGC 6656 − Clues on the dynamical status
Nov 2018
Gaurav Singh
R K S Yadav
Department of Physics and Astrophysics
Aryabhatta Research Institute of Observational Sciences
Manora Peak
263002NainitalIndia
University of Delhi
110007Delhi
The radial distribution of Blue stragglers in Galactic Globular Cluster NGC 6656 − Clues on the dynamical status
MNRAS
000Nov 2018Preprint 2 November 2018 Compiled using MNRAS L A T E X style file v3.0Galaxy: globular clusters: individual: NGC 6656 -stars: blue stragglers
We present dynamical status of the Galactic globular cluster NGC 6656 using spatial distribution of Blue Straggler Stars (BSSs). A combination of multi-wavelength highresolution space and ground-based data are used to cover a large cluster region. We determine the centre of gravity (C grav ) and construct the projected density profile of the cluster using the probable cluster members selected from HST and Gaia DR2 proper motion data sets. The projected density profile in the investigated region is nicely reproduced by a single mass King model, with core (r c ) and tidal (r t ) radius as 75 ′′ .2 ± 3 ′′ .1 and 35 ′ .6 ± 1 ′ .1 respectively. In total, 90 BSSs are identified on the basis of proper motion data in the region of radius 625 ′′ . An average mass of the BSSs is determined as 1.06 ± 0.09 M ⊙ and with an age range of 0.5 to 7 Gyrs. The BSS radial distribution shows a bimodal trend, with a peak in the centre, a minimum at r ∼ r c and a rising tendency in the outer region. The BSS radial distribution shows a flat behaviour in the outermost region of the cluster. We also estimate A + rh parameter as an alternative indicator of the dynamical status of the cluster and is found to be 0.038 ± 0.016. Based on the radial distribution and A + rh parameter, we conclude that NGC 6656 is an intermediate dynamical age cluster.
INTRODUCTION
Globular clusters (GCs) are one of the oldest systems, where stellar interactions occur due to the high stellar density of the system. These stellar interactions lead to several dynamical processes, such as stellar collisions, core collapse, stellar mergers, two-body relaxation and mass segregation from equipartition of energy (Meylan & Heggie 1997). These dynamical processes result in several exotic populations, like low mass millisecond pulsars, cataclysmic variables, and blue straggler stars (BSSs) (Bailyn 1995;Ferraro et al. 2001). Among these exotic populations, BSSs are found in bulk and therefore play a crucial role in understanding the GC internal dynamics.
The BSSs were discovered in 1953 by Sandage in the external regions of the GC M3. Sandage (1953) found that these stars appear as an extension of the main sequence and are hotter and luminous than the main sequence turnoff point. The actual nature of these peculiar objects is still ⋆ E-mail: [email protected] † E-mail: [email protected] not clear. Observational evidences (Shara et al. 1997) have shown that BSSs are more massive (M ∼ 1.2 M ⊙ ) than average stars in the GCs (M ∼ 0.3 M ⊙ ). Hence, they can be affected by dynamical friction, which segregates the BSSs towards the cluster centre (Ferraro et al. 2006).
The construction of a complete sample of the BSSs has always been a challenging task. Because of stellar crowding in the central region and the dominant contribution of the luminous cool and bright giants (e.g, RGB, SGB), it is difficult to identify the BSSs in optical bands. However, in ultraviolet (UV) bands, RGB stars are among the faint ones, whereas BSSs are the brightest objects to be easily identified in this plane (Paresce et al. 1991). Therefore, with the help of UV observations taken with the Hubble Space Telescope (HST ), it has become possible to study BSSs in the central regions of the dense clusters.
The selection of BSSs is a difficult task on the basis of only photometric data. After the second Gaia Data Release, Gaia DR2, it has now become possible to separate the genuine BSSs from the field stars using proper motion information. This allows the study of BSS radial distribution from the very central out to large cluster region using c The Authors HST and wide-field imagers mounted on ground-based telescopes. Various studies have been done on the distribution of BSS in the literature. On the basis of the shape of observed BSS radial distribution, Ferraro et al. (2012) (hereafter F12) grouped GCs into three families;
In Family I, the BSS radial distribution shows flat distribution, suggesting that dynamical friction has not yet segregated the BSSs located in the innermost cluster region. Various investigations for many GCs e.g., ω Centauri studied by Ferraro et al. (2006), Palomar 14 by Beccari et al. (2011) and NGC 2419 by Dalessandro et al. (2008b) show flat BSS radial distribution and are classified as Family I clusters.
Several clusters show bimodality in their BSS radial distribution and are classified as Family II clusters. The example of Family II clusters are: M53 (Beccari et al. 2008), NGC 6388 (Dalessandro et al. 2008a), 47 Tuc , M5 (Lanzoni et al. 2007a), M55 (Lanzoni et al. 2007c), NGC 6752 ) and NGC 5824 (Sanna et al. 2014). In these clusters bimodality is seen with a peak in the inner region, a clear minimum at (rmin), and an external rising trend. The rmin tells the distance up to which the role of dynamical friction can be effectively seen.
Family III clusters show a monotonically decreasing radial distribution of BSS, with only a central peak followed by a rapid decline and no signs of an external upturn. Many GCs e.g., M80 & M30 studied by F12, M79 by Lanzoni et al. (2007b) and M75 by Contreras Ramos et al. (2012) have shown unimodal behavior in BSS radial distribution. In these clusters, the dynamical friction has already moved the most distant BSS toward the cluster centre.
As mentioned in F12, these different families correspond to the different dynamical age of the clusters. The Family I, II and III are termed as dynamically young, dynamically intermediate and the dynamically old clusters respectively.
To measure the segregation level of BSSs, Alessandrini et al. (2016) proposed a new parameter (A + ), which is defined as the area between the cumulative radial distribution curves of BSSs (φBSS(x)) and the reference population (φREF (x)):
A + (x) = x x min φBSS(x ′ ) − φREF (x ′ )dx ′(1)
where x (= log(r/r h )) is the logarithmic distance from the cluster centre in units of the half-mass radius r h of the cluster. For meaningful cluster to cluster comparison, Lanzoni et al. (2016) (hereafter L16) defined the measure of this parameter (as A + rh ) within one half-mass radius (r h ). They found a tight correlation between rmin and A + rh for a sample of 25 Galactic GCs. The value of A + rh is, therefore, an alternative indicator to understand the dynamical status of the cluster.
In this paper, we present the dynamical status of the Galactic GC NGC 6656 (M22) using radial distribution of BSS population. NGC 6656 (αJ2000 = 18 h 36 m 23 s .94, δJ2000 = −23 • 54 ′ 17 ′′ .1 , Harris (2010) 1 (hereafter HA10) is one of the nearest globular cluster at a distance of 3.2 kpc from the Sun. The core radius of NGC 6656 is ∼ 1.3 pc and this is a 1 http://physwww.physics.mcmaster.ca/~harris/mwgc.dat low density cluster (< 10 4 M ⊙ pc −3 ). Baldwin et al. (2016) studied the BSS kinematic profile of 19 GCs including NGC 6656 using HST proper-motion catalogue and they identified 34 BSSs in the central region of NGC 6656. They calculated the average mass of BSSs as 1.49 +0.47 −0.28 M ⊙ . Present analysis of the BSS is performed in larger region (625 ′′ in radius) of the cluster. In addition to this BSSs are selected from the HST proper motion catalogue in the central region and Gaia DR2 catalogue in the outer region of the cluster.
DATA SETS AND ANALYSIS
The data sets
In order to select the BSSs in the entire sample, we need high resolution (HR) data set in the central region and wide field (WF) data set in the outer region of the cluster. Therefore, we used HST data in the centre and ground-based CCD data taken from 2.2-m ESO/MPI telescope in the outer region of the cluster.
For the HR data set, we used the astro-photometric catalogue 2 provided by Soto et al. (2017). The data set used in this catalogue are taken from the Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters (GO-13297) and from two other projects GO-12311 and GO-12605 (Piotto et al. 2015). The photometric data in this catalogue are in F275W, F336W and F438W filters which are taken from Wide Field Camera 3 (WFC3) through Ultraviolet-Visible (UVIS) channel. The WFC3/UVIS camera consists two chips, each of 4096 × 2051 pixels with a pixel scale of 0 ′′ .0395 pixel −1 provides a field of view (FOV) of ∼ 162 ′′ × 162 ′′ . This catalogue also provides photometric data in F606W and F814W filters taken from ACS/WFC (GO-10775, PI-Sarajedini). The ACS/WFC camera provides a total FOV of ∼ 202 ′′ × 202 ′′ with a pixel scale of 0 ′′ .05 pixel −1 . The respective circular FOV is shown in the upper right panel of Fig. 1. In addition to the photometric data, the catalogue also contains relative proper motion information of stars which is calculated by using the ACS and WFC3 UV positions as the first and second epoch, respectively.
For WF sample, we used archival images 3 taken with the Wide-Field Imager (WFI) mounted at 2.2-m ESO/MPI telescope (La Silla, Chile). A set of wide-field B, V, I, and U images are used to sample the BSSs over larger FOV. The WFI data set in B, V, I were observed on 1999 May 15, and U data set was observed on 2012 April 20. The aim of selecting U filter was to fill the gaps between the CCD chips. The WFI consists of eight CCD chips placed together (each composed of 2048 × 4096, with a pixel scale ∼ 0 ′′ .238 pixel −1 ), with a total FOV of ∼ 34 ′ × 33 ′ as shown in Fig. 1.
The data reduction and calibration of 2.2-m ESO data set
The WF images are reduced with the procedures explained in Anderson et al. (2006) (hereafter A06). The preprocessing are done using MSCRED package under IRAF. As mentioned in A06, the shape of the PSF in the WF images varies significantly with the position. Therefore, an array of 9 PSFs per CCD chip (3 × 3) is constructed to minimize the spatial variability down to 1 per cent. The PSFs are constructed from an empirical grid of the size of a quarter pixel. Each PSF is represented by a 201 × 201 grid, centred at (101,101) and extend out to a radius of 25 pixels. In order to find the instrumental magnitudes and positions from the array of PSFs, an automated code described in A06 is used. The code starts by finding the brightest stars and goes deeper into the fainter stars. Also, to minimize the effect of geometric distortion, the distortion solution derived in A06 is used. However, the geometric distortion solution, particularly for U is not appropriate for astrometry (A06) and the astrometric solutions can be affected due to systematic errors.
To transform the averaged X, Y coordinates to the corresponding right ascension (αJ2000) and declination (δJ2000) we used CCMAP and CCTRAN tasks available in IRAF. An accuracy of ∼ 0 ′′ .1 is obtained in RA and DEC transformation. The U , B, V and I instrumental magnitudes (exposure time and airmass corrected) of WF images are converted into the F336W, F438W, F606W, and F814W VEGAMAG photometric system by using the transformations derived on the basis of about thousand common stars between HR and WF data sets. Only stars brighter than 20 mag in the F 606W filter are used for calibration purposes.
In order to derive the photometric zero-points and colour terms, we used the following transformation equations;
F 336W std = Uins + Cu * (Uins − Bins) + Zu F 438W std = Bins + C b * (Bins − Vins) + Z bF 606W std = Vins + Cv * (Vins − Iins) + Zv F 814W std = Iins + Ci * (Vins − Iins) + Zi
where instrumental magnitudes and standard magnitudes are represented by the subscript "ins" and "std", respectively. In the above equations, Cu, C b , Cv, and Ci denote the colour coefficients, while Zu, Z b , Zv, and Zi are the zero-points. The corresponding values of the colour coefficients are 0.37, 0.47, -0.30, and 0.028 and zero-points are 21.09, 24.77, 24.06, and 23.44 mag respectively.
FIELD STAR DECONTAMINATION
The photometric data sets are contaminated by the presence of field stars. The field star contamination can be reduced by using proper motion information of the stars. Soto et al. (2017) contains relative proper motions information for the stars common in ACS and WFC3 FOV. An analysis of the VPD shown in Soto et al. (2017) exhibits that the stars lying outside the radius 0.35 pixel can be considered as field stars. We have adopted the same criteria as discussed in Soto et al. (2017) for selection of members for the HR sample.
The Gaia DR2 catalogue contains the astrometric and photometric information for all the stars down to G ∼ 21 mag (Gaia Collaboration et al. 2016Collaboration et al. , 2018. The Gaia DR2 proper motion data are used to select the members outside the HR sample. The VPD for all the stars except the HR sample is shown in Fig. 2. The stars plotted in the VPD are having an accuracy of ≤ 1 mas yr −1 in both the proper motion directions. From the VPD we can see that cluster stars are clearly separated from field stars. The centres of the cluster and field star distributions are (9.81,-5.57) and (-1.72, -4.35) mas yr −1 respectively. We define the selection criteria for cluster members located within the circle of radius 3.0 mas yr −1 around the centre of cluster distribution as shown in Fig. 2. For further analysis, we used only those stars which are lying within the radius of 3.0 mas yr −1 .
RESULTS AND DISCUSSIONS
Centre of Gravity
Stellar evolutionary theories say that stellar luminosities are not directly proportional to stellar masses in a cluster (Montegriffo et al. 1995). The estimation of geometrical centre (Cgrav) for the cluster may differ significantly from previously derived centre of luminosity (C lum ) values using surface brightness distribution. By considering the centre reported in HA10 as initial guess, the centre of gravity Cgrav is estimated by taking the average of α and δ of stars for HR data set. Using the iterative method described in Contreras Ramos et al. (2012), we constructed four circular areas with different radii (10 ′′ , 11 ′′ , 12 ′′ and 13 ′′ ) and for each of these radii, three different magnitude limits (mF 606W = 20.5, 21 and 21.5 mag) are consider to reduce the effect of incompleteness. We estimated Cgrav as αJ2000 = 18 h 36 m 24 s .0 and δJ2000 = −23 • 54 ′ 16. ′′ 4 by averaging the values of Cgrav computed over all the 12 different combinations. Our calculated Cgrav value differs with 0 ′′ .5 in α and δ from the value estimated by Goldsbury et al. (2010), using density contour method.
Projected density profile
To construct the Projected density profile (PDP) of the cluster, we made a catalogue of stars with HST in the centre and Gaia DR2 in the outer region. We considered Gaia DR2 stars because the catalogue is complete up to G = 18 mag (Arenou et al. 2018). We also transformed the HST F606W to Gaia G-band magnitude using the relation derived in Jordi et al. (2010).
In order to obtain the structural parameters of the cluster, we constructed the PDP of NGC 6656, starting from Cgrav out to ∼ 36 ′ (tidal radius). We divided the area into 19 concentric annuli with varying radii centred on Cgrav, and each annulus is divided into four equivalent sub-sectors. The density estimation for each sub-sector is done by counting the stars present in the region divided by the area of subsector. The resulting density for each annulus is estimated by averaging the densities of the sub-sectors and standard deviation in the average is considered as the uncertainty in the annulus densities. The radius of each annulus is considered as the midpoint value of the corresponding radial bin.
The resultant PDP of NGC 6656 is shown in Fig. 3. The PDP is nicely fitted with an isotropic single-mass King model given by King (1962) . The fitting provides the core (rc) and tidal (rt) radii as 75 ′′ .2 ± 3 ′′ .1 and 35 ′ .6 ± 1 ′ .1 respectively. In HA10 and Kunder et al. (2014), the (rc) and (rt) are listed as rc ≃ 79 ′′ .8 and rt ≃ 31 ′ .9. Trager et al. (1995) have also estimated rc and rt as 85 ′′ .1 and 28 ′ .9 respectively, using surface brightness profile of the cluster. The present estimate of rc and rt are quite different with the previous studies. However, the current estimate is based on Figure 3. Observed projected density profile, plotted over ∼ 36 ′ for the cluster NGC 6656. The density profile is nicely reproduced by an isotropic single-mass King model with the parameters shown in the box. Continuous line shows the King (1962) profile.
proper motion data and considered to be more reliable than the earlier investigations.
Selection of BSSs and reference populations
To study the BSS radial distribution, the first step is to select genuine BSSs and reference populations (HB or GB (RGB+SGB)) carefully. For this purpose, we used the selected cluster members as described in Sec. 3.
The BSS population
We adopted UV-CMD (mF 275W , mF 275W − mF 336W ) as our primary selection criteria for selecting the BSSs. In this plane, BSSs are easily distinguishable from cooler giants (RGB or AGB). As shown in Fig. 4, the BSS populations follow a vertical sequence in the UV-CMD. The contamination from the sub-giant branch and MS-TO is minimized by adopting mF 275W =18.65 mag as the limiting magnitude of the BSS selection box. A total of 26 BSSs are identified in the inner sample (r ≤ 104 ′′ ), using UV-CMD. These BSSs are also checked for membership as described in Sec. 3 and found to be genuine members of the cluster.
The BSSs once selected from the UV-CMD are replotted in the Optical-CMD (mF 606W , mF 606W − mF 814W ) to define the size of the selection box in optical band. Based on the position of 26 BSSs in Optical-CMD, we defined the selection box criteria (15.65 < mF 606W < 17.12 and 0.36 < mF 606W − mF 814W < 0.65) as shown in left panel of Fig. 5. Six more BSSs in the ACS FOV, not covered by the WFC3 FOV are identified in the inner sample using Optical-CMD. Based on the member selection norms, they are bonafide members of the cluster. Finally, a total of 32 BSSs are identified in the inner sample. Baldwin et al. (2016) found 34 BSSs in the region of 202×202 arcsec 2 using (mF 814W , mF 606W − mF 814W ) CMD. The investigated area by Baldwin et al. (2016) is more than the present analysis.
For the outer sample (104 ′′ < r ≤ 625 ′′ ), we adopted the same selection box as used for the inner sample and shown in the right panel of Fig. 5. In this way, we found 58 BSSs in the outer sample. Therefore, a total of 90 BSSs are identified from the entire (inner+outer) sample on the basis of photometric and proper motion data.
The selection of reference populations
In order to carry out a qualitative study of BSS in terms of radial distribution and specific frequency, the definition of the reference population is very necessary. The reference population should show a non-peculiar radial trend within the cluster. According to Lanzoni et al. (2007c) and Renzini & Fusi Pecci (1988), the number of stars in any post-MS stage is proportional to the duration of the evolutionary phase itself. The specific frequencies (NHB/NGB) are expected to be constant and equal to the ratio between the evolutionary timescales of the HB and GB phase. Therefore, we considered these branches as reference populations. We set the same fainter magnitude limit for BSSs and reference populations so that our selection is not affected by the completeness of the sample. For the selection of the reference population, we used Optical-CMD as shown in Fig. 5. We made a box around the HB population in such a way that it should contain most of the HB stars. In this way a total of 428 (119 in the inner sample and 309 in the outer sample) HB reference stars are identified. For GB population selection, we defined a mean ridgeline by taking the average in colour (mF 606W − mF 814W ) in the interval of 0.4 mag in mF 606W magnitude. For each sample (inner and outer), we used a 3σ selection box criteria defined from the mean ridge line. Here, σ is the standard deviation in the mean colour. This is done to reduce the effects of differential reddening and error in photometric calibration. In this way, we found 1233 and 2779 GB reference stars in the inner and outer sample respectively.
Finally, we found 428 HB stars and 4012 GB stars as reference population in the entire sample.
The BSS Mass Distribution
The location of BSSs in Optical-CMD suggest that they are massive in comparison to the normal stars in the GCs. Their masses are estimated by comparing their location in the CMD with theoretical isochrones taken from Girardi et al. (2002). Fig. 6 shows the set of theoretical isochrones fitted in the population of 90 BSSs with an age range of 0.5 to 7 Gyrs, in a step of 0.5 Gyr. The metallicity ([F e/H] = −1.72) Table 1. It is clear from the table that the maximum number of BSS stars are in the range of 0.95-1.00 M ⊙ . The average mass of BSS is found to be 1.06 ± 0.09 M ⊙ . Lanzoni et al. (2007b) has obtained the average mass of the BSS as 1.2 M ⊙ in the cluster NGC 1904. Baldwin et al. (2016) has provided the BSS kinematic profile for 19 GCs using ACS/HST sample and estimated the average mass as 1.49 +0.47 −0.28 M ⊙ for the BSSs lying near the centre of the cluster NGC 6656. The present estimate of the average mass of BSSs in NGC 6656 is similar to the value derived by previous authors.
The radial distribution of BSS, GB, and HB
In this section, we present the radial distribution of BSS, GB, and HB populations. The cumulative radial distribution of BSS, GB, and HB are shown with continuous, dashed and dotted lines respectively in Fig. 7. A comparison of the BSS cumulative radial distribution with GB and HB distribution indicates that BSSs are more centrally concentrated than reference populations. To get the more clear picture about the distributions we performed the Kolmogorov-Smirnov test 4 . The probabilities with which the BSSs have a different radial distribution than the HB and GB populations are 98.3 and 99.9 per cent respectively. This shows 4 http://www.physics.csbsju.edu/stats/KS-test.html In order to investigate the radial distribution behavior of BSSs, GB, and HB further, the cluster region is divided into seven concentric annuli centred on Cgrav. We counted the number of BSSs, GB and HB populations in each bin and listed in Table 2 as NBSS, NGB and NHB. We then computed the specific frequencies F BSS GB = NBSS/NGB , F BSS HB = NBSS/NHB and F HB GB = NHB/NGB . These Specific frequencies are plotted with respect to the radius in Fig. 8. The specific frequency F HB GB shows a flat behavior across the entire field of investigation of the cluster. On the other hand, the specific frequencies of BSS show a bimodal trend with both the reference populations. A peak in the centre, a minimum at r ∼ rc and a rising trend in the outer region of the cluster are present. A flattening is also seen in the outermost region of the cluster.
To recheck the bimodality in BSS radial distribution, we estimated the doubly normalized ratio for BSS. It is defined as the number of BSSs observed in a region to the total number of BSSs divided by the fraction of light sampled in the same region with respect to the total measured luminosity (in L ⊙ ) (Ferraro et al. 1993). It is written as,
RBSS = NBSS/N tot BSS Lsamp/L tot samp(2)
We also computed this ratio for reference stars. By adopting the parameters obtained in Sec. 4.2, we calculated the sampled luminosity for each annulus by integrating the isotropic single-mass King profile and scaled it to the area covered between the annular regions. We considered Poisson error for the different populations and luminosities. The error in double normalized ratios is considered as the propagation of errors . The computed luminosity ratios in each annulus are listed in Table 2. Fig. 9 shows the variation of double normalized ratios of BSS (RBSS), HB (RHB) and GB (RGB) populations with the radial distance normalized to rc. The upper panel represents RBSS and RHB, while lower panel represents RBSS and RGB with respect to r/rc. The value of RBSS show a peak in the centre, a dip at r ∼ rc and an external rising trend followed by flattening in the outskirt of the cluster. The nature of this plot is very similar to Fig. 8. The resulting radial distributions of RGB and RHB follow the cluster luminosity and is almost a constant value (∼ 1) as expected from the theory of stellar evolution (Renzini & Fusi Pecci 1988).
Discussion on the BSS radial distribution
An analysis of Fig. 8 and 9 shows that BSS radial distribution in NGC 6656 is bimodal. Dynamical friction plays a very crucial role in shaping the BSS radial distribution. It segregates the massive objects like BSS that are rotating close to the centre of the cluster. As a result, a peak in the centre and a dip in smaller radii is visible in the BSS radial distribution. The BSSs located in the outskirts of the cluster have not yet been influenced by the action of dynamical friction and hence show a flat distribution. The BSS radial distribution found for NGC 6656 is similar to the previously studied GC M5 (Lanzoni et al. 2007a). The features in the BSS radial distribution can be used as a measure of dynamical age. Based on the shape of BSS radial distribution, this cluster seems to be dynamically intermediate age cluster.
A + rh determination and dynamical status of the cluster
In the previous section, we discussed the dynamical state of the cluster based on BSS radial distribution in the region of the cluster. Despite several observational confirmations, it has not been possible to reproduce the BSS radial distribution profile from Monte Carlo and N -body simulations of GCs. Also, the bimodality seems to be unstable and temporary feature (see Miocchi et al. 2015;Hypki & Giersz 2017). Therefore, to reconfirm the dynamical age, we estimated A + rh parameter proposed by Alessandrini et al. (2016), as a measure of the dynamical status of the cluster. A + rh parameter is defined as the area between the curves of cumulative radial distribution of reference and BSSs populations over the half-mass radius (r h ). The value r h = 201 ′′ .6 is taken from HA10.
In Fig. 10, we plot the cumulative radial distributions of BSSs and HB in the left panel and BSSs and GB in the right panel. Using these distributions the values of A + rh are estimated as 0.027 and 0.049 for HB and GB sample respectively. The average value of A + rh = 0.038 ± 0.016. As discussed in L16, A + rh can be adopted as an alternative indicator for measuring the level of dynamical evolution experienced by the cluster from the beginning. They have shown that A + rh can be related to core relaxation time (trc/tH ). Here, tH = 13.7 Gyr is the age of the Universe and trc = 0.34 Gyr is taken from HA10. We plotted A + rh versus log(trc/tH) in lower panel of Fig. 11. The values of A + rh and trc for cluster NGC 6656 is shown with the filled circle and these values for other clusters are adopted from L16 and shown with empty circles. As discussed in L16, a decreasing trend of A + rh with relaxation time is seen. It is clear from the plot that the cluster NGC 6656 follows the trend.
In the framework of the empirical dynamical clock relation defined in F12, the position of rmin/rc can be used as an indicator to measure the extent of dynamical evolution of the cluster. Therefore, we plot rmin/rc against trc/tH in the upper panel of Fig. 11. Filled circle represents the cluster NGC 6656. We adopted the value of trc, rmin and rc for other clusters from F12. The filled triangles are dynamically old clusters while open circles are the dynamically intermediate age clusters. The dynamically young clusters are plotted as lower-limit arrows at rmin ∼ 0.1. An inspection of this figure exhibits that the cluster NGC 6656 lies in the region of dynamically intermediate age clusters. Hence, we suggest that this is an intermediate dynamical age cluster of Family II classification.
Therefore, BSS radial distribution and A + rh parameter indicates that NGC 6656 is a dynamically intermediate age cluster.
SUMMARY AND CONCLUSIONS
In this paper, a combination of multi-wavelength highresolution space and ground-based data are used to probe the dynamical status of NGC 6656 based on BSS radial distribution. The BSS and reference populations are selected from both the photometric and kinematic data. The important findings of the present analysis are the following:
(i) The centre of gravity, Cgrav of the cluster is determined as αJ2000 = 18 h 36 m 24 s .0 and δJ2000 = −23 • 54 ′ 16 ′′ .4, with an accuracy of ∼ 0.05 ′′ in both α and δ using HR data set. The derived PDP can be nicely reproduced by an isotropic single-mass King model and provides rc = 75 ′′ .2 ± 3 ′′ .1 and rt = 35 ′ .6 ± 1 ′ .1. The stars used for the determination of Cgrav and PDP are selected using proper motion data.
(ii) We identified a total of 90 BSSs in the entire sample, with 32 BSSs lying in the inner sample and 58 BSSs in the outer sample. These BSSs are selected using proper motion data taken from HST and Gaia DR2 catalogues. We estimated an average BSSs mass as 1.06 ± 0.09 M ⊙ and with an age range of 0.5 to 7 Gyrs.
(iii) The BSS radial distribution shows a bimodal trend with a peak in the centre, a minimum at r ∼ rc, and an external rising trend followed by a flattening in the outermost region of the cluster. We also determined A + rh parameter, which is found to be 0.038 ± 0.016. This newly determined parameter along with the bimodal trend in the BSS radial distribution indicate that this cluster is an intermediate dynamical age cluster. Our results are consistent with the empirical dynamical clock relation defined in F12 and L16, further confirming that NGC 6656 is an intermediate dynamical-aged cluster belonging to Family II classification of GCs.
Figure 1 .
1The wide-field data set covering the large cluster region. The map of all the stars found in the WF images are plotted with respect to Cgrav and the open region located towards the cluster centre is taken from the HR sample. The map of highresolution data set is shown on the top right of the figure.
Figure 2 .
2The Vector-point diagram (VPD) for the stars having the proper motion information in the Gaia DR2 catalogue. The criteria for selection of cluster members is indicated by a circle of radius 3.0 mas yr −1 around the cluster centre (9.81, -5.57) mas yr −1 .
Figure 4 .
4The selection criteria for the BSSs in UV-CMD (m F 275W , m F 275W − m F 336W ) for the HR sample is shown with box. The selected BSSs are shown with filled dots.
Figure 5 .
5lef t panel: the Optical-CMD (m F 606W , m F 606W − m F 814W ) of the inner sample (r ≤ 104 ′′ ). right panel: the Optical-CMD of the outer sample (104 ′′ < r ≤ 625 ′′ ). The selection boxes for selecting the BSSs, HB, and GB are also shown. The BSSs are represented by filled dots.
Figure 6 .
6The Optical-CMD for 90 BSS identified in the present analysis. The continuous lines are the theoretical isochrones taken fromGirardi et al. (2002) and fitted to the BSS. and the distance modulus ((m − M )V = 13.60) values of the cluster are adopted from HA10 catalogue. The projection of the BSSs on the nearest isochrone is used to derive their masses. In this way, BSS masses are estimated in a range of 0.90−1.35 M ⊙ . The frequency distribution of BSSs in the different mass interval is listed in
Figure 7 .
7Cumulative radial distribution plot of the BSSs (continuous line), HB stars (dotted line) and GB stars (dashed line) with respect to Cgrav of the cluster.
Figure 8 .
8The radial distribution of the specific frequencies F BSS HB (top), F BSS GB (middle), and F HB GB (bottom) plotted with respect to distance from the cluster centre normalized by the core radius. The dotted line shows the mean of all the points plotted in the bottom panel.
Figure 9 .
9The BSSs radial distribution (filled circles) and the reference populations with double normalized ratios are plotted with respect to the distance from the cluster centre, normalized to rc. The shaded portion in the upper and lower panel shows the distribution of HB and GB population respectively, which is almost constant (∼ 1). The width of the shaded regions represent the error bars.
Figure 10 .
10Left panel : cumulative radial distribution of BSSs and reference population (HB) plotted over one half-mass radius (r h ). Right panel : cumulative radial distributions of BSSs and GB as reference population over one r h . The curve of reference populations are shown with a dashed line and the BSSs is shown with a continuous line.
Figure 11 .
11U pper panel : The plot shows the empirical dynamical clock relation defined in F12, as a function of r min /rc and trc/t H . The open circles show the position of dynamically intermediate clusters and filled triangles for the dynamically old clusters. The dynamically young systems with the lower-limit arrows at r min = 0.1. The position of NGC 6656 is marked as the filled circle. Lower panel : show the position of NGC 6656 as the filled circle and other clusters are marked as open circles.
Table 1 .
1The frequency distribution of BSSs in different mass intervalsMass range
Frequency
0.90-0.95
3
0.95-1.00
24
1.00-1.05
18
1.05-1.10
14
1.10-1.15
12
1.15-1.20
5
1.20-1.25
9
1.25-1.30
4
1.30-1.35
1
Table 2 .
2The log of the Number counts for BSSs and reference populationsRadial bin (in arcsec)
N BSS
N HB
N GB
Lsamp/L tot
samp
0 -25
7
16
146
0.03
25 -50
8
38
365
0.07
50 -75
5
39
406
0.09
75 -150
30
111
1005
0.24
150 -250
21
95
828
0.22
250 -410
11
79
740
0.20
410 -625
8
50
517
0.14
that BSS population is extracted from different parent pop-
ulation than the reference populations.
http://www.astro.uda.cl/public_release/globularclusters41.html 3 http://archive.eso.org/eso/eso_archive_main.html
MNRAS 000, 1-9 ()
ACKNOWLEDGEMENTWe thank the anonymous referee for the useful comments that helped us to improve the scientific content of the paper. We would like to acknowledge ESO, for the archival data observed with ESO Telescope at the La Silla Observatory under program ID 163.O-0741(C) and 088.A-9012(A). This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. We are thankful to Dr. Andrea Bellini for the description of the astro-photometric catalogue.
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| [] |
[
"Prepared for submission to JHEP Two Dimensional Renormalization Group Flows in Next to Leading Order",
"Prepared for submission to JHEP Two Dimensional Renormalization Group Flows in Next to Leading Order"
] | [
"Rubik Poghossian [email protected] \nYerevan Physics Institute Alikhanian\nBr. 20036YerevanArmenia\n"
] | [
"Yerevan Physics Institute Alikhanian\nBr. 20036YerevanArmenia"
] | [] | Zamolodchikov's famous analysis of the RG trajectory connecting successive minimal CFT models M p and M p−1 for p 1, is improved by including second order in coupling constant corrections. This allows to compute IR quantities with next to leading order accuracy of the 1/p expansion. We compute in particular, the beta function and the anomalous dimensions for certain classes of fields. As a result we are able to identify with a greater accuracy the IR limit of these fields with certain linear combination of the IR theory M p−1 . We discuss the relation of these results with Gaiotto's recent RG domain wall proposal. | 10.1007/jhep01(2014)167 | [
"https://arxiv.org/pdf/1303.3015v3.pdf"
] | 27,061,150 | 1303.3015 | 7c1e14a010a6eccd0ccebbce45d82bc842238736 |
Prepared for submission to JHEP Two Dimensional Renormalization Group Flows in Next to Leading Order
9 Feb 2014
Rubik Poghossian [email protected]
Yerevan Physics Institute Alikhanian
Br. 20036YerevanArmenia
Prepared for submission to JHEP Two Dimensional Renormalization Group Flows in Next to Leading Order
9 Feb 2014
Zamolodchikov's famous analysis of the RG trajectory connecting successive minimal CFT models M p and M p−1 for p 1, is improved by including second order in coupling constant corrections. This allows to compute IR quantities with next to leading order accuracy of the 1/p expansion. We compute in particular, the beta function and the anomalous dimensions for certain classes of fields. As a result we are able to identify with a greater accuracy the IR limit of these fields with certain linear combination of the IR theory M p−1 . We discuss the relation of these results with Gaiotto's recent RG domain wall proposal.
Introduction
In his famous paper [1] A. Zamolodchikov has investigated Renormalization Group (RG) flow from the minimal model M p to the M p−1 for large p 1 caused by the relevant operator φ 1,3 . Two main circumstances made it possible to investigate this RG flow using a single coupling constant perturbation theory. First, the conformal dimension of the field φ 1,3 , ∆ 1,3 = 1 − 2 p+1 ≡ 1 − (see Appendix A) is nearly marginal when p 1 and second, the Operator Product Expansion (OPE) of this field with itself produces no relevant field besides the initial one and the unit operator. The method of A. Zamolodchikov not only allowed to identify the IR theory with M p−1 , but also provided detailed description how several classes of local fields behave along the RG trajectory. The analogous RG flow for the N = 1 super minimal models has been investigated in [2].
The main purpose of this paper is a sharpening of Zamolodchikov's analysis, by the inclusion of second order perturbative corrections. It is interesting to note that in all cases we have investigated, the rotation matrix (in the space of fields), that diagonalizes the matrix of anomalous dimensions, does not receive 1/p or 1/p 2 corrections. So an interesting question arises, if any higher order corrections appear at all.
As intermediate results, in this paper we have found several four-point correlation functions in large p limit (see formulae (C.1)).
The initial motivation to carry out these computations came from the recent approach to this RG flow by D. Gaiotto [3]. Using Goddard-Kent-Olive construction, Gaiotto has constructed a non-trivial conformal interface between two successive minimal models and made a striking conjecture, that this interface is the exact RG domain wall which encodes the map between the UV and IR fields. Gaiotto's conjecture survives a strong test: it is fully compatible with the first order parturbative calculations of the mixing amplitudes performed by Zamolodchikov. In this paper we show that this mixing coefficients computed with the help of the perturbation theory up to the second order, unlike those obtained from the Gaiotto's conjecture, do not receive any corrections up to the order 1/p 2 . Nevertheless, this discrepancy might be attributed to the renormalization scheme which is adopted here following Zamolodchikov. Presently the author of this paper does not have any clue how to take into account possible dependencies on the renormalization schemes in order to be able to make any conclusive statement about Gaiotto's conjecture beyond the leading order.
The paper is organized as follows.
In Section 1, we develope some technical tools, necessary to carry out second order in coupling constant calculations.
In Section 2 the β-function and Zamolodchikov's c-function [4] are computed with next to leading order accuracy. The critical value of the renormalized coupling constant, the slope of the β-function as well as the c-function at the critical point are calculated. The results of these computations confirm that also the second order contributions perfectly match with the Zamolodchikoved's conclusion that the IR fixed point corresponds to the CFT M p−1 and that the UV field φ 1,3 flows to the field φ 3,1 of the IR theory. Section 3 is devoted to the renormalization of several series of local fields and to the calculation of their anomalous dimensions. Thus: in Section 3.1 we investigate the renormalization of the fields φ n,n .
In Section 3.2 the renormalization of the fields φ n,n+1 and φ n,n−1 is discussed and the matrix of anomalous dimensions is found. At the fixed point the matrix of anomalous dimensions is diagonalized and its eigenvalues are calculated. In Section 3.3 the same steps are performed for the fields φ n,n+2 , ∂∂φ n,n and φ n,n+2 .
In all cases the predictions of Zamolodchikov successfully withstand the next to leading order test.
In Appendix A some basic facts about the minimal models of 2d CFT are reviewed. The Appendices B and C are devoted to computation of the integrals used in the main text. The Appendix D comments how to calculate the large p limit of those four point correlation functions used in the main text.
Perturbation theory in second order
Suppose the (Euclidean) action density is given by
H(x) = H 0 (x) + λφ(x) (1.1)
with H 0 being the UV CFT action density, φ a relevant local spinless field and λ the coupling constant. Then for a two-point function up to second order we'll have
φ 1 (y 1 )φ 2 (y 2 ) λ = φ 1 (y 1 )φ 2 (y 2 ) 0 − λ φ 1 (y 1 )φ 2 (y 2 )φ(x) 0 d 2 x + λ 2 2 φ 1 (y 1 )φ 2 (y 2 )φ(x 1 )φ(x 2 ) 0 d 2 x 1 d 2 x 2 + O(λ 3 ) (1.2)
In this paper we consider a theory, whose UV limit is given by the minimal CFT model M p with p 1 and the perturbing field is φ ≡ φ 1,3 . Leading order corrections in this theory has been investigated by A. Zamolodchikov [1]. Second order computations are more complicated. Indeed, not only the knowledge of four point correlation functions which in general are quite non-trivial in a CFT [5], but also their integrals over two insertion points is required. Fortunately, as we demonstrate below, the conformal invariance allows to perform integration over one of the insertion points explicitly. First let us notice that translational and scale invariance can be exploited to locate the points y 1 , y 2 at y 1 = 1 and y 2 = 0 without loss of generality:
φ 1 (y 1 )φ 2 (y 2 )φ(x 1 )φ(x 2 ) 0 d 2 x 1 d 2 x 2 = (y 12ȳ12 ) 2−2∆−∆ 1 −∆ 2 φ 1 (1)φ 2 (0)φ(x 1 )φ(x 2 ) 0 d 2 x 1 d 2 x 2 (1.3)
(here and below I frequently use the shorthand notation x 12 = x 1 − x 2 , y 12 = y 1 − y 2 et.al.). Any four-point function of primary fields in a CFT essentially depends only on the cross ratio x = x 12 x 34 x 14 x 32 of the insertion points and its conjugate [5]
φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 )φ 4 (x 4 ) = (x 14x14 ) −2∆ 1 (x 24x24 ) ∆ 1 +∆ 3 −∆ 2 −∆ 4 (1.4) × (x 34x34 ) ∆ 1 +∆ 2 −∆ 3 −∆ 4 (x 23x23 ) ∆ 4 −∆ 1 −∆ 2 −∆ 3 G(x,x),
where it is assumed that the fields are spin-less (i.e. ∆ i =∆ i ). Specifying the insertion points as
x 1 = x, x 2 = 0, x 3 = 1 and x 4 = ∞ we get G(x,x) = lim x 4 →∞ (x 4x4 ) 2∆ 4 φ 1 (x)φ 2 (0)φ 3 (1)φ 4 (x 4 ) ≡ φ 1 (x)φ 2 (0)φ 3 (1)φ 4 (∞) (1.5)
Alternatively specifying x 1 = 1/x, x 2 = ∞, x 3 = 1 and x 4 = 0 and comparing with (1.5) we get the identity
φ 1 (x)φ 2 (0)φ 3 (1)φ 4 (∞) = (xx) −2∆ 1 φ 1 (1/x)φ 4 (0)φ 3 (1)φ 2 (∞) (1.6)
which is useful when investigating the correlation functions at large x. After application of (1.4), (1.5) to the four-point function φ(x 1 )φ 2 (0)φ 1 (1)φ(x 2 ) 0 and introduction of the new integration variables
x 1 (1 − x 2 ) x 12 → x 1 ; 1 − x 2 → x 2 .
two integrations on the r.h.s. of eq. (1.3) become partly disentangled
φ 1 (1)φ 2 (0)φ(x 1 )φ(x 2 ) 0 d 2 x 1 d 2 x 2 = I(x 1 ) φ(x 1 )φ 2 (0)φ 1 (1)φ(∞) 0 d 2 x 1 (1.7)
where
I(x) = (yȳ) a−1 ((1 − y)(1 −ȳ)) b−1 ((x − y)(x −ȳ)) c d 2 y,(1.I(x) = πγ(b)γ(a + c) γ(a + b + c) |F (1 − a − b − c, −c, 1 − a − c, x)| 2 (1.10) + πγ(1 + c)γ(a) γ(1 + a + c) |x a+c F (a, 1 − b, 1 + a + c, x)| 2 = πγ(a)γ(b + c) γ(a + b + c) |F (1 − a − b − c, −c, 1 − b − c, 1 − x)| 2 + πγ(1 + c)γ(b) γ(1 + b + c) |(1 − x) b+c F (b, 1 − a, 1 + b + c, 1 − x)| 2 = πγ(a)γ(b) γ(a + b) |x c F (a, −c, a + b, 1/x)| 2 + πγ(1 + c)γ(a + b − 1) γ(a + b + c) |x a+b+c−1 F (1 − a − b − c, 1 − b, 2 − a − b, 1/x)| 2 where γ(x) = Γ(x)/Γ(1 − x) and F (a, b, c, x)
is the Gaussian hypergeometric function. Above three expressions for I(x) are convenient when exploring the regions x ∼ 0, x ∼ 1 and x ∼ ∞ respectively. Note also that these expressions make explicit the single-valuedness of I(x). Specifying the choice of parameters to (1.9) and applying the identity
F (a, b, c, x) = (1 − z) c−a−b F (c − a, c − b, c, x)
to the second term of the second equality, the eqs. (1.10) can be rewritten as
I(x) = πγ(2 + 21 )γ( 12 ) γ(2 ) |F (1 − 2 , 2 , 1 + 21 , x)| 2 + πγ(2 + 12 )γ( 21 ) γ(2 ) |(x/(1 − x)) 12 F (2 , 1 − 2 , 1 + 12 , x)| 2 = πγ(2 + 12 )γ( 21 ) γ(2 ) |F (1 − 2 , 2 , 1 + 12 , 1 − x)| 2 + πγ(2 + 21 )γ( 12 ) γ(2 ) |(x/(1 − x)) 12 F (2 , 1 − 2 , 1 + 21 , 1 − x)| 2 = πγ(2 + 12 )γ(2 + 21 ) γ(4 ) |x −2 F (2 + 12 , 2 , 4 , 1/x)| 2 + πγ(4 − 1) γ 2 (2 ) |x 2 −1 F (1 − 2 , 1 − 2 + 12 , 2 − 4 , 1/x)| 2 (1.11)
It is worth noting that in the case when 12 ≡ 1 − 2 = 0 only the third expression is manifestly nonsingular, the first two expressions require a subtle limiting procedure.
Thus for this case it is better to employ the third expression:
I(x) = πγ 2 (2 ) γ(4 ) |x −2 F (2 , 2 , 4 , 1/x)| 2 + πγ(4 − 1) γ 2 (2 ) |x 2 −1 F (1 − 2 , 1 − 2 , 2 − 4 , 1/x)| 2 (1.12)
Let us investigate the behaviour of (1.12) at x ∼ 1. Using standard formulae for the analytic continuation of the hypergeometric function with parameters satisfying the condition a + b − c ∈ Z (see e.g. [7]) one can get convinced that
I(x) ≈ π(x +x − 4) + π (1 + 2 (2 − 1)(x +x − 2)) × 2 − log |x − 1| 2 − 2π cot(2π ) − 4ψ(2 ) − 4γ (1.13)
where γ = 0.577216 · · · is the Euler constant and the omitted terms are at most of order |x − 1| 2 log |x − 1| in x → 1 limit. There is no need to investigate the limit x → 0 separately since the obvious symmetry of I(x) with respect to x ↔ 1 − x at x ∼ 0 immediately ensures
I(x) ≈ −π(2 + x +x) + π (1 + 2 (1 − 2 )(x +x)) × 2 − log |x| 2 − 2π cot(2π ) − 4ψ(2 ) − 4γ (1.14) 2 β-function
In this section we calculate the β-function up to 1/p 4 ∼ 4 corrections for the small values of the (renormalized) coupling constant (of order or smaller). As it will become quite clear later for this purpose one should evaluate the integral (1.7) in the special case φ 1 = φ 2 = φ and I(y) given by (1.12) with the accuracy ∼ 1/ . Our strategy will be as follows: separate in the integration region the discs D l,0 = {x ∈ C | |x| < l}, D l,1 = {x ∈ C | |x − 1| < l} and D l,∞ = {x ∈ C | |x| > 1/l} where l is an intermediate length scale such that 0 < l 0 exp(−1/ ) l 1 and l 0 is the ultraviolet scale. For the integral outside these discs we will safely use the small limits of the correlation functions given in the appendix while inside the discs we'll explore (exact in ) OPE. We will see that the trace of the intermediate scale l will be washed out entirely from the final result. In present case the = 0 limit of the four-point function is given by (see appendix C)
φ(x)φ(0)φ(1)φ(∞) (2.1) = 1 − 2x + 3x 2 − 2x 3 + x 4 3 x 2 (1 − x) 2 2 + 16 3 1 − 3x 2 + x 2 − x 3 4 x(1 − x) 2 2 + 5 9 x 1 − x 4
With required accuracy I(x) ≈ π/ . It is convenient to carry out the integration in radial coordinates x = r exp(iϕ),x = r exp(−iϕ), d 2 x = rdrdϕ. The result of integration over the angular variable ϕ will depend on the region where the radial coordinates takes its value
R(x,x)dϕ = Res x=0 R(x, r 2 /x)/x + Res x=r 2 R(x, r 2 /x)/x, if r < 1 Res x=0 R(x, r 2 /x)/x + Res x=1 R(x, r 2 /x)/x, if r > 1 (2.2)
for arbitrary rational function R(x,x) with poles located at x = 0 or x = 1. In particular when R(x,x) is the r.h.s. of the eq. (2.1) we get
3r 10 − 9r 8 + 25r 6 − 23r 4 + 7r 2 + 3 3r 4 (1 − r 2 ) 3 , if r < 1 3r 10 + 7r 8 − 23r 6 + 25r 4 − 9r 2 + 3 3r 4 (r 2 − 1) 3 , if r > 1 (2.3)
After performing the remaining elementary integration over r we finally get
Ω l,l 0 φ(x)φ(0)φ(1)φ(∞) d 2 x = 2π l 2 + π 2l 2 0 − 33π 8 − 32π 3 log(2l 0 l 2 ) + · · · (2.4)
where (see Fig. 1 )
Ω l,l0 = (D 1−l 0 ,0 \D l,0 ) ∪ (D 1/l,0 \D 1+l 0 ,0 )
and the dots stand for negligible terms of order l or l 0 /l. There is a subtlety to be treated carefully here. The fact that the part of the white narrow ring (of width 2l 0 ) outside of the red circle is missing from the integration region Ω l,l 0 is insignificant since its inclusion would produce only negligible terms of order l 0 . Instead we have to subtract the contribution of two lens-like regions of Ω l,l 0 included in the red circle (as already said, the contribution coming from the regions around the singular points will be computed separately exploring OPE). a) Integration over lens-like regions Expanding (2.1) around x ∼ 1 we get
1 |x − 1| 4 + 2 (x − 1) 2 + 2 (x − 1) 2 + 16 3|x − 1| 2 + · · · (2.5)
where only the singular terms, whose integrals over the region around x = 1 diverge, are presented. The integrals of such terms have been evaluated in Appendix D. As a result the contribution of the lens-like regions, to be subtracted from the r.h.s. of eq. (2.4), is equal to
π − π l 2 − π 8 + π 2l 2 0 + 2 × 2π + 16 3 2π log l 2l 0 (2.6)φ(x)φ(0) = (xx) −2∆ (1 + · · · ) + C (1,3) (1,3)(1,3) (xx) −∆ (φ(0) + · · · ) (2.7)
Taking into account (1.14) and that
C (1,3) (1,3)(1,3) C (1,3)(1,3)(1,3) = 16 3 (1 − 3 + O( 2 )) (2.8) one easily gets D l,0 \D l 0 ,0 I(x) φ(x)φ(0)φ(1)φ(∞) d 2 x ≈ π 2 l 2−4 0 − π 2 l 2 + 32π 2 (log(l) − 3) 3 + 32π 2 3 2 (2.9)
where the first two terms come from the identity and the last two terms from the φ field channel of OPE (2.7) respectively. c) Contribution of x ∼ ∞
At large x we first make use of eq. (1.6) to pass to the inverse variable 1/x ∼ 0 and then we apply OPE. The calculation is similar to the previous case, the main difference being the fact that at this limit I(x) becomes simply π (xx) −2 . The result is
D l,∞ \D l 0 ,∞ I(x) φ(x)φ(0)φ(1)φ(∞) d 2 x ≈ π 2 l 2−4 0 − π 2 l 2 + 16π 2 3 1 2 − 3 − 2 log l (2.10)
Let's pick up all the ingredients: (2.4) times I(x) ≈ 2π , minus (2.6), plus twice (2.9), and plus (2.10). We get
3π 2 l 2−4 0 − 88π 2 + 80π 2 3 2 + O( 0 ) (2.11)
As expected, the l dependence disappeared. The presence of the divergent term 3π 2
l 2−4 0
also is not surprising. In our naive regularization scheme we could cancel this infinity by adding an appropriate, proportional to the area counter-term in action. In fact, if we would have been able to treat the integral (1.7) analytically as continuation from a region of parameters where integral converges, then this kind of non-analytic divergent term couldn't emerge at all. In what follows we'll simply drop out such terms without further ado. Thus for the two point function
G(x, λ) = φ(x)φ(0) λ we get (c.f. (1.2)) G(x, λ) = (xx) −2+2 1 − λ 4π √ 3 2 − 3 + O( ) (xx) + λ 2 2 80π 2 3 2 − 88π 2 + O( 0 ) (xx) 2 + · · · (2.12)
Following A. Zamolodchikov let us introduce a new coordinate g ("renormalized" coupling constant) in the space of one-parameter family of theories (1.1) instead of the initial coupling λ and introduce the local field φ (g) = ∂ g H (according to (1.1) the initial "bare" perturbing field φ = ∂ λ H). The new coupling g is fixed by the requirement that the two point function
G(x, g) = φ (g) (x)φ (g) (0) λ satisfies the normalization condition G(1, g) = 1 (2.13)
Then the β-function can be computed from the identity (see [1])
Θ(x) = λφ(x) = β(g)φ (g) (x) (2.14)
where Θ is the trace of the energy-momentum tensor. Combining (2.13) and (2.14) one easily finds
∂ λ g = G(1, λ) (2.15)
and
β(g) = λ G(1, λ) (2.16)
The equation (2.15) allows one to express g in terms of λ (the integration constant can be set to zero so that the unperturbed CFT will corresponds to g = 0)
g = λ − πλ 2 √ 3 2 − 3 + O( ) + 2π 2 λ 3 3 2 2 − 7 + O(1) + O(λ 4 ) (2.17)
or, inversely
λ = g + πg 2 √ 3 2 − 3 + O( ) + 2π 2 g 3 3 2 2 − 5 + O(1) + O(g 4 ) (2.18)
Inverting and replacing in (2.16) λ in favour of g we get
β(g) = g − πg 2 √ 3 2 − 3 + O( 2 ) − 4π 2 g 3 3 (1 + O( )) + · · · (2.19)
The equation
β(g * ) = 0 (2.20)
admits a nonzero solution
2πg * = √ 3 + √ 3 2 2 + O( 3 ) (2.21)
so we have a non-trivial infrared fixed point. In [1] this fixed point has been identified with the minimal model M p−1 and the local field φ (g * ) with the field φ (p−1)
3,1 . Now we are in a position to check this identification more accurately. The anomalous dimension of φ (g * ) is related to the slope of β-function
∆ * = 1 − ∂ g β(g) g=g * = 1 + + 2 + O( 3 ) (2.22) which matches to the conformal dimension of φ (p−1) 3,1
computed from the Kac formula (A.2). Also the shift of the central charge [4]
c * − c p = −12π 2 g * 0 β(g)dg = − 3 3 2 − 9 4 4 + O( 5 ) (2.23)
neatly matches to the exact expression
c p−1 − c p = − 12 p(p 2 − 1) = − 3 3 (2 − )(1 − ) .
Field renormalization and the UV -IR map
In this section we calculate the matrices of anomalous dimensions for several classes of fields. Diagonalization of these matrices at the IR fixed point provides a detailed map between the UV local fields and their image under RG flow in the IR theory.
Primary fields φ n,n
This is the simplest case to analyze since the fields φ n,n never get mixed with other fields [1]. This follows from the structure of the OPE involving the perturbing field φ 1,3 . The subspace of fields which is generated by the field φ n,n and is closed w.r.t. OPE with φ 1,3 , doesn't contain any other field with a dimension close to ∆ n,n = O( 2 ). We are going to calculate corrections to the anomalous dimension up to the order 4 . That is why for the present purpose the knowledge of the four point function φ(x)φ n,n (0)φ n,n (1)φ(∞) up to 2 correction is required. As in previous case, to find this correlation function we first used AGT relation to find the relevant conformal blocks up to sufficiently large level (actually the computations were performed up to the order x 6 terms). Expanding a conformal block up to 2 and examining first few coefficients of the resulting power series in x it is possible to guess the entire power series and identify it with some elementary function. Having in our disposal the expression for the correlation function we then checked that it satisfies all the nontrivial physical requirements: the single-valuedness and the compatibility with OPE around the points x ∼ 1 and x ∼ ∞. Here is the final expression (see Appendix C)
φ(x)φ n,n (0)φ n,n (1)φ(∞) (3.1) = 1 + 2 (n 2 − 1) 12 1 2x(x − 1) + 1 2x(x − 1) + 4 log 2 1 − x x + O( 3 )
From eq. (1.12), up to order , I(x) is equal to
I(x) = π − 2π log |(1 − x)x| + 8π log |x| log |1 − x| + O( 2 ) (3.2)
Now we are ready to perform integration over the region Ω l 0 ,l (see Fig. 1). Since the singularities at x ∼ 0 and x ∼ 1 are integrable, we can put l 0 = 0. As in Section 2 the integration over the angular variable should be performed separately for the cases 0 < |x| < 1 and |x| > 1. Integration of rational expressions we have already discussed earlier. As about the logarithmic terms, they can be easily handled first expanding into power series in x if |x| < 1 or in 1/x if |x| > 1. Then we proceed with the radial integration. Both steps are elementary and we present only the final result:
Ω l,l 0 I(x) φ(x)φ n,n (0)φ n,n (1)φ(∞) d 2 x ≈ π 2 (n 2 − 1) 3 log 1 l + 1 (3.3) + π 2 l 2 (2 + 4 + (4 + 8 ) log l + 8 log 2 l) − π 2 (1 + 4 ) − π 2 (n 2 − 1) 12 + π 2 l 2
Due to the already mentioned mildness of singularities at 0 and 1 the only remaining contribution to be taken into account comes from the neighbourhood of ∞ i.e. from D l,∞ \D l 0 ,∞ .
At large x it is convenient to employ eq. (1.6)
φ(x)φ n,n (0)φ n,n (1)φ(∞) = (xx) −2∆ φ(1/x)φ(0)φ n,n (1)φ n,n (∞) (3.4)
and apply the OPE (2.7) with x replaced by 1/x. The correlation function decomposes into a sum of two partial amplitudes one corresponding to the identity and the other to the field φ. a) Contribution of identity The prefactor (xx) −2∆ in (3.4) compensates the factor (xx) 2∆ accompanying the identity operator in OPE and with sufficient accuracy we can replace this partial amplitude by 1. It is straightforward to expand I(x) given by (1.12) at large x keeping only those terms which after integration may produce non-vanishing terms in small l limit
I(x) ≈ πγ 2 (2 ) γ(4 ) (xx) −2 1 + x 1 + x + πγ(4 − 1) γ(2 ) 2 (xx) 2 −1 (3.5)
Integrating this expression over the region D l,∞ \D l 0 ,∞ , dropping out, as earlier, all singular in l 0 terms and expanding the result up to the linear in terms we get
− π 2 l 2 + π 2 − 2π 2 (2 log(l) + 1) l 2 + 4π 2 l 2 − 2 log 2 (l) − 2 log(l) − 1 l 2 (3.6) b) Contribution of the field φ 1,3 This contribution is D l,∞ \D l 0 ,∞ π C 2 (1,3)(n,n)(n,n) (xx) −2 −2+2 +1− d 2 x (3.7)
where π (xx) −2 is just the function I(x) with required accuracy, (xx) −2+2 is the prefactor of (3.4), (xx) 1− comes from OPE and the squared structure constant is equal to
C 2 (1,3)(n,n)(n,n) = 2 (1 + )(n 2 − 1) 6 + O( 4 ) (3.8)
The integral is converging at the limit l 0 → 0, so we may perform integration over the entire region D l,∞ . The result reads
π 2 (n 2 − 1)(1 − + 2 log(l)) 6 (3.9)
The sum of all contributions (3.3), (3.6), and (3.9) is
π 2 (n 2 − 1) (2 + 5 ) 12 + O( 2 ) (3.10)
Combining this with the first order in coupling constant contribution
φ n,n (1)φ n,n (0)φ(x) d 2 x = π (n 2 − 1) (2 + 5 ) 8 √ 3 + O( 4 ) (3.11)
where the value
C (1,3)(n,n)(n,n) = (n 2 − 1) (2 + 5 ) 2 16 √ 3 + O( 4 ) (3.12)
for the structure constant is inserted, we get
G n (x, λ) ≡ φ n,n (x)φ n,n (0) λ = (xx) −2∆n,n 1 − λ π (n 2 − 1) (2 + 5 + O( 2 )) 8 √ 3 (xx) + λ 2 2 π 2 (n 2 − 1) (2 + 5 + O( 2 )) 12 (xx) 2 + · · · (3.13)
Let's introduce the renormalized field φ (g) n,n = B(λ)φ n,n by requiring that the two point
function G n (x, g) = φ (g) n,n (x)φ (g)
n,n (0) λ satisfies the normalization condition G n (1, g) = 1 (3.14) so that
B(λ) = 1 G n (1, λ) (3.15)
Then for the anomalous dimension we get (cf. eq. (3.48), derived for a more general situation)
∆ (g) n,n = ∆ n,n + λ ∂ λ log B = ∆ n,n − λ 2 ∂ λ G n (1, λ) (3.16)
In view of (2.18) we find ∆ (g) n,n = ∆ n,n (3.17)
+ πg (n 2 − 1) 2 (2 + 5 + O( 2 )) 16 √ 3 − π 2 g 2 (n 2 − 1) 2 (1 + O( )) 8 + O(g 3 )
So that at the fixed point
∆ (g * ) n,n = (n 2 − 1)(4 2 + 6 3 + 7 4 + O( 5 )) 64 (3.18)
which completely agrees with the dimension ∆ is robust also against our second order test.
3.2 Renormalization of the fields φ n,n+1 and φ n,n−1 Already in this case one encounters with the phenomenon of mixing. The OPE φ 1,3 φ n,n+1 produces besides φ n,n+1 also the primary field φ n,n−1 , both having dimensions close to 1/4 in large p limit. Thus we have to consider the correlation functions φ(x)φ n,n±1 (0)φ n,n±1 (1)φ(∞) with all four possible choices of signs. The strategy is exactly the same as in previous sections and for each choice we will follow the steps performed in Section 3.1.
3.2.1 Correlation function φ n,n+1 (1)φ n,n+1 (0) λ a) Contribution of the region Ω l,l 0 This is given by the integral
Ω l,l 0 I(x) φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) d 2 x (3.19)
The large p limit of the four-point function found from AGT relation is (See Appendix C):
φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) = 2(n + 2) 3n x − 1 2 x(x − 1) 2 + x 2 − x + 1 2 x(x − 1) 2 + O( ) (3.20)
With required accuracy I(x) can be replaced by π . The integral (3.19) can be performed using the technique already explored in computing (2.4) or (3.3). The result is
Ω l,l 0 I(x) φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) d 2 x ≈ π l 2 − π − π(13n + 20) log(l) 6n − π(5n + 4) log (2lφ(x)φ n,n+1 (0) = (xx) −∆ C (n,n+1) (1,3)(n,n+1) (φ n,n+1 + · · · ) + C (n,n−1) (1,3)(n,n+1) (xx) ∆ n,n−1 −∆ n,n+1 −∆ (φ n,n−1 (0) + · · · ) (3.23)
Taking into account that in this region I(x) ≈ π − π log (|x| 2 ) (see (1.14)) and that
C 2 (1,3)(n,n+1)(n,n+1) = (n + 2) 2 (1 − (2n − 1) ) 12n 2 + O( 2 ) C 2 (1,3)(n,n+1)(n,n−1) = (n 2 − 1) (1 + ) 3n 2 + O( 2 ) (3.24) we get D l,0 \D l 0 ,0 I(x) φ(x)φ(0)φ(1)φ(∞) d 2 x ≈ π 2 (n + 2) 2 6n 2 1 2 + 1 − 2n + log l + 2π 2 (n 2 − 1) 3n 2 (n + 2) 2 n + 4 2 + n + 4 + (n + 2) 2 log l (3.25)
Above two terms come from two primaries φ n±1 appearing on the r.h.s. of the OPE (3.23). The contribution from the region D l,∞ is completely analogous to the case of the correlation function φφ λ discussed in Section 2. The only difference is that now the contribution of the field φ which appears in u-channel OPE is proportional to
C (1,3) (1,3)(1,3) C (1,3)(n,n+1)(n,n+1) = 2(n + 2) (1 − (n + 1) ) 3n + O( 2 ) (3.26) instead of C 2 (1,3)(1,3)(1,3) ≈ 16(1−3 ) 3
. The result is (c.f. eq. (2.10))
D l,∞ \D l 0 ,∞ I(x) φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) d 2 x ≈ − π 2 l 2 + 2π 2 (n + 2) 3n 1 2 − n + 1 − 2 log l (3.27)
Picking up all the contributions: (3.21) minus (3.22) plus twice (3.25) and plus (3.27), we get π 2 (3n 3 + 24n 2 + 64n + 44) 3n(n + 2) 2 2 − 4π 2 (n + 1) (n 3 + 7n 2 + 14n + 5) 3n(n + 2) 2 + O( 0 ) (3.28)
3.2.2 Correlation function φ n,n−1 (1)φ n,n+1 (0) λ a) Contribution of the region Ω l,l 0 is given by the integral
Ω l,l 0 I(x) φ(x)φ n,n+1 (0)φ n,n−1 (1)φ(∞) d 2 x (3.29)
The large p limit of the four-point function now is (see Appendix C):
φ(x)φ n,n+1 (0)φ n,n−1 (1)φ(∞) = √ n 2 − 1 3n 2x − 1 x(x − 1) 2 + O( ) (3.30)
Using the last equality in (1.11) we see that I(x) ≈ 16π (16−n 2 ) and for the result of the integral (3.29) we get Ω l,l 0 (1,3)(n,n+1) (xx) −∆ (φ n,n+1 + · · · ) + C (n,n−1) (1,3)(n,n+1) (xx) ∆ n,n−1 −∆ n,n+1 −∆ (φ n,n−1 (0) + · · · ) It follows from (1.11) that in this region with sufficient accuracy
I(x) φ(x)φ n,n+1 (0)φ n,n−1 (1)φ(∞) d 2 x ≈ 32π 2 √I(x) ≈ 8π n − 1 n + 4 − (xx) ∆ n,n+1 −∆ n,n−1 n − 4 From (A.5) C (n,n+1) (1,3)(n,n+1) C (1,3)(n,n+1)(n,n−1) = √ n 2 − 1 (n + 2)(1 − (n − 1) ) 6n 2 + O( 2 ) C (n,n−1) (1,3)(n,n+1) C (1,3)(n,n−1)(n,n−1) = √ n 2 − 1 (n − 2)(1 + (n + 1) ) 6n 2 + O( 2 ) (3.33)
and for the D l,0 contribution we get
D l,0 \D l 0 ,0 I(x) φ(x)φ(0)φ(1)φ(∞) d 2 x ≈ − 4π 2 (n + 2) √ n 2 − 1((n − 8)(1 − (n − 1) ) + 4(n − 2) log l) 3(n 2 − 16)(n − 2)n 2 2 − 4π 2 (n − 2) √ n 2 − 1((n + 8)(1 + (n + 1) ) + 4(n + 2) log l) 3(n 2 − 16)n 2 (n + 2) 2 (3.34)
where two terms correspond to the two intermediate primaries
φ n±1 . ii) D l,1 contribution. The relevant OPE is φ(x)φ n,n−1 (1) = C (n,n+1)
(1,3)(n,n−1) ((x − 1)(x − 1)) ∆ n,n+1 −∆ n,n−1 −∆ (φ n,n+1 + · · · ) +C (n,n−1)
(1,3)(n,n−1) ((x − 1)(x − 1)) −∆ (φ n,n−1 (1) + · · · ) Considering x → 1 limit of (1.11) we get
I(x) ≈ 8π n − 1 n − 4 − ((x − 1)(x − 1)) ∆ n,n−1 −∆ n,n+1 n + 4
The combinations of structure constants relevant for this case are those already presented in (3.33). For the D l,1 contribution we get
D l,1 \D l 0 ,1 I(x) φ(x)φ(0)φ(1)φ(∞) d 2 x ≈ − 4π 2 (n + 2) √ n 2 − 1((n − 8)(1 − (n − 1) ) + 4(n − 2) log l) 3(n 2 − 16)(n − 2)n 2 2 − 4π 2 (n − 2)
√ n 2 − 1((n + 8)(1 + (n + 1) ) + 4(n + 2) log(l)) 3(n 2 − 16)n 2 (n + 2) 2 (3.35)
Again the two terms correspond to two intermediate primaries φ n±1 . Notice that due to some subtle interplay among quantities involved, for the contribution of D l,1 we got exactly the same result as for the contribution of D l,0 . iii) D l,∞ contribution.
Since the structure constant C (1,1)(n,n−1)(n,n+1) = 0 only the field φ which appears in the u-channel OPE gives a nonzero contribution. This contribution is proportional to
C (1,3) (1,3)(1,3) C (1,3)(n,n−1)(n,n+1) = 4 √ n 2 − 1 (1 − ) 3n + O( 2 ) (3.36) Approximating I(x) by I(x) ≈ π |x| −4 we get D l,∞ \D l 0 ,∞ I(x) φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) d 2 x ≈ − 64π 2 √ n 2 − 1(1 − (1 − 2 log l)) 3n (n 2 − 16) 2 (3.37)
It remains to collect all the contributions together to get
− 16π 2 √ n 2 − 1 (5n 2 − 44 + (n 2 + 20) ) 3n (n 2 − 16) (n 2 − 4) 2 + O( 0 ) (3.38)
The matrix of anomalous dimensions
There is no need to calculate the remaining two point functions φ n,n+1 (1)φ n,n−1 (0) λ and φ n,n−1 (1)φ n,n−1 (0) λ since the former is identical with φ n,n−1 (1)φ n,n+1 (0) λ and the latter can be obtained from φ n,n+1 (1)φ n,n+1 (0) λ by simply replacing n → −n. For simplicity of notation let us denote φ n,n+1 ≡ φ 1 and φ n,n−1 ≡ φ 2 , then the two-point functions can be represented as
G α,β (x, λ) ≡ φ α (x)φ β (0) λ (3.39) = (xx) −∆a−∆ β δ α,β − λC (1) α,β (xx) + λ 2 2 C (2) α,β (xx) 2 + · · ·
The first order coefficients C
α,β are given by
C (1) α,β = φ α (1)φ β (0)φ(x) d 2 x = C (1,3)(α)(β) πγ( + ∆ α − ∆ β )γ( + ∆ β − ∆ α ) γ(2 ) (3.40)
From eq. (A.2) for the dimensions we have
∆ 1 ≡ ∆ n,n+1 = 1 4 − n 4 + 1 8 + 1 16 n 2 − 1 2 + O( 3 ) ∆ 2 ≡ ∆ n,n−1 = 1 4 + n 4 − 1 8 + 1 16 n 2 − 1 2 + O( 3 ) (3.41)
Explicitly, up to O( ) terms we get
C (1) 1,1 = π(n + 2)(2 − (2n − 1) ) 2 √ 3 n + O( ); C (1) 2,2 = π(n − 2)(2 + (2n + 1) ) 2 √ 3 n + O( ) C (1) 1,2 = C (1) 2,1 = − 4π √ n 2 − 1( + 2) √ 3 n(n 2 − 4) + O( ) (3.42)
and for the second order coefficients we have (see (3.28), (3.38) ) C
1,1 = π 2 (3n 3 + 24n 2 + 64n + 44) 3n(n + 2) 2 2 − 4π 2 (n + 1) (n 3 + 7n 2 + 14n + 5) 3n(n + 2) 2
+ O( 0 ) C (2) 2,2 = π 2 (3n 3 − 24n 2 + 64n − 44) 3n(n − 2) 2 2 + 4π 2 (n − 1) (n 3 − 7n 2 + 14n − 5) 3n(n − 2) 2 + O( 0 ) C (2) 1,2 = C (2) 2,1 = − 16π 2 √ n 2 − 1 (5n 2 − 44 + (n 2 + 20) ) 3n (n 2 − 16) (n 2 − 4) 2 + O( 0 ) (3.43)
Obviously the correlation function (3.39) satisfies the Callan-Symanzik equation
(x∂ x + ∆ α + ∆ β − λ∂ λ ) G α,β (x, λ) = 0 (3.44)
As in Section 3.1 let us introduce renormalized fields
φ (g) α = B α,β (λ)φ β
and require that the two point functions G In matrix notations we may write
G (g) (x) = B · G(x) · B T (3.46)
Comparing with (3.44) we see that the renormalized two-point function satisfies the equation
(x∂ x − β(g)∂ g )G (g) α,β + 2 ρ=1 (Γ α,ρ G (g) ρ,β + Γ β,ρ G (g) α,ρ ) = 0 (3.47)
where the β function and the renormalized coupling g have been introduced in Section 2 and the matrix of anomalous dimensions Γ is defined as
Γ = B∆B −1 − λB∂ λ B −1 (3.48) where∆ = ∆ 1 0 0 ∆ 2 (3.49)
Expanding the matrix B up to second order in λ
B = 1 + λB 1 + λ 2 B 2 + O(λ 3 ) B −1 = 1 − λB 1 + λ 2 (B 2 1 − B 2 ) + O(λ 3 ) (3.50)
imposing the normalization condition (3.45) and requiring that the matrix of the anomalous dimensions (3.48) be symmetric, we find
B 1 = 1 2 C (1) + 1 2 ∆ , C (1) B 2 = 3 8 C (1) 2 − 1 4 C (2) − 1 8 ∆ , C (2) + 1 8 ∆ , C (1) , C (1) + 1 4 ∆ , C (1) 2 + 1 8 2 ∆ , C (1) 2 (3.51)
Now all the ingredients to calculate the matrix of anomalous dimensions (3.48) are at our disposal. Taking also into account the λ-g relation (2.18), we get
Γ 1,1 = ∆ 1 + πg(n + 2)(2 − (2n − 1) ) 4 √ 3 n + π 2 g 2 2 Γ 1,2 = Γ 2,1 = πg √ n 2 − 1 (2 + ) 2 √ 3 n Γ 2,2 = ∆ 2 + πg(n − 2)(2 + (2n + 1) ) 4 √ 3 n + π 2 g 2 2 (3.52)
Notice that all the matrix elements are regular at = 0, all double and single poles in disappeared. At the fixed point g = g * (see (2.21)) Γ (g * )
1,1 = 1 4 − (2n 2 − n − 4) 8n + (n 3 − 4n 2 + n + 8) 2 16n Γ (g * ) 1,2 = Γ (g * ) 2,1 = √ n 2 − 1 (1 + ) 2n Γ (g * ) 2,2 = 1 4 + (2n 2 + n − 4) 8n + (n 3 + 4n 2 + n − 8) 2 16n (3.53)
It is easy to get the eigenvalues of this matrix n−1,n of the IR CFT M p−1 . We can easily identify also the corresponding normalized eigenvectors and establish the explicit map
φ (p−1) n+1,n = 1 n φ (g * ) 1 + √ n 2 − 1 n φ (g * ) 2 φ (p−1) n−1,n = − √ n 2 − 1 n φ (g * ) 1 + 1 n φ (g * ) 2 (3.55)
Remarkably the coefficients in (3.55) did not receive neither nor 2 corrections. Thus it is quite perceivable that under the renormalization scheme (3.45), which we have adopted following A. Zamolodchikov, the relation (3.55) is exact. The same phenomenon we will encounter in the next section where a more involved case of mixing of the three fields φ n,n±2 and ∂∂φ n,n will be considered.
3.3 Renormalization of the fields φ n,n+2 , ∂∂φ n,n and φ n,n−2
The OPE φ 1,3 φ n,n+2 includes fields from the conformal families [φ n,n+4 ], [φ n,n+2 ] and [φ n,n ]. Similarly the product φ 1,3 φ n,n−2 produces fields from the families [φ n,n−4 ], [φ n,n−2 ] and [φ n,n ]. Since the dimensions of the primary fields φ n,n±2 and the descendant field ∂∂φ n,n are close to 1 in large p limit, we have a situation when these three fields effectively get mixed along the RG flow 1 [1]. To find the matrix of anomalous dimensions one has to calculate all the two point correlators of these fields.
Correlation function φ n,n+2
(1)φ n,n+2 (0) λ a) Contribution of the region Ω l,l 0 is given by the integral
Ω l,l 0 I(x) φ(x)φ n,n+2 (0)φ n,n+2 (1)φ(∞) d 2 x (3.56)
At large p from the AGT relation we have found that (see Appendix C)
φ(x)φ n,n+2 (0)φ n,n+2 (1)φ(∞) = 3x 4 − 6x 3 + 9x 2 − 6x + 1 3(x − 1) 2 x 2 2 (3.57) + 8(3 + n) 3(1 + n) (2x − 1) (2x 2 − 2x + 1) 4(x − 1) 2 x 2 2 + (3 + n)(4 + n) 18n(n + 1) (x − 1) 2 x 2 −2 + O( )
For present purposes I(x) can be simply replaced by π . Performing the integration (3.56) we get
Ω l,l 0 I(x) φ(x)φ n,n+2 (0)φ n,n+2 (1)φ(∞) d 2 x (3.58)
≈ − 16π 2 (2n 2 + 5n + 1) log(l) 3n(n + 1) − 16π 2 (n + 1) log(l 0 ) 3n − π 2 (66 + 33n + 128(n + 1) log(2)) 24n + 2π 2 (2n + 1) 3nl 2 + π 2 (n + 2) 6nl 2 0 b) Contribution of lens-like regions Expanding (3.57) near x ∼ 1 we get
2 + n 3n|x − 1| 4 + 2(n 2 + n − 2)(x +x − 2) 3n(n + 1)|x − 1| 4 + (n 2 + 5n + 6) ((x − 1) 2 + (x − 1) 2 ) 3n(n + 1)|x − 1| 4 + 8(n + 1 3n|x − 1| 2 (3.59)
So the contribution of the lens-like regions near x ∼ 1 is (see Appendix D)
π 2 + n 3n π 2l 2 0 − π l 2 − π 8 + 2(n 2 + n − 2)π 3n(n + 1) + 2π(n 2 + 5n + 6) 3n(n + 1) (1,3)(n,n+2) (xx) ∆n,n−∆ n,n+2 −∆ × 1 + ∆ + ∆ n,n − ∆ n,n+2 2∆ n,n xL −1 1 + ∆ + ∆ n,n − ∆ n,n+2 2∆ n,nxL −1 φ n,n (0)
+ 8(n + 1) 3n 2π log l 2l 0 (3.+ · · · (3.61)
where L k ,L k are the left and right Virasoro generators. To proceed let us notice that the effect of the Virasoro generator L −1 orL −1 in the three point function is rather simple. Namely the relation
L −1 φ n,n (0)φ n,n+2 (1)φ(∞) = (∆ n,n + ∆ n,n+2 − ∆) φ n,n (0)φ n,n+2 (1)φ(∞) (3.62)
and a similar relation with L −1 replaced byL −1 hold. We see from (1.14) that in this region I(x) can be approximated by
I(x) ≈ π + π x +x − log |x| 2 − 2π (x +x) log |x| 2 (3.63)
We need also the combination of structure constants C 2 (1,3)(n,n+2)(n,n+2) = 4(n + 3) 2 (1 − (n + 2) ) 3(n + 1) 2
+ O( 2 ) C (n,n) (1,3)(n,n+2) C (n,n)(n,n+2)(1,3) = n + 2 3n + O( 2 ) (3.64)
Taking into account that ∆ n,n − ∆ n,n+2 − ∆ = −2 + n + 3 2 (∆ + ∆ n,n − ∆ n,n+2 ) (∆ n,n + ∆ n,n+2 − ∆) 2∆ n,n = − 2(n − 1) n + 1 + (n − 1)(n + 3) 2(n + 1)
we get that the piece of integrand corresponding to the contribution of the family [φ n,n ] is equal to
2 + n 3n |x| −4+(3+n) 1 − 2(n − 1) n + 1 − (n − 1)(n + 3) 2(n + 1) x 2 × π + π x +x − log |x| 2 − 2π (x +x) log |x| 2 (3.65)
The part corresponding to the intermediate field φ n,n+2 is simpler. For this one we can restrict us with a less accurate expression for I(x)
I(x) ≈ π − π log |x| 2 (3.66) and the integrand is simply 4(n + 3) 2 (1 − (n + 2) ) 3(n + 1) 2 π − π log |x| 2 |x| −2+2
Performing integrations we get
D l,0 \D l 0 ,0 I(x) φ(x)φ n,n+2 (0)φ n,n+2 (1)φ(∞) d 2 x = 8π 2 (n + 2)(n + 5)(n − 1) 2 3n(n + 1) 2 (n + 3) 2 2 − π 2 (n + 2) 3l 2 n + 4π 2 (n + 2)(n − 1) ((2n 3 + 10n 2 + 6n − 18) log(l) − n 3 − 9n 2 − 23n + 1) 3n(n + 1) 2 (n + 3) 2 + 8π 2 (n + 3) 2 3(n + 1) 2 2 − 8π 2 (n + 3) 2 (n + 2 − log l) 3(n + 1) 2 (3.67)
where the second and the third lines come from the family [φ n,n ] and the last line, from the φ n,n+2 field of the OPE (3.61).
Also in this case the contribution coming from the region D l,∞ is quite similar to the case discussed in Section 2. We should simply take into account that the contribution of the field φ appearing in the u-channel OPE is proportional to
C (1,3) (1,3)(1,3) C (1,3)(n,n+2)(n,n+2) = 4(n + 3)(2 − (n + 5) ) 3(n + 1) + O( 2 ) (3.68)
The result (c.f. eq. (2.10)) is I(x) can be replaced by
D l,∞ \D l 0 ,∞ I(x) φ(x)φ n,n+1 (0)φ n,n+1 (1)φ(∞) d 2 x ≈ − π 2 l 2 + 4π 2 (n + 3) 3(n + 1) 2 2 − n + 5 − 4 log lI(x) ≈ − 2π(n + 1) n + 5 1 + 4x (n + 1)(1 − x) 2 − 2π(n − 3) n + 1 x 1 − x 2 (3.73)
and the result of integration is
Ω l,l 0 I(x) φ(x)φ n,n+2 (0)φ n,n (1)φ(∞) d 2 x ≈
16π 2 3(n + 5)
L −1 φ n,n (0)φ n,n (1)φ(∞) = (2∆ n,n − ∆) φ n,n (0)φ n,n (1)φ(∞) (3.76)
The function I(x) will be determined using the first equality in (1.11). For the calculation of the contribution of the field φ n,n+2 it is safe to replace the hypergeometric functions simply by 1. Instead for the contribution of the family [φ n,n ] also the first order in x and inx terms should be taken into account. Below we present expressions for the relevant combinations of structure constants with required accuracy where the two lines correspond to the [φ n,n ] and φ n,n+2 contributions respectively. ii) D l,1 contribution. The relevant OPE is
I(x) φ(x)φ n,n+2 (0)φ n,n (1)φ(∞) d 2 x ≈ − 8π 2 (n − 1)φ(x)φ n,n (1) = |x − 1| 2(∆ n,n+2 −∆n,n−∆) C (n,n+2) (1,3)(n,n) φ n,n+2 (1) (3.79) +C (n,n) (1,3)(n,n) |x − 1| −2∆ 1 + ∆(x − 1) 2∆ n,n L −1 1 + ∆(x − 1) 2∆ n,nL −1 φ n,n (1) + · · ·
The relation between the three point functions relevant for this case is
L −1 φ n,n (1)φ n,n+2 (0)φ(∞) = (∆ − ∆ n,n − ∆ n,n+2 ) φ n,n (1)φ n,n+2 (1)φ(∞) (3.80)
Notice the flip of sign compared to (3.62) due to the rearrangement of the points 0 and 1. For the function I(x) the second equality in (1.11) should be used. The hypergeometric functions should be expanded around x = 1. When calculating the contribution of φ n,n+2 it would suffice to keep the constant term only while for the contribution of [φ n,n ] also the terms linear in x − 1 (orx − 1) should be taken into account. During the calculation one encounters the same combinations of the structure constants as in (3.77). Finally we get for the D l,1 contribution a result identical to that of D l,0 given by (3.34). Remember that a similar phenomenon we have encountered earlier in Section 3.2.2. iii) D l,∞ contribution Only the field φ appearing in u-channel OPE gives a nonzero contribution. Since and, from the third equality in (1.11),
I(x) ≈ − 4π (n − 3) n + 5 (1 + (n + 5) ) |x| −4 (3.82) we get D l,∞ \D l 0 ,∞ I(x) φ(x)φ n,n+2 (0)φ n,n (1)φ(∞) d 2 x ≈ − n + 2 nI(x) φ(x)φ n,n−2 (0)φ n,n+2 (1)φ(∞) d 2 x (3.85)
The large p limit of the four-point function is (see Appendix C)
φ(x)φ n,n−2 (0)φ n,n+2 (1)φ(∞) = √ n 2 − 4 3n|x(1 − x)| 4 + O( ) (3.86)
I(x) can be replaced by
I(x) ≈ − 4π (n 2 − 4) (3.87)
and the result of the integration is
Ω l,l 0 I(x) φ(x)φ n,n−2 (0)φ n,n+2 (1)φ(∞) d 2 x ≈ π 2 6n √ n 2 − 4 64 log (2l l 0 ) + 33 − 8 l 2 −− 4π (1 − 2(x +x − 2) + (x − 1) 2 + (x − 1) 2 + 4|x − 1| 2 ) 3n √ n 2 − 4 |x − 1| 4
Consequently, from the Appendix D, we see that the contribution of the lens-like regions near x ∼ 1 is 1 + ∆ + ∆ n,n − ∆ n,n−2 2∆ n,n xL −1 1 + ∆ + ∆ n,n − ∆ n,n−2 2∆ n,nxL −1 φ n,n (0) + · · ·
− 4π 3n √ n 2 − 4 π 2l 2 0 − π l 2 − π 8 − 2(π) + (2π) + 8π log l 2l 0 (3.
The impact of L −1 on the three-point function:
L −1 φ n,n (0)φ n,n+2 (1)φ(∞) = (∆ n,n + ∆ n,n+2 − ∆) φ n,n (0)φ n,n+2 (1)φ(∞) (3.91) In the first expression of (1.11) for I(x), the hypergeometric functions should be expanded up to the linear order in x (orx) terms. The relevant combination of the structure constants:
C (n,n) (1,3)(n,n−2) C (1,3)(n,n)(n,n+2) = √ n 2 − 4 3n + O( 2 ) (3.92)
So, the final result for the D l,0 contribution is
D l,0 \D l 0 ,0 I(x) φ(x)φ n,n−2 (0)φ n,n+2 (1)φ(∞) d 2 x ≈ 16π 2 3n (n 2 − 9) √ n 2 − 4 10 2 + n 2 − 9 4l 2 − n 2 + 1 + 2(n 2 − 9) log l (3.93)
ii) D l,1 contribution The relevant OPE:
φ(x)φ n,n+2 (1) = |x − 1| 2(∆n,n−∆ n,n+2 −∆ )C (n,n) (1,3)(n,n+2) (3.94)
× 1 + ∆ + ∆ n,n − ∆ n,n+2 2∆ n,n (x − 1)L −1 × 1 + ∆ + ∆ n,n − ∆ n,n+2 2∆ n,n (x − 1)L −1 φ n,n (1) + · · ·
The impact of L −1 on the three-point function:
L −1 φ n,n (1)φ n,n−2 (0)φ(∞) = (∆ − ∆ n,n − ∆ n,n−2 ) φ n,n (1)φ n,n−2 (1)φ(∞) (3.95)
The combination of structure constants required for this computation coincides with that given by eq. (3.92). The explicit calculation shows that in this case too, the D l,1 contribution is identical to that of D l,0 given by (3.93). Note also that the contribution of D l,∞ is negligible. Combining all the contributions for the case at hand we get
320π 2 (1 − ) 3 2 n(n 2 − 9) √ n 2 − 4 + O( 2 ) (3.96)
The matrix of anomalous dimensions
The remaining two point functions φ n,n−2 (1)φ n,n−2 (0) λ and φ n,n (1)φ n,n−2 (0) λ can be obtained from φ n,n+2 (1)φ n,n+2 (0) λ and φ n,n (1)φ n,n+2 (0) λ replacing n by −n. So we have all necessary material to repeat the steps of Section 3.2.3 and calculate the matrix of anomalous dimensions for the fields φ 1 ≡ φ n,n+2 ; φ 2 ≡ (2∆ n,n (2∆ n,n + 1)) −1 ∂∂φ n,n ; φ 3 ≡ φ n,n−2
The two-point functions of these fields can be represented as in (3.39), but the indices now take the values α,β = 1, 2, 3. The replacement of the field φ n,n by φ 2 in a two point function, at a given order k of the perturbation theory, results in an extra multiplier which is easy to calculate. Here is the rule: the coefficients C
(k − ∆ α − ∆ 2 ) 2 2∆ n,n (2∆ n,n + 1) and (k − 2∆ 2 ) (k − 2∆ 2 − 1)
2∆ n,n (2∆ n,n + 1) 2 respectively. The numerators come from the derivatives and the denominators from the normalization factor, present in the definition of the field φ 2 . The dimensions at the zero coupling λ = 0 are
∆ 1 = ∆ n,n+2 = 1 − n + 1 2 + n 2 − 1 16 2 + O( 3 ) ∆ 2 = 1 + ∆ n,n = 1 + n 2 − 1 16 2 + O( 3 ) ∆ 3 = ∆ n,n−2 = 1 + n − 1 2 + n 2 − 1 16 2 + O( 3 ) (3.97)
Computation of the first order coefficients as in previous cases is quite easy and with desired accuracy we get
C (1) 1,1 = 2π(n + 3)(2 − (n + 2) ) √ 3(n + 1) + O( ) C (1) 1,2 = C (1) 2,1 = − 8π n+2 3n (2 − ) (n + 1)(n + 3) + O( ) C (1) 1,3 = C (1) 3,1 = 0; C (1) 2,2 = 4π (4 − (n 2 + 1) ) √ 3(n 2 − 1) + O( ) C (1) 2,3 = C (1) 3,2 = − 8π n−2 3n (2 − ) (n − 3)(n − 1) + O( ) C (1) 3,3 = 2π(n − 3)(2 + (n − 2) ) √ 3(n − 1) + O( ) (3.98)
From (3.70), (3.84), (3.10), (3.96) and the above presented considerations, for the second order coefficients C
(2) α,β = C (2) β,α we find C(2)
1,1 = 8π 2 (3n 4 + 33n 3 + 121n 2 + 143n + 20) 3n(n + 1)(n + 3) 2 2 − 4π 2 (n + 5) (5n 4 + 45n 3 + 143n 2 + 151n + 8) 3n(n + 1)(n + 3) 2
+ O( 0 ) C(2)
1,2 = − 64π 2 n+2 n (3n + 11) 3(n + 1)(n + 3)(n + 5) 2 + 32π 2 n+2 n (n 2 + 18n + 57) 3(n + 1)(n + 3)(n + 5)
+ O( 0 ) C (2) 1,3 = 320π 2 3n(n 2 − 9) √ n 2 − 4 2 − 320π 2 3n(n 2 − 9) √ n 2 − 4 + O( 0 ) C (2) 2,2 = 128π 2 3 (n 2 − 1) 2 − 16π 2 (n 2 + 19) 3 (n 2 − 1) + O( 0 ) C (2) 2,3 = − 64π 2 n−2 n (3n − 11) 3(n − 1)(n − 3)(n − 5) 2 − 32π 2 n−2 n (n 2 − 18n + 57) 3(n − 1)(n − 3)(n − 5) + O( 0 ) C(2)3,3 = 8π 2 (3n 4 − 33n 3 + 121n 2 − 143n + 20) 3n(n − 1)(n − 3) 2 2 + 4π 2 (n − 5) (5n 4 − 45n 3 + 143n 2 − 151n + 8) 3n(n − 1)(n − 3) 2 + O( 0 ) (3.99)
With this input we can repeat the procedure of the Section 3.2.3 and compute the matrix of anomalous dimensions. Here is the final result: Γ 1,1 ≈ ∆ 1 + πg(n + 3)(2 − (n + 2) ) √ 3 (n + 1) + 8π 2 g 2 (n + 2) 3(n + 1)
Γ 1,2 = Γ 2,1 ≈ πg(n − 1) n+2 3n (2 − ) n + 1 + 4π 2 g 2 (n − 1) n+2 n 3(n + 1) Γ 1,3 = Γ 3,1 ≈ 0 Γ 2,2 ≈ ∆ 2 + 2πg(4 − (n 2 + 1) ) √ 3 (n 2 − 1) + 4π 2 g 2 (n 2 + 3) 3(n 2 − 1) Γ 2,3 = Γ 3,2 ≈ πg(n + 1) n−2 3n (2 − ) n − 1 + 4π 2 g 2 (n + 1) n−2 n 3(n − 1) Γ 3,3 ≈ ∆ 3 + πg(n − 3)(2 + (n − 2) ) √ 3 (n − 1) + 8π 2 g 2 (n − 2) 3(n − 1) (3.100)
Again we see that all matrix elements are regular at = 0. All double and single poles in disappeared. At the fixed point g = g * (see (2.21)) Γ (g * )
1,1 = 1 − (n 2 − 5) 2(n + 1) + (n 3 − 7n 2 − n + 39) 2 16(n + 1) + O( 3 ) Γ (g * ) 1,2 = Γ (g * ) 2,1 = (n − 1) n+2 n ( + 1) n + 1 + O( 3 ) Γ (g * ) 1,3 = Γ (g * ) 3,1 = O( 3 ) Γ (g * ) 2,3 = Γ (g * )
3,2 = (n + 1) n−2 n ( + 1)
n − 1 + O( 3 ) Γ (g * ) 3,3 = 1 + (n 2 − 5) 2(n − 1) + (n 3 + 7n 2 − n − 39) 2 16(n − 1) + O( 3 ) (3.101)
Here are the eigenvalues of this matrix ∆ (g * ) 1 = 1 + (n + 1) 2 + (n + 1)(n + 7) 2 16 n+2,n = dimension 0 is the identity operator. The operator product expansions satisfy the fusion rules φ n 1 ,m 1 φ n 2 ,m 3 ∈ n 1 +n 2 −1
+ O( 3 ) ∆ (g * ) 2 = 1 + (n 2 − 1) 2 16 + O( 3 ) ∆ (g * ) 3 = 1 − (n − 1) 2 + (n − 1)(n − 7) 2 16 + O( 3 )(n 3 =|n 1 −n 2 |+1 n 1 +n 2 −n 3 ∈2Z+1 m 1 +m 2 −1 m 3 =|m 1 −m 2 |+1 m 1 +m 2 −m 3 ∈2Z+1 [φ n 3 ,m 3 ] (A.3)
The main subject of this paper is the minimal model M p perturbed by the relevant field φ 1,3 . It's dimension
∆ 1,3 = 1 − 2 p + 1 ≡ 1 − < 1 (A.4)
For large p this field becomes nearly marginal which is the main reason why in this region the non-trivial RG behavior can be investigated by means of the perturbation theory. The structure constants of the OPE have been computed in [9]. A slightly more compact expression which we present below is taken from [10]
C (n 1 ,m 1 )(n 2 ,m 2 )(n 3 ,m 3 ) = ρ 4st+2t−2s−1 (A.5) × γ(ρ − 1)γ(m 1 − n 1 ρ −1 )γ(m 2 − n 2 ρ −1 )γ(−m 3 + n 3 ρ −1 ) γ(1 − ρ −1 )γ(−n 1 + m 1 ρ)γ(−n 2 + m 2 ρ)γ(n 3 − m 3 ρ) × s i=1 t j=1 ((i − jρ)(i + n 3 − (j + m 3 )ρ)(i − n 1 − (j − m 1 )ρ)(i − n 2 − (j − m 2 )ρ)) −2 × s i=1 γ(iρ −1 )γ(−m 3 + (i + n 3 )ρ −1 )γ(m 1 + (i − n 1 )ρ −1 )γ(m 2 + (i − n 2 )ρ −1 ) × t j=1 γ(jρ)γ(−n 3 + (j + m 3 )ρ)γ(n 1 + (j − m 1 )ρ)γ(n 2 + (j − m 2 )ρ) where γ(x) ≡ Γ(x) Γ(1 − x) ; s = n 1 + n 2 − n 3 − 1 2 ; t = m 1 + m 2 − m 3 − 1 2
B Computation of I(x)
One way to get the result (1.10) for the integral (1.8) is to notice that I(x) satisfies the hypergeometric differential equation independently with respect to the both variables x andx. The starting point is the identity
x(1 − x)∂ 2 x + (1 − a − c + (a + b + 2c − 2)x)∂ x (B.1) +c(1 − a − b − c)] y a−1 (1 − y) b−1 (y − x) c = ∂ y c y a (1 − y) b (y − x) c−1
which shows that as a function of the variable x, I(x) is a linear combination of the hypergeometric functions
F (1 − a − b − c, −c, 1 − a − c, x) and x a+c F (a, 1 − b, 1 + a + c, x)
The same conclusion is true also for the conjugate variablex. The condition that the function I(x) is single valued around the points x = 0 a x = 1 fixes a specific combination of holomorphic and anti-holomorphic parts up to a constant which in its turn can be easily evaluated considering the special case x = 0. The final result is presented in eq. (1.10).
C Four-point functions at large p limit
Since the structure constants of OPE for the minimal models are known (see A.5), to construct the correlation functions it remains to calculate related conformal blocks. According to AGT relation [11] this conformal blocks in a simple fashion are related to the instanton part of the Nekrasov partition function of N = 2 SYM theory with the gauge group SU (2) and with four fundamental hypermultiplets. In the large p limit the minimal models approach to a free theory (the central charge c ≈ 1), so it is not surprising that in this limit conformal blocs of degenerated primary fields become very simple and can be expressed in terms of rational (and also logarithmic in the cases when the leading corrections in 1/p is required to be taken into account) functions of the the cross ratio of the coordinates. It is straightforward to compute Nekrasov partition [12] function up to desired order in instanton expansion using combinatorial formula found in [13] and extended to the case with extra hypermultiplets in [14]. Computing the first few coefficients of the instanton expansion (for more confidence we made calculations up to 6th order ), adjusting appropriately the parameters in order to get the required conformal block and finally taking the large p limit one can easily guess the exact dependence of the conformal block on the cross ratio of the insertion points (which is the same as the instanton counting parameter, from the gauge theory point of view).
In this way we got expressions 3 for the correlation functions φ 1,3 (x)φ 1,3 (0)φ n,n (1)φ n,n (∞) = |x| 4 −4 + (n 2 − 1) 2 12|x| 4
x 2 2(1 − x) +x 2 2(1 −x) + log 2 |1 − x| 2 + O( 3 ) φ 1,3 (x)φ 1,3 (0)φ n,n+1 (1)φ n,n+1 (∞) = 1 − x + x 2 2 x 2 (1 − x) 2 + 2(n + 2) 3n 1 − x 2 x(1 − x) 2 + O( ) φ 1,3 (x)φ 1,3 (0)φ n,n−1 (1)φ n,n+1 (∞) = 4 √ n 2 − 1 3n 1 − x 2 x(1 − x) 2 + O( ) φ 1,3 (x)φ 1,3 (0)φ n,n+2 (1)φ n,n+2 (∞) = 1 − 2x + 3x 2 − 2x 3 + x 4 3 x 2 (1 − x) 2 2 + 8(n + 3) 3(n + 1) 1 − 3x 2 + x 2 − x 3 4
x(1 − x) 2 2 + (n + 3)(n + 4) 18n(n + 1) Fig.1 and also the discussion coming after the eq. (2.4)). Using Green's theorem the integrals over lens-like regions can be easily transformed to the contour integrals over their boundaries. The integrals over the arcs which belong to the circle |x − 1| = l are trivial. Instead, the integrals along remaining parts of the boundary which lay on |x| = 1 + l 0 (for the right lens-like region D R ) or on |x| = 1 − l 0 (for the left lens-like region D L ) seem more complicated, but fortunately these contour integrals too (with an exception to be considered later) admit exact treatment. Below we give the details on the integration along the arc |x| = 1 + l 0 . The formulae for the other arc |x| = 1 − l 0 can be found by a simple replacement l 0 ↔ −l 0 . During the calculations we heavily employ the formulae (we use the notation r ≡ |x − 1| and the angles φ, α are depicted in Fig.2 (1 + l 0 ) 2 cos 2 (α) + cos 2 (ϕ) + 2(1 + l 0 ) cos(α) cos(ϕ) l 2 0 (1 + l 0 ) 2 dϕ = (1 + l 0 ) 2 d(α + ϕ) + (1 + l 0 )d sin(α + ϕ) l 2 0 (2 + l 0 ) 2 sin(2ϕ)dr r = (1 + l 0 ) 2 2 d(2α − sin(2α)) 2d sin(ϕ) r = d (1 + l 0 ) 2 α + ϕ + (1 + l 0 ) sin(α + ϕ) l 0 (l 0 + 2) (D.5)
x 1 − x 4 + O( )
With these formulae at hand it is easy to evaluate the integrals over lens-like regions in the limit l 0 /l → 0 and l → 0
D L ∪D R d 2 x |x − 1| 4 ≈ − π l 2 − π 8 + π 2l 2 0 (D.6) D L ∪D R d 2 x 1 (x − 1) 2 + 1 (x − 1) 2 ≈ 2π D L ∪D R d 2 x |x − 1| 2 1 x − 1 + 1 x − 1 ≈ π (D.7)
We need also the integral Unlike the previous cases this integral can not be evaluated exactly in terms of elementary functions. Nevertheless it is not difficult to show that up to terms vanishing in the limit l 0 /l → 0 and l → 0 it is equal to
D L ∪D R d 2 x |x − 1| 2 ≈ 2π log l 2l 2 0 (D.9)
Figure 1 .
1Ω l,l 0 is the gray region b) Contributions of the regions around singularities It remains to calculate the contributions of the regions D l,0 , D l,1 , D l,∞ to the integral (1.7). Evidently the first two regions give identical contributions, so let's concentrate on the region D l,0 for definiteness. To calculate the four-point function in this region we apply the OPE (all the structure constants we use in this paper can be extracted from the general formula (A.5))
Contribution of lens-like regions Near x ∼ 1 the r.h.c. of eq. (3.20) becomes 5n+4 12n|x−1| 2 , hence the contribution of the lens-like regions near x ∼ 1 to be subtracted from the r.h.s. of eq. Contributions of the regions D l,0 , D l,1 and D l,∞ Contributions of D l,0 and D l,1 obviously are identical so we will concentrate on D l,0 only. The relevant OPE is
Contribution of lens-like regions Near x ∼ 1 the r.h.s. of eq. (3.30) behaves as √ n 2 −1 3n |x − 1| −2 and the contribution of the lens-like regions near x ∼ 1, which should be subtracted from the r.h.s. of eq. (3.31) is (see (D.Contributions of the regions D l,0 , D l,1 and D l,∞ We will treat contributions of D l,0 and D l,1 separately. i) D l,0 contribution. The relevant OPE is φ(x)φ n,n+1 (0) = C (n,n+1)
Contributions of the regions D l,0 , D l,1 and D l,∞ The contributions of D l,0 and D l,1 are identical so we compute only the D l,0 part. Here are the relevant terms of the OPE φ(x)φ n,n+2 (0) = (xx)
function φ n,n (1)φ n,n+2 (0) λ a) Contribution of the region Ω l,l 0 is given by the integralΩ l,l 0 I(x) φ(x)φ n,n+2 (0)φ n,n (1)φ(∞) d 2 x (3.71)The large p limit of the four-point function is very simple (see Appendix C)
((3n − 5) log(l) + (n + 1) log (2l 0 )) (3.74) b) Contribution of lens-like regions We see from (3.72), (3.73) that near x ∼ 1 the integrand in eq. So according to the Appendix D the contribution of the lens-like regions near x ∼ 1 which should be subtracted from the r.h.s. of eq. Contributions of the regions D l,0 , D l,1 and D l,∞ Let us compute the contributions of D l,0 and D l,1 separately. i) D l,0 contribution. The relevant OPE has already appeared in (3.61). Instead of (3.62) we now need the analogous relation
final result of the D l,0 contribution we get D l,0 \D l 0 ,0
Contribution of lens-like regions It follows from (3.86) and (3.87) that near x ∼ 1 the integrand in eq. (3.85) up to less singular terms behaves as
Contributions of the regions D l,0 , D l,1 and D l,∞ i) D l,0 contribution The relevant OPE:φ(x)φ n,n−2 (0) = (xx) ∆n,n−∆ n,
be endowed with the extra multipliers
nontrivial check, we have tested the crossing invariance of all these correlation functions. Of course the interested reader can get convinced in correctness of our expressions also by examining the third order differential equation satisfied by any conformal bloc which includes the degenerated field φ 1,3[5].Performing the conformal map x → 1/x with the help of the eq.(1.6) we get the correlation functions (2.1), (3.2), (3.20), (3.2), (3.30), (3.57), (3.72), (3.86) used in the main text.D Integrations over lens-like regionsHere we compute the contributions of the lens-like regionsD L = D 1−l 0 ,0 ∩ D l,1 D R = D 1+l 0 ,0 ∩ D
Figure 2 .
2The angles φ and α; |x| = 1 + l 0 (1 + l 0 ) sin(α) = sin(ϕ); r = (1 + l 0 ) cos(α) − cos(on the circle |x| = 1 + l 0 the one forms appearing on the r.h.s.
3.102) which, up to O( 3 ) terms coincide with the dimensions ∆ of the IR CFT M p−1 . It is easy to find the orthogonal matrix which diagonalizes the matrix of anomalous dimensions (3.101) and to establish the explicit map(p−1)
n+2,n , 1 + ∆
(p−1)
n,n
and ∆
(p−1)
n−2,n
φ
(p−1)
The fields φ n,n±4 have larger dimensions ∼ 4 and do not get mixed with these three fields.
We consider here the so called diagonal series only.
Some particular conformal blocs in large p limit have been computed earlier in[15] using more traditional approach.
AcknowledgementsIts a pleasure to thank R. Flume for interesting discussions. This work was partly supported by European Commission FP7 Programme Marie Curie IIF Return Phase Grant Agreement 908571, by Volkswagen foundation of Germany, by a grant of the Armenian State Council of Science and by Armenian-Russian grant "Common projects in Fundamental Scientific Research"-2013.In this case too we see that the coefficients in (3.103) do not receive or 2 corrections.A Minimal modelsFor the readers convenience we present here few facts about unitary series of the minimal models[5,8]denoted by M p , p = 3, 4, . . . . The central charge is given byThis theory contains finitely many spinless 2 primary fields denoted by φ n,m with conformal dimensions (the famous Kac spectrum[6])where n ∈ {1, 2, . . . , p − 1}, m ∈ {1, 2, . . . , p}. There is an identification φ p−n,p+1−m ≡ φ n,m so that the number of primary fields is equal to p(p − 1)/2. The field φ 1,1 with
Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov. A Zamolodchikov, J.Nucl.Phys. 461090A. Zamolodchikov, Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory, Sov.J.Nucl.Phys. 46 (1987) 1090.
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Domain Walls for Two-Dimensional Renormalization Group Flows. D Gaiotto, arXiv:1203.1052hep-thD. Gaiotto, Domain Walls for Two-Dimensional Renormalization Group Flows, [arXiv:1203.1052 [hep-th]].
Irreversibility" of the flux of the renormalization group in a 2D field theory. A B Zamolodchikov, JETP Lett. 4312A. B. Zamolodchikov, "Irreversibility" of the flux of the renormalization group in a 2D field theory, JETP Lett.43(12), 730-732 (1986).
Infinite conformal symmetry in two-dimensional quantum field theory. A Belavin, A Polyakov, A Zamolodchikov, Nucl.Phys. 241A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory , Nucl.Phys. B241 (1984). 333-380.
Highest weight representations of infinite dimensional Lie algebras. V G Kac, Proc. Internat. Congress Mathematicians. Internat. Congress MathematiciansHelsinkiV. G. Kac, Highest weight representations of infinite dimensional Lie algebras, Proc. Internat. Congress Mathematicians (Helsinki, 1978).
Higher transcendental functions. A Erdelyi, McGraw-Hill Book Co., Inc1New York, N.Y.A. Erdelyi et al.,Higher transcendental functions, vol. 1, (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).
Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions. D Friedan, Z Qiu, S Shenker, Phys.Rev.Lett. v. 52D. Friedan, Z. Qiu and S. Shenker, Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions , Phys.Rev.Lett. v. 52 (1984) 1575-1578.
Operator algebra of two-dimensional conformal theories with central charge C ≤ 1. Vl, V Dotsenko, Fateev, Phys.Lett. 154Vl. Dotsenko, V. Fateev, Operator algebra of two-dimensional conformal theories with central charge C ≤ 1 , Phys.Lett. B154 (1985) 291-295.
Fields with spin in the minimal models M (p) (c < 1) of two-dimensional conformal field theory, preprint YERPHI-1198-75-89. R G Pogosian, KEK library linkR. G. Pogosian, Fields with spin in the minimal models M (p) (c < 1) of two-dimensional conformal field theory, preprint YERPHI-1198-75-89. [KEK library link]
Liouville Correlation Functions from Four-dimensional Gauge Theories. L Alday, D Gaiotto, Y Tachikawa, arXiv:0906.3219Lett. Math. Phys. 91hep-thL. Alday, D. Gaiotto, and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167-197, [arXiv:0906.3219 [hep-th]].
Seiberg-Witten prepotential from instanton counting. N Nekrasov, hep-th/0206161Adv.Theor.Math.Phys. 7N. Nekrasov, Seiberg-Witten prepotential from instanton counting, in Adv.Theor.Math.Phys.7 (2004): 831-864, [hep-th/0206161].
An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. R Flume, R Poghossian, hep-th/0208176Int.J.Mod.Phys. 18R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int.J.Mod.Phys. A18 (2003) 2541, [hep-th/0208176].
Multi-instanton calculus and equivariant cohomology. U Bruzzo, F Fucito, J F Morales, A Tanzini, hep-th/0211108JHEP. 030554U. Bruzzo, F. Fucito, J. F. Morales, and A. Tanzini, Multi-instanton calculus and equivariant cohomology, JHEP 0305:054,2003, [hep-th/0211108].
Renormalization group defects for boundary flows. A Konechny, arXiv:1211.3665J.Phys.A. 46145401hep-thA. Konechny, Renormalization group defects for boundary flows, J.Phys.A 46, 145401, (2013), [arXiv:1211.3665 [hep-th]].
| [] |
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"Detecting Cancer Metastases on Gigapixel Pathology Images",
"Detecting Cancer Metastases on Gigapixel Pathology Images"
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"Yun Liu [email protected] \nGoogle Brain\n\n",
"Krishna Gadepalli \nGoogle Brain\n\n",
"Mohammad Norouzi [email protected] \nGoogle Brain\n\n",
"George E Dahl [email protected] \nGoogle Brain\n\n",
"Timo Kohlberger \nGoogle Brain\n\n",
"Aleksey Boyko \nGoogle Brain\n\n",
"Subhashini Venugopalan \nGoogle Inc\n\n",
"Aleksei Timofeev \nGoogle Inc\n\n",
"Philip Q Nelson \nGoogle Inc\n\n",
"Greg S Corrado \nGoogle Brain\n\n",
"Jason D Hipp \nVerily Life Sciences\nMountain ViewCAUSA\n",
"Lily Peng [email protected] \nGoogle Brain\n\n",
"Martin C Stumpe [email protected] \nGoogle Brain\n\n"
] | [
"Google Brain\n",
"Google Brain\n",
"Google Brain\n",
"Google Brain\n",
"Google Brain\n",
"Google Brain\n",
"Google Inc\n",
"Google Inc\n",
"Google Inc\n",
"Google Brain\n",
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] | [] | Each year, the treatment decisions for more than 230, 000 breast cancer patients in the U.S. hinge on whether the cancer has metastasized away from the breast. Metastasis detection is currently performed by pathologists reviewing large expanses of biological tissues. This process is labor intensive and error-prone. We present a framework to automatically detect and localize tumors as small as 100 × 100 pixels in gigapixel microscopy images sized 100, 000×100, 000 pixels. Our method leverages a convolutional neural network (CNN) architecture and obtains state-of-the-art results on the Camelyon16 dataset in the challenging lesion-level tumor detection task. At 8 false positives per image, we detect 92.4% of the tumors, relative to 82.7% by the previous best automated approach. For comparison, a human pathologist attempting exhaustive search achieved 73.2% sensitivity. We achieve image-level AUC scores above 97% on both the Camelyon16 test set and an independent set of 110 slides. In addition, we discover that two slides in the Came-lyon16 training set were erroneously labeled normal. Our approach could considerably reduce false negative rates in metastasis detection. | null | [
"https://arxiv.org/pdf/1703.02442v2.pdf"
] | 15,895,922 | 1703.02442 | cec1bf05f2410312054299a23396e3e90ab30b5f |
Detecting Cancer Metastases on Gigapixel Pathology Images
Yun Liu [email protected]
Google Brain
Krishna Gadepalli
Google Brain
Mohammad Norouzi [email protected]
Google Brain
George E Dahl [email protected]
Google Brain
Timo Kohlberger
Google Brain
Aleksey Boyko
Google Brain
Subhashini Venugopalan
Google Inc
Aleksei Timofeev
Google Inc
Philip Q Nelson
Google Inc
Greg S Corrado
Google Brain
Jason D Hipp
Verily Life Sciences
Mountain ViewCAUSA
Lily Peng [email protected]
Google Brain
Martin C Stumpe [email protected]
Google Brain
Detecting Cancer Metastases on Gigapixel Pathology Images
neural networkpathologycancerdeep learning
Each year, the treatment decisions for more than 230, 000 breast cancer patients in the U.S. hinge on whether the cancer has metastasized away from the breast. Metastasis detection is currently performed by pathologists reviewing large expanses of biological tissues. This process is labor intensive and error-prone. We present a framework to automatically detect and localize tumors as small as 100 × 100 pixels in gigapixel microscopy images sized 100, 000×100, 000 pixels. Our method leverages a convolutional neural network (CNN) architecture and obtains state-of-the-art results on the Camelyon16 dataset in the challenging lesion-level tumor detection task. At 8 false positives per image, we detect 92.4% of the tumors, relative to 82.7% by the previous best automated approach. For comparison, a human pathologist attempting exhaustive search achieved 73.2% sensitivity. We achieve image-level AUC scores above 97% on both the Camelyon16 test set and an independent set of 110 slides. In addition, we discover that two slides in the Came-lyon16 training set were erroneously labeled normal. Our approach could considerably reduce false negative rates in metastasis detection.
Introduction
The treatment and management of breast cancer is determined by the disease stage. A central component of breast cancer staging involves the microscopic examination of lymph nodes adjacent to the breast for evidence that the cancer has spread, or metastasized [3]. This process requires highly skilled pathologists and is fairly time-consuming and error-prone, particularly for lymph nodes with either no or small tumors. Computer assisted detection of lymph node metastasis could increase the sensitivity, speed, and consistency of metastasis detection [16].
Work done as a Google Brain Resident (g.co/brainresidency). Work done as a Google intern.
In recent years, deep CNNs have significantly improved accuracy on a wide range of computer vision tasks such as image recognition [14,11,19], object detection [8], and semantic segmentation [17]. Similarly, deep CNNs have been applied productively to improve healthcare (e.g., [9]). This paper presents a CNN framework to aid breast cancer metastasis detection in lymph nodes. We build on [23] by leveraging a more recent Inception architecture [20], careful image patch sampling and data augmentations. Despite performing inference with stride 128 (instead of 4), we halve the error rate at 8 false positives (FPs) per slide, setting a new state-of-the-art. We also found that several approaches yielded no benefits: (1) a multi-scale approach that mimics the human cognition of a pathologist's examination of biological tissue, (2) pretraining the model on ImageNet image recognition, and (3) color normalization. Finally, we dispense with the random forest classifier and feature engineering used in [23] and find that the maximum function is an effective whole-slide classification procedure.
Related Work Several promising studies have applied deep learning to histopathology. The Camelyon16 challenge winner [1] achieved a sensitivity of 75% at 8 FP per slide and a slide-level classification AUC of 92.5% [23]. The authors trained a Inception (V1, GoogLeNet) [20] model on a pre-sampled set of image patches, and trained a random forest classifier on 28 hand-engineered features to predict the slide label. A second Inception model was trained on harder examples, and predicted points were generated using the average of the two models' predictions. This team later improved these metrics to 82.7% and 99.4% respectively [1] using color normalization [4], additional data augmentation, and lowering the inference stride from 64 to 4. The Camelyon organizers also trained CNNs on smaller datasets to detect breast cancer in lymph nodes and prostate cancer biopsies [16]. [12] applied CNNs to segmenting or detecting nuclei, epithelium, tubules, lymphocytes, mitosis, invasive ductal carcinoma and lymphoma. [7] demonstrated that CNNs achieved higher F1 score and balanced accuracy in detecting invasive ductal carcinoma. CNNs were also used to detect mitosis, winning the ICPR12 [6] and AMIDA13 [22] mitosis detection competitions. Other efforts at leveraging machine learning for predictions in cancer pathology include predicting prognosis in non-small cell lung cancer [25].
Methods
Given a gigapixel pathology image (slide 1 ), the goal is to classify if the image contains tumor and localize the tumors for a pathologist's review. This use case and the difficulty of pixel-accurate annotation (Fig. 2) renders detection and localization more important than pixel-level segmentation. Because of the large size of the slide and the limited number of slides (270), we train models using smaller image patches extracted from the slide (Fig. 1). Similarly, we perform inference over patches in a sliding window across the slide, generating a tumor probability heatmap. For each slide, we report the maximum value in the heatmap as the slide-level tumor prediction.
We utilize the Inception (V3) architecture [20] with inputs sized 299 × 299 (the default) to assess the value of initializing from existing models pre-trained on another domain. For each input patch, we predict the label of the center 128×128 region. A 128 pixel region can span several tumor cells and was also used in [16]. We label a patch as tumor if at least one pixel in the center region is annotated as tumor. We explored the influence of the number of parameters by reducing the number of filters per layer while keeping the number of layers constant (e.g., depth multiplier = 0.1 in TensorFlow). We denote these models "small". We also experimented with multi-scale approaches that utilize patches at multiple magnifications centered on the same region (Fig. 3). Because preliminary experiments did not show a benefit from using up to four magnifications, we present results only for up to two magnifications.
Training and evaluating our models was challenging because of the large number of patches and the tumor class imbalance. Each slide contains 10, 000 to 400, 000 patches (median 90, 000). However, each tumor slide contains 20 to 150, 000 tumors patches (median 2, 000), corresponding to tumor patch percentages ranging from 0.01% to 70% (median 2%). Avoiding biases towards slides containing more patches (both normal and tumor) required careful sampling. First, we select "normal" or "tumor" with equal probability. Next, we select a slide that contains that class of patches uniformly at random, and sample patches from that slide. By contrast, some existing methods pre-sample a set of patches from each slide [23], which limits the breadth of patches seen during training.
To combat the rarity of tumor patches, we apply several data augmentations. First, we rotate the input patch by 4 multiples of 90 • , apply a left-right flip and repeat the rotations. All 8 orientations are valid because pathology slides do not have canonical orientations. Next, we use TensorFlow's image library (tensorflow.image.random X ) to perturb color: brightness with a maximum delta of 64/255, saturation with a maximum delta of 0.25, hue with a maximum delta of 0.04, and contrast with a maximum delta of 0.75. Lastly, we add jitter to the patch extraction process such that each patch has a small x,y offset of up to 8 pixels. The magnitudes of the color perturbations and jitter were lightly tuned using our validation set. Pixel values are clipped to [0, 1] and scaled to [−1, 1].
We run inference across the slide in a sliding window with a stride of 128 to match the center region's size. For each patch, we apply the rotations and left-right flip to obtain predictions for each of the 8 orientations, and average the 8 predictions.
Implementation Details We trained our networks with stochastic gradient descent in TensorFlow [2], with 8 replicas each running on a NVIDIA Pascal GPU with asynchronous gradient updates and batch size of 32 per replica. We used RMSProp [21] with momentum of 0.9, decay of 0.9 and = 1.0. The initial learning rate was 0.05, with a decay of 0.5 every 2 million examples. For refining a model pretrained on ImageNet, we used an initial learning rate of 0.002.
Evaluation and Datasets
We use the two Camelyon16 evaluation metrics [1]. The first metric, the area under receiver operating characteristic, (Area Under ROC, AUC) [10] evaluates slide-level classification. This metric is challenging because of the potential for FPs when 10 5 patch-level predictions are obtained per slide. We obtained 95% confidence intervals using a bootstrap approach 2 .
The second metric, FROC [5], evaluates tumor detection and localization. We first generate a list of coordinates and corresponding predictions from each heatmap. Among all coordinates that fall within each annotated tumor region, the highest prediction is retained. Coordinates falling outside tumor regions are FPs. We use these values to compute the ROC. The FROC is defined as the sensitivity at 0.25, 0.5, 1, 2, 4, 8 average FPs per tumor-negative slide [16]. This metric is challenging because reporting multiple points per FP region can quickly erode the score. We focused on the FROC as opposed to the AUC because there are approximately twice as many tumors as slides, which improves the reliability of the evaluation metric. Similar to the AUC, we report 95% confidence intervals by computing the FROC over 2000 bootstrap samples of the predicted points. In addition, we report the sensitivity at 8 FP per slide ("@8FP") to assess the false negative rate.
To generate points for FROC computation, the Camelyon winners [23,1] thresholded the heatmap to produce a bit-mask, and reported a single prediction for each connected component in the bit-mask. By contrast, we use a non-maxima suppression method similar to [6] that repeats two steps until no values in the heatmap remain above a threshold t: (1) report the maximum and corresponding coordinate, and (2) set all values within a radius r of the maximum to 0. Because we apply this procedure to the heatmap, r has units of 128 pixels. t controls the number of points reported and has no effect on the FROC unless the curve plateaus before 8 FP. To avoid erroneously dropping tumor predictions, we used a conservative threshold of t = 0.5.
Datasets Our work utilizes the Camelyon16 dataset [1], which contains 400 slides: 270 slides with pixel-level annotations, and 130 unlabeled slides as a test set. 3 We split the 270 slides into train and validation sets (Appendix) for hyperparameter tuning. Typically only a small portion of a slide contains biological tissue of interest, with background and fat comprising the remainder (e.g., Fig. 2). To reduce computation, we removed background patches (gray value > 0.8 [12]), and verified visually that lymph node tissue was not discarded.
Additional Evaluation: NHO-1 We digitized another set of 110 slides (57 containing tumor) from H&E-stained lymph nodes extracted from 20 patients (86 biological tissue blocks 4 ) as an additional evaluation set. These slides came with patient-or block-level labels. To determine the slide labels, a board-certified pathologist blinded to the predictions adjudicated any differences, and briefly reviewed all 110 slides.
Experiments & Results
To perform slide-level classification, the current state-of-the-art methods apply a random forest to features extracted from a heatmap prediction [1]. Unfortunately, we were unable to train slide-level classifiers because the 100% validationset AUC ( to 10 − 20% variance), and can confound evaluation of model improvements by grouping multiple nearby tumors as one. By contrast, our non-maxima suppression approach is relatively insensitive to r between 4 and 6, although less accurate models benefited from tuning r using the validation set (e.g., 8). Finally, we achieve 100% FROC on larger tumors (macrometastasis), indicating that most false negatives are comprised of smaller tumors. Previous work (e.g., [24,9]) has shown that pre-training on a different domain improves performance. However, we find that although pre-training significantly improved convergence speed, it did not improve the FROC (see Table 1: 40X vs. 40X-pretrained). This may be due to a large domain difference between pathology images and natural scenes in ImageNet, leading to limited transferability. In addition, our large dataset size (10 7 patches) and data augmentation may have enabled the training of accurate models without pre-training.
Next, we studied the effect of model size. Although we were originally motivated by improved experiment turn-around time, we surprisingly found that slimmed-down Inception architectures with only 3% of the parameters achieved similar performance to the full version (Table 1: 40X vs. 40X-small). Thus, we performed the remaining experiments using this smaller model.
We also experimented with a multi-scale approach inspired by pathologists' workflow of examining a slide at multiple magnifications to get context. However, we find no performance benefit in combining 40X with an additional input at lower magnification (Fig. 3). However, these combinations output smoother heatmaps (Fig. 4), likely because of translational invariance of the CNN and overlap in adjacent patches. These visual improvements can be deceptive: some of the speckles in the 40X models reveal small non-tumor regions surrounded by tumor. Figures 1 and 3 highlight the variability in the images. Although the current leading approaches report improvements from color normalization, our experi- ments revealed no benefit (Appendix). This could be explained by our extensive data augmentations causing our models to learn color-invariant features.
Finally, we experimented with ensembling models in two ways. First, averaging predictions across the 8 rotations/flips yielded a few percent improvement in the metrics. Second, ensembling across independently trained models yield additional but smaller improvements, and gave diminishing returns after 3 models.
Additional Validation We also tested our models on another 110 slides that were digitized on different scanners, from different patients, and treated with different tissue preparation protocols. Encouragingly, we obtained an AUC of 97.6 (93.6, 100), on-par with our Camelyon16 test set performance.
Qualitative Evaluation We discovered tumors in two "normal" slides: 086 and 144. Fortunately, the challenge organizers confirmed that both were data processing errors, and the patients were unaffected. Remarkably, both slides were in our training set, suggesting that our model was relatively resilient to label noise. In addition, we discovered an additional 7 tumor slides with incomplete annotations: 5 in train, 2 in validation (Appendix). Samples of our predictions and corresponding patches are shown in the Appendix.
Limitations Our errors were related to out-of-focus tissues (macrophages, germinal centers, stroma), and tissue preparation artifacts. These errors could be reduced by better scanning quality, tissue preparation, and more comprehensive labels for different tissue types. In addition, we were unable to exhaustively tune our hyperparameters owing to the near-perfect FROC and AUC on our validation set. We plan to further develop our work on larger datasets.
Conclusion
Our method yields state-of-the-art sensitivity on the challenging task of detecting small tumors in gigapixel pathology slides, reducing the false negative rate to a quarter of a pathologist and less than half of the previous best result. We further achieve pathologist-level slide-level AUCs in two independent test sets. Our method could improve accuracy and consistency of evaluating breast cancer cases, and potentially improve patient outcomes. Future work will focus on improvements utilizing larger datasets.
A Appendix
A.1 Dataset Details
A.2 Soft Labels
Our experiments used binary labels: a patch is positive if at least one pixel in the center 128 x 128 region is annotated as tumor. We also explored an alternative "soft label" approach in preliminary experiments, assigning as the label the fraction of tumor pixels in the center region. However, we found that the thresholded labels yielded substantially better performance. Because the FROC rewards detecting tumors of all size equally, this might reflect the model being trained to assign lower values to smaller tumors (where on average, a smaller portion of each patch contains tumor cells).
A.3 Image Color Normalization
As can be seen in Fig. 1 & 3, the (H&E) stained tissue vary significantly in color. These variations arise from differences in the underlying biological tissue, physical and chemical preparation of the slide, and scanner adjustments. Because reducing these variations have improved performances in other automated detection systems [4,13], we experimented with a similar color normalizing approach. However, we have not found this normalization to improve performance, and thus we detail our approach for reference only. This lack of improvement likely stems from our extensive color perturbations encouraging our models to learn color-insensitive features, and thus the color normalization was unnecessary.
First, we separate color and intensity information by mapping the raw RGB values to a Hue-Saturation-Density (HSD) space [15], and then normalize each component separately. This maps each color channel (I R , I G , I B ) ∈ [0, 255] 3 to a corresponding optical density value: D ν = − ln((I ν + 1)/257), ν ∈ {R, G, B}, followed by applying a common Hue-Saturation-Intensity color space transformation with D = (D R + D B + D G )/3 being the intensity value, and c x = D R D − 1 and c y = (D G − D B )/( √ 3 · D) denoting the Cartesian coordinates that span the two-dimensional hue-saturation plane. We chose the HSD mapping over a direct HSI mapping of RGB values [15], because it is more compatible with the image acquisition physics and yields more compact distributions in general.
Next, we fit a single Gaussian to the color coordinates (c x , c y ) i of the pixels in all tissue-containing patches, i.e. compute their empirical mean µ = (µ x , µ y ) T and covariance Σ ∈ R 2×2 , and then determine the transformation T ∈ R 2×2 of the covariance Σ to a reference covariance matrix Σ R using the Monge-Kantorovitch approach presented in [18]: T = Σ −1/2 Σ 1/2 Σ R Σ 1/2 Σ −1/2 . Subsequently, we normalize the color values by applying the mapping:
c x c y = T c x c y − µ x µ y + µ R x µ R y .(1)
Intensity values, D i , are normalized in the same manner, i.e. by applying the one-dimensional version of Equation 1 in order to transform the empirical mean and variance of all patch intensities to a reference intensity mean and variance.
As reference means and variances for the color and intensity component, respectively (i.e. µ R v , Σ R for color), we chose the component-wise medians over the corresponding statistical moments of all the training slides.
Finally, we map the normalized (c x , c y , D ) values back to RGB space by first applying the inverse HSI transform [15], followed by inverting the nonlinear mapping, i.e. by applying I ν = exp(−D ν ) · 257 − 1 to each component ν ∈ {R, G, B}.
We applied this normalization in two ways. First, we applied this at inference only, by testing a model ("40X-small" in Table 1) on color-normalized slides. Unfortunately, this resulted in a few percent drop in FROC. Next we trained two models on color-normalized slides, both with and without the color perturbations. We then tested these models on color-normalized slides. Neither approach improved the performance.
A.4 Sample Results
Tumor slides with incomplete annotations At the outset, 11 tumor slides were known to have non-exhaustive pixel level annotations: 015, 018, 020, 029, 033, 044, 046, 051, 054, 055, 079, 092, and 095. Thus, we did not use non-tumor patches from these slides as training examples of normal patches. Over the course of our experiments, we discovered several more such cases that we verified with a pathologist: 010, 025, 034, 056, 067, 085, 110.
Fig. 1 .
1Left: three tumor patches and right: three challenging normal patches.
Fig. 2 .
2Difficulty of pixel-accurate annotations for scattered tumor cells. Ground truth annotation is overlaid with a lighter shade. Note that the tumor annotations include both tumor cells and normal cells e.g.,white space representing adipose tissue (fat).
Fig. 3 .
3The three colorful blocks represent Inception (V3) towers up to the second-last layer (PreLogit). Single scale utilizes one tower with input images at 40X magnification; multi-scale utilizes multiple (e.g.,2) input magnifications that are input to separate towers and merged.
Fig. 4 .
4Left to right: sample image, ground truth (tumor in white), and heatmap outputs (40X-ensemble-of-3, 40X+20X, and 40X+10X). Heatmaps of 40X and 40Xensemble-of-3 look identical. The red circular regions at the bottom left quadrant of the heatmaps are unannotated tumor. Some of the speckles are either out of focus patches on the image or non-tumor patches within a large tumor.
Fig. 5 .
5Left: a patch from a H&E-stained slide. The darker regions are tumor, but not the lighter pink regions. Right: the corresponding predicted heatmap that accurately identifies the tumor cells while assigning lower probabilities to the non-tumor regions.
Fig. 6 .
6Left: a patch from a H&E-stained slide, "Normal" 086. The larger pink cells near the top are tumor, while the smaller pink cells at the bottom are macrophages, a normal cell. Right: the corresponding predicted heatmap that accurately identifies the tumor cells while ignoring the macrophages.
Fig. 7 .
7Left: a patch from a H&E-stained slide, "Normal" 144. The cluster of larger, dark purple cells in the bottom right quadrant are tumor, while the smaller dark purple cells are lymphocytes. The pink areas are connective tissue, with interspersed tumor cells. Right: the corresponding predicted heatmap that accurately identifies the tumor cells while ignoring the connective tissue and lymphocytes.
Table 1 )
1rendered internal evaluation of improvements impossible. Nonetheless, using the maximum value of each slide's heatmap achieved AUCs > 97%, statistically indistinguishable from the current best results.For tumor-level classification, we find that the connected component approach[23] provides a 1−5% gain in FROC when the FROC is modest (< 80%), by masking FP regions. However, this approach is sensitive to the threshold (upInput &
Validation
Test
model size
FROC @8FP AUC
FROC
@8FP
AUC
40X
98.1 100 99.0 87.3 (83.2, 91.1) 91.1 (87.2, 94.5) 96.7 (92.6, 99.6)
40X-pretrained
99.3 100 100 85.5 (81.0, 89.5) 91.1 (86.8, 94.6) 97.5 (93.8, 99.8)
40X-small
99.3 100 100 86.4 (82.2, 90.4) 92.4 (88.8, 95.7) 97.1 (93.2, 99.8)
ensemble-of-3
-
-
-88.5 (84.3, 92.2) 92.4 (88.7, 95.6) 97.7 (93.0, 100)
20X-small
94.7 100 99.6 85.5 (81.0, 89.7) 91.1 (86.9, 94.8) 98.6 (96.7, 100)
10X-small
88.7 97.2 97.7 79.3 (74.2, 84.1) 84.9 (80.0, 89.4) 96.5 (91.9, 99.7)
40X+20X-small
94.9 98.6 99.0 85.9 (81.6, 89.9) 92.9 (89.3, 96.1) 97.0 (93.1, 99.9)
40X+10X-small
93.8 98.6 100 82.2 (77.0, 86.7) 87.6 (83.2, 91.7) 98.6 (96.2, 99.9)
Pathologist [1]
-
-
-
73.3*
73.3*
96.6
Camelyon16 winner [1,23]
-
-
-
80.7
82.7
99.4
Table 1. Results on Camelyon16 dataset (95% confidence intervals, CI). Bold indicates
results within the CI of the best model. "Small" models contain 300K parameters per
Inception tower instead of 20M. -: not reported. *A pathologist achieved this sensitivity
(with no FP) using 30 hours.
Each slide contains human lymph node tissue stained with hematoxylin and eosin (H&E), and is scanned at the most common high magnification in a microscope, "40X". We also experimented with 2-and 4-times down-sampled patches ("20X" and "10X").
Sample with replacement n slides from the dataset/split, where n is the number of slides in the dataset/split, and compute the AUC. Repeat for a total of 2000 bootstrap samples, and report the 2.5 and 97.5 percentile values.
The test slides labels were released recently as part of the training dataset for Came-lyon17. We used these labels for evaluation, but not for parameter tuning.4 A tissue block can contain multiple slides that vary considerably at the pixel level.
Fig. 8. Left: a patch from a H&E-stained slide in our additional validation set, NHO-1. The tumor cells are a lighter purple than the surrounding cells. A variety of artifacts are visible: the dark continuous region in the top left quadrant is an air bubble, and the white parallel streaks in the tumor and adjacent tissue are cutting artifacts. Furthermore, the tissue is hemorrhagic, necrotic and poorly processed, leading to color alterations to the typical pink and purple of a H&E slide. Right: the corresponding predicted heatmap that accurately identifies the tumor cells while ignoring the various artifacts, including lymphocytes and the cutting artifacts running through the tumor tissue.
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Rich feature hierarchies for accurate object detection and semantic segmentation. R Girshick, Comput. Vis. and Pattern Recognit. Girshick, R., et al.: Rich feature hierarchies for accurate object detection and se- mantic segmentation. In: Comput. Vis. and Pattern Recognit. (2014)
Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs. V Gulshan, J. of the Am. Medical Soc. 31622Gulshan, V., et al.: Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs. J. of the Am. Medical Soc. 316(22), 2402-2410 (2016)
The meaning and use of the area under a receiver operating characteristic (roc) curve. J A Hanley, B J Mcneil, Radiology. 1431Hanley, J.A., McNeil, B.J.: The meaning and use of the area under a receiver operating characteristic (roc) curve. Radiology 143(1), 29-36 (1982)
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Hue-saturation-density model for stain recognition in digital images from transmitted light microscopy. J A Van Der Laak, Cytometry. 394van der Laak, J.A., et al.: Hue-saturation-density model for stain recognition in digital images from transmitted light microscopy. Cytometry 39(4), 275-284 (2000)
Deep learning as a tool for increased accuracy and efficiency of histopathological diagnosis. G Litjens, Sci. Reports. 6Litjens, G., et al.: Deep learning as a tool for increased accuracy and efficiency of histopathological diagnosis. Sci. Reports 6 (2016)
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The linear monge-kantorovitch linear colour mapping for example-based colour transfer. F Pitié, A Kokaram, Pitié, F., Kokaram, A.: The linear monge-kantorovitch linear colour mapping for example-based colour transfer (2007)
Imagenet large scale visual recognition challenge. O Russakovsky, Int. J. of Comput. Vis. 1153Russakovsky, O., et al.: Imagenet large scale visual recognition challenge. Int. J. of Comput. Vis. 115(3), 211-252 (2015)
C Szegedy, Going deeper with convolutions. Comput. Vis. and Pattern Recognit. Szegedy, C., et al.: Going deeper with convolutions. Comput. Vis. and Pattern Recognit. (2015)
Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. T Tieleman, G Hinton, Tieleman, T., Hinton, G.: Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude (2012)
Assessment of algorithms for mitosis detection in breast cancer histopathology images. M Veta, Medical image analysis. 201Veta, M., et al.: Assessment of algorithms for mitosis detection in breast cancer histopathology images. Medical image analysis 20(1), 237-248 (2015)
D Wang, arXiv:1606.05718Deep learning for identifying metastatic breast cancer. arXiv preprintWang, D., et al.: Deep learning for identifying metastatic breast cancer. arXiv preprint arXiv:1606.05718 (2016)
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Predicting non-small cell lung cancer prognosis by fully automated microscopic pathology image features. K H Yu, Nat. Commun. 7Yu, K.H., et al.: Predicting non-small cell lung cancer prognosis by fully automated microscopic pathology image features. Nat. Commun. 7 (2016)
| [] |
[
"Detecting Structural Breaks in Foreign Exchange Markets by using the group LASSO technique *",
"Detecting Structural Breaks in Foreign Exchange Markets by using the group LASSO technique *"
] | [
"Mikio Ito \nKeio University\nMinato-ku2-15-45, 108-8345Mita, TokyoJapan\n"
] | [
"Keio University\nMinato-ku2-15-45, 108-8345Mita, TokyoJapan"
] | [] | This article proposes an estimation method to detect breakpoints for linear time series models with their parameters that jump scarcely. Its basic idea owes the group LASSO (group least absolute shrinkage and selection operator). The method practically provides estimates of such time-varying parameters of the models. An example shows that our method can detect each structural breakpoint's date and magnitude. | null | [
"https://arxiv.org/pdf/2202.02988v1.pdf"
] | 246,634,067 | 2202.02988 | b7d1278eba1e9cb618c4257c33eb181c9b51517c |
Detecting Structural Breaks in Foreign Exchange Markets by using the group LASSO technique *
February 8, 2022
Mikio Ito
Keio University
Minato-ku2-15-45, 108-8345Mita, TokyoJapan
Detecting Structural Breaks in Foreign Exchange Markets by using the group LASSO technique *
February 8, 2022Vector Error Correction ModelTime-Varying Parametersgroup LASSOStructural Break
This article proposes an estimation method to detect breakpoints for linear time series models with their parameters that jump scarcely. Its basic idea owes the group LASSO (group least absolute shrinkage and selection operator). The method practically provides estimates of such time-varying parameters of the models. An example shows that our method can detect each structural breakpoint's date and magnitude.
Introduction
Recent development in data science has enabled us to handle 'big' datasets, including high dimensional time series; such data have been increasingly common in economics and finance. At the same time, we can adopt models with more than ten thousand parameters to estimate, as this paper addresses. A sparse regression technique based on the least absolute shrinkage and selection operator (LASSO) or a similar one is typical to extract useful information from such big data. (See Tibshirani et al. (2015) for general discussion on LASSO.) There exists recent literature about the theory and application of the LASSO technique to time-series datasets. Articles by Michailidis and d'Alché Buc (2013) and Basu et al. (2015) study vector autoregressive model using the LASSO technique. Chan et al. (2014) adopt a group LASSO model to solve the problem in a variable selection for a structural break autoregression model Ito (2019) shows a method to estimate state parameters that jump rarely based on the group LASSO.
On the other hand, recent economists use econometric models with parameters varying over time when they are interested in structural breaks of economic or financial systems. For example, Primiceri (2005) studies a macroeconomic model with parameters varying over time by adopting a Bayesian technique to estimate a state-space model. The author and his coworkers apply a conventional least square method to estimate time-varying parameters of time series models such as a VAR model. (See Ito et al. (2021).) This article proposes an estimation method to detect breakpoints for linear time series models with their parameters that jump scarcely. One usually supposes non-Gaussian processes like the combination of Gaussian and Poisson ones for the models; the author supposes no statistical specific noises in his estimation. His basic idea owes a sparse regression technique to regard them as linear regression models with possibly quite many parameters. Its most significant point is that scarcely changes in state variables are equivalent to a sparse structure of solutions for a transformed linear equation system obtained from the linear time series model by differencing parameters. The transformation allows the researchers to employ group LASSO techniques when they estimate quite many parameters supposed to jump.
The article's organization is as follows. Section 2 reviews a basic idea. It shows how to transform a linear time series model into another with differenced parameters allowing group LASSO techniques. Finally, the section proposes the practical method to estimate a time series model with multiple parameters scarcely jumping over time. Section 3 shows an application of how we detect several structural changes in foreign exchange markets by using our method. In practice, the author employs the technique for the same data that he and his co-authors studied the foreign exchange markets in Ito et al. (2021). The result shows the periods in which structural changes happened in the markets in a sharper way than Ito et al. (2021). Section 4 concludes and shows several points for future research tasks.
Linear Time series Model with parameters that jump scarcely
There is literature about the estimation of time-varying coefficients of economic models. P. Swamy has published many articles on broadly theoretical frameworks providing several models with time-varying parameters since the 1970's, for example, Swamy et al. (2010). However, we focus on a narrower class of models. Suppose m × n matrix X t and mdimensional vector y t are given at each t.
y t = X t β t + u t , (t = 1 · · · T ),(1)
where β t is a n-dimensional vector coefficient to be estimated. Let t denote time. Thus T is the sample size when the data is set. We can regard the above model as a linear regression model whose each coefficient β varies over time. At this time, we do not have to consider u t 's are gausian. We can rewrite (1) in the following matrix form.
y 1 y 2 y 3 . . . y T = X 1 O X 2 X 3 . . . O X T β 1 β 2 β 3 . . . β T + u 1 u 2 u 3 . . . u T (2)
The data matrix in the regression (2) is mT × nT and the coefficient vector is of nT dimension. Thus, usual regression solver such as OLS is inapplicable since the coefficient matrix in (2) is singular. Now,we suppose that (1) is a regression model whose coefficients β t 's quite rarely change. Then, we rewrite the linear equation system (1) into another linear system with respect to ∆β t in place of β t to focus on sparsity. The following relations are my key idea.
β 1 = β 0 + ∆β 1 β 2 = β 0 + ∆β 1 + ∆β 2 . . . β T = β 0 + ∆β 1 + ∆β 2 + · · · + ∆β T ,(3)
where we regard β 0 as the starting value given of a sequence of ∆β s.
From (3), we obtain its matrix form.
β 1 β 2 β 3 . . . β T = I O O · · · O I I O · · · O I I I O . . . . . . . . . . . . I I I · · · I ∆β 1 ∆β 2 ∆β 3 . . . ∆β T + β 0 β 0 β 0 . . . β 0 (4)
One can set β 0 as the coefficient vector for the usual time-invariant regression. (Please see Appendix for detail.) We apply the group LASSO method to the following equation.
r = X∆β + u,(5)
where r is the residuals of a VEC model with time invariant parameters and X is the matrix as shown in the following two slides.
r = r 1 r 2 r 3 . . . r T = y 1 y 2 y 3 . . . y T − X 1 O X 2 X 3 . . . O X T β 0 β 0 β 0 . . . β 0 (6)
and
X = X 1 O X 2 X 3 . . . O X T I O O · · · O I I O · · · O I I I O . . . . . . . . . . . . I I I · · · I ,(7)
Where I is the n × n identity matrix. At this point, we provide our breakpoint detecting procedure.
Step 1 We apply a group Lasso solver to the equation (5) by grouping variables according to periods.
• Both r and each column of X should be "normalized" before the above operation.
Step 2 The group Lasso solver finds a series of ∆β t whose all components are zeros for almost all periods.
Step 3 We regard the period such that t such that ∆β t = 0 as a break point.
Note that we can obtain each β t through the following equation
β t =β 0 + t τ =1 ∆β τ .
We call the linear equation system (5) a difference form of the linear time series model. Note that the transformation above from the original model (1) to the different form is quite algebraic and that the error terms are identical. Thus, the linear equation (5) as a statistical model is equivalent to the original state-space model corresponding to the equation (1). Irrespective of an estimation method, the above two statistical models provide essentially the same results. One should contemplate which kind of noises they choose, Gaussian or non-Gaussian, when they employ a linear time series model given data y observed.
As the author stressed in Ito (2019), when we address state-space models covering the linear time series model, we have assumed Gaussian noises in many applications. At the same time, their error terms are not always supposed to follow Gaussian noises. Intuitively, under the assumption of Gaussian noise for the models, the sequence of estimated parameters tends to be smooth. In contrast to Gaussian noises, under the assumption of non-Gaussian noise, for instance, the Poisson process for a state-space model, the sequence of estimated state variables tends to be almost invariant and jump scarcely.
Potentially supposing broader classes of errors, the author regards LASSO (or other sparse regression solvers) as a suitable solver for a linear time series model with parameters that jump in scarce timings. LASSO effectively solves the corresponding difference form (5) even if the number of parameters amounts to more than ten thousand, as is shown in the next section. In this case, the group LASSO or generalized LASSO is applicable. (See Yuan and Lin (2006) and Tibshirani (2011) for detail of the two LASSO.)
Application to Foreign Exchange Markets
The author shows an application in which the method in Section 2 possibly detects several structural changes in a real financial market. The data is the same as Ito et al. (2021). They used monthly nominal data from the Thomson Reuters Datastream on the spot and forward exchange rates for three developed countries (Canada, Japan, and the UK) from May 1990 to July 2015 taken. (See Ito et al.(2021) for detail) They estimated a time-varying VEC model's parameters, supposing them varying like random walks over time. They argue that the estimated degree of market comovement has increased over sample periods. At the same time, the behavior is not monotonous. One can regard the degree as the speed of adjustment if the deviation from the long-run equilibrium exists. Figures 1 and 2 This section aims to show possible breakpoints sharper than Ito et al. (2021) and in what periods they occurred. Ito et al.(2021) suggests that the exchange markets vary over time and that the changes are gradual. Assuming there are some cointegrating relationships, we consider a vector error correction (VEC) model for m-vector time series X t .
∆X t = Γ 1 ∆X t−1 + · · · + Γ k ∆X t−k + Π k X t−k + µ + u t ,(8)
where ∆X t = X t − X t−1 , µ is a vector of intercepts, and u t is a vector of error terms. The
Coefficient matrix Π k represents some long-run relationships among the components of X t 's. Regarding (8) as a linear time series regression (2) and supposing Π k varying over time, we apply our group LASSO technique. At this point, we show a practical procedure applied in this section as follows.
Step 1 First, we estimate a usual VEC model supposing any coefficient time-invariant after some cointegration test.
Step 2 Using its loading matrix Π k = αβ, we specify β matrix, say, β * .
Step 3 Considering Π k time-varying, we estimate ∆Π k,t for each period t.
Step 4 We obtain Π k,t through the cumulated sum of ∆Π k,t 's.
Step 5 Supposing Π k,t = α t β * , we calculate α t , t = 1, · · ·
Step 6 We obtain our degree of comovement from α t 's in the same way as Ito et al.(2021).
We provide here practical information about our estimation. The sample size = 306; the number of variables = 6. According to Ito et al. (2021), we choose k = 2 in (8), the total number of parameters is 11016 when we attempt to obtain time-varying loading matrices to calculate our time-varying degree. We need to try many values of the hyperparameter of the group LASSO model to find some breaks. It takes about 1800 seconds on my Mac mini (Late 2014) with Intel Core i7 (16 GB 1600 MHz DDR3 memory).
Figures 3 and 4 summarize this paper's findings as Figures 1 and 2. Moreover, the results of breakpoint are summarized as follows.
1st Impact The degree is non-zero between May 1990 and July 1994 and attains a local maximum at December 1993.
Concluding Remarks
Here the author provides some remarks. We have shown a method to estimate linear time series models with time-varying parameters (TVP) that jump in very few periods. Using group LASSO allows us to detect breakpoints when addressing multivariate linear models. The robustness relies on not any statistical technique but a simple, sparse technique that guarantees sparse structure parameters. This paper's result does not reflect the Asian financial crisis from July 1997. It could come from our choice of data: Canada, Japan, UK. Ito et al.(2021) also detect two peaks of the changes in the Degree of comovement while the "resolution" is lower than this paper. The two estimates of Ito et al.(2021) and this paper, which estimate is better? It depends on what information the users want. Finally, we show some future tasks: more statistical inference more applications to evaluate our framework's effectiveness. We should study correspondences between our detected breaks and possible actual shocks in practice. We are searching for better forecast performance by focusing on the periods after the last breakpoint by our technique.
Figure 1 :Figure 2 :
12Degree of comovement in foreign exhange rates (1990-2015) (from Ito et al. Changes in the Degree of comovement (1990-2015) (from Ito et al. (2021))
Figure 3 :Figure 4 :
342nd Impact The degree is non-zero between May 2007 and July 2011 and attains a local maximum at August 2008.Notice that the first break possibly occurred before May 1990, and the left peak in the figure 4 possibly reflects the UK pound crisis in September 1992 as well as the UK's rapid economic recovery after the crisis. The right peak in the figure demonstrates Lehman Brothers Holdings Inc.'s bankruptcy on September 15, 2008. Degree of comovement in foreign exhange rates Changes in the Degree of comovement
AppendixOne can regard β 0 as a prior if they are a Bayesian. But, I propose the least square estimate of the following linear regression model to obtain β 0 conveniently with our givens.
Regularized estimation in sparse high-dimensional time series models. S Basu, G Michailidis, The Annals of Statistics. 43Basu, S., Michailidis, G., et al. (2015), "Regularized estimation in sparse high-dimensional time series models," The Annals of Statistics, 43, 1535-1567.
Group LASSO for structural break time series. N H Chan, C Y Yau, R.-M Zhang, Journal of the American Statistical Association. 109Chan, N. H., Yau, C. Y., and Zhang, R.-M. (2014), "Group LASSO for structural break time series," Journal of the American Statistical Association, 109, 590-599.
An Estimation Method for State Space Models using generalized LASSO Techniques. M Ito, Reported in 94th annual conference of WEAIIto, M. (2019), "An Estimation Method for State Space Models using generalized LASSO Techniques," Reported in 94th annual conference of WEAI.
Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach. M Ito, A Noda, T Wada, Mathematics. 9849Ito, M., Noda, A., and Wada, T. (2021), "Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach," Mathematics, 9, 849.
Autoregressive models for gene regulatory network inference: Sparsity, stability and causality issues. G Michailidis, F Buc, Mathematical biosciences. 246Michailidis, G. and d'Alché Buc, F. (2013), "Autoregressive models for gene regulatory network inference: Sparsity, stability and causality issues," Mathematical biosciences, 246, 326-334.
Estimation of parameters in the presence of model misspecification and measurement error. P Swamy, G S Tavlas, S G Hall, G Hondroyiannis, Studies in Nonlinear Dynamics & Econometrics. 14Swamy, P., Tavlas, G. S., Hall, S. G., and Hondroyiannis, G. (2010), "Estimation of parameters in the presence of model misspecification and measurement error," Studies in Nonlinear Dynamics & Econometrics, 14.
Statistical learning with sparsity: the lasso and generalizations. R Tibshirani, M Wainwright, T Hastie, Chapman and Hall/CRCTibshirani, R., Wainwright, M., and Hastie, T. (2015), Statistical learning with sparsity: the lasso and generalizations, Chapman and Hall/CRC.
The solution path of the generalized lasso. R J Tibshirani, Stanford UniversityTibshirani, R. J. (2011), The solution path of the generalized lasso, Stanford University.
Model selection and estimation in regression with grouped variables. M Yuan, Y Lin, Journal of the Royal Statistical Society: Series B (Statistical Methodology). 68Yuan, M. and Lin, Y. (2006), "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68, 49-67.
| [] |
[
"**TITLE** ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR** **EDITORS** A Solar Neighborhood Search for Tidal Debris from ω Centauri's Hypothetical Parent Galaxy",
"**TITLE** ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR** **EDITORS** A Solar Neighborhood Search for Tidal Debris from ω Centauri's Hypothetical Parent Galaxy"
] | [
"Dana I Dinescu \nDepartment of Astronomy\nUniversity of Virginia\n530 McCormick Road22903CharlottesvilleVA\n"
] | [
"Department of Astronomy\nUniversity of Virginia\n530 McCormick Road22903CharlottesvilleVA"
] | [] | Recent stellar population and chemical abundance studies point to an accreted origin of ω Cen. In this light, and given the retrograde, small size orbit of ω Cen, we search for a kinematical signature left by its hypothetical parent galaxy in the Solar neighborhood. We analyze the largest-to-date sample of metal poor stars and we find that, in the metallicity range −2.0 < [Fe/H] ≤ −1.5, a retrograde signature that departs from the characteristics of the inner halo, and that resembles ω Cen's orbit, can be identified. | null | [
"https://arxiv.org/pdf/astro-ph/0112364v1.pdf"
] | 118,231,285 | astro-ph/0112364 | 9d16aa1825df021dbd8587165b95e34b91b47830 |
**TITLE** ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR** **EDITORS** A Solar Neighborhood Search for Tidal Debris from ω Centauri's Hypothetical Parent Galaxy
14 Dec 2001
Dana I Dinescu
Department of Astronomy
University of Virginia
530 McCormick Road22903CharlottesvilleVA
**TITLE** ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR** **EDITORS** A Solar Neighborhood Search for Tidal Debris from ω Centauri's Hypothetical Parent Galaxy
14 Dec 2001
Recent stellar population and chemical abundance studies point to an accreted origin of ω Cen. In this light, and given the retrograde, small size orbit of ω Cen, we search for a kinematical signature left by its hypothetical parent galaxy in the Solar neighborhood. We analyze the largest-to-date sample of metal poor stars and we find that, in the metallicity range −2.0 < [Fe/H] ≤ −1.5, a retrograde signature that departs from the characteristics of the inner halo, and that resembles ω Cen's orbit, can be identified.
Introduction
Recent advances in understanding the nature and origin of the highly unusual globular cluster ω Centauri (see Majewski et al. 2000 for a summary of properties), are due primarily to the following findings: the multiple-peak metallicity distribution seen in the structure of the giant branch (Lee et al. 1999;Pancino et al. 2000;Frinchaboy et al. 2001), the correlation between age and metallicity (e. g., Hughes & Wallerstein 2000, Hilker & Richtler 2000, and the s-process enhanced enrichment in cluster stars compared to halo stars of similar metallicity Vanture, Wallerstein & Brown 1994). These findings suggest that ω Cen underwent self-enrichment with at least three primary enrichment peaks (Pancino et al. 2000, Frinchaboy et al. 2001, over a period of at least 3 Gyr (Hughes & Wallerstein 2000). The s-process heavy-elements are primarily synthesized in low-mass (1.5 to 3.0 M ⊙ ) asymptotic giant branch (AGB) stars (see e.g., Travaglio et al. 1999 and references therein). In order to enrich the cluster in s-process elements, the ejecta from low-mass stars that evolve on timescales of 10 9 years had to be retained by the cluster and incorporated in the next generations of stars. This long and complex star formation history is inconsistent with the cluster originating on its current orbit, which is of low energy and confined to the disk. With a period of only 120 Myr (Dinescu, Girard, & van Altena 1999 -hereafter DGvA), the frequent disk crossings would have certainly swept out all of the intracluster gas soon after its formation, and the result would be a single-metallicity system that would resemble most of the Galactic globular clusters.
Dinescu
It appears thus that ω Cen evolved somewhere away and independently from the Milky Way, in a system that was massive enough to retain ejecta from previous generations of stars, and to undergo multiple episodes of star formation. Its current orbit can be reconciled with the complex star formation history only if it represents a strongly decayed orbit. This, in turn, requires a massive enough system such that dynamical friction was able to drag it to the inner regions of the Galaxy. This system must have also been rather dense in order to survive the tidal field of the Milky Way and continue to loose orbital energy due to dynamical friction down to an orbit with an apocenter of the order of the Solar circle radius. The current mass of ω Cen (5 10 6 M ⊙ ; Meylan et al. 1995) can not generate sufficient dynamical friction to modify its orbit to its current small size (DGvA).
Following these arguments, the debris from the massive putative parent galaxy of ω Cen may be expected to imprint a kinematical feature in large samples of local, metal-poor stars. The purpose of this investigation is to search for such a feature in the kinematically hot halo. We have used three data sets: the largest, kinematically unbiased sample of metal poor stars (∼ 1200) provided by Beers et al. (2000) (hereafter B2000), the sample of globular clusters with measured absolute proper motions (DGvA updated with new distances from Harris 1996, and with a few more clusters; see Dinescu et al. 2001), and a small sample of stars with complete kinematics and abundance measurements for O, Na, Mg, Si, Ca, Ti, Cr, Fe, Ni, Y and Ba (Nissen & Schuster 1997, hereafter NS97).
ω Cen and the Globular Cluster System
DGvA pointed out that ω Cen's orbit is markedly retrograde for its low inclination and low orbital energy, when compared to the orbits of clusters with similar metallicity, horizontal branch morphology and orbital energy (see Fig. 6 and 7 in DGvA). Along with ω Cen, two other clusters were identified to lie in the same region of the orbital parameter space: NGC 362, and NGC 6779 (M56) (DGvA). They both have low-inclination, retrograde orbits, with pericentric distances similar to that of ω Cen (∼ 1 kpc). However, NGC 362 and NGC 6779 have apocentric distances larger than ω Cen: 11 and 14 kpc respectively, compared to 6 kpc for ω Cen.
Some controversy regarding NGC 362's absolute proper motion is apparent: Geffert reported at this conference that his new determination with respect to T ycho stars implies an orbit less similar to that of ω Cen. In our paper (DGvA) we have used an average of two proper-motion measurements for NGC 362. These measurements were: a) calibrated to Hipparcos stars (Odenkirchen et al. 1997), and b) calibrated to stars in the Small Magellanic Cloud (SMC) (Tucholke 1992), to which was applied the absolute proper motion of the SMC as measured by Kroupa & Bastian (1997). While the absolute proper motion determination may remain debatable, another piece of evidence was presented at this conference regarding a possible common origin of NGC 362 and ω Cen. Smith et al. (2000) showed that the [Cu/Fe] abundance of ω Cen's stars is low ([Cu/Fe] = -0.6) and remains remarkably constant with increasing metallicity (their Fig. 9). In field stars, the Cu abundance increases with metallicity (Sneden, Gratton, & Crocker APS Conf. Ser. Style 3 1991, Castro, Porto de Mello, & da Silva 1999 from subsolar values to 0 at solar metallicity. This trend is explained if a substantial contribution to Cu comes from supernovae Type Ia (Matteucci et al. 1993). Cuhna et al. (2001) reported at this conference that NGC 362 too shows the deficit in [Cu/Fe] much like ω Cen's stars, and in contradistinction with clusters NGC 288 and M4. These latest clusters have a similar metallicity with NGC 362, but do not show an underabundance in Cu, when compared to field stars of similar metallicity.
NGC 6779's absolute proper motion determination was based only on the calibration to Hipparcos stars (Odenkirchen et al. 1997). This measurement seems reliable as diffuse X-ray emission detected in the area of the cluster is found to be aligned with the direction of the proper motion (Hopwood et al. 2000). This emission is interpreted as heated interstellar medium in the wake of the cluster as a result of the interaction of the intracluster gas with the halo gas.
Inner Halo Properties as Derived from the Beers et al. Catalog
Chiba characterized the local halo from the analysis of the B2000 catalog. The sample they analyzed comprised stars within 4 kpc of the Sun, and with Galactocentric distance along the plane between 7 and 10 kpc. From their results we summarize here those that are relevant to our investigation. A relatively "pure" sample of halo stars can be found at [Fe/H] ≤ −2.0 (their Fig. 9). Thick disk stars begin to contribute to the overall metal poor population ([Fe/H] < −1), with a fraction that increases with increasing metallicity (their Fig. 10). In the metallicity range −2.0 < [Fe/H] ≤ -1.5, this fraction is ∼ 10%. The mean rotation velocity of halo stars decreases with increasing distance from the Galactic plane: −52 ± 6 km/s/kpc). At a mean distance from the Galactic plane |z| ∼ 0.5 kpc, the rotation velocity is ∼ 55 km/s (their Fig. 4). They have also found that, at a metallicity of -1.7, the mean rotation of the low (|z| < 1 kpc) halo drops to 20 km/s, from 60 km/s at lower metallicities. At higher metallicities, the mean rotation increases with increasing metallicity, as more thick disk stars begin to contribute to the overall population (see their Fig. 3). At the same value of [Fe/H] ∼ −1.7, also find a higher concentration of stars at large eccentricities (e ∼ 0.9) (their Fig. 6). They speculate that stars at [Fe/H] ∼ −1.7 formed from infalling gas rather than from the somewhat more organized material of the inner halo. Interestingly, the mean metallicity of ω Cen is -1.6 (Harris 1996).
Here, we will look in more detail at the stars that produce the drop in the rotation velocity dependence with metallicity. Using the B2000 distances, radial velocities, and absolute proper motions we derive velocities using R 0 = 8 kpc, Θ LSR = 220 km/s , and a peculiar Solar motion (U, V, W ) = (-11.0, 14.0, -7.5) km/s, where U is positive toward the Anticenter. We integrate the orbits in the axi-symmetric analytic potential described in Paczyński (1990), and we derive orbital parameters in the manner described in DGvA.
Rotation Velocity Distributions
In Figure 1 we show the distribution of the rotation velocity Θ (in a cylindrical coordinate system, where the Π component is positive outward from the Galactic Dinescu Figure 1.
Rotation velocity distributions for four metallicity intervals, and for |z| ≤ 1 kpc. The shaded areas represent the observed distributions, while the line shows the distributions derived from a simple kinematical model (see text).
center, Θ is positive toward Galactic rotation, and W is positive toward the North Galactic Pole), for four metallicity groups, and within |z| ≤ 1 kpc. The shaded histograms represent the observed distribution, and the continuous line represents the distribution derived from a simple kinematical model. Velocities in this model were drawn from Gaussian velocity distributions of means and dispersions derived from the B2000 catalog, and including both halo and thick disk populations. Specifically, for [Fe/H] ≤ −2.0 we used < Θ halo >= 50 km/s, σ halo = 106 km/s, and the fraction of thick disk stars f
T D = 0.0; for −2.0 < [Fe/H] ≤ −1.5, < Θ halo >= 60 km/s, σ halo = 106 km/s, f T D = 0.1, < Θ T D >= 190 km/s, σ T D = 50 km/s; for −1.5 < [Fe/H] ≤ −1.2, < Θ halo >= 60 km/s, σ halo = 106 km/s, f T D = 0.3, < Θ T D >= 190 km/s, σ T D = 50 km/s; and for −1.2 < [Fe/H] ≤ −1.0, < Θ halo >= 60 km/s, σ halo = 106 km/s, f T D = 0.5, < Θ T D >= 190 km/s, σ T D = 50 km/s.
The total number of stars in each sample is indicated in the respective panel. The model distributions were drawn for 10000 points, and normalized to the area of the observed distributions. We have chosen only one mean rotation velocity to represent the halo, for each metallicity interval in the following way. For all the stars within |z| ≤ 1 kpc, the median z was determined, and for this value of z, the rotation velocity was read directly from the linear fit derived in Fig. 4 of .
For the "pure" halo sample (top, left panel of Fig. 1) the simple, one-Gaussian model does not represent the data well. The data peak at Θ ∼ 120 km/s -a higher velocity than the value we chose in the model -and they also show more stars at strongly retrograde velocities (Θ ∼ −200 km/s) than the model. A superposition of two or more Gaussians with appropriately chosen mean rotation velocities and relative number of stars with respect to the majority of stars that peak at Θ = 120 km/s, can be envisaged however, to better describe the data. This composite kinematics can be thought of as arising from the fact that some of the stars within |z| ≤ 1 kpc are visiting this region rather than being confined to it. These stars actually reside on orbits of various, mean z larger than 1 kpc, and therefore belong to various populations of accordingly lower mean Θ.
The next two metallicity samples (upper right and lower left panels) show the expected peaks at prograde velocities that are approximately matched by the models, and a second peak at modest retrograde velocities (Θ ∼ −30 km/s). The most metal rich sample (lower right panel) shows a rather good fit to the simple kinematical model, but there are few stars in this sample. The second peak seen at retrograde velocities is intriguing not necessarily because it is not reproduced by the kinematical model, but for the following reason. The sample with −2.0 < [Fe/H] ≤ −1.5 has a minor contamination from the thick disk; therefore, it can be compared in a meaningful way to the "pure" halo sample. They both show a tail toward strongly retrograde velocities -that is not reproduced by our one-Gaussian models -but there is no second peak in the "pure" halo sample. It is thus difficult to explain the second peak at retrograde velocities even in a composite kinematical model of the halo, as sketched above and perhaps seen in the "pure" halo sample. One would have to imagine a significant population that comes from orbits of one particular mean z value that corresponds to Θ = −30 km/s, rather than a population drawn from a mixture of various mean z orbits. In what follows we will focus on two metallicity samples: the "pure" halo [Fe/H] ≤ −2.0, and the sample within −2.0 < [Fe/H] ≤ −1.5, which has a relatively small fraction of thick disk stars (10%).
Orbital Angular Momentum Distributions
We have constructed the orbital angular momentum L z distributions for the [Fe/H] ≤ −2.0 sample and for the −2.0 < [Fe/H] ≤ −1.5 sample, and for stars with maximum excursions above/below the Galactic plane |z max | ≤ 4 kpc, to represent the low halo, and |z max | ≤ 20 kpc for the whole halo. These are shown in Figure 2, where the thin line represents the metal-poor sample (referred to as sample A hereafter), and the thick line the less-metal-poor sample (referred to as sample B hereafter); the range of z max is specified in each panel. The distributions were constructed by passing a moving box of half-width equal to the dispersion in L z : 900 kpc km/s. They were also normalized to the total number of stars, in order to be intercompared. A thick-disk component of ∼ 10% (Section 3) of the total number of stars is expected in sample B, roughly at L z = 8 kpc ×190 km/s = 1520 kpc km/s. For this reason, sample B was normalized to a slightly larger number than the number corresponding to a "pure" halo population. Therefore, the distributions were slightly shifted in L z (up to 100 kpc km/s) in an attempt to best match them, and to keep the sample B distribution slightly under the sample A distribution. For the |z max | ≤ 20 kpc samples (top right panel), the distributions are remarkably similar with the following exceptions. They differ in the thick-disk range (L z = 1500 to 2000 kpc km/s), which was expected, at strongly retrograde orbits (L z ∼ −1800 kpc km/s), but there are few stars here, and -most significantly -at modest retrograde orbits L z ∼ −400 kpc km/s. At this latter value of L z , sample B shows an excess of stars, that produce a shoulder in the distribution when compared with sample A. For the low halo (|z max | ≤ 4 kpc; top left panel in Fig. 2), sample B again shows the shoulder, or the excess of stars at L z ∼ −400, when compared to sample A. Sample B also seems to have a slightly sharper peak than sample A.
The bottom panels of Fig. 2 show -for the low halo -the observed distributions for sample A (left panel) and sample B (right panel) compared to a model of the halo (shaded curves). This model was designed to describe the halo only, and to input the kinematics only and not the spatial distribution. Thus, for each star in the appropriate metallicity sample, a velocity drawn from a Gaussian distribution was assigned, while preserving the positional information from the B2000 catalog. Each of the velocity components were drawn from Gaussian distributions specific for the halo. We have used (< Π >, < Θ >, <W>) = (0, 60, 0) km/s, and (σ Π , σ Θ , σ W ) = (141, 106, 94) km/s. Ten sets of halo velocity components were assigned to each star, and for each the orbits were integrated in order to obtain the model distributions. The models agree well with the data in the prograde regime; however, they fail to reproduce the tail toward retrograde orbits, likely due to the use of a single value of the mean Θ. They can not reproduce the shoulder structure seen in sample B. Therefore, preserving the current spatial distribution of the stars and statistically assigning halo-like velocities will not produce the excess of stars at L z ∼ −400 kpc km/s. The conclusion is that the shoulder is due to stars on particular orbits; the excess of such orbits is not seen in sample A. For Solar neighborhood stars, this L z excess corresponds to that seen in the rotation velocity distribution at modest retrograde velocities (Section 3.1).
Orbit Characteristics
We look now at the detailed characteristics of the orbits in samples A and B. In Figure 3, top panels, we show the orbital energy E as a function of L z . All left panels represent sample A, while the right panels sample B. The number of stars is indicated in each panel. Open symbols represent all stars in B2000, while highlighted (filled) symbols show those classified as RR Lyrae variables in B2000. For sample B, an almost vertical structure can be seen at L z ∼ −400 kpc km/s and orbital energy −1.2 10 5 ≤ E < −0.5 10 5 (km/s) 2 . This structure is seen in the RR Lyrae population as well. We divide now each metallicity sample into two groups: prograde (L z > 0) and retrograde (L z ≤ 0) orbits. For each 5 (sample B). The filled circles represent the RR Lyrae variables in the B2000 catalog. The top panels show the orbital energy E as a function of orbital angular momentum. The rest of the panels show the orbital inclination as a function of eccentricity for prograde orbits (L z > 0; middle panels), and for retrograde orbits (L z ≤ 0; bottom panels). group we plot the orbit inclination Ψ as a function of orbital eccentricity e. The middle panels of Fig. 3 show the prograde group, and the bottom panels the retrograde group. A higher density of stars, reproduced as well in the RR Lyrae population, can be seen clumping at e ∼ 0.85 and Ψ ∼ 25 • , for the retrograde group of the B sample of stars.
By selecting stars with high eccentricities, we safely discard the poorly known fraction of stars that have rotational support, be it halo or thick disk. Therefore, in each metallicity sample, one would expect the same number of highly eccentric stars in the prograde sample, as in the retrograde sample, in a completely pressure-supported halo. Selecting stars with e > 0.8 we find 33 stars in the prograde sample A, and 37 in the retrograde sample A, while in sample B we find 45 stars in the prograde group and 70 in the retrograde group. The 2.3σ excess of highly eccentric stars in the retrograde group of sample B resides at moderate to very low orbit inclinations. Interestingly enough, the RR Lyrae population in sample B follows the same pattern at e > 0.8: there are 15 stars in the prograde group, and 30 stars in the retrograde group.
We plot now all of the three integrals of motion, L z , E, and total angular momentum L in Figure 4. In addition to the B2000 sample, we use the NS97 sample of stars with chemical abundance measurements, and the globular cluster sample (DGvA). Distances and absolute proper motions for the NS97 sample are from the Hipparcos catalog (ESA 1997), while the radial velocities are from the SIMBAD database. The highlighted (filled) symbols show objects of special interest. Among the globular clusters, ω Cen, NGC 362, and NGC 6779 are highlighted, as they are hypothesized to belong to the same parent galaxy (Section 2). One star in the NS97 sample, namely HD 106038, is particularly interesting, and it is highlighted in Fig. 4 4 and 5). At a metallicity of -1.26, HD 106038 displays canonical α enhancement for metal-poor, halo stars produced in a type II supernovae-dominated environment (NS97). HD 106038's abundance patterns, in particular the enhanced s-process elements are unusual among halo stars (NS97); however, they agree very well with the abundance patterns of ω Cen stars shown by Smith et al. (2000). HD 106038 is a main sequence star, with no radial-velocity evidence of a companion likely to pollute the star with AGB ejecta (NS97).
In the plot of E as a function of L z , the clusters ω Cen, NGC 362, and NGC 6779 and HD 106038 lie in the retrograde region where the excess of stars was found in sample B of the B2000 catalog (Fig. 2, and 3). The plots of L versus E, and L versus L z show that, indeed the three clusters and HD 106038 lie in the same volume of the phase space. Guided by the distribution of the three clusters, we define a box in the L-L z plot (shaded area in Fig. 4) aimed at selecting candidate objects in the NS97 and B2000 data sets. The corresponding area is overplotted in the E-L z and L-E plots. It is interesting to note that, although the box was defined by the L-L z pair to cover a restricted region, in E, this region covers a large range of values. Figure 5 shows the orbit inclination as a function of eccentricity for stars and clusters in the L-L z region defined above, and for stars in a symmetrically defined region at prograde orbits Dinescu Figure 4.
Orbital parameters for Beers et al. (2000) and Nissen & Schuster (1997) stars, and for globular clusters (DGvA). Candidate stars from ω Cen's host galaxy are chosen to lie in the shaded zone defined in the L-L z plot (bottom panel). The same area is also represented in the E-L z plot (middle panel) and the L-E plot (top panel). This area is defined by the three globular clusters NGC 362, ω Cen and NGC 6779 (see text). In the L-L z plot, a similar zone in the prograde domain is marked. This region has the same area, and is symmetrically located to the one in the retrograde domain. Figure 5.
Orbit inclination as a function of eccentricity for the stars and clusters selected in the prograde and retrograde areas defined in Fig. 4. Symbols are as described in Fig. 4. The highlighted (filled) symbols represent objects of particular interest, as follows. The star symbols are the globular clusters ω Cen, NGC 362, and NGC 6779, the triangle is HD 106038, and the square is V 716 Oph (see text).
(shown by the contour in the bottom panel of Fig 4). There is not a significantly larger number of stars at e > 0.8 in the retrograde group (56 stars) than in the prograde group (50 stars) in this latter plot. This is because we have included all stars from B2000. However, selecting RR Lyrae with e > 0.8, we obtain 15 stars in the prograde box, and 29 in the retrograde box. Also, in the retrograde domain, a structure that starts from very high eccentricities (e ≥ 0.9) and very low inclinations (Ψ ∼ 8 − 10 • ) toward e ∼ 0.8 and Ψ ∼ 24 • is apparent. This structure is also seen in the RR Lyrae population (Fig. 3, bottom right panel). Although the region of e > 0.8 is well populated in the prograde domainone globular cluster, and two NS97 stars are found here -there is no structure apparent here, and there are very few stars at e ≥ 0.9.
RR Lyrae in Beers et al. Catalog
We have seen that the excess population at Lz ∼ −400 kpc km/s, e ∼ 0.85, and Ψ ∼ 20 • is enhanced when one considers the RR Lyrae stars in B2000 (Section 4). In Figure 6 we show the periods as a function of metallicity for RRab and RRc stars in B2000, and in ω Cen, for comparison. The periods and the RR type were taken from the General Catalogue of Variable Stars 4th edition (Kholopov et al. 1998, hereafter GCVS) for the B2000 stars. For ω Cen, periods, RR type and metallicities (based on the hk index) were taken from Rey et al. (2000). The top panel shows the RRab (filled circles) and the RRc (open triangles) stars in the prograde box defined in Fig. 4. The middle panel shows those in the retrograde box, and the bottom panel those in ω Cen.
The presence of RRab variables with P ≥ 0.8 days, and RRc variables with P ≥ 0.45 days is characteristic of ω Cen (Fig. 6 with data from Rey et al. 2000, see also, Clement & Rowe 2000. These long period RR Lyrae stars are seen in a larger abundance than in ω Cen only in two metal rich, bulge clusters (NGC 6388 and NGC 6441, Pritzl et al. 2000). There are no such long period RR Lyrae stars in either the prograde or the retrograde sample.
The prograde sample shows a rather random distribution, with periodson the mean -lower than those in ω Cen. The retrograde sample distinctly shows a high concentration of stars at P ∼ 0.5 days and [Fe/H] ∼ −1.5, that is not seen in either the prograde sample or the ω Cen sample. It is predominantly these latter stars that comprise the excess of RR Lyrae stars in the retrograde sample, when compared to the prograde sample ( Fig. 6 and Fig. 3, also Section 4). The information from Fig. 6 is however insufficient to either support or reject confidently the notion that part of the population of the RR Lyrae in the retrograde sample may resemble that in ω Cen.
Two stars that were classified as RR Lyrae in B2000, are in fact W Virginis variables of short periods (P = 1.1 and 1.3 days), also known as BL Herculis variables, according to GCVS. This type of variable is found in globular clusters. ω Cen is the Galactic cluster with the largest population of BL Her objects: it has 5 such stars according to Nemec, Linnell-Nemec, & Lutz (1994). More recently, Kaluzny et al. (1997) have found three new BL Her candidates in ω Cen. Interestingly, one of the two BL Her stars in B2000, namely V 716 Oph, resides in the retrograde region defined in Section 4, has an eccentricity e = 0.82, an inclination Ψ = 13 • , and a metallicity [Fe/H] = -1.55. V 716 Oph is highlighted in the e − Ψ plot of Fig. 5 with a filled square symbol. The second BL Her star in B2000 is XX Vir. It has a metallicity value [Fe/H] = -2.4, that is much lower than the metal-poor limit of stars in ω Cen.
A Simple Disruption Model
The stars and globular clusters that have similar integrals of motion as ω Cen, tend to have larger eccentricities than ω Cen (Fig. 4, Section 4). Obviously, these candidates will have larger apocentric radii than ω Cen at a fixed pericenter radius, in other words, they reside on orbits that are slightly more energetic than ω Cen's. Under the hypothesis that these candidates belong to ω Cen or the disrupted system that once contained ω Cen, they ought to lie on trailing streams if they are on orbits slightly more energetic than ω Cen's (e. g., Johnston 1998). We have used the tidal disruption model developed by Johnston (1998) in order to see whether tidal debris from a system that now has the orbit of ω Cen can attain orbits such as those of our candidates. We note that the Johnston (1998) model was developed for the disruption of Galactic satellites that reside mostly in the outer Galaxy; therefore, quantitatively it may not be an accurate description of the event. However, here we will present only a qualitative inspection.
We start with a system of M = 5 10 6 M ⊙ that has the orbit of ω Cen. We integrate back in time for 1 Gyr; at each pericenter passage, the system is assumed to have lost 30% of it's mass. In Figure 7 we show the spatial distribution in the Galactic plane (left panel) and perpendicular to the Galactic plane (right panel) of the B2000 stars (dots), ω Cen's orbit (continuous line), and two trailing streams (grey bands). The trailing streams correspond to pericenter passage n p = -11 (light gray band; 918 Myr ago), and to pericenter passage n p = -10 (dark gray band; 835 Myr ago). The pericenter passage n p = 0 corresponds to the passage closest in time to present time. At n p = -11, the system had a total mass of 3.6 10 8 M ⊙ . The spatial location of the trails shows that indeed, in the Solar neighborhood we can expect to find debris from this system. These tidal streams are distributed along orbits that have the shape of the orbits of our candidates: high eccentricity and low orbital inclination. We also note that Dinescu Figure 7.
The spatial distribution of the B2000 data sample (dots), ω Cen's orbit (thin dark line), and two trailing tidal streams: pericenter passage = -11 (918 Myr ago; 0 is the pericenter passage closest in time to present; light gray band), and pericenter passage = -10 (835 Myr ago; dark gray band).
all of the corresponding leading (lower energy) tidal streams are located in and close to the bulge.
A detailed N-body simulation of the disruption event has yet to be explored in order to better understand the process, and to be able to quantitatively describe it. For instance, the mass of tidal debris expected in the Solar neighborhood would be particularly useful in order to correlate with the excess of stars that we see in a given domain of phase space (Section 4). Similarly, the predicted kinematics of tidal debris, when compared to that of candidate stars, can help demonstrate how viable, and under what conditions, the accreted scenario for ω Cen is.
Summary
We have shown that a distinct population of stars with a metallicity range that inludes the mean metallicity of ω Cen, and with ω Cen-like phase-space characteristics emerges from the B2000 data. Choosing a metallicity and orbitalparameter range (Section 4) such that we maximize the "signal" of this population with respect to the "noise" of the halo, we obtain an excess population at 2.3-σ level. By considering the RR Lyrae stars in B2000, we also see this population. We find that the excess RR Lyrae population is predominantly of RRab type, with periods of 0.5 days, and [Fe/H] ∼ −1.5.
The candidates to have been torn from the system that once contained/was ω Cen have one main orbit property: they have a larger eccentricity (e ∼ 0.8) (i. e. orbital energy) than that of ω Cen (e = 0.67). Using the disruption model developed by Johnston (1998) for the orbit of ω Cen, we find that trailing tidal debris with orbit characteristics of those of the candidates are found in the Solar neighborhood.
DinescuFigure 2 .
2Orbital angular momentum distributions. The thin line represents the sample with [Fe/H] ≤ −2.0 (sample A), while the thick line that with −2.0 < [Fe/H] ≤ −1.5 (sample B). The shaded areas represent the distributions as derived from a kinematical model (see text). The top left panel, and the bottom panels show stars that are more confined to the Galactic plane than the stars in the top right panel.
DinescuFigure 3 .
3Orbital parameters for Beers et al. (2000) stars. All left side panels show stars with [Fe/H] ≤ −2.0 (sample A), while the right side ones show those with −2.0 < [Fe/H] ≤ −1.
with a filled triangle. HD 106038 is overabundant in Si ([Si/Fe] = 0.57), Ni ([Ni/Fe] = 0.18), and in s-process elements Y ([Y/Fe = 0.49) and Ba ([Ba/Fe] = 0.49), when compared to halo stars of similar metallicity (NS97, their Fig
DinescuFigure 6 .
6The distribution of RR Lyrae periods with metallicities, for the prograde and retrograde regions defined in Fig. 4 (top and middle panel respectively), and for stars in ω Cen (bottom panel).
Astronomical Institute of the Romanian Academy, Str. Cutitul de Argint 5, RO-75212, Bucharest 28, Romania
We also find that HD 106038, a single, main sequence star with a chemical abundance pattern very similar to that in ω Cen stars, in particular enhanced s-process elements (NS97), has ω Cen-like orbital properties. Similarly, V716 Oph in B2000 (a BL Her-type variable found in globular clusters of which ω Cen is most abundant) has ω Cen-like orbital properties, and a metallicity close to the mean metallicity of ω Cen.We identify two globular clusters as candidates for belonging to the system that once contained/was ω Cen, NGC 362 and NGC 6779. The more metal rich cluster, NGC 362 shows a deficiency in [Cu/Fe] when compared to globular clusters of similar metallicity, a deficiency seen so far only in ω Cen stars.Acknowledgments. I am grateful to Márcio Catelan for his suggestions regarding the RR Lyrae stars, and to both Márcio Catelan and Terry Girard for numerous helpful discussions concerning this work. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
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| [] |
[
"Highly Stretchable MoS 2 Kirigami",
"Highly Stretchable MoS 2 Kirigami"
] | [
"Paul Z Hanakata ",
"David K Campbell ",
"Zenan Qi ",
"Harold S Park ",
"\nDepartment of Physics\nDepartment of Mechanical Engineering\nBoston University\n02215BostonMAUSA\n",
"\nBoston University\n02215BostonMAUSA\n"
] | [
"Department of Physics\nDepartment of Mechanical Engineering\nBoston University\n02215BostonMAUSA",
"Boston University\n02215BostonMAUSA"
] | [] | We report the results of classical molecular dynamics simulations focused on studying the mechanical properties of MoS2 kirigami. Several different kirigami structures were studied based upon two simple non-dimensional parameters, which are related to the density of cuts, as well as the ratio of the overlapping cut length to the nanoribbon length. Our key finding is significant enhancements in tensile yield (by a factor of four) and fracture strains (by a factor of six) as compared to pristine MoS2 nanoribbons. These results in conjunction with recent results on graphene suggest that the kirigami approach may be a generally useful one for enhancing the ductility of two-dimensional nanomaterials. | 10.1039/c5nr06431g | [
"https://arxiv.org/pdf/1511.02961v3.pdf"
] | 35,636,264 | 1511.02961 | c85173664ec6db355aa093aa33ed6708094b7de6 |
Highly Stretchable MoS 2 Kirigami
17 Dec 2015
Paul Z Hanakata
David K Campbell
Zenan Qi
Harold S Park
Department of Physics
Department of Mechanical Engineering
Boston University
02215BostonMAUSA
Boston University
02215BostonMAUSA
Highly Stretchable MoS 2 Kirigami
17 Dec 2015(Dated: December 18, 2015)
We report the results of classical molecular dynamics simulations focused on studying the mechanical properties of MoS2 kirigami. Several different kirigami structures were studied based upon two simple non-dimensional parameters, which are related to the density of cuts, as well as the ratio of the overlapping cut length to the nanoribbon length. Our key finding is significant enhancements in tensile yield (by a factor of four) and fracture strains (by a factor of six) as compared to pristine MoS2 nanoribbons. These results in conjunction with recent results on graphene suggest that the kirigami approach may be a generally useful one for enhancing the ductility of two-dimensional nanomaterials.
Molybdenum disulfide (MoS 2 ) has been intensely studied in recent years as an alternative two-dimensional (2D) material to graphene. This interest has arisen in large part because (i) MoS 2 exhibits a direct band gap of nearly 2 eV in monolayer form which is suitable for photovoltaics [1]; and (ii) it has recently been explored for many potential applications, ranging from energy storage to valleytonics [2][3][4][5].
The mechanical properties of MoS 2 have also been explored recently, through both experimental [6][7][8] and theoretical methods [9][10][11][12]. That MoS 2 has been reported experimentally to be more ductile than graphene [8] naturally raises the critical issue of developing new approaches to enhancing the ductility of 2D materials.
One approach that has recently been proposed towards this end is in utilizing concepts of kirigami, the Japanese technique of paper cutting, in which cutting is used to change the morphology of a structure. This approach has traditionally been applied to bulk materials and recently to micro-scale materials [13][14][15], though recent experimental [16] and theoretical [17] works have shown the benefits of kirigami for the stretchability of graphene.
Our objective in the present work is to build upon previous successes in applying kirigami concepts to graphene [17] to investigate their effectiveness in enhancing the ductility of a different 2D material, MoS 2 , which is structurally more complex than monolayer graphene due to its three-layer structure involving multiple atom types. We accomplish this using classical molecular dynamics (MD) with a recently developed Stillinger-Weber potential [18]. We find that kirigami can substantially enhance the yield and fracture strains of monolayer MoS 2 , with increases that exceed those previously seen in monolayer graphene [17].
We performed MD simulations using the Sandiadeveloped open source code LAMMPS [19,20] using the Stillinger-Weber potential for MoS 2 of Jiang [18]. All simulations were performed on single-layer MoS 2 sheets. Of relevance to the results in this work, we note that while the Stillinger-Weber potential does not have a term explicitly devoted to rotations, it does contain two and three-body terms including angular dependencies, which is important for out-of-plane deformations. Furthermore, the Stillinger-Weber potential of Jiang [18] was fit to the phonon spectrum of single-layer MoS 2 , which includes both in and out-of-plane vibrational motions. As a result, the Stillinger-Weber potential should do a reasonable job of capturing out-of-plane deformations that involve angle changes, such as rotations. The MoS 2 kirigami was made by cutting an MoS 2 nanoribbon, which had free edges without additional surface treatment or termination. A schematic view of the kirigami structure and the relevant geometric parameters is shown in Fig. 1. The key geometric parameters are the nanoribbon length L 0 , the width b, the height of each interior cut w, the width of each interior cut c, and the distance between successive cuts d. L 0 ∼ 450Å, width b ∼ 100Å, height of each interior cut w ∼ 70Å, width of each interior cut c ∼ 11Å, and distance between successive cuts d ∼ 55Å is shown in Fig. 1. The MD simulations were performed as follows. The kirigami was first relaxed for 200 ps within the NVT (constant number of atoms N , volume V and temperature T ) ensemble at low temperature (4.2 K), while non-periodic boundary conditions were used in all three directions. The kirigami was subsequently deformed in tension by applying uniform displacement loading on both ends, such that the kirigami was pulled apart until fracture occurred. We note that in actual applications, the MoS 2 kirigami will likely lie on a substrate, and thus adhesive interactions with the substrate may impact the deformation characteristics. In the present work, we focus on the intrinsic stretchability of the MoS 2 kirigami while leaving the interactions with a substrate for future work.
In addition, we simulated MoS 2 sheets (defined as monolayer MoS 2 with periodic boundary conditions in the plane) and pristine nanoribbons with no cuts for comparative purposes. The calculated fracture strains f , fracture stresses σ 3D f , and Young's modulus Y 3D are tabulated in Table I. The results are in reasonably good agreement with the experimental and firstprinciples studies of MoS 2 monolayer sheets [6,8]. [21] In Figure 2 (a), we plot a representative stress-strain curve of MoS 2 kirigami. For this, and the subsequent discussion, we introduce two non-dimensional geometric parameters α = (w − 0.5b)/L 0 and β = (0.5d − c)/L 0 , which were also previously used to describe graphene kirigami [17]. α represents the ratio of the overlapping cut length to the nanoribbon length, while β represents the ratio of overlapping width to the nanoribbon length. Put another way, α describes the geometry orthogonal to the loading direction, while β describes the geometry parallel to the loading direction. Figure 2(a) shows the stress-strain for the specific choices of α = 0.0866, and β = 0.0375, which were obtained by choosing b=101.312Å, L 0 =438.693Å, w=88.648Å, c=10.967Å, and d=54.837Å. In contrast, Figure 2(b) shows the change in the stress-strain response if β = 0.0375 is kept constant while α changes. This is achieved by changing w while keeping other geometric parameters constant. We also note that the 2D stress was calculated as stress times simulation box size perpendicular to the plane σ ×t to remove any issues in calculating the thickness [10], where the stress was obtained using the virial theorem, as is done in LAMMPS.
It can be seen that there are generally three major stages of deformation for the kirigami, as separated by the dashed lines in Fig. 2(a). In the first stage (region I), the deformation occurs via elastic bond stretching, and neither flipping nor rotation of the monolayer MoS 2 sheet is observed as shown in Fig. 3. In previous work, it was found that graphene kirigami rotates and flips in the first stage instead of bond stretching [17]. This does not occur for kirigami in MoS 2 in this first stage because the bending modulus of MoS 2 is nearly seven times higher than that of graphene [10].
In the second stage (region II), for tensile strains ( ) exceeding about 10%, further strain hardening occurs. Kirigami patterning allows the MoS 2 monolayer to exhibit out-of-plane deflections, as shown in Fig. 3, which allows the MoS 2 monolayer to undergo additional tensile deformation, which is in contrast to the brittle fracture
Top View
Side View observed for the pristine nanoribbon immediately following the initial yielding event as shown in Fig. 2(b). Furthermore, the out-of-plane deflections cause the slope of the stress-strain curve in region II to be smaller than that in region I. This is because of the change in deformation mechanism from purely elastic stretching of bonds in region I, to a combination of stretching and out of plane buckling in region II. Local bond breaking near the edges starts to occur at the tensile strain of = 35%. The occurrence of bond breaking is usually defined as the yield point, and signifies the demarkation between regions II and III. This local bond breaking occurs due to the concentrated stress at the edges connecting each slab, as previously observed in graphene kirigami [17]. At this stage, each kirigami unit is held by a small connecting ribbon which allows the monolayer to be almost foldable. Fig. 3 (stages 1 to 3) shows how the inner cut surface area having initial area w×c and the height of the monolayer (largest out-ofplane distance between S atoms) can change significantly during the tensile elongation.
In the final stage, after more than 62.5% tensile strain, fracture and thus failure of the kirigami nanoribbon is observed. Unlike the pristine nanoribbon, the yield point can differ substantially from the fracture strain and the difference increases with increasing cut-overlap, which was described previously as shown in Fig. 2(b). Thus, it is important to quantify the yield point of the kirigami as it defines the beginning of the irreversible deformation regime. Note that these regions vary depending on the kirigami structure as shown in Fig. 2(b). We also show, in Fig. 4, the von Mises stress distribution prior to fracture at a tensile strain of 62%. In Fig. 4, the stress values were scaled between 0 and 1, and the stress distributions in the top S layer and single Mo layer were plotted separately for ease of viewing as MoS 2 has a tri-layer structure. We found that the largest stresses are concentrated near the edges of the each kirigami unit cell similar to that previously observed in graphene kirigami [17].
Having established the general deformation characteristics for MoS 2 kirigami, we now discuss how the yield and failure characteristics are dependent on the specific kirigami geometry. We discuss the yield and fracture stresses and strains in terms of the two geometric parameters α and β that were previously defined. The yield strain as a function of α is shown in Fig. 5(a), while the yield stress as a function of α is shown in Fig. 5(b). In these, and all subsequent figures, the stresses and strains are normalized by those for pristine MoS 2 nanoribbons of the same width such that the effect of the kirigami parameters can be directly quantified. As shown in Fig. 5, the MoS 2 kirigami becomes significantly more ductile for α > 0, where the zigzag chirality reaches a yield strain that is about a factor of 6 larger than the pristine nanoribbon. In contrast, Fig. 5 (b) shows that the yield stress for kirigami correspondingly decreases dramatically for increasing α. We also note that the kirigami patterning appears to have a similar effect on the ductility of zigzag and armchair MoS 2 kirigami (shown in Fig. 5(a)) as the fracture strain and bending modulus of MoS 2 monolayer sheet in zigzag and armchair direction are similar [10,18].
The increased ductility occurs because α = 0 corresponds to the case when the edge and interior cuts begin to overlap. Increasing α above zero corresponds to when the edge and interior cuts do overlap, and thus it is clear that increasing the overlap increases the ductility of the MoS 2 kirigami. In contrast, the yield stress is higher for smaller α because for negative α, the edge and interior cuts do not overlap, and thus the deformation of the kirigami more closely resembles that of the cut-free nanoribbon.
In addition to the results of α, the effect of β on the kirigami ductility is shown in Figs. 6(a) and 6(b). Specifically, β is varied by changing d while keeping other geometric parameters constant. For both the yield stress and strain, β does impact the yield stress and strain. Increasing β corresponds to an increase in the overlapping region width, which thus results in a smaller yield strain, and increased yield stress as compared to a pristine nanoribbon. For β ≥ 0.03, we do not observe large differences between the AC and ZZ behavior in the case of varying β because increasing β (or decreasing the cut density) makes the kirigami more pristine, leading to similar values of fracture stress and strain in the AC or ZZ direction (see Table I). Our results suggest that the failure strain can be maximized by increasing the overlapping cut (increasing α) and increasing density of the cuts (decreasing β).
Recently, Guo et al. showed stretchability of metal electrodes can be enhanced by creating geometries similar to the ones illustrated in Fig. 1 [15]. Adopting the geometric ratios determining fracture strain described in Ref. [15], we found similar trends: the fracture strain increases with decreasing (b−w) c and increases with increasing b d . It is interesting to see that a similar trend is operant at a different length scale (an atomically-thin monolayer in this work as compared to a ≈40 nm thin film in the work of Guo et al.), and for a different material system (MoS 2 in this work, nanocrystalline gold in the work of Guo et al.), which suggests that the fracture strain in patterned membranes can be described entirely by geometric parameters.
It is also interesting to note that the yield and fracture strain enhancements shown in Fig. 5(a) exceed those previously reported for monolayer graphene kirigami [17]. The main reason for this is that the failure strain for the normalizing constant, that of a pristine nanoribbon of the same width, is smaller for MoS 2 . As shown in Table I, this value is about 13%, whereas the value for a pristine graphene nanoribbon was found to be closer to 30% [17]. However, the largest failure strain for the MoS 2 and graphene kirigami were found to be around 65%, so the overall failure strains for graphene and MoS 2 kirigami appear to reach similar values.
In addition to the yield and fracture behavior, we also discuss the elastic properties, or Young's modulus. For the kirigami system, we expect the Young's modulus to decrease with increasing width of the cut w due to edge effects [9]. Fig. 7 shows the dependence of Young's modulus with effective width b eff = b − w.
As can be seen for both armchair and zigzag orientations, the modulus decreases nonlinearly with decreasing effective width, reaching a value that is nearly 200 times smaller than the corresponding bulk value for the smallest effective width value we examined. Furthermore, the trend of the decrease differs from that previously seen in graphene nanoribbons based on first principles calculations [22] and MoS 2 nanoribbons based on atomistic simulations [9], where a significantly more gradual decrease in stiffness was observed. This is due to the fact that for a given nanoribbon width b, the kirigami has significantly more edge area than a nanoribbon, leading to significant decreases in elastic stiffness even for effective widths b eff that are close to the corresponding nanoribbon width. Before concluding, we note that we have used the more recent Stillinger-Weber (SW15) potential of Jiang [18] rather than the earlier SW potential also developed by Jiang and co-workers [9] (SW13). This is because in comparing the tensile stress-strain curves, the SW15 potential more closely captured the trends observed in DFT calculations [8]. We have also performed simulations of many kirigamis, nanoribbons, and monolayer sheets using the old SW potential. We have found qualitatively similar results with the very important difference that the SW13 potential predicts a tensile phase transition in pristine nanoribbon and monolayer sheet [12] that is not observed in the SW15 potential [18]. A comparison of the tensile stress-strain curve for monolayer MoS 2 is shown in Fig. 8 for the potentials of Jiang (SW15) [18], and Jiang et al. (SW13) [9].
In summary, we have applied classical molecular dynamics simulations to demonstrate that the kirigami pat- MoS2 sheet under tensile loading along the armchair direction using two different SW potentials. The newer SW potential [18] matches better with the trends observed in DFT calculations [8] than the first SW potential of Jiang et al. [9]. No phase transition is observed with the more recent SW potential of Jiang [18]. For SW13, breaking of bonds between the Mo and S layers occur at ∼0.2 and ∼0.3 as observed in Ref. [12] terning approach can be used to significantly enhance the tensile ductility of monolayer MoS 2 , despite the much higher bending modulus and rather more complex trilayer structure of MoS 2 compared to graphene. The resulting enhancements in tensile ductility are found to exceed those previously reported for graphene [17]. These results suggest that kirigami may be a broadly applicable technique for increasing the tensile ductility of twodimensional materials generally, and for opening up the possibility of stretchable electronics and photovoltaics using monolayer MoS 2 . P.Z.H. and Z.Q. acknowledge the support of the Physics and Mechanical Engineering Departments at Boston University. D.K.C. is grateful for the hospitality of the Aspen Center for Physics which is supported by NSF Grant #PHY-1066293, and of the International Institute for Physics of the Federal University of Rio Grande do Norte, in Natal, Brazil, where some of this work was completed. * [email protected]
FIG. 1 .
1(Color online) Schematic of the MoS2 kirigami, with key geometric parameters labeled. The kirigami is deformed via tensile displacement loading that is applied at the two ends in the direction indicated by the arrows. Top image represents a top view of the kirigami.
FIG. 2 .
2We considered kirigami for both zig-zag (ZZ) and armchair (AC) edges. A representative AC MoS 2 kirigami consisting a number of N ∼ 12, 000 atoms with a nanoribbon length arXiv:1511.02961v3 [cond-mat.mtrl-sci] (Color online) Stress-strain curves of AC MoS2 kirigami, where the 2D stress was calculated as the stress σ times simulation box t. (a) Stress-strain curve for constant α = 0.0866, β = 0.0375. (b) Stress-strain curve for AC kirigami keeping β = 0.0375 constant and varying α. Note the brittle fracture of the pristine MoS2 nanoribbon. In general, the strain in region III increases substantially for α > 0.
FIG. 3 .
3Side and top views of kirigami during deformation. FIG. 4. Von Mises stress prior to the fracture at a tensile strain of 62% in (a) Mo layer and (b) top S layer of kirigami inFig. 3. We plot the stress distribution layer by layer to give a clear picture of the stress distribution. The von Mises stress were scaled between 0 and 1.
FIG. 5 .
5(Color online) (a) Influence of α on yield and fracture strain for zigzag (ZZ) and armchair (AC) MoS2 kirigami, with constant β = 0.0375 for AC and constant β = 0.0417 for ZZ. (b) Influence of α on yield and fracture stress for zigzag (ZZ) and armchair (AC) MoS2 kirigami. Data are normalized by MoS2 nanoribbon results with the same width.
FIG. 6 .
6(Color online) Influence of β on the kirigami yield and fracture strain (a) and stress (b), with constant α = 0.0186 for AC and constant α = 0.0157 for ZZ. Data are normalized by MoS2 nanoribbon results with the same width.
FIG. 7 .
72D Young's modulus E 2D of armchair (AC) and zigzag (ZZ) kirigami, pristine nanoribbons (PNR), and sheets. Inset shows E 2D of kirigami normalized by PNRs. The fitting dashed line (colored blue) is given as a guide to the eye.
FIG. 8 .
8(Color online) Stress-strain curve of a monolayer
[ 1 ]
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TABLE I .
IComparison of mechanical properties of MoS2 sheets and pristine nanoribbons in the armchair (AC) and zigzag (ZZ) direction.System
f
σ 3D
f (GPa)
Y 3D (GPa)
Sheet (AC)
0.178
16.8
154.0
Sheet (ZZ)
0.175
15.6
150.7
NR (AC)
0.130
14.6
145.8
NR (ZZ)
0.129
13.6
130.0
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| [] |
[
"p − air production cross-section and uncorrelated mini-jets processes in pp-scattering",
"p − air production cross-section and uncorrelated mini-jets processes in pp-scattering"
] | [
"D A Fagundes [email protected]†[email protected]‡[email protected]§[email protected]¶[email protected] ",
"A Grau ",
"G Pancheri ",
"Y N Srivastava ",
"O Shekhovtsova ",
"\nDepartamento de Física Teórica y del Cosmos\nInstituto de Física Gleb Wataghin\nUniversidade Estadual de Campinas\nUNICAMP\n13083-859Campinas SPBrazil\n",
"\nUniversidad de Granada\n18071GranadaSpain\n",
"\nPhysics Department\nINFN Frascati National Laboratories\nVia E. Fermi 4000444Italy\n",
"\nKharkov Institute of Physics and Technology\nInstitute of Nuclear Physics\nUniversity of Perugia\n61108, Akademicheskaya,1PAN ul. Radzikowskiego 15206123, 31-342Perugia, Kharkov, KrakowItaly, Ukraine, Poland\n"
] | [
"Departamento de Física Teórica y del Cosmos\nInstituto de Física Gleb Wataghin\nUniversidade Estadual de Campinas\nUNICAMP\n13083-859Campinas SPBrazil",
"Universidad de Granada\n18071GranadaSpain",
"Physics Department\nINFN Frascati National Laboratories\nVia E. Fermi 4000444Italy",
"Kharkov Institute of Physics and Technology\nInstitute of Nuclear Physics\nUniversity of Perugia\n61108, Akademicheskaya,1PAN ul. Radzikowskiego 15206123, 31-342Perugia, Kharkov, KrakowItaly, Ukraine, Poland"
] | [] | For the p − air production cross-section, we use a Glauber formalism which inputs the pp inelastic cross-section from a mini-jet model embedded in a one-channel eikonal expression, which provides the needed contribution of uncorrelated processes. It is then shown that current LO parton density functions for the pp mini-jet cross-sections, with a rise tempered by collinearity induced by soft gluon re-summation, are well suited to reproduce recent cosmic ray results. By comparing results for GRV, MRST72 and MSTW parametrizations, we estimate the uncertainty related to the low-x behavior of these densities. | 10.1051/epjconf/20159003002 | [
"https://arxiv.org/pdf/1408.2921v2.pdf"
] | 53,336,502 | 1408.2921 | acfb75c6fdab8a626367a66e75394c242c158efa |
p − air production cross-section and uncorrelated mini-jets processes in pp-scattering
13 Aug 2014
D A Fagundes [email protected]†[email protected]‡[email protected]§[email protected]¶[email protected]
A Grau
G Pancheri
Y N Srivastava
O Shekhovtsova
Departamento de Física Teórica y del Cosmos
Instituto de Física Gleb Wataghin
Universidade Estadual de Campinas
UNICAMP
13083-859Campinas SPBrazil
Universidad de Granada
18071GranadaSpain
Physics Department
INFN Frascati National Laboratories
Via E. Fermi 4000444Italy
Kharkov Institute of Physics and Technology
Institute of Nuclear Physics
University of Perugia
61108, Akademicheskaya,1PAN ul. Radzikowskiego 15206123, 31-342Perugia, Kharkov, KrakowItaly, Ukraine, Poland
p − air production cross-section and uncorrelated mini-jets processes in pp-scattering
13 Aug 20142numbers: 1375Cs1385-t Keywords: Cosmic raysp − air production cross-sectionpp total and inelastic collisions *
For the p − air production cross-section, we use a Glauber formalism which inputs the pp inelastic cross-section from a mini-jet model embedded in a one-channel eikonal expression, which provides the needed contribution of uncorrelated processes. It is then shown that current LO parton density functions for the pp mini-jet cross-sections, with a rise tempered by collinearity induced by soft gluon re-summation, are well suited to reproduce recent cosmic ray results. By comparing results for GRV, MRST72 and MSTW parametrizations, we estimate the uncertainty related to the low-x behavior of these densities.
I. INTRODUCTION
In this paper we address the problem of how to relate p − air production cross-section measurements from cosmic rays, to accelerator data for proton − proton scattering. This is a very old question and very ingenious ways to do so have been developed through the years [1]. The issue is often obfuscated by the need to estimate the contribution from elastic and diffractive processes, both in p − air, but mostly in pp collisions.
The question of whether it is σ pp total or σ pp inel which is input to the Glauber formalism was discussed in [2]. Presently, most current analyses define a σ p−air prod through the inelastic cross-section, and a σ p−air inel through the total pp cross-section. In either case, elastic and quasi-elastic contributions need to be subtracted and a degree of uncertainty can arise from their parametrization. The definition of inelastic cross-section is also affected by uncertainties, both theoretically and experimentally, as seen in LHC experiments with different cuts in the forward region [3].
Here we shall show that the total p − air production cross-section can be obtained in a very direct way through the inelastic pp cross-section resulting from one-channel eikonal models. This formalism for the inelastic cross-section provides a description of non-correlated inelastic processes [4], and thus avoids the problem of how to model diffraction and elastic cross-section. The description of the latter, including the elastic differential cross-section, is still not resolved, and is obtained through various parametrizations. A recent suggestion by the Telaviv group [5] has made efforts in this direction. Here we shall follow a different path.
It is important to stress that in the case of cosmic rays, first and foremost one needs an eikonal function which gives a description of the total pp cross-section, through a good understanding of the underlying physics. In this paper we describe proton − air production cross-section up to the recent AUGER measurement [6], using the inelastic pp cross-section obtained from a QCD mini-jet model with soft gluon re-summation [7,8].
We have long advocated QCD mini-jets as the driving mechanism for the rise of all total crosssections [9] and have proposed a saturation mechanism based on infrared gluon resummation to tame the excessive rise with energy of the mini-jet cross-sections [10]. Thus, the emphasis of the present work is two fold. First to provide a good phenomenological description of cosmic p − air production cross-sections through a successful well accepted formalism, such as in the Glauber theory [11]. The second is to reconfirm that the rise of all total, elastic and inelastic cross-sections of protons on protons, or protons on nuclei and other hadrons, have the same origin: a rising contribution from the increasing number (with energy) of low-x gluons excited in the collision [12].
Since the '80s, many models have used mini-jets in total cross-section physics [13][14][15] and more recently in [16]. In most cases, the parton density functions [PDFs] are chosen or parametrized ad hoc. However, we believe that mini-jets can give interesting information only if used in connection with current LO parton densities, such as available through updated PDF libraries. As in any perturbative QCD calculation, this LO effect needs then to be complemented by other QCD effects, such as that of very soft gluons arising from the QCD confinement potential [10].
We shall use the Glauber model [17], with the following basic hypothesis when the target is a nucleus: i) for low transverse momentum collisions p t (1 ÷ 2) GeV , the incoming proton does not penetrate the air nucleus and basically scatters off the surface, whereas, as the transverse momentum increases, the proton penetrates the nucleus of atomic number A and scatters off all the protons in the volume occupied by the nucleus. Thus the nuclear density seen by the incoming protons will only be proportional to A 2/3 for the soft collisions, and to A for the hard part. (ii) For interactions with transverse momenta p t (1 ÷ 2) GeV , we shall employ QCD effects in the form of mini-jets and soft gluon emission as in the model developed in [7,8,10].
Neglecting momentarily the above surface/volume effect, we begin with the usual Glauber expression for the production cross-section in the impact parameter representation, as given by
σ p−air prod (E lab ) = d 2 b[1 − e −np−air(b,s) ](1)
with
n p−air (b, s) = T N (b)σ pp inel (s)(2)
wherein T N (b) is the nuclear density, for which we start by choosing a standard gaussian distribution,
T N (b) = A πR 2 N e −b 2 /R 2 N ,(3)
properly normalized to
d 2 bT N (b) = A.(4)
The parameters used in the profile (3), namely the average mass number of an "air" nucleus, A, and the nuclear radius, R N , are the following:
A = 14.5, R N = (1.1f ermi)A 1/3 .(5)
The inelastic pp cross-section, σ pp inel , is obtained from pp scattering, with
σ pp inel = d 2 b[1 − e −2χ I (b,s) ](6)σ pp tot = 2 d 2 b[1 − e iχ(b,s) ](7)
where χ I (b, s) = mχ(b, s) is the imaginary part of the eikonal function that defines the elastic amplitude. At high energy, it is a good approximation to neglect a possible real part of the eikonal function in Eq. (7) and write
σ pp tot = 2 d 2 b[1 − e −χ I (b,s) ](8)
This formalism gives both the total and the inelastic non-correlated cross-section, once the quantity χ I (b, s) is known. The latter is an important point in the discussion of p − air processes. The one-channel eikonal formalism for the inelastic cross-section given by Eq. (6) includes only noncorrelated, Poisson distributed independent collisions. This can be seen easily by comparing this equation with a sum over all independent Poisson like distributions, as discussed in [4]. Thus the above one-channel eikonal has the virtue of identifying all non-correlated processes, which we argue (and later verify phenomenologically) are all the non-diffractive processes contributing to the p − air production cross-section. We notice here that this property of the one-channel eikonal is a hindrance when one wants to separate the purely elastic from the diffractive part, but it is exactly what one needs for p-air shower initiated measurements. We shall return to this point again later. In the following, we shall first consider pp scattering and give a brief summary of the physics content of our model and determine the parameters which give an optimal description of pp data up to LHC. We shall then use the one-channel eikonal to calculate the inelastic non-diffractive pp cross-section and obtain the p − air production cross-section to compare with data.
II. PROTON-PROTON TOTAL AND INELASTIC NON-DIFFRACTIVE CROSS-SECTION
The eikonal function of the mini-jet model of [7,8] is given by
2χ I (b, s) = n pp sof t (b, s) + n pp jet (b, s) = A F F σ pp sof t (s) + A pp BN (p; b, s)σ jet (P DF, p tmin ; s)(9)
where A F F , the impact parameter distribution in the non perturbative term, is obtained through a convolution of two proton form factors, whereas for the perturbative term, the distribution A pp BN (p; b, s), multiplying the mini-jet contribution, is given by the Fourier transform of overall soft gluon re-summation, i.e. we have
A pp BN (p; b, s) = e −h(p;b,s) d 2 be −h(p;b,s) (10) h(p; b, s) = constant qmax 0 dk t k t α s (k t ) log 2q max k t [1 − J 0 (bk t )] (11) α s (k t ) ( k t Λ QCD ) −2p k t → 0(12)
We have discussed this distribution in many publications, its main characteristic is to include soft gluon re-summation down to k t = 0, and regulate the infrared singularity so as correspond to a dressed gluon potential V ∼ r 2p−1 for r → ∞. We have also shown an important consequence of an expression such as the above for α s (k t → 0) [18], namely that asymptotically the regularized and integrated soft gluon spectrum of Eq. (11) is seen to rise as
h(p; b, s) → (bΛ) 2p(13)
and thus the b− distribution exhibits a cut-off in b−space strongly dependent on the parameter p, i.e.
A BN → e −(bΛ) 2p b → ∞(14)
withΛ ∝ Λ QCD . Since the mini jet cross-sections at low-x are parametrized so as to rise as s , the behavior of Eq. (14) leads to a high energy behavior for the total cross-section given as
σ pp tot ∼ 2π (Λ) 2 [ log s] 1/p(15)
The parameter 1/2 < p < 1: the lower limit so as to have a confining potential, the upper limit to insure convergence of the integral over the soft gluon spectrum of Eq. (11). An immediate consequence of this model is that the cross-section will never rise more than [log s] 2 , the saturation of the Froissart limiting behavior being obtained for p = 1/2. Notice, that, in this model, the minijet contribution, just as in hard Pomeron models [19], rises as σ jet ∼ s , with ∼ 0.3−0.4 depending on the low-x parametrization of the PDF. However the strong cut-off in b-space (saturation) brought in by the singular, but integrable, effective quark − sof t − gluon coupling constant leads only upto a (logarithmic) 2 rise with energy. For more details, we refer the reader to Ref. [18]. The low energy term includes collision with p t ≤ p tmin ∼ 1 − 2 GeV , and the cross-section σ pp sof t (s) is not predicted by this model so far, thus we parametrize it here with a constant and one or more decreasing terms. The result is shown in Fig. 1. The perturbative, mini-jet, part is defined with p parton t ≥ p tmin and is determined through a set of perturbative parameters for the jet cross-section, namely a choice of PDF and p tmin . Since the soft gluon re-summation includes all order terms in soft gluon emission, as in previous publications we have used only LO densities. An important point of our approach is that we use the same, library distributed PDF, as used for jet physics. Previously used PDFs were GRV [20][21][22], or MRST72 [23]. Both still give a good description of data up to LHC results, as shown here and in the next section. In Fig. 2 we show the results obtained through a more recent set of LO densities, MSTW [24], for both the total and the inelastic pp cross-sections. [27], CMS [28] and ALICE [29].
The parameter p, whose value is explicitly given in this figure, is related to the amount of saturation due to soft gluon emission, as discussed in [18]. Its value lies in the range 0.6 p 0.8 depending on the PDF used. For MSTW, we find that the parameter set {p tmin = 1.3 GeV , p=0.66} best reproduces the pp cross-section up to LHC8.
We note the important result that the inelastic cross-section predicted by the parametrization of the total cross-section through a one-channel eikonal, reproduces very well the LHC data for non-diffractive collisions by ATLAS [27], CMS [28] and ALICE [29]. Such agreement had already been highlighted in [4]. We shall return to comment on this point at the end of the paper.
II.1. A comment on the model parameters
The present focus of our model is the parametrization of the high energy behavior described by QCD processes. To this aim, we need a set of PDFs, a lower cut-off dividing the perturbative and non-perturbative regions, p tmin , and a saturation, parameter p, which we also referred to as singularity parameter. The higher this parameter, the more saturation is present. Phenomenologically, its value is fixed in relation to the low-x behavior of the densities. The parameter p thus appears to be unrelated to the perturbative expression for the QCD coupling constant α s (Q 2 ). We however believe it be of more fundamental interest, and have made the ansatz [30] that the actual expression to use in the integrand of Eq. (11) is
α BN s (Q 2 ) = 1 ln[1 + ( Q 2 Λ 2 ) b0 ](16)
where b 0 = (33 − 2N f )/12π and the suffix BN is used to indicate its applicability into the infrared region (the one first explored in QED by Bloch and Nordsieck [31]), while coinciding with the usual one-loop asymptotic freedom expression at high Q 2 . The above ansatz would imply that the infrared region description does not require introduction of an extra parameter p: the behavior from Q 2 = 0 to Q 2 → ∞ is dictated only by the anomalous dimension factor. However, the present uncertainty about a fundamental calculation for the low-x behavior of the parton densities, prevents a full use of Eq. (16). Suffice to say that our phenomenological values for p are in the same range of variability of the anomalous dimension factor b 0 .
III. THE PRODUCTION CROSS-SECTION FOR p − air
With the low energy part parametrized as shown in Fig. 1, and the mini-jet part, we can calculate the inelastic pp cross-section and thus the production p − air cross-section. The result is shown in Fig. 3 where our model is compared with cosmic ray data [6,[32][33][34][35][36][37][38]. In this figure we have reduced the constants σ 0,1 in the pp cross-section so as to comply with the surface/volume effect for the low transverse momentum collisions. Because of the uncertainty in this low energy region, the soft term in the pp cross-section has been included openly as a constant. However, we have also considered the full low-energy parametrization of Figs. 1,2, but in the energy range of Fig. 3 such low energy decreasing term makes no difference whatsoever.
To estimate the error of this procedure as well as check the stability of the model and its application to both pp and p − air cross-sections, we have done the following checks:
• after parametrizing the low energy part of pp data, the rise has been described through other available LO PDFs, namely MRST72 and GRV in addition to MSTW. For a given PDF set, the parameters p min and p have been chosen to best reproduce LHC results for σ pp tot [25,26] . • we have done an actual fit to both the low energy data and LHC (excluding cosmic rays extracted data), using GRV and MRST72, and with free saturation parameter p.
• We have changed the nuclear density model, applying a Wood-Saxon potential, as in [16].
The results of this exercise for different densities are shown in the two panels of Fig. 4, where the bands highlight the uncertainty related to the the low-x behavior of the parton densities used for the mini jet calculation. As we are not so much interested in understanding right now the low energy part, the constant σ 0 has simply been reduced adjusting it to the data. As expected, the contribution from the low energy part gets weaker and weaker for very high energies. The results are also shown in Table I. In the table, the low energy part of the eikonal function, n sof t , is fitted to the low energy data alone, as in Fig. 1, whereas the QCD part n hard is chosen so as to best describe the pp accelerator data. As for the other check, non reproduced in this table, namely fitting at the same time both the low and the high energy accelerator data in order to determine the best p-value, for a given choice of PDF and p tmin , we have found the result to be consistent with above, for p 0.6 for MRST72 densities and p tmin 1.3 − 1.4 GeV . Using the Wood-Saxon potential slightly lowers the curves for p − air with respect to the standard nuclear potential of Eq. (3).
Before concluding this paper, we would like to return to an important physics point, namely that the experimentally measured σ prod p−air differs from the total σ tot p−air through the exclusion of elastic σ el p−air as well as quasi-elastic σ q−el p−air . An example of σ q−el p−air is given by processes such as p + p → ∆(1238) + p. In general, the measured cosmic cross-section does not include (single as well as double) diffractive processes. This acquires a particular significance (and endows a certain simplicity) to (one-channel) mini-jet models when applied to an analysis of cosmic ray cross-sections. As has been noticed by several authors [4,39], single-channel mini-jet models overestimate the elastic cross-section by including in it the diffractive processes. Otherwise said, the inelastic cross-section in such models does not include the diffractive part. It is thus best suited for calculations of the production p − air cross-section from cosmic ray measurements. For this purpose, we have employed parameters (such as p) suitable for describing the total cross-section well and by default giving us the inelastic part devoid of diffraction. A posteriori, such a description seems to work quite well.
The most remarkable result that we find is that we reproduce very well the AUGER point, in addition to have a reasonably good description of all the more recent cosmic ray measurements.
IV. CONCLUDING REMARKS
In this paper, we have seen that the Glauber formula in conjunction with an inelastic pp crosssection obtained through a one-channel eikonal formalism provides a very good description of the cosmic ray extracted (p − air) cross-section. Thus, we might ask, whether a one-channel eikonal expression adequately representing the pp total cross-section is also sufficient to describe high energy elastic scattering. Obviously not, unfortunately. It is fair to say that the momentum transfer (t)-dependence of the elastic differential cross-section from the forward (t = 0) up to after the dip still escapes a fundamental QCD explanation. For this, and thus for the diffractive part of the cross-section, a multi channel formalism [39][40][41] is still required. However, it is our ansatz that a viable multichannel formalism must be geared to reproduce the results from a single term at the optical point (that is at t = 0).
For the present, we may reiterate that a good one-channel eikonal representation for the total cross-section should be sufficient to describe the cosmic ray p − air production cross-section data and conversely, that models which reproduce σ p−air production can be trusted to extrapolate correctly σ pp inel−non−dif f ractive , and thus the total σ pp tot in a one-channel eikonal model. However, very high energy predictions are affected by an uncertainty related to the low-x behavior of the PDFs used in the phenomenological calculation of the mini-jet cross-sections. It may thus be very important to include the forthcoming LHC data at √ s > 10 T eV to reduce such uncertainty and hopefully be able to extract information on σ pp tot/inel from the even higher energy cosmic ray measurements to be expected from cosmic rays.
FIG. 1 .σFIG. 2 .
12Low energy parametrization of pp total cross-section Fly's Eye and AKENO , ARGO from cosmic rays -QCD mini-jet with soft gluon resummation model and pp total cross-section (full line). Accelerator data at LHC include TOTEM [25, 26] and ATLAS measurements, as from [ATLAS-CONF-2014-040, ATLAS-COM-CONF-2014-054]. The inelastic uncorrelated cross-section is given by the dashed curve and compared with central collisions results at LHC by ATLAS
FIG
al. (AKENO), PRL 70 (1993) 525 Baltrusaitis et al. (Fly's Eye), PRL 52 (1984) 1380 Abreu et al. (AUGER), PRL 109 (2012) 062002 Aielli et al. (ARGO), PRD 80 (2009) 092004 Aglietta et al. (EAS-TOP), PRD 79 (2009) 032004 Mielke et al. (KASCADE), JPG 20 (1994) 637 Belov et al. (HiRes), NPB Proc. Suppl. 151 (2006) 197 Knurenko et al., Proc. of 26th ICRC (1999) Vol.. 3. p − air production cross-section using MSTW2008 parton densities in a one-channel eikonal mini-jet model with infrared gluon resummation.
σFIG. 4 .
4Fly's Eye and AKENO , ARGO from cosmic rays -Left panel: total and uncorrelated inelastic pp cross-sections for different PDF sets. Right panel: the production p − air cross-section following the results from the left panel. The green and yellow bands indicate the uncertainty related to the low-x behavior of the PDF used. Symbols for p − air data as inFig. 3.
TABLE I .
ITotal and inelastic uncorrelated pp cross-sections (second and third column). Fourth column is the uncorrelated inelastic pp cross-section for input to the Glauber formula for σ p−air , with low energy part reduced for nuclear area/volume effect. Last column shows the resulting p − air cross-section. Different parameter sets are as indicated.Parameter set : MSTW, p tmin = 1.3 GeV , p=0.66 with σ 0 = 47.9 mb with σ 0 = 47.9 mb with σ 0 = 30 mb σ 0 = 30 mb.Parameter set : GRV, p tmin = 1.2 GeV , p=0.69
√
s (GeV )
σ
pp
tot
σ
pp−uncorr
inel
σ
pp−uncorr
inel
σ
p−air
prod
with σ 0 = 48 mb
with σ 0 = 48 mb with σ 0 = 32 mb σ 0 = 32 mb
5
39.9
33.2
24.9
255.8
10
38.2
32.0
24.0
248.9
50
41.9
34.0
26.7
268.7
100
46.7
36.1
29.7
288.6
500
63.2
43.0
38.6
340.9
1000 .
71.7
46.9
43.1
364.1
1800.
79.5
50.5
47.2
383.5
7000
98.9
59.8
57.4
426.1
8000
100.9
60.7
58.4
430.0
14000
109.3
64.8
62.8
445.9
30000
121.3
70.7
69.0
467.0
60000
132.0
76.0
74.6
484.3
Parameter set : MRST72, p tmin = 1.25 GeV , p=0.62
5
39.9
33.2
24.9
255.8
10
38.3
32.0
24.0
249.1
50
43.1
34.6
27.6
274.5
100
48.4
36.9
30.8
295.8
500
63.8
43.7
39.3
344.6
1000
71.3
47.1
43.3
365.1
1800
78.1
50.3
46.9
382.3
7000
98.2
60.4
58.0
428.3
8000
100.7
61.7
59.4
433.6
14000
112.2
67.7
65.7
456.2
30000
129.1
76.5
75.0
485.7
60000
144.2
84.4
83.3
509.1
5
39.21
32.7
23.7
246.8
10
38.60
32.3
23.1
242.6
50
42.2
34.2
25.9
263.4
100
46.9
36.4
29.2
285.5
500
62.0
43.3
38.1
338.6
1000
71.0
47.5
43.1
364.4
1800
77.5
50.5
46.6
381.2
7000
98.3
60.5
57.8
428.0
8000
101.3
62.0
59.4
434.0
14000
113.7
68.2
66.1
457.7
30000
129.4
76.0
74.3
483.8
60000
150.3
86.3
85.1
514.3
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| [] |
[
"arXiv:math-ph/0501012v1 6 Jan 2005 Weak Coupling and Continuous Limits for Repeated Quantum Interactions",
"arXiv:math-ph/0501012v1 6 Jan 2005 Weak Coupling and Continuous Limits for Repeated Quantum Interactions"
] | [
"Stéphane Attal \nInstitut Girard Desargues\nInstitut Fourier\nUniversité de Lyon\n1 21 Av. Claude Bernard69622Villeurbanne CedexFrance\n",
"Alain Joye \nUniversité de Grenoble\n1, BP 7438402St.-Martin d'Hères CedexFrance\n"
] | [
"Institut Girard Desargues\nInstitut Fourier\nUniversité de Lyon\n1 21 Av. Claude Bernard69622Villeurbanne CedexFrance",
"Université de Grenoble\n1, BP 7438402St.-Martin d'Hères CedexFrance"
] | [] | We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical sub-systems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ , the reference system is coupled to one new element of the chain only, by means of an interaction of strength λ. We consider three asymptotic regimes of the parameters λ and τ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales T and whose generator can be computed: the weak coupling limit regime λ → 0, τ = 1, the regime τ → 0, λ 2 τ → 0 and the critical case λ 2 τ = 1, τ → 0. The first two regimes are perturbative in nature and the effective generators they determine is such that a non-trivial invariant sub-algebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator. | 10.1007/s10955-006-9085-z | [
"https://arxiv.org/pdf/math-ph/0501012v1.pdf"
] | 14,415,445 | math-ph/0501012 | f04e29348f2f7c85c3cfba38de2cb4c190cd34ef |
arXiv:math-ph/0501012v1 6 Jan 2005 Weak Coupling and Continuous Limits for Repeated Quantum Interactions
Stéphane Attal
Institut Girard Desargues
Institut Fourier
Université de Lyon
1 21 Av. Claude Bernard69622Villeurbanne CedexFrance
Alain Joye
Université de Grenoble
1, BP 7438402St.-Martin d'Hères CedexFrance
arXiv:math-ph/0501012v1 6 Jan 2005 Weak Coupling and Continuous Limits for Repeated Quantum Interactions
We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical sub-systems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ , the reference system is coupled to one new element of the chain only, by means of an interaction of strength λ. We consider three asymptotic regimes of the parameters λ and τ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales T and whose generator can be computed: the weak coupling limit regime λ → 0, τ = 1, the regime τ → 0, λ 2 τ → 0 and the critical case λ 2 τ = 1, τ → 0. The first two regimes are perturbative in nature and the effective generators they determine is such that a non-trivial invariant sub-algebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator.
Introduction
This paper is concerned with the study of the weak coupling limit, and variations thereof, of open quantum systems consisting in a small quantum system defined by a Hamiltonian h 0 on a Hilbert space H 0 coupled to a field or heat bath modelled by an infinite chain of identical independent n+1-level sub-systems on ⊗ N * H, with n finite. The coupling between the distinguished system and the chain is provided by a discrete sequence of interactions of the small system with one individual sub-system of the chain, in the following way: if τ > 0 is a microscopic time scale, over a macroscopic time interval ]0, kτ ], k ∈ N * , the small system is coupled with elements 1, 2, ..., k of the chain, in sequence, for the same time τ and with the same interaction of strength λ. The interactions we consider are of the linear minimal coupling type n j=0 V * j ⊗ a j + V j ⊗ a * j , where the a * j 's and a j 's are creation and annihilation operators relative to the levels of the sub-system and the V j 's are arbitrary operators on H 0 . Such models of repeated quantum interactions are used in physics, e.g. in quantum optics, in the theory of quantum measurement or in decoherence. The lack of coupling, and thus of coherence, between the elements of the chain allows to expect that an effective continuous dissipative dynamics for pure states or observables on the small system of the form e tΓ should emerge when the number k of discrete interactions goes to infinity and the coupling λ with the chain elements is weak, in the familiar weak coupling regime. Recall that this corresponds to choosing t ∈ R and considering N ∋ k = t/λ 2 so that the macroscopic time scale equals T = τ t/λ 2 . When τ is fixed and λ → 0, both k and T go to infinity as 1/λ 2 . Moreover, in the setting adopted here, we have another parameter at hand which is the microscopic interaction time τ of the small system with each individual element of the chain. It allows us to explore different asymptotic regimes, as τ goes to zero as well, which characterizes the continuous limit, over suitable macroscopic time scales T .
One goal of this paper is to establish the existence of effective continuous Markovian dynamics in weak and/or continuous limits defining three asymptotic regimes. We consider successively the effective Schrödinger evolution on the small system at zero temperature and, when the chain is at equilibrium at zero or positive temperature, the Heisenberg evolution of observables on the small system. While the existence of an effective dynamics obtained by a weak limit procedure (τ = 1) is proven for a large class of time-independent Hamiltonian systems, as well as in certain time-dependent situations, see e.g. [D1], [D2], [DS], [LS], [DJ], this question is not addressed in the literature for the case under study. Note also that a Hamiltonian formulation of our system necessarily involves a piecewise constant time-dependent generator. The analysis relies on the following property of the model, which is inherent to its definition. The effective dynamics on the small system of pure states or of observables from time 0 to time kτ is shown to be given by the k th power of a linear operator, which depends on the parameters λ and τ . This expresses the Markov property in a discrete setting.
The first part of the paper is devoted to the usual weak limit regime λ → 0, τ fixed and T = tτ /λ 2 → ∞, 0 < t finite. We show the existence an effective dynamics driven by a τ dependent generator which we determine. This first result is obtained by adapting the arguments developed in the study of the weak coupling regime for stationary Hamiltonians to our discrete quantum dynamics framework. The method is then extended to accommodate the whole range τ → 0, λ 2 τ → 0 over macroscopic time scales T = t/(τ λ 2 ) → ∞, which defines our second regime. This gives rise to an effective dynamics driven by a τ independent generator we compute as well and to which we come back below. The analysis of these first two regimes is strongly related to regular perturbation theory in the parameter λ 2 τ and we refer to these regimes as perturbative regimes. Technically, the study of the second regime relies on an asymptotic analysis in the two parameters λ and τ of the discrete evolution of our system. The divergence of the macroscopic time scale imposes, as usual, some renormalization of the dynamics by the restriction of the uncoupled dynamics. Finally, note that in the second regime, the interaction strength λ is not required to go to zero and can even diverge. The common feature of the generators of the dynamics of observables obtained in these two regimes is that they commute with the generator i[h 0 , ·] of the uncoupled unitary evolution restricted to H 0 . In other words, the corresponding effective dynamics admits the commutant of h 0 as a non trivial invariant sub-algebra of observables. This property is well known in the weak coupling regime for time-independent Hamiltonians, [D2], [LS], [DJ].
Our primary motivation actually comes from the recent paper [AP] where such repeated interactions models are shown to converge in some subtle limiting procedure to open quantum systems with a heat bath consisting in continuous fields of quantum noises, at zero temperature. These limiting systems give rise in a natural and spontaneous way to effective dynamics on the Hilbert space H 0 of the small system governed by quantum Langevin equations. The above mentioned limit involves at the same time the time scale τ , the strength of interaction λ as well as a notion of spacing between the sub-systems forming the chain in an intricate way. While reminiscent of weak coupling methods in spirit, the limiting procedure of [AP] is nevertheless distinct from the weak coupling limit. Indeed, while τ → 0, the product λ 2 τ is kept constant in [AP], which leads us beyond the perturbative regime. Another goal of the present work is to consider as our third regime the critical scaling λ 2 τ = 1 in our repeated quantum interactions model and to derive an effective evolution of observables for a chain at inverse temperature β. The relevant macroscopic time scale in this regime is T = t/(λ 2 τ ) = t, which is finite.
With this scaling, we show that an effective Heisenberg dynamics for observables on H 0 emerges at any temperature. It is generated by a general Lindblad operator whose dissipative part is explicitly constructed in terms of the V j 's defining the coupling in the Hamiltonian, whereas its conservative part is simply i[h 0 , ·]. At zero temperature, we recover the effective Heisenberg dynamics of observables on H 0 of [AP] obtained by means of quantum noises. At positive temperature, our generator coincides with a construction proposed in [LM] for certain models using an a priori modelization of the heat bath by some thermal quantum noises, generalizing those used at zero temperature. For any temperature, the effective dynamics is distinct from that obtained in the previous two perturbative regimes. In particular, the generator obtained does not commute with i[h 0 , ·] anymore, the generator of the uncoupled evolution restricted to H 0 . Hence, there is no obvious subalgebra of observables left invariant by the effective dynamics of observables. The analysis of this critical case makes use of Chernoff's Theorem, rather than perturbative methods.
Let us compare the generator of the effective dynamics of observables obtained in the regime τ → 0, τ λ 2 → 0, and the general Lindblad operator obtained as τ → 0 with τ λ 2 = 1. In the former case, the generator is obtained from the dissipative part of the Lindblad operator of the latter case by retaining its diagonal terms only with respect to the spectral decomposition of the uncoupled evolution restricted to H 0 . Or, in an equivalent way, by performing a time average of the Lindblad operator with respect to the uncoupled evolution restricted to H 0 . This defines the so-called # operation that makes the commutant of h 0 invariant under the effective dynamics in the regime τ → 0, τ λ 2 → 0 (and in the weak coupling regime as well). Our results show that the # operation is present as long as λ 2 τ → 0, whereas it disappears in the critical regime τ λ 2 . In other words, in the regime τ → 0, τ λ 2 → 0, a non-trivial distinguished invariant sub-algebra of observables exists, whereas in the critical case τ → 0, τ λ 2 = 1, there is a priori no sub-algebra left invariant by the effective dynamics, since its generator takes the form of a generic Lindblad operator.
We finally note here that from a practical point of view, the modelization of the dynamics of observables (or states) of a small system in contact with a reservoir at a certain temperature often starts with a choice of a certain Lindblad generator suited to the physical phenomena to be discussed. Our analysis allows to assign to any Lindblad generator a simple model of repeated quantum interactions, with explicit couplings constructed from the Lindblad generator, whose effective dynamics in the limit τ → 0, λ = 1/ √ τ , is generated by the chosen Lindblad operator.
The paper is organized as follows. The general setup and definition of the model are provided in the next section. Section 3 is devoted to the analysis of the weak limit of the model at zero temperature, in the Schrödinger picture. Our main results in this setup are expressed as Corollary 3.2 for the weak coupling regime and Corollary 3.3 for the regime λ 2 τ → 0, τ → 0. This section also contains the technical basis underlying our perturbative analysese in both the Schrödinger and Heisenberg pictures. The main technical result, of independent interest, is actually valid in a Banach space framework and is stated as Theorem 3.1. The positive temperature case, in the Heisenberg picture is dealt with in Section 4. The generators of the effective dynamics of observables in the two perturbative regimes are given in Theorem 4.1 and Corollary 4.1. The analysis of the critical regime λ 2 τ = 1 is presented in Section 5, for both the Schrödinger and Heisenberg pictures. Section 6 is devoted to a thorough analysis of the first non-trivial case where both the small system and the elements of the chain consist in two-level systems.
A Repeated Interaction Model
Consider the following setup to start with. Our small system, described by the Hilbert space H 0 of dimension d + 1 > 1 and a self-adjoint Hamiltonian h 0 , interacts with an infinite chain of identical finite dimensional sub-systems modelling a field or heat bath, by means of a time dependent Hamiltonian. The total Hilbert space is H 0 ⊗ H, where H = ⊗ j≥1 C n+1 , n ≥ 1. We will call the jth Hilbert space C n+1 j ≡ C n+1 , the Hilbert space at site j, j = 1, 2, · · · and, following the usage when n = 1, we will call the subsystem at site j the spin at site j. We adopt the following convenient notations used in [AP]. The vacuum Ω ∈ H is defined as the infinite tensor product of the vacuum vector ω = ( 0 · · · 0 1 ) T in C n+1 ,
Ω = ω ⊗ ω ⊗ ω ⊗ · · · ∈ C n+1 ⊗ C n+1 ⊗ C n+1 ⊗ · · · .
(2.1)
Denoting the ith excited vectors x i = ( 0 · · · 0 1 0 · · · 0 ) T , where the 1 sits at the ith line, starting from the bottom, i = 1, 2, · · · , d, the corresponding excited state at site j ≥ 1 is given by
x i (j) = ω ⊗ · · · ⊗ ω ⊗ x i ⊗ ω ⊗ · · · , (2.2)
where x i sits at site j ≥ 1. More generally, given a finite set
S = {(k 1 , i 1 ), (k 2 , i 2 ), · · · , (k m , i m )} ⊂ (N * ×{1, 2, · · · , d}) m with all k j 's distinct, (2.3)
we define X S as the vector given by an infinite tensor product as above, with i j th excited vectors x i j (k j ) at all sites k j ≥ 1, j = 1, · · · , m, and ground state vectors ω everywhere else. This construction together with the vacuum Ω ≡ X ∅ yield an orthonormal basis of H, when S runs over all finite sets of the type above. Let us introduce creation and annihilation operators associated with the vectors x i (j). Let a i and a * i , i = 1, 2, · · · , n, denote the operators corresponding to {ω, x 1 , · · · , x n } in C n+1 , i.e. such that
a i x i = ω, a i ω = a i x j = 0, if j = i, a *
i ω = x i , a * i x j = 0 for any j = 1, 2, · · · , n.
(2.4) Note that these operators do not coincide with the familiar creation and annihiliation, however, for i fixed, they satisfy the anti-commutation rules when restricted to the two dimensional subspace < ω, x i > and are zero on the orthogonal complement of this subspace. Then, for j ≥ 1, the operators a i (j) and a i (j) * on H are defined as acting as a i and a * i on the jth copy of C n+1 at site j, and as the identity everywhere else. Therefore, when acting on different copies of C n+1 , these operators commute. In keeping with the notations for the reservoir, we introduce a basis of eigenvectors of h 0 for H 0 of the form
{ω, x 1 , x 2 , · · · , x d }, where d = dim(H 0 ) − 1.
(2.5)
Note that d = n in general, but we shall nevertheless use sometimes the notation ω(0) and {x i (0)} i=1,2,···,d to denote these vectors. No confusion should arise with vectors of H above, since we labelled the sites of the spins by positive integers. In some cases, H 0 will be an infinite dimensional separable Hilbert space, which corresponds formally to d = ∞.
Our formal time dependent Hamiltonian H(t, λ) on H 0 ⊗ H has the form
H(t, λ) = H 0 + H F + λH I (t), (2.6) where H 0 = h 0 ⊗ I, H F = ∞ j=1 n i=1 I ⊗ δ i a i (j) * a i (j), with δ i ∈ R, (2.7)
and, for t ∈ [τ (k − 1), τ k[,
H I (t) = n i=1 V * i ⊗ a i (k) + V i ⊗ a i (k) * ≡ I(k), (2.8)
where the V i 's and h 0 are bounded operators on H 0 , in case H 0 is a separable infinite dimensional Hilbert space. These operators describe the interaction between the small system with the different levels of energy δ i of the spin at site k, during the time interval ]τ (k − 1), τ k] of length τ . The form of H F makes it an unbounded operator, but, as we will see in the sequel, we will only make use of the unitary evolution it generates and, moreover, it will always be sufficient to work with subspaces containing finitely many excited states only.
In order to make the notations more compact, we introduce vectors with operator valued entries that allow to get rid of the indices i = 1, · · · , n. Let a(j) ♯ = ( a 1 (j) ♯ a 2 (j) ♯ · · · a n (j) ♯ ) T (2.9)
V ♯ = ( V ♯ 1 V ♯ 2 · · · V ♯ n ) (2.10)
where ♯ denotes either nothing or * . Then, using the rules of matrix composition, we can write
V ♯ 1 ⊗ a(j) ♯ 2 = n i=1 V ♯ 1 i ⊗ a ♯ 2 i (j),(2.11)
so that we can rewrite the interaction Hamiltonian for t ∈]τ (k − 1), τ k] as
I(k) = V * ⊗ a(k) + V ⊗ a(k) * .
(2.12) Similarly, with a(j) ♯ a(j) = ( a 1 (j) ♯ a 1 (j) a 2 (j) ♯ a 2 (j) · · · a n (j) ♯ a n (j) ) T (2.13) δ = ( δ 1 δ 2 · · · δ n ) , (2.14)
we can write
H F = I ⊗ j≥1
δa(j) * a(j).
(2.15)
We will denote the corresponding evolution operator between the time τ (k − 1) and τ k by U k , so that
U k = e −iτ (H 0 +H F +λI(k)) ,
(2.16) and the evolution from 0 to τ n is given by
U (n, 0) = U n U n−1 · · · U k · · · U 1 . (2.17)
Although not explicited in the notation, the operator U (n, 0) depends on λ and τ .
We will first be interested in the weak coupling limit of this evolution operator characterized by the familiar scaling n = t/λ 2 , λ → 0 and τ fixed.
(2.18)
Hence, the macroscopic time scale T is given by
T = τ n = τ t/λ 2 → ∞. (2.19)
Note, however, that in contrast with the usual set up, we have here a non-smooth time dependent Hamiltonian H(t, λ).
Remarks:
i) In order not to bury the main points of our analysis under technical subtleties, we have chosen to work in a simple framework where all relevant operators are bounded or matrix valued. Nevertheless, some of our results below hold if we consider our heat bath to live in a tensor product of infinite dimensional separable Hilbert spaces and make further assumptions so that the field Hamiltonian and interaction are bounded. ii) In some cases we shall allow H 0 to be a separable Hilbert space. This will be explicitly stated in the hypotheses. Otherwise, we will work on the model defined above, under the general assumption H0: The Hamiltonian is defined on the Hilbert space H 0 ⊗ H, where H 0 = C d+1 , H = ⊗ j≥1 C n+1 , for d, n finite, and is given by (2.6), (2.7), (2.8). The evolution it generates is given by (2.17) and (2.16).
3 Weak limit of the Schrödinger representation at zero temperature As a warm up, and in order to derive some preliminary estimates, we prove here the existence of the weak limit for our model at zero temperature in the Schrödinger picture, and compute this limit. We first prove a key lemma that reduces the computation of the projected part of the evolution U (n, 0) (2.17) to the n th power of a single matrix. Then we perform a general analysis of large powers of operators based on perturbative expansions which appear in the computations of weak limits. These technical results are expressed in Proposition 3.2 and Theorem 3.1 under different sets of hypotheses. Their applications to our model are given in Corollaries 3.2 and 3.3.
Markov Properties
Let P be the projection from H 0 ⊗ H to the subspace H 0 ⊗ CΩ defined by P = I ⊗ |Ω Ω|.
(3.1)
The object of interest to us in this Section will thus be the limit lim λ→0 P (U (t/λ 2 , 0))P, (3.2) as an operator from P H 0 ⊗ H to P H 0 ⊗ H, identified with H 0 , the Hilbert space of the small system. Note that
U j = e −iτĤ j e −iτH j , (3.3) with H j = h 0 ⊗ I + I ⊗ δa(j) * a(j) + λ(V * ⊗ a(j) + V ⊗ a(j) * ) H j = I ⊗ k =j δa(k) * a(k), (3.4)
two operators that commute. We observe the following property of products of operators U k , which shows the Markovian nature of the reduced evolution.
Lemma 3.1 Let us write the restriction of U j to H 0 ⊗ C n+1 j as a block matrix with respect to the ordered basis of
H 0 ⊗ C n+1 j {ω ⊗ ω, x 1 ⊗ ω, · · · , x d ⊗ ω, ω ⊗ x 1 , x 1 ⊗ x 1 , · · · x d ⊗ x 1 , . . . ω ⊗ x n , x 1 ⊗ x n , · · · x d ⊗ x n } (3.5) as U j | H 0 ⊗C n+1 j = A B C D , (3.6) where A is a (d + 1) × (d + 1) matrix, B is (d + 1) × n(d + 1), C is (d + 1)n × (d + 1) and D is n(d + 1) × n(d + 1).
Then, for any m ≥ 0,
P U (m, 0)P = A m ⊗ |Ω Ω| ≃ A m . (3.7)
Proof: Follows from the fact that
U j (I ⊗ |Ω Ω|) = e −iτ H j (I ⊗ |Ω Ω|) (3.8) where, if H 0 ∋ v = v 0 ω(0) + d i=1 v i x i (0) ≃ v, e −iτ H j v ⊗ Ω = A v ⊗ Ω + n+1 i=1 (C v) i ⊗ x i (j), (3.9) where ( w) i denotes the i'th component of the vector w. Hence, due to the fact that different U j 's act on different C n+1 j 's, U m U m−1 · · · U 1 v ⊗ Ω = A m v ⊗ Ω + m(d+1) i=1 w i ⊗ X S i , (3.10)
where w i are some vector in C d+1 and the excited sets S i are never empty. Therefore their contribution vanishes in the computation
(I ⊗ |Ω Ω|)U m U m−1 · · · U 1 v ⊗ Ω = A m v ⊗ |Ω Ω|. (3.11)
As is easy to check along the same lines, in case H 0 is infinite dimensional, we can generalize the above Lemma as follows.
Lemma 3.2 Let H 0 be a separable Hilbert space and h 0 , V j , j = 1, · · · , n be bounded on H 0 . We set P j = |x j x j | : C n+1 → C n+1 , j = 1, · · · , n, P 0 = |ω ω| and Q 0 = I − P 0 , (3.12) so that
H 0 ⊗ C n+1 = (H 0 ⊗ P 0 C n+1 ) ⊕ (H 0 ⊗ Q 0 C n+1 ) ≃ (H 0 ⊗ C) ⊕ (H 0 ⊗ C n ). (3.13) We can decompose U j | H 0 ⊗C n+1 j = A B C D , (3.14) where A : H 0 → H 0 , B : H 0 ⊗ C n → H 0 , C : H 0 → H 0 ⊗ C n and D : H 0 ⊗ C n → H 0 ⊗ C n .
Then, for any m ≥ 1,
P U (m, 0)P = A m ⊗ |Ω Ω| ≃ A m . (3.15)
The above Lemmas thus lead us to consider a reduced problem on H 0 . We need to compute the matrix A in the decomposition (3.6) of e −iτ (h 0 +δa * a+λ(V * a+V a * )) , where we dropped the indices j, the I and the ⊗ symbol in the notation. Recall however that a summation over the excited states of H F is implicit in the notation.
Preliminary Estimates
In order to apply perturbation theory as λ → 0 and, later on, in other regimes involving τ → 0 as well, we derive below estimates to be used throughout the paper.
We rewrite the generator as
H(λ) = H(0) + λW, with H(0) = h 0 + δa * a and W = V * a + V a * . (3.16)
With a slight abuse of notations, the projector P takes the form
P = I − a * a.
(3.17)
We can slightly generalize the setup and work under the following hypothesis:
H1:
Let P be a projector on a Banach space B and H(λ) be an operator in of the form
H(λ) = H(0) + λW, (3.18)
where H(0) and W are bounded and 0 ≤ λ ≤ λ 0 for some λ 0 > 0. Further assume that
[P, H(0)] = 0 and W = P W Q + QW P where Q = I − P. (3.19)
We consider
U τ (λ) = e −iτ H(λ) . (3.20)
For later purposes, we also take care of the dependence in τ of the error terms. As this parameter will eventually tend to zero in some applications to come below, we consider the error terms as both λ and τ tend to zero, independently of each other. We have a first easy perturbative result Lemma 3.3 Let H1 be true. Then, as λ and τ go to zero,
e −iτ (H(0)+λW ) = e −iτ H(0) + λF (τ ) + λ 2 G(τ ) + O(λ 3 τ 3 ) (3.21) P e −iτ (H(0)+λW ) P = P e −iτ H(0) P + λ 2 P G(τ )P + P O(λ 4 τ 4 )P, (3.22) where F (τ ) = n≥1 (−iτ ) n n! m j ∈N m 1 +m 2 =n−1 H(0) m 1 W H(0) m 2 = −ie −iτ H(0) τ 0 ds 1 e is 1 H(0) W e −is 1 H(0) (3.23) G(τ ) = n≥2 (−iτ ) n n! m j ∈N m 1 +m 2 +m 3 =n−2 H(0) m 1 W H(0) m 2 W H(0) m 3 = −e −iτ H(0) τ 0 ds 1 s 1 0 ds 2 e is 1 H(0) W e −i(s 1 −s 2 )H(0) W e −is 2 H(0) . (3.24) Moreover d dτ G(τ ) = −iH(0)G(τ ) − iW F (τ ), G(0) = 0 (3.25) F (−τ ) = −e iτ H(0) F (τ )e iτ H(0) (3.26) G(−τ ) = −e iτ H(0) G(τ )e iτ H(0) + e iτ H(0) F (τ )e iτ H(0) F (τ )e iτ H(0) . (3.27)
Remark: Formula (3.21) is true without assuming that W is off-diagonal with respect to P and Q.
Proof: First note that U τ (λ) = e −iτ H(λ) is analytic in both variables λ and τ in C 2 . Then, we compute the exponential of −iτ times H(λ) as a convergent series. Consider terms of the form
(H(0) + λW ) n = H(0) n + λ n−1 k=0 H(0) k W H(0) n−1−k + λ 2 m j ∈N m 1 +m 2 +m 3 =n−2 H(0) m 1 W H(0) m 2 W H(0) m 3 + O(λ 3 C n ). (3.28)
The error term in C n comes from the boundedness of the operators involved. Multiplication by (−iτ ) n /n! and summation over n ≥ 0 yields the first result with our definition of F (τ ) and G(τ ). The second result follows from taking into account that P W P = QW Q = 0, hence only the terms with an even number of W 's survive and we get
P (H(0) + λW ) n n! P = (3.29) P H(0) n n! + λ 2 m j ∈N m 1 +m 2 +m 3 =n−2 H(0) m 1 W H(0) m 2 W H(0) m 3 n! + O(λ 4 C n /n!) P.
The overall error in τ 4 λ 4 comes from the fact that it takes at least four terms in (3.28) to get a contribution of order λ 4 . The computation above was conducted to order λ 2 because of the scaling (2.18). The order λ term F (τ ) doesn't contribute, being off diagonal with respect to P .
An alternative derivation of a perturbation series of e −iτ (H(0)+λW ) in λ yields the other expressions for F (τ ) and G(τ ). It is obtained via Dyson series in the familiar interaction picture. We have the identity
i d dτ e −iτ (H(0)+λW ) = (H(0) + λW )e −iτ (H(0)+λW ) , e −iτ (H(0)+λW ) τ =0 = I. (3.30) Introducing Θ(λ, τ ) = e iτ H(0) e −iτ (H(0)+λW ) , (3.31) this operator satisfies i d dτ Θ(λ, τ ) = λe iτ H(0) W e −iτ H(0) Θ(λ, τ ), Θ(λ, τ )| τ =0 = I. (3.32)
Hence we have the convergent expansion
Θ(λ, τ ) = ∞ n=0 (−iλ) n τ 0 ds 1 s 1 0 ds 2 · · · s n−1 0 ds n e is 1 H(0) W × ×e −i(s 1 −s 2 )H(0) W e −i(s 2 −s 3 )H(0) · · · e −is n−1 −sn)H(0) W e −isnH(0) . (3.33)
Therefore, focusing on the terms of order λ and λ 2 , we get the alternative expressions for F (τ ) and G(τ ).
The differential equation yielding G(τ ) as a function of F (τ ) follows from explicit computations on the expressions above, as the identities for τ → −τ .
Let us give some more properties of the expansion of U τ (λ) for λ > 0 small, τ > 0 in the Hilbert space context that will be used later on.
Corollary 3.1 Assume B is a Hilbert space, H(0), W and P are self-adjoint and λ, τ are real. As λ → 0, the operator U τ (λ) = e −iτ H(λ) satisfies
U τ (λ) = e −iτ H(0) + λF (τ ) + λ 2 G(τ ) + O(λ 3 τ 3 ) (3.34) U τ (λ) −1 = U τ (λ) * = U −τ (λ) = e iτ H(0) + λF (−τ ) + λ 2 G(−τ ) + O(λ 3 τ 3 ) (3.35) with the identities for all τ ∈ R F (−τ ) = F * (τ ) (3.36) G(−τ ) = G * (τ ). (3.37)
Proof: Follows from the fact that H(λ) is self-adjoint.
Weak Limit Results
The technical basis underlying all our weak limit results is contained in the next two Lemmas and the Proposition following them. They are stated in a general framework that will suit both our analysese of the Schrödinger and Heisenberg representations. This is why we use independ notations.
Lemma 3.4 Let V (x), x ∈ [0, x 0 ), and R be bounded linear operators on a Banach space B such that, in the operator norm, V (x) = V (0) + xR + O(x 2 )
, and V (0) is an isometry which admits the following spectral decomposition
V (0) = r j=0 e −iE j P j where r < ∞, E j ∈ R, {e −iE j } j=0,···,r distinct. (3.38) Let h = r j=0 E j P j so that V (0) = e −ih and J = r j,k=0 α jk P j RP k where α jk = E j −E k e −iE j −e −iE k if j = k ie iE j if j = k. (3.39) Then, for any 0 ≤ t ≤ t 0 , where t 0 finite, and t/x ∈ N, V (x) t x − e −i(h+xJ) t x = O(x), as x → 0, s.t. t/x ∈ N.
(3.40)
Remarks: i) Expressing the projectors P j by Von Neumann's ergodic theorem as
P j = lim N →∞ 1 N N −1 n=0 (e iE j V (0)) n (3.41)
shows that they are of norm one.
ii) The operator J = J(R, h) is defined as the solution to the equation (3.43). This equation is a particular case of i 1 0 e ish Xe −ish ds = Y which is solved in a similar fashion.
Proof: With m = t/x ∈ N, V (x) m − e −i(h+xJ) m = m−1 k=0 V (x) k (V (x) − e −i(h+xJ) )e −i(h+xtJ) m−1−k (3.42)
where, by hypothesis and Lemma 3.3
V (x) − e −i(h+xJ) = x R + ie −ih 1 0 e ihs Je −ihs ds + O(x 2 ). (3.43)
Moreover, note also
V (x) = 1 + O(x), e −i(h+xJ) = 1 + O(x). (3.44)
Our definition (3.39) of J is designed to make the term of order x in (3.43) vanish. Therefore, there exists positive constants c 0 , c 1 such that we can estimate for any
0 ≤ t ≤ t 0 < ∞ V (x) m − e −i(h+xJ) m ≤ cx 2 m−1 k=0 V (x) k e −i(h+xJ) m−1−k (3.45) ≤ c 0 x 2 m(1 + c 0 x) m ≤ c 0 txe t x ln(1+c 0 x) ≤ xc 0 t 0 e c 1 t 0 = O(x).
It will be necessary to control the dependence of such estimates on a parameter τ → 0 later on. This will cause no serious difficulty, since all steps are explicited in the argument. To achieve sufficient control in τ , we need to revisit the proof of a well known lemma, which holds under weaker hypothesese than ours, see Davies [D2].
Lemma 3.5 Let e −ih = r j=0 e −iE j P j be the isometry (3.38) on the Banach space B and let K be a bounded operator on B. There exists a constant c depending on r and t 0 only, such that for any t ∈ [0, t 0 ], t 0 finite,
e ith/x e −i t x (h+xK) − e −itK # ≤ c x K (1 + K )e 2 K t 0 inf j =k |E j − E k | , as x → 0, (3.46) where K # = r j=0 P j KP j = lim T →∞ 1 T T 0 e ish Ke −ish ds.
Remark: The expression of K # as a Cesaro mean is a classical computation which shows that K # ≤ K .
Proof: We follow [D2]. Let f ∈ B and
f x (t) = e ith/x e −i t x (h+xK) f, f (t) = e −itK # f. (3.47)
By the fundamental Theorem of calculus, we can write
i(f x (t) − f (t)) = t 0 e ish/x Ke −ish/x f x (s) − K # f (s) ds (3.48) = t 0 e ish/x Ke −ish/x (f x (s) − f (s)) + e ish/x Ke −ish/x − K # f (s) ds.
Hence,
f x (t) − f (t) ≤ K t 0 f x (s) − f (s) ds + F(x, t 0 ) (3.49) where F(x, t 0 ) = sup 0≤t≤t 0 t 0 e ish/x Ke −ish/x − K # e −isK # f ds . (3.50) Now, e ish/x Ke −ish/x − K # = e ish/x (K − K # )e −ish/x = j =k e ish/x P j KP k e −ish/x = j =k e is(E j −E k )/x P j KP k , (3.51)
so that we can integrate (3.50) by parts to obtain
t 0 e ish/x Ke −ish/x − K # e −isK # f ds = (3.52) j =k t 0 x i(E j − E k ) d ds e is(E j −E k )/x P j KP k e −isK # f ds = j =k x i(E j − E k ) e is(E j −E k )/x P j KP k e −isK # f t 0 + j =k t 0 x (E j − E k ) e is(E j −E k )/x P j KP k K # e −isK # f ds. Hence, using K # ≤ K , we can bound (3.52) by j =k x K (2 + t K )e K t |E j − E k | f . (3.53) Thus, F(x, t 0 ) ≤ max(2, t 0 )(r 2 − r) x(1 + K ) K e K t 0 inf j =k |E j − E k | . (3.54)
At this point we can invoke Gronwall's Lemma, the above estimate and (3.49) to finish the proof.
From these two Lemmas, we immediately get the
Proposition 3.1 Let V (x), x ∈ [0, x 0 ) and R be bounded operators on a Banach space B such that, in the operator norm, V (x) = V (0) + xR + O(x 2 ), where V (0) is an isometry admitting the spectral decomposition V (0) = r j=0 e −iE j P j and let h = r j=0 E j P j . Then, for any 0 ≤ t ≤ t 0 , if x → 0 in such a way that t/x ∈ N, V (0) −t/x V (x) t/x = e te ih R # + O(x), in norm,(3.
55)
where K # = r j=0 P j KP j , for any K ∈ L(B).
Remarks:
i) The operator in the exponent can be rewritten as
e ih R # = e ih R # = (e ih R) # = (Re ih ) # .
( 3.56) ii) The hypotheses are made on the isometry V (0), not on the operator h.
We can now derive our first results concerning the weak limit in the Schrödinger picture. We do so in the general setup described in H1. We further assume:
H2:
The restriction H P (0) of H(0) to P B is diagonalizable and reads
H P (0) = r j=0 E j P j , with dim(P j ) ≤ ∞, r finite .
(3.57)
Moreover, the operator P e −iτ H(0) = P e −iτ H P (0) is an isometry on P B.
Note that this implies P e −iτ H P (0) is invertible and
P = r j=0 P j , E j ∈ R ∀j = 0, · · · , r, and P e −iτ H(0) = r j=0 e −iτ E j P j ,(3.58)
where the projectors P j are eigenprojectors of P e −iτ H(0) iff the e −iτ E j 's are distinct. In case B is a finite dimensional Hilbert space and H(0) is self adjoint, H2 is automatically true.
Proposition 3.2 Let H(λ) and P on B satisfy H1 and H2. Further assume τ > 0 is such that the values are {e −iτ E j } r j=0 are distinct. Then, for any 0 ≤ t < ∞,
lim λ→0 t/λ 2 ∈N e iτ tH(0)/λ 2 P e −iτ H(λ) P t/λ 2 = e tΓ w (τ ) on P B (3.59) where Γ w (τ ) = e iτ H(0) G(τ ) # = − τ 0 ds s 0 dtW e −it(H(0)−E j ) W # ,
(3.60) and K # = r j P j KP j for any K ∈ L(P B).
Remarks:
i) In case some values among {e −iτ E j } r j=0 coincide, the result holds whith the P j 's replaced by Π j 's, the spectral projectors of P e −iτ H(0) | P B . ii) If, for any j = 0, . . . , r, the reduced resolvents
R Q (E j ) = (H(0) − E j )| −1 QB all exist, with Q = I − P , then Γ w (τ ) = − r j=0 P j W R Q (E j ) R Q (E j ) − R Q (E j )e −iτ (H(0)−E j )| QB − iτ I W P j (3.61)
iii) If B is a Hilbert space, and H(λ) is self-adjoint with dim P j = 1, we can express Γ w in yet another way. We write P j = |ϕ j ϕ j | and introduce dµ W j (E), j = 0, · · · , r, the spectral measures of the vectors W ϕ j = QW ϕ j , with respect to H(0)| QB . Then, if · denotes the Fourier transform,
Γ w (τ ) = − r j=0 τ 0 ds s 0 dt µ W j (t)e itE j P j . (3.62)
Proof of Proposition 3.2:
As we are to work in P B, we will write A P for P AP etc... Our assumption on τ makes the eigenvalues of e −iτ H P (0) distinct so that the P ′ j s are eigenprojectors of both H P (0) and e i τ H P (0) . Then, Lemma 3.3 shows that V (x) := P (e −iτ (H(0)+ √ xW ) P , x = λ 2 , satisfies the hypotheses of Proposition 3.1 with h = τ H P (0), R = G P (τ ) and τ > 0 fixed. Hence the result, making use of e −iτ H P (0) = e −iτ H(0) P . The last statement follows from (3.24).
We are now in a position to state the existence of a contraction semi-group on P B obtained by means of a weak limit for our specific time dependent Hamiltonian model. The following is a direct application of Proposition 3.2. by (2.17, 2.16, 2.6), let P = I ⊗ |Ω Ω|, and let {E j } j=0,···,r be the eigenvalues of h 0 associated with eigenprojectors {P j } j=0,···,r . Assume the values {e −iτ E j } j=0,···,r are distinct. Then, for any fixed 0 ≤ t < ∞,
Corollary 3.2 Let U (n, 0) be defined on H 0 ⊗ H, where H 0 is separable,lim λ→0 t/λ 2 ∈N e iτ tH(0)/λ 2 P U (t/λ 2 , 0)P = e tΓ w (τ ) on P B (3.63) where Γ w (τ ) = e iτ H(0) G(τ ) # = − τ 0 ds s 0 dt r j=0 n m=1 P j V * m e −it(h 0 +δm−E j ) V m P j (3.64)
generates a contraction semi-group and # corresponds to the set of eigenprojectors {P j } j=0,···,r .
Remarks:
0) The macroscopic time scale at which we observe the system is T = τ t/λ 2 → ∞. i) There are cases where Γ w (τ ) generates a group of isometries. ii) Again, if the e −iτ E j 's are not distinct, we have to take the spectral projectors of e −iτ h 0 instead of the P j 's in the definition of the operation #. iii) Note that the effective dynamics commutes with h 0 , so that no transition between the eigenspaces of h 0 can take place. However, if the e −iτ E j 's are not distinct, transitions between different eigenspaces of h 0 corresponding to the same eigenvalue of e −iτ h 0 are possible. iv) In case h 0 is non degenerate, r = d and we can write P j = |x j x j |, with x j the eigenvector associated with E j , and
Γ w (τ ) = − d j=0 d k=0 n m=1 | x k |V m x j | 2 τ 0 ds s 0 dt e −it(E k −E j +δm) |x j x j |,(3.65)
where the double integral equals
τ 0 ds s 0 dt e −itα = τ 2 /2 α = 0 1 α 2 (1 − e −iτ α ) − i α τ α = 0 (3.66)
Proof of Corollary 3.2: By Lemma 3.1 above, e iτ tH P (0)/λ 2 P U (t/λ 2 , 0)P = e iτ tH(0)/λ 2 P e −iτ (H(0)+λW ) P
t/λ 2 ,(3.67)
where conditions H1 and H2 are met and Proposition 3.2 applies. The fact that Γ w (τ ) generates a contraction semigroup in that case stems from the a priori bound, uniform in t, τ, λ, e iτ tH(0)/λ 2 P U (t/λ 2 , 0)P ≤ 1.
(3.68)
The expression for Γ w (τ ) comes from the explicit evaluation of (3.60) in our model.
Different Time Scales
Looking at the dependence in τ of the result in Corollary 3.2, we observe that we can obtain a different non-trivial effective evolution with our conventional weak limit approach, provided one further makes the time scale τ → 0 and, at the same time, increases the parameter t to t/τ 2 . This yields a macroscopic time scale given by T = t/(τ λ 2 ) → ∞. We'll come back to this point also, when we deal with the Heisenberg evolution of observables.
Using the first expression (3.24), one immediately gets
lim τ →0 lim λ→0 t/(τ λ) 2 ∈N e iτ tH(0)/(τ λ) 2 P U (t/(τ λ) 2 , 0)P = lim τ →0 e tΓ w (τ )/τ 2 ≡ e tΓ 1 ,(3.69)
where
Γ 1 = Γ 0 # = r j=0 P j Γ 0 P j , Γ 0 = − 1 2 n i=1 V i V * i . (3.70)
Note that under the hypotheses of Corollary 3.2, the spectral projectors of h 0 and e −iτ h 0 coincide when τ → 0.
This calls for a redefinition of the scaling, right from the beginning of the calculation, in order to arrive at the same result, without resorting to iterated limits, as above. This is at this point that we need to consider the dependence in τ of the previous steps.
We state below is our main theorem regarding this issue in the general Banach space framework under hypotheses H1 and H2. Actually, the application above is a consequence of the theorem to come. The study at positive temperature in Heisenberg picture of the forthcoming Sections will rely on this result as well.
Theorem 3.1 Suppose Hypotheses H1 and H2 hold true and further assume the spectral projectors P j , j = 0, · · · , r, of e −iτ H P (0) coincide with those of H P (0) on P B. Set
K # = r j=0 P j KP j , for K ∈ L(B).
A) Then, for any 0 < t 0 < ∞, there exists 0 < c < ∞ such that for any 0 ≤ t ≤ t 0 , the following estimate holds in the limit λ 2 τ → 0, λ 2 τ 2 → 0, and t/(λτ ) 2 ∈ N: e iH(0)t/(λ 2 τ ) P e −iτ (H(0)+λW ) P t/(λτ ) 2 − e t e iτ H(0) G P (τ ) # /τ 2 ≤ c(λ 2 τ 2 + λ 2 τ ). (3.71) B) Then, for any 0 < t 0 < ∞, there exists 0 < c < ∞ such that for any 0 ≤ t ≤ t 0 , the following estimate holds in the limit λ 2 τ → 0, τ → 0, and t/(λτ ) 2 ∈ N: e iH(0)t/(λ 2 τ ) P e −iτ (H(0)+λW ) P t/(λτ ) 2 − e −t(W 2 ) # /2 ≤ c(τ + λ 2 τ ).
(3.72)
Remarks: 0) If τ is small enough, the spectral projectors of e −iτ H P (0) and H P (0) on P B coincide. i) If τ is fixed, part A) of the Theorem coincides with Proposition 3.2 witht := t/τ 2 in place of t.
Proof: We only need to consider the case τ > 0 small, where Remark 0) applies. We proceed in two steps, using Lemmas 3.4 and 3.5 in sequence. Let x = λ 2 τ 2 . The expansions provided in Lemma 3.3 yield As e −iτ H P (0) = r j e −iτ E j P j , with P j independent of τ , the operator J(τ ) defined in (3.39) reads
P e −iτ (H(0)+λW ) P = e −iτ H P (0) + xG P (τ )/τ 2 + O(x 2 ),(3.J(τ ) = r j,k=0 P j G P (τ ) τ 2 P k α jk (τ ), where α jk (τ ) = τ (E j −E−k) e −iτ E j −e −iτ E k j = k ie iτ E j j = k. (3.75) Hence, α jk (τ ) = i + O(τ ), G P (τ ) τ 2 = − W 2 P 2 + O(τ ) and J(τ ) = O(1) as τ → 0.(3.76)
Now, using (3.21) with coupling constant x/τ (and the first remark following Lemma 3.3), we can write for x/τ small, uniformly in τ ,
e −iτ (H P (0)+ x τ J(τ )) = (3.77) e −iτ H P (0) + x τ −ie −iτ H P (0) τ 0 e isH P (0) J(τ )e −isH P (0) ds + O((x/τ ) 2 τ 2 ) = e −iτ H P (0) + x −ie −iτ H P (0) 1 0 e isτ H P (0) J(τ )e −isτ H P (0) ds + O(x 2 ),
where the operator in the bracket above is O(1) as τ → 0. Hence e −iτ (H P (0)+ x τ J(τ )) = 1 + O(x), uniformly in τ.
(3.78)
Thus, we apply Lemma 3.4, to get
P e −iτ (H(0)+λW ) P t x − e −i t x (τ H P (0)+xJ(τ )) = O(x 2 ),(3.79)
as x → 0, and x τ → 0, with a remainder uniform in τ . We now turn to the second step. We can write
e −i t x (τ H P (0)+xJ(τ )) = e −i t λ 2 τ (H P (0)+λ 2 τ J(τ )) ≡ e −i t y (H P (0)+yJ(τ ))
with y = λ 2 τ. (3.80) Therefore, by Lemma 3.5 and the last statment of (3.76),
e i t y H P (0) e −i t y (H P (0)+yJ(τ )) − e −itJ # (τ )) = O(y),(3.81)
uniformly in τ . Hence, for any given t 0 , we get the existence of a constant 0 < c < ∞, uniform in τ , such that for all 0 < t ≤ t 0 < ∞,
e i t λ 2 τ H P (0) P e −iτ (H(0)+λW ) P t (λτ ) 2 − e −itJ # (τ )) ≤ c(λ 2 τ + λ 2 τ 2 ), (3.82)
as λ 2 τ and λ 2 τ 2 go to zero in such a way that t/(λτ ) 2 ∈ N, which is part A) of the Theorem. Part B) follows from the first statements in (3.76) and of the fact that the projectors P j 's are independent of t.
As a direct Corollary, we get,
0 ≤ t ≤ t 0 , lim τ →0,λ 2 τ →0 t/(τ λ) 2 ∈N e iτ tH(0)/(τ λ) 2 P U (t/(τ λ) 2 , 0)P = e tΓ # 0 , (3.83) where Γ 0 # = r j=0 P j Γ 0 P j , and Γ 0 = − 1 2 n i=1 V i V * i .
Heisenberg representation for non-zero temperature
From now on, we stick to our model Hamiltonian characterized by hypothesis H0. We first express the evolution at positive temperature of observables B of the small system (4.4) after k repeated interactions as the action of the k-th power of an operator U β (λ, τ ) on H 0 . This reflects the Markovian nature of our model. This is done in Proposition 4.1. This allows us to apply Theorem 3.1 again to compute the weak limit in Theorem 4.1. Let us mention here already that we perform a complete analysis of the special case where both the small system and the individual spins of the chain live in C 2 in the last Section of the paper.
Let us define the equilibrium state ω(β) N of a chain of N spins at inverse temperature β by a tensor product of individual diagonal density matrices of the form
r(β) = 1 1 + n j=1 e −βδ j 1 0 · · · 0 0 e −βδ 1 · · · 0 . . . . . . . . . 0 · · · 0 e −βδn = e −βδa * a Z(β) , (4.1) in the basis {ω, x 1 , · · · , x n } of C n+1 j , i.e. ω(β) N = r(β) ⊗ r(β) ⊗ · · · ⊗ r(β). (4.2)
The individual density matrices r(β) are defined by Gibbs prescription for the Hamiltonians at each site
n i=1 δ i a * i a i (4.3)
corresponding to our model (2.7) Our spin chain is of finite length N , but, as we will see below, only the first k spins matter to study the time evolution up to time k. This will allow us to take the thermodynamical limit by hand. If ρ is any state on C d+1 , the initial state of the small system plus spin chain is ρ ⊗ ω(β) N . We shall study the Heisenberg evolution of observables of the form
B ⊗ I H , where B ∈ M d+1 (C), defined by B β (k, λ, τ ) = Tr H ((I ⊗ ω N (β)) U (k, 0) −1 (B ⊗ I H )U (k, 0)), (4.4)
where, for any A ∈ L(H 0 ⊗ H),
Tr H (A) = S x i ⊗ x S | A x j ⊗ x S i,j∈{0,···,d} with x 0 = ω, (4.5)
denotes the partial trace taken on the spin variables only. Hence, the expectation in the state ρ of the observable B after k interactions over a time interval of length kτ with the chain at inverse temperature β is given by
B(k, β) ρ = Tr C d+1 (ρB β (k, λ, τ )).
(4.6)
Remark:
In case H 0 is infinite dimensional, the definitions (4.4) and (4.5) hold, mutatis mutandis. For instance, consider B ∈ L(H 0 ) in (4.4), where (4.5) should be read as
Tr H (A) = S I H 0 ⊗ x S | A I H 0 ⊗ |x S , (4.7)
with a slight abuse of notations.
Markov Properties
Recall that
U (k, 0) −1 (B ⊗ I H )U (k, 0) = U * 1 U * 2 · · · U * k (B ⊗ I H )U k U k−1 · · · U 1 , (4.8)
where U j is non-trivial on C d+1 ⊗ C n+1 j only. Let us specify a bit more the partial trace operator Tr H ((I ⊗ ω N (β)) A), where A is an operator on C d+1 ⊗ Π N j=1 C n+1 j .
Lemma 4.1 Let us denote the matrix elements of A as follows
A i,j S,S ′ = x i ⊗ X S |A x j ⊗ X S ′ , (4.9)
where i, j belong to {0, · · · , d}, and S, S ′ run over subsets of {{1, · · · , N } × {1, · · · , n}} N as in (2.3) Then
Tr H ((I ⊗ ω N (β)) A) i,j = S e −β n l=1 δ l |S| l (1 + n l=1 e −βδ l ) N A i,j S,S (4.10)
where, for
S = {(k 1 , i 1 ), (k 2 , i 2 ), · · · , (k m , i m )} ⊂ (N × {1, 2, · · · , n}) m (4.11)
with all 1 ≤ k j ≤ N distinct and m = 0, · · · , N ,
|S| l = #{k r s.t. i r = l}.
(4.12)
Proof: Follows directly from
ω N (β)X S = Π m r=1 e −βδ ir (1 + j e −δ j β ) N X S = e −β n l=1 δ l |S| l (1 + j e −δ j β ) N X S .
(4.13)
We now further compute the action of U (k, 0) given by the product of U ′ j s. Let us denote the vectors ω ⊗ X S and x j ⊗ X S by n 0 ⊗ |n 1 , n 2 , · · · , n N ≡ n 0 ⊗ | n , where n 0 ∈ {0, 1, · · · d}, and n j ∈ {0, 1, · · · n}, for any j = 1, · · · N , with ω ≃ 0 and x k ≃ k and X {(1,n 1 ),···,(N,n N )} ≃ | n .
Recall that (4.14) where e −iτ H j is diagonal. More precisely, with the convention δ 0 = 0, e −iτ H j n 0 ⊗ |n 1 , n 2 , · · · , n N = e −iτ N k=1 k =j δn k n 0 ⊗ |n 1 , n 2 , · · · , n N .
U j = e −iτ H j e −iτ H j ,
(4.15)
Lemma 4.2 Denoting the k-independent matrix elements of e −iτ H k | C d+1 ⊗C n+1 k =Ũ k | C d+1 ⊗C n+1 k by U n,n ′ m,m ′ = n ⊗ m|Ũ k n ′ ⊗ m ′ , (4.16) we have for any N ≥ k U k U k−1 · · · U 2 U 1 n 0 ⊗ |n 1 , · · · , n N = (4.17) m 0 ∈{0,···,d} k m∈{0,···,n} k e −iτ ϕ( m, n) U m k 0 ,m k−1 0 m k ,n k · · · U m 2 0 ,m 1 0 m 2 ,n 2 U m 1 0 ,n 0 m 1 ,n 1 m k 0 ⊗ |m 1 , m 2 , · · · , m k , n k+1 , · · · , n N , where ϕ( m, n) = k j=1 j<l≤N δ n l + l<j δ m l (4.18)
Proof: Consequence of the iteration of formulae of the type U 1 n 0 ⊗ |n 1 , · · · , n N = m 1 0 =0,1,···,d m 1 =0,1,···,n e −iτ j>1 δn j U m 1 0 ,n 0 m 1 ,n 1 m 1 0 ⊗ |m 1 , n 2 , n 3 , · · · , n N .(4.19)
A consequence of these formulae is that we can consider spin chains consisting in k spins only:
Lemma 4.3 For any N ≥ k, Tr H (I ⊗ ω N (β) U * 1 U * 2 · · · U * k (B ⊗ I H )U k U k−1 · · · U 1 ) = Tr H (I ⊗ ω k (β) U * 1 U * 2 · · · U * k (B ⊗ I H )U k U k−1 · · · U 1 )(4.20)
Proof: Obvious from the tensor product structure of ω N (β).
To proceed, let us adopt the following block notation
U = e −iτ (H(0)+λW ) = U 0,0 U 0,1 · · · U 0,n U 1,0 U 1,1 · · · U 1,n . . . . . . . . . . . . U n,0 U n,1 · · · U n,n (4.21) where U m,m ′ = U 0,0 m,m ′ U 0,1 m,m ′ · · · U 0,d m,m ′ U 1,0 m,m ′ U 1,1 m,m ′ · · · U 1,d m,m ′ . . . . . . . . . . . . U d,0 m,m ′ U d,1 m,m ′ · · · U d,d m,m ′ . (4.22)
In terms of the notations of the previous Section, U = P U P P U Q QU P QU Q , (4.23)
we have the identifications
P U P ≃ U 0,0 , QU Q ≃ U 1,1 · · · U 1,n . . . . . . . . . U n,1 · · · U n,n , P U Q ≃ ( U 0,1 · · · U 0,n ) , QU P ≃ ( U 1,0 · · · U n,0 ) T .{n 0 ⊗ |n 1 , · · · , n k } = {n 0 ⊗ | n } read ñ 0 ⊗ ñ|(U k · · · U 1 ) * B ⊗ I H (U k · · · U 1 ) n 0 ⊗ n = (4.27) e −iτ (ϕ(0, n)−ϕ(0, ñ))
m∈{0,···,n} k (Vñ 1 ,m 1 · · · Vñ k ,m k BU m k ,n k · · · U m 1 ,n 1 )ñ 0 ,n 0 Proof: Expand the products and make use of Lemma 4.2 and (4.18). (1 + n j=1 e −δ j β ) k V n 1 ,m 1 · · · V n k ,m k BU m k ,n k · · · U m 1 ,n 1 (4.28) in various limiting cases as λ and/or τ go to zero, with the notation | n| l = ♯{n r s.t. n r = l} = |S| l .
(4.29)
We introduce operators on the Hilbert space M d+1 (C) equipped with the scalar product A|B = Tr(A * B), for any A, B ∈ M d+1 (C) by
U m,m ′ (A) := V m ′ ,m A U m,m ′ , (m, m ′ ) ∈ {0, 1, · · · , n} 2 .
(4.30)
These operators are linear and one has with respect to the above scalar product,
U * m,m ′ (·) = (V m ′ ,m · U m,m ′ ) * = U m,m ′ · V m ′ ,m . (4.31)
The composition of such operators will be denoted as follows
U m ′ ,n ′ U m,n (A) = V n ′ ,m ′ V n,m A U m,n U m ′ ,n ′ .
(4.32)
We are now in a position to express the Markovian nature of the evolution of our observables:
Proposition 4.1 In terms of the operators defined above, we can write (1 + n j=1 e −δ j β ) k U m 1 ,n 1 · · · U m k ,n k (B).
B β (k, λ, τ ) = 1 (1+ n j=1 e −δ j β ) k U 0,0 + e −βδ 1 U 0,1 + · · · + e −βδn U 0,n +U 1,0 + e −βδ 1 U 1,1 + · · · + e −βδn U 1,n + U n,0 + e −βδ 1 U n,1 + · · · + e −βδn U n,n k (B) ≡ U β (λ, τ ) k (B).
(4.34)
Furthermore introducing Y m,n = e −δnβ U m,n , we get B β (k, λ, τ ) = 1 (1 + n j=1 e −δ j β ) k n=(n 1 ,···,n k ) m=(m 1 ,···,m k ) Y m 1 ,n 1 · · · Y m k ,n k (B).
(4.35)
There are (n + 1) 2 distinct operators Y m,m ′ in that expression, and the set of vectors n, m in the sum yields all different ways of composing k of them. Therefore B β (k, λ, τ ) = 1 (1 + n j=1 e −δ j β ) k (Y 0,0 + · · · + Y 0,n + · · · + Y n,0 + · · · + Y n,n ) k (B).(4.36)
Remark:
The formula of Proposition 4.1 holds if H 0 is a separable Hilbert space, provided the decomposition of operators A in (4.21) is interpreted as A pq ∈ L(H 0 ), q, p ∈ {1, · · · , n}, with (4.37) and the identification H 0 ⊗ C|q ≃ H 0 , for all q.
A pq = I H 0 ⊗ |p p| A I H 0 ⊗ |q q|,
Weak Limit in the Heisenberg Picture
The λ-dependence in B β (k, λ, τ ) comes from the definition (4.38) which implies that the U n,m 's depend on λ as well, in an analytic fashion, and will be denoted U n,m (λ). Expliciting the λ dependence in B β (k, λ, τ ), the weak limit corresponds to taking k = t/λ 2 and computing the behavior of B β (t/λ 2 , λ, τ ), as λ → 0 (keeping τ fixed). We shall use the same strategy as in the previous Section and Lemma 3.1 to identify the weak limit by means of perturbation theory. We shall also eventually consider the possibility of letting τ → 0, therefore we explicit the behavior in τ of the expansions below.
U = U τ (λ) = e −iτ (H(0)+λW ) ,
Consequently, with (4.24) and Corollary 3.1, we get Lemma 4.5 Let U be given by (4.38), with H(0), W self adjoint and satisfying H1, and further assume H(0) is diagonal with respect to the basis (3.5). If U m,m ′ (λ) is defined by (4.30) As λ → 0, we get the expansions with the convention δ 0 = 0 and Z(β) = n j=0 e −δ j β . Recall that We have thus shown the Lemma 4.6 Assume the hypotheses of Lemma 4.5. Then
U 0,0 (λ) = U 0,0 (0) + λ 2 U (2) 0,0 + O(λ 4 τ 4 ) (4.39) U m,m ′ (λ) = U m,m ′ (0) + λ 2 U (2) m,m ′ + O(λ 4 τ 4 ), m, m ′ ≥ 1 (4.40) U 0,m (λ) = λ 2 U (1) 0,m + O(λ 4 τ 4 ), m ≥ 1 (4.41) U m,0 (λ) = λ 2 U (1) m,0 + O(λ 4 τ 4 ), m ≥ 1 (4.42) where, for all 0 ≤ m, m ′ ≤ n U m,m ′ (0)(B) = δ m,m ′ e iτB β (k, λ, τ ) = U β (λ, τ ) k (B).U β (λ, τ ) = U 0,0 (0) + λ 2 Z(β) n m=1 e −βδm U (1) 0,m + U (2) m,m + U (1) m,0 + U (2) 0,0 + O(λ 4 τ 4 ) ≡ U 0,0 (0) + λ 2 Z(β) −1 T β + O(λ 4 τ 4 ), (4.51) with T β = T β (τ ) = O(τ 2 ).
The above operator enjoys the following symmetry property and to the structure of T β , the result will be proven once we show that for all A, B, C ∈ M d+1 (C)
Tr(B * ABC + B * C * BA * ) = Tr(BAB * C + BC * B * A * ).
(4.55)
But this follows from TrB = TrB T , where · T denotes the transpose, and from the cyclicity of the trace again.
Recall also the property (4.56) and the fact that in case the spectrum {E j } j=0,···,d of h 0 is non-degenerate and {|x j } j=0,···,d denotes the corresponding eigenvectors, the unitary U 0,0 (0) has degenerate spectrum:
U β (λ, τ )(I) = I ⇒ T β (I) = 0,U 0,0 (0)(|x j x k |) = e iτ (E j −E k ) |x j x k |, ∀ 0 ≤ j, k ≤ d. (4.57)
That is, σ(U 0,0 (0)) = {e iτ (E j −E k ) } 0≤j,k≤d , so that 1 is d + 1 times degenerate at least.
We are in the same position as in the proof of Proposition (3.2). Therefore, we can compute the weak limit from Proposition 3.1 immediately to get the following Theorem 4.1 Let U β (λ, τ ) be given by (4.47), and U 0,0 (0), T β by (4.51). Let {e iτ ∆ l } l=1,···,r be the set of distinct eigenvalues of U 0,0 (0) and denote by P l the corresponding orthogonal projectors. Then
lim λ→0 t/λ 2 ∈N U 0,0 (0) −t/λ 2 B β (t/λ 2 , λ, τ ) = (4.58) lim λ→0 t/λ 2 ∈N U 0,0 (0) −t/λ 2 U β (λ, τ ) t/λ 2 (B) = e tΓ w β (B), were Γ w β (B) = 1 Z(β) U 0,0 (0) −1 T β # (B), (4.59)
with # corresponding to the set of projectors {P l } l=1,···,r .
Remarks: 0) In order to make the generator Γ w β completely explicit, one needs to analyse the properties T β , i.e. of the operators V j defining the coupling, within the eigenspaces of U 0,0 (0). A non trivial example is worked out in Section 6, see Proposition 6.1. i) The degeneracy of the eigenvalue 1 of U 0,0 (0) is responsible for the existence of a nontrivial invariant sub-algebra of observables which is the commutant of h 0 .
As in Section 2, we generalize our result to the regime λ 2 τ → 0, τ → 0, by switching to the macroscopic time scale T = t/(λ 2 τ ) → ∞. We first compute
Γ β (B) = lim τ →0 U 0,0 (0) −1 T β Z(β)τ 2 (B) = − 1 2Z(β) (W 2 0,0 B + BW 2 0,0 ) + (4.60) 1 Z(β) n m=1 e −δmβ W m,0 BW 0,m − 1 2 (W 2 m,m B + BW 2 m,m ) + W 0,m BW m,0 ,
which, using the following formulas for m ≥ 1 (4.61) to express the operators W mm ′ in terms of V m , eventually becomes
W 0,m = V * m , W m,0 = V m , W 2 m,m = V m V * m , W 2 0,0 = n j=1 V * j V j ,Γ β (B) = 1 Z(β) n m=1 e −βδm V m BV * m − 1 2 (V m V * m B + BV m V * m ) +V * m BV m − 1 2 (V * m V m B + BV * m V m ). (4.62)
We note here that this operator has the form of the dissipative part of a Lindblad generator. We'll come back to this operator Γ β in connection to the modelization in terms of Quantum Noises proposed in [AP] and [LM], in the next Section.
Corollary 4.1 Assume the hypotheses of Theorem 4.1. Then with t/(τ λ) 2 = k ∈ N,
lim τ →0,λ 2 τ →0 t/(τ λ) 2 ∈N U 0,0 (0) −t/(τ λ) 2 B β (t/(τ λ) 2 , λ, τ )) = (4.63) lim τ →0,λ 2 τ →0 t/(τ λ) 2 ∈N U 0,0 (0) −t/(τ λ) 2 U β (λ, τ ) t/λ 2 (B) = e tΓ β # (B),
were Γ β (B) is defined in (4.62).
Proof: We can simply repeat the arguments of the proof Theorem 3.1 once we note the following facts: i) The operator U 0,0 (0) = e iτ [h 0 ,·] is unitary on M d+1 (C), with spectral projectors that are independent of τ as τ → 0 and eigenvalues of the form e iτ ∆ j . ii) Introducing x = (λτ ) 2 , (4.51) states that uniformly in τ ,
U β (λ, τ ) = U 0,0 (0) + xT β (τ )/(τ 2 Z(β)) + O(x 2 ), (4.64)
where T β (τ )/τ 2 → Γ β as τ → 0.
Evolution of states
Let us close this Section by briefly recalling some consequences of these results about the evolution of states, i.e. trace one positive matrices. This is conveniently done in our setup by using duality with respect to the scalar product A|B = Tr(A * B). If Γ is the generator of the dynamics of observables, B is an observable and ρ is a state, then for any t ∈ R, Tr(ρe tΓ (B)) = Tr(e tΓ * (ρ)B) (4.65)
where the generator of the dynamics of the states is Γ * such that for all states ρ and observables B,
Tr((Γ * (ρ)) * B) = ρ|Γ(B) = Γ * ρ|B . (4.66)
In the particular case where the observables P jk = |x j x k |, with the notations of (4.57), form an orthonormal basis of eigenvectors of the restricted uncoupled evolution U 0,0 , the corresponding eigenprojectors are denoted by Π jk and act as
Π jk (B) = P jk Tr(|x k x j |B) = P jk x j |Bx k H 0 , (4.67)
where the subscript H 0 denotes the scalar product within H 0 . Hence, the # operation on the operator Γ with respect to the projectors Π jk is given by
Γ # (B) = j,k Π jk ΓΠ jk (B) = j,k |x j x k | x j |Γ(|x j x k |)x k H 0 x j |Bx k H 0 . (4.68)
Therefore, one computes that the corresponding generator of states, (Γ # ) * is given by
Γ # * (ρ) = j,k |x j x k | x k |Γ(|x j x k |) * x j H 0 x j |ρx k H 0 . (4.69)
Consequently,
(Γ # ) * = j,k Π jk Γ * Π jk = (Γ * ) # .
(4.70)
We note that states defined as functions of the Hamiltonian h 0 of the small system form an invariant subspace of sets whose Markovian dynamics is characterized by the scalars { x j |Γ(|x j x j |)x j H 0 } j=0,···,d .
Schrödinger Evolution
Let us start with the Schrödinger effective evolution under the following assumptions:
H3: Hypothesis H1 holds with B a Hilbert space and P , H(λ) = H(0)+λW self-adjoint.
In the scaling adopted here, the number of interactions n has to grow like n = t/τ. This is in keeping with by the fact that in all cases considered so far, n = t/(λτ ) 2 = t/τ . Note that the macroscopic time T = τ n = t is finite here. Therefore, according to the analysis of Section 3, we are led to study
P U (t/τ, 0)P = P e −i(τ H(0)+ √ τ W ) P t/τ , as τ → 0, t/τ ∈ N * . (5.1)
This limit is easily computed by applying the following version of Chernoff's Theorem, see e.g. [BR], [D2] or [Paz], which suffices for our purpose:
S(τ ) = I − iτ H(0) P − τ 2 (W 2 ) P + O(τ 2 ) . (5.4) It thus implies S ′ (τ )| τ =0 = −iH(0) P − (W 2 ) P 2 = Γ ∈ L(P B). (5.5) Now Γ is dissipative, since ∀ϕ ∈ P B ℜ ϕ|Γϕ = −ℜ ϕ|P W QW P ϕ /2 = − QW P ϕ B /2 ≤ 0. (5.6)
Hence, by Lumer-Phillips, see [Paz], Γ generates a contraction semigroup. Therefore
Theorem 5.2 Under the hypothesis H3, for any t > 0 fixed,
s − lim τ →0 t/τ ∈N P U (t/τ, 0)P = s − lim τ →0 t/τ ∈N P e −i(τ H(0)+ √ τ W ) P t/τ = e −t(iH(0) P + (W 2 ) P 2 ) . (5.7)
Remark: Specializing to our model Hamiltonian, we get that the effective dynamics on P B is
e −t(ih 0 + 1 2 j V * j V j ) . (5.8)
Apart from the self-adjoint part h 0 stemming from the uncoupled evolution, the main difference with respect to the corresponding weak coupling result in Corollary 3.3, lies in the absence of the # operation on the dissipative part 1 2 j V * j V j of the generator. This prevents the spectral subspaces of h 0 from being invariant under the effective dynamics.
Heisenberg Evolution
Let us now turn to the more interesting case of the Heisenberg dynamics of observables when the spins are at equilibrium at inverse temperature β. We assume the general hypothesis H0, i.e. we stick to our matrix model, even though certain results below hold for more general situations.
The analysis of Section 4 shows that the evolution of an observable B ∈ M d+1 (C) after k repeated interactions reads
B → B β (k, λ, τ ) = U β (λ, τ ) k (B) (5.9)
with U β (λ, τ ) defined by (4.47), where we explicited the dependence in τ in the notation. We want to apply Chernoff's Theorem again to the operator valued function τ → U β (1/ √ τ , τ ) on L(M d+1 (C)). In order to check the first hypotheses we recall the formula (see (4.10))
U β (λ, τ )(B) = Tr H (I ⊗ ω 1 (β))U −1 (1, 0)(B ⊗ I)U (1, 0) (5.10) = n q=0 e −βδq Z(β) B(τ ) qq , where B(τ ) qq = (U −1 (1, 0)(B ⊗ I)U (1, 0)) qq = P q U −1 (1, 0)(B ⊗ I)U
(1, 0)P q according to the block notation (4.21), with the corresponding orthogonal projectors P q . Identifying P q C (n+1)(d+1) with H 0 = C d+1 , we deduce from the above formula that U β (λ, τ ) is a contraction for any value of the parameters:
U β (λ, τ )(B) H 0 ≤ n q=0 e −βδq Z(β) B(τ ) qq H 0 (5.11) ≤ n q=0 e −βδq Z(β) P q U −1 (1, 0)(B ⊗ I)U (1, 0)P q C (n+1)(d+1) ≤ n q=0 e −βδq Z(β) (B ⊗ I) C (n+1)(d+1) = B H 0 .
Moreover, U β (1/ √ τ , τ )| τ =0 = I, so we are left with the computation of the derivative w.r.t.
τ at the origin. This involves the control of the operator U τ (λ) (3.20) as τ → 0 and λ = 1/ √ τ → ∞, as in the previous paragraph. Let us get estimates in a more systematic way than above. So far, all our estimates are derived for both λ and τ going to zero or at most finite. However, the expansion of U τ (λ) in powers of λ is convergent, with τ dependent coefficients we control sufficiently well. Indeed, (3.33) yields
U τ (λ) = e −iτ H(0) Θ(λ, τ ) = n≥0 e −iτ H(0) Θ n (λ, τ ),(5.12)
where Θ n contains n operators W and satisfies Θ n (λ, τ ) = O((τ λ) n /n!).
(5.13)
Using the fact that (λτ ) n = τ n/2 → 0 and that W is off-diagonal with respect to P and Q, we get that the replacement of λ by 1/ √ τ doesn't spoil the estimates as τ → 0 given in Proposition 4.1 and Lemma 4.5. Those together with the computation (4.62) yield
U β (1/ √ τ , τ )(B) = e iτ h 0 Be −iτ h 0 + (Z(β)τ ) −1 T β (τ )(B) + O(τ 2 ) ≡ e iτ h 0 Be −iτ h 0 + τ Γ β (B) + O(τ 2 ),(5.14)
where, see (4.62),
Γ β (B) = 1 Z(β) n m=1 e −βδm V m BV * m − 1 2 (V m V * m B + BV m V * m ) +V * m BV m − 1 2 (V * m V m B + BV * m V m ). (5.15)
Hence, the derivative at the origin exists and is given by
U β (1/ √ τ , τ ) ′ (B)| τ =0 = i[h 0 , B] + Γ β (B). (5.16)
We recognize at once that Γ β (B) is the dissipative part of a Lindblad operator of the form
2m j=1 L j BL * j − 1 2 L j L * j B + BL j L * j (5.17) with L j = e −βδ j /2 Z(β) V j , 1 ≤ j ≤ m and L j = 1 Z(β) V * j , m + 1 ≤ j ≤ 2ms − lim τ →0 t/τ ∈N B β (t/τ, 1/ √ τ , τ ) = s − lim τ →0 t/τ ∈N U β (1/ √ τ , τ ) t/τ (B) = e t(i[h 0 ,·]+Γ β (·)) (B) (5.20)
with a Lindblad generator i[h 0 , ·] + Γ β (·) explicited in (5.17)
Remarks:
i) Let us make a comparison of the above with the results of [AP], Section IV.2, which concern similar generators as ours. More precisely, (2.6) corresponds to a particular case of the Hamiltonian of eq. (15) in [AP], with D ij = 0, ∀i, j. In [AP], the choice of time scale τ and coupling λ is such that λ 2 τ = 1, τ → 0. A supplementary structure is present in that work which consists in making the suitably renormalized spins forming the chain merge in the limit τ → 0 to yield a heat bath represented by a Fock space of quantum noises. The limit τ → 0 performed in the language adopted in [AP] exists and yields a quantum Langevin equation for the whole limiting system consisting in the original small system in interaction with a field of quantum noises. When restricted to H 0 , the effective dynamics of observables at zero temperature corresponds to a contraction semigroup generated by
Γ ∞ (·) = i[h 0 , ·] + n m=1 V * m · V m − 1 2 (V * m V m · + · V * m V m ) ,(5.21)
which coincides with Theorem 5.3 at β = ∞.
ii) The generator Γ β coincides with the generator (4.62) obtained in Corollary 4.1 in the scaling λ 2 τ → 0, τ → 0, modulo the # operation, which appears as a trade mark of the perturbative regime.
The Continuous Limit
For completeness, we mention here the easier cases of continuous limit characterized by τ → 0 and λ constant. The omitted proof are quite analogous to those of the previous Section.
First considering the Schrödinger picture, we get Proposition 5.1 Assume the hypothesis H3 holds and fix λ = 1. Then,
s − lim τ →0 t/τ ∈N P U (t/τ, 0)P = s − lim τ →0 t/τ ∈N P e −iτ (H(0)+W ) P t/τ = e −itH(0) P . (5.22)
The Heisenberg evolution also yields a unitary effective evolution in the continuous limit:
Proposition 5.2 Consider the matrix model of Section 2 and fix λ = 1. Then,
lim τ →0 t/τ ∈N B β (t/τ, τ, 1) = e i[h 0 ,·] (B). (5.23)
In order to make explicit the results of Section 4, we provide below a detailed analysis of the case d = n = 1.
6 The case d = n = 1
In that Section, we focus on the first non-trivial case where the small system lives on C 2 and the heat bath is formed by a chain of spins 1/2. We provide explicit formulas for T β and T # β which are valid for any coupling operator V appearing in (2.8). We further diagonalize the restriction of T β to the degenerate subspace Ker(U 0,0 (0) − 1) in order to determine the subalgebra of observables invariant under the effective dynamics in the weak coupling limit (keeping τ fixed).
For H 0 = C 2 , H = ⊗ j≥1 C 2 , we write for t ∈ [τ k − 1, τ k[ in H 0 ⊗ C 2 k , H(t, λ) = H(λ) = H(0) + λW, (6.1)
where H(0) = h 0 ⊗ I + I ⊗ δa * a, W = V * ⊗ a + V ⊗ a * . (6.2)
We choose, without loss of generality, h 0 = ǫσ z , ǫ = 0, so that we have in the ordered basis {ω ⊗ ω, x ⊗ ω, ω ⊗ x, x ⊗ x} H(λ) = ǫσ z λV * λV δI + ǫσ z , with the convention σ z = −1 0 0 1 . (6.3)
Specifying the results of the previous sections to the case under study, we can write, uniformly in β, as λ → 0, U β (λ) = U 0,0 (0) + λ 2 T β 1 + e −δβ + O(λ 4 ), (6.4) with T β (B) = F 0,1 (−τ )BF 1,0 (τ ) + G 0,0 (−τ )Be −iτ H 0,0 (0) + e iτ H 0,0 (0) BG 0,0 (τ ) (6.5) + e −δβ (F 1,0 (−τ )BF 0,1 (τ ) + G 1,1 (−τ )Be −iτ H 1,1 (0) + e iτ H 1,1 (0) BG 1,1 (τ )).
We use the norm induced by the scalar product A, B = Tr(A * B), i.e. the Hilbert-Schmidt norm. As easily verified, an orthonormal basis of eigenvectors for the unitary operator U 0,0 (0)(·) = e iτ ǫσz · e −iτ ǫσz , with associated eigenvalues, is provided by {Î,σ z , σ − , σ + } ←→ {1, 1, e −2iτ ǫ , e 2iτ ǫ }, (6.6) where σ + = 0 0 1 0 , σ z = 0 1 0 0 ,Î = I/ √ 2 andσ z = σ z / √ 2. (6.7)
Let us compute T β restricted to the subspace Ker (U 0,0 (0) − 1) appearing in T # β .
Lemma 6.1 With respect to the orthonormal basis {Î,σ z }, and with the notation
A OD = 0 b c 0 if A = a b c d ∈ M 2 (C) (6.8)
we have T β | {Î,σz} = 0 T β 1,2 0 T β 2,2 , (6.9)
where T β 1,2 = |(F OD 1,0 ) 2,1 | 2 − |(F OD 1,0 ) 1,2 | 2 (1 − e −δβ ) T β 2,2 = −( F OD Proof: The first column is proportional to T β (I) = 0. The second column of the matrix is given by 1 2
Tr(T β (σ z )) Tr(σ z T β (σ z )) , (6.12)
where, dropping the positive argument τ in F and further making use of (3.36) and (3.37),
T β (σ z )) = F * 1,0 σ z F 1,0 + G * 0,0 σ z e −iτ H 0,0 (0) + e iτ H 0,0 (0) σ z G 0,0 + e −δβ F * 0,1 σ z F 0,1 + G * 1,1 σ z e −iτ H 1,1 (0) + e iτ H 1,1 (0) σ z G 1,1 . (6.13)
Further making use of the cyclicity of the trace, [σ z , H n,n (0)] = 0, σ 2 z = I and of (3.36) and (3.37) again, we can write Tr(σ z T β (σ z ))) = Tr(σ z F * 1,0 σ z F 1,0 ) − Tr(F * 1,0 F 1,0 ) + e −δβ (Tr(σ z F * 0,1 σ z F 0,1 ) − Tr(F * 0,1 F 0,1 )). (6.14)
Explicit computations on 2 × 2 matrices yields the first equality in (6.10). Let us turn to (6.11). From the definitions (3.23) and (2.8), we have which yields the expression for T β i,j . By similar manipulations we get Tr(IT β (σ z )) = Tr(F * 1,0 σ z F 1,0 ) − Tr(σ z F * 1,0 F 1,0 ) + e −δβ (Tr(F * 0,1 σ z F 0,1 ) − Tr(σ z F * 0,1 F 0,1 )). (6.20)
F (τ ) = O F 0,1 (τ ) F 1,0 (τ ) O ,(6.
Now, for any F ∈ M 2 (C), Tr(σ z (F F * − F * F )) = 2(|F 21 | 2 − |F 12 | 2 ), (6.21) so that we get the first line of (6.10).
We also need to compute Tr(σ − T β (σ + )) and Tr(σ + T β (σ − )) to get T # β . Lemma 6.2 By explicit computation and Lemma 4.7, we have Tr(σ − T β (σ + )) = Tr(σ + T β (σ − )) = (F 1,0 ) 1,1 (F 1,0 ) 2,2 + e iτ ǫ ((G 0,0 ) 1,1 + (G 0,0 ) 2,2 ) (6.22) + e −δβ (F 0,1 ) 1,1 (F 0,1 ) 2,2 + e iτ ǫ (e iτ δ (G 1,1 ) 1,1 + e −iτ δ (G 1,1 ) 2,2 ) .
It remains to diagonalize the restriction of T β to span(Î,σ z ) to have a complete description of the generator of the effective evolution. Introducing µ = T β 1,2 , ν = T β 2,2 , (6.23) we actually get by perturbation theory, Lemma 6.3 Assume ǫτ / ∈ Zπ. Then, for λ > 0, there exists a continuous set of eigenprojectors and eigenvalues of U β (λ, τ ) denoted respectively by {Π j (λ)} j=1,···,4 and {u j (λ)} j=1,···,4 such that u 1 (λ) = 1 + O(λ 4 ), u 2 (λ) = 1 − λ 2 F OD 1,0 2 + O(λ 4 ), u 3 (λ) = e 2iτ ǫ + λ 2 Tr(σ − T β (σ + )) + O(λ 4 ), u 4 (λ) = e −2iτ ǫ + λ 2 Tr(σ + T β (σ − )) + O(λ 4 ), (6.24) and Π 1 (λ)(B) = Tr((I − µ ν σ z )B) 2 I + O(λ 2 ), Π 2 (λ)(B) = Tr(σ z B) 2 ( µ ν I + σ z ) + O(λ 2 ) Π 3 (λ)(B) = Tr(σ − B)σ + + O(λ 2 ), Π 4 (λ)(B) = Tr(σ + B)σ − + O(λ 2 ). (6.25)
Moreover, Π 0 := Π 1 (0) + Π 2 (0), Π 3 (0) and Π 4 (0) are the spectral projectors of U 0,0 (0) and {Π j (0)} j=1,···,4 are those of T β .
Hence, we obtain the Proposition 6.1 Let t/λ 2 = k ∈ N, and consider the Hamiltonian (6.3). Then lim λ→0 U 0,0 (0) −t/λ 2 B(t/λ 2 , β, λ) = lim λ→0 U 0,0 (0) −t/λ 2 U β (λ, τ ) t/λ 2 (B) = e tΓ w β (B), (6.26)
were Γ w β = 1 1 + e −δβ (− F OD 1,0 2 Π 2 (0) (6.27) + e −2iτ ǫ Tr(σ − T β (σ + ))Π 3 (0) + e 2iτ ǫ Tr(σ + T β (σ − ))Π 4 (0)).
The dynamics of any observable is thus fully determined from these formulas.
73)with G P (τ )/τ 2 = O(1) and reminder uniform in τ → 0. Hence,P e −iτ (H(0)+λW ) P = 1 + O(x),uniformly in τ.(3.74)
finally denote the inverse of U = (U n,n ′ m,m ′ ) by V = (V n,n ′ m,m ′ ) = U −1 = (U −1 n,n ′ m,m ′ ) ∈ M (1+d)(1+n) (C),(4.25) so that we have for any m and n U * n,m = V m,n ∈ M 1+d (C). (4.26) With these notations, we have Lemma 4.4 The matrix elements of U (k, 0) −1 (B ⊗ I H ) U (k, 0) in the orthonormal basis
The above Lemmas and (4.4) lead us to the study of the matrix in M d+1 (C) B β (k, λ, τ ) = n=(n 1 ,···,n k ) m=(m 1 ,···m k ) e −β n l=0 δ l | n| l
( 4 . 33 )
433Proof: By definition of U m,n we have B β (k, λ, τ ) = n=(n 1 ,···,n k ) m=(m 1 ,···m k )
m ′ (B) = δ m,m ′ (G m,m (−τ )Be −iτ Hm,m(0) + e iτ Hm,m(0) BG m,m (τ )allows us to perform the analysis of the operator defined in Proposition 4.1 U β (λ, τ ) = Z(β) −1 0≤m≤n 0≤l≤n U l,m (λ)e −δmβ , as λ → 0, (4.47)
,m (0) = H 0,0 (0) + δ m ≃ h 0 + δ m , (4.49) we get for all 0 ≤ m ≤ n U m,m (0)(B) = U 0,0 (0)(B) ≃ e iτ h 0 Be −iτ h 0 = e iτ [h 0 ,·] (B).(4.50)
Lemma 4. 7
7For any B ∈ M d+1 (C),Tr(BT β (B * )) = Tr(B * T β (B)).(4.52)Proof: Due to F n,m (−τ ) = F m,n (τ ) * , m = n (4.53) G n,n (−τ ) = G n,n (τ ) * (4.54)
Theorem 5. 1
1Let S(τ ) defined on a Banach space B be such that S(0) = I, and S(τ ) ≤ 1, for all τ ≥ 0. If, lim τ →0 τ −1 (S(τ ) − I) = Γ in the strong sense exists in L(B) and generates a contraction semi-group, then s − lim S(t/n) n = e tΓ . (5.2) Now, it is easily checked that S(τ ) := P e −i(τ H(0)+ √ τ W ) P on the subspace P B (5.3) satisfies the first requirements. Then, by expanding the exponent and making use of the properties of H(0) and W , we can write
F
1,0 (τ ) = −i e iτ (ǫ−δ) τ 0 e isδ ds a e iτ (ǫ−δ) τ 0 e is(δ−2ǫ) ds b e −iτ (ǫ+δ) τ 0 e is(δ+2ǫ) ds c e −iτ (ǫ+δ) τ 0 e isδ ds d .(6.11)
computations with V as in the statement, we obtainF 0,1 (τ ) = −i e iτ ǫ τ 0 e −isδ ds a e iτ ǫ τ 0 e −is(δ+2ǫ) ds c e −iτ ǫ τ 0 e −is(δ−2ǫ) ds b e −iτ ǫ τ 0 e −isδ ds d (6.18) F 1,0 (τ ) = −i e iτ (ǫ−δ) τ 0 e isδ ds a e iτ (ǫ−δ) τ 0 e is(δ−2ǫ) ds b e −iτ (ǫ+δ) τ 0 e is(δ+2ǫ) ds c e −iτ (ǫ+δ) τ 0 e isδ ds d ,(6.19)
By the Theorem of Lindblad, see e.g.[AF], we know that generates a completely positive semigroup of contractions. Therefore, we are in a position to apply Chernoff's theorem to eventually getTheorem 5.3 Assume hypothesis H0 where H 0 is a separable Hilbert space and h 0 , the V j 's and B are bounded on H 0 . Let B β (t/τ, 1/ √ τ , τ ) be defined by (4.4), U β (λ, τ ) is defined by proposition 4.1 and the Remark following it. Then.
(5.18)
i[h 0 , B] + Γ β (B)
(5.19)
5 Beyond the perturbative regime: λ 2 τ = 1We consider here the regime λ 2 τ = 1, and τ → 0 used in [AP] in their construction of the field of quantum noises. It can be viewed as a regime where the weak limit scaling holds at the microscopic level, while, at the macroscopic level, T = t/(τ λ 2 ) is kept finite.As we saw in Corollaries 3.3 and 4.1 in the Schrödinger and Heisenberg pictures respectively, the small parameter that allows to make use of perturbation theory to compute the effective evolution is the combination λ 2 τ . Therefore, we have to resort to a different technique since our scaling imposes a non-perturbative regime. Our main tool will be Chernoff's Theorem as we now explain.Acknowledgements:We wish to thank Laurent Bruneau for a careful and critical reading of the manuscript and Claude-Alain Pillet for useful discussions.
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Open Quantum Systems with Time-Dependent Hamiltonians and Their Linear Response. E B Davies, H Spohn, J.Stat.Phys. 19Davies, E.B., Spohn, H.: "Open Quantum Systems with Time-Dependent Hamiltonians and Their Linear Response", J.Stat.Phys. 19, 511-523, (1978).
Perturbation Theory for Linear Operators. T Kato, SpringerKato, T.: Perturbation Theory for Linear Operators, Springer, (1980).
Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs. J Lebowitz, H Spohn, Adv.Chem.Phys. 39Lebowitz, J. and Spohn, H., "Irreversible Thermodynamics for Quantum Sys- tems Weakly Coupled to Thermal Reservoirs", Adv.Chem.Phys. 39, 109-142, (1978).
Stochastic Calculus for Quantum Brownian Motion of a non-minimal variance" In: Mark Kac Seminar of probability in Physics. J M Lindsay, H Maassen, CWI Syllabus. 32J.M. Lindsay and H. Maassen, "Stochastic Calculus for Quantum Brownian Motion of a non-minimal variance" In: Mark Kac Seminar of probability in Physics, Syllabus 1987-1992, CWI Syllabus 32, Amsterdam, (1992).
Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. SpringerPazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
| [] |
[
"THE ELLIPTIC FUNCTIONS IN A FIRST-ORDER SYSTEM",
"THE ELLIPTIC FUNCTIONS IN A FIRST-ORDER SYSTEM"
] | [
"P L Robinson "
] | [] | [] | We investigate the first-order system 's ′ = c 3 , c ′ = −s 3 ; s(0) = 0, c(0) = 1'. Its solutions have the property that s c, s 2 and c 2 extend to simply-poled elliptic functions, which we explicitly identify in terms of the lemniscatic Weierstrass ℘ function. | null | [
"https://arxiv.org/pdf/1903.07147v1.pdf"
] | 119,300,471 | 1903.07147 | e51bcb556b982b3112f61089aa8153c8f7890610 |
THE ELLIPTIC FUNCTIONS IN A FIRST-ORDER SYSTEM
17 Mar 2019
P L Robinson
THE ELLIPTIC FUNCTIONS IN A FIRST-ORDER SYSTEM
17 Mar 2019
We investigate the first-order system 's ′ = c 3 , c ′ = −s 3 ; s(0) = 0, c(0) = 1'. Its solutions have the property that s c, s 2 and c 2 extend to simply-poled elliptic functions, which we explicitly identify in terms of the lemniscatic Weierstrass ℘ function.
Introduction
The first-order initial value problem s ′ = c, c ′ = −s; s(0) = 0, c(0) = 1 needs no introduction: its solutions are entire, being the familiar trigonometric functions s = sin and c = cos.
The first-order initial value problem s ′ = c 2 , c ′ = −s 2 ; s(0) = 0, c(0) = 1 is less well-known: its solutions are the elliptic functions s = sm and c = cm of Dixon [2]. Long after their discovery, these Dixonian elliptic functions were found to have interesting connexions to combinatorics [1] and geometry [5].
Here we propose to study the 'next' first-order initial value problem s ′ = c 3 , c ′ = −s 3 ; s(0) = 0, c(0) = 1.
The standard Picard theorem ensures that this system has a unique holomorphic solution pair in a suitably small disc about 0. It is easy to see that solutions to this system cannot be meromorphic throughout the plane; in particular, the solutions to the system are not elliptic. However, elliptic functions are still present: in fact, the quadratic combinations sc, s 2 and c 2 are elliptic; indeed, we shall here identify them in terms of the lemniscatic Weierstrass ℘ function and the associated lemniscatic sine function sl of Gauss. For these lemniscatic functions, see Chapter XXII of [6].
A First-Order System
This paper concerns the initial value problem
s ′ = c 3 , c ′ = −s 3 ; s(0) = 0, c(0) = 1
which will henceforth be referred to as IVP. In this section, the discussion will be largely local: we primarily consider properties of the unique solutions to IVP in an appropriate disc about the origin. The global questions involved in extending these solutions will be settled in the following section: as we shall see, the solutions themselves do not extend meromorphically, but their product and their squares actually extend as elliptic functions.
First of all, we observe that solutions to IVP satisfy an identity akin to the 'Pythagorean' trigonometric identity.
Proof. Differentiation yields
(s 4 + c 4 ) ′ = 4s 3 (s ′ ) + 4c 3 (c ′ ) = 4s 3 (c 3 ) + 4c 3 (−s 3 ) = 0
and evaluation at 0 places the constant value at 1.
More generally, setting aside the initial conditions, if functions s and c satisfy the differential equations s ′ = c 3 and c ′ = −s 3 then s 4 + c 4 is constant on any connected open set.
The Picard existence-uniqueness theorem ensures that IVP has a unique pair of holomorphic solutions (s, c) in a sufficiently small disc about the origin. For the sake of completeness, we shall offer an appropriate disc; however, the size of such a disc is immaterial for our present purposes. As references for the Picard theorem we cite [3]
(modified for complex use as mentioned in Section 12.1).
Theorem 2. The system IVP has a unique holomorphic solution pair (s, c) in the disc D r of radius r = 2 2 3 3 about 0.
Proof. Fix b > 0: for s ⩽ b and c − 1 ⩽ b we have s 3 ⩽ b 3 and c 3 ⩽ (b + 1) 3 ; as the system IVP is autonomous, the Picard theorem ensures that this system has a unique holomorphic solution in the disc of radius b (b + 1) 3 about the origin. The radius of this disc is maximized by taking b = 1 2: thus IVP has a unique holomorphic solution in the disc of radius r = 2 2 3 3 about 0.
Incidentally, we may use the 'Pythagorean' identity of Theorem 1 to extract from IVP a corresponding initial value problem for s alone: namely,
s ′ = (1 − s 4 ) 3 4 , s ′ (0) = 1.
Here, the initial condition s ′ (0) = 1 both forces s(0) = 0 and specifies the (principal) branch of the power; the initial condition s(0) = 0 alone would be inadequate. A similar application of the Picard theorem to this initial value problem furnishes a unique holomorphic solution in the larger disc of radius (2 2 The solutions s and c to IVP in the disc D r of radius r = 2 2 3 3 about 0 satisfy certain symmetry properties. Under multiplication by the imaginary unit i they behave as follows.
Theorem 3. If z < r then s(iz) = is(z) and c(iz) = c(z).
Proof. Define functions f and g by f (z) = −is(iz) and g(z) = c(iz). By differentiation, f ′ = g 3 and g ′ = −f 3 ; by evaluation, f (0) = 0 and g(0) = 1. Now appeal to the uniqueness clause in Theorem 2 to conclude that f = s and g = c.
A similar argument shows that s and c are 'real' in the sense that if z < r then s(z) = s(z) and c(z) = c(z). As an immediate corollary to Theorem 3, s is an odd function and c is an even function.
We now wish to explore the possibility of extending the solutions s and c of IVP beyond the disc D r provided by Theorem 2 or beyond the larger disc provided by the subsequent remark. Of course, if these functions are extended to a suitably symmetric connected open set then the symmetries just discussed will continue to hold by the principle of analytic continuation.
The fundamental question to be answered is whether the extension of s and c runs up against singularities (be they poles or otherwise) or the necessity of forming branches. We can answer this question with regard to poles decisively and at once. This proof suggests the possibility that one or more of the products sc, s 2 and c 2 might admit extension as a meromorphic function with simple poles. In the next Section we realize this possibility and improve upon it, thereby simultaneously addressing both singularities and branches.
The Elliptic Functions
We begin with the product of the functions s and c appearing in Theorem 2: thus, let p = s c.
By differentiation, p ′ = s ′ c + s c ′ = c 4 − s 4 so that (p ′ ) 2 = (c 4 + s 4 ) 2 − 4s 4 c 4 = 1 − 4 p 4
(by Theorem 1) and
p ′′ = 4c 3 c ′ − 4s 3 s ′ = −8s 3 c 3 = −8 p 3 . Rescale: define P by the rule that if z < r √ 2 then P (z) = √ 2 p( 1 √ 2 z)
. We conclude that P satisfies the following second-order initial value problem:
P ′′ = −2P 3 ; P (0) = 0, P ′ (0) = 1.
This enables us to relate the product s c to the lemniscatic sine function sl of Gauss and so in turn to the Glaisher quotient sd = sn dn of the Jacobian functions with self-complementary modulus 1 √ 2.
Theorem 5. If z < r then s(z)c(z) = 1 √ 2 sl( √ 2z) = 1 2 sd(2z). Proof. The second-order initial value problem displayed immediately prior to the Theorem characterizes the lemniscatic sine function: P coincides with sl on a neighbourhood of 0 by the Picard existence-uniqueness theorem and thence throughout the disc D r √ 2 of radius r √ 2 about 0 by the principle of analytic continuation; all that remains is to undo the rescaling and recall the expression for sl in terms of sd.
That is, the product s c agrees with the elliptic function z ↦ 1 2 sd(2 z) on the disc D r ; in other words, s c extends to this elliptic function by analytic continuation. In short, we may simply say that s c is this elliptic function, taking similar liberties in Theorem 6 and Theorem 7.
We now turn to the squares S = s 2 and C = c 2 of s and c. We take C first, as this case is a little more straightforward. By differentiation,
C ′ = 2c c ′ = −2c s 3 whence Theorem 1 yields (C ′ ) 4 = 16c 4 (s 4 ) 3 = 16c 4 (1 − c 4 ) 3 .
Thus, C satisfies the first-order differential equation
(C ′ ) 4 = 16 C 2 (1 − C 2 ) 3
of fourth degree, along with the initial condition C(0) = 1. This is a differential equation of Briot-Bouquet type: see page 423 of [3] and page 314 of [4]; it may be solved as follows.
First, the substitution E = C −1 has the effect of removing the quadratic factor from the right side: explicitly,
E ′ = −C −2 C ′ so that (E ′ ) 4 = C −8 (C ′ ) 4 = C −8 16 C 2 (1 − C 2 ) 3 = 16 C −6 (1 − C 2 ) 3 = 16 (C −2 − 1) 3 and therefore (E ′ ) 4 = 16 (E 2 − 1) 3 = 16 (E − 1) 3 (E + 1) 3 . Next, the substitution F = (E − 1) −1 yields (F ′ ) 4 = 128 F 2 (F + 1 2 ) 3 and the substitution G 2 = F + 1 2 leads to (G ′ ) 4 = 8 (G 2 − 1 2 ) 2 G 2 whence (G ′ ) 2 = 2 √ 2G(G 2 − 1 2
) after a choice of square-root. Now make a final substitution H = 1 √ 2 G: reversal of the various substitutions reveals that
C = 4 H 2 − 1 4 H 2 + 1 .
On the one hand, H satisfies the first-order differential equation
(H ′ ) 2 = 4 H 3 − H;
on the other hand, H has a pole at the origin since C(0) = 1. These conditions force H to be the lemniscatic Weierstrass function ℘ with invariants g 2 = 1 and g 3 = 0.
Theorem 6. The square C = c 2 is given by
C = ℘ 2 − 1 4 ℘ 2 + 1 4
where ℘ is the Weierstrass function with invariants g 2 = 1 and g 3 = 0.
Proof. Essentially a reversal of the arguments that led to the theorem. In the result, the Weierstrass function may be replaced by its negative, since only its square appears. In the derivation, this ambiguity arises from the substitution G 2 = F + 1 2 and the attendant choice of square-root.
The analysis for S runs parallel to that for C until the point at which the initial condition intervenes. From S ′ = 2s c 3 it follows that
(S ′ ) 4 = 16 S 2 (1 − S 2 ) 3
and a repetition of the foregoing argument shows that
S = 4 H 2 − 1 4 H 2 + 1 where (H ′ ) 2 = 4 H 3 − H.
This requires that H be a translate of the Weierstrass function ℘ that appears in Theorem 6: thus, for some k and for all z,
S(z) = ℘(z − k) 2 − 1 4 ℘(z − k) 2 + 1 4 .
The initial condition S(0) = 0 forces ℘(k) 2 = 1 4 so that ℘(k) is a mid-point value ±1 2 of the lemniscatic ℘: modulo periods, k is either the real half-period
ω = 2 1 0 dτ (1 + τ 4 ) 1 2 = 1.85407467730...
for which ℘(ω) = 1 2 or the purely imaginary half-period i ω for which ℘(i ω) = −1 2; we claim that the former half-period is appropriate, rather than the latter. Direct calculation (for instance, using the ℘ addition formula) reveals that
℘(z − ω) = 1 2 ℘(z) + 1 2 ℘(z) − 1 2 whence ℘(z − ω) 2 − 1 4 ℘(z − ω) 2 + 1 4 = ℘(z) ℘(z) 2 + 1 4 and therefore ℘(z − i ω) 2 − 1 4 ℘(z − i ω) 2 + 1 4 = − ℘(z) ℘(z) 2 + 1 4
because the lemniscatic ℘ satisfies ℘(i z) = −℘(z). Finally, we see that k = ω must be chosen: for if t is real then (as noted after Theorem 3) s(t) is real so that S(t) = s(t) 2 ⩾ 0.
Theorem 7. The square S = s 2 is given by
S = ℘ ℘ 2 + 1 4
where ℘ is the Weierstrass function with invariants g 2 = 1 and g 3 = 0.
Proof. Above.
It is perhaps barely worth noting that the expressions for C and S in Theorem 6 and Theorem 7 satisfy S 2 + C 2 = 1 as they should.
It is certainly worth noting that s 2 and c 2 satisfy the second-order system S ′′ = 2 C 3 − 6 S 2 C and C ′′ = 2S 3 − 6 C 2 S with S(0) = 0, S ′ (0) = 0 and C(0) = 1, C ′ (0) = 0 as do the rational functions of the lemniscatic ℘ displayed in Theorem 6 and Theorem 7. The right members of these differential equations being polynomial in S and C, the uniqueness clause in the Picard theorem applies to justify the conclusions of Theorem 6 and Theorem 7. We may instead proceed to fourth order, each of S and C being a solution to the differential equation
F ′′′′ = −12 F (32F 4 − 40F 2 + 9).
It is also worth noting that the results displayed in Theorem 5, Theorem 6 and Theorem 7 are consistent. That this is so follows from special properties of the lemniscatic ℘: for this Weierstrass function, the Glaisher quotient sd satisfies sd 2 = 1 ℘ and the duplication formula for ℘ gives
℘(2z) = (℘(z) 2 + 1 4) 2 ℘ ′ (z) 2 whence 1 2 sd(2z) 2 = 1 4 ℘ ′ (z) ℘(z) 2 + 1 4 2 = ℘(z) 2 − 1 4 ℘(z) 2 + 1 4 ℘(z) ℘(z) 2 + 1 4 .
In fact, we can go further and extract the square-root: an examination of behaviour as z → 0 reveals that
sd(2z) = − ℘ ′ (z) ℘(z) 2 + 1 4 .
As a consequence, we may now return to Theorem 5 and express the product s c directly in terms of ℘: thus,
s c = − 1 2 ℘ ′ ℘ 2 + 1 4 .
Finally, we realize the possibility that opened up at the close of the previous Section. It follows from Theorem 5 (or the reformulation just given), Theorem 6 and Theorem 7 that each of s c, c 2 and s 2 extends to an elliptic function having (simple) poles exactly at the points congruent to 1 2 (±1±i) ω modulo the periods 2 ω and 2 i ω of the lemniscatic Weierstrass function ℘. Notice that each of these three elliptic functions is holomorphic in the open disc of radius ω √ 2 about 0. Within this disc, the function ℘ (℘ 2 + 1 4) has a double zero at 0 (originating as a removable singularity) but is otherwise nonzero; in this disc it therefore has two holomorphic square-roots, which extend the functions s and −s. The simple poles of the function ℘ (℘ 2 +1 4) at the points 1 2 (±1±i) ω on the boundary of this disc sprout branches when the attempt is made to extend the holomorphic function s beyond them. The situation as regards the function c is slightly simpler, the function (℘ 2 − 1 4) (℘ 2 + 1 4) being zero-free in the same disc.
Theorem 1 .
1Solutions s and c to IVP in any connected open set containing 0 satisfy s 4 + c 4 = 1.
3 3 ) 1 4 about 0; this solution s takes values in the open unit disc, so a companion holomorphic c is provided by (1 − s 4 ) 1 4 with principal-valued power.
Theorem 4 .
4Isolated singularities of functions s and c that satisfy s ′ = c 3 and c ′ = −s 3 cannot be poles. Proof. The functions s and c are plainly copolar, in the sense that if either has a pole at some point then so does the other. Consider a pole: of order m for s and of order n for c. From s = c 3 follows m + 1 = 3n; from c ′ = −s 3 follows n + 1 = 3m. These two equations for m and n have 1 2 as their unique solution.
The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion. E Conrad, P Flajolet, Séminaire Lotharingien de Combinatoire. 5454E. Conrad and P. Flajolet, The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion, Séminaire Lotharingien de Combinatoire 54 (2006) Article B54g.
On the doubly periodic functions arising out of the curve x 3 + y 3 − 3αxy = 1. A C Dixon, The Quarterly Journal of Pure and Applied Mathematics. 24A.C. Dixon, On the doubly periodic functions arising out of the curve x 3 + y 3 − 3αxy = 1, The Quarterly Journal of Pure and Applied Mathematics, 24 (1890) 167-233.
E Hille, Ordinary Differential Equations in the Complex Domain. Wiley-InterscienceE. Hille, Ordinary Differential Equations in the Complex Domain, Wiley-Interscience (1976);
. E L Ince, Ordinary Differential Equations, Longman, Green and Company. E.L. Ince. Ordinary Differential Equations, Longman, Green and Company (1926);
The Trefoil. J C Langer, D A Singer, Milan Journal of Mathematics. 82J.C. Langer and D.A. Singer, The Trefoil, Milan Journal of Mathematics, 82 (2014) 161-182.
E T Whittaker, G N Watson, A Course of Modern Analysis. Cambridge University PressSecond EditionE. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Second Edition, Cambridge University Press (1915).
| [] |
[
"Partial-envelope stripping and nuclear-timescale mass transfer from evolved supergiants at low metallicity",
"Partial-envelope stripping and nuclear-timescale mass transfer from evolved supergiants at low metallicity"
] | [
"Jakub Klencki [email protected] \nEuropean Southern Observatory\nKarl-Schwarzschild-Strasse 285748Garching bei MünchenGermany\n\nDepartment of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands\n",
"Alina Istrate \nDepartment of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands\n",
"Gijs Nelemans \nDepartment of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands\n\nInstitute of Astronomy, KU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium\n\nInstitute for Space Research\nSRON\nSorbonnelaan 23584 CAUtrechtNLNetherlands, The Netherlands\n",
"Onno Pols \nDepartment of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands\n"
] | [
"European Southern Observatory\nKarl-Schwarzschild-Strasse 285748Garching bei MünchenGermany",
"Department of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands",
"Department of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands",
"Department of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands",
"Institute of Astronomy, KU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium",
"Institute for Space Research\nSRON\nSorbonnelaan 23584 CAUtrechtNLNetherlands, The Netherlands",
"Department of Astrophysics/IMAPP\nRadboud University\nP O Box 90106500 GLNijmegenNLThe Netherlands"
] | [] | Stable mass transfer from a massive post-main sequence (post-MS) donor is thought to be a short-lived event of thermal-timescale mass transfer (∼ 10 −3 M yr −1 ) which within 10 4 yr strips the donor star of nearly its entire H-rich envelope, producing a hot, compact helium star. This long-standing picture is based on stellar models with rapidly-expanding Hertzprung gap (HG) donor stars. Motivated by a finding that in low-metallicity binaries, post-MS mass transfer may instead be initiated by donors already at the corehelium burning (CHeB) stage, we use the MESA stellar-evolution code to compute grids of detailed massive binary models at three metallicities: those of the Sun, the Large Magellanic Cloud (LMC, Z Fe;LMC /Z Fe; ≈ 0.36), and the Small Magellanic Cloud (SMC, Z Fe;SMC /Z Fe; ≈ 0.2). Our grids span a wide range in orbital periods (∼ 3 to 5000 days) and initial primary masses (10 M to 36-53 M , depending on metallicity). We find that metallicity strongly influences the course and outcome of mass-transfer evolution. We identify two novel types of post-MS mass transfer: (a) mass exchange on the long nuclear timescale (∆T MT 10 5 yr,Ṁ ∼ 10 −5 M yr −1 ) that continues until the end of the CHeB phase, and (b) rapid mass transfer leading to detached binaries with mass-losers that are only partially stripped of their envelopes. At LMC and SMC compositions, the majority of binary models with donor masses ≥ 17 M follow one of these two types of evolution. In neither (a) or (b) does the donor become a fully stripped helium star by the end of CHeB. Boundaries between the different types of post-MS mass transfer evolution are associated with the degree of rapid post-MS expansion of massive stars and, for a given metallicity, are sensitive to the assumptions about internal mixing. At low metallicity, due to partial envelope stripping, we predict fewer hot fully stripped stars formed through binary interactions as well as higher compactness of the pre-supernova core structures of mass losers. Nuclear-timescale post-MS mass transfer suggests a strong preference for metal-poor host galaxies of ultra-luminous X-ray sources with black-hole (BH) accretors and massive donors, some of which might be the immediate progenitors of binary BH mergers. It also implies a population of interacting binaries with blue and yellow supergiant donors. Partially-stripped stars could potentially explain the puzzling nitrogen-enriched slowly-rotating (super)giants in the LMC. | 10.1051/0004-6361/202142701 | [
"https://arxiv.org/pdf/2111.10271v2.pdf"
] | 244,463,210 | 2111.10271 | 6a30eadcab112b7db2a0c089f679970d5fa39ec6 |
Partial-envelope stripping and nuclear-timescale mass transfer from evolved supergiants at low metallicity
April 6, 2022
Jakub Klencki [email protected]
European Southern Observatory
Karl-Schwarzschild-Strasse 285748Garching bei MünchenGermany
Department of Astrophysics/IMAPP
Radboud University
P O Box 90106500 GLNijmegenNLThe Netherlands
Alina Istrate
Department of Astrophysics/IMAPP
Radboud University
P O Box 90106500 GLNijmegenNLThe Netherlands
Gijs Nelemans
Department of Astrophysics/IMAPP
Radboud University
P O Box 90106500 GLNijmegenNLThe Netherlands
Institute of Astronomy, KU Leuven
Celestijnenlaan 200DB-3001LeuvenBelgium
Institute for Space Research
SRON
Sorbonnelaan 23584 CAUtrechtNLNetherlands, The Netherlands
Onno Pols
Department of Astrophysics/IMAPP
Radboud University
P O Box 90106500 GLNijmegenNLThe Netherlands
Partial-envelope stripping and nuclear-timescale mass transfer from evolved supergiants at low metallicity
April 6, 2022Received November, 2021; accepted...Astronomy & Astrophysics manuscript no. mainstars: massive -stars: binaries: general -stars: evolution
Stable mass transfer from a massive post-main sequence (post-MS) donor is thought to be a short-lived event of thermal-timescale mass transfer (∼ 10 −3 M yr −1 ) which within 10 4 yr strips the donor star of nearly its entire H-rich envelope, producing a hot, compact helium star. This long-standing picture is based on stellar models with rapidly-expanding Hertzprung gap (HG) donor stars. Motivated by a finding that in low-metallicity binaries, post-MS mass transfer may instead be initiated by donors already at the corehelium burning (CHeB) stage, we use the MESA stellar-evolution code to compute grids of detailed massive binary models at three metallicities: those of the Sun, the Large Magellanic Cloud (LMC, Z Fe;LMC /Z Fe; ≈ 0.36), and the Small Magellanic Cloud (SMC, Z Fe;SMC /Z Fe; ≈ 0.2). Our grids span a wide range in orbital periods (∼ 3 to 5000 days) and initial primary masses (10 M to 36-53 M , depending on metallicity). We find that metallicity strongly influences the course and outcome of mass-transfer evolution. We identify two novel types of post-MS mass transfer: (a) mass exchange on the long nuclear timescale (∆T MT 10 5 yr,Ṁ ∼ 10 −5 M yr −1 ) that continues until the end of the CHeB phase, and (b) rapid mass transfer leading to detached binaries with mass-losers that are only partially stripped of their envelopes. At LMC and SMC compositions, the majority of binary models with donor masses ≥ 17 M follow one of these two types of evolution. In neither (a) or (b) does the donor become a fully stripped helium star by the end of CHeB. Boundaries between the different types of post-MS mass transfer evolution are associated with the degree of rapid post-MS expansion of massive stars and, for a given metallicity, are sensitive to the assumptions about internal mixing. At low metallicity, due to partial envelope stripping, we predict fewer hot fully stripped stars formed through binary interactions as well as higher compactness of the pre-supernova core structures of mass losers. Nuclear-timescale post-MS mass transfer suggests a strong preference for metal-poor host galaxies of ultra-luminous X-ray sources with black-hole (BH) accretors and massive donors, some of which might be the immediate progenitors of binary BH mergers. It also implies a population of interacting binaries with blue and yellow supergiant donors. Partially-stripped stars could potentially explain the puzzling nitrogen-enriched slowly-rotating (super)giants in the LMC.
Introduction
Massive stars do not live alone. The majority are formed in close binary or higher-order systems in which they are destined to strongly interact with their companions by transferring mass and angular momentum (Sana et al. 2012;Moe & Di Stefano 2017). Mass transfer has significant consequences for basic properties and final fates of both stars. Many mass losers become heliumrich stripped stars, from hot subdwarfs to Wolf-Rayet (WR) stars, (depending on the mass ;Paczyński 1967;van den Heuvel 1975;Vanbeveren 1991;Podsiadlowski et al. 1992;De Loore & De Greve 1992;Petrovic et al. 2005;Eldridge et al. 2008;Götberg et al. 2017;Laplace et al. 2020), and a UV-bright prominent source of ionizing photons . Mass gainers and stellar mergers, on the other hand, are the likely explanation for the existence of blue stragglers (McCrea 1964;Pols 1994;Braun & Langer 1995;Glebbeek et al. 2013;Schneider et al. 2015), may dominate the population of rapidly rotating stars (de Mink et al. 2013;Renzo & Gotberg 2021) in particular Be stars (Pols et al. 1991;Shao & Li 2014;Hastings et al. 2021), lead to formation of magnetic fields in stars (Ferrario et al. 2009;Schneider et al. 2019), and a peculiar class of supernovae from blue-supergiant progenitors (including the most recent naked-eye supernova SN 1987A, see Podsiadlowski et al. 1990;Justham et al. 2014). Prior mass transfer interactions have been shown to leave clear signatures in the properties of stellar cores at core collapse, affect lightcurves and yields of the resulting supernovae, and play an important role in deciding whether the final remnant will be a neutron star (NS) or a black hole (BH, Morris & Podsiadlowski 2007;Justham et al. 2014;Woosley 2019;Schneider et al. 2021;Laplace et al. 2021;Vartanyan et al. 2021). Through mass transfer, binaries enrich the interstellar medium with processed material (de Mink et al. 2009b), shine as X-ray binaries (van den Heuvel & De Loore 1973;van den Heuvel 1975;Verbunt 1993), and bring two BHs or neutron stars NSs close together, driving them to merge and blaze in gravitational waves (GW; Tutukov & Yungelson 1993;Portegies Zwart & Yungelson 1998;Belczynski et al. 2002;Voss & Tauris 2003) in the most energetic spectacles observed to date (Abbott et al. 2016).
Article number, page 1 of 24 arXiv:2111.10271v2 [astro-ph.SR] 5 Apr 2022 A&A proofs: manuscript no. main In most massive binary systems, a phase of mass transfer is initiated when one of the stars evolves off the main sequence (MS) and expands significantly leading to the so-called case B Roche-lobe overflow (RLOF, Kippenhahn & Weigert 1967;Paczyński 1971). 1 The textbook view of case B mass transfer is that the donor star looses nearly its entire H-rich envelope in a short-lived phase of thermal-timescale mass exchange ( 10 4 yr Kippenhahn & Weigert 1967;Paczyński 1971;van den Heuvel 1975;Podsiadlowski et al. 1992;Vanbeveren et al. 1998). This is because the donor is thought to usually be a rapidly-expanding H-shell burning giant, a Hertzprung gap (HG) star. It is out of thermal-equilibrium even before any RLOF occurs and would need to expand all the way to the red giant branch in order to regain a stable thermal structure. In a close binary, this continuous rapid expansion of the donor star is what causes the mass transfer to proceed on the thermal timescale until only a thin envelope layer remains. By the time the binary detaches, the donor becomes a stripped star, composed predominantly of helium.
This paradigm may no longer be true for massive binaries in low-metallicity (low-Z) environments. As pointed out by de Mink et al. (2008) and recently shown in much detail by Klencki et al. (2020), metallicity has a strong effect on what is the typical evolutionary state of donors when they initiate a case B mass transfer phase in a population of massive binaries. While at high (Solar-like) metallicity a post-MS interaction nearly always occurs when the donor is expanding as a HG star, it has been shown that at low metallicity the parameter space for RLOF from core-He burning donors becomes very significant (and possibly dominant above a certain mass, see Klencki et al. 2020). In contrast to HG donors, stars that are already at the core-He burning stage are in thermal equilibrium and slowly expanding on the nuclear timescale of ∼ 10 6 yr. Mass transfer from such donors has never been studied with detailed binary evolution models and its understanding is still lacking.
At the same time, various observational clues suggest that metallicity has a strong influence on the evolution of massive stars and binaries. Long gamma-ray bursts (Graham & Fruchter 2013), superluminous supernovae (Gal-Yam 2012), and ultralumionous X-ray (ULX) sources (Kovlakas et al. 2020) all preferentially occur in low-Z galaxies. Similarly, Type Ic supernovae with broad lines in the spectra are typically found in metal-poor hosts, whereas normal Type Ic supernovae avoid dwarf galaxies (Modjaz et al. 2011). It has been suggested that BHs with masses above ∼ 30 M , frequently found in binary BH mergers detected in GW by LIGO/Virgo (Abbott et al. 2020), may originate from metal-poor environments, where the reduced strength of line-driven winds allows for the formation of more massive BHs (e.g. Belczynski et al. 2010;Vink et al. 2021). Thanks to their close proximity to the Milky Way, the Small and the Large Magellanic Cloud (SMC and LMC, respectively) serve as excellent test-beds of massive star evolution at low metallicity. Largescale spectroscopic surveys of the Magellanic Clouds have revealed populations of stars that cannot be explained by current models, in particular the slowly-rotating nitrogen-enriched (super) giants (Evans et al. 2006;Hunter et al. 2008;McEvoy et al. 2015;Grin et al. 2017). The population of high-mass X-ray binaries (HMXBs) also appears to be metallicity-dependent, with a surprisingly large number of Be HMXBs found in the Magellanic Clouds (Dray 2006).
Motivated by the importance of low metallicity on the evolution of massive stars, evidenced on both theoretical and observational grounds, in this work we follow up on Klencki & Nelemans (2019); Klencki et al. (2020) and explore mass-transfer evolution in massive binaries of different metallicities.
The paper is organized as follows. In Section 2 we describe our computational method and physical assumptions as well as the parameter space explored with our models. In Section 3 we present the results from our binary evolution sequences, focusing on the mass-transfer evolution from post-MS donors. In Section 4 we take an in-depth look at the origin of different types of mass-transfer evolution found in Sec. 3. In Section 5 we discuss the caveats of our models as well as various implications of the findings. We conclude in Section 6.
Method: binary stellar evolution models
Physical ingredients: single and binary evolution
We employ the MESA stellar evolution code (Paxton et al. 2011;; 2019) 2 . Convection is modeled using the mixing-length theory (Böhm-Vitense 1958) with the mixing length of α = 1.5. We employ the Ledoux criterion for convection and account for semiconvective mixing with an efficiency of α SC = 33. We account for convective core-overshooting by applying step overshooting with an overshooting length σ ov = 0.33 (Brott et al. 2011a). Such choices of overshooting and semiconvection efficiency were shown to be in good agreement with the observed ratio of blue and red supergiants in the SMC (Schootemeijer et al. 2019;Klencki et al. 2020).
Models are computed at three different initial chemical compositions: either at Solar metallicity with Z = 0.017 and abundance ratios from Grevesse et al. (1996) or with non-Solar abundance ratios of the Magellanic Clouds following Brott et al. (2011a). In the case of the LMC that yields Z LMC ≈ 0.0048 and the relative iron abundance Z Fe;LMC /Z Fe; ≈ 0.36, whereas in the case of SMC: Z SMC ≈ 0.0022 and the relative iron abundance Z Fe;SMC /Z Fe; ≈ 0.2. Similarly to , we use custom-made OPAL opacity tables (Iglesias & Rogers 1996) corresponding to the adopted initial abundances of the SMC or the LMC.
The winds of stars on the cool side of the bi-stability jump with T eff < 0.95 T eff;jump ≈ 25 kK (see Eqn. 15 in Vink et al. 2001) are modeled as the larger of the mass-loss rates from Vink et al. (2001) and Nieuwenhuijzen & de Jager (1990). The winds of stars on the hot side of the bi-stability jump (with T eff > 1.05 T eff;jump ) are modeled as a combination of several different prescriptions, depending on the surface hydrogen abundance X. For stars with X > 0.45, we follow Vink et al. (2001). For stars with 0.1 < X < 0.35, we apply the empirical mass-loss rates from Nugis & Lamers (2000). For (nearly) hydrogen-free stars with X < 0.05, we follow Yoon (2017), whose prescriptions for WNE stars are based on the results from Hainich et al. (2014) and for WC/WO stars were derived by Tramper et al. (2016). In the intermediate X regimes, as well as in the temperature range around the bi-stability jump 0.95 < T eff /T eff;jump < 1.05, we linearly interpolate between the above prescriptions to provide smooth transitions.
In addition, when using the Vink et al. (2001) prescription at T eff > T eff;jump (optically thin winds of OB supergiants), we account for the possible transition to the optically thick WR-type mass loss. Both theoretical (Gräfener & Hamann 2008;Vink Fig. 1: Distribution of the main grid of binary models at each metallicity over the parameter space of initial orbital periods and primary masses. Background color indicates the evolutionary stage of the primary star at the onset of mass transfer through RLOF. We differentiate between core-H burning donors (blue), H-shell burning donors (i.e. donors that experience the Hertzprung-gap phase of rapid expansion; green), as well as core-He burning donors (yellow). The initial mass ratio is q = M 2 /M 1 = 0.6. Colored rectangles indicate which models from the grid are shown in figures throughout the paper (see legend). Sander et al. 2020) and empirical studies (Gräfener et al. 2011;Bestenlehner et al. 2014) suggest that this transition takes place when the Eddington factor Γ e = 10 −4.813 (1 + X)(L/L )(M/M ) −1 becomes sufficiently large, which typically happens when a substantial amount of mass has already been lost (either in winds or as a result of mass transfer). Here, we follow Gräfener & Hamann (2008) and assume that this threshold is at Γ e ≈ 0.1 + Γ 0 where Γ 0 is metallicity (iron) dependent and equals to 0.326, 0.468, and 0.514 for our Solar, LMC, and SMC iron abundances, respectively (Eqn. 6 of Gräfener & Hamann 2008). We maintain the Vink et al. (2001) mass-loss rate for Γ e − Γ 0 < 0.08, whereas for Γ e − Γ 0 > 0.12 we apply the theoretical recipe for optically thick winds from Gräfener & Hamann (2008). For intermediate Eddington factors, we linearly interpolate between the two mass-loss rates.
We include rotationally-induced mixing of elements due to Eddington-Sweet circulation, secular shear instabilities, and the Goldreich-Schubert-Fricke instability, with an efficiency factor f c = 1/30 (Heger et al. 2000;Brott et al. 2011a). We model rotationally-enhanced mass loss as in Langer (1998), see implementation in Paxton et al. (2013). In most of our models, we avoid using the MLT++ option in MESA (Paxton et al. 2013) as it was shown to artificially reduce the stellar radii during the giant phases of evolution and therefore could affect the behavior of stars during mass transfer. In the few cases of the most massive donors in our grid (36 M , 44 M , and 53 M at Solar, LMC, and SMC compositions, respectively) we resort to a limited use of MLT++ by gradually reducing superadiabacity in outer layers (T < 10 6 K) of donors once they have reached the mass-transfer rate of log(Ṁ) = −2.5. These models would have not converged otherwise due to numerical difficulties arising at the bottom of a subsurface convective zone located at the iron opacity peak.
We follow the formalism of Kolb & Ritter (1990) to calculate the mass-transfer rate through the L1 Lagrangian point. This includes both the optically-thin regime of mass transfer when R don is still slightly smaller than R RL , where R don is the radius of the donor star and R RL is the size of its Roche lobe, as well as the main phase of mass transfer when R don > R RL . The mass transfer is assumed to remain stable for as long as the binary model computes, i.e. unstable cases are when the mass transfer rate clearly diverges to infinity and the simulation stops.
The main focus of the paper is to study the behavior of the donor star during a phase of mass transfer and to understand how the slowed down post-MS expansion of low-metallicity massive stars influences their evolution as donors in interacting binaries. Therefore, for simplicity, in our binary models we only evolve the primary (donor) star and treat the companion as a point mass. We also assume that the accretion efficiency is Eddingtonlimited (calculated for a BH accretor), which in practice will mean nearly fully non-conservative mass transfer. We further assume that the specific angular momentum of the mass ejected from the system is that of the accretor on its binary orbit (i.e. the isotropic re-emission mode). These assumptions make our models well suited to represent systems with compact-object accretors (specifically BHs, given that our accretors are always more massive than 6 M ). However, as we argue in Sec. 5.2, all our main findings and conclusions also hold in the case of systems with stellar accretors. Throughout the paper we will thus not assume anything about the nature of the companion unless explicitly stated otherwise.
We model the spin-up of the primary due to tidal interactions following the synchronization timescale for radiative envelopes from Hurley et al. (2002). In MESA the angular momentum from tides is being deposited in such a way that each layer is being synchronized on the same timescale. We include the effect of isotropic wind mass-loss on the evolution of orbital parameters as well as the spin of the mass-losing star. The computed binary models as well as MESA inlists (input files) used in this work are available at https://zenodo.org/record/6412508.
Initial parameter space and stopping conditions
We compute three main grids of binary models: at Solar, LMC, and SMC composition. Each grid spans 20 different initial orbital periods P ini ranging from ≈ 3.16 to 5012 days, spaced in equal logarithmic steps. The SMC grid spans 10 different initial primary masses M 1 : 10, 12, 14, 17, 20, 25, 30, 36, 44, and 53 M . The LMC grid ends at 44 M whereas at Solar metallicity we compute models until 36 M . The default initial mass ratio q = M 2 /M 1 is q = 0.6 for our main grids. For the case of M 1 = 25 M and P ini ≈ 47 days we compute additional models with the mass ratio q varying from 0.1 to 1.5 (15 models). We do this by preserving the structure of the donor at RLOF taken from Top panels show the mass-transfer rate, bottom panels show the mass of the envelope (hydrogen content X H > 10 −3 ). Different colors correspond to different mass ratios between the binary components q = M 2 /M 1 , varying from 0.1 and 1.5 (the donor stays the same). Bold part of each curve indicates when the donor overflows its Roche lobe. In each case, the initial RLOF takes place when the primary radius is around R RLOF ≈ 90 R , see the radius-age diagram on the right-hand side. In models with q ≤ 0.225 (not shown in the figure) the mass transfer became dynamically unstable. Models terminate at the end of core-He burning. the standard binary model with q = 0.6 and manually adjusting the mass ratio and orbital separation in such a way that at the moment of RLOF the donor has the same radius for any q. This approximation is justified because the tidal spin-up of the donor star, which would have been different in the evolution prior to RLOF for different mass ratios, is insignificant for binaries this wide. The primaries are initialized with a small initial rotational velocity of 30 km/s. All the binaries are circular. Figure 1 illustrates how our grids of binary models cover the parameter space for a mass transfer interaction initiated by donors at different evolutionary stages, similarly to figures in Klencki et al. (2020). At each composition, the grid covers the entire range of periods for a post-MS mass transfer interaction and fully encloses the range of masses at which donors begin the core-He burning stage as blue supergiants (at LMC and SMC compositions). In addition, each grid includes a small number of binaries that evolved through a case-A mass-transfer phase (i.e. initiated by a MS star). These models are beyond the main scope of the paper and will not be discussed in detail.
All the binary sequences terminate at central helium depletion (Y center < 10 −6 ). The remaining lifetimes of massive stars from that point on until the final core-collapse are short compared to hydrogen-and helium-burning lifetimes (< 10 5 yr or even < 10 4 yr for the more massive among our primaries).
Results
Nuclear-timescale mass transfer and partial envelope stripping
We begin describing the results by taking a close look at binary evolution models from the middle of our grids: systems with initial primary masses of M 1 = 25 M and orbital periods P ini ≈ 47 days. For this selected case of M 1 and P ini , apart from the default initial mass ratio q = M 2 /M 1 = 0.6, we computed additional mass transfer sequences with q varying from 0.1 to 1.5, as described in Sec. 2.2. In each case, the size of the primary (donor) star at the moment of RLOF was about R RLOF ≈ 90 R . At both Solar and LMC compositions, the primary star of that radius was still a rapidly-expanding HG giant, whereas at SMC composition the donor was already at the core-He burning stage. For this selected set of models, in Fig. 2 we plot the time evolution of the mass transfer rateṀ (upper panels) and envelope masses (lower panels). Phases of RLOF are marked in bold.
Other phases with log(Ṁ) > −8 indicate mass transfer through stellar winds from a donor that is close to Roche-lobe filling. In all the SMC models, there is a small peak of wind mass transfer with log(Ṁ) ≈ −7.2 at the age of 7.55 Myr, followed by a brief drop inṀ before RLOF starts at 7.6 Myr. This is due to a temporary slight contraction of the primary after it regains equilibrium as a core-He burning star at ∼ 7.55 Myr.
We find a remarkable difference in the mass-transfer evolution between the Solar case and the low-Z models of the LMC and SMC compositions. At Solar Z, irrespective of the mass ratio, the mass transfer interaction is a short-lived and rapid event. Over the course of only several thousand years the donor star is stripped of its nearly entire H-rich envelope. This is a phase of thermal-timescale mass transfer with the rate reachinġ M ∼ 10 −2 M yr −1 . The thin remaining envelope layer (∼ 1 M ) is further quickly lost in winds and the primary becomes a fully stripped helium star. This type of evolution is the canonical picture of how a case-B mass transfer interaction can lead to the envelope loss and formation of stripped stars of various kinds, from subdwarfs to WR stars (see Sec. 1 and the references therein).
However, the evolution of LMC and SMC models in Fig. 2 turns out to be substantially different. In these low-Z cases the initial rapid (thermal) phase of mass transfer slows down or terminates while a significant part of the envelope is still retained (M env;left from ∼ 1 up to even 7 M , depending on the mass ratio and metallicity). In models with M env;left 2.5 M , when the partially-stripped donor continues its evolution as a core He-burning star, its slow expansion leads to a long phase of nuclear-timescale mass transfer (a few times 10 5 yr,Ṁ ∼ 10 −6 − 10 −5 M yr −1 ). The slow mass transfer dominates the rest of the evolution: it persists at least until the end of core-He burning, at which point we terminate our models (the remaining lifetime is relatively short, < 0.1 Myr). Occasionally the mass exchange may be interrupted by temporary detachments, in particular around the age of 7.6 (7.7) Myr of the LMC (SMC) models. These are caused by (typically subtle) contractions of the donor star in response to changes of the hydrogen abundance in the moving location of the H-burning shell. 3 Oscillations in the mass transfer rate during the nuclear-timescale phase are associated with changes in the helium abundance of the outer donor layers as it is being stripped deeper into layers that used to be convective during MS (see Sec.4). The few low-Z models with the lowest envelope masses M env;left 2.5 M left after the initial rapid mass transfer, (q 0.4 at LMC and q = 0.25 at SMC composition) show a somewhat different type of evolution. These binaries remain detached for most of their He-burning lifetime, with possibly only a relatively brief episode of nuclear-timescale mass transfer close to the end of this phase.
Notably, in none of the LMC or SMC models in Fig. 2 is the envelope fully lost in winds or mass transfer (by the end of core-He burning), which is partly due to weaker winds at low Z. In fact, at no point does the primary become a hot ( log(T eff /K) 4.6) stripped helium star, in sharp contrast to the Solar metallicity case. See Besides the mass ratio, another factor that affects the degree of envelope stripping at low Z is the initial orbital period P ini . In Fig. 3, we show this for the case of models with a M 1 = 25 M primary, the mass ratio q = 0.6, and the SMC metallicity (see Fig. A.1 for the LMC case). As the top and the middle panel illustrate, the shorter the orbital period, the lower the mass of the remaining envelope after the initial rapid mass transfer phase (M env;left ). We notice that above a certain period (or equivalently mass ratio), the donors tend to slow down the mass transfer rate at similar M env;left values and then continue to transfer mass on the nuclear timescale. In the case shown in Fig. 3, this happens for P ini 30 days and leads to M env;left ≈ 6 − 7 M . Notably, the donors in those models are already near the end of the HG expansion or more evolved at the onset of RLOF (see the botom panel).
As previously mentioned and illustrated in the donor radius panel of Fig. 3, temporary detachments happening at ∼ 7.7 Myr are caused by relatively modest contractions of donors during the core-He burning phase (by 10 − 20% in this particular case).
We discuss the above findings in view of the donor envelope structure in Sec. 4. We note that although not explored in the current study, the degree of envelope stripping is expected to also depend on the assumptions about the accretion efficiency and angular momentum of the non-accreted matter, i.e. all the factors that influence the evolution of the Roche lobe size of the donor star.
3 A similar transition in the H shell was previously found to trigger a blue-loop evolution in 15 M stellar models by Langer et al. (1985). Fig. 3: Mass transfer evolution in binaries with a 25 M primary (initial mass), SMC composition, and various initial orbital periods P ini . The shorter the orbital period, the lower the envelope mass M env;left that remains after the initial phase of rapid (thermal) mass transfer.
The reason why the mass transfer evolution is so different between the Solar and low-Z cases in Fig. 2 is related to radial expansion of mass-losing stars and the size that a partially-stripped donor of a given metallicity would need to have in order to regain thermal equilibrium. We discuss this is detail in Sec. 4 and Fig. 12. In summary, we find that a partially-stripped donor of Solar Z would need to expand to a size of a red supergiant to regain equilibrium, ∼ 1800 R , which is much larger than a size of its Roche lobe. As a result, the donor keeps rapidly expanding and the thermal-timescale mass transfer continues until it strips nearly entire envelope, at which point the donor finally contracts to become a hot stripped star. In contrast, the donors of LMC and SMC compositions (M 1 = 25 M ) that are partially-stripped in the initial phase of rapid mass transfer can be in thermal equilibrium at a much smaller size ∼ 50 R , which is similar to that of their Roche lobes. As a result, they can regain equilibrium and slow down the mass transfer interaction or even detach. In a similar way, single massive low-Z stars may remain relatively compact before the onset of the core-He burning, avoiding a rapid HG expansion until the red giant branch (in particular the 25 M models, see the radius-age diagram on the right-hand side of Fig. 2). In fact, as we will show in Sec. 3.3, we find nucleartimescale mass transfer evolution in models with donor masses roughly from the mass range in which the halted HG expansion happens. Fig. 4: Three different types of mass-transfer evolution found across our binary models, illustrated with cases with M 1 = 30 M primaries and different compositions (metallicities). Going from top to bottom, subsequent panels show: masstransfer rate, envelope mass, surface He abundance, and surface nitrogen enrichment. Bold parts of the curves in the second panel indicate phases of RLOF. Letters mark distinctive events that may occur during the evolution as following. A: a phase of rapid thermal-timescale mass transfer, B: a point when the donor regains thermal equilibrium and the mass transfer transitions to the nuclear timescale, C: phases of detachment, D: temporary masstransfer rate increase when the donor becomes stripped down to layers that used to be in the intermediate convective zone, and E: mass transfer peak associated with the end of core-He burning and re-expansion of the donor.
Three types of post-MS mass-transfer evolution
Throughout our binary grid, we find three qualitatively different types of mass-transfer evolution, which we illustrate in Fig. 4 (using the M 1 = 30 M case as an example). In Fig. 5, we plot the corresponding donor tracks in the HR diagram, with the phase of RLOF shown in bold and crosses indicating where the donor spends most of its core-He burning lifetime (spaced every 50,000 yr).
The Solar metallicity donor transfers nearly its entire envelope (M env;left ≈ 1.7 M ) during a phase of thermal-timescale mass transfer (labeled 'A'). After detachment it rapidly contracts and moves leftwards in the HR diagram to spend most of its Heburning lifetime (∼ 85%∆t He;burn ) as a hot UV-bright stripped star ('C' in Fig. 4 in the HR diagram. Letters correspond to the same events in the mass transfer evolution as before (see caption of Fig. 4 or text). Diamond (star) symbols mark the end of core H (core He) burning. These models illustrate the three main types of stable post-MS mass transfer evolution: thermaltimescale mass transfer producing hot (fully) stripped stars (top panel), thermal-timescale mass transfer producing much cooler, partially stripped stars (middle panel, case on the left-hand side), as well as binaries that evolve through phases of both thermaland nuclear-timescale mass transfer (the remaining two models).
perature of initially log(T eff /K) ≈ 4.85 and then subsequently log(T eff /K) ≈ 5.1. The transition to the higher T eff takes place at the age of ∼ 6.2 Myr. At that point the stellar wind removes the remaining part of the envelope with X He;surf ≈ 0.6 and reveals deeper layers with a higher He abundance and a much steeper He/H gradient, leading to a further contraction (see The LMC donor in a binary with P ini ≈ 22 days (dashed line) also experiences only a brief phase of thermal-timescale mass exchange ('A'). In contrast to the Solar case, it retains a bigger envelope (M env;left ≈ 3.7 M ) that is never fully lost in winds. The donor burns He as a partially stripped star in a detached binary ('C'). It is much cooler and bigger than a stripped star: log(T eff /K) ≈ 4.4 and R ≈ 30 R . In fact, its locus in the HR diagram is similar to that of a single 30 M star at the end in all the binary models computed across different primary masses M 1 , orbital periods P ini , and metallicities. Each binary evolution model is represented by a rectangle colored according to its corresponding log(∆T MT /yr) value. Missing rectangles are non-converged models. The binary models are mapped onto the parameter space of varying M 1 and P ini , with different background areas indicating the evolutionary state of the donor star at the moment of the initial RLOF (similar to Fig. 1): the MS donors (dashed), rapidly expanding HG donors (light grey), and core-He burning donors (densely dotted). Rectangles colored in various shades of yellow and orange correspond to models that evolve through a phase of nuclear-timescale mass transfer (log(∆T MT /yr) 5) stages of MS or early stages of post-MS evolution. The postinteraction nature of a partially-stripped donor may be revealed through measurements of increased helium and nitrogen abundances (see the bottom two panels of Fig. 4) as well as its low spectroscopic mass, see Sec. 3.4 for a further discussion. At the very end of core-He burning, the donor expands again leading to another phase of mass transfer (the so-called case BB RLOF, 'E'). The outcome of this phase and the final evolutionary stages before the core-collapse of the donor are outside the scope of this work.
The evolution of the LMC model with P ini ≈ 153 days as well as the SMC model is dominated by long phases of nucleartimescale mass transfer. After the initial rapid mass-transfer (labeled 'A'), these donors regain thermal-equilibrium and the mass-transfer rate decreases ('B'). During the subsequent phase of slow and long-lived mass transfer, their position in the HR diagram hardly changes at all. Note that the temporary detachment in the LMC model ('C') is associated with only a slight contraction of the donor star and therefore no significant displacement in the HR diagram. We find that binaries evolving through a nuclear-timescale mass transfer often experience a secondary ('delayed') peak of mass transfer rate which we label with 'D' in Fig. 4 and Fig. 5. This peak inṀ is associated with the donor being stripped down to a helium abundance plateau left by layers that used to be fully convective as part of the so-called intermediate convective zone. We discuss this in more detail in Sec. 4. Once again, at the final stages of He-burning the donor stars expand leading to a phase of thermal-timescale mass transfer at the point of central-He exhaustion ('E').
In the following sections we discuss the parameter space for the different types of mass transfer evolution illustrated above.
Parameter space for nuclear-timescale mass transfer
In the previous section we demonstrated that a post-MS mass transfer in massive low-Z binaries can be long and slow, proceeding on a nuclear-timescale and leading to only partial envelope stripping. In this section we analyze the parameter space for this type of evolution. Fig. 6 includes results from our entire grid of binary models (initial mass ratio q = 0.6). Each model is represented by a rectangle that is colored according to the integrated duration of RLOF mass transfer in that model, on a logarithmic scale from 10 4 to 10 6 yr. Non-interacting models (i.e. the widest binaries) are not plotted. Empty spaces without a rectangle indicate non-converged models (due to numerical issues). In addition, similarly to Fig. 1, in the background of Fig. 6 we indicate the parameter ranges for RLOF corresponding to different evolutionary stages of the donor star: MS donors (dashed), rapidly-expanding HG donors (light grey), and core-He burning donors (densely dotted). Note that each mass includes at least one case-A mass transfer model, i.e. with a donor star that is still on the MS. In what follows we focus only on wider binaries: those that interact during the post-MS evolution of the donor.
Rectangles colored in various shades of orange in Fig. 6 correspond to models that evolve through a phase of nucleartimescale mass transfer (with duration log(∆T MT /yr) > 5). In a similar Fig. 8, which maps the major types of mass transfer evolution found across the grid of models, we clearly distinguish the nuclear-timescale cases in magenta. We find that the parameter space for this type of evolution is related to the parameter space for RLOF from core-He burning donors, but not identical. This is the mass range in which primaries experience a low degree of rapid post-MS expansion and burn He as blue or yellow supergiants. Intuitively, it makes sense that those are the donors that can regain thermal equilibrium while only partially stripped and transfer mass on the nuclear timescale of core-He burning. Somewhat surprisingly, we find slow mass transfer and partial- Fig. 7: Mass-transfer rate and the envelope mass as a function of time in binary models evolving through a case B mass-transfer. Different panels correspond to different initial masses of the donor (M 1 ). Colors indicate different metallicities. All the models were computed with initial orbital period P ≈ 225 days and initial mass ratio q = 0.6. At Solar metallicity, the case B mass transfer is always a short-lived phase (< 10 4 yr) that strips the (nearly) entire hydrogen envelope leaving behind a stripped helium core. At subsolar metallicities of the LMC and SMC, above a certain donor mass a significant part of the envelope may be stripped on a much longer nuclear timescale of core He burning. Such donors remain only partially stripped for the duration of core-He burning. Fig. 6, each binary model is represented by a rectangle and the background areas indicate the evolutionary state of the donor at the moment of the initial RLOF (Main Sequence, Hertzprung gap, or core-He burning). Missing rectangles are non-converged models. Black rectangles correspond to models in which the mass transfer takes place already during the MS. Among binaries with a post-MS interaction, we differentiate between those that evolve through phase(s) of nuclear-timescale mass transfer (∆T MT > 10 5 yr, in magenta) and those in which the interaction happens on a short thermal timescale (∆T MT < 10 5 yr, typically < 10 4 yr, in blue). In addition, in hatch are models in which the post-interaction donor phase with T eff < T eff;ZAMS is long (> 0.75 the core-He burning lifetime), i.e. models leading to partially-stripped rather than hot stripped stars. envelope stripping also in various models in which the donor at RLOF is still a rapidly-expanding HG star. For example: models with P ini ≈ 10 − 30 days and M 1 = 20 − 36 M in the SMC grid and models with M 1 = 36 M and M 1 = 44 M donors in the LMC and SMC cases, respectively. Notably, in all these cases the detachment or nuclear-timescale mass transfer occurs only after the core-He burning phase is reached. Fig. 8 illustrates that the parameter space for nuclear-timescale mass transfer (and partialenvelope stripping in general) is significantly larger than simply the parameter space for mass transfer initiated at the core-He burning stage.
The effect of the primary mass and metallicity on the mass transfer duration is also evident in Fig. 7, in which we take a slice through the grid of binary models selecting those with ini- Fig. 9: Distribution of post-interaction donors in detached binaries with Roche-lobe filling factors < 0.6 (shown in greyscale) from all the grid models interacting after the end of MS, i.e. through case B mass transfer, plotted for the Solar metallicity (left) and the LMC composition (right). Overplotted are several donor tracks from selected binary models (colored lines) as well as single stellar tracks for comparison (grey lines). The mass transfer phase is shown in bold. Crosses are spaced by 50,000 yr starting from the onset of RLOF. Note that models were not weighted by the initial-mass function or the initial period distribution to construct the distribution. In all the Solar metallicity models, thermal-timescale mass transfer leads to the formation of hot stripped stars. In the LMC models, primaries above the initial mass of M 1 ≥ 17 M become only partially stripped and as post-interaction stars can largely overlap with single-stellar tracks. tial periods of P ini ≈ 225 days and varying the initial primary mass M 1 . Above a certain primary mass, the low-Z donors retain a substantially larger fraction of their envelopes from the initial rapid interaction and experience episodes of slow mass transfer afterwards. Notably, LMC systems with M 1 = 17 M and 44 M as well as the SMC model with M 1 = 53 M evolve through mostly rapid (thermal) mass transfer and yet they retain a higher envelope fraction compared to Solar metallicity models. Those model lead to the formation of long-lived partially stripped stars in detached binaries, similarly to the M 1 = 30 M LMC model with P ini ≈ 22 days shown earlier in detail in Fig. 4 and Fig. 5.
Formation and properties of partially-stripped stars in detached binaries
In this section we take a closer look at models that lead to formation of long-lived partially-stripped stars in detached binaries (e.g. Fig. 4 and Fig. 5). We define those as binaries in which a donor star spends at least 75% of its core-He burning lifetime as a post-interaction star that stays on the cooler side of the ZAMS temperature in the HR diagram. Such donors are partially-stripped of their envelopes and overlap with single stellar tracks, as opposed to fully stripped stars that become hot helium stars (with T eff T ZAMS ), see Fig. 5. In Fig. 8, with hatches we mark which models in our grid lead to partially-stripped stars. It should be noted that various non-hatched magenta rectangles (representing models that evolve through nuclear-timescale mass transfer) can also produce partially-stripped donors but only for relatively short detachment phases (that do not make our 0.75∆T He;burn cut). For more details see Fig. A.4 in the appendix. Importantly, in those cases the donor star never shrinks significantly with respect to its Roche lobe (usually R don /R RL > 0.6). 4 Fig. 8 makes it clear that in the M 1 − P ini plane models leading to partially-stripped stars are an intermediate stage between rapid mass transfer models and models experiencing nucleartimescale mass transfer. Similarly to nuclear-timescale interaction models, we find that partial-envelope stripping is roughly associated with the primary mass range in which the HG expansion is halted at the blue/yellow supergiant stage. Additionally, for a given primary mass among the LMC and SMC models with M 1 ≥ 17 M , the type of mass transfer evolution depends on the orbital period, where long detachments with partially-stripped stars usually take place in binaries with relatively short orbital periods. This is because the shorter the orbital period, the more of the envelope is transferred in the initial mass transfer phase (which favors long detachments), see also Fig. 3. A similar effect would be observed in models with a more extreme mass ratio or a higher specific angular momentum of the non-accreted matter. We also find relatively fewer binaries with long detachment phases among the SMC models compared to the LMC grid.
In the following, we discuss the basic properties of partiallystripped stars. Fig. 9 compares the distribution of donors stripped in the LMC metallicity models with their Solar metallicity counterparts in the HR diagram. In each panel, the coloured lines show donor tracks from a selection of binary models with different primary masses in which the mass transfer was initiated The colors correspond to the few selected donor tracks from binary models shown in the HR diagram in Fig. 9, with crosses (spaced by 50,000 yr) marking the position occupied for most of the evolution since the onset of RLOF. Grey lines show the evolution of single stars with the corresponding masses. Similarly to Fig. 9, in greyscale we plot the distribution of postinteraction stars in detached systems from all the grid models (except those interacting on MS). The figure demonstrates that mass transfer can produce partially-stripped stars that may appear significantly undermassive for their apparent evolutionary mass (as one would deduce from single tracks). Note, however, that models with M 1 ≤ 14 M produce post-interaction stars that are hot stripped stars (T eff > T eff;ZAMS ). soon after the end of MS. Crosses indicate where a donor spends most of its core-He burning lifetime since the onset of RLOF (spaced by 50,000 yr). Grey lines show single-stellar models of the same masses. In addition, in greyscale we plot the distribution of post-interaction donors in a detached state with a Rochelobe filling factor R/R RL < 0.6 from all our binary models. The 0.6 cut makes sure that we do not include most of binaries that evolve primarily through phases of nuclear-timescale mass transfer (and temporary detachments with R/R RL > 0.6) with blue or yellow supergiant donors at log(T eff /K) < 4.0. We discuss those in Sec. 5.4. We caution that the binary models used to construct the greyscale distribution were not weighted by either the initial mass function or the initial period distribution (for a weighted HRD distribution we refer to Fig. 14).
At Solar metallicity (left panel in Fig. 9), all the binary models evolve through thermal-timescale mass transfer which rapidly strips nearly the entire envelope of the donor. As a result, most of the post-interaction lifetime of donors is spend in the hot region of stripped helium stars, to the left of the T eff range of ZAMS. It is noteworthy that hot stripped stars originating from ≥ 17 M donors (luminosities log(L/L ) 4.9) tend to be somewhat cooler than those originating from lower mass stars. This is because, in general, more massive donors tend to retain a more massive envelope after mass transfer. As a result, above a certain donor mass (∼ 17−20 M in our grid), stripped stars still retain some of the envelope layers characterized by a relatively low He abundance (Y ∼ 0.6) and a nearly flat abundance profile, see Fig. A.2. Such remnant envelopes lead to stripped stars with cooler effective temperatures compared to those stripped more Fig. 11: Properties of hot stripped stars (T eff > T eff;ZAMS , in blue) compared to those of partially-stripped (T eff < T eff;ZAMS , in red) in detached binaries with R/R RL < 0.6 (as in Fig. 9) based on LMC models with a post-MS mass transfer interaction (i.e. excluding case A models). Note that partially-stripped stars originate primarily from models with primary masses M 1 > 17 M .
deeply into layers with a steep He/H abundance gradient (see also Schootemeijer & Langer 2018). This effect is present until luminosity log(L/L ) ≈ 5.3, above which stellar winds are strong enough to quickly strip these remaining envelope layers and increase the effective temperature to log(T eff /K) > 5.0.
In the LMC case, models with primaries with M 1 ≥ 17 M and a post-MS mass-transfer interaction no longer produce hot stripped stars. Instead, they lead to partially-stripped postinteraction donors that populate the HR diagram region with T eff < T eff;ZAMS in which they occupy the same region as pre-interaction stars or single stars. Interestingly, we not only find post-interaction core-He burning stars in the region of MS (where, for most realistic star formation histories, they would be vastly outnumbered by core-H burning stars) but also in the T eff range of the early post-MS stage ( log(T eff /K) between ∼ 4.2 and 4.4), where single stellar tracks predict virtually not stars and a sharp gap in the HR diagram distribution, in contrast to observations (see Sec. 5.4).
Partially-stripped stars, while potentially mixed with single or pre-interaction stars, have several distinctive characteristics. First, they are undermassive for they apparent luminosities compared to single stellar tracks. We illustrate this in Fig. 10 where, similarly to Fig. 9, in greyscale we plot the mass-luminosity distribution of post-interaction stars in detached binaries from our entire grid of LMC models (excluding case A systems). With colored lines, we overplot donor tracks from a few selected binary models (the same as in Fig. 9), and with grey lines we show single stellar tracks for comparison. The figure demonstrates that mass transfer can produce partially-stripped stars that may appear significantly undermassive for their apparent evolutionary mass (as one would deduce from their luminosity and based on single tracks). Wind-mass loss can reduce the mass of post-MS single stars in a similar way only for the most massive stars in our grid ( 44 M ).
In Fig. 11, we illustrate several other basic properties of partially-stripped stars (in red) compared to hot stripped stars (in blue). In addition to their high luminosity-to-mass ratio, partially-stripped stars are also expected to be helium and nitrogen enriched on the surface, as prior mass loss has revealed deep envelope layers which have been mixed with products of CNO burning. At the same time, they are not expected to be fast rotators: they originate from wide binaries (P ini > 10 days) so that tidal synchronization leads to rotational velocities 10 km/s. As we find them all across the HR diagram, they have surface gravities in a wide range, from ∼ 2 to 4.3 (in log(g)), and can reside in systems with very different orbital periods (from tens to thousands of days).
In Sec. 5.4 we discuss implications of our findings on the distribution and observables of core-He burning stars in the HR diagram.
Why partially stripped? Radial response of stars to mass loss
In the previous sections we found that in many low-Z massive binaries the post-MS mass transfer may include phases of long nuclear-timescale mass exchange (Ṁ ∼ 10 −5 M yr −1 , ∆T MT > 10 5 yr) and produce post-interaction core-He burning supergiants that are only partially stripped of their envelopes. This is in stark contrast to Solar metallicity models, all of which predict the thermal-timescale mass transfer to strip nearly the entire donor envelope (Ṁ ∼ 10 −3 M yr −1 , ∆T MT 10 4 yr) and produce hot stripped stars. In this section, we seek to understand the origin of these canonically different types of post-MS mass transfer evolution. To do so, we study the behavior of donor stars of different metallicities in response to mass loss. We define R th;eq as the radius that a star needs to have in order to be in thermal equilibrium. In the case of singe-star evolution, most of the time R th;eq = R where R is the actual radius of a star. The exception are short-lived phases such as the HG expansion or a phase of He-shell burning right after the central He exhaustion. In the case of binary evolution, the radius of a star is restricted by the size of its Roche lobe (R R RL ), which may prevent it from reaching thermal equilibrium. For instance, during thermal-timescale mass transfer the donor star is thermally unstable having radius R don ≈ R RL << R th;eq . Its continuous fast expansion in a futile attempt to equate R don = R th;eq is what leads to high mass transfer rates.
Knowing R th;eq of a donor star and how it changes over the course of mass transfer is the key to understand the transition from thermal-timescale mass transfer to detachment or to a nuclear-timescale mass exchange found in our SMC and LMC models. To obtain the evolution of R th;eq as a function of the remaining donor mass M don , we proceed in a similar way to Quast et al. (2019). We begin by evolving a binary system until the onset of RLOF. Normally, from that point onward, all of our binary models would enter a phase of thermal-timescale mass transfer (at least initially). Here, however, we terminate the binary model, extract the donor star, and subject it to the following mass-stripping experiment. We switch off stellar winds and instead apply a constant mass-loss rate 10 −5 M yr −1 which is low enough to guarantee that the star remains at thermal equilibrium. At the same time, we switch off any composition changes due to nuclear burning. This mimics the rapid evolution through a Fig. 12: Origin of partial envelope stripping of the LMC and SMC models explained based on the response of donor stars to mass loss, using the 20 M donor case as an example. Quantities are plotted as a function of the remaining donor mass M don , i.e. the time direction in the top two panels is to the left. Top panel: mass transfer rate from binary models (P ini ≈ 32 days). Middle panel: equilibrium radius R th;eq that the donor star would need to have in order to be in thermal equilibrium, compared with its Roche lobe size R RL . For as long as R th;eq > R RL , the thermaltimescale mass transfer continues to strip the envelope of the donor. Vertical dotted lines mark the points when R th;eq = R RL , the donor can regain equilibrium, and the mass transfer may stop or slow down to the nuclear timescale. Bottom panel: internal He abundance profile of the donor at the onset of RLOF.
phase of thermal-timescale mass transfer when there is no time for any significant burning to occur. The radius of a star that is being stripped in this way is a good representation of the equilibrium radius R th;eq that a donor star would need to have in order to regain stability and stop the thermal-timescale mass transfer.
In Fig. 12, we plot the results for the case of 20 M donors in binaries with an initial period of P ini ≈ 32 days and three different metallicities. By the onset of RLOF (R RLOF ≈ 60 R ), the SMC donor has nearly regained equilibrium from the HG phase as a core-He burning star, the LMC donor would still continue the HG expansion up to R ≈ 100 R and the Solar donor up to R ≈ 1000 R . The panels are plotted as a function of the remaining donor mass, i.e. the time direction in the top and middle panel is to the left. The bottom panel shows the internal helium abundance (X He ) profile of the donor at the onset of RLOF. The He profiles are characterized by a plateau with a constant He abundance X He ≈ 0.5 left by an intermediate convective zone and a constant X He gradient above the plateau left by a retreating convective core during the MS (e.g. Langer et al. 1985;Langer 1991). The middle panel shows the evolution of R th;eq obtained from the mass-stripping experiment described above as well as the size of the donor's Roche lobe R RL in the corresponding binary model (initial mass ratio q = 0.6). For as long as R th;eq > R RL ≈ R don , the donor star is out-of-equilibrium and the mass transfer proceeds on the thermal timescale. Vertical dotted lines mark the donor mass when R th;eq = R RL . For comparison, in the top panel we plot the actual mass transfer rate evolution from the corresponding binary models. Note that beside the initial thermal phase withṀ ≈ 10 −2.5 M yr −1 , the SMC model also experiences a nuclear-timescale mass transfer witḣ M ≈ 10 −5 M yr −1 .
The middle panel of Fig. 12 illustrates the key difference between the Solar-metallicity and the low-Z donors. All throughout the mass transfer, the Solar-metallicity donor would need to expand to R th;eq ≈ 1000 R (the size of a red supergiant) to regain thermal equilibrium. This is much larger than the size of the Roche lobe, not only in the P ini ≈ 32 days and q = 0.6 example shown in Fig. 12 but in nearly all the possible orbital configurations in an interacting binary. Only when the donor is stripped of nearly its entire envelope (M don ≈ 7.5 M ), does the R th;eq begin to rapidly decrease towards the typical size of a fully-stripped helium star (a few R ) and the binary detaches. The R th;eq evolution of the Solar model in Fig. 12 is very well representative for all the post-MS donor stars that become fully stripped during thermal-timescale mass transfer, i.e. the entire Solar-metallicity grid as well as the 10-14 M donors at SMC and LMC compositions.
The SMC and LMC donors in Fig. 12 behave in a distinctively different way. The initial increase of R th;eq is slower, which leads to slightly smaller thermalṀ rates found in low-Z compared to Solar binary models. More importantly, R th;eq begins to decrease much earlier, when the donors are still far from being fully stripped. 5 This allows the low-Z donors to regain thermal equilibrium and detach when only partially stripped, as seen in the binary models in the top panel (and found all throughout the low-Z grids of binary models). In addition, the fact that R th;eq decreases gradually over a wide range of remaining donor masses is what leads to a large variety of envelope masses of partially stripped low-Z donors, depending on the orbital period and the mass ratio (e.g. Fig. 2 and Fig. 3).
One may notice that the moment when R th;eq = R RL in the mass-stripping experiment is not always exactly aligned with thė M drop in the binary model (top panel), e.g. the LMC model in Fig. 12. This is because as the donor star relaxes to regain equilibrium at R = R th;eq , it may still continue to expand relative to its Roche lobe, leading to 'over-stripping'. We find that this is especially the case when the R th;eq = R RL vertical line falls in the region of the He abundance plateau (M don range between ∼ 7 and 10 M in the SMC and LMC models in Fig. 12).
In general, the behavior of R th;eq in Fig. 12 could be described as a combination of a steep increase, that may happen right after the RLOF (cf. the LMC and Solar cases), followed by a gradual increase, maximum, and then a gradual decrease of R th;eq as a function of the decreasing donor mass. 6 The initial steep increase takes R th;eq to the radius that a normal single stellar model would expand to by the end of the HG phase. The SMC donor in Fig. 12 was already near that size when RLOF occurred, hence no steep 5 In fact, the R th;eq behavior of low-Z models in Fig. 12 is somewhat similar to that of MS donors. 6 Note that any phase of R th;eq increase will saturate if a model reaches the Hayashi track (at ∼ 1000 R ) and may no longer expand, leading to a R th;eq plateau in Fig. 12. Fig. 13: Similar to Fig. 12, but for the case of a 25 M donor at LMC composition and five different initial orbital periods. The legend on top details what was the evolutionary state of the donor at the onset of RLOF (HG phase or core-He burning) as well as the main outcome of the mass transfer interaction (hot stripped star, partially stripped star, or nuclear-timescale mass transfer). The mass-radius exponent in the third panel is defined as ζ th = dlogR th;eq /dlogM don as derives directly from the results in the second panel. Horizontal lines show the mass-radius exponent of the donor's Roche lobe ζ RL = dlogR RL /dlogM don assuming a fully non-conservative mass transfer. The behavior of ζ th as a function of the remaining donor mass can be linked to various features found in binary models, see text.
increase in R th;eq . The subsequent gradual increase followed by a gradual decrease in R th;eq can be understood as a result of the changing envelope to core mass ratio, as described in Sec. 5.4 and Fig. 9 of Farrell et al. (2022). Fig. 12 suggests that the equilibrium radius R th;eq is to some extent affected by the underlying He abundance profile of the donor (as also found by Quast et al. 2019), in particular at the location of the already mentioned He plateau (where the R th;eq -M don slope of the low-Z models temporarily flattens). We illustrate this better in Fig. 13, where we repeat the mass-stripping experiment for the case of a 25 M donor at LMC metallicity and five different initial orbital periods. The corresponding binary models have led to various outcomes (hot stripped star, partially stripped star, or nuclear-timescale mass transfer), as explained in the legend. In the third panel we now show the mass-radius exponent ζ th = dlogR th;eq /dlogM don . The value of ζ th is a useful magnifying glass to expose any changes in the R th;eq -M don slope. For reference, with horizontal lines we show the mass-radius exponent of the donor's Roche lobe ζ RL = dlogR RL /dlogM don for a few different mass ratios and fully non-conservative mass trans-fer (Eqn. 16.25-26 from Tauris & van den Heuvel 2006). While the values of ζ th and ζ RL can be compared to assess whether the mass transfer will proceed on the thermal timescale (i.e. ζ th > ζ RL ), we stress that this comparison is only valid when the star is in thermal equilibrium, e.g. during a nuclear-timescale expansion or nuclear-timescale mass transfer.
The double-peaked behavior of ζ th in Fig. 13 can be linked to some of the features found in binary models throughout Sec. 3. In most binaries that evolve through phase(s) of nucleartimescale mass transfer, the donor regains thermal-equilibrium when R th;eq drops strongly as a function of M don (between ∼ 14 and 17.5 M in Fig. 13). In turn, in binary models in which the donor remains detached for most of the core-He burning lifetime (either as hot-or partially-stripped star) the thermal-timescale mass transfer tends to strip the donor down to deeper layers of constant X He ( 14 M in Fig. 13). The abundance pattern of these layers sets the surface composition of partially-stripped stars discussed in Sec. 3.4.
We always find a ζ th peak in layers above the He abundance plateau. Its saw-like shape is related to the step-like He abundance gradient above the region of He plateau. It manifests itself in oscillations of the mass transfer rate during the nucleartimescale mass transfer phase when a donor is being stripped from those envelope layers (see e.g. the SMC models in Fig. 2). The local minimum of ζ th around M don ∼ 14 M in Fig. 13, on the other hand, is responsible for the secondary majorṀ peak that in some models is a very prominent feature (see Fig. 4 and Fig. 5, label 'D'). 7 Stripping through the bottom envelope layers of constant He abundance (M don 14 M ) leads to a smooth ζ th increase and also results in a smoothṀ behavior found in late phases of mass transfer in binary models. 8 It is noteworthy that the values of ζ th found in the midenvelope region can be very high, with ζ th > ζ RL for even extreme mass ratios of q < 0.1. We discuss this in the context of ULX sources with NS accretors in Sec. 5.7.
In summary, we find that partial envelope stripping and nuclear-timescale mass transfer occurs in binary models when the donor can be in thermal equilibrium as a partially-stripped star of an intermediate size (R th;eq of a few tens or hundreds R ). Such donors, when evolved as single stars, are characterized by a relatively modest HG expansion and a long blue/yellow supergiant lifetime during the core-He burning phase. The post-MS expansion of a massive star is thus the key factor affecting its evolution in an interacting binary. A secondary role is played by detailed features of the He abundance profile, some of which may be causing small variations in the mass transfer rate or temporary detachments. The abundance profile in massive star envelopes is a particularly uncertain prediction of stellar models. Consequently, details of the nuclear-timescale mass transfer sequences should be treated with much caution. 7 If the mass ratio is close to unity then ζ th may temporarily become smaller than ζ RL , leading to a brief phase of another thermal-timescale mass transfer. 8 We notice a slight shift in the values ζ th of those layers between the HG and more evolved core-He burning donors in Fig. 13. This is likely associated with changes in the H abundance and He/H gradient in the moving location of the H-burning shell, which has been found to affect the radii of core-He burning massive stars (Langer 1991).
Discussion
The importance of post-MS expansion and its uncertainty due to mixing
Throughout the paper, we have found that the course and outcome of stable mass transfer evolution initiated by a post-MS donor is closely related to the way in which massive stars expand after the end of MS when they transition to the core-He burning phase. Stars that rapidly expand all the way to the red (super)giant branch (i.e. the HG phase), when transferring mass in interacting binaries, become nearly fully stripped of their H envelopes in a short phase of thermal-timescale mass transfer (Ṁ ∼ 10 −3 M yr −1 , ∆T MT 10 4 yr). In contrast, stars that remain relatively compact in the transition to the core-He burning phase and burn He as blue or yellow supergiants, when in binaries, remain only partially stripped of their H envelopes (at least by the end of core-He burning) and can evolve through phases of nuclear-timescale mass transfer (Ṁ ∼ 10 −5 M yr −1 , ∆T MT > 10 5 yr). Such a halted HG expansion of massive stars has been found in single low-Z models computed with different codes over the years (being more and more prominent the lower the metallicity, e.g. Brunish & Truran 1982;Baraffe & El Eid 1991;Langer 1991;Georgy et al. 2013;Tang et al. 2014;Groh et al. 2019;Klencki et al. 2020). The fact that low-Z massive giants can remain much more compact compared to their high (Solar) metallicity counterparts is a result of a complicated interplay between at least two different Z-dependent factors: higher temperatures and densities of low-Z helium cores at TAMS as well as lower opacities of low-Z envelopes. It is not until the current study that this phenomenon has gained special significance as the key factor affecting the evolution of stars through mass transfer in binaries.
It is essential to realize that the post-MS expansion of massive stars is notoriously model sensitive due to being highly dependent on the efficiency of internal mixing. This wellestablished fact (e.g. Langer et al. 1985;Langer & Maeder 1995;Maeder & Meynet 2001) is especially clear in the recent studies in which the increased computational power has allowed for a more comprehensive exploration of various mixing coefficients (Schootemeijer et al. 2019;Klencki et al. 2020;Kaiser et al. 2020;Higgins & Vink 2020). Among the factors that were identified to play a role are convective-core overshooting during the MS (Stothers & Chin 1992;Langer & Maeder 1995), semiconvection (Langer et al. 1985;Langer 1991), rotational mixing (Georgy et al. 2013), as well as past accretion phases (especially if the accretor is non-rejuvenated, see Hellings 1984;Braun & Langer 1995;Dray & Tout 2007). We refer to an extensive discussion of the subject in Sec. 5.1 and App.B in Klencki et al. (2020). For example, if we were to compute our grids of binary models with lower efficiency of semiconvection (α SC = 1 instead of α SC = 33) then even the SMC-metallicity models would all experience a rapid HG expansion all the way until the red supergiant stage (and be subject to full envelope stripping through thermal-timescale mass transfer).
For the time being, the key piece of evidence in support of the models presented in this paper are large populations of blue and yellow supergiants identified in the LMC and the SMC (Hunter et al. 2008;Neugent et al. 2010;2012;Urbaneja et al. 2017;Kalari et al. 2018, see also HR diagrams in Ramachandran et al. 2019;Gilkis et al. 2021). Their existence can only be reconciled with models that predict a significant fraction of core-He burning taking place in the middle of the HR diagram (Schootemeijer et al. 2019;Klencki et al. 2020) such as the models adopted here (with a halted HG expansion at the LMC and SMC composi-tions). A robust comparison between theory and observation to calibrate the post-MS expansion is challenged by the fact that the population of blue and yellow supergiants may also include stars in a post-red supergiant stage (either stripped in binaries or through cool-star winds, or stars experiencing blue-loops, e.g. Ekström et al. 2012;Meynet et al. 2015;Farrell et al. 2019) as well as potentially some of the accretors and mergers from past binary interaction phases (e.g. Podsiadlowski et al. 1992; Vanbeveren et al. 2013;Glebbeek et al. 2013;Justham et al. 2014). Extended grids of binary models, exploring various mixing assumptions and including the evolution of accretors, are likely necessary to further calibrate the post-MS expansion of LMC and SMC stars in the future studies.
Instead, a promising way to constrain the mass transfer evolution in low-Z binaries and verify our findings is to search for signatures of partial-envelope stripping and nuclear-timescale mass transfer among the populations of massive stars in metalpoor galaxies. We outline the main observational predictions from our models in Sec. 5.4.
Notably, while details of the post-MS expansion of massive stars at a given metallicity are very uncertain in stellar models, the overall trend with metallicity appears to be a robust prediction (Sec. 5.2 in Klencki et al. 2020). As such, we predict that at some sufficiently low metallicity (possibly already at the LMC composition), the thermal-timescale mass transfer and full envelope stripping will yield ground to nuclear-timescale mass exchange and partial-envelope stripping of massive post-MS donors.
The case of stellar-accretor binaries
In this work, for simplicity, we treated the secondary star as a point mass and we set the accretion efficiency β at the Eddington limit. This makes our models directly applicable to BH binaries (assuming that the Eddington limit is not substantially exceeded). However, as we argue below, we predict that most of our results and conclusions will also hold for the case of binaries with stellar accretors.
It is important to realize that Eddington-limited accretion in massive binaries is similar to the assumption of a fully nonconservative mass transfer (i.e. β = 0). For the default initial mass ratio q = 0.6, the accretor masses range from 6 to 31.8 M across our models. The corresponding Eddington accretion rates for BHs (assumed as the accretion limit in our models) range from ∼ 1.3 × 10 −7 M yr −1 to ∼ 7 × 10 −7 M yr −1 . Even during phases of slow nuclear-timescale mass transfer, with typical rates of 10 −5 M yr −1 , this yields accretion efficiencies of only a few percent. We thus expect that a grid of binary models with stellar accretors and a small accretion efficiency would produce mass-transfer sequences very similar to our models.
As we discuss in Sec. 5.3, the true accretion efficiency in stellar-accretor binaries corresponding to our models is currently unknown and so ideally any β value between 0 and 1 should be considered. Varying β, and similarly considering different values for the specific angular momentum of the non-accreted matter, would affect the evolution of binary separation during mass transfer and consequently the size of the donor's Roche lobe. These effects are degenerate with changing the mass ratio between the binary components. In Fig. 2, using the 25 M donor example, we showed that for a wide range of initial mass ratios (q between 0.25 and 1.5), the essential differences between the Solar and low-Z models remain unaffected. In particular, binaries in which thermal-timescale mass transfer produces fully stripped donors (the Solar-Z example) evolve towards the same outcome no matter the mass ratio. In the case of low-Z models with partial envelope stripping, variations in factors that affect the Roche lobe of the donor will likely affect the ratio between systems that remain primarily in the detached stage and those that maintain the nuclear-timescale mass transfer.
Besides the possibility of high accretion efficiencies, the presence of a stellar accretor could also lead to some of the binaries evolving towards a contact phase (Pols 1994;Wellstein et al. 2001) and potentially dynamical instability and a merger (Marchant et al. 2016). Details of this process remain highly uncertain.
Increased accretion efficiency in low-Z binaries?
It is interesting to speculate about the impact of long and slow mass transfer phases in our low-Z models on the fraction of the transferred matter that gets accreted by the companion. A clear coherent picture of accretion efficiencies β in binaries of different orbital periods and component masses is still missing. Observational clues from double-lined eclipsing SMC binaries (de Mink et al. 2009b), WR-O star systems (Petrovic et al. 2005), the well studied sdO+Be binary system ϕ Persei (Schootemeijer et al. 2018), or Be-X ray binaries in the SMC (Vinciguerra et al. 2020) yield very different accretion efficiencies, ranging from highly non-conservative (β ≈ 0) to nearly fully conservative cases (β ≈ 1). 9 Since the emergence of binary evolution models with rotation (Langer et al. 2003b), a promising way of obtaining β self-consistently from models has been to assume that the material can be accreted conservatively up to the point when the accretor becomes spun-up to critical rotation, at which point β drastically decreases and is further controlled by the timescale of internal angular momentum transport in the accretor (Langer et al. 2003a). 10 Because little mass accretion is needed to reach the critical rotation of the accretor (Packet 1981), and the thermal timescale of mass transfer is typically much shorter than the timescale for angular momentum transport in the accretor, such models tend to predict a highly non-conservative post-MS mass transfer evolution (Petrovic et al. 2005;de Mink et al. 2013). The exception are cases with a close to equal mass ratio due to their relatively lower mass transfer rates (e.g. Cantiello et al. 2007).
Based on those arguments, we expect that long nucleartimescale mass transfer phases in low-Z binaries may result in considerably higher accretion efficiencies compared to systems evolving through only thermal-timescale mass transfer. This might possibly bring such models into agreement with β ∼ 0.5 inferred from Be-X-ray binaries of the SMC composition (Vinciguerra et al. 2020), although a detailed study of the issue is certainty needed before any conclusions could be drawn. Longer mass transfer phases could potentially also extend the duration of the Be phenomenon in rapidly-rotating accretors, which might help in resolving the apparent overabundance of Be stars in the SMC compared to the Galactic environment (Dray 2006).
Binary-interaction products in the HR diagram
Here, we discuss one of the most interesting implications of binary evolution models: the predicted distribution of post-MS Fig. 14: Distribution of post-MS primaries in the HR diagram (the vast majority being at the core-He burning stage) inferred from the grid of q = 0.6 binary models at the LMC composition. For the SMC version of this figure see A.5. Models have been weighted by the initial mass function and orbital period distribution of early-type binaries, and normalized to a constant star formation rate of 1 M /yr (see text). The overall distribution is showed in greyscale. In addition, we plot a random sampling of stars from the distribution, color-coded according to their pre-or post-interaction state. Several single stellar tracks are plotted with solid lines for comparison. Around 30% of primaries reside in wide non-interacting systems. The four Boxes indicate four characteristic regions: hot stripped stars from both case A + case AB evolution as well as those formed through case B mass transfer (Box I), hot stripped stars from case A + case AB mass transfer only (Box II), a region where most of the stars are donors in currently mass-transferring systems (Box III), and finally a region where most of the long-lived partially-stripped stars in detached binaries reside (Box IV). stars, including the binary-interaction products, in the HR diagram. In the current study, for each initial composition we computed only a single grid of models (with initial mass ratio q = 0.6) and evolved only the primary stars. For that reason, we are unable to construct a complete population model of massive stars and binaries. Even a single q grid, however, is sufficient to showcase the main subpopulations of post-MS primaries predicted by the binary models and discuss their main observational characteristics. To this end, we choose the LMC grid as the most illustrative example.
In order to construct a simple population model, we weight all our binary models by the initial mass function dN/dM 1 ∝ M −2.3 1 (Salpeter 1955;Bastian et al. 2010), making the usual assumption that it well describes the initial mass function of primaries (Kroupa et al. 2013). In addition, we weight each model according to an initial orbital period distribution dN/d logP ini ∝ logP −0.55 ini (Sana et al. 2012). We normalize the period distribution to the range logP ini = [0.15, 5.5]. While wider early-type binaries do exist, they are very rare for q > 0.3 (see Fig. 37 in Moe & Di Stefano 2017). For simplicity, we only consider massive stars that are formed in binaries, which neglects that a significant fraction of early-type stars are formed in triples or higher order systems (see Fig. 39 in Moe & Di Stefano 2017). For each primary mass, we use our widest binary model (i.e. the non-interacting one) to represent all the wide non-interacting binaries in the weighting procedure described above. Likewise, we extrapolate to the shortest orbital periods of very close binaries (P ini ∼ 1.41 days) using the model with the shortest initial orbital period in our grid (P ini ≈ 3.16 days), which is relevant for our estimates of the products of case A mass transfer evolution. We caution that this is a rather crude approximation: very close binaries will produce somewhat less massive and luminous stripped helium stars than our P ini ≈ 3.16 days models. 11 Based on these assumptions, we are able to estimate the distribution of post-MS primaries in the HR diagram. The result is shown in greyscale in Fig. 14, normalized to show a number of stars per bin assuming a constant star-formation rate of 1M /yr. 12 In addition, we plot a random sampling of 770 primaries from the distribution (the expected number, given the normalization) and color-code the stars according to their preor post-interaction state. For comparison, with grey lines we plot several single stellar tracks.
Post-MS primaries from wide non-interacting binaries (around 30% of stars in Fig. 14) occupy regions in the HR diagram that would also be populated by single stellar tracks, primarily the red (super) giant branch at log(T eff /K) ≈ 3.6. Binaryinteraction products, on the other hand, can occupy nearly any location in the HR diagram (although in the region of the MS they would be a small minority compared to core-H burning objects). Below we describe four most prominent categories of post-interaction and interacting primaries, with Boxes I-IV marking regions where they can typically be found in Fig. 14.
Box I is populated by hot fully stripped stars originating from both the primaries that interacted already during the MS (case A mass transfer) as well as those that are stripped solely through post-MS (case B) mass transfer. As the case A mass transfer systems are not the focus of the paper and have not been presented as part of the results, we clarify that in each of our case A models the primary becomes a fully stripped helium star after a sequence of case A and case AB mass transfer phases. This is in agreement with all the previous detailed mass transfer models of such systems (e.g. Podsiadlowski et al. 1992;Pols 1994;Wellstein et al. 2001;Petrovic et al. 2005;de Mink et al. 2007;Wang et al. 2020). In general, we find that stars stripped in case A + case AB mass transfer are slightly hotter in our models compared to those fully stripped in post-MS mass transfer (by about 0.1 dex in log(T eff /K)). A more detailed discussion of differences between the two classes, including potential surface abundance signatures (e.g. Schootemeijer & Langer 2018), is beyond the scope of this paper. Notably, Box I is populated by primaries with initial masses ≤ 14 M from our LMC grid, i.e. the mass range in which post-MS mass transfer always leads to full envelope stripping on the thermal timescale (see Fig. 8 for the summary of model outcomes).
In contrast to Box I, the hot stripped stars populating Box II (WR stars above a certain luminosity) are only formed in case A mass transfer systems (to compare, see the right-hand panel of Fig. 9 which excludes the case A systems). This is because in the M 1 ≥ 17 M primary mass range the majority of LMC models evolve to either produce long-lived partially stripped stars or spend most of the core-He burning lifetime in nuclear-timescale mass transfer. This naturally leads to a prediction that the formation efficiency of hot stripped stars in interacting binaries decreases with increasing luminosity around log(L/L ) ≈ 4.8 for the LMC composition relative to Solar metallicity models (which always lead to full envelope stripping in mass transfer). We discuss this further in Sec. 5.5.
Box III is where the majority of donors in nuclear-timescale mass transferring systems reside. They originate from systems with primaries in the mass range where single stellar models predict a halted HG expansion and core-He burning blue or yellow supergiants (M 1 between ∼ 20 and 36 M in our LMC grid). In fact, about 40% of stars in Box III are currently mass transferring, i.e. they are donors in semi-detached or contact systems. Their presence or absence in a population of late-blue/yellow supergiants in the LMC (or similarly SMC) will serve as a strong observational test to verify whether nuclear-timescale post-MS evolution can happen already at the LMC (or SMC) metallicity. Currently, the census of yellow supergiants in the Magellanic Clouds comes from single-epoch spectroscopic surveys by Neugent et al. (2010;2012) and offers little insights into their binary fraction. The period distribution of mass-transferring systems in Box III is roughly flat in log(P) in the range between ∼ 50 and 1000 days. The long periods of these systems may make it challenging to assess their binary nature in typical spectroscopic surveys (unless a long-term monitoring campaign is possible, e.g. Sperauskas et al. 2014). An alternative approach could be to systematically search for eclipsing binaries with yellow supergiants in photometric variability surveys. One such system was serendipitously discovered in a dwarf galaxy Holmberg IX by Prieto et al. (2008) with the Large Binocular Telescope. The authors also pointed out to another candidate system in the SMC that had already been present in the All Sky Automated Survey data.
Finally, Box IV is largely populated by the class of long-lived partially-stripped core-He burning stars in detached binaries that is unique to our low-Z models, see also Sec. 3.4 for details about their formation. 13 The properties of stars in Box IV are quite interesting. First of all, they are under-massive for their luminosity when compared to single stellar tracks (by about a factor of 2, depending on luminosity, see Fig. 10). They are also helium and nitrogen enriched while being slow rotators (Fig. 11), which distinguishes them from fast rotators in which the surface enrichment is the result of rotational mixing in their interiors. These properties in principle make them similar to MS donors interacting in case A mass transfer binaries. However, in case A systems the donor spends most of its lifetime in a semi-detached or nearly Roche-lobe filling stage in a close orbit (Wang et al. 2020). Partially-stripped stars from the post-MS mass transfer models, on the other hand, can be found in binaries with much wider orbits, with their period distribution stretching from tens to thousands of days (Fig. 11). Together with the supposed presence of a rapidly-spinning companion (spun-up as a result of mass transfer), these set of characteristics makes them a unique class of objects.
It is tempting to view partially-stripped stars as a promising explanation for at least some of the puzzling nitrogen-enriched slowly-rotating B-and O-type (super)giants that have been identified in VLT-Flames spectroscopic surveys of the LMC (Hunter et al. 2008;McEvoy et al. 2015;Grin et al. 2017). Even more interestingly, Box IV stretches out to temperatures cooler than the TAMS, i.e. log(T eff /K) < 4.35. This is the region of the HR diagram that remains virtually unpopulated by single stellar tracks. In reality, a surprisingly large number of B (super)giants living apparently next to the MS have been found in the LMC (e.g. Evans et al. 2006), many of which being N-rich and with spectroscopic masses systematically lower than their evolutionary masses (McEvoy et al. 2015), i.e. with properties resembling those of partially-stripped stars. In the past, binary-interaction channels involving the accretors or stellar mergers have been put forth as a possible explanation of this population (Brott et al. 2011b;Glebbeek et al. 2013).
The elephant in the room of the discussion so far is the nature of the companion star. Mass losers that become hot stripped stars (Box I and Box II) emit most of the light in the UV, which is why irrespective of the relative bolometric luminosity of the components, the secondary (if a star and not a compact object) is nearly always the brighter source in optical bands . This is no longer the case for the much cooler partiallystripped stars with T eff < T eff;ZAMS produced in our LMC and SMC binary models (e.g. Box III and Box IV). Among those, depending on the initial mass ratio and the accretion efficiency, there could be systems in which the secondary becomes more massive and luminous than the primary as well as such in which the primary remains the brighter and more easily detectable component. In the first case, the partially-stripped companion may remain hidden in the presence of a (most likely) rapidly spinning companion with broad emission lines that make it challenging to detect orbital velocity variations. Such two components may be especially difficult to disentangle from their spectra in the case of the Box IV population, where both stars are likely to have a similar spectral type (type B). They may appear as more massive analogues of the recently reported LB-1 system (Liu et al. 2019), in which case the presence of a B-and Be-type components in the spectra has proven to be misleading and challenging to unravel without a dedicated technique (Shenar et al. 2020a;Bodensteiner et al. 2020). On the other hand, binaries in which the partially-stripped donor remains the more luminous of the two components are the most promising to explain some of the peculiar nitrogen-enriched (super)giants mentioned above.
Fewer stripped and WR stars from interacting binaries at low-Z?
Mass transfer interaction in binary systems is thought to be the main formation channel for stripped stars and, alongside the single-star channel in which the envelope is stripped through strong winds (Conti 1975;Smith 2014), one of the two main channels for their most luminous subclass: WR stars (Paczyński 1967) 14 . While the single-star WR channel is expected to become less efficient the lower the metallicity (on the basis that line-driven mass loss decreases with metallicity, see Vink et al. 2001;Vink & de Koter 2005), the mass-transfer channel has so far been predicted to have an efficiency that is roughly independent of metallicity (e.g. Maeder & Meynet 1994). The binary models computed in this work suggest something different. At LMC and SMC metallicities, most of our post-MS donors with masses above 17 M never become fully stripped in mass transfer during the core-He burning phase (see Fig. 8 with the summary of model outcomes). As such, we predict the efficiency of the binary formation channel for stripped and WR stars to decrease at low metallicity. We illustrate this in Fig. 15, where for each metallicity, we estimate what fraction of primaries of a given initial mass will evolve to form a hot stripped star as a result of a mass-transfer interaction. The rough approximate luminosity of the resulting stripped stars (top X axis) is estimated as the TAMS luminosity of the corresponding single star model. While at Solar metallicity this fraction is nearly independent of the primary mass and very high (70% − 80%, with the remaining 20% − 30% being primaries in non-interacting wide systems), at lower LMC and SMC metallicities it drops down by a factor of two (to 30%−40%) for stars with initial masses 20 M , which corresponds to stripped-star luminosities log(L/L ) 4.8. For this calculation, similarly to that in Sec. 5.4, we assumed an initial mass function dN/dM 1 ∝ M −2.3 1 and an initial orbital period distribution dN/d logP ini ∝ logP −0.55 ini spanning the range logP ini = [0.15, 5.5]. Based on the few case A mass transfer models in our grid as well as previous detailed studies of case A binary evolution (e.g. Pols 1994;Petrovic et al. 2005;Wang et al. 2020), we predict that all the primaries interacting during the MS will become hot stripped stars after an episode of case AB mass transfer. Note that the result in Fig. 15 is based on a single grid of binary models with the initial mass ratio q = 0.6. The relative fraction of case A compared to post-MS (case B) 14 WR stars are essentially stripped stars with high enough luminosity to mass ratio to launch strong optically-thick winds (e.g. Gräfener et al. 2011). Fig. 15: Fraction of primaries that become hot stripped stars as a result of a mass transfer interaction, estimated based on the results from binary models as well as the assumed distributions for initial masses and orbital periods (see text). At Solar metallicity, any primary in an interacting binary eventually becomes a hot stripped star (the remaining 20%-30% are in wide noninteracting systems). At LMC and SMC metallicities, post-MS donors with masses above 17 − 20 M become only partially rather than fully stripped in mass transfer, leading to a predicted decrease in the efficiency of formation of stripped stars from low-Z interacting binaries. mass transfer systems is subject to uncertainty in the degree of envelope inflation of massive MS stars (Sanyal et al. 2015;2017;Klencki et al. 2020), a phenomenon that occurs in some of the most massive primaries in our grid of models. Fig. 15 constitutes a prediction that the number of stripped stars above a certain mass (and luminosity: log(L/L ) ≈ 4.8 in the case of our models) should be lower in metal-poor galaxies compared to high-metallicity environments. An observational test may not be straightforward. Stripped stars that are not luminous enough to appear as WR stars ( log(L/L ) < 5.25 in the LMC and log(L/L ) < 5.6 in the SMC case, following Shenar et al. 2020b) are difficult to detect in optical surveys. The most promising strategy so far has been far-UV spectroscopy targeting Galactic Be stars and it has recently led to a discovery of ten new subdwarfs (sdO), thus tripling the number of known Be+sdO systems . Future similar campaigns focused on the LMC and SMC could help verify the metallicity trend predicted by our binary models.
In the WR regime, the binary fraction of WR stars has long been predicted to increase with metallicity (due to the decreasing efficiency of the single-star channel, e.g. Maeder & Meynet 1994). Surprisingly, this does not seem to be the case as the binary fractions of classical WRs is found to be about ∼ 40% in the Milky Way and Magellanic Clouds alike (Bartzakos et al. 2001;van der Hucht 2001;Foellmi et al. 2003a;. Shenar et al. (2020b) has recently pointed out that this tension may be reduced taking into account the metallicity-dependent minimum luminosity for the WR phenomenon. Our results, suggesting a metallicity-depended formation of WRs in binaries, may also help to solve this apparent discrepancy.
Implications for explodability and SN progenitors
In this section, we discuss the impact of the newly found partial envelope stripping of low-Z massive donors on their carbon-oxygen (CO) cores, pre-SN structure, and explodability. Whether a massive star loses its H-rich envelope shortly after the end of MS as a result of mass transfer (or strong winds) or still retains at least part of its envelope during the core-He burning phase has implications for the evolution of the core. In the first case, the naked helium core decreases in mass due to wind mass-loss. In the second case, the helium core grows in mass due to continuous H-shell burning. This effect was shown to lead to lower CO core masses (M CO ) and higher carbon mass fractions (X C ) in the CO cores of fully stripped stars, two factors that largely determine the further core evolution throughout the advanced burning stages until the core collapses (Timmes et al. 1996;Brown et al. 2001;Patton & Sukhbold 2020). 15 Consequently, pre-SN stellar models computed from naked helium stars (Woosley et al. 1995;McClelland & Eldridge 2016;Woosley 2019;Ertl et al. 2020) and from fully stripped stars alike (Schneider et al. 2021;Laplace et al. 2021;Vartanyan et al. 2021) were found to have systematically smaller compactness and to be more prone to explode rather than to collapse into a BH. In particular, Schneider et al. (2021) argues that envelope loss due to mass transfer increases the minimum initial mass to form a BH from ∼20−25 M up to ∼ 70 M . Among other implications, their work suggests a drastic reduction of the formation rate of binary BH and BH-NS systems from binary evolution. The only exception in Schneider et al. (2021) were cases of case C mass transfer, i.e. the RLOF initiated after the end of core-He burning, which the authors found to have very little effect on the pre-SN core structure with respect to single stellar models. However, in the case of massive stars with initial masses above 20−25 M , case C evolution is extremely rare except in very metal-poor environments (Z 0.04 Z in Klencki et al. 2020).
Importantly, the key CO core properties (M CO and X C ) that were found to determine the compactness are set already at the end of core-He burning (when we terminate our models). This allows us to make a comparison between the single and binary evolution in Fig. 16. Binary models, mapped onto the parameter space of different primary masses and orbital periods, are colored according to the ratio M CO /M CO;single at central He depletion. At Solar metallicity (the left panel), mass transfer interaction and the envelope loss leads to systematically lower CO core masses in primaries compared to single stars of the same initial mass. Meanwhile, at the subsolar metallicities of LMC and SMC, we find a large parameter space in which binary models produce CO cores with masses very close to those formed in single stars. Those are the models in which the primaries experience only partial rather than full envelope stripping. Similarly, in Fig. A.6 we compare the central C mass fractions (X C ) between single and binary models, finding similarities between partiallystripped primaries and single stars.
Based on Fig. 16 and Fig. A.6 and the results of the abovementioned studies, it is well-founded to expect that partiallystripped stars will produce pre-SN core structures that are similar to those of single stars: more compact and prone to BH formation compared to pre-SN cores of fully stripped stars. Partial envelope stripping is common among our models: the vast majority of low-Z donors with masses between ∼20 and 50 M are never fully stripped in post-MS mass transfer (see Fig. 8) and we esti-mate that fully stripped stars are the minority in this mass range (originating mostly from case A mass transfer, see Fig. 15). Taken together with the results of Schneider et al. (2021), this could mean that most of the BHs in close binaries in the LMC and SMC-like environments are formed from partially-stripped progenitors. This may also 'save' the binary BH and BH-NS formation scenario from low-Z massive binaries.
In the context of SN light curves and stripped SN in general, we point out that our binary models are terminated at the end of core-He burning and at that point, in most of the massive low-Z systems the mass transfer is still ongoing. This means that even though many of the low-Z primaries have not been fully stripped in our simulations (see Fig. A.3 for the final envelope masses), they could still lose the remaining hydrogen (or possibly also helium) in the short remaining evolution until the core collapse (several thousand yr, depending on mass). Such H-rich layers, if not-accreted by the companion, could still reside in the proximity of the star at the core collapse and the possible SN, leading to a transient with a circumstellar medium interaction.
Implications for ultra-luminous X-ray sources
Here, we discuss the implications of our binary models assuming that the accretor is a compact object: a stellar-mass BH. For the default mass ratio q = 0.6, the BH mass would range from 6 to ∼ 32 M with the corresponding Eddington accretion rates ranging from ∼ 1.3×10 −7 M yr −1 to ∼ 7×10 −7 M yr −1 . This is much lower than the mass transfer rates found in models when they are at the RLOF stage (even during the phases of slow nucleartimescale mass transfer, whenṀ ∼ 10 −5 M yr −1 ). Such BH binaries would thus be HMXBs with highly super-Eddington mass transfer rates. It was shown that in this supercritical regime the thin disk model is no longer valid (Shakura & Sunyaev 1973). Instead, the accretion is thought to proceed through a thicker (slim) disk, possibly with a super-Eddington accretion rate and most of the energy being advected into the BH or used to power strong disk winds and jets (Lipunova 1999;King et al. 2001;Poutanen et al. 2007;Lasota et al. 2016). The disk together with optically-thick outflows make the X-ray emission geometrically beamed. Depending on the viewing angle, such systems may appear as ULXs (defined as point X-ray sources with sphericallyequivalent X-ray luminosity L X > 10 39 erg/s, see the review by Kaaret et al. 2017) or X-ray bright microquasars such as the Galactic SS433 system (Fabrika 2004).
In this work, we found that a significant fraction of massive low-Z binaries evolve through nuclear-timescale post-MS mass transfer. In such models, the super-Eddington mass exchange phase lasts a few times 10 5 yr, which is more than an order of magnitude longer than the thermal-timescale mass transfer phase found in the high-metallicity models ( 10 4 yr). This leads to a prediction that HMXBs and ULXs with BH accretors, post-MS donors (blue and yellow supergiants), and periods of at least a few tens of days should be much more common in metal-poor galaxies.
It is interesting to discuss whether that is indeed observed. At first glace, our results might seem in tension with the fact that no such BH-HMXB system is known in the LMC (the LMC X-1 BH binary contains a MS donor in a short-period orbit Orosz et al. 2009). However, the lack of such systems in the LMC may in fact be statistically consistent with our results. Based on detailed binary models, estimate that about 100 BH-OB star binaries should be present in the LMC. Taking that the MS lasts for about 90% the stellar lifetime, this corresponds to ∼ 10 BH binaries with post-MS companions. Roughly 1/3 of Fig. 16: Comparison of the CO core masses at the end of core He burning between primaries in binary models (M CO ) and single stellar models (M CO;single ). Each binary evolution model, defined by the initial primary mass (M 1 ) and the initial orbital period (P ini ), is represented by a rectangle colored according to the ratio M CO /M CO;single . The initial mass ratio is q = 0.6. Single stellar models correspond to P ini > 10 4 days. Similarly to Fig. 6, areas in the background indicate the evolutionary state of the donor stare at the onset of RLOF. We find that the CO core masses in low-Z binary models can be very close to those formed in single stars, despite the mass-transfer interaction. This is because only partial envelope stripping occurs in those models.
BH binaries predicted by have companion masses of 20 M and above and roughly 1/2 have orbital periods above ∼ 50 days (the minimum donor mass and orbital period for nuclear-timescale mass transfer in our LMC grid). This estimation leads to an approximate prediction of 10/6 ≈ 1.7 BH binaries with post-MS donors that are currently in a masstransferring state: a number that is not inconsistent with no such systems being observed.
Instead, the effect of nuclear-timescale mass transfer on the number of ULX sources may be more evident when looking at larger scales, across more distant and diverse galaxies. The recent census of ULX sources in the Local Universe amounts to 629 ULX candidates in 309 galaxies with distance smaller than 40 Mpc (Kovlakas et al. 2020). Interestingly, the number of ULXs per unit of star formation rate is found to increase with decreasing metallicity of the host galaxy (as pointed out by Zampieri & Roberts 2009;Mapelli et al. 2009, see also observational studies of the ULX host galaxies by Walton et al. 2011;Swartz et al. 2011). Qualitatively, this agrees with the results from out models. Unfortunately, the potential to use the observed number of ULXs to constrain our findings is at the moment hindered by the fact that in the vast majority of ULXs the nature of neither the accretor (whether a BH or a NS) nor the donor (whether or not a high-mass star) is known.
In the context of ULX systems, it is interesting to highlight that in Sec. 4 and Fig. 13 we found that low-Z donors can be characterized by very high ζ th values in the mid-envelope region (with ζ th > ζ RL for even extreme mass ratios of q < 0.1). This means that mass transfer from a partially-stripped star could be thermally stable even in binaries with a NS accretor (M NS ≈ 1 − 2.5 M ), provided that the mass transfer is also dynamically stable, i.e. ζ ad > ζ RL (which may well be the case for radiativeenvelope supergiants Ge et al. 2015). Quast et al. (2019) suggested this as a possible explanation for ultra-luminous X-ray sources with blue-supergiant donors and NS accretors such as the NGC 7793 P13 system (Israel et al. 2017). In their models, the high stability region could be reached after stripping most but not quite all of the donor envelope (up to the point with high ζ th values found near the helium core also in our models). Interestingly, in our stellar models we find another region of high ζ th located higher up inside the envelope, meaning that a smaller fraction of the outer envelope would have to be lost in previous evolution before reaching the high-stability regime. A possible agent for such prior stripping in systems with extreme mass radios could be common-envelope evolution with a partialenvelope ejection (see discussion in Sec. 4.6 in Klencki et al. 2021).
Summary
In this paper, we studied the mass transfer evolution in massive binaries and the effect played by the metallicity of the donor star, as motivated by Klencki et al. (2020). To this end, using the MESA stellar evolution code, we computed grids of detailed binary models at three different metallicities (Solar, LMC, and SMC compositions) spanning a wide range of orbital periods (from ∼ 3 to 5000 days) and initial primary masses (from 10 M to 36-53 M , depending on metallicity). Our main focus was on the mass transfer initiated by a post-MS donor star. Due to the challenging numerical nature of such models, we treated the secondary as a point mass. Most of the models were computed with an initial mass ratio of q = 0.6 although we also explored the effect of varying q for a few cases. Our conclusions can be summarized as follows.
− We reveal that metallicity has a substantial effect on the course and outcome of mass transfer evolution of massive binaries. While at high (Solar) metallicity a post-MS mass transfer is always a short-lived phase (∆T MT 10 4 yr) of thermal-timescale mass transfer (Ṁ ∼ 10 −3 M yr −1 ) with the mass loser becoming a stripped helium star (in agreement with the long-standing Helium abundance profiles near the core-envelope boundary of stripped primaries from a few selected binary models at Solar metallicity. The selected binary models are the same as the ones shown in the left panel of Fig. 9. Diamonds mark the point to which each donor was stripped in mass transfer. Solid lines are taken from donor structures just after the end of mass transfer. Dashed lines show the extension of He profiles taken from the structure from the onset of mass transfer. The figure illustrates that the more massive the primary (M 1 ), the more massive the envelope that remains after mass transfer. As a result, stripped stars from models with M 1 17 M primaries may retain some envelope layers from the plateau of He abundance, leading to cooler effective temperatures (see Fig. 9 and the associated text). Fig. 6 but the rectangles representing the binary models are colored according to the mass of the H-rich envelope (H content X H > 10 −3 ) that remains at the end of core-He burning. The colorscale focuses on the range between 0 and 1 M but in some of the low-Z binary models the remaining envelope mass is even greater than 1 M . Fig. 6 but the rectangles representing the binary models are colored according to the integrated duration of a post-interaction detachment phase during which the core-He burning mass loser remains on the cooler side of the ZAMS line in the HR diagram. Rectangles corresponding to models with post-MS donors (either at the Hertzprung gap or the core-He burning stage, see the background colors) that are colored in various shades of green and yellow are models that produce partially-stripped long-lived stars (see Sec. 3.4 and Sec. 5.4 for a discussion of this population). Fig. 16 but comparing the central carbon mass fraction X C between the single and binary models, rather than the CO core mass.
Fig. A.4: Similar to
Fig. 2 :
2Mass transfer evolution in binaries with a 25 M primary (initial mass), compared between the three different metallicities.
Fig. A.3 for the amount of envelope left at the end of core-He burning across all our models.
Fig. 5) before central He exhaustion ('E' in Fig. 5). For most of its lifetime, the stripped donor has an effective tem-Fig. 5: Evolution of 30 M primaries (donors) from models shown in
Fig. A.2 for a few examples of He abundance profiles in stripped stars).
Fig. 6 :
6Integrated duration of mass transfer through RLOF (∆T MT )
Fig. 8 :
8Major types of mass transfer evolution found across the grid of binary models. Similarly to
Fig. 10 :
10Luminosity-mass relation of post-interaction donor stars in detached binaries with R/R RL < 0.6 at the LMC metallicity.
Fig. A. 1 :
1Same asFig. 3but for the LMC metallicity. Mass transfer evolution in binaries with a 25 M primary (initial mass), LMC composition, and various initial orbital periods P ini .Appendix A: Additional figures
Fig. A.2: Helium abundance profiles near the core-envelope boundary of stripped primaries from a few selected binary models at Solar metallicity. The selected binary models are the same as the ones shown in the left panel of Fig. 9. Diamonds mark the point to which each donor was stripped in mass transfer. Solid lines are taken from donor structures just after the end of mass transfer. Dashed lines show the extension of He profiles taken from the structure from the onset of mass transfer. The figure illustrates that the more massive the primary (M 1 ), the more massive the envelope that remains after mass transfer. As a result, stripped stars from models with M 1 17 M primaries may retain some envelope layers from the plateau of He abundance, leading to cooler effective temperatures (see Fig. 9 and the associated text).
Fig
Fig. A.3: Similar to Fig. 6 but the rectangles representing the binary models are colored according to the mass of the H-rich envelope (H content X H > 10 −3 ) that remains at the end of core-He burning. The colorscale focuses on the range between 0 and 1 M but in some of the low-Z binary models the remaining envelope mass is even greater than 1 M .
Fig. A. 5 :
5Same as Fig. 14 but for the SMC composition.
Fig
Fig. A.6: Same as Fig. 16 but comparing the central carbon mass fraction X C between the single and binary models, rather than the CO core mass.
There is some ambiguity in the literature as to whether the term case B mass transfer refers to only the cases initiated before the onset of core-He burning or also during that phase of evolution. Throughout this paper, we will adapt the latter naming convention.
MESA version r11554, http://mesa.sourceforge.net/ Article number, page 2 of 24 Klencki et al.: Nuclear-timescale mass transfer from evolved massive stars
The detachment phases in nuclear-timescale mass transfer models are associated with changes in the H abundance in the moving location of the H-burning shell. Given the uncertainties in the detailed features of the H abundance profile of the bottom envelope layers, these detachments are not a robust prediction of evolutionary models.
Note that these different types of post-interaction systems are probing different mass transfer regimes, so the large spread in β values is perhaps not unexpected. 10 A model of efficient accretion at breakup rotation velocity has been proposed by(Popham & Narayan 1991). Article number, page 14 of 24 Klencki et al.: Nuclear-timescale mass transfer from evolved massive stars
Additionally, their orbits would typically be synchronized at ZAMS, which could lead to chemically-homogeneous evolution of the most massive LMC or SMC cases in that period range(de Mink et al. 2009a).12 The normalization assumes that all massive stars are formed in binaries and that the initial mass ratio is 0.6. Article number, page 15 of 24 A&A proofs: manuscript no. main
The grey region just below Box IV are stripped donors contracting on the thermal-timescale to become hot stripped stars.
In particular whether or not carbon burning will trigger convection, which is especially consequential for the final compactness of the pre-SN core (e.g. Chieffi & Limongi 2020).
Acknowledgements. We thank the referee for taking the time and effort to carefully review our work. It is a pleasure to acknowledge valuable discussions and suggestions from Stephen Justham, Selma de Mink, Manos Zapartas, Lida Oskinova, Tomer Shenar, Julia Bodensteiner, Pablo Marchant, Ylva Götberg, Thomas Tauris, and David Aguilera-Dena. The authors acknowledge support from the Netherlands Organisation for Scientific Research (NWO). JK acknowledges support from an ESO Fellowship.A&A proofs: manuscript no. main paradigm, e.g.Paczyński 1971;van den Heuvel 1975;Podsiadlowski et al. 1992;Vanbeveren et al. 1998), this turns out to not be the case in our LMC and SMC models with donor masses 17 M . For such massive low-Z donors, the post-MS mass transfer is much less violent: leading either to evolution through long nuclear-timescale mass exchange, that continues until the end of core-He burning (∆T MT 10 5 yr,Ṁ ∼ 10 −5 M yr −1 ), or to detached binaries with mass-losers that are only partially stripped of their envelopes.− The origin of the metallicity effect found in the mass transfer models lies in the different response of low-Z donors to mass loss and their small equilibrium radii as partially-stripped core-He burning stars. This in turn is related to the post-MS expansion of massive stars when they transition to the core-He burning stage. Stars in which the rapid HG expansion continues until the red (super)giant branch (a common feature of high-Z models), when donors in binaries, become fully stripped in thermaltimescale mass transfer. Partial envelope stripping and slow mass transfer, on the other hand, occur in the mass range in which massive stars begin to burn He already as blue or yellow supergiants. Such a halted HG expansion is often found in low-Z models of massive stars. Although uncertain due to its sensitivity to mixing (in particular semiconvection), it is supported observationally by the large populations of yellow supergiants in the LMC and the SMC.− Based on a simple population model, we predict that at SMC and LMC metallicities, fewer (by a factor of ∼2−2.5) stripped (WR) stars with 4.8 < log(L/L ) < 6.0 are produced by binary interactions compared to a Solar-metallicity environments. This is because among our LMC and SMC models, only ∼0−20% of post-MS donors with M 1 20 M become hot stripped stars by the end of core-He burning (∼100% at Solar metallicity). Case A mass transfer evolution leads to full envelope stripping irrespective of metallicity.− We find a significantly longer average duration of post-MS mass transfer in low-Z binary systems (by more than an order of magnitude). In the case of BH accretors, this implies longer lifetimes of high-mass X-ray binaries, which at face value agrees with the large numbers of ULXs found in metal-poor galaxies (although the nature of ULX accretors is usually unknown). This also suggests that the immediate progenitors of binary BH systems could be in the mass-transferring state rather than being detached BH-WRs. In the case of stellar accretors, our models provide a testable prediction that many of the blue and yellow supergiants with log(L/L ) 5 in the LMC and SMC should be in semi-detached binaries (∼30−40% from our q = 0.6 grid). We also speculate that lower mass transfer rates of low-Z models could lead to higher accretion efficiencies.− We predict a population of partially-stripped stars in detached binaries in the LMC and SMC. Unlike stripped stars, such mass losers are relatively cool (typically 4.1 < log(T eff /K) < 4.5 and 2.0 < log(g) < 4.3) and thus overlap with the MS and blue supergiants. They are expected to be undermassive for their luminosity (by a factor of ∼1.5−2), He and N rich, slowly rotating, and reside in binaries with a wide range of periods (from tens to thousands of days). Potentially, they could explain some of the puzzling N-rich slow rotators observed in the LMC(Hunter et al. 2008;McEvoy et al. 2015;Grin et al. 2017).− Guided by the carbon-oxygen core properties, we expect the pre-SN core structure of most of the low-Z 20 M primaries to be similar to that of single stars in terms of their higher compactness (and lower explodability) compared to fully stripped binary interaction products. Our results therefore suggest that the recently obtained high initial masses needed for the BH formation in case A and case B binaries( 70 M Schneider et al. 2021), at low metallicity, could potentially be reduced to the often quoted ∼ 25 M BH-formation threshold.
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| [] |
[
"Coherent population pumping in a bright state",
"Coherent population pumping in a bright state"
] | [
"Sumanta Khan \nDepartment of Physics\nIndian Institute of Science\nBangalore-560012India\n",
"Vineet Bharti \nDepartment of Physics\nIndian Institute of Science\nBangalore-560012India\n",
"Vasant Natarajan \nDepartment of Physics\nIndian Institute of Science\nBangalore-560012India\n"
] | [
"Department of Physics\nIndian Institute of Science\nBangalore-560012India",
"Department of Physics\nIndian Institute of Science\nBangalore-560012India",
"Department of Physics\nIndian Institute of Science\nBangalore-560012India"
] | [] | We demonstrate resonances due to coherent population pumping in a bright state (CBS), using magnetic sublevels of the closed F g = 2 → F e = 3 transition in 87 Rb. The experiments are performed at room temperature vapor in two kinds of cells-one that is pure and the second that contains a buffer gas of Ne at 20 torr. We also present the effect of pump power variation on the CBS linewidth, and explain the behavior by using a power-dependent scattering rate. The experimentally observed CBS resonances are supported by a density-matrix analysis of the system. | null | [
"https://arxiv.org/pdf/1706.04384v1.pdf"
] | 118,855,250 | 1706.04384 | 7830734bce9a72a255a36f583f5751523fe16d1c |
Coherent population pumping in a bright state
14 Jun 2017
Sumanta Khan
Department of Physics
Indian Institute of Science
Bangalore-560012India
Vineet Bharti
Department of Physics
Indian Institute of Science
Bangalore-560012India
Vasant Natarajan
Department of Physics
Indian Institute of Science
Bangalore-560012India
Coherent population pumping in a bright state
14 Jun 2017arXiv:1706.04384v1 [physics.atom-ph]numbers: 4250Gy4250Md3270Jz3280Qk * vasant@physicsiiscernetin
We demonstrate resonances due to coherent population pumping in a bright state (CBS), using magnetic sublevels of the closed F g = 2 → F e = 3 transition in 87 Rb. The experiments are performed at room temperature vapor in two kinds of cells-one that is pure and the second that contains a buffer gas of Ne at 20 torr. We also present the effect of pump power variation on the CBS linewidth, and explain the behavior by using a power-dependent scattering rate. The experimentally observed CBS resonances are supported by a density-matrix analysis of the system.
INTRODUCTION
Coherent population trapping (CPT) is a well studied phenomenon in many atoms. It is a phenomenon in which atoms get optically pumped into a dark non-absorbing state by two phase-coherent beams. Once they are pumped, the atoms get trapped in the dark state and cannot fluoresce because the coupling of this superposition state to the excited state cancels [1]. The easiest way to observe this experimentally is to use magnetic sublevels of a degenerate transition. The required phase coherence is then achieved by deriving both beams from the same laser. A narrow absorption dip then appears at line center when one of the two beams is scanned; the line center being the point at which the two-photon Raman resonance condition is satisfied. The linewidth of the dip is much smaller than the natural linewidth of the excited state, and is limited by decoherence among the ground sublevels.
A similar arrangement with two phase-coherent beams can be used to create a bright superposition state. The result is enhanced absorption at line center, exactly opposite to the dip seen in CPT. The linewidth is similar to that obtained in CPT, and is again limited by decoherence among the magnetic sublevels of the ground state. However, unlike in CPT, the population does not get trapped in this state because it can decay by coupling to the excited state. The conditions for observing this in a F g → F e transition are:
(i) It is a closed transition, so that there is no decay out of the system.
(ii) F e = F g + 1, so that the correct superposition state can be formed.
(iii) F g = 0, so that there are multiple magnetic sublevels in the ground state.
All these conditions are met for the F g = 3 → F e = 4 transition in 85 Rb, which was therefore used for the first observation of such increased absorption in Ref. [2,3]. The authors called the phenomenon electromagnetically induced absorption (EIA), in order to highlight the fact that there was increased absorption at line center. However, we feel that a more appropriate term would be CBS standing for coherent population pumping in a bright state, while the term EIA is better used for enhanced absorption of a weak probe beam in the presence of two or more strong pump beams in a multilevel system [4][5][6][7][8].
In this work, we study CBS resonance satisfying the above conditions but in the other isotope of Rb, namely the F g = 2 → F e = 3 transition in 87 Rb. We experimentally study these resonances in two kinds of vapor cells-one that is pure and contains both isotopes in their natural abundances, and the second that contains only 87 Rb and has a buffer gas of Ne at 20 torr. The presence of the buffer gas is advantageous because it increases the coherence time among the magnetic sublevels, and hence results in a smaller linewidth for the resonance. The explanation of enhanced absorption at line center for this transition is borne out by a numerical density-matrix analysis, which takes into account Doppler averaging in room temperature vapor. We also study the effect of power variation on the linewidth of the CBS resonances, and find that it follows the power-dependent scattering rate from the excited state.
I. EXPERIMENTAL DETAILS
The experimental setup is shown schematically in Fig. 1. The probe and pump beams are derived from the same laser to achieve the required phase coherence. The laser consists of a grating stabilized diode laser system, as described in reference [9]. The linewidth of the laser after stabilization is 1 MHz. The size of the output beam is 3 mm × 4 mm. The power in the beams is controlled using λ/2 waveplates in front of the respective PBSs.
The two beams are made to have orthogonal linear polarizations so that they can be mixed on a PBS. The experiment requires them to have circular polarizations, which is achieved by using a λ/4 waveplate before entering the cell. The laser is locked to the F g = 2 → F e = 3 transition using a saturated absorption (SAS) signal from another vapor cell. The orthogonal circular polarizations for the two beams means that the probe beam couples m Fg → m Fg + 1 transitions, while the pump beams couples m Fg → m Fg −1 transitions. As mentioned before, the probe beam frequency is fixed while that of the pump beam is scanned. This scanning is achieved by using two AOMs in its path-one with a downshift of 180 MHz, and the other compensating for this shift by a double-passed AOM with an upshift of 90 MHz. The double passing ensures that the direction of the beam does not change when the frequency is scanned. The frequency of the AOM driver is set using a commercial function generator.
Two kinds of vapor cells were used for the experiment-one pure and the second with a buffer gas of Ne (at a pressure of 20 torr). Both cells are cylindrical with dimensions of 25 mm diameter × 50 mm length. The cell is inside a 3-layer µ-metal magnetic shield. The shield reduces stray external fields to less than 1 mG.
The polarizations after the cell are made linear using a second λ/4 waveplate, and the beams are separated using another PBS. The probe beam alone is detected using a photodiode; therefore, the photodiode signal is proportional to probe transmission. Since the SAS signal used for locking corresponds to absorption by zero-velocity atoms, detecting the non-scanning probe beam allows us to have a flat Doppler-free background for the CBS signal.
II. CBS IN A PURE CELL
A. Experimental results
An experimental spectrum for CBS in the F g = 2 → F e = 3 transition obtained in a pure cell is shown in Fig. 2. Probe transmission as a function of detuning of the pump beam shows a dip-the CBS resonance-at line center; the photodiode signal is scaled so that the percentage absorption is about 8%. This behavior is opposite to the CPT resonance seen in the F g = 1 → F e = 1 transition in the same isotope [10]. The difference is because the 1 → 1 transition does not satisfy the requirements for a CBS resonance (mentioned earlier).
B. Theoretical analysis
The experimental spectrum can be explained from a detailed density-matrix analysis of the sublevel structure for this transition. The calculations were carried out using the atomic density matrix (ADM) package written by Simon Rochester [11]. It solves numerically the following time evolution equation for the density-matrix elements involved:
ρ = − ī h [H, ρ] − 1 2 {Γ, ρ} + repopulation terms(1)
where Γ is the relaxation matrix-its diagonal terms gives the total decay rate (radiative and non-radiative) of the respective populations, and its off-diagonal terms represent the decoherence between states |i and |j , such that
Γ ij = Γ ii + Γ jj 2 (2)
The repopulation terms take care of decay of atoms from the excited state to the ground state.
The magnetic sublevel structure for the transition is shown in Fig. 3. The pump beam is σ − polarized-hence it couples sublevels with the selection rule ∆m = −1. The probe beam is σ + polarized and couples sublevels with the selection rule ∆m = +1. The probe beam has no detuning for zero-velocity atoms while the pump beam has a detuning for the same atoms, but the actual detuning seen in the atom's frame depends on its velocity.
The following parameters are input to the calculation:
(i) The F values for the ground and excited state of the transition.
(ii) The proper polarizations for the probe and pump beams.
(iii) A uniform intensity, equal for both beams.
(iv) A decay rate among ground sublevels of 10 kHz.
(v) A decay rate from an excited sublevel to a ground sublevel of 6 MHz.
(vi) A repopulation term for a particular ground sublevel equal to the 6 MHz decay rate multiplied by the appropriate branching ratio.
The probe transition spectrum is Doppler averaged over atomic velocities corresponding to the Maxwell-Boltzmann distribution for Rb atom at room temperature.
The results of the simulation are shown in Fig. 4. The calculated spectrum reproduces the experimental one quite well, in terms of linewidth. The only difference is that the calculation assumes a constant intensity of 21 µW/cm 2 , which only appears in the wings of the Gaussian distribution for the 30 µW power used in the experiment.
C. Effect of pump power
In a CBS experiment-like in CPT-the pump beam causes decoherence through the excited state. The scattering rate is intensity dependent, and is given by
R = Γ 2 I/I s 1 + I/I s(3)
where Γ is the natural linewidth of the state, I is the intensity, and I s is the saturation intensity (the intensity at which the transition gets power broadened by a factor of √ 2).
Since the intensity is directly proportional to the power through a geometric factor, I = gP , the scattering rate can be rewritten as
R = Γ 2 gP/I s 1 + gP/I s(4)
This equation shows that the scattering rate will increase initially but asymptote to a saturation value at high powers. Thus the linewidth of the CBS resonance will also show the same behavior.
The results are shown in Fig. 5. The solid line is a fit to Eq. (4), with an offset to account for linewidth from experimental noise. The fit describes the experimental results quite well.
III. CBS IN A BUFFER CELL
Before concluding, we turn to experimental results in a buffer cell-one filled with 20 torr of Ne as buffer gas. The role of the buffer gas is to increase the coherence time among magnetic sublevels of the ground state. This will result in a smaller linewidth for the CBS resonance. The results shown in Fig. 6 bear out this expectation-the linewidth reduces to 9 kHz in such a cell. In this case, the absorption is a factor of 2 lower than that in a pure cell, and the photodiode signal is scaled to reflect this. In summary, we have studied an enhanced absorption or CBS resonance in a closed transition in room-temperature vapor of 87 Rb atoms. The observation requires the proper superposition state to be formed, which is achieved by using magnetic sublevels of the ground state and a phase coherence between the probe and pump beams by deriving them from the same laser. The observed linewidth is limited by decoherence among the magnetic sublevels.
This explanation is borne out by a density-matrix analysis of the sublevels involved in the transition. The calculation takes into account Doppler averaging in room temperature 87 Rb vapor. We study the effect of pump power on the CBS linewidth, and find that the behavior follows a power-dependent scattering rate from the excited state. We also study the same CBS resonance in a buffer-gas filled cell, and find that it reduces the linewidth because it increases the coherence time among the magnetic sublevels. and Harish Ravi and Mangesh Bhattarai for helpful discussions.
FIG. 1 .
1(Color online) Experimental setup for CBS experiment. The required phase coherence is achieved by deriving both beams from a single laser. The probe beam is locked while the pump beam is scanned by scanning the frequency of the double-passed AOM. Figure key: λ/2 -half wave retardation plate; λ/4 -quarter wave retardation plate; PBS -polarizing beam splitter cube; AOM -acousto-optic modulator; PD -photodiode.
FIG. 2 .
2(Color online) CBS resonance obtained in a pure cell.
FIG. 3 .FIG. 4 .
34(Color online) Magnetic sublevels of F g = 2 → F e = 3 transition in the D 2 line of 87 Rb. (Color online) Simulated probe transmission spectrum versus Raman detuning for F g = 2 → F e = 3 transition with probe and pump beam intensity 21 µW/cm 2 .
FIG
. 5. (Color online) Effect of pump power on the linewidth of the CBS resonance showing increase in linewidth due to increased decoherence through the upper level. The solid line is a fit to the scattering-rate expression in Eq. (4)
FIG
. 6. (Color online) CBS resonance obtained in a buffer gas filled cell.
ACKNOWLEDGMENTS
This work was supported by the Department of Science and Technology, India. S K acknowledges financial support from INSPIRE Fellowship, Department of Science and Technology, India. The authors thank S Raghuveer for help with the manuscript preparation;
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| [] |
[
"Multiphonon Raman Scattering in Graphene",
"Multiphonon Raman Scattering in Graphene"
] | [
"Rahul Rao [email protected] \nMaterials and Manufacturing Directorate\nAir Force Research Laboratory\nWPAFB\n43433OHUSA\n",
"Derek Tishler \nDepartment of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA\n\nHonda Research Institute\n1381 Kinnear RdColumbusOH\n",
"Jyoti Katoch \nDepartment of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA\n\nHonda Research Institute\n1381 Kinnear RdColumbusOH\n",
"Masa Ishigami \nDepartment of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA\n\nHonda Research Institute\n1381 Kinnear RdColumbusOH\n"
] | [
"Materials and Manufacturing Directorate\nAir Force Research Laboratory\nWPAFB\n43433OHUSA",
"Department of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA",
"Honda Research Institute\n1381 Kinnear RdColumbusOH",
"Department of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA",
"Honda Research Institute\n1381 Kinnear RdColumbusOH",
"Department of Physics and Nanoscience Technology Center\nUniversity of Central Florida\n32816OrlandoFLUSA",
"Honda Research Institute\n1381 Kinnear RdColumbusOH"
] | [] | We report multiphonon Raman scattering in graphene samples. Higher order combination modes involving 3 phonons and 4 phonons are observed in single-layer (SLG), bi-layer (BLG), and few layer (FLG) graphene samples prepared by mechanical exfoliation. The intensity of the higher order phonon modes (relative to the G peak) is highest in SLG and decreases with increasing layers. In addition, all higher order modes are observed to upshift in frequency almost linearly with increasing graphene layers, betraying the underlying interlayer van der Waals interactions. | 10.1103/physrevb.84.113406 | [
"https://arxiv.org/pdf/1108.3502v1.pdf"
] | 119,260,781 | 1108.3502 | 83cddbd7ce6931c6c469a5d45a40bae88d97b982 |
Multiphonon Raman Scattering in Graphene
Rahul Rao [email protected]
Materials and Manufacturing Directorate
Air Force Research Laboratory
WPAFB
43433OHUSA
Derek Tishler
Department of Physics and Nanoscience Technology Center
University of Central Florida
32816OrlandoFLUSA
Honda Research Institute
1381 Kinnear RdColumbusOH
Jyoti Katoch
Department of Physics and Nanoscience Technology Center
University of Central Florida
32816OrlandoFLUSA
Honda Research Institute
1381 Kinnear RdColumbusOH
Masa Ishigami
Department of Physics and Nanoscience Technology Center
University of Central Florida
32816OrlandoFLUSA
Honda Research Institute
1381 Kinnear RdColumbusOH
Multiphonon Raman Scattering in Graphene
1 * Coresponding Author,
We report multiphonon Raman scattering in graphene samples. Higher order combination modes involving 3 phonons and 4 phonons are observed in single-layer (SLG), bi-layer (BLG), and few layer (FLG) graphene samples prepared by mechanical exfoliation. The intensity of the higher order phonon modes (relative to the G peak) is highest in SLG and decreases with increasing layers. In addition, all higher order modes are observed to upshift in frequency almost linearly with increasing graphene layers, betraying the underlying interlayer van der Waals interactions.
Single to multilayer graphene display a variety of unusual electronic and transport properties such as the Dirac physics, tunable band gap due to the broken symmetry by additional layers, and electron interaction effects. 1 Light-interaction with graphene is highly important for both graphene science and technology as optical techniques like Raman spectroscopy reveal fundamental properties of graphene such as doping levels, defect concentrations, 2 and the utility of graphene in optoelectronics seems promising.
Raman spectroscopy is also the standard technique for characterizing graphene samples due to distinct features that depend strongly on the number of layers, 3,4 as well as the stacking order in few-layer graphene. [5][6][7][8] In particular, the second order 2D (also called the G') peak, which occurs due to a double resonance Raman process involving inter-valley scattering of an electron by two transverse optical (iTO) phonons, is the most intense feature in the Raman spectrum of single layer graphene (SLG) on SiO 2 and can be fitted with a single Lorentzian peak. 2,3,9 On the other hand, the 2D peak in bilayer graphene (BLG) is composed of four Lorentzian peaks, and reflects the hyperbolic electronic band structure due to the stacking between two graphene layers. As the number of layers increase, the 2D peak evolves into a two-peak structure due to coupling between graphene layers in a three dimensional crystal. 2, 10 Furthermore, the intensity of the 2D peak diminishes with respect to the G peak with increasing graphene layers, and has been used to identify the number of layers in graphene samples. 11 While the two-phonon 2D peak has received much attention in graphene samples, higher order modes involving multiple phonons remain unexplored. Multi-phonon Raman scattering is generally weaker in bulk materials due to a vanishing DOS for higher order phonons. 12 Yet, multiphonon Raman scattering can be observed in single-(SWNTs), 13,14 multi-walled carbon nanotubes (MWNTs), 15 and highly oriented pyrolytic graphite (HOPG). 14, 16 Wang et al. 13 reported intense higher order combination modes involving up to 6 phonons (between 2500 -8500 cm -1 ) in individual SWNTs, which was made possible by large resonance enhancements due to coupling with the excitation laser. On the other hand, the multiphonon modes in MWNTs 15 and HOPG 14,16 are much weaker in intensity compared to their one-phonon or two-phonon modes, and are difficult to observe. In the same vein, it is of considerable interest to determine whether such higher order modes can also be observed in graphene, and how they evolve with increased layer stacking. Here we show higher order combination modes (up to 4 phonons) from single and multiple graphene layers prepared by exfoliation from HOPG on SiO 2 substrates. 17 We find that the multiphonon modes are most intense in single layer graphene (SLG) and decrease in intensity with increasing layers. In addition, the three-phonon modes are observed to upshift in frequency with increasing number of layers, presumably due to van der Waals interactions caused by layer stacking.
Raman spectra (E laser = 2.33 eV) from single-layer (SLG), bi-layer (BLG), and few-layer (FLG) graphene samples are shown along with a spectrum from HOPG in Fig. 1. The first order E 2g mode (G peak) at ~1585 cm -1 and the overtone of the iTO phonon mode (2D peak) at ~2700 cm -1 exhibit frequencies and lineshapes similar to what have been described previously; 3, 5 the 2D peaks in SLG, BLG and FLG can be fitted with 1, 4, and 2 Lorentzian peaks, respectively. Beyond the 2D peak frequency, several weak intensity modes can be observed between 3000 -6000 cm -1 . The sharp peak (single Lorentzian) at ~3230 cm -1 is an overtone of the D' peak and is called the 2D' peak. The D' peak is a disorder-induced peak occurring at ~1620 cm -1 in sp 2 carbon samples, and is caused by double resonance intra-valley scattering of a photo-excited electron by a phonon along with elastic scattering by a defect. The 2D' peak at ~3230 cm -1 is its overtone and like the 2D peak, does not need defects for activation. The overtone of the G peak (2G) at ~3160 cm -1 is generally not observed in the graphite Raman spectrum, 18 although it has been observed previously in the resonance Raman spectra from SWNTs. 13 Hence as described below, we assign some of the higher order modes above 4500 cm -1 to combinations involving the 2D' peak, rather than the 2G peak as assigned by earlier reports. 15,16 Beyond 4000 cm -1 , a peak at ~4250 cm -1 is the most intense feature in the Raman spectra shown in Fig. 1. This feature is assigned to a combination of the G and 2D modes (G+2D) as explained further below. 19 The intensity of the G+2D peak is highest for SLG and decreases in intensity with increasing layers, tracking the intensity of the 2D peak, which is also the most intense for SLG compared to multi-layered graphene. A weak intensity peak at ~4030 cm -1 is also observed in all samples and is assigned to a combination of the D and 2D modes (D+2D). Both the G+2D and D+2D peaks are shown more clearly in Fig. 2 where the spectra have been normalized by the intensity of the G+2D peak and fitted with Lorentzian peaks. The G+2D peak in SLG can be fit with a single Lorentzian peak; hence its lineshape is similar to that of the 2D peak in SLG.
However, the linewidth of the G+2D peak (FWHM ~ 85 cm -1 ) is greater than the 2D peak (FWHM ~30 cm -1 ). On the other hand, the G+2D peaks in BLG, FLG, and HOPG can be deconvoluted into multiple peaks, reflecting the changes in the electronic band structure brought about by coupling between graphene layers; this is also observed in the 2D peak from BLG and FLG samples. 2,3,9 The assignments of the G+2D and D+2D peaks can be confirmed by the dispersion of the peaks with increasing laser energy (E Laser = 2.33, 2.54, and 3.81 eV), as shown in Fig. 3a. The dispersion of the G+2D peak is ~100 cm -1 /eV, which should be similar to the dispersion of the 2D peak (~95 cm -1 /eV) since the G peak is dispersionless. On the other hand, the dispersion of the D+2D peak should approximately equal the sum of the dispersions of the D (typically ~50 cm -1 /eV) 20 and 2D peaks (~95 cm -1 /eV), and is found to be ~130 cm -1 /eV. Similar dispersions can also be observed for the G+2D peak in BLG, as shown in Fig. 4. The G+2D peak in BLG can be deconvoluted into 4 peaks, similar to the 2D peak in BLG 3,9 . The dispersions of the four components within the 2D peak have been reported to vary between 80 -100 cm -1 /eV. [21][22][23] As shown in Fig. 4, the dispersion of the highest frequency component within the G+2D peak in BLG is ~98 cm -1 /eV, and the dispersions of the other peaks within the G+2D as well as 2D peaks are similar. Moreover, the differences in frequencies between the four components of the G+2D peak (~20-40 cm -1 ) are similar to those in between the four components in the 2D peak in BLG.
Another interesting feature that can be observed in the spectra shown in Fig. 2 (indicated by the dotted lines) is that the G+2D and D+2D peaks appear to upshift in frequency with increasing graphene layers. Such upshifts with increasing graphene layers have been observed recently for combination modes between 1700 -2300 cm -1 involving optical and acoustic phonons, as well as for the G peak phonons in exfoliated graphene. 5 As shown in the plot of peak frequencies versus number of layers (1/n) in Fig. 3b, the G+2D and D+2D peaks exhibit an almost linear dependence on 1/n and the data can be fitted by an equation of the form ω(n) = β/n + ω(∞) , where n is the number of graphene layers, and β is a constant. 4 The values of β (50 cm -1 and 25 cm -1 for the G+2D and D+2D peaks respectively) obtained from the linear fits in Fig. 3b are comparable to those reported for combination modes in graphene. 5 Frequency upshifts of the G band in SLG (up to 13 cm -1 ) have been found to occur due to unintentional doping. 24 However, in addition to an upshift in the G peak frequency, there is a corresponding narrowing of the G peak and decrease in the 2D peak intensity with unintentional (and inhomogeneous) doping. 24 In all the spectra used in this study we have confirmed the consistency of the G peak linewidth (for example, FWHM ~ 9-11 cm -1 in SLG) as well as the ratio of intensities between the 2D and G peaks between all samples and multiple spots measured Finally, we turn our attention to several weak intensity modes observed above 4500 cm -1 in the Raman spectrum of SLG (Fig. 1). The first two peaks occur at ~4600 cm -1 and ~4800 cm -1 (see magnified peaks in Fig. 1), and are assigned to combinations of the D and 2D' (D+2D'), and G and 2D' peaks (G+2D'), respectively. These peaks have previously been assigned to the D+2G 15, 16 and 3G 13 peaks, respectively. Tan et al. did observe a peak at 4800 cm -1 in SWNTs (with 488 nm laser excitation) and assigned it to the G+2D' peak. 14 As mentioned above and also seen in the spectra in Fig. 1, we do not observe the overtone of the G peak (2G), which occurs at ~3160 cm -1 . In addition, the expected sum of peak frequencies for the D (~1345 cm -1 ) and 2D' (~3232 cm -1 ) peaks is ~4577 cm -1 , which is ~50 cm -1 higher than the sum of frequencies of the D and 2G peaks.
Our observed peak frequency in SLG is ~4600 cm -1 , which is closer to the sum of frequencies for the D and 2D' peaks. Hence for the above reasons, we assign the two peaks at ~4600 and ~4800 cm -1 to the D+2D' and G+2D' peaks, respectively. The next weak intensity mode in SLG appears at ~5330 cm -1 , and is the fourth harmonic of the D peak. Note that the occurrence of a three-phonon (3D) peak is improbable due to the requirement of momentum conservation in the scattering process. 13 Thus the next observable overtone of the D peak phonon is the four-phonon overtone of the D peak (four iTO phonons with equal and opposite momentum) at ~5330 cm -1 and is called the 4D peak. 14,16,26 The final four-phonon peak appears at ~5850 cm -1 and has been previously assigned to a combination of 2G and 2D phonons in graphite and SWNTs. 14,16 This peak is very weak in intensity and is shown magnified 50 x in Fig. 1. In order to be consistent with our reasoning concerning the 2G and 2D' peaks above, and because the intensity of the peak at 5300 cm -1 is very weak, we assign the four phonon peak at ~5850 cm -1 to the 2D'+2D mode. Due to increasing background scattering from the SiO 2 substrate, higher order combination modes beyond 6000 cm -1 involving 5 or 6 phonons are difficult to observe in our samples. Furthermore, the weak intensity three and four phonon modes above 4500 cm -1 are only observed from SLG and very difficult to resolve in multilayered graphene samples. This is due to larger resonance enhancement for phonon modes in SLG. For example, the high intensity 2D peak in SLG has been attributed to occur due to a triple resonance process, where all the steps in the typical double resonance process are resonant. 2 Thus one would expect the intensities of higher order combination modes involving the 2D phonon to also be higher than the corresponding peaks in BLG, FLG, and HOPG samples. This might explain why the higher order combination modes are observed in SLG while they are very weak in intensity in multi-layer graphene. All the multiphonon peaks described above are listed in Table 1 along with peak assignments and expected frequencies (based on the sum of the individual components). Also included in Table 1 are experimentally observed multiphonon peaks in HOPG for comparison.
In summary, we have observed multiphonon Raman modes (3 and 4 phonon modes) in SLG, BLG and FLG samples on SiO 2 substrates. Peak assignments are made based on the dispersion of the peaks versus laser excitation energy, as well as the expected frequency obtained from the sum of the individual components. The D+2D and G+2D peaks frequencies are observed have an almost linear dependence on the number of graphene layers, suggesting an influence of interlayer van der Waals interactions on the peak frequencies. The distinct layer-dependence of the G+2D peak frequency provides another metric for the correct identification of the number of layers in graphene samples. Higher order four-phonon modes are also observed in SLG, while these peaks are much weaker in intensity in graphene samples with 2 or more layers due to the large resonance enhancement in SLG. The peak frequencies of all the combination modes in SLG are also shifted relative to the same modes observed previously in SWNTs; for example the G+2D peak in SWNTs has been observed at ~4300 cm -1 (with E laser = 2.41 eV), 13, 14 while the same peak in SLG occurs at ~4260 cm -1 with the same laser excitation. This confirms the uniqueness of the phonon band structure of this one-atom thick two-dimensional material, which is different from both the one-dimensional SWNTs and three-dimensional graphite.
FIG. 1: Higher order Raman spectra from graphene samples collected with E Laser = 2.33 eV. All spectra have been normalized by the G peak intensity. Some of the weak intensity combination modes appearing above 4500 cm -1 are magnified for clarity.
on each sample. The frequency upshifts shown in Fig 3b are likely due to van der Waals interactions in layered systems and, in fact, have been observed recently in the phonon modes of few-layered MoS 2 samples.25
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Higher order combination modes between 4000 -4400 cm -1 in graphene samples collected with E Laser = 2.33 eV. All spectra have been normalized with respect to the G+2D peak intensity for clarity and fitted with Lorentzian peaks. The D+2D and G+2D peaks upshift in frequency with increasing layers. FIG. 2: Higher order combination modes between 4000 -4400 cm -1 in graphene samples collected with E Laser = 2.33 eV. All spectra have been normalized with respect to the G+2D peak intensity for clarity and fitted with Lorentzian peaks. The D+2D and G+2D peaks upshift in frequency with increasing layers.
Dispersion of the G+2D and D+2D peaks in SLG versus laser energy. (b) Peak frequencies of the G+2D and D+2D peaks versus 1/n. The error bars for the FLG samples were obtained from AFM measurements, which confirmed the presence of 3-5 layers. FIG. 4: Dispersion of four components within the G+2D and 2D peaks in BLG versus laser energy. The dispersion of the highest frequency component in the G+2D peak. ~98 cm -1 /eV) is indicated in the figureFIG. 3: (a) Dispersion of the G+2D and D+2D peaks in SLG versus laser energy. (b) Peak frequencies of the G+2D and D+2D peaks versus 1/n. The error bars for the FLG samples were obtained from AFM measurements, which confirmed the presence of 3-5 layers. FIG. 4: Dispersion of four components within the G+2D and 2D peaks in BLG versus laser energy. The dispersion of the highest frequency component in the G+2D peak (~98 cm -1 /eV) is indicated in the figure.
| [] |
[
"Ultrafast quantum spin-state switching in the Co-octaethylporphyrin molecular magnet with a terahertz pulsed magnetic field",
"Ultrafast quantum spin-state switching in the Co-octaethylporphyrin molecular magnet with a terahertz pulsed magnetic field"
] | [
"Oleg V Farberovich \nSchool of Physics and Astronomy, Beverly and Raymond Sackler Faculty of Exact Sciences\nTel Aviv University\n69978Tel AvivIsrael\n\nInternational Center \"Smart Materials\"\nSouthern Federal University\nZorge 5344090Rostov-on-DonRussia\n\nVoronezh State University\n394000VoronezhRussia\n",
"Victoria L Mazalova \nInternational Center \"Smart Materials\"\nSouthern Federal University\nZorge 5344090Rostov-on-DonRussia\n",
"Valeri S Stepanyuk \nMax Planck Institute of Microstructure Physics\nHalleGermany\n"
] | [
"School of Physics and Astronomy, Beverly and Raymond Sackler Faculty of Exact Sciences\nTel Aviv University\n69978Tel AvivIsrael",
"International Center \"Smart Materials\"\nSouthern Federal University\nZorge 5344090Rostov-on-DonRussia",
"Voronezh State University\n394000VoronezhRussia",
"International Center \"Smart Materials\"\nSouthern Federal University\nZorge 5344090Rostov-on-DonRussia",
"Max Planck Institute of Microstructure Physics\nHalleGermany"
] | [] | Molecular spin crossover switches are the objects of intense theoretical and experimental studies in recent years. This interest is due to the fact that these systems allow one to control their spin state by applying an external photo-, thermo-, piezo-, or magnetic stimuli. The greatest amount of research is currently devoted to the study the effect of the photoexcitation on the bi-stable states of spin crossover single molecular magnets (SMMs). The main limitation of photo-induced bi-stable states is their short lifetime. In this paper we present the results of a study of the spin dynamics of the CoOEP molecule in the Low Spin (LS) state and the High Spin (HS) state induced by applying the magnetic pulse of 36.8T . We show that the spin switching in case of the HS state of the CoOEP molecule is characterized by a long lifetime and is dependent on the magnitude and duration of the applied field. Thus, after applying an external stimuli the system in the LS state after the spin switching reverts to its ground state, whereas the system in the HS state remains in the excited state for a long time. We found that the temperature dependency of magnetic susceptibility shows an abrupt thermal spin transition between two spin states at 40K. The proposed here theoretical approach opens the way to create modern devices for spintronics with the controllable spin switching process.PACS numbers: | 10.1016/j.jmmm.2015.12.038 | [
"https://arxiv.org/pdf/1506.07315v1.pdf"
] | 119,183,392 | 1506.07315 | 8bb46255dec6e0b1909bd43ec22bcbf80a427c2f |
Ultrafast quantum spin-state switching in the Co-octaethylporphyrin molecular magnet with a terahertz pulsed magnetic field
24 Jun 2015
Oleg V Farberovich
School of Physics and Astronomy, Beverly and Raymond Sackler Faculty of Exact Sciences
Tel Aviv University
69978Tel AvivIsrael
International Center "Smart Materials"
Southern Federal University
Zorge 5344090Rostov-on-DonRussia
Voronezh State University
394000VoronezhRussia
Victoria L Mazalova
International Center "Smart Materials"
Southern Federal University
Zorge 5344090Rostov-on-DonRussia
Valeri S Stepanyuk
Max Planck Institute of Microstructure Physics
HalleGermany
Ultrafast quantum spin-state switching in the Co-octaethylporphyrin molecular magnet with a terahertz pulsed magnetic field
24 Jun 2015(Dated: June 25, 2015)PACS numbers:
Molecular spin crossover switches are the objects of intense theoretical and experimental studies in recent years. This interest is due to the fact that these systems allow one to control their spin state by applying an external photo-, thermo-, piezo-, or magnetic stimuli. The greatest amount of research is currently devoted to the study the effect of the photoexcitation on the bi-stable states of spin crossover single molecular magnets (SMMs). The main limitation of photo-induced bi-stable states is their short lifetime. In this paper we present the results of a study of the spin dynamics of the CoOEP molecule in the Low Spin (LS) state and the High Spin (HS) state induced by applying the magnetic pulse of 36.8T . We show that the spin switching in case of the HS state of the CoOEP molecule is characterized by a long lifetime and is dependent on the magnitude and duration of the applied field. Thus, after applying an external stimuli the system in the LS state after the spin switching reverts to its ground state, whereas the system in the HS state remains in the excited state for a long time. We found that the temperature dependency of magnetic susceptibility shows an abrupt thermal spin transition between two spin states at 40K. The proposed here theoretical approach opens the way to create modern devices for spintronics with the controllable spin switching process.PACS numbers:
I. INTRODUCTION
Spin engineering mostly concentrates on the manipulation of spin degrees of freedom in various magnetic materials and the exploration of their potential applications in spintronics 1 . Current examples for the applications of spin engineering in magnetic nanostructures and devices are increasingly ample, ranging from the development of single molecular magnet as qubits 2 . Therefore, the spin engineering in magnetic nanostructures, including theoretical modeling, experimental demonstration, and device design for applications, is demanding multidisciplinary backgrounds of knowledge and technology which create great challenges for researchers.
Magnetic or spin logic appears as an appealing alternative due to its nonvolatile character, which can boost up switching on/off, its possibility to reduce the size of the element down to the several-atoms scale on/off spin per atom instead of one elementary charge per 10 4 atoms in semiconductors, and speed increase as a secondary size effect 3 . The modern experiments are promising, they still move in the micrometer regime 4 , thus not fully exploiting the possible quantum nature of molecular magnetism. In other hand, small molecules perform logic operations using as input cation concentrations. The latter is fast with respect to the logic operation but slow with respect to repeatability. Thus a need for magnetic-logic devices on the molecular scale emerges. At the same time the experimental evidence of laser-driven ultrafast magnetic(spin) manipulation in the antiferromagnetic materials motivates the design of a cluster with more than one spin center which allows for spin manipulation 5,6 .
The problem of the gigahertz magnetization(spin) switching speed of todays magnetic logic and a magnetic memory devices into the terahertz regime underlies the entire field of information processing 7,8 . The physics of present-day devices imposes serious limitations on this technological transformation, so we must invent new paradigms based on the quantum spin dynamics in the picosecond regime. This challenge could be met by a simulation of the quantum spin switching in a picosecond pulsed magnetic field 9,10 . Currently, no easily accessible method is available to generate intense sub-picosecond magnetic pulses localized at the nanoscale. However in the work 11 , show that bimetallic nanorings can act as nanoscale sources of intense ultrashort magnetic pulses. In this instance rely on the enhanced light absorption associated with the plasmons of metallic rings to generate transient thermoelectric currents that in turn produce sub-picosecond pulses with magnetic fields as high as a few tenths of a Tesla in the vicinity of the rings. The ability to generate strong magnetic fields localized on the nanoscale is of interest for elucidating spin and magnetization dynamics at sub-picosecond time and nanometer length scales 12 , and it holds great potential for materials characterization, terahertz radiation generation, and magnetic recording.
Magnetic molecules have been intensively studied so far because of their potential technological applications and for the possibility they offer to investigate fundamental properties of matter at the nanoscopic scale 13 . An intensively studied class of magnetic molecules is constituted by the so called single molecule magnets or molecular nanomagnets. These systems are characterized by a slow relaxation of magnetization at low T and give rise to magnetic hysteresis. This is one condition for storing information in a molecule. Therefore, since a SMM represents an isolated magnetic entity, it might be considered as the smallest practical unit for magnetic memories. The studying of a possibility of the switching of the spin states would incredibly increase the amount of information storable with respect to current devices. In order to have a deeper understanding of fundamental features and a major control on technological aspects, the time-dependent spin dynamics constitutes the key point. The micromagnetic model describes appropriately the dynamics of the magnetization(spin) in a magnetic nanostructures. In this model the magnetization(spin) is assumed to be a spatial-and time-dependent continuous function. The magnetization(spin) dynamics are described by the Landau-Lifshitz-Gilbert (LLG) equation including the energy contributions of the anisotropy, the exchange interaction, the magnetostatic interaction, and the Zeeman energy 14 . The microscopic result of the quantum modeling is the cluster spin models, which have been proved to be a powerful tool for approaching the extreme phenomenology of the ultrafast magnetization(spin) dynamics. Ab initio models for the general spin Hamiltonian can be efficient in providing insight on the local atomic scale values such as the local magnetic moment µ s , the local anisotropy D, or a pairwise exchange J ij in the magnetic molecule. The parameters of the theory, the exchange integrals and the anisotropy, are usually fitted to experiments or calculated from static density functional theory (DFT) 15 .
Switching between the LS and HS states can be done by varying the temperature, applying pressure or by light irradiation. In the first and the second case the population of the LS and HS states is fully determined by thermodynamic equilibrium and such transitions are called temperature-and pressure-induced spin crossover transitions 16 . In the case of light-induced spin crossover, after initial excitation to d-states or to metal-to-ligand charge transfer states, the system undergoes a complex cascade of intermediate transitions and eventually gets trapped in a non-equilibrium metastable state. The whole process is called light-induced excited spin state trapping (LIESST). The details of the LIESST process are still under debate 17 .
We report here the possibility of a commutation of the CoOEP SMM between the two states of a spincrossover system by a pulsed magnetic field 18 . Choos-ing a cobalt-based spin-crossover system follows the idea that cobalt(II)[CoOEP] materials 21 as well as iron(III) ones have metal-ligand bonds shorter than those in iron(II) materials, leading to faster relaxation dynamics between the HS and LS states. The molecule cobalt(II) ([Ar]3d 7 4s 2 ) may be either in the LS state (↑↓↑↓↑↓↑ S = 1/2) or in the HS state (↑↓↑↓↑↑↑ S = 3/2), both of which are paramagnetic.
II. THEORETICAL ASPECTS
A. The spin-dynamics simulations
The dynamic behavior of a spin is determined by the equation of motion, which can be derived from the quantum theory with the general spin Hamiltonian H spin that calculated the spin structure of a magnetic molecule with Hamiltonian
H spin = H ex + H an + H ZEE + H a (t).
(1)
The first termĤ ex is the Heisenberg-Dirac Hamiltonian, which represents the isotropic exchange interaction, H an is the exchange Hamiltonian the term due to the axial single-ion anisotropy, and H ZEE is the interaction between the spin system and the external magnetic field. In order to give a theoretical description of a magnetic molecule we exploit the irreducible tensor operator technique 20 . For the study of the CoOEP magnetic properties, it is a good approximation to treat the electrons of one Co-atom as a single atomic spin which is interacting with its surroundings. Thus, one can be content with H spin without the first termĤ ex . We will consider an approximate treatment, restricting ourselves to the simplest, but very useful, approximation, known as mean field theory. This "mean field" Hamiltonian 21 , which describes the interaction of the spin S with the external magnetic field, given by its flux H ef f , can be expressed as:
H spin = −γH ef f S.(2)
The generally shaped formula is
H ef f = − ∂ H spin ∂ S .(3)
Here the effective magnetic field H ef f is an external magnetic field H z , the anisotropy fields H an , the exchange interaction H ex and external magnetic pulse field H pulse (t). We use the "mean field" 21 approximation for
H ef f ⇒ H mean ef f with replacement S ⇒ M s = γ S .
We call "mean" the mean field. (Historically it was often called the "molecular field"). It includes the average effect of the neighbors but neglects correlations between the spin and its neighbors. Since we have the one magnetic atom, this approach us quite suitable. Use (2) we obtain that 22
∂ S ∂t = 1 1 + λ 2 S × H mean ef f − λ 1 + λ 2 S × ( S × H mean ef f )(4)H mean ef f (M s , t) = − δF δM s ,
where F is the free energy of the magnetic nanosystem
F = −N k B T ln Z(H z )
with the partition function
Z(H z ) = Ms,µ exp[−ǫ µ (M s )/kT ] Ms exp[−g e M s H z /kT ](5)
. Here we have the energy levels ǫ µ (M s ) of the spin-Hamiltonian H spin , which stay we diagonalizied H spin .
A spin structure is defined only proceeding from the spin model of a molecule. Here we use to calculate a spin structure by the ITO method within the generalized spin Hamiltonian H spin .
Once we have the energy levels of the spin-Hamiltonian, we can evaluate a different thermodynamic properties of the system as the magnetization, the magnetic susceptibility, and the magnetic specific heat. Since in further researches the anisotropic part of a molecule will be only scalar, the magnetic properties of the anisotropic system do not depend on the direction of the magnetic field. Thus we can consider the external magnetic field H z directed along arbitrary axis z of the molecule coordinate frame that is chosen as a spin quantization axis. In this case the energies of the system will be ǫ µ (M s )+g e M s H z , where ǫ µ (M s ) are the eigenvalues of the spin-Hamiltonian containing the magnetic exchange and the double exchange contributions (index µ runs over the energy levels with given total spin protection M s ). Using this expression one can evaluate the magnetization M s by the standart thermodinamic definitions:
M s = ∂F (M s , H z ) ∂H = N kT ∂ln Z ∂H(6)
The effective field H mean ef f can be derived from the free energy functional
H mean ef f = − δ(F (M s , H z ) + F (t)) δM s = − ∂F (M s , H z ) ∂M s +H x pulse (t),(7)
We have derived a general form of the time-dependend spin equation for a system of the spins precessing in an effective magnetic field with specifying the interactions in the magnetic molecule.
B. Calculations of spin-Hamiltonian parameters
The present research is devoted to study a time-dependent behavior of the Co-octaethylporphyrin molecule after applying an external magnetic field in a picoseconds time span. In order to optimize the molecule geometry and determinate the spin-Hamiltonian parameters -zero-field splitting (ZFS, D-tensor), g-tensor, the exchange parameters J ij , and the single-ion anisotropy parameters, we performed Density Functional Theory (DFT) calculations with taking into account the spinorbit coupling effects using ADF package 23,24 . The calculations were performed with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and the ZORA scalar relativistic Hamiltonian, using an all-electron valence triple zeta with a one polarization function (TZP) basis set and a 'Good Becke' grid. The zero-field splitting (ZFS) of the Co OEP ground state was found to be equal 1.32meV. ZFS is the breaking of degeneracy of the ground state that is not described by a standard non-relativistic Hamiltonian. As calculated by ADF, the ZFS is that exhibited by molecules whose ground state characterized by the spin S>1/2 and absence of a spatial degeneracy. This type of ZFS has two contributions, second-order spin-orbit coupling and spin-spin coupling. In the present implementation only the spin-orbit coupling term is included. In our study the ZFS was calculated in combination with GGA PBE functional. In order to calculate ZFS the relativistic scalar Zero Order Regular Approximation (ZORA) option 25 was included. The ZFS Hamiltonian is given by
H ZF S = S · D · S (8)
where S is the vector operator for the effective spin of Co atom, and D is the second-order anisotropy tensor. After diagonalization in the magnetic axis frames, the Hamiltonian becomes
H ZF S = D S 2 z + E( S 2 x − S 2 y )(9)
The parameter D is called the axial ZFS parameter and E is the rhombic ZFS parameter, which vanishes for high symmetry molecules. Parameter D has to be negative for the magnetic bistability assosiated with SMMs. When a molecule is placed in a magnetic field H, the electron's energies will depend on its magnetic moment value M s . To account for the anisotropy of the Zeeman response to an applied magnetic field, an "effective" Zeeman Hamiltonian using a so-called "g tensor" is used
H ZEE = µ B H·g· S = µ B H x H y H z g x 0 0 0 g yy 0 0 0 g zz S x S y S z (
10) The g-tensor was calculated in a spin-orbit coupled spin unrestricted relativistic ZORA approach. The degenerate perturbation theory was used with the external magnetic field as perturbation.
III. RESULTS AND DISCUSSION
The most important aspect to the establishment of real devices based on SMMs, e.g. high density rewritable memories, is to achieve such conditions under that externally driven system could be switched between two stable LS and HS spin states and would remain in each of these states for a certain period of time. The LS/HS bi-stability is connected with small energy (∆E HS−LS =2.133 meV ) magnitudes involved in the switching between these two spin states (Figure 1a). Much research is currently focused on the study of photoswitching mechanism between LS and HS states, changes in the geometric and electronic structure of molecules in these states, the magnetic properties of SMMs. However, we found no studies on the time-dependent spin dynamics of spin-crossover systems. In this paper we focus on the behavior of the spin of the molecules under the influence of external stimuli and finding the conditions under which the system would be able to stay in one of the bistable states arbitrarily long time. Since the photo-induced states of SMMs are characterized by a short lifetime, we effect on the system being studied by a short magnetic pulse of high amplitude. In the experimental study 18 , it was shown that the application of an external magnetic field of 32T to spin-crossover solid system Co(H 2 (f sa) 2 en)(py) 2 takes place an irreversible and quasicomplete transition from the HS to the LS state. This also means that after applying a perturbation the system can remain in HS state for a long time. In this paper, we have developed an approach for the study of the spin dynamics of spin-crossover systems and apply it to one of the intensively studied objects from a number of SMMs the CoOEP molecule. The chemical structure of CoOEP molecule is exhibited in Fig. 1(c). Fig.2 shows the magnetic susceptibility plotted as the product χT versus temperature for the CoOEP molecule. The temperature variation of the magnetic susceptibility χT showing the abrupt of the thermal spin transition with the jump between two spin states occurs at temperature of T 1/2 =40K.
By irradiating LS molecules with visible light, HS molecule can be generated because of the efficient electronic decay of excited state via Inter System Crossing towards matastable HS state. This effect is now referred as LIESST (Light Induced Electronic Spin State Trapping) in the literature 26 . Once irradiation is stopped, the system relaxes to the thermal equilibrium state LS. One important LIESST-result is the decoupling of different degrees of freedom during the electronic spin state trapping and the transfer of excess energy to molecular vibration modes. Contrary to the LIESST, here considered in some detail the quantum dynamics of the HS spin state trapping and impacts on it of a short magnetic pulse.
We present here a theoretical approach to the organic magnetic CoOEP-molecule in terms of numerically solvable the time-dependent LLG-model with a general spin Hamiltonian in the context of the ITO model. In practice, these models apply to the quantum spin-state switching in the Co-octaethylporphyrin molecular magnet with a picosecond pulsed magnetic field and provide an understanding of a spin-crossover phenomenon in these molecule. In the calculations by LLG-method was used the next parameters: T pulse =0.17ps (ultrashort THz pulse); time of start pulse is 37.4ps; the height of pulse is 36.8T . We solved the LLG equation for two different states S=1/2, t 6 2g e 1 g (LS) and S=3/2, t 5 2g e 2 g (HS). Under the influence of the constant Zeeman magnetic field 0.1T , after a certain period of time, which is determined in case of a single molecule by the size of ZFS, the time-dependent spin structure is formed. In the time of "saturation" of the system we apply the external stimuli of a magnetic pulse. The system in the LS state behaves classically without the application of the pulse. Under the influence of the 36.8T pulse a spin switching on the pulse duration of 100f s is happened, after which the system returns to the ground state (Fig.3). In the case of the HS state of the system the situation is changing drastically. Under the influence of the pulse a spin switching occurs as well, but the system remains in this state for a long time (Fig.6). With increasing magnetic field up to 36.8T a spin switch occurs and the H ef f sign is changed. In our approach H ef f is the S derivative of the spin-Hamiltonian. In our case, the Hamiltonian (1) has no H ex term, because there is only one transition metal atom in CoOEP molecule and, accordingly, the spin-spin interaction is absent. Thus, the anisotropic H an term is only a nonconstant one, which is quadratic in S (9) and, therefore, H ef f is proportional to H ef f is the S . Qualitatively, the H ef f behavior is the same as the S , which is seeing from the Fig .5 and 6.
IV. CONCLUDING REMARKS
In the paper we present the results of the theoretical study of the spin behavior in the CoOEP molecule under influence of high magnetic pulse. Based on the LLG equation we described the spin dynamics of the molecule in the LS and HS states. We found that the spin-crossover CoOEP molecule undergo the spin switching between LS and HS states after applying the external stimuli of a short magnetic pulse of 36.8T. The temperature dependency of magnetic susceptibility shows an abrupt thermal spin transition between two spin states at 40K. We also found the difference in the spin dynamics for two spin states of CoOEP molecule. Thus, after the spin switching under the magnetic pulse the system in the LS state reverts to its ground state, whereas the system in the HS state after the spin switching remains in the excited state for a long time. This behavior gives us the possibility to assign such system as an appropriate candidate for the SMM and opens a way to realize a controlled spin switching in real devices for spintronics.
FIG. 1 :
1(Color online) (a)Spin structure of CoOEP molecule in a spin crossover system HS-LS states;(b)HS and LS states with 3d 7 configuration of Co II in a tetrahedral field;(c)The molecular structure of CoOEP.
FIG
. 3: (Color online) The spin dynamics for the CoOEP molecule with magnetic pulse (S=1/2).
FIG
. 5: (Color online) The spin dynamics for CoOEP molecule without magnetic pulse (S=3/2).
online) The spin dynamics for the CoOEP molecule with magnetic pulse (S=3/2).
The effective magnetic field H mean ef f is given by the free magnetic energy variational with magnetization:
AcknowledgmentsThe part of this research has been supported by Rus-
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. S F Matar, Ph Guionneau, G Chastanet, Int. J. Mol. Sci. 164007S.F. Matar, Ph. Guionneau and G. Chastanet, Int. J. Mol. Sci., 16, 4007 (2015).
| [] |
[
"SUM-OF-SQUARES RESULTS FOR POLYNOMIALS RELATED TO THE BESSIS-MOUSSA-VILLANI CONJECTURE",
"SUM-OF-SQUARES RESULTS FOR POLYNOMIALS RELATED TO THE BESSIS-MOUSSA-VILLANI CONJECTURE"
] | [
"Benoît Collins ",
"Kenneth J Dykema ",
"Francisco Torres-Ayala "
] | [] | [] | We show that the polynomial S m,k (A, B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R X, Y , where X 2 = A and Y 2 = B, for all even values of m and k with 6 ≤ k ≤ m − 10, and also for (m, k) = (12, 6). This leaves only the case (m, k) = (16,8)open. This topic is of interest in connection with the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture, which asks whether Tr(S m,k (A, B)) ≥ 0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using "descent + sum-of-squares" to prove the BMV conjecture.We also show that S m,4 (A, B) is equal to a sum of commutators and Hermitian squares in R A, B when m is even and not a multiple of 4, which implies Tr(S m,4 (A, B)) ≥ 0 holds for all Hermitian matrices A and B, for these values of m. | 10.1007/s10955-010-9959-y | [
"https://arxiv.org/pdf/0905.0420v3.pdf"
] | 16,542,793 | 0905.0420 | e0ab8a9941452ccab22399747192f8859df6e43d |
SUM-OF-SQUARES RESULTS FOR POLYNOMIALS RELATED TO THE BESSIS-MOUSSA-VILLANI CONJECTURE
27 Mar 2010
Benoît Collins
Kenneth J Dykema
Francisco Torres-Ayala
SUM-OF-SQUARES RESULTS FOR POLYNOMIALS RELATED TO THE BESSIS-MOUSSA-VILLANI CONJECTURE
27 Mar 2010
We show that the polynomial S m,k (A, B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R X, Y , where X 2 = A and Y 2 = B, for all even values of m and k with 6 ≤ k ≤ m − 10, and also for (m, k) = (12, 6). This leaves only the case (m, k) = (16,8)open. This topic is of interest in connection with the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture, which asks whether Tr(S m,k (A, B)) ≥ 0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using "descent + sum-of-squares" to prove the BMV conjecture.We also show that S m,4 (A, B) is equal to a sum of commutators and Hermitian squares in R A, B when m is even and not a multiple of 4, which implies Tr(S m,4 (A, B)) ≥ 0 holds for all Hermitian matrices A and B, for these values of m.
Introduction
While working on quantum statistical mechanics, Bessis, Moussa and Villani [1] conjectured in 1975 that for any positive semidefinite Hermitian matrices A and B, the function t → Tr(e A−tB ) is the Laplace transform of a positive measure supported in R + . This is referred to as the Bessis-Moussa-Villani or BMV conjecture. In 2004, Lieb and Seiringer [9] proved that the BMV conjecture is equivalent to the following reformulation: for every A and B as above, all of the coefficients of the polynomial
p(t) = Tr((A + tB) m ) ∈ R[t](1)
are nonnegative. Recently, there has been much activity around this algebraic reformulation, (see [5], [4], [8], [2], [6]). The latest state of knowledge is summarized in [6], and we'll review this here. Let S m,k (A, B) denote the sum of all words of length m in A and B having k letters equal to B and m − k equal to A. Thus, the coefficient of t k in the polynomial p(t) of (1) is equal to the trace of S m,k (A, B), and the Lieb-Seiringer reformulation of the BMV conjecture is that this trace is always nonnegative. An important result, due to Hillar [4], is that if this conjecture fails for some (m, k), then it fails for all (m ′ , k ′ ) satisfying k ′ ≥ k and m ′ − k ′ ≥ m − k. We'll refer to this as Hillar's descent theorem.
One strategy that has been used to show that the trace of S m,k (A, B) is nonnegative for certain values of m and k is to let X and Y be formal square roots of A and B, respectively and, working in the algebra R X, Y of polynomials in noncommuting variables X and Y , to show that S m,k (A, B) is equal to a sum of commutators [g, h] = gh − hg and Hermitian squares f * f . Here, the algebra R X, Y is endowed with the involutive * -operation that is anti-multiplicative and so that X = X * and Y = Y * are Hermitian. We adopt the notation of [6] and say that two elements a, b ∈ R X, Y are cyclically equivalent (written a cyc ∼ b) if they differ by a sum of commutators. We will use repeatedly Proposition 2.3 of [6], which states that two words v and w in X and Y are cyclically equivalent if and only if they can be written v = u 1 u 2 and w = u 2 u 1 for words u 1 and u 2 in X and Y , and that two polynomials a, b ∈ R X, Y are cyclically equivalent if and only if for each cyclic equivalence class [w] of words in X and Y , the sum over all v in [w] of the coefficients a v of a agrees with the sum over all v in [w] of the coefficients b v of b. It is clear that any element of R A, B that is cyclically equivalent in R X, Y to a sum of Hermitian squares in R X, Y must have nonnegative trace whenever A and B are replaced by positive semidefinite matrices, and this has been the strategy used to show that S m,k (A, B) has nonnegative trace, for certain values of m and k. We will adopt the terminology of [6] and write Θ 2 to denote the set of elements of R X, Y that are cyclically equivalent to sums of Hermitian squares in R X, Y . (It is not difficult to see that Θ 2 C ∩ R X, Y = Θ 2 , where Θ 2 C is the analogous quantity in C X, Y .) Clearly, S m,k (A, B) ∈ Θ 2 if and only if S m,m−k (A, B) ∈ Θ 2 . Due to work of Hägele [3], Landweber and Speer [8], Burgdorf [2] and Klep and Schweighofer [6], it is known that S m,k (A, B) ∈ Θ 2 holds • whenever k ∈ {0, 1, 2, 4} • for m = 14 and k = 6 • for m ∈ {7, 11} and k = 3
These cases together with Hillar's descent theorem implied that the Lieb-Seiringer formulation of the BMV-conjecture holds for m ≤ 13 (see [6]). On the other hand, it is known that S m,k (A, B) / ∈ Θ 2 holds • whenever m ≥ 12 or m ∈ {6, 8, 9, 10} and k = 3 • whenever m ≥ 10 and 5 ≤ k ≤ m − 5 and either k or m is odd.
It was hoped that proofs of S m,k (A, B) ∈ Θ 2 for other values of m and k would be posible, so as to prove the conjecture for more values of m, and possibly even to prove the BMV conjecture itself.
These results left open the cases (m, k) = (12, 6) and m ≥ 16, 6 ≤ k ≤ m − 6 with both m and k even. In this paper (see Section 2), we prove S m,k (A, B) / ∈ Θ 2 whenever m and k are even and 6 ≤ k ≤ m − 10. Using S m,k (A, B) = S m,m−k (B, A), this leaves open only the cases (m, k) = (12, 6) and (m, k) = (16, 8). We resolve the first of these cases by showing, via an easier argument, S 12,6 (A, B) / ∈ Θ 2 . The case of (m, k) = (16, 8) remains open, though, as indicated in [6], numerical evidence seems to suggest it does not lie in Θ 2 .
Our results, thus, show that it is impossible to prove the BMV conjecture by showing that S m,k (A, B) is cyclically equivalent to a sum of Hermitian squares for sufficiently many values of m and k. However there are other plausible approaches to showing Tr(S m,k (A, B)) ≥ 0 must always hold.
Though our proofs are straightforward and easy to check by hand, to find them we calculated with Mathematica 7.0 [10], on an Apple MacBook running OS X version 10.4.11.
While exploring, we found (see Proposition 3.3) that if m is even and is not a multiple of 4, then S m,4 (A, B) is equal to a sum of commutators and Hermitian squares in R A, B . Thus, we do not need the square roots of A and B: for these values of m we have Tr(S m,4 (A, B)) ≥ 0 whenever A and B are Hermitian matrices. Using Hillar's descent theorem, a positive answer to Question 1.1 would imply the Lieb-Seiringer formulation of the BMV conjecture.
We will prove the following theorem in Section 4. It shows that Question 1.1 has an equivalent formulation that seems easier to satisfy, and is analogous to Theorem 1.10 of [4]. Note that S m,k (A, B) is Hermitian whenever A and B are Hermitian. In Section 3 we also show (Proposition 3.8) that S 8,4 (A, B) is not cyclically equivalent to a sum of Hermitian squares in R A, B . This makes the case (m, k) = (8, 4) of particular interest for Question 1.1.
Our interest in Question 1.1 has two motivations. One is its relation to the BMV conjecture. Although the question is known to be stronger than the BMV conjecture and we have no particular reason to think it will be easier to prove than the BMV conjecture itself, it is clearly related to the BMV conjecture and it may be helpful to explore it. A second motivation is the relation to Connes' embedding problem. For positive semidefinite matrices A and B, the trace of S 6,3 (A, B) is always nonnegative, though it is not cyclically equivalent to a sum of squares in C X, Y ; as was pointed out in [7], this makes S 6,3 (A, B), with A and B positive operators in a II 1 -factor, an interesting test case for Connes' embedding problem. In a similar way, if Question 1.1 turns out to have a positive answer for S 8,4 (A, B), then because of Proposition 3.8, then it will provide another interesting test case for Connes' embedding problem, involving self-adjoint operators. At this point, it seems important to generate such test cases.
After a first version of this paper was circulated, we learned that S. Burgdorf (see Remarks (b) and (c) of Section 4 of [2]) had, long previously to us, also found that if m is not a multiple of 4, then S m,4 (A, B) is cyclically equivalent to a sum of Hermitian squares in R A, B ; no proof was given in [2].
Some non-sum-of-squares results
In this section, we show that S m,k (A, B) is not cyclically equivalent to a sum of Hermitian squares in R X, Y for various values of m and k, all of which are even.
Let W q,p (A, B) denote the set of all words in A and B containing q A's and p B's. Let Z denote the column vector whose entries are all words in W ℓ,k (A, B) in some fixed order, and similarly let Z X and, respectively, Z Y be column vectors containing all elements of XW ℓ−1,k (A, B)X, respectively, Y W ℓ,k−1 (A, B)Y . Klep and Schweighofer have shown (Proposition 3.3 of [6]) that, for integers k and ℓ, S 2(k+ℓ),2k (A, B) is cyclically equivalent to a sum of Hermitian squares in R X, Y if and only if there are real, positive semidefinite matrices H, H X and H Y such that
Z * HZ + Z * X H X Z X + Z * Y H Y Z Y cyc ∼ S 2(k+ℓ),2k (A, B),(2)
where Z * denotes the row vector whose entries are the adjoints of the entries of Z, etc. Let us denote the matrix entry of H corresonding to words u, v ∈ W ℓ,k (A, B) by H(u, v), and similarly for H X and H Y . Thus, we have
Z * HZ = u,v∈W ℓ,k (A,B) H(u, v)u * v,(3)
and similarly for the other two terms.
Remark 2.1. If H is a matrix as appearing in (3), and if H is the matrix defined by
H(u, v) = H(u * , v * ), then Z * HZ = u,v H(u * , v * )u * v = u,v H(u, v)uv * cyc ∼ u,v H(v, u)v * u = Z * HZ,
where the last equality uses that H is symmetric. In a similar way, defining
H X (u, v) = H X (u * , v * ) and H Y (u, v) = H Y (u * , v * ), we have Z * X H X Z X cyc ∼ Z * X H X Z X Z * Y H Y Z Y cyc ∼ Z * Y H Y Z Y .H(u, v) = H(u * , v * ), (u, v ∈ W ℓ,k (A, B)),(4)H X (u, v) = H X (u * , v * ), (u, v ∈ XW ℓ−1,k (A, B)X),(5)H Y (u, v) = H Y (u * , v * ), (u, v ∈ Y W ℓ,k−1 (A, B)Y ).(6)
Suppose, furthermore, we have k = ℓ. Let σ is the map on words that exchanges A and B and exchanges X and Y , extended by linearity to R X, Y . Then σ(Z * HZ) =
Z * H σ Z, where H σ (u, v) = H(σ(u), σ(v)), and, similarly, σ(Z * X H X Z X ) = Z * Y H σ X Z Y and σ(Z * Y H Y Z Y ) = Z * X H σ Y Z X , where H σ X (u, v) = H X (σ(u), σ(v)) and H σ Y (u, v) = H Y (σ(u), σ(v)H(σ(u), σ(v)) = H(u, v), (u, v ∈ W ℓ,k (A, B)),(7)H Y (σ(u), σ(v)) = H X (u, v), (u, v ∈ XW ℓ−1,k (A, B)X).(8)
Since σ(u * ) = σ(u) * , we can assume that (4)-(6) and (7)-(8) hold simultaneously. We note that the relation (4) will be used in this section, while (7) will be used only in the proof of Proposition 3.8, and the conditions on H X and H Y won't be needed at all in this paper.
Remark 2.2. For a given word w ∈ W 2ℓ,2k (A, B), we are interested in the different ways we can have
w cyc ∼ u * v, (u, v ∈ W ℓ,k (A, B)),(9)w cyc ∼ u * X v X , (u X , v X ∈ XW ℓ−1,k (A, B)X),(10)w cyc ∼ u * Y v Y , (u Y , v Y ∈ Y W ℓ,k−1 (A, B)Y ).(11)
Indeed, if |[w]| denotes the number of different elements of W 2ℓ,2k (A, B) that are cyclically equivalent to w, and assuming (2) holds, then we have
|[w]| = {(u,v)|u * v cyc ∼ w} H(u, v) + {(u X ,v X )|u * X v X cyc ∼ w} H X (u X , v X ) + {(u Y ,v Y )|u * Y v Y cyc ∼ w} H Y (u Y , v Y ) (12)
where the respective sums are over all pairs (u, v) such that (9) holds, all pairs (u X , v X ) such that (10) holds and all pairs (u Y , v Y ) such that (11) holds. To find all the ways we have (9), we can write down all the cyclic permutations of w and record those for which the first k + ℓ letters consists of ℓ A's and k B's. Furthermore, if we have an instance of (10) with
u X = Xu ′ X and v X = Xv ′ X, u ′ , v ′ ∈ W ℓ−1,k (A, B), then w cyc ∼ X(u ′ ) * Av ′ X cyc ∼ A(u ′ ) * Av ′ ;
this yields an instance of (9), where both u * and v start with A, and clearly each such instance corresponds in this manner to an instance of (10). Similarly, the instances of (11) are in one-to-one correspondence with those instances of (9) where both u * and v start with B.
We will apply (in a finite dimensional setting) the following elementary lemma, whose proof we provide for completeness.
T = T 11 T 12 T 21 T 22 , where T ij : H j → H i . Suppose v ∈ ker T 11 ⊆ H 1 . Then v ∈ ker T 21 .
Proof. If T 21 v = 0, then there is w ∈ H 2 such that T 21 v, w < 0. Letting t > 0 and using
T 12 = T * 21 , we have T (v ⊕ tw), v ⊕ tw = 2tRe T 21 v, w + t 2 T 22 w, w .(13)
But taking t small enough forces the right-hand-side of (13) to be negative, which contradicts T ≥ 0.
Proposition 2.4. Let k and ℓ be integers, k ≥ 3 and ℓ ≥ 5. Then S 2(ℓ+k),2k (A, B) is not cyclically equivalent to a sum of Hermitian squares in R X, Y .
Proof. Suppose the contrary, to obtain a contradiction. Let H, H X and H Y be real, positive semidefinite matrices so that (2) holds, and without loss of generality assume also the property (4) in Remark 2.1 holds. We consider five elements of W 2ℓ,2k (A, B) and the different ways of writing them as in (9). These elements are
w 1 = A 2ℓ B 2k w 2 = A 2ℓ−2 B k−1 A 2 B k+1 w 3 = A ℓ+1 B 2 A ℓ−1 B 2k−2 w 4 = A 2ℓ−4 B k−1 A 2 B 2 A 2 B k−1 w 5 = A ℓ−1 B 2 A ℓ−1 B k−1 A 2 B k−1
and their factorizations will be in terms of the elements
u 1 = A ℓ B k , v 1 = u * 1 = B k A ℓ u 2 = A ℓ−2 B k−1 A 2 B, v 2 = u * 2 = BA 2 B k−1 A ℓ−2 u 3 = AB 2 A ℓ−1 B k−2 , v 3 = u * 3 = B k−2 A ℓ−1 B 2 A u 4 = AB k−1 A ℓ−1 B, v 4 = u * 4 = BA ℓ−1 B k−1 A of W ℓ,k (A, B)
. Note that these are all distinct if k ≥ 4; in the case k = 3, the six elements u 1 , u 2 , u 3 , v 1 , v 2 , v 3 are distinct but we have u 4 = u 3 and v 4 = v 3 . This will not bother us.
We begin with the easiest of the w j to factorize, namely, w 1 . In the Table 1 are listed all the cyclically equivalent forms of w 1 and it is indicated which of these can be factored as in (9). Table 1. Forms of w 1 = A 2ℓ B 2k and factorizations as in (9).
cyclically equivalent form j value factorization A j B 2k A 2ℓ−j (1 ≤ j ≤ 2ℓ) j = ℓ v * 1 v 1 B j A 2ℓ B 2k−j (1 ≤ j ≤ 2k) j = k u * 1 u 1
This also shows that there are no factorizations as in (10) or (11) (see Remark 2.2). Since w 1 has 2(k + ℓ) cyclically equivalent forms, by (12) we must have H(u 1 , (9).
u 1 ) + H(v 1 , v 1 ) = 2(k + ℓ). Since we have H(v 1 , v 1 ) = H(u 1 , u 1 ), we get H(u 1 , u 1 ) = k + ℓ.(14)
cyclically equivalent form j value factorization
A j B k−1 A 2 B k+1 A 2ℓ−2−j (1 ≤ j ≤ 2ℓ − 2) j = ℓ − 2 v * 2 v 1 B j A 2 B k+1 A 2ℓ−2 B k−1−j (1 ≤ j ≤ k − 1) none A j B k+1 A 2ℓ−2 B k−1 A 2−j (1 ≤ j ≤ 2) none B j A 2ℓ−2 B k−1 A 2 B k+1−j (1 ≤ j ≤ k + 1) j = k u * 1 u 2
The cyclically equivalent forms and all factorizations of w 2 , w 3 , w 4 and w 5 as in (9) are given in Tables 2-5. (Note that the assertions in rows 2, 3 and 6 of Table 4 do require ℓ ≥ 5.) Table 3. Forms of w 3 = A ℓ+1 B 2 A ℓ−1 B 2k−2 and factorizations as in (9). cyclically equivalent form j value factorization Table 4. Forms of w 4 = A 2ℓ−4 B k−1 A 2 B 2 A 2 B k−1 and factorizations as in (9).
A j B 2 A ℓ−1 B 2k−2 A ℓ+1−j (1 ≤ j ≤ ℓ + 1) j = 1 v * 3 v 1 B j A ℓ−1 B 2k−2 A ℓ+1 B 2−j (1 ≤ j ≤ 2) none A j B 2k−2 A ℓ+1 B 2 A ℓ−1−j (1 ≤ j ≤ ℓ − 1) none B j A ℓ+1 B 2 A ℓ−1 B 2k−2−j (1 ≤ j ≤ 2k − 2) j = k u * 1 u 3
cyclically equivalent form j value factorization
A j B k−1 A 2 B 2 A 2 B k−1 A 2ℓ−4−j (1 ≤ j ≤ 2ℓ − 4) j = ℓ − 2 v * 2 v 2 B j A 2 B 2 A 2 B k−1 A 2ℓ−4 B k−1−j (1 ≤ j ≤ k − 1) none A j B 2 A 2 B k−1 A 2ℓ−4 B k−1 A 2−j (1 ≤ j ≤ 2) none B j A 2 B k−1 A 2ℓ−4 B k−1 A 2 B 2−j (1 ≤ j ≤ 2) j = 1 u * 2 u 2 A j B k−1 A 2ℓ−4 B k−1 A 2 B 2 A 2−j (1 ≤ j ≤ 2) none B j A 2ℓ−4 B k−1 A 2 B 2 A 2 B k−1−j (1 ≤ j ≤ k − 1) none
From these, we see that each of the words w j , 2 ≤ j ≤ 5 has 2(k + ℓ) different cyclically equivalent forms, and none have factorizations involving X or Y , as in (10) or (11). Looking at the two factorizations of w 2 , and using (12) and H(v 2 , v 1 ) = H(v 1 , v 2 ) = H(u 1 , u 2 ), we conclude (9).
H(u 1 , u 2 ) = k + ℓ.(15)
cyclically equivalent form j value factorization
A j B 2 A ℓ−1 B k−1 A 2 B k−1 A ℓ−1−j (1 ≤ j ≤ ℓ − 1) j = 1 v * 3 v 2 B j A ℓ−1 B k−1 A 2 B k−1 A ℓ−1 B 2−j (1 ≤ j ≤ 2) j = 1 u * 4 u 4 A j B k−1 A 2 B k−1 A ℓ−1 B 2 A ℓ−1−j (1 ≤ j ≤ ℓ − 1) j = ℓ − 2 v * 2 v 3 B j A 2 B k−1 A ℓ−1 B 2 A ℓ−1 B k−1−j (1 ≤ j ≤ k − 1) j = 1 u * 2 u 3 A j B k−1 A ℓ−1 B 2 A ℓ−1 B k−1 A 2−j (1 ≤ j ≤ 2) j = 1 v * 4 v 4 B j A ℓ−1 B 2 A ℓ−1 B k−1 A 2 B k−1−j (1 ≤ j ≤ k − 1) j = k − 2 u * 3 u 2
Similarly, considering all the factorizations of w 3 , w 4 and w 5 we get, respectively,
H(u 1 , u 3 ) = k + ℓ (16) H(u 2 , u 2 ) = k + ℓ (17) 2H(u 2 , u 3 ) + H(u 4 , u 4 ) = k + ℓ.(18)
Now from equations (14)-(17), for the 3 × 3 submatrix of H corresponding to the entries u 1 , u 2 , u 3 , we have
H(u 1 , u 1 ) H(u 1 , u 2 ) H(u 1 , u 3 ) H(u 1 , u 2 ) H(u 2 , u 2 ) H(u 2 , u 3 ) H(u 1 , u 3 ) H(u 2 , u 3 ) H(u 3 , u 3 ) = k + ℓ k + ℓ k + ℓ k + ℓ k + ℓ H(u 2 , u 3 ) k + ℓ H(u 2 , u 3 ) H(u 3 , u 3 ) .(19)
From ( Proof. This is like the proof of Proposition 2.4, but easier. Again we assume, to obtain a contradiction, that H, H X and H Y are real, positive semidefinite matrices such that (2) holds (with k = ℓ = 3) and that the properties (9)-(11) hold. We need only consider the words
w 6 = A 6 B 6 , w 7 = A 4 B 2 A 2 B 4 , w 8 = A 2 B 2 A 2 B 2 A 2 B 2
in W 6,6 (A, B) and their factorizations, which will be in terms of the elements 3 (A, B). These factorizations are given in Tables 6-8. Again, w 6 , w 7 and w 8 have no factorizations as in (10) or (11). From Table 6, we see that w 6 has 12 distinct cyclically equivalent forms, and since H(u 5 , u 5 ) = H(v 5 , v 5 ), from (12) we get H(u 5 , u 5 ) = 6. From Table 7 and H(v 6 , v 5 ) = H(u 6 , u 5 ) = H(u 5 , u 6 ), Table 6. Forms of w 6 = A 6 B 6 and factorizations as in (9). cyclically equivalent form j value factorization Table 7. Forms of w 7 = A 4 B 2 A 2 B 4 and factorizations as in (9).
u 5 = A 3 B 3 , v 5 = u * 5 = B 3 A 3 u 6 = AB 2 A 2 B, v 6 = u * 6 = BA 2 B 2 A of W 3,A j B 6 A 6−j (1 ≤ j ≤ 6) j = 3 v * 5 v 5 B j A 6 B 6−j (1 ≤ j ≤ 6) j = 3 u * 5 u 5
cyclically equivalent form j value factorization (9).
A j B 2 A 2 B 4 A 4−j (1 ≤ j ≤ 4) j = 1 v * 6 v 5 B j A 2 B 4 A 4 B 2−j (1 ≤ j ≤ 2) none A j B 4 A 4 B 2 A 2−j (1 ≤ j ≤ 2) none B j A 4 B 2 A 2 B 4−j (1 ≤ j ≤ 4) j = 3 u * 5 u 6
cyclically equivalent form j value factorization
A j B 2 A 2 B 2 A 2 B 2 A 2−j (1 ≤ j ≤ 2) j = 1 v * 6 v 6 B j A 2 B 2 A 2 B 2 A 2 B 2−j (1 ≤ j ≤ 2) j = 1 u * 6 u 6
we get H(u 5 , u 6 ) = 6, while from Table 8 we see that w 8 has only four distinct cyclically equivalent forms, and we get H(u 6 , u 6 ) = 2. The 2 × 2 submatrix of H corresponding to {u 5 , u 6 } is, therefore, H(u 5 , u 5 ) H(u 5 , u 6 ) H(u 6 , u 5 ) H(u 6 , u 6 ) = 6 6 6 2 , which is not positive semidefinite. This gives a contradiction.
Sums of squares in R A, B
In this section, we prove some results related to Question 1.1. As per the discussion in the introduction (see Proposition 2.3 of [6]), we say f, g ∈ R A, B are cyclically equivalent if and only if f − g is a sum of commutators of elements from R A, B . This holds if and only if, for every word w in A and B, the sum over words v that are cyclic permutations of w of the coeefficients in f of v agrees with the same sum for g.
Clearly, if S m,k (A, B) is cyclically equivalent to a sum i f * i f i of Hermitian squares, for f i ∈ R A, B , then Question 1.1 has a positive answer for this particular pair (m, k).
Of course, S 2m,0 (A, B) = A 2m is a Hermitian square in R A, B , for every integer m ≥ 0.
Verification of the following two lemmas is straightforward.
Lemma 3.1. Let m ∈ N. Then S 4m,2 (A, B) cyc ∼ mf * m f m + 2m m−1 j=0 f * j f j , where f 0 = BA 2m−1 f j = A j−1 BA 2m−j + A j BA 2m−j−1 , (1 ≤ j ≤ m). Lemma 3.2. Let m ∈ N. Then S 4m+2,2 (A, B) cyc ∼ (2m + 1) m j=0 f * j f j , where f 0 = BA 2m f j = A j−1 BA 2m−j+1 + A j BA 2m−j , (1 ≤ j ≤ m).
The next proposition shows that S 2q,4 (A, B) is cyclically equivalent to a sum of Hermitian squares in R A, B , when q is odd. Note that Klep and Schweighofer in Section 5 of [6] proved this in the case q = 7. In fact, we found the expression (20) below by exploration using Mathematica [10] and checked it by computation for all values of m ≤ 20. The best proof we could find, which is given below, turned out to be surprisingly intricate.
where
f 0 = 2m−1 s=0 BA 2m−s−1 BA s , f p = p i=p−1 2m−i−1 s=p A i BA 2m−s−i−1 BA s , (1 ≤ p ≤ m − 1) f m = A m−1 B 2 A m .
As before W q,4 (A, B) denotes the set of all words in A and B with exactly q A's and four B's.
Let N 0 = N ∪ {0}. For ι = (ι 1 , ι 2 , ι 3 , ι 4 .ι 5 ) ∈ N 5 0 let E(ι) = A ι 1 BA ι 2 BA ι 3 BA ι 4 BA ι 5
and take I = {ι ∈ N 5 0 | ι 1 + ι 2 + ι 3 + ι 4 + ι 5 = 4m − 2}.
Note that the map ι → E(ι) gives a bijection from I onto W 4m−2,4 (A, B). With this notation we may write
S 4m+2,4 (A, B) = ι∈I E(ι).
The proof of Proposition 3.3 will use the following three lemmas. The first of these is readily verified, and a proof will be omitted. 4 (A, B) is cyclically equivalent to a unique word of the form
Lemma 3.4. Each word in W 4m−2,BA k 1 BA k 2 BA k 3 BA k 4
where κ = (0, k 1 , k 2 , k 3 , k 4 ) ∈ I satisfies either
k 1 ≤ k 3 and k 2 < k 4 (21) or k 1 = k 3 ≤ k 2 = k 4 .(22)
We will call the words (or indices) described in (21) and (22) canonically ordered and those of the form (21) will be called type I while those given by (22) will be called type II. Since the first letter of any canonically ordered word is a B, canonically ordered words are parameterized by only four non-negative integers, and we'll frequently omit to write the first element of a canonically ordered index κ, since it is always zero. #{κ ∈ I | κ is canonically ordered of type I} = 2m(2m − 1)(2m + 1) 3 .
#{κ ∈ I | κ is canonically ordered of type II} = m.
Proof. We recall that a partition of n ∈ N into k parts is a k-tuple (a 1 , a 2 , · · · , a k ) such that 1 ≤ a 1 ≤ a 2 ≤ · · · ≤ a k and a 1 + a 2 + · · · + a k = n. We denote it as (a 1 , a 2 , · · · , a k ) ⊢ n.
Consider the sets
B = {(a, b, a 1 , a 2 , b 1 , b 2 ) ∈ N 6 | a + b = 4m + 1, (a 1 , a 2 ) ⊢ a, (b 1 , b 2 ) ⊢ b}
and A = {κ ∈ I | κ is canonically ordered of type I}. Take the function from A into B given by (k 1 , k 2 , k 3 , k 4 ) → (k 1 + k 3 + 2, k 2 + k 4 + 1, k 1 + 1, k 3 + 1, k 2 + 1, k 4 ).
One can show this function is a bijection onto B. Thus,
#A = (a,b)∈N 2 a+b=4m+1 a 2 b 2 = 2 3 m(2m − 1)(2m + 1).
Similarly, the function
(k 1 , k 2 , k 3 , k 4 ) → (k 1 + 1, k 2 + 1)
is a bijection from {κ ∈ I | κ is canonically ordered of type II} onto the set {(a, b) ∈ N 2 | (a, b) ⊢ 2m + 1}. Hence #{κ ∈ I | κ is canonically ordered of type II} = 2m + 1 2 = m.
The following lemma is easily verified by writing out the cyclically equivalent forms of words; see Tables 1-8 for other exercises of this sort. For g ∈ R A, B and w a word in A and B, we let c w (g) denote the coefficient of w in g. By Lemmas 3.4 and 3.6 it will suffice to show, for every canonically ordered word w ∈ W 4m−2,4 (A, B), i.e., for each such w, there is only one representative in m p=0 f * p f p if w is type II and exactly two representatives if w is type I.
We begin by taking a closer look at each f * p f p . We have
f * 0 f 0 = 0≤s,t≤l A s BA l−s B 2 A l−t BA t = ι∈I 0 E(ι),
where
I 0 = {ι = (s, l − s, 0, l − t, t) | 0 ≤ s, t ≤ l} and for 1 ≤ p ≤ m − 1, f * p f p = p−1≤i,j≤p p≤s≤l−i p≤t≤l−j A s BA l−i−s BA i+j BA l−j−t BA t = ι∈Ip(p−1,p−1) E(ι) + ι∈Ip(p−1,p) E(ι) + ι∈Ip(p,p−1) E(ι) + ι∈Ip(p,p) E(ι),
where
I p (i, j) = {ι = (s, l − i − s, i + j, l − j − t, t) | p ≤ s ≤ l − i, p ≤ t ≤ l − j}, while f * m f m = ι∈Im E(ι),
where I m = {(m, 0, 2m − 2, 0, m)}. We also write I 0 (0, 0) = I 0 and I m (m − 1, m − 1) = I m .
Let J be the disjoint union ) is a copy of the corresponding I p (i, j) and similarly for J 0 = J 0 (0, 0) and J m = J m (m − 1, m − 1). Formally, given 0 ≤ p ≤ m and max{0, p − 1} ≤ i, j ≤ min{p, m − 1}, we set
J 0 ⊔ m−1 p=1 p−1≤i,j≤p J p (i, j) ⊔ J m where each J p (i, jJ p (i, j) = {(p, i, j, ι) | ι ∈ I p (i, j)}
and we let α j, s, t), if (i = j and t > s) or (i > j and t − 1 > s) or (j > i and t > s − 1) L(i, j, s, t), if (i = j and t ≤ s) or (i > j and t ≤ s − 1) or (j > i and t ≤ s − 1),
(i, j) (1 ≤ p ≤ m − 1, p − 1 ≤ i, j ≤ p) as follows. For ι = (s, l − i − s, i + j, l − j − t, t) ∈ I p (i, j) we have O(ι) = U(i,
where U and L are given by
U(i, j, s, t) = (0, l, 0, l) + 1 1 0 0 0 −1 0 −1 0 0 1 1 −1 0 −1 0 i j s t , L(i, j, s, t) = (l, 0, l, 0) + −1 0 −1 0 1 1 0 0 0 −1 0 −1 0 0 1 1 i j s t .
The canonical form of an element of J is naturally taken to be the same as the canonical form of the element of I to which it corresponds and we denote the "canonical form map" also by O : J → I. We now work on proving (23). For 0 ≤ p ≤ m − 1 define
ι p = (p, l − 2p, 2p, l − 2p, p) ∈ I p (p, p).
Then O(ι p ) = (l − 2p, 2p, l − 2p, 2p), which is of type II. We will show that there are no other words of type II in J. Since we have m different values of O, Lemma 3.5 will imply (23) in the case w is of type II.
Let K = J \ {α (p,p) p (ι p ) | 0 ≤ p ≤ m − 1}.
We will find a partition of K into two sets, B and C, both with cardinality 2m(2m − 1)(2m + 1)/3, and a bijection β : B → C such that O(β(ι)) = O(ι) and check that O restricted to B is injective and its values are of type I. From this it will follow that (23) holds in the case w is of type I, and this will complete the proof of (23) in the case w is of type II.
The partition and bijection are defined below in several parts. In all cases, it is straightforward to check the identity O(β(i)) = O(i).
(i) For 0 ≤ p ≤ m − 1 take
B 1 (p) = I p+1 (p, p), C 1 (p) = {(s, l − p − s, 2p, l − p − t, t) ∈ I p (p, p) | p + 1 ≤ s, t}.
We notice B 1 (p) = C 1 (p) for all 0 ≤ p ≤ m − 1. This identification is used to define the restriction of β to J p+1 (p, p) by β • α (p,p)
p+1 = α (p,p) p . For ι = (s, l − p − s, 2p, l − p − t, t) ∈ I p+1 (p, p) we have O(ι) = (l − p − s, 2p, l − p − t, s + t), p + 1 ≤ t ≤ s ≤ l − p (2p, l − p − t, s + t, l − p − s), p + 1 ≤ s < t ≤ l − p,
and this element is of type I.
Let B 1 = m−1 p=0 α (p,p) p+1 (B 1 (p)) and C 1 = m−1 p=0 α (p,p) p (C 1 (p)). We have #B 1 = m−1 p=0 (2(m − p) − 1) 2 . (ii) For 1 ≤ p ≤ m − 1, let B 2 (p) = {(s, l − (p − 1) − s, 2p − 1, l − p − t, t) ∈ I p (p − 1, p) | p + 1 ≤ s}, C 2 (p) = {(s, l − p −s, 2p − 1, l − (p − 1) −t,t) ∈ I p (p, p − 1) | p + 1 ≤t}. For ι = (s, l − (p − 1) − s, 2p − 1, l − p − t, t) ∈ B 2 (p) let β(α (p−1,p) p (ι)) = α (p,p−1) p (s − 1, l − p − (s − 1), 2p − 1, l − (p − 1) − (t + 1), t + 1).
Then β : α
(p−1,p) p (B 2 (p)) → α (p,p−1) p (C 2 (p))
is a bijection and a computation shows
O(β(α (p−1,p) p (ι))) = O(α (p−1,p) p (ι)) = (l − p − s − 1, 2p − 1, l − p − t, s + t), p ≤ t ≤ s − 1 ≤ l − p, (2p − 1, l − p − t, s + t, l − p − s − 1), p ≤ s − 1 < t ≤ l − p
and this is a word of type I. Take
B 2 = m−1 p=1 α (p−1,p) p (B 2 (p)), C 2 = m−1 p=1 α (p,p−1) p (C 2 (p)).
By disjointness, we have
#B 2 = m−1 p=1
(2(m − p)) 2 .
(iii) In I 0 (0, 0), the cases (s, t) = (0, l) and (s, t) = (l, 0) have the same value under O, namely (0, 0, l, l), which is type I. Take
B 4 (0) = {(0, l, 0, l − t, t) : 1 ≤ t ≤ l − 1} ⊂ I 0 (0, 0). For ι = (0, l, 0, l − t, t) ∈ B 4 (0), take β(α (0,0) 0 (ι)) = α (q,q) q (l − q, 0, 2q, l − 2q, q), l − t even, q = l−t 2 , α (q,q−1) q (l − q, 0, 2q − 1, l − 2q + 1, q), l − t odd, q = l−t+1 2 . Let B 4 = α (0,0) 0 (B 4 (0)) and let C 4 be the image of B 4 under β. A direct compu- tation shows O(β(α (0,0) 0 (ι))) = O(α (0,0) 0 (ι)) = (0, l − t, t, l),
which is type I. We also have #B 4 = 2(m − 1).
(v) Consider the set is of type I. We also have #B 5 = 2(m − 1). (vi) Let
B 5 (0) = {(s, l − s, 0, l, 0) : 1 ≤ s ≤ l − 1} ⊂ I 0 (0, 0). For ι = (s, l − s, 0, l, 0) ∈ B 5 (0) define β(α (0,0) 0 (ι)) = α (q,q) q (q, l − q, 2q, 0, l − q), l − s even, q = l−s 2 , α (q−1,q) q (q, l − 2q + 1, 2q − 1, 0, l − q), l − s odd, q = l−s+1 2 . LetB 1 6 = m−1 p=1 {α (p−1,p) p (p, l − 2p + 1, 2p − 1, l − p − t, t) : p ≤ t ≤ l − p − 1}, B 2 6 = m−2 p=1 {α (p,p) p (p, l − 2p, 2p, l − p − t, t) : p + 1 ≤ t ≤ l − p − 1} and let B 6 = B 1 6 ∪ B 2 6 . For η = α (p−1,p) p (p, l − 2p + 1, 2p − 1, l − p − t, t) ∈ B 1 6 ,(24)
let
β(η) = α (q,q) q (2m − 2p − q, 2p − 1, 2q, l − 2q, q), p + t odd α (q,q−1) q (2m − 2p − q, 2p − 1, 2q − 1, l − 2q + 1, q), p + t even, where q = m − ⌊ p+t+1 2 ⌋. For η = α (p,p) p (p, l − 2p, 2p, l − p − t, t) ∈ B 2 6 (25) let β(η) = α (q,q) q (2m − 2p − q − 1, 2p, 2q, l − 2q, q), p + t odd, α (q,q−1) q (2m − 2p − q − 1, 2p, 2q − 1, l − 2q + 1, q) p + t even,
where q = m − ⌊ p+t+1 2 ⌋. Take C 6 to be the image of B 6 under β. Then β : B 6 → C 6 is a bijection and (25) is of type I. We also have
O(β(η)) = O(η) = (2p − 1, l − p − t + 1, p + t, l − 2p), η as in (24) (2p, lp − t, p + t, l − 2p), η as in#B 6 = m−1 p=1 (2(m − p) − 1) + m−2 p=1 (2(m − p) − 2) = (2m − 3)(m − 1).
Lastly, we take
B = 6 k=1 B k , C = 6 k=1 C k . A computation shows #B = m−1 p=1 (2(m − p) − 1) 2 + m−1 p=1 (2(m − p)) 2 + 1 + 4(m − 1) + (2m − 3)(m − 1) = 1 + (2m − 1) 2 + 4(m − 1) + (2m − 3)(m − 1) + 2(m−1) j=1 j 2 = 2m(2m − 1)(2m + 1) 3 .
We have, thus, constructed a bijection β : B → C that satisfies O(β(η)) = O(η) and, as can be checked, the restriction of O to B is injective and takes values that are all of type I. Lastly the sets B and C form a partition of K. This completes the proof of Proposition 3.3.
The bijection we have defined may be better understood using some pictures, which are contained in Figures 1 and 2. We parameterize I 0 by the square {(s, t) ∈ Z 2 : 0 ≤ s, t ≤ l} and I m by the single point (m, m). Likewise for fixed 1 ≤ p ≤ m − 1 and i, j ∈ {p − 1, p}, the set
I p (i, j) is parameterized by {(s, t) ∈ Z 2 : p ≤ s ≤ l − i, p ≤ t ≤ l − i}.
We show the case m = 3.
In these figures, • The points that give words of type II are marked with diamonds.
• The light circles in the right column are matched with the circles in the left. Likewise the solid circles. These correspond to cases 1 and 2.
In the case 2 the bijection is implemented by (s, t) → (s − 1, t + 1), form the rightmost sub-square of side l − 2p + 1 in I p (p − 1, p) to the uppermost sub-square of side l − 2p + 1 in I P (p, p − 1), for 1 ≤ p ≤ m − 1.
• Case 3 is marked with a solid square.
• The remaining points (which correspond to the most complicated part of the bijection), plotted in light squares, correspond the the cases 4,5 and 6.
The following theorem summarizes the results obtained so far in this section. cyc ∼ Z * HZ for H a 6 × 6 real, positive semidefinite matrix. So suppose, to obtain a contradiction, that such exists. There are ten cyclic equivalence class of words in W 4,4 (A, B). We've chosen one representative for each and we have listed them in Table 9 with their orders, where we say the order of a word is the number of cyclically equivalent forms that it has. If we denote the Table 9. Representatives of cyclic equivalence classes in W 4,4 (A, B). name word order
w 1 A 4 B 4 8 w 2 A 3 BAB 3 8 w 3 A 3 B 2 AB 2 8 w 4 A 3 B 3 AB 8 w 5 A 2 BA 2 B 3 8 w 6 A 2 BABAB 2 8 w 7 A 2 BAB 2 AB 8 w 8 A 2 B 2 A 2 B 2 4 w 9 A 2 B 2 ABAB 8 w 10 ABABABAB 2
ith element of the vector Z by z i , then the matrix whose (i, j)th entry is the symbol k ∈ {1, . . . , 10} such that w k is cyclically equivalent to z * i z j is the matrix found below. 1 2 3 5 6 8 4 7 9 9 10 6 3 6 8 7 9 3 5 6 7 8 9 5 9 10 6 6 7 2 8 9 3 5 4 1
.
The hypothesis Z * HZ ∼ S 8,4 (A, B) is, therefore, equivalent to the ten linear equations 8
in the entries of the matrix H. However, H is real symmetric. Moreover, we may assume without loss of generality that the relations (4) and (7) from Remark 2.1 hold, and we find, therefore, that H commutes with the permutation matrices corresponding to the order-two permutations τ : 1 ↔ 6, 2 ↔ 5. σ : 1 ↔ 6, 2 ↔ 5, 3 ↔ 4.
Since x 3 ≥ 1 and x 2 ≥ 4, the first two factors are strictly positive. So the third factor must be nonnegative, and we conclude
x 2 (x 3 − 6) ≥ (x 3 − 4)(x 3 + 2).
Since x 3 ≤ 6 we must have x 3 ≤ 4 and
x 2 ≤ (4 − x 3 )(x 3 + 2) 6 − x 3 .
But combining this with x 2 ≥ 4, we get 24 − 4x 3 ≤ 8 + 2x 3 − x 2 3 , so x 2 3 − 6x 3 + 16 ≤ 0, which is impossible. This is the desired contradiction.
Proof of Theorem 1.2
In this section, we prove Theorem 1.2 using a straightforward application of the method of Lagrange multipliers.
Question 1 . 1 .
11Do we have Tr(S m,k (A, B)) ≥ 0 whenever A and B are Hermitian matrices and m and k are even integers, m ≥ k?
Theorem 1 . 2 .
12Fix n, m, k ∈ N with m and k even and m ≥ k. Then the following are equivalent: (i) for all n × n Hermitian matrices A and B, we have Tr(S m,k (A, B)) ≥ 0, (ii) for all n × n Hermitian matrices A and B, either S m,k (A, B) = 0 or S m,k (A, B) has a strictly positive eigenvalue.
Lemma 2 . 3 .
23Let H = H 1 ⊕ H 2 be an orthogonal direct sum decomposition of a Hilbert space and let T ∈ B(H) be a positive operator: T ≥ 0. With respect to the given decomposition of H, write T in block form
19), the positivity of H and Lemma 2.3, we obtain also H(u 2 , u 3 ) = k + ℓ. But then, from (18), we must have H(u 4 , u 4 ) = −(k + ℓ), which contradicts the positive semidefiniteness of H. Proposition 2.5. S 12,6 (A, B) is not cyclically equivalent to a sum of squares in R X, Y .
Proposition 3 . 3 .
33Let m ∈ N. Then S 4m+2,4 (A,
Lemma 3 . 6 .
36Let w ∈ W 4m−2,4 (A, B) be a canonically ordered word. If w is of type I, then there are 4m+2 words in W 4m−2,4 (A, B) that are cyclically equivalent to w, while if w is of type II, then there are 2m + 1 words in W 4m−2,4 (A, B) that are cyclically equivalent to w. Proof of Proposition 3.3. Let l = 2m − 1.
:
I p (i, j) → J p (i, j) be the bijection given by ι → (p, i, j, ι). Consider the function O : I → I, where O(ι) is the index of the canonically ordered word that is cyclically equivalent to E(ι). This function O is explicitly given on I 0 and on each I p
C 5 be the image of B 5 under β. Then β : B 5 → C 5 is a bijection and
Theorem 3. 7 .Figure 1 .Figure 2 .
712If k = 2 and m ≥ 2 is even, or if k = 4 and m ≥ 6 is even but not a multiple of 4, then S m,k (A, B) is cyclically equivalent to a sum of Hermitian squares Some sets in K with m More sets in K with m = R A, B . Therefore, for these values of m and k, Tr(S m,k (A, B)) ≥ 0 whenever A and B are Hermitian matrices. Below is a non-sum-of-squares result for S 8,4 (A, B). However, Question 1.1 for m = 8 and k = 4 is still open.
Proposition 3. 8 .
8The polynomial S 8,4 (A, B) is not cyclically equivalent to a sum of Hermitian squares in R A, B . Proof. We order the elements of W 2,2 (A, B) in the column vector Z = (A 2 B 2 , ABAB, AB 2 A, BA 2 B, BABA, B 2 A 2 ) t . If S 8,4 (A, B) were equivalent to a sums of squares in R A, B , then by Proposition 3.3 of [6], we would have S 8,4 (A, B)
Lemma 4 . 1 .
41Let n, m, k ∈ N and fix an n × n Hermitian matrix B. Consider the function A → Tr(S m,k (A, B)) with domain consisting of the n×n Hermitian matrices A such that Tr(A 2 ) = 1. Suppose A 0 is a point where this function has a relative extrumum. ThenS m−1,k (A 0 , B) = m − k m Tr(S m,k (A 0 , B))A 0 .(36)for some µ ∈ R and all H, and this implies2µA 0 = mS m−1,k (A 0 , B)Multiplying both sides byA 0 , taking the trace and using Lemma 2.1 of [4], we get 2µ = 2µTr(A 2 0 ) = mTr(A 0 S m−1,k (A 0 , B)) = (m − k)Tr(S m,k (A 0 , B)), and (36) follows. Proof of Theorem 1.2. The implication (i) =⇒ (ii) is clear. Suppose (i) does not hold. Let A 0 and B 0 be a Hermitian n × n matrices where Tr(S m,k (A, B)) takes its absolute minimum subject to Tr(A 2 ) = Tr(B 2 ) = 1. By assumption, we have Tr(S m,k (A 0 , B 0 )) < 0. By Lemma 4.1 and the analogue obtained by switching A and B, we haveS m−1,k (A 0 , B 0 ) = m − k m Tr(S m,k (A 0 , B 0 ))A 0 S m,k−1 (A 0 , B 0 ) = k m Tr(S m,k (A 0 , B 0 ))B 0 .Therefore, the Hermitian matrixS m,k (A 0 , B 0 ) = A 0 S m−1,k (A 0 , B 0 ) + B 0 S m−1,k−1 (A 0 , B 0 )= Tr(S m,k (A 0 , B 0 )) nonpositive eigenvalues. Thus, (ii) does not hold.
Consequently, if H, H X and H Y are such that (2) holds, then by replacing H with (H + H)/2, if necessary, and similarly for H X and H Y , we may without loss of generality assume
). Consequently, if H, H X and H Y are such that (2) holds, then since S 2(k+ℓ),2k (A, B) is σ-invariant and since σ respectscyc
∼ , by replacing H with (H +
H σ )/2, H X with (H X + H σ
Y )/2 and H Y with (H Y + H σ
X )/2, if necessary, we may
without loss of generality assume
Table 2 .
2Forms of w 2 = A 2ℓ−2 B k−1 A 2 B k+1and factorizations as in
Table 5 .
5Forms of w 5 = A ℓ−1 B 2 A ℓ−1 B k−1 A 2 B k−1 and factorizations as in
Table 8 .
8Forms of w 8 = A 2 B 2 A 2 B 2 A 2 B 2 and factorizations as in
= H 11 + H 66 = H 14 + H 41 + H 46 + H 64 (30) 8 = H 15 + H 26 + H 32 + H 42 + H 53 + H 54 (31) 8 = H 22 + H 34 + H 43 + H 55 (32) 4 = H 16 + H 33 + H 44 + H 61 (33) 8 = H 23 + H 24 + H 35 + H 45 + H 51 + H 62(26)
8 = H 12 + H 56
(27)
8 = H 13 + H 31 + H 36 + H 63
(28)
8 = H 21 + H 65
(29)
8 (34)
2 = H 25 + H 52
Acknowledgement. The authors thank an anonymous referee for suggestions that improved the exposition.We will show that there is no positive semidefinite real matrix of this form. To make the formulas slightly more readable, we will use the symbols x 2 = H 22 and x 3 = H 33 . Of course, we must have x 2 ≥ 0 and x 3 ≥ 0. We will consider compressions of H obtained by restricting to rows and columns in subsets of {1, . . . , 6}. The compression to {1, 2} is ( 4 4 4 x 2 ), and from positivity we obtain x 2 ≥ 4. Compression to {1, 6} yields |2 − x 3 | ≤ 4, so x 3 ≤ 6. Compression to {1, 3} yields x 3 ≥ 1. The determinant of the compression of the matrix H to {1, 3, 4, 6} is the polynomial with factorizationProof. This is an application of the method of Lagrange multipliers to the problem of optimizing Tr(S m,k (A, B)) subject to the constraint Tr(A 2 ) = 1. (Compare to Appendix A of[6].) The space of Hermitian n × n matrices is a real vector space of dimension n 2 . If H and A are Hermitian matrices, thenLetting H run through the same basis as taken above, the list of values (38) forms the gradient of the objective function with respect to the n 2 variables. By the method of Lagrange multipliers, we conclude that at a relative extremum A 0 , these two gradients must be parallel. In other words, we must have 2µTr(HA 0 ) = mTr(HS m−1,k (A 0 , B))
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Hägele's approach to the Bessis-Moussa-Villani conjecture. P S Landweber, E R Speer, D On, Lin. Alg. Appl. 431P.S. Landweber, E.R. Speer, On D. Hägele's approach to the Bessis-Moussa-Villani con- jecture, Lin. Alg. Appl. 431 (2009), 1317-1324.
Equivalent forms of the Bessis-Moussa-Villani conjecture. E H Lieb, R Seiringer, J. Stat. Phys. 115E.H. Lieb, R. Seiringer, Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys. 115 (2004), 185-190.
Mathematica Version 7.0. Wolfram Research, IncChampaign, IllinoisWolfram Research, Inc., Mathematica Version 7.0, Wolfram Research, Inc., Champaign, Illinois, 2008.
| [] |
[
"Non dissipative decoherence of Rabi oscillations",
"Non dissipative decoherence of Rabi oscillations"
] | [
"Rodolfo Bonifacio \nDip. di Fisica\nINFN and INFM\nUniversità di Milano\nSezione di Milano\nvia Celoria 1620133MilanoItaly\n",
"Stefano Olivares \nDip. di Fisica\nINFN and INFM\nUniversità di Milano\nSezione di Milano\nvia Celoria 1620133MilanoItaly\n",
"Paolo Tombesi \nDip. di Matematica e Fisica and Unità INFM\nUniversità di Camerino\nvia Madonna delle Carceri 62032CamerinoItaly\n",
"David Vitali \nDip. di Matematica e Fisica and Unità INFM\nUniversità di Camerino\nvia Madonna delle Carceri 62032CamerinoItaly\n"
] | [
"Dip. di Fisica\nINFN and INFM\nUniversità di Milano\nSezione di Milano\nvia Celoria 1620133MilanoItaly",
"Dip. di Fisica\nINFN and INFM\nUniversità di Milano\nSezione di Milano\nvia Celoria 1620133MilanoItaly",
"Dip. di Matematica e Fisica and Unità INFM\nUniversità di Camerino\nvia Madonna delle Carceri 62032CamerinoItaly",
"Dip. di Matematica e Fisica and Unità INFM\nUniversità di Camerino\nvia Madonna delle Carceri 62032CamerinoItaly"
] | [] | We present a simple theoretical description of two recent experiments where damping of Rabi oscillations, which cannot be attributed to dissipative decoherence, has been observed. This is obtained considering the evolution time or the Hamiltonian as random variables and then averaging the usual unitary evolution on a properly derived, model-independent, probability distribution. | 10.1080/09500340008235142 | [
"https://arxiv.org/pdf/quant-ph/9906115v3.pdf"
] | 118,899,247 | quant-ph/9906115 | 4dd9bdd27a012a6a655a100af7e394c6cf4da150 |
Non dissipative decoherence of Rabi oscillations
25 Oct 1999 (October 13, 2018)
Rodolfo Bonifacio
Dip. di Fisica
INFN and INFM
Università di Milano
Sezione di Milano
via Celoria 1620133MilanoItaly
Stefano Olivares
Dip. di Fisica
INFN and INFM
Università di Milano
Sezione di Milano
via Celoria 1620133MilanoItaly
Paolo Tombesi
Dip. di Matematica e Fisica and Unità INFM
Università di Camerino
via Madonna delle Carceri 62032CamerinoItaly
David Vitali
Dip. di Matematica e Fisica and Unità INFM
Università di Camerino
via Madonna delle Carceri 62032CamerinoItaly
Non dissipative decoherence of Rabi oscillations
25 Oct 1999 (October 13, 2018)arXiv:quant-ph/9906115v3
We present a simple theoretical description of two recent experiments where damping of Rabi oscillations, which cannot be attributed to dissipative decoherence, has been observed. This is obtained considering the evolution time or the Hamiltonian as random variables and then averaging the usual unitary evolution on a properly derived, model-independent, probability distribution.
Even though decoherence is a very general phenomenon [1], it is very difficult to verify it experimentally because most often the physical nature of the environmental degrees of freedom responsible for the decoherence process remains unknown. The only controlled experimental verification of decoherence has been given by the experiment of Ref. [2], in which the progressive transformation of a linear superposition of two coherent states of a microwave cavity mode into the corresponding statistical mixture has been monitored. In this case, the environmental decoherence has been checked with no fitting parameters because its physical origin, i.e., photon leakage out of the cavity, was easily recognizable and measurable. In this case, it is even possible to control decoherence, i.e., to considerably suppress its effects, for example by using appropriately designed feedback schemes [3].
In some cases however, the mechanisms responsible for decoherence are not easily individuated and examples are provided by two recent experiments which observed Rabi oscillations between two circular states of a Rydberg atom in a high-Q cavity [4], and of two internal states of a 9 Be + ion coupled with the vibrations in a trapping potential [5]. In both cases one observes damped oscillations to a steady state in which the population of each of the two levels approaches 1/2. A number of candidates have been already considered as possible physical sources of decoherence in these cases. In the trapped ion case of Ref. [5], fluctuations of classical parameters such as the intensity of the laser beams used to couple internal and vibrational degrees of freedom, or the voltage and the frequency of the trapping potentials have been suggested. In the case of the Rydberg atom in a high-Q cavity, dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity [4,6] have been proposed as possible sources of decoherence. Despite this, a complete quantitative explanation of the observed decay rate of the Rabi oscillations (see Ref. [7]) in the two experiments is still lacking. The only established fact is that, differently from Ref. [2], in both cases, decoherence has a non-dissipative origin. In fact, the observed decay of the Rabi oscillations is much faster than the energy relaxation rate in these experimental configurations. Moreover, the fact that in both cases the population of each of the two levels asymptotically approaches 1/2 cannot be explained in terms of dissipative mechanisms as the photon leakage out of the cavity.
A different approach to decoherence has been proposed in [8], where a model-independent formalism has been derived to describe decoherence. Here we shall adopt a more pragmatic point of view and we shall use this formalism to explain in simple terms both Rabi oscillation experiments, even though they are realized in different physical situations. The idea underlying the approach of Ref. [8] is the fact that the interaction time, i.e. the time interval in which the effective Hamiltonian evolution takes place, is a random variable. This randomness can have different origins depending on the studied system. For example, in the case of the Rydberg atom experiment [4], the interaction time is determined by the transit time of the velocity-selected atom through the high-Q microwave cavity. This interaction time is random, due to fluctuations of the atomic velocities. This randomness implies having random phases e −iEnt/h in the energy eigenstates basis. The experimental results unavoidably average over these random phases and this leads to decoherence, i.e., to the decay of off-diagonal matrix elements of the density operator in the energy basis. Notice however that one would have the same phase fluctuations if the Hamiltonian (and therefore the eigenvalues E n ) fluctuates instead of the interaction time. Therefore, as we shall see, our approach will give a generalized phasedestroying master equation, able to describe many situations in which decoherence is associated with random phases, originating for example from some frequency or interaction time fluctuations.
Let us consider an initial state ρ(0) and consider the case of a random evolution time. The experimentally observed state is not described by the usual density matrix of the whole system ρ(t), but by its time averaged counterpart [8]
ρ(t) = ∞ 0 dt ′ P (t, t ′ )ρ(t ′ ) ,(1)
where ρ(t ′ ) = exp{−iLt ′ }ρ(0) is the usual unitarily evolved density operator from the initial state and L . . . = [H, . . .]/h. Hence one can writē
ρ(t) = V (t)ρ(0) ,(2)
where
V (t) = ∞ 0 dt ′ P (t, t ′ )e −iLt ′ . In Ref. [8]
, the function P (t, t ′ ) has been determined so to satisfy the following conditions: i)ρ(t) must be a density operator, i.e. it must be self-adjoint, positive-definite, and with unit-trace. This leads to the condition that P (t, t ′ ) must be non-negative and normalized, i.e a probability density in t ′ so that Eq. (1) is a completely positive mapping. ii) V (t) satisfies the semigroup property
V (t 1 + t 2 ) = V (t 1 )V (t 2 )
, with t 1 , t 2 ≥ 0. These requirements are satisfied by [8]
V (t) = (1 + iLτ ) −t/τ (3) P (t, t ′ ) = e −t ′ /τ τ (t ′ /τ ) (t/τ )−1 Γ(t/τ ) ,(4)
so that V (t) and P (t, t ′ ) are connected by the so-called Γ-function integral identity [9,10]. The parameter τ characterizes the strength of the evolution time fluctuations.
When τ → 0, P (t, t ′ ) → δ(t − t ′ ) so thatρ(t) = ρ(t)
and V (t) = exp{−iLt} is the usual unitary evolution. However, for finite τ , the evolution operator V (t) of Eq. (3) describes a decay of the off-diagonal matrix elements in the energy representation, whereas the diagonal matrix elements remain constant, i.e. the energy is still a constant of motion. In fact, by differentiating with respect to time Eq. (2) and using (3), one gets the following master equation forρ(t)
ρ(t) = − 1 τ log (1 + iLτ )ρ(t)(5)
If one expands the logarithm at second order in τ , one obtainsρ
(t) = − ī h [H,ρ(t)] − τ h 2 [H, [H,ρ(t)]] ,(6)
which is the well-known phase-destroying master equation [11]. Hence Eq. (5) appears as a generalized phasedestroying master equation taking into account higher order terms in τ . Notice, however, that the present approach is different from the usual master equation approach in the sense that no perturbative and specific statistical assumptions are made. We now apply this formalism to the two experiments of Refs. [4,5]. In the experiment of Ref. [4], the resonant interaction between a quantized mode in a high-Q microwave cavity (with annihilation operator a) and two circular Rydberg states (|e and |g ) of a Rb atom is studied. This interaction is well described by the usual Jaynes-Cummings [12] model, which in the interaction picture reads
H =hΩ R |e g|a + |g e|a † ,(7)
where Ω R is the Rabi frequency. The Rabi oscillations describing the exchange of excitations between atom and cavity mode are studied by injecting the velocity-selected Rydberg atom, prepared in the excited state |e , in the high-Q cavity and measuring the population of the lower atomic level g, P eg (t) as a function of the interaction time t, which is varied by changing the Rydberg atom velocity. In the case of vacuum state induced Rabi oscillations, the decoherence effect is particularly evident and the Hamiltonian evolution according to Eq. (7) predicts
P eg (t) = 1 2 (1 − cos (2Ω R t)) .(8)
Experimentally instead, damped oscillations are observed, which are well fitted by
P exp eg (t) = 1 2 1 − e −γt cos (2Ω R t) ,(9)
where the decay time fitting the experimental data is γ −1 = 40µsec [6] and the corresponding Rabi frequency is Ω R /2π = 25 Khz. This decay of quantum coherence cannot be associated with photon leakage out of the cavity because the cavity relaxation time is larger (220 µsec) and also because in this case one would have an asymptotic limit P exp eg (∞) = 1. The damped behavior of Eq. (9) is instead easily obtained if one applies the approach described above. In fact, from the linearity of Eq. (1), one has that the time averaging procedure is also valid for mean values and matrix elements of each subsystem. Therefore one has
P eg (t) = ∞ 0 dt ′ P (t, t ′ )P eg (t ′ ) .(10)
Using Eqs.
where γ = 1 2τ
log 1 + 4Ω 2 R τ 2 (12) ν = 1 τ arctg (2Ω R τ )(13)
We note that in general the time averaging procedure introduces not only a damping of the probability oscillations but also a frequency shift. However, if the characteristic time τ is sufficiently small, i.e. Ω R τ ≪ 1, there is no phase shift, ν ≃ 2Ω R , and
γ = 2Ω 2 R τ(14)
The fact that in Ref. [4] the Rabi oscillation frequency essentially coincides with the theoretically expected one, suggests that the time τ characterizing the fluctuations of the interaction time is sufficiently small so that it is reasonable to use Eq. (14). Using the above values for γ and Ω R , one can derive an estimate for τ , so to get τ ≃ 0.5 µsec. This estimate is consistent with the assumption Ω R τ ≪ 1 we have made, but, more importantly, it turns out to be comparable to the experimental value of the uncertainty in the interaction time. In fact, the fluctuations of the interaction time are mainly due to the experimental uncertainty of the atomic velocity v. In fact, one has t = √ πw/v, where w is the cavity mode waist. Since w = 0.6 cm, the mean velocity isv ≃ 300 m/sec and the velocity uncertainty is δv/v = 1% (see Ref. [4]), one hast = √ πw/v ≃ 50 µsec and τ ≃ δt =tδv/v = 0.5 µsec, which is just the estimate we have derived from the experimental values. This simple argument supports the interpretation that the decoherence observed in [4] is essentially due to the randomness of the interaction time.
Let us now consider the case of the trapped ion experiment of Ref. [5], in which the interaction between two internal states (| ↑ and | ↓ ) of a Be ion and the center-of-mass vibrations in the z direction, induced by two driving Raman lasers is studied. In the interaction picture with respect to the free vibrational and internal Hamiltonian, this interaction is described by the following Hamiltonian [7] H
=hΩ| ↑ ↓ | exp i η ae −iωzt + a † e iωz t − δt + φ + H.C. ,(15)
where a denotes the annihiliation operator for the vibrations along the z direction, ω z is the corresponding frequency and δ is the detuning between the internal transition and the frequency difference between the two Raman lasers. The Rabi frequency Ω is proportional to the two Raman laser intensities, and η is the Lamb-Dicke parameter [5,7]. When the two Raman lasers are tuned to the first blue sideband, i.e. δ = ω z , Hamiltonian (15) predicts Rabi oscillations between | ↓, n and | ↑, n + 1 (|n is a vibrational Fock state) with a frequency [7] Ω n = Ω e −η 2 /2 √ n + 1
ηL 1 n (η 2 ) ,(16)
where L 1 n is the generalized Laguerre polynomial. These Rabi oscillations have been experimentally verified by preparing the initial state | ↓, n , (with n ranging from 0 to 16) and measuring the probability P ↓ (t) as a function of the interaction time t, which is varied by changing the duration of the Raman laser pulses. Again, as in the cavity QED experiment of [4], the experimental Rabi oscillations are damped and well fitted by [5,7] P ↓ (n, t) = 1 2 1 + e −γnt cos (2Ω n t) ,
where the measured oscillation frequencies Ω n are in very good agreement with the theoretical prediction (16) corresponding to the measured Lamb-Dicke parameter η = 0.202 [5]. As concerns the decay rates γ n , the experimental values are fitted in [5] by
γ n = γ 0 (n + 1) 0.7(18)
where γ 0 = 11.9 Khz. This power-law scaling has been investigated in Refs. [13,14], but a clear explanation of this behavior of the decay rates is still lacking. On the contrary, the scaling law (18) can be accounted for in the previous formalism if we consider the small τ limit of Eq. (14), which is again suggested by the fact that the experimental and theoretical predictions for the frequencies Ω n agree. In fact, the n-dependence of the theoretical prediction of Eq. (16) for η = 0.202 is well approximated, within 10 %, by the power law dependence
Ω n ≃ Ω 0 (n + 1) 0.35 ,(19)
so that, using Eq. (14) with Ω R replaced by Ω n , one has immediately the power law dependence (n + 1) 0.7 of Eq. (18). The value of the parameter τ can be obtained by matching the values corresponding to n = 0, and using Eq. (14), that is τ = γ 0 /2Ω 2 0 ≃ 1.5 · 10 −8 sec, where we have used the experimental value Ω 0 /2π = 94 Khz. However, this value of the parameter τ cannot be explained in terms of some interaction time uncertainty, such as the time jitter of the Raman laser pulses, which is experimentally found to be much smaller [15]. In this case, instead, the observed decoherence can be attributed, as already suggested in [7,13,14], to the fluctuation of the Raman laser intensities, yielding a fluctuating Rabi frequency parameter Ω(t) of the Hamiltonian (15). In this case the evolution is driven by a fluctuating Hamiltonian H(t) =hΩ(t)H, whereH = H/Ω in Eq. (15), so that , which is proportional to the pulse area. It is now easy to understand that the physical situation is analogous to that characterized by a random interaction time considered above, with L replaced byL and t ′ by A(t). One has again phase fluctuations in the energy basis representation and, in analogy with Eq. (1), one considers an averaged density matrix
ρ(t) = exp −iLρ(t) = ∞ 0 dAP (t, A)e −iLA ρ(0) .(21)
Imposing again thatρ(t) must be a density operator and the semigroup property, one finds results analogous to Eqs. (3) and (4)
V (t) = 1 + iLΩτ −t/τ (22) P (t, A) = e −A/Ωτ Ωτ (A/Ωτ ) (t/τ )−1 Γ(t/τ ) ,(23)
where, the parameter Ω of Eq. (15) plays now the role of a mean Rabi frequency. In fact, consistently with the probability distribution of Eq. (23), one has Ω = A /t.
The scaling time τ characterizes in this case the strength of the pulse area fluctuations, since from Eq. (23), one has σ 2 (A) = A 2 − A 2 = Ω 2 tτ . The estimated value of τ is reasonable since it corresponds to a fractional error of the pulse area σ 2 (A)/ A = τ /t of 10% for a pulse duration of t = 1 µsec, and which is decreasing for increasing pulse durations. The present analysis shows many similarities with that of Ref. [13] which also tries to explain the decay of the Rabi oscillations in the ion trap experiments of [5] in terms of laser intensity fluctuations. The authors of Ref. [13] in fact use a phase destroying master equation coinciding with the second-order expansion (6) of our generalized master equation of Eq. (5) (see Eq. (16) of Ref. [13] with the identifications G ↔ H/h and Γ ↔ τ ) and moreover derive the same numerical estimate for the pulse area fluctuation strength Γ ↔ τ . Despite this similarities, they do not recover the scaling (18) of the decay rates γ n only because they do not use the general expression of the Rabi frequency (16), (and which is well approximated by the power law (19)) but its Lamb-Dicke limit Ω n = Ω 0 (n + 1) 0.5 , which is valid only when η ≪ 1. There is however another, more fundamental, difference between our approach and that of Ref. [13]. They assume from the beginning that the laser intensity fluctuations have a white and gaussian character, while we make no a priori assumption on the statistical properties of the pulse area A. We derive these properties, i.e. the probability distribution (23), only from the very general semigroup condition, and it is interesting to note that this condition yields a gaussian probability distribution for the pulse area only as a limiting case. In fact, from Eq. (23) one can see that P (t, A) tends to become a gaussian with the same mean value Ωt and the same variance Ω 2 τ t only in the large time limit t/τ ≫ 1. This non-gaussian character of P (t, A) can be traced back to the fact that P (t, A) must be definite and normalized in the interval 0 ≤ A < +∞ and not in −∞ < A < +∞. Notice that at t = τ , Eq. (23) assumes the exponential form P (t = τ, A) = e −A/Ωτ /Ωτ . Only at large times t the random variable A becomes the sum of many independent contributions and assumes the gaussian form. Due to the non-gaussian nature of the random variable A, we find that the more generally valid phase-destroying master equation is given by Eq. (5) (with L replaced by ΩL), whose predictions significantly depart from its second order expansion (6) (corresponding to the gaussian limit) as soon as τ becomes comparable with the typical timescale of the system under study, which, in the present case, is the inverse of the Rabi frequency.
In conclusion, we have presented a model-independent theory for non-dissipative decoherence, able to provide a simple and unified description of the same decoherence phenomenon observed in two Rabi oscillations experiments which were performed under different situations. A simple way to test experimentally our prediction is to check that the dependence of the decay rate as a function of the Rabi frequency is given by Eq. (12). One should observe a transition from a quadratic dependence to a logarithmic dependence, increasing the value of the Rabi frequency, or of τ .
Discussions with J.M. Raimond and D. Wineland are greatly acknowledged. This work has been supported by MURST under the "Cofinanziamento".
( 2 )
2, (3), (4) and (8), Eq. (10) can be rewritten in the same form of Eq. e −γt cos (νt) ,
= e −iLA(t) ρ(0) (20) whereL = [H, . . .]/h and we have defined the positive dimensionless random variable A(t) = t 0 dξΩ(ξ)
. W H Zurek, Phys. Today. 441036and references thereinW.H. Zurek, Phys. Today 44(10), 36 (1991), and refer- ences therein.
. M Brune, Phys. Rev. Lett. 774887M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996).
. D Vitali, P Tombesi, G J Milburn, Phys. Rev. Lett. 792442D. Vitali, P. Tombesi, G.J. Milburn, Phys. Rev. Lett. 79 2442 (1997);
. Phys. Rev. A. 574930Phys. Rev. A 57 4930 (1998).
. M Brune, Phys. Rev. Lett. 761800M. Brune et al., Phys. Rev. Lett. 76, 1800 (1996).
. D M Meekhof, Phys. Rev. Lett. 761796D.M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996).
. J M Raimond, private communicationJ.M. Raimond, private communication.
. D J Wineland, J. Res. Natl. Inst. Stand. Technol. 103259D.J. Wineland et al., J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).
LANL e-print archive quant-ph/9901063; in Mysteries, Puzzles and Paradoxes in Quantum Mechanics. R Bonifacio ; R. Bonifacio, Aip, Woodbury, pag. 122Nuovo Cimento 114B. 473R. Bonifacio, Nuovo Cimento 114B, 473 (1999); LANL e-print archive quant-ph/9901063; in Mysteries, Puz- zles and Paradoxes in Quantum Mechanics, edited by R. Bonifacio, AIP, Woodbury, 1999, pag. 122.
. I S Gradshteyn, I M Ryzhik, pag. 317Table of Integrals and Series. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals and Series, Academic, Orlando, 1980, pag. 317.
V (t), depending on two parameters τ1 and τ2 is derived. In Ref, We choose τ1 = τ2 = τ because in the experiments considered here, the effective interaction time t ′ = tτ1/τ2 [8] has to coincide with the "laboratory time" t, implying therefore τ1 = τ2In Ref. [8] a more general expression for P (t, t ′ ) and V (t), depending on two parameters τ1 and τ2 is derived. We choose τ1 = τ2 = τ because in the experiments considered here, the effective interaction time t ′ = tτ1/τ2 [8] has to coincide with the "laboratory time" t, implying therefore τ1 = τ2.
D F Walls, G J Milburn, Quantum Optics. BerlinSpringerD.F. Walls and G.J. Milburn, Quantum Optics, Springer, Berlin, 1996.
E T Jaynes, F W Cummings, Proc. IEEE. IEEE5189E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 89 (1963).
. S Schneider, G J Milburn, Phys. Rev. A. 573748S. Schneider and G.J. Milburn, Phys. Rev. A 57, 3748 (1998).
. M Murao, P L Knight, Phys. Rev. A. 58663M. Murao, P.L. Knight, Phys. Rev. A 58, 663 (1998).
. D Wineland, private communicationD. Wineland, private communication.
| [] |
[
"A NEW FAMILY OF EXCEPTIONAL POLYNOMIALS IN CHARACTERISTIC TWO",
"A NEW FAMILY OF EXCEPTIONAL POLYNOMIALS IN CHARACTERISTIC TWO"
] | [
"Robert M Guralnick ",
"ANDJoel E Rosenberg ",
"Michael E Zieve "
] | [] | [] | We produce a new family of polynomials f (X) over fields k of characteristic 2 which are exceptional, in the sense that f (X) − f (Y ) has no absolutely irreducible factors in k[X, Y ] except for scalar multiples of X −Y ; when k is finite, this condition is equivalent to saying that the map α → f (α) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2 e−1 (2 e − 1), where e > 1 is odd. We also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. | 10.4007/annals.2010.172.1361 | [
"https://arxiv.org/pdf/0707.1837v2.pdf"
] | 14,331,589 | 0707.1837 | a5b1fce4268decc0ef88805264eb9f60dd260021 |
A NEW FAMILY OF EXCEPTIONAL POLYNOMIALS IN CHARACTERISTIC TWO
8 May 2008
Robert M Guralnick
ANDJoel E Rosenberg
Michael E Zieve
A NEW FAMILY OF EXCEPTIONAL POLYNOMIALS IN CHARACTERISTIC TWO
8 May 2008
We produce a new family of polynomials f (X) over fields k of characteristic 2 which are exceptional, in the sense that f (X) − f (Y ) has no absolutely irreducible factors in k[X, Y ] except for scalar multiples of X −Y ; when k is finite, this condition is equivalent to saying that the map α → f (α) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2 e−1 (2 e − 1), where e > 1 is odd. We also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic.
Introduction
Let k be a field of characteristic p ≥ 0, let f (X) ∈ k[X]\k, and let k be an algebraic closure of k. A polynomial in k[X, Y ] is called absolutely irreducible if it is irreducible in k[X, Y ]. We say f is exceptional if f (X) − f (Y ) has no absolutely irreducible factors in k[X, Y ] except for scalar multiples of X − Y . If k is finite, this condition is equivalent to saying that the map α → f (α) induces a bijection on an infinite algebraic extension of k [3,6]. Via this property, exceptional polynomials have been used to construct remarkable examples of various types of objects: curves whose Jacobians have real multiplication [33], Galois extensions of number fields with group PSL 2 (q) [5], maximal curves over finite fields [2,28], families of character sums with small average value [6], difference sets [9,11], binary sequences with ideal autocorrelation [9], almost perfect nonlinear power functions [13,14,10], bent functions [35,11], and double-error correcting codes [10].
Trivially any linear polynomial is exceptional. The simplest nontrivial examples are the multiplicative polynomials X d (which are exceptional when k contains no d-th roots of unity except 1) and the additive polynomials α i X p i (which are exceptional when they have no nonzero root in k). Dickson [7] showed that certain variants of these polynomials are also exceptional in some situations: the Dickson polynomials D d (X, α) (with α ∈ k), which are defined by D d (Y + α/Y, α) = Y d + (α/Y ) d ; and the subadditive polynomials S(X), which satisfy S(X m ) = L(X) m with L an additive polynomial and m a positive integer. For nearly 100 years, the only known exceptional polynomials were compositions of these classical examples.
We thank the referee for useful advice on notation. The first author was partially supported by NSF grant DMS 0653873.
Klyachko [21] showed that compositions of these polynomials yield all exceptional polynomials of degree not divisible by p, and also all exceptional polynomials of degree p. A vast generalization of this result was proved by Fried, Guralnick and Saxl [15], which greatly restricted the possibilities for the monodromy groups of exceptional polynomials. We recall the relevant terminology: let x be transcendental over k. We say f (X) ∈ k[X] \ k is separable if the field extension k(x)/k(f (x)) is separable, or equivalently f ′ (X) = 0. For a separable f (X) ∈ k[X], let E be the Galois closure of k(x)/k(f (x)). The arithmetic monodromy group of f (over k) is Gal(E/k(f (x))); the geometric monodromy group of f is Gal(E/ℓ(f (x))), where ℓ is the algebraic closure of k in E. If k is finite, then the composition b • c of two polynomials b, c ∈ k[X] is exceptional if and only if both b and c are exceptional [6]. Thus, the study of exceptional polynomials over finite fields reduces to the case of indecomposable polynomials, i.e., polynomials which are not compositions of lower-degree polynomials. For extensions of these results to infinite fields and to maps between other varieties, see [18,19,22,24]. Fried, Guralnick and Saxl proved the following result about the monodromy groups of an indecomposable exceptional polynomial [15,18]: Theorem 1.1. Let k be a field of characteristic p, and let f (X) ∈ k[X] be separable, indecomposable, and exceptional of degree d > 1. Let A be the arithmetic monodromy group of f . Then one of the following holds.
(i) d = p is prime, and A is solvable.
(ii) d = p e and A has a normal elementary abelian subgroup V of order p e . (iii) p ∈ {2, 3}, d = p e (p e − 1)/2 with e > 1 odd, and A ∼ = PΓL 2 (p e ) = PGL 2 (p e ) ⋊ Gal(F p e /F p ).
It remains to determine the polynomials corresponding to these group theoretic possibilities. Case (i) is completely understood: up to compositions with linear polynomials, one just gets the Dickson polynomials D d (X, α) (see [26,Appendix] or [21]). In case (ii), we have G = V G 1 for some G 1 ; this case includes the additive polynomials (where G 1 = 1) and the subadditive polynomials (where G 1 is cyclic). In joint work with Müller [16,17], we have found families of case (ii) examples in which G 1 is dihedral [16,17]. Moreover, in all known examples in case (ii), the fixed field E V has genus zero; conversely, we show in [17] that there are no further examples in which E V has genus zero or one. We suspect there are no other examples in case (ii): for if E V has genus g > 1 then G 1 will be a group of automorphisms of E V whose order is large compared to g, and there are not many possibilities for such a field E V . We hope to complete the analysis of case (ii) in a subsequent paper. The present paper addresses case (iii).
In the two years following [15], examples were found in case (iii) for each p ∈ {2, 3} and each odd e > 1 [4,23,25]. In the companion paper [20], we show that twists of these examples comprise all examples in case (iii), except possibly in the following situation: p = 2, G = SL 2 (2 e ), and the extension k(x)/k(f (x)) is wildly ramified over at least two places of k(f (x)). In the present paper we conclude the treatment of case (iii) by handling this final ramification setup. In particular, we find a new family of exceptional polynomials. Our main result is the following, in which we say polynomials b, c ∈ k[X] are k-equivalent if there are linear polynomials ℓ 1 , ℓ 2 ∈ k[X] such that b = ℓ 1 • c • ℓ 2 : Theorem 1.2. Let k be a field of characteristic 2. Let q = 2 e > 2. For
α ∈ k \ F 2 , define f α (X) := T(X) + α X q · T(X) + T(X) + α α + 1 · T X(α 2 + α) (T(X) + α) 2 ,
where T(X) = X q/2 + X q/4 + · · · + X. Then the map α → f α defines a bijection from k \ F 2 to the set of k-equivalence classes of separable polynomials
f ∈ k[X] of degree q(q − 1)/2 satisfying (i) the geometric monodromy group of f is SL 2 (q); and (ii) the extension k(x)/k(f (x)) is wildly ramified over at least two places of k(f (x)). Every f α is indecomposable. Moreover, f α is exceptional if and only if e is odd and k ∩ F q = F 2 .
The strategy of our proof is to identify the curve C corresponding to the Galois closure E of k(x)/k(f (x)), for f a polynomial satisfying (i) and (ii). It turns out that C is geometrically isomorphic to the smooth plane curve y q+1 + z q+1 = T(yz) + α.
A key step in our proof is the computation of the automorphism groups of curves of the form v q + v = h(w), with h varying over a two-parameter family of rational functions. Our method for this computation is rather general, and applies to many families of rational functions h.
As noted above, Theorem 1.2 completes the classification of non-affine indecomposable exceptional polynomials: Corollary 1.3. Let k be a field of characteristic p ≥ 0. Up to k-equivalence, the separable indecomposable exceptional polynomials over k which lie in cases (i) or (iii) of Theorem 1.1 are precisely:
(i) for any p, the polynomial X d where d = p is prime and k contains no d-th roots of unity except 1; (ii) for any p, the polynomial
D d (X, α) := ⌊d/2⌋ i=0 d d − i d − i i (−α) i X d−2i
where d = p is prime, α ∈ k * , and k contains no elements of the form ζ + 1/ζ with ζ being a primitive d-th root of unity in k; (iii) for p = 2 and q = 2 e > 2 with e odd and k ∩ F q = F 2 , the polynomial f α (X) where α ∈ k \ F 2 ;
(iv) for p = 2 and q = 2 e > 2 with e odd and k ∩ F q = F 2 , the polynomial
X e−1 i=0 (αX n ) 2 i −1 (q+1)/n
where n divides q + 1 and α ∈ k * ; (v) for p = 3 and q = 3 e > 3 with e odd and k ∩ F q = F 3 , the polynomial
X(X 2n − α) (q+1)/(4n) (X 2n − α) (q−1)/2 + α (q−1)/2 X 2n (q+1)/(2n)
where n divides (q + 1)/4 and α ∈ k * has image in k * /(k * ) 2n of even order.
The contents of this paper are as follows. In the next section we prove some useful results about ramification groups. In Section 3 we record results from [20] which describe the ramification (including the higher ramification groups) in E/k(f (x)). In Section 4 we classify curves which admit B, the group of upper triangular matrices in SL 2 (q), as a group of automorphisms with our desired ramification configuration. There is a two-parameter family of such curves. In Sections 5 and 6 we determine the automorphism groups of the curves in this family (which turn out to be either B or SL 2 (q)). The curves with automorphism group SL 2 (q) form a one-parameter subfamily. The group theoretic data yields the existence and uniqueness of the desired polynomials. In particular, it shows we cannot have k = F 2 ; in Section 9 we give a different, more direct proof of this fact. In the final two sections we consider different forms of the curves, and in particular we determine a smooth plane model. We then use this model to explicitly compute the polynomials, and we conclude the paper by proving Theorem 1.2 and Corollary 1.3.
Notation. Throughout this paper, all curves are assumed to be smooth, projective, and geometrically irreducible. We often define a curve by giving an affine plane model, in which case we mean the completion of the normalization of the stated model. Also, in this case we describe points on the curve by giving the corresponding points on the plane model.
A cover is a separable nonconstant morphism between curves. If ρ : C → D is a cover of curves over a field k, by a 'branch point' of ρ we mean a point of D(k) which is ramified in ρ × k k (for k an algebraic closure of k). In particular, branch points need not be defined over k, but the set of branch points is preserved by the absolute Galois group of k. If f ∈ k[X] is a separable polynomial, then we refer to the branch points of the corresponding cover f : P 1 → P 1 as the branch points of f .
If ρ : C → D is a Galois cover, and P is a point of C, then the ramification groups at P (in the lower numbering, as in [30]) are denoted I 0 (P ), I 1 (P ), . . ., or simply I 0 , I 1 , . . .. Here I 0 is the inertia group and I 1 is its Sylow psubgroup. We refer to I 1 as the first ramification group, I 2 as the second, and so on.
We reserve the letter x for an element transcendental over the field k. Throughout this paper we write q = 2 e where e > 1. We use the following notation for subgroups of SL 2 (q). The group of diagonal matrices is denoted T . The group of upper triangular matrices is denoted B. The group of elements of B with 1's on the diagonal is denoted U . The two-element group generated by 1 1 0 1 is denoted W . Finally, T(X) denotes the polynomial X q/2 + X q/4 + · · · + X.
Ramification in Galois p-power covers
In this section we prove a useful result (Corollary 2.2) about ramification groups in Galois covers of degree a power of the characteristic. We give two proofs, each of which provides additional information. Throughout this section, ℓ is an algebraically closed field of characteristic p > 0.
Proposition 2.1. Let ρ : C → D and ρ ′ : C → B be Galois covers of curves over ℓ. Let n and r be positive integers. Suppose that B ∼ = P 1 and the degree of ρ ′ is p r . If all n-th ramification groups of ρ ′ are trivial, then the same is true of ρ.
Proof. Since P 1 has no nontrivial unramified covers, and any ramified Galois cover of p-power degree has a nontrivial first ramification group, the hypotheses imply n ≥ 2. Without loss, we may assume ρ has degree p. First assume C has genus greater than 1, so that Aut C is finite. Let H be a Sylow p-subgroup of Aut C which contains Gal(ρ ′ ). By replacing ρ by one of its (Aut C)-conjugates, we may assume that H contains Gal(ρ) as well. Then the cover B → C/H induced from C → C/H and C → B is the composition of a sequence of Galois degree-p covers B = B 0 → B 1 → · · · → B m = C/H. Since B ∼ = P 1 , each B i → B i+1 has trivial second ramification groups (by Riemann-Hurwitz). Thus, each n-th ramification group of C → B i+1 is also an n-th ramification group of C → B i (by [30,Prop. IV.3]). By induction, C → B m = C/H has trivial n-th ramification groups, whence the same is true of C → D.
If C has genus 0, then ρ is a degree-p cover between genus-0 curves and hence has trivial second ramification groups.
Finally, assume C has genus 1. Pick a point of C with nontrivial inertia group under ρ ′ , and let J be an order-p subgroup of this inertia group. Then C/J ∼ = P 1 , so by replacing B by C/J we may assume ρ ′ has degree p. If p > 3 then no such ρ ′ exists (e.g., by Riemann-Hurwitz). If p = 3 then any Galois degree-p map C → D is either unramified (with D of genus 1) or has a unique branch point (with I 2 = I 3 and D of genus 0). Henceforth assume p = 2. Then a degree-p map C → D is either unramified (with D of genus 1) or has precisely two branch points (each with I 1 = I 2 and D of genus 0) or has a unique branch point (with I 3 = I 4 and D of genus 0). If there is a unique branch point then C is isomorphic to the curve y 2 + y = z 3 . Since the corresponding elliptic curve has trivial 2-torsion, it follows that a degree-2 function on this curve cannot have two branch points. This completes the proof.
We will use the following result, which follows from Proposition 2.1 and standard results about ramification groups (cf. [30,§IV2]).
Corollary 2.2. Let ρ : C → D and ρ ′ : C → P 1 be Galois covers of curves over ℓ. Suppose that ρ ′ has degree a power of p, and that all second ramification groups of ρ ′ are trivial. If I 1 and I 2 are the first and second ramification groups of ρ at some point of C, then I 1 is elementary abelian, I 2 = 1, and I 1 is its own centralizer in I.
We now give a different proof of this corollary, which generalizes the corollary in a different direction than does Proposition 2.1. For a curve C over ℓ, let p C denote the p-rank of C (i.e., the rank of the p-torsion subgroup of the Jacobian of C). Let g C denote the genus of C. These quantities are related by p C ≤ g C . Recall that C is called ordinary if p C = g C . We first record a standard basic fact.
p C − 1 |V | = p D − 1 + Q∈D 1 − 1 e Q ,
where e Q is the ramification index of θ at the point Q.
The Riemann-Hurwitz formula yields
g C − 1 |V | = g D − 1 + Q∈D 1 − 1 e Q + s,
where s is the contribution from the second and higher ramification groups. Note that s ≥ 0, with equality if and only if all second ramification groups are trivial. Since p D ≤ g D , we conclude that p C = g C holds if and only if p D = g D and s = 0.
Alternate proof of Corollary 2.2. By applying the previous result with θ = ρ ′ , we see that C is ordinary. Applying it with θ = ρ shows that I 2 = 1, and then the remaining assertions follow from standard properties of the higher ramification groups.
Remark. It would be interesting to refine the above alternate proof of Corollary 2.2 to prove Proposition 2.1. Such a refinement would likely require a refinement of the Deuring-Shafarevich formula that involves finer invariants than just the p-rank. However, we do not know such a refined formula. We thank Hendrik Lenstra for suggesting this possibility.
Previous results
We will use the following result from the companion paper [20,Lemma 2.7]. Recall our convention that x is transcendental over k; also B is the group of upper triangular matrices in SL 2 (q), and W = B ∩ SL 2 (2). Lemma 3.1. Let k be a perfect field of characteristic 2, and let q = 2 e with e > 1. Suppose f ∈ k[X] is a separable polynomial of degree q(q − 1)/2 which satisfies conditions (i) and (ii) of Theorem 1.2. Let E be the Galois closure of k(x)/k(f (x)), and let ℓ be the algebraic closure of k in E. Then E/ℓ(f (x)) has precisely two ramified places, both of degree one, and the corresponding inertia groups are B and W (up to conjugacy). Moreover, the second ramification group over each ramified place is trivial, and f is indecomposable. The degree [ℓ : k] divides e, and f is exceptional if and only if e is odd and [ℓ : k] = e. Finally, there is a curve C 0 over k such that ℓ.k(C 0 ) ∼ = ℓ E.
The following consequence of Lemma 3.1 describes the ramification in C → C/B, where C = C 0 × k ℓ. This too was proved in the companion paper [20,Cor. 2.8]. Here T is the group of diagonal matrices in SL 2 (q).
Corollary 3.2. If C is a curve over ℓ for which ℓ(C) = E, then the following hold:
(i) B acts as a group of ℓ-automorphisms on C;
(ii) the quotient curve C/B has genus zero; (iii) the cover C → C/B has exactly three branch points;
(iv) the inertia groups over these branch points are B, T , and W (up to conjugacy); and (v) all second ramification groups in the cover C → C/B are trivial.
We now record some standard facts about subgroups of SL 2 (q); see for instance [8, §260], [32, §3.6], or [20,App.]. Here U is the group of elements of B whose diagonal entries are 1.
(q). The normalizer of W in SL 2 (q) is U .
There is no group strictly between B and SL 2 (q).
B-curves
Let ℓ be a perfect field of characteristic 2. In this section we describe the curves C over ℓ which admit a B-action as in Corollary 3.2. We will show that the only such curves C are the curves C α,β defined as follows. For any α, β ∈ ℓ * , let C α,β be the curve defined by
(4.1) v q + v = (α + β)w + w q T β 1 + w q−1 ,
where T(X) := X q/2 + X q/4 + · · · + X. Note that C α,β is geometrically irreducible, since the left side of (4.1) is a polynomial in v and the right side is a rational function in w with a simple pole (at w = ∞, with residue α).
Theorem 4.2. Suppose C is a curve over ℓ satisfying the five properties in Corollary 3.2. Then ℓ ⊇ F q and C ∼ = C α,β for some α, β ∈ ℓ * .
Indeed, suppose C satisfies the properties of Corollary 3.2. Since the inertia groups B, T , and W are not conjugate, the corresponding branch points are ℓ-rational, so for a suitably chosen coordinate t on C/B they are ∞, 0, and 1, respectively. Note that C/U → C/B is a cyclic cover of degree q − 1 which is totally ramified over ∞ and 0, and unramified elsewhere. By Riemann-Hurwitz, C/U has genus zero. Each of the q − 1 order-2 subgroups of U is conjugate to W , and is thus an inertia group in C → C/B, hence also in C → C/U . Thus there must be at least q − 1 distinct places of C/U lying over the place t = 1 of C/B, so all of these places must be rational. Choose a coordinate w on C/U such that, in the cover C/U → C/B, the points ∞, 0, and 1 map to ∞, 0, and 1, respectively. Then ℓ(C/U ) = ℓ(w) and
ℓ(C/B) = ℓ(t) where t = w q−1 . Since C/U → C/B is Galois, ℓ contains F q .
In these coordinates, the branch points of the cover C → C/U are ∞ and the q − 1 elements of F * q (i.e., the points over t = 1) with the corresponding inertia groups being U and its q − 1 subgroups of order 2.
Let
C 1 = C/H, where H is a maximal subgroup of U .
Since C → C/U has no nontrivial second ramification groups, the same is true of
C 1 → C/U , so (since ℓ is perfect) C 1 is defined by an equation of the form (4.3) y 2 + y = αw + ζ∈F * q β ζ ζ w + ζ + γ
for some y ∈ ℓ(C) and α, β ζ , γ ∈ ℓ. Note that α = 0 (since w = ∞ is a branch point). Clearly β ζ = 0 if and only if w = ζ is a branch point of the cover C 1 → C/U , and the latter holds if and only if H does not contain the inertia group of w = ζ in C → C/U . Thus, β ζ is nonzero for precisely q/2 values ζ.
Let Γ be the set of elements z ∈ ℓ(C) for which
z 2 + z = α(z)w + ζ∈F * q β ζ (z)ζ w + ζ + γ(z) with α(z), β ζ (z), γ(z) ∈ ℓ.
Note that α(z), β λ (z), and γ(z) are uniquely determined by z, and each of them defines a homomorphism Γ → ℓ. Let Γ 0 = Γ ∩ ℓ(w); considering orders of poles, we see that Γ 0 = ℓ.
Since B = U T , restriction to C/U induces an isomorphism T ∼ = B/U , so T = {φ η : η ∈ F * q } where φ η (w) = ηw. Clearly Γ is T -invariant.
The following lemma enables us to choose y so that T y ∪ {0} is a group. Proof. The map θ : z + Γ 0 → ℓ(w, z) defines a surjective T -set homomorphism between Γ/Γ 0 \ {0} and the set Λ of degree-2 extensions of ℓ(C/U ) contained in ℓ(C). We first prove injectivity of θ:
suppose z 1 , z 2 ∈ Γ \ Γ 0 sat- isfy ℓ(w, z 1 ) = ℓ(w, z 2 ). Then the nonidentity element of Gal(ℓ(w, z 1 )/ℓ(w)) maps z 1 → z 1 + 1 and z 2 → z 2 + 1, hence fixes z 1 + z 2 , so z 1 + z 2 ∈ Γ 0 = ℓ.
Hence θ is injective. Since Λ is a transitive T -set of size q − 1, it follows that |Γ/Γ 0 | = q and T acts transitively on Γ/Γ 0 \{0}. Finally, since |T | is odd and both Γ and Γ 0 are T -invariant elementary abelian 2-groups, Maschke's theorem ([1, 12.9]) implies there is a T -invariant group Γ 1 such that Γ = Γ 0 ⊕ Γ 1 , and
|Γ 1 | = |Γ/Γ 0 | = q.
By replacing y by y + δ for some δ ∈ Γ 0 , we may assume that y is in Γ 1 . Applying φ η to (4.3), we see that y η := φ η (y) satisfies
y 2 η + y η = αηw + ζ∈F * q β ζ η −1 ζ w + η −1 ζ + γ.
Thus, γ(y η ) = γ and α(y η ) = αη and β η −1 ζ (y η ) = β ζ . Since the homomorphism z → γ(z) is constant on the nonzero elements of the group Γ 1 , it follows that γ = 0.
Since Γ 1 = {y η } ∪ {0}
is closed under addition, y η + y η ′ = y η ′′ for some η ′′ . Comparing images under α yields that
y η + y η ′ = y η+η ′ . Thus, β ζ + β η = β 1 (y ζ ) + β 1 (y η ) = β 1 (y ζ+η ) = β ζ+η .
Since β ζ = 0 for exactly q/2 − 1 choices of ζ ∈ F * q , this implies that β ζ = 0 for ζ in some hyperplane (i.e., index-2 subgroup) H of F q , and β ζ = β ζ ′ for ζ, ζ ′ ∈ H. Hence, β ζ (y η ) = 0 for ζ ∈ η −1 H. The hyperplanes η −1 H comprise all q − 1 hyperplanes in F q , so there is some η for which η −1 H is the set of roots of T(X) := X q/2 + X q/4 + · · · + X. Replacing y by y η , the equation for C 1 becomes
y 2 + y = αw + β ζ∈F * q T(ζ)ζ w + ζ .
Note that α and β are nonzero elements of ℓ.
Since ℓ(C) is the Galois closure of ℓ(C 1 )/ℓ(w q−1 ), it is uniquely determined by the choice of α and β. Thus, to conclude the proof of Theorem 4.2, it suffices to show that (for each choice of α, β ∈ ℓ * ) the curve C α,β satisfies the hypotheses of the theorem, and that the quotients of C α,β by B and by some order-q/2 subgroup induce the above cover C 1 → P 1 w q−1 . The following lemma is clear:
Lemma 4.5. If ℓ contains F q (α, β), then for any γ −1 δ 0 γ ∈ B there is a unique ℓ-automorphism of C α,β mapping w → γ 2 w and v → γ 2 v + γδ. This correspondence defines an embedding B ֒→ Aut ℓ (C α,β ).
We now show that C α,β (together with this action of B) has the desired properties.
Lemma 4.6. The curve C := C α,β has genus q(q − 1)/2. Moreover, the fixed fields ℓ(C) U and ℓ(C) B equal ℓ(w) and ℓ(w q−1 ), and the cover C → C/B has precisely three branch points. The inertia groups over these points are (up to conjugacy) B, T , and W . Also, the second ramification groups at all three points are trivial.
Finally, if H = { 1 δ 0 1 : T(δ) = 0}, then ℓ(C) H = ℓ(w, y) where (4.7) y 2 + y = αw + β ζ∈F * q T(ζ)ζ w − ζ .
Proof. It is clear that ℓ(C) U = ℓ(w) and ℓ(C) B = ℓ(t), where t := w q−1 . Also, both w and y := T(v) are fixed by H, and a straightforward calculation yields
y 2 + y = (α + β)w + w q T β t + 1 = αw + β ζ∈F * q T(ζ)ζ w − ζ + h + h 2
for an appropriate h ∈ ℓ(w). Thus, H fixes w and y := y + h, and y satisfies (4.7). Since y / ∈ ℓ(w), it follows that ℓ(C) H = ℓ(w, y). Note that the genus of ℓ(w, y) is q/2, since the right hand side of (4.7) has precisely 1 + q/2 poles and they are all simple.
Let D = C/U and B = C/B, so ℓ(D) = ℓ(w) and ℓ(B) = ℓ(t). The cover D → B is only ramified at w = 0 and w = ∞, and is totally ramified at both of these points. The cover C → D can only be ramified at points with v = ∞, hence at points with w ∈ F * q or w = ∞. The point w = ∞ of D is totally ramified in C → D, since w is a simple pole of the right hand side of (4.1). The points w ∈ F * q of D all lie over the point t = 1 of B, and precisely q/2 of these points are ramified in C/H → D. Since T permutes transitively both the q − 1 points in D over t = 1 and the q − 1 index-2 subgroups of U , we see that each such point ramifies in precisely q/2 of the covers C/V → D as V ranges over the q − 1 index-2 subgroups of U . This implies that each w ∈ F * q has ramification index 2 in C → D. Thus, the only branch points of the cover C → C/B are ∞, 0, and 1, and the corresponding ramification indices are q(q − 1), q − 1, and 2. Hence, up to conjugacy, the corresponding inertia groups are B, T , and W . Moreover, since the second ramification groups of C/H → D are trivial, the same is true of every C/V → D, and hence of C → D. It follows from Riemann-Hurwitz that C has genus q(q − 1)/2.
This concludes the proof of Theorem 4.2.
Automorphism groups of B-curves
Let ℓ be an algebraically closed field of characteristic 2, let α, β ∈ ℓ * , and put C := C α,β as in (4.1). By Lemma 4.6, C admits an action of B satisfying the five properties of Corollary 3.2. In this section we prove that the automorphism group of C is either B or SL 2 (q). Let P 1 , P 2 , and P 3 be points of C whose stabilizers (in B) are B, W , and T , respectively. Let G be the automorphism group of C.
Lemma 5.1. Let V ≤ U be a subgroup with |V | > 2. Then N G (V ) ≤ B and |G : B| is odd. Moreover, B is the stabilizer of P 1 in G.
Proof. Let B be the stabilizer of P 1 in G, and let U be the Sylow 2-subgroup of B. Corollary 2.2 implies that U is elementary abelian and that C → C/U has trivial second ramification groups.
Write |U | = qq. Since the q − 1 order-2 subgroups of U are all conjugate under B, they are all inertia groups in C → C/U . These subgroups are nonconjugate in the abelian group U , so C → C/U has at least q − 1 distinct branch points not lying under P 1 . By Riemann-Hurwitz, 2(qq + q(q − 1)/2 − 1) = Q ind(Q) where Q varies over the branch points of C → C/U and ind(Q) is the sum of the different exponents (in the cover C → C/U ) of the points over Q. If Q lies under P 1 , then Q is totally ramified so ind(Q) = 2(qq −1). Any branch point satisfies ind(Q) ≥ qq (since C → C/U is a Galois cover with Galois group a 2-group). Thus,
2 qq + q(q − 1) 2 − 1 ≥ 2(qq − 1) + (q − 1)qq, or q(q − 1) ≥ (q − 1)qq. Hence q = 1, so U = U .
By Corollary 2.2, U is its own centralizer in B, so conjugation induces a faithful action of B/U on U and thus also on
U \ {0}. Since B/U is cyclic, it follows that |B/U | ≤ |U \ {0}| = |B/U |, so B = B.
Since we know the inertia groups of C → C/B, we see that P 1 is the only point of C fixed by V . Thus, N G (V ) fixes P 1 , so N G (V ) ≤ B. In particular, In this case C/W is dihedral of order 6, so (since C contains U ) the group C is dihedral of order 12.
N G (U ) = N B (U ) = B. If U is
Let T ′ be the order-3 subgroup of C. Since T ′ is normal in C, no subgroup of C properly containing T ′ can be an inertia group in C → C/C (by Corollary 2.2). Thus, every orbit of C/T ′ on the set Γ of fixed points of T ′ is regular, so |Γ| is divisible by 4. Since T fixes precisely two points of C, it follows that T ′ and T are not conjugate in G, so a Sylow 3-subgroup of G is noncyclic, and thus contains an elementary abelian subgroup of order 9 [1, 23.9]. By Lemma 4.6, C has genus q(q − 1)/2 = 6. But Riemann-Hurwitz shows that (in characteristic not 3) an elementary abelian group of order 9 cannot act on a genus-6 curve, contradiction.
Theorem 5.4. If G = B then G = SL 2 (q) and C → C/G has precisely two branch points, with inertia groups B and W .
Proof. Assume that G = B. Consider the cover C → C/G. By Lemmas 5.1 and 5.3, the inertia groups of P 1 and P 2 in this cover are B and W , respectively. Since these groups are nonconjugate, P 1 and P 2 lie over distinct branch points Q 1 and Q 2 . By Lemma 4.6 and Corollary 2.2, every branch point of C → C/G has trivial second ramification group.
For a point Q of C/G, let ind(Q) denote the sum of the different exponents (in the cover C → C/G) of the points lying over Q. Note that ind(Q 1 )/|G| = 1 − 2/|B| + |U |/|B| = 1 + (q − 2)/|B| and ind(Q 2 ) = |G|. The Riemann-Hurwitz formula gives
q(q − 1) − 2 = −2|G| + ind(Q 1 ) + ind(Q 2 ) + Q / ∈{Q 1 ,Q 2 } ind(Q) = (q − 2)|G : B| + Q ind(Q);
since any branch point Q satisfies ind(Q) ≥ 2|G|/3 > q(q − 1), it follows that Q 1 and Q 2 are the only branch points in C → C/G, and we must have |G : B| = q + 1. By Lemma 5.2, T has index 2 in H := N G (T ). Thus H preserves the set {P 1 , P 3 } of fixed points of T . Lemma 5.2 implies P 1 ∈ GP 3 , so |GP 3 | = |GP 1 | = |G : B| = q + 1. Since |BP 3 | = q, it follows that GP 3 = BP 3 ∪ {P 1 }. Pick an involution ν ∈ H. If ν fixes P 1 then Lemma 5.1 implies ν ∈ B; but ν must also fix P 3 , which is impossible since the stabilizer of P 3 in B is T (and T contains no involutions). Thus ν must swap P 1 and P 3 . By Lemma 5.1, U is a Sylow 2-subgroup of G; since all involutions of U are conjugate in B, it follows that ν is conjugate in G to the nonidentity element of W , and thus fixes a unique point of GP 1 .
The orbits of B on Λ := GP 1 are the fixed point P 1 and the q-element orbit BP 3 . Since B has a unique conjugacy class of index-q subgroups, this determines Λ as a B-set. The same orbit sizes occur in the action of B on P 1 (F q ) induced by the usual action of PSL 2 (q) on P 1 (F q ). Thus, Λ and P 1 (F q ) are isomorphic B-sets. We will show below that, up to T -conjugacy, there is a unique involution in the symmetric group of Γ which normalizes T and has a unique fixed point. Since SL 2 (q) contains such an involution, we can extend our isomorphism of B-sets Λ ∼ = B P 1 (F q ) to an isomorphism of B, ν -sets, and in particular SL 2 (q) has a subgroup isomorphic to B, ν . Since B is a maximal subgroup of SL 2 (q), we have B, ν ∼ = SL 2 (q), whence (since |G| ≤ | SL 2 (q)|) we conclude G ∼ = SL 2 (q).
It remains to show that, up to T -conjugacy, there is a unique involution ν in the symmetric group of Λ which normalizes T and has a unique fixed point. Note that T fixes P 1 and P 3 , and T is transitive on the other q − 1 points of Λ. Thusν permutes {P 1 , P 3 }, and the fixed point hypothesis impliesν interchanges P 1 and P 3 . Henceν fixes a unique point of T P 3 , so we may may identify this orbit with T and assume the fixed point is 1 ∈ T . The only order-2 automorphism of T with no nontrivial fixed points is the automorphism inverting all elements of T , whenceν is unique up to T -conjugacy.
G-curves and hyperelliptic quotients
Let ℓ be an algebraically closed field of characteristic 2, let α, β ∈ ℓ * , and let C := C α,β be as in (4.1). We use the embedding B → Aut C from Lemma 4.5. By Theorem 5.4, the automorphism group of C is either B or G := SL 2 (q). In this section we determine when the latter occurs. Set t := w q−1 and y := v/w. Since T fixes t and y, we have ℓ(t, y) ⊆ ℓ(v, w) T = ℓ(C/T ). Clearly w has degree at most q − 1 over ℓ(t, y), and also ℓ(v, w) = ℓ(y, w). Thus, ℓ(C/T ) = ℓ(t, y).
The curve C/T is defined by the equation
y q + y t = α + β t + T β t + 1 , which is irreducible because [ℓ(y, t) : ℓ(t)] = q. Putting z := y 2 + y + β t + 1 , we compute T(z) = y q + y + T β t + 1 = y 1 + 1 t + α + β t ,
and thus y = t T(z) + α + β t + 1 .
It follows that
Lemma 6.2. ℓ(C/T ) = ℓ(t, z) and ℓ(C) = ℓ(w, z).
Our next result gives further information about C/T . Lemma 6.3. C/T is hyperelliptic of genus q/2, and the hyperelliptic involution ν fixes z and maps t → (α 2 + α + β 2 + z)/(z q t).
Proof. Substituting our expression for y (in terms of t and T(z)) into the definition of z gives
z = (t T(z) + α + β) 2 + (t T(z) + α + β)(t + 1) + β(t + 1) (t + 1) 2 ,
so 0 = t 2 z q + t(T(z) + α) + (z + α 2 + α + β 2 ). By considering the order of the pole at the point z = ∞ in this equation, we see that t / ∈ ℓ(z). Thus, [ℓ(t, z) : ℓ(z)] = 2. Our hypothesis on the ramification in C → C/B implies that ℓ(t, z) has genus q/2. Hence C/T is hyperelliptic, and the hyperelliptic involution ν fixes z and maps t → (α 2 + α + β 2 + z)/(z q t).
Suppose in this paragraph that Aut ℓ (C) ∼ = G, and choose the isomorphism so that it extends our previous embedding B ֒→ Aut ℓ (C). By Theorem 5.4, there are points P 1 , P 2 on C whose stabilizers in G are B and W , respectively, and moreover the corresponding points Q 1 , Q 2 on C/G are the only two branch points of C → C/G. By Lemma 5.2, H := N G (T ) has order 2(q − 1), so Lemma 3.3 implies H is dihedral, hence contains q −1 involutions. But all involutions in G are conjugate, and each fixes q/2 points of GP 2 , so C/T → C/H is ramified over q/2 points lying over Q 2 . Likewise, C/T → C/H is ramified over a unique point lying over Q 1 , so C/T → C/H has 1+q/2 branch points and thus (since C/T has genus q/2) we find that C/H has genus zero. By uniqueness of the hyperelliptic involution, we must have ℓ(C) H = ℓ(z), and each element µ ∈ H \ T is an involution whose restriction to C/T is the hyperelliptic involution ν. Now, (wµ(w)) q−1 = tρ(t) = (α 2 +α+β 2 +z)/z q is in ℓ(z), so ℓ(wµ(w), z)/ℓ(z) is cyclic of order dividing q − 1; but the dihedral group of order 2(q − 1) has no proper normal subgroups of even order, so wµ(w) ∈ ℓ(z). Thus (α 2 + α + β 2 + z)/z q is a (q − 1)-th power in ℓ(z), so
β 2 = α + α 2 .
Conversely, we now assume that β 2 = α 2 + α (with α / ∈ F 2 , since β = 0). By Lemma 6.2, there are precisely q − 1 extensions of ν to an embedding of ℓ(C) into its algebraic closure, one for each (q − 1)-th root of ρ(t) (this root will be ρ(w)). Since tρ(t) = 1/z q−1 , each of these extensions maps w → ζ/(zw) with ζ ∈ F * q and so in particular leaves ℓ(C) = ℓ(w, z) invariant (and thus is an automorphism of ℓ(C)). Since Aut ℓ (C) properly contains B, Theorem 5.4 implies that Aut ℓ (C) ∼ = SL 2 (q). This completes the proof of Proposition 6.1.
Forms of C α,β
In this section we study isomorphisms between curves of the shape C α,β , and isomorphisms between these curves and other curves.
Proposition 7.1. Let ℓ be an algebraically closed field of characteristic 2. For α, β, α ′ , β ′ ∈ ℓ * , the curves C α,β and C α ′ ,β ′ are isomorphic if and only if α = α ′ and β = β ′ .
Proof. Let C = C α,β and C ′ = C α ′ ,β ′ , and let G = Aut C and G ′ = Aut C ′ . Write the equations of C and C ′ as v q +v = (α+β)w+w q T(β/(1+w q−1 )) and Since B is the normalizer of U in both G and G ′ (by Lemma 5.1), it follows from θ(U ) = U that θ(B) = B. The only points of C/U which ramify in C/U → C/B are w = 0 and w = ∞, so ρ must map these to w ′ = 0 and w ′ = ∞ in some order. Thus, ρ(w) is a constant times either w ′ or 1/w ′ . Since also ρ preserves {δ : T(δ) = 1} ∪ {∞}, we must have ρ(w) = w ′ . Since θ(H) = H and the right hand side of (4.7) has only simple poles, by applying ρ to this equation we see that α = α ′ and β = β ′ . Proposition 7.2. Let k be a perfect field of characteristic 2, and let k be an algebraic closure of k. Let C = C α,β where α, β ∈ k * . Let C ′ be a curve over k which is isomorphic to C over k. Let ℓ be an extension of k such that Aut ℓ (ℓ(C ′ )) ∼ = Aut k (k(C ′ )). Then:
(v ′ ) q +v ′ = (α ′ +β ′ )w ′ +(w ′ ) q T(β ′ /(1+(w ′ ) q−1 )),(i) k contains F 2 (α, β); (ii) C is defined over k; and (iii) C is isomorphic to C ′ over ℓ. Proof. Note that k(C) = k(v, w) where v, w satisfy v q + v = (α + β)w + w q T β 1 + w q−1 .
If ρ is any k-automorphism of k(C), then k(C) = k(v 1 , w 1 ) where v 1 := ρ(v) and
w 1 := ρ(w) satisfy v q 1 + v 1 = (ρ(α) + ρ(β))w 1 + w q 1 T ρ(β) 1 + w q−1 1 .
Thus, C α,β ∼ = C ρ(α),ρ(β) , whence (by the previous result) ρ fixes α and β.
Hence F 2 (α, β) is fixed by the full group of k-automorphisms of k(C). By hypothesis, there is a k-isomorphism θ between k(C) and k(C ′ ). Conjugation by θ induces an isomorphism Aut k (k(C)) ∼ = Aut k (k(C ′ )), so in particular both of these groups fix the same subfield of k. Since k is perfect and C ′ is defined over k, the subfield of k fixed by Aut k (k(C ′ )) is just k, so
F 2 (α, β) ⊆ k.
Clearly C is defined over F 2 (α, β), hence over k. Finally, by Theorem 4.2 and Proposition 7.1 there is an ℓ-isomorphism ℓ(C) ∼ = ℓ(C ′ ).
Existence and uniqueness of polynomials
Let k be a perfect field of characteristic 2, and let q = 2 e > 2. In this section we prove a preliminary version of Theorem 1.2, in which we describe the Galois closure of k(x)/k(f (x)) rather than describing the polynomials f . Here x is transcendental over k, and we say b, c ∈ k[X] are k-equivalent if there are linear polynomials ℓ 1 ,
ℓ 2 ∈ k[X] such that b = ℓ 1 • c • ℓ 2 . Theorem 8.1. If f ∈ k[X]
is a separable polynomial of degree (q 2 − q)/2 such that (i) the geometric monodromy group of f is SL 2 (q); and (ii) the extension k(x)/k(f (x)) is wildly ramified over at least two places of k(f (x)), then there is a unique pair (α, β) ∈ k * × k * with β 2 = α + α 2 for which the Galois closure of k(x)/k(f (x)) is isomorphic to (k.F q )(C α,β ). Conversely, each such pair (α, β) actually occurs for some f with these properties, and two such polynomials are k-equivalent if and only if they correspond to the same pair (α, β). Finally, every such f is indecomposable, and f is exceptional if and only if e is odd and k ∩ F q = F 2 .
Our proof uses a corollary of the following simple lemma (cf. [12,Thm. 4
.2A]):
Lemma 8.2. Let G be a transitive permutation group on a set ∆, and let G 1 be the stabilizer of a point π ∈ ∆. Let C be the centralizer of G in the symmetric group on ∆. Then C ∼ = N G (G 1 )/G 1 , and C acts faithfully and regularly on the set of fixed points of G 1 . In particular, C is trivial if G 1 is self-normalizing in G.
Proof. Note that an element τ ∈ C is determined by the value τ (π) (since τ (ν(π)) = ν(τ (π)) for every ν ∈ G).
If G acts regularly on ∆, then we can identify the action of G on ∆ with the action of G on itself by left multiplication. Clearly right multiplication commutes with this action, so the map τ → τ (1) induces an isomorphism C ∼ = G, and C acts regularly on ∆.
Let Λ be the set of fixed points of G 1 . Then N G (G 1 )/G 1 acts regularly on Λ. LettingĈ be the centralizer of N G (G 1 ) in Sym(Λ), the previous paragraph shows thatĈ ∼ = N G (G 1 )/G 1 acts regularly on Λ. Since C acts on Λ and C centralizes N G (G 1 ), restriction to Λ induces a homomorphism θ : C →Ĉ. We see that θ is injective, since τ ∈ C is determined by τ (π). It remains only to prove that θ is surjective. For µ ∈Ĉ, ν ∈ G and λ ∈ G 1 , note that ν(λ(µ(π))) = ν(µ(λ(π))) = ν(µ(π)); hence the image of µ(π) is constant on each coset in G/G 1 , so the map ν(π) → ν(µ(π)) defines a permutation φ of ∆. Plainly φ centralizes G and θ(φ) = µ, so the proof is complete.
Corollary 8.3. Let f ∈ k[X]
be a separable polynomial, let E be the Galois closure of k(x)/k(f (x)), and let ℓ be the algebraic closure of k in E. Put A := Gal(E/k(f (x))), G := Gal(E/ℓ(f (x))), and G 1 := Gal(E/ℓ(x)). If
N G (G 1 ) = G 1 , then C A (G) = 1. Proof of Theorem 8.1. Suppose f ∈ k[X]
is a separable polynomial of degree (q 2 − q)/2 which satisfies conditions (i) and (ii) of Theorem 8.1. Let E be the Galois closure of k(x)/k(f (x)), and let ℓ be the algebraic closure of k in E. Then there is an ℓ-isomorphism between E and ℓ(C α,β ) for some α, β ∈ ℓ * , and also ℓ ⊇ F q (by Corollary 3.2 and Theorem 4.2). This uniquely determines the pair (α, β) (Proposition 7.1). By Theorem 5.4, the geometric monodromy group G := Gal(E/ℓ(f (x))) equals Aut ℓ ℓ(C α,β ), so Proposition 6.1 implies β 2 = α 2 + α. By Lemma 3.1 and Proposition 7.2, both α and β are in k. By Lemma 3.3, the hypotheses of the above corollary are satisfied, so no nontrivial element of Gal(E/k(f (x))) centralizes G. Since every ℓautomorphism of ℓ(C α,β ) is defined over k.F q , we see that G commutes with Gal(E/(k.F q )(C α,β )), so L = (k.F q )(C α,β ). We have proven the first sentence of Theorem 8.1.
Conversely, suppose α, β ∈ k * satisfy β 2 = α + α 2 , and put ℓ := kF q . Let E = ℓ(C α,β ). We have shown that G := Aut ℓ E satisfies G ∼ = SL 2 (q), and that there are degree-one places P 1 and P 2 of E whose stabilizers in G are B and W , respectively. Moreover, E has genus q(q − 1)/2, and the second ramification groups at P 1 and P 2 are trivial. By Riemann-Hurwitz, the only places of E G which ramify in E/E G are the places Q 1 and Q 2 which lie under P 1 and P 2 . Let G 1 be a subgroup of G of index q(q − 1)/2. Then G 1 is dihedral of order 2(q + 1), and hence contains q + 1 involutions. Each of the q + 1 conjugates of U contains precisely one of these involutions. Hence there is a unique place of E G 1 lying over Q 1 , and its ramification index in E/E G 1 is 2. Also there are precisely q/2 places of E G 1 which lie over Q 2 and ramify in E/E G 1 , and each has ramification index 2. Thus E G 1 has genus zero, and Q 1 is totally ramified in E G 1 /E G . Next, A := Aut k E satisfies A = G. Gal(E/k(C α,β )). Since G is normal in A, and G 1 is conjugate (in G) to all (q 2 − q)/2 subgroups of G having order 2q + 2, it follows that |N A (G 1 ) : G 1 | = |ℓ : k| and N A (G 1 )G = A. Thus, E N A (G 1 ) is a genus-zero function field over k which contains a degree-one place that is totally ramified over E A . We can write E N A (G 1 ) = k(x) and E A = k(u), and by making linear fractional changes in x and u we may assume that the unique place of k(x) lying over the infinite place of k(u) is the infinite place. In other words,
u = f (x) for some f ∈ k[X]
. Separability of f follows from separability of k(x)/k(u). The degree of f is (q 2 −q)/2, and its geometric monodromy group is SL 2 (q) (since G 1 contains no nontrivial normal subgroup of SL 2 (q)). The extension k(x)/k(f (x)) is totally ramified over infinity, and also is wildly ramified over another place of k(f (x)).
Next we show that the Galois closure of k(x)/k(f (x)) is E, or equivalently that N A (G 1 ) contains no nontrivial normal subgroup of A. Let J be a proper normal subgroup of A. Since G is normal in A (and simple), J must intersect G trivially. Thus each element of J has shape νσ, where ν ∈ G and σ ∈ Gal(E/k(C α,β )) satisfy | νσ | = | σ |. In particular, J is cyclic; let νσ be a generator of J. Since G and J normalize one another and intersect trivially, they must commute. + 1, w). Since τ commutes with both J and σ, it also must commute with ν. Hence ν maps (v, w) → (v + α, w) for some α ∈ F q . For ζ ∈ F * q , let λ ζ ∈ G map (v, w) → (ζv, ζw). Then λ ζ νσ(w) = ζw, but νσλ ζ (w) = σ(ζ)w, so σ fixes ζ. Hence σ fixes both F q and k(C α,β ), so it fixes E, whence J = 1. Thus the arithmetic monodromy group of f is A. Since G has a unique conjugacy class of subgroups of index (q 2 − q)/2, all of which are self-normalizing, any two index-(q 2 − q)/2 subgroups of A which surject onto A/G are conjugate. Since A = Aut k E, it follows that there is a unique k-equivalence class of polynomials f which satisfy all our hypotheses for a given pair (α, β). Conversely, k-equivalent polynomials have isomorphic Galois closures, hence correspond to the same pair (α, β). Finally, the indecomposability and exceptionality criteria follow from Lemma 3.1. Corollary 8.5. There exists a separable exceptional polynomial f ∈ k[X] of degree q(q − 1)/2 with two wild branch points and geometric monodromy group SL 2 (q) if and only if e is odd, k ∩ F q = F 2 , and k properly contains F 2 .
Write E = ℓ(v, w) where v q + v = (α + β)w + w q T(β/(1 + w q−1 )). Let τ ∈ G map (v, w) → (v
9. Another nonexistence proof over F 2 One consequence of Corollary 8.4 is that there is no separable polynomial f over F 2 of degree q(q−1)/2 such that the cover f : P 1 → P 1 has at least two wildly ramified branch points and has geometric monodromy group SL 2 (q).
In this section we give a more direct proof of this fact, by showing that the Galois closure of such a cover f : P 1 → P 1 would be a curve having more rational points than is permitted by the Weil bound.
Theorem 9.1. There is no separable polynomial f ∈ F 2 [X] of degree q(q − 1)/2 satisfying the following conditions:
(i) the geometric monodromy group of f is G := SL 2 (q); (ii) the extension F 2 (x)/F 2 (f (x)) has precisely two branch points, and in the Galois closure E/F 2 (f (x)) their ramification indices are q(q − 1) and 2; and (iii) all second ramification groups in E/F 2 (f (x)) are trivial.
Remark. By Lemma 3.1, conditions (ii) and (iii) follow from (i) if we assume that f has two wild branch points. Thus, the combination of Theorem 9.1 and Lemma 3.1 implies the 'only if' implication in Corollary 8.4.
Proof.
Suppose there is an f satisfying the above conditions. The Riemann-Hurwitz formula implies that the genus of E is q(q − 1)/2.
Since the two branch points of E/F 2 (f (x)) have nonconjugate inertia groups, these points must be F 2 -rational. Let Q be the point with ramification index 2.
Let A := Gal(E/F 2 (f (x))) be the arithmetic monodromy group of f . By Corollary 8.3, G ≤ A ≤ Aut(G) = SL 2 (q).e. Thus, A = G.e ′ for some e ′ | e. It follows that the algebraic closure of F 2 in E is ℓ := F 2 e ′ . Let P be a place of E lying over Q. Let H be the decomposition group of P in the extension E/F 2 (f (x)). We know that the inertia group W of P has order 2, so U := N G (W ) has order q. Thus, H ≤ N A (W ) = U, ν , where ν ∈ A has order e ′ and maps to a generator of A/G. Since Q is F 2 -rational, H/W surjects onto A/G, or equivalently A = GH. Since H/W is cyclic, it follows that |H/W | is either e ′ or 2e ′ .
Suppose that e ′ < e. Letl be the quadratic extension of ℓ. Then |l| ≤ q. LetP be a place oflE lying over P (there are one or two such places). Since |H/W | divides [l : F 2 ], the placeP is rational overl. Moreover, the ramification index ofP inlE/l(f (x)) is 2. Thus, Q lies under |G|/2 rational places oflE. SincelE has genus q(q − 1)/2, this violates the Weil bound for the number of rational points on a curve over a finite field. Now suppose that e ′ = e. As noted above, H ≤ N A (W ) = U, ν . For any µ ∈ U , the element (µν) e ∈ H lies in U and centralizes µν, hence it centralizes ν. However, the centralizer of ν in U is W . Since |C U (ν)| = 2, it follows that no element of N A (W )/W has order 2e, so H/W is cyclic of order e. Now, as in the previous case, we obtain a contradiction by counting points.
Construction of polynomials
In this section we use the results proved so far in order to compute explicit forms of the polynomials whose existence was proved in Theorem 8.1.
Let k be a perfect field of characteristic 2, and let q = 2 e > 2. Let α, β ∈ k * satisfy β 2 = α + α 2 .
Theorem 10.1. The polynomial
(10.2)f (X) := (T(X) + α + 1) ζ q−1 =1 ζ =1 e−1 i=0 ζ 2 i + ζ ζ 2 i + 1 X 2 i + ζα + 1
is in the k-equivalence class corresponding to (α, β) in Theorem 8.1.
Proof. Let ℓ = k.F q and E = ℓ(C α,β ). Write E = ℓ(v, w), where v q + v = (α + β)w + w q T(β/(1 + w q−1 )). Letŵ = 1/w andv = v 2 /w + v + βw/(1 + w q−1 ). Then T(vŵ) = v w q + v w + T β 1 + w q−1 = v 1 w + 1 w q + α + β w q−1 , so k(v,ŵ) = k(v, w). Next, v qŵ = v 2q w q+1 + v q w + β q w q−1 1 + w q 2 −q = v 2 w q+1 + α 2 + β 2 w q−1 + w q−1 T β 1 + w q−1 2 + v w + α + β + w q−1 T β 1 + w q−1 + β q w q−1 1 + w q 2 −q = v 2 w q+1 + α w q−1 + w q−1 β 1 + w q−1 + v w + α + β = T(vŵ) +ŵ qv + α.
Since [ℓ(v,ŵ) : ℓ(ŵ)] = [ℓ(v, w) : ℓ(w)] = q, the polynomialŵX q + T(ŵX) + w q X +α is irreducible over ℓ(ŵ), so alsovX q +T(vX)+v q X +α is irreducible over ℓ(v). Letl = F q 2 .ℓ, andÊ =l.E. Pick γ ∈l * of multiplicative order q + 1, and let δ = γ + 1/γ ∈ F * q ⊆ ℓ * . Let y = (vγ +ŵ/γ + 1)/δ and z = (v/γ +ŵγ + 1)/δ. ThenÊ =l(v,ŵ) =l(y, z). For η ∈ F q 2 with η q+1 = 1, there is a unique elementν η ∈ AutlÊ which maps (y, z) → (yη, z/η). Moreover, ν η :=ν η | E is in Aut ℓ E.
Proof. We compute
y q+1 + z q+1 = (vγ +ŵ γ + 1)(v q γ +ŵ q γ + 1) δ q+1 + (v γ +ŵγ + 1)(v q γ +ŵ q γ + 1) δ q+1 =ŵ qv +ŵv q +ŵ q +v q +ŵ +v δ = T(ŵv) + α +ŵ q +v q δ q +ŵ +v δ = T ŵv +ŵ +v δ +ŵ 2 +v 2 + 1 δ 2 + α + T 1 δ 2 = T(yz) + α + T 1 δ 2 . Since 1/δ = γ/(γ 2 + 1) = γ/(γ + 1) + γ 2 /(γ 2 + 1), we have T 1 δ 2 = T 1 δ = γ q γ q + 1 + γ γ + 1 = 1 γ 1 γ + 1 + γ γ + 1 = 1.
SinceÊ =l(y, z) has genus q(q − 1)/2 where y and z satisfy equation (10.4) of total degree q + 1, this equation must define a smooth (projective) plane curve, and in particular must be irreducible. Thus [l(y, z) :l(z)] = q+1. Now existence and uniqueness ofν η are clear. A straightforward computation yields thatν η mapŝ
w → 1 δ 2 δ + γ η + η γ +ŵ η γ 2 + γ 2 η +v η + 1 η v → 1 δ 2 δ + γη + 1 γη +ŵ η + 1 η +v γ 2 η + 1 γ 2 η .
Since ℓ(ν η (v),ν η (ŵ)) = E, it follows thatν η induces an automorphism of E.
We now compute the subfield of E fixed by an index-(q 2 − q)/2 subgroup of G := Aut ℓ E ∼ = SL 2 (q). There is a unique element τ ∈ Aut ℓ E such that τ : (v,ŵ) → (ŵ,v). Note that τ maps (y, z) to (z, y), and the group G 1 := τ, {ν η : η q+1 = 1} is dihedral of order 2q + 2. Hence the subfield of E fixed by G 1 containsl(yz). Multiplying equation ( lies in E, the subfield of E fixed by G 1 is ℓ(yz).
Next we compute an invariant of G. Recall that SL 2 (q) can be written as CT U , where T is the diagonal subgroup, U is a unipotent subgroup, and C is a cyclic subgroup of order (q + 1). We can choose U to be the set of maps σ ξ : (v, w) → (v + ξ, w) with ξ ∈ F q , so E U = ℓ(w). We can choose T to be the set of maps µ ζ : (v, w) → (ζ −1 v, ζ −1 w) with ζ q−1 = 1, and C to be the set of maps ν η defined in the above lemma. Hence the product
η q+1 =1 ζ q−1 =1 ξ∈Fq ν η µ ζ σ ξ δ w + 1 is G-invariant.
Since this product is the q-th power of
u := η q+1 =1 ζ q−1 =1 ν η µ ζ δ w + 1 ,
also G fixes u. Since 1/w = (zγ + 1 + yγ −1 )/δ, we have
u = ζ q−1 =1 η q+1 =1 (ηζyγ −1 + ζ + 1 + η −1 ζγz).
By the following lemma,
u = ζ q−1 =1 η q+1 =1 (ηζy + ζ + 1 + η −1 ζz) = (y q+1 + z q+1 ) ζ q−1 =1 ζ =1 ζ 2 (y q+1 + z q+1 ) + (ζ 2 + 1) 1 + T yz ζ 2 ζ 2 + 1 = (T(yz) + α + 1) ζ∈Fq F 2 ζ T(yz) + α + 1 + (ζ + 1) 1 + T yz ζ ζ + 1 = (T(yz) + α + 1) ζ∈Fq F 2 e−1 i=0 ζ 2 i + ζ ζ 2 i + 1 (yz) 2 i + ζα + 1 .
Thus u =f (yz) wheref is the polynomial defined in (10.2). It follows that [ℓ(yz) : ℓ(u)] = deg(f ) = (q 2 − q)/2. Since E G 1 = ℓ(yz) and E G ⊇ ℓ(u) and [E G 1 : E G ] = (q 2 − q)/2, it follows that E G = ℓ(u). Now, G 1 contains no nontrivial normal subgroup of G, so E is the Galois closure of ℓ(yz)/ℓ(u), whence G is the geometric monodromy group off . Clearlyf is fixed by Gal(ℓ/k), sof ∈ k[X]. By Theorem 5.4, the extension E/E G has two wildly ramified branch points, so Theorem 8.1 implies thatf is in the k-equivalence class corresponding to the pair (α, β).
Lemma 10.5. The following identity holds in k[Y, Z]:
ω q+1 =1 (ωY + 1 + ω −1 Z) = Y q+1 + Z q+1 + T(Y Z) + 1.
Proof. By applying the transformation (Y, Z) → (ωY, Z/ω), we see that
(ωY + 1 + ω −1 Z) − Y q+1 − Z q+1 is a polynomial h(Y Z) ∈ k[Y Z]
, with degree at most q/2 and constant term 1. If we substitute Y = Z = ω/(ω 2 +1) (where ω q+1 = 1 and ω = 1), we see that Y Z = ω 2 /(ω 4 + 1) is a root of h. These roots of h are precisely the trace 1 elements of F q , namely the roots of T(Y Z) + 1. Hence h(Y Z) and T(Y Z) + 1 have the same roots and the same constant term, and T(Y Z)
+ 1 is squarefree with deg(T +1) ≥ deg(h), so h(Y Z) = T(Y Z) + 1.
Remark. Once one knows 'where to look' for these polynomials -especially, what should be the Galois closure E of k(x)/k(f (x)) -one can give direct proofs of their properties. But such proofs would seem unmotivated, since we know no way to guess what E should be besides appealing to the results in this paper.
Another form for the polynomials
In the previous section we computed the polynomials whose existence was proved in Theorem 8.1. Our expression for the polynomials was concise, but involved a product. In this section we prove Theorem 1.2 and Corollary 1.3 by writing the polynomials without any sums or products other than the usual T(X) = X q/2 + X q/4 + · · · + X. Here q = 2 e > 2 and k is a perfect field of characteristic 2. Also α, β ∈ k * satisfy β 2 = α 2 + α.
Theorem 11.1. The expression f (X) := T(X) + α X q · T(X) + T(X) + α α + 1 · T X(α 2 + α) (T(X) + α) 2 defines a polynomial which lies in the k-equivalence class corresponding to (α + 1, β) in Theorem 8.1.
Proof. First we show that f is a polynomial. Writing h(X) := X q f (X), we have
h = (T(X) + α) q · T(X) + (T(X) + α) q+1 α + 1 · T X(α 2 + α) (T(X) + α) 2 = (T(X) + α) q · T(X) + 1 α + 1 e−1 i=0 X 2 i (α 2 + α) 2 i (T(X) + α) q+1−2 i+1 .
Thus h is a polynomial divisible by X · (T(X) + α), and moreover h is monic of degree q(q + 1)/2. We now determine the multiplicity of X as a divisor of h. This multiplicity is unchanged if we replace h bŷ h := h · h + (T(X) + α) q+1 α + 1 ;
writing c := X(α 2 + α)/(T(X) + α) 2 , we computê
h = h 2 + h · (T(X) + α) q+1 α + 1 = (T(X) + α) 2q · T(X) 2 + T(X) · (T(X) + α) 2q+1 α + 1 + (T(X) + α) 2q+2 α 2 + 1 · T(c 2 + c).
Substituting T(c 2 + c) = c q + c, and reducing mod X 2q , we find that
h ≡ α 2q T(X) 2 + α 2q α + 1 (T(X) 2 + α T(X)) + (T(X) + α) 2 α 2 + 1 X q (α 2 + α) q + α 2q α 2 + 1 X(α 2 + α) (mod X 2q ) = α 2q α + 1 T(X) 2 (α + 1) + T(X) 2 + α T(X) + αX + (T(X) + α) 2 α 2 + 1 X q (α 2 + α) q = α q α 2 + 1 X q α q+1 (α + 1) + (T(X) + α) 2 (α + 1) q ,
so X q dividesĥ, whence f is a polynomial divisible by (T(X) + α). Furthermore, X divides f (equivalently X q+1 dividesĥ) precisely when α ∈ F q , in which case X 2 exactly divides f . Since h is monic of degree q(q + 1)/2, it follows that f is monic of degree q(q − 1)/2.
We now show that f /(T(X) + α) is in k[X 2 ]. It suffices to show that f := X q f /(T(X) + α) is in k[X 2 ]. We compute f = T(X)(T(X) + α) q−1 + (T(X) + α) q α + 1 · T X(α 2 + α) (T(X) + α) 2 = (T(X) + α) q + α(T(X) + α) q−1 + 1 α + 1 e−1 i=0 (X(α 2 + α)) 2 i (T(X) + α) q−2 i+1 .
The summands with i > 0 are polynomials in X 2 . Thus, there exists b ∈ k[X] such that
f = b(X 2 ) + α(T(X) + α) q−1 + αX(T(X) + α) q−2 = b(X 2 ) + α(T(X) + α) q−2 (T(X) + α + X), so indeed f ∈ k[X 2 ], whence f /(T(X) + α) is in k[X 2 ].
By Theorem 10.1, the polynomial f (X) := (T(X) + α)
ζ q−1 =1 ζ =1 e−1 i=1 ζ 2 i + ζ ζ 2 i + 1 X 2 i + ζ(α + 1) + 1
is in the k-equivalence class corresponding to (α + 1, β) in Theorem 8.1. By Lemma 3.1, the extension k(x)/k(f (x)) has precisely two branch points; one of these points is totally ramified, and the ramification index at any point of k(x) lying over the other branch point is at most 2. Since k(x)/k(f (x)) is totally ramified over the infinite place, there is a unique finite branch point.
But plainlyf (X) = (T(X) + α)b(X) 2 for some nonconstantb ∈ k[X], sô f (x) = 0 is the finite branch point, and thusb(X) is squarefree and coprime to (T(X) + α). We will show that every root δ off is a root of f ; it follows that the multiplicity of δ as a root of f is at least as big as the corresponding multiplicity forf . Since f andf have the same degree and the same leading coefficient, we conclude that f =f . It remains to prove that every root off is a root of f . Recall that, in the function field k(y, z) where y q+1 + z q+1 = T(yz) + α, we have the identitŷ
f (yz) = ζ q−1 =1 η q+1 =1 (ηζy + ζ + 1 + ζ η z) = (y q+1 + z q+1 ) ζ q−1 =1 ζ =1 η q+1 =1
(ηζy + ζ + 1 + ζ η z).
Let δ be a root off . Pickŷ ∈ k * andẑ ∈ k such that δ =ŷẑ andŷ q+1 + z q+1 = T(ŷẑ) + α: suchŷ,ẑ exist because substitutingẑ = δ/ŷ into the latter equation (and clearing denominators) gives a polynomial inŷ which is not a monomial, and thus has a nonzero root. If T(δ) = α then we already know that f (δ) = 0. If δ = 0 thenẑ = 0 andŷ q+1 = α, so 0 =f (0) = ζ q−1 =1 η q+1 =1
(ηζŷ + ζ + 1) = ζ q−1 =1
(ζ q+1ŷq+1 + (ζ + 1) q+1 ) = ζ q−1 =1
(ζ 2 α + ζ 2 + 1) = (α + 1) q−1 + 1.
Thus α ∈ F q , so X 2 divides f . Henceforth we assume α = T(δ) and δ = 0. This implies ηζŷ + ζ + 1 + zζ/η = 0 for some ζ, η with ζ ∈ F q \ F 2 and η q+1 = 1. By replacingŷ andẑ with ηŷ andẑ/η, we may assume η = 1, sô z =ŷ + 1 + 1 ζ .
Writeζ := 1 + 1/ζ, and note thatζ ∈ F q \ F 2 . Since δ =ŷẑ, we compute T(δ) + α =ŷ q+1 +ẑ q+1 =ŷ q+1 +ŷ q+1 +ζŷ q +ζ qŷ +ζ q+1 =ζŷ q +ζŷ +ζ 2 and T(δ) = T(ŷ 2 +ζŷ) =ζŷ q +ζŷ + T(ŷ 2 +ζ 2ŷ2 ).
Thus α +ζ 2 = T(ŷ 2 +ζ 2ŷ2 ), so √ α +ζ = T(ŷ +ζŷ).
Adding the last two equations gives α + √ α +ζ 2 +ζ =ŷ q +ŷ +ζŷ q +ζŷ = (1 +ζ)(ŷ q +ŷ), so T(δ) + α =ζ 2 +ζ(ŷ q +ŷ) =ζ 2 +ζ 1 +ζ α + √ α +ζ 2 +ζ =ζ 1 +ζ α + √ α and T δ ζ 2 = T ŷ 2 ζ 2 +ŷ ζ =ŷ q ζ +ŷ ζ = 1 + α + √ α ζ 2 +ζ .
Writingf (X) := X q f (X)/(T(X) + α) q , we havẽ f (δ) = T(δ) + T(δ) + α α + 1 · T δ(α 2 + α)
(T(δ) + α) 2 = α +ζ √ α 1 +ζ +ζ √ α (1 +ζ)( √ α + 1) · T δ(α 2 + α)(1 +ζ) 2 ζ 2 (α 2 + α) = α +ζ √ α 1 +ζ +ζ √ α (1 +ζ)( √ α + 1) · T δ + δ ζ 2 = α +ζ √ α 1 +ζ +ζ √ α (1 +ζ)( √ α + 1) ·ζ + α + √ α(1 +ζ) ζ = α +ζ √ α + √ α( √ α +ζ)
1 +ζ , sof (δ) = 0 and thus f (δ) = 0, which completes the proof.
Remark. The above proof is not completely satisfying, since it is a verification that f (X) has the desired property, rather than a derivation of the simple expression for f (X). We do not have a good explanation why the polynomial in Theorem 8.1 can be written in such a simple form.
We conclude the paper by proving the results stated in the introduction.
Proof of Theorem 1.2. In case k is perfect, the result follows from Theorem 8.1 and Theorem 11.1. For general k, letk denote the perfect closure of k. Let f ∈ k[X] satisfy properties (i) and (ii) of Theorem 1.2. Then f satisfies the same properties over the perfect fieldk, so f isk-equivalent to f α for some α ∈k \ F 2 . We will show that this implies f is k-equivalent to f α , and that α ∈ k. Since the monodromy groups of f over k are the same as those overk, indecomposability and exceptionality of f over k are equivalent to the corresponding properties overk. Sincek ∩ F q = k ∩ F q (because F q /F 2 is separable), the result follows.
It remains to prove that if f (X) := δ + ηf α (ζX + γ) is in k[X], where δ, η, α, ζ, γ ∈k with ηζ = 0 and α / ∈ F 2 , then δ, η, α, ζ, γ are in k. The terms of f α (X) of degree at least (q 2 − 3q)/2 are T(X)X (q 2 −2q)/2 + αX (q 2 −3q+2)/2 and (if q = 4) (α + 1)X 2 . Hence the coefficients of X q 2 /2−q+2 and X q 2 /2−q+1 in f (X) are ηζ q 2 /2−q+2 and ηζ q 2 /2−q+1 , and since these are in k * , we must have ζ, η ∈ k * . The coefficients of X (q 2 −3q+2)/2 and X (q 2 −2q)/2 in f (X) are αηζ (q 2 −3q+2)/2 and ηζ (q 2 −2q)/2 T(γ), so α ∈ k * and T(γ) ∈ k, whence T(γ) 2 + T(γ) = γ q + γ is in k. The coefficient of X (q 2 −3q)/2 in f (X) is ηζ (q 2 −3q)/2 (αγ + γ q ) (plus η(α + 1)ζ 2 if q = 4), so γ is in k. Finally, we conclude that δ = f (0) − ηf α (γ) is in k.
Proof of Corollary 1. 3. First assume f ∈ k[X] is a separable indecomposable exceptional polynomial in case (i) of Theorem 1.1. Then the geometric monodromy group G of f is solvable, and the degree d of f is prime and not equal to p. By [26,Thm. 4], it follows that f is k-equivalent to either X d or D d (X, 1). By [34,Lemma 1.9], f is k-equivalent to either X d or D d (X, a) with a ∈ k * . These polynomials f (X) are separable and indecomposable. We verify exceptionality by examining the factorization of f (X) − f (Y ) in k[X, Y ], given for instance in [34,Prop. 1.7]. Now consider case (iii) of Theorem 1.1. In this case, Corollary 1.3 for p = 3 is [20, Thm. 1.3]. So suppose p = 2, and let f ∈ k[X] be a separable indecomposable exceptional polynomial of degree d = q(q − 1)/2 where q = 2 e > 2 with e > 1 odd. By Theorem 1.1, the arithmetic monodromy group A of f is PΓL 2 (q), and thus G has a transitive normal subgroup isomorphic to PSL 2 (q). The desired result follows from [20,Thm. 4.3] if k(x)/k(f (x)) has no finite branch points, or if the Galois closure E of this extension does not have genus (q 2 −q)/2. If neither of these conditions hold, then [20,Thm. 2.1] implies that G = PSL 2 (q) and E/k(f (x)) has precisely one finite branch point, whose inertia group has order 2 and whose second ramification group is trivial. In particular, f satisfies conditions (i) and (ii) of Theorem 1.2, so in this case the result follows from Theorem 1.2.
Lemma 2 . 3 .
23Let θ : C → D be a cover of curves over ℓ. If C is ordinary, then D is ordinary. This lemma and the next one are proved in [29, Thm. 1.2]. The strategy for proving the next lemma comes from [27, Thm. 2]. Lemma 2.4. Let θ : C → D be a Galois cover (of curves over ℓ) whose Galois group H is a p-group. Then C is ordinary if and only if both (i) D is ordinary; and (ii) every branch point of θ has trivial second ramification group. Proof. We use the Deuring-Shafarevich formula ([31, Thm. 4.2]):
Lemma 3.3. B = U ⋊ T is the semidirect product of the normal subgroup U by the cyclic subgroup T . All involutions in B are conjugate. All subgroups of B of order q − 1 are conjugate. For j ∈ {1, −1}, all subgroups of SL 2 (q) of order 2(q + j) are conjugate, and these subgroups are dihedral and are maximal proper subgroups of SL 2
Lemma 4 . 4 .
44There exists an order-q subgroup Γ 1 of Γ such that Γ = Γ 0 ⊕Γ 1 and the nonzero elements of Γ 1 comprise a single T -orbit.
not a full Sylow 2-subgroup of G, then (since 2-groups are nilpotent) |N G (U ) : U | is even, a contradiction. Thus, |G : B| is odd.Lemma 5.2. The following are equivalent:(i) G = B; (ii) P 1 and P 3 are in distinct G-orbits; (iii) T = N G (T ); and (iv) |N G (T ) : T | = 2.Proof. By Lemma 5.1, the intersection of the stabilizers (in G) of P 1 and P 3 is T . Since distinct B-conjugates of T intersect trivially, any nontrivial element of T fixes precisely two points of C (namely P 1 and P 3 ). Thus, N G (T ) preserves Λ := {P 1 , P 3 }, so either it acts transitively on Λ (and |N G (T ) : T | = 2) or else N G (T ) = T . Hence conditions (iii) and (iv) are equivalent, and they both follow from (ii). IfνP 3 = P 1 with ν ∈ G, then T ν is contained in B, so T ν = T µ for some µ ∈ B; but then µ −1 ν ∈ N G (T ) \ T .Hence (ii) and (iii) are equivalent. Clearly if G = B, all the remaining conditions are true. So we assume the last three conditions and show that G = B.Suppose P 1 and P 3 are in distinct G-orbits. Let I be the stabilizer of P 3 in G, so I = V T where V is a normal 2-subgroup and T is a cyclic group of odd order. Since I contains T , by Schur-Zassenhaus T is contained in an I-conjugate T ′ of T , so I = V T ′ . Since T ′ is cyclic, it normalizes T , so (by (iii)) T ′ = T . Since U is a Sylow 2-subgroup of G (Lemma 5.1), some conjugate V ′ of V is contained in U ; by our hypothesis on the inertia groups of C → C/B, either |V | ≤ 2 or V ′ = U . But V ′ = U because P 1 and P 3 are in distinct G-orbits, and |V | = 2 since |I : T | = 2 contradicts (iii). Hence I = T . Since any nontrivial element of T fixes no point of GP 3 \ {P 3 }, it follows that G acts on GP 3 as a Frobenius group with Frobenius complement T ; let K be the Frobenius kernel. Since K is a normal subgroup of G that contains a Sylow 2-subgroup, K contains every Sylow 2-subgroup, so U ≤ K. By Lemma 5.1, N G (U ) = B, so N K (U ) = B ∩ K = U . Nilpotence of the Frobenius kernel implies K = U , so G = KT = B.Lemma 5.3. W is the stabilizer of P 2 in G.
Proposition 6. 1 .
1C has automorphism group G if and only if β 2 = α + α 2 .
respectively. Suppose there is an isomorphism ρ : C → C ′ . Conjugation by ρ induces an isomorphism θ : G → G ′ . By replacing ρ by its compositions with automorphisms of C and C ′ , we can replace θ by its compositions with arbitrary inner automorphisms of G and G ′ .We use the embeddings B → G and B → G ′ from Lemma 4.5. By Lemma 5.1, U is a Sylow 2-subgroup of G and G ′ , so (by composing ρ with automorphisms) we may assume θ(U ) = U . Since all index-2 subgroups of U are conjugate under B, we may assume in addition that θ(H) = H where H is a prescribed index-2 subgroup of U . Then ρ induces an isomorphism between C/U and C ′ /U which maps the set of branch points of C/H → C/U to the corresponding set in C ′ /U . For definiteness, choose H to be the subgroup defined in Lemma 4.6, and choose the coordinates w and w ′ on C/U and C ′ /U . The branch points of each of C/H → C/U and C ′ /H → C ′ /U (in the coordinates w and w ′ ) are {δ : T(δ) = 1} ∪ {∞}.
Corollary 8. 4 .
4There exists a separable polynomial f ∈ k[X] of degree q(q − 1)/2 with two wild branch points and geometric monodromy group SL 2 (q) if and only if k properly contains F 2 .
Lemma 10 . 3 .
103We have [l(y, z) :l(z)] = q + 1 and (10.4)y q+1 + z q+1 = T(yz) + α + 1.
10.4) by y q+1 , we see that [l(y, z) :l(yz)] = [l(y, yz) :l(yz)] ≤ 2q + 2, sol(yz) is the subfield ofÊ fixed by G 1 .
Proof. LetŴ be the stabilizer of P 2 in G. Let W be the Sylow 2-subgroup of W . By Corollary 2.2, W is elementary abelian and is its own centralizer in W . Thus,Ŵ /W embeds in Aut(W ). If W = W , this implies thatŴ = W . Now assume that W strictly contains W ; we will show that this leads to a contradiction. Note that Lemma 5.2 implies P 3 ∈ GP 1 .Let C be the centralizer of W in G. Then C contains U and W , where W ∩ U = W . Let Λ = CP 2 . Since W and C commute, W acts trivially on Λ, so Λ ⊆ {P 1 } ∪ U P 2 . But U is a Sylow 2-subgroup of G, so it contains a conjugate W ν of W in G, and since |W | > 2 we must have νP 2 = P 1 .Hence Λ = {P 1 } ∪ U P 2 . The stabilizer of P 1 in G is B, and the stabilizer in B of any element of U P 2 is W . Thus any two-point stabilizer of C on Λ is conjugate in C to W , hence equals W , so C/W is a Frobenius group on Λ. A Frobenius complement is U/W (since B ∩ C = U ). It is well known (and elementary in this case: cf.[12, Thm. 3.4A]) that an abelian subgroup of a Frobenius complement must be cyclic. Hence U/W is cyclic, so q = 4.
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| [] |
[
"Entropy Conserving Binarization Scheme for Video and Image Compression",
"Entropy Conserving Binarization Scheme for Video and Image Compression"
] | [
"Madhur Srivastava \nDepartment of Biomedical Engineering\nCornell University\nIthacaNew YorkUSA\n"
] | [
"Department of Biomedical Engineering\nCornell University\nIthacaNew YorkUSA"
] | [] | The paper presents a binarization scheme that converts non-binary data into a set of binary strings. At present, there are many binarization algorithms, but they are optimal for only specific probability distributions of the data source. Overcoming the problem, it is shown in this paper that the presented binarization scheme conserves the entropy of the original data having any probability distribution of m-ary source. The major advantages of this scheme are that it conserves entropy without the knowledge of the source and the probability distribution of the source symbols. The scheme has linear complexity in terms of the length of the input data. The binarization scheme can be implemented in Context-based Adaptive Binary Arithmetic Coding (CABAC) for video and image compression. It can also be utilized by various universal data compression algorithms that have high complexity in compressing non-binary data, and by binary data compression algorithms to optimally compress non-binary data. | null | [
"https://arxiv.org/pdf/1408.3083v1.pdf"
] | 16,122,894 | 1408.3083 | a9bd89cd45ed6f3d6a0d3f1f1c555376afe0526e |
Entropy Conserving Binarization Scheme for Video and Image Compression
13 Aug 2014
Madhur Srivastava
Department of Biomedical Engineering
Cornell University
IthacaNew YorkUSA
Entropy Conserving Binarization Scheme for Video and Image Compression
13 Aug 2014BinarizationSource CodingData CompressionImage CompressionVideo CompressionBinary Arithmetic CodingContext-based Adaptive Binary Arithmetic Coding (CABAC)
The paper presents a binarization scheme that converts non-binary data into a set of binary strings. At present, there are many binarization algorithms, but they are optimal for only specific probability distributions of the data source. Overcoming the problem, it is shown in this paper that the presented binarization scheme conserves the entropy of the original data having any probability distribution of m-ary source. The major advantages of this scheme are that it conserves entropy without the knowledge of the source and the probability distribution of the source symbols. The scheme has linear complexity in terms of the length of the input data. The binarization scheme can be implemented in Context-based Adaptive Binary Arithmetic Coding (CABAC) for video and image compression. It can also be utilized by various universal data compression algorithms that have high complexity in compressing non-binary data, and by binary data compression algorithms to optimally compress non-binary data.
Introduction
Data compression is performed in all types of data requiring storage and transmission. It preserves space, energy and bandwidth, while representing the data in most efficient way [1][2][3][4]. There are numerous coding algorithms used for compression in various applications [1][2][3][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Some of them are optimal [4,19] in all cases, whereas others are optimal for a specific probability distribution of the source symbols. All of these algorithms are mostly applied on m-ary data source. However, most of the universal compression algorithms substantially increase their coding complexity and memory requirements when the data changes from binary to m-ary source. For instance, in arithmetic coding, the computational complexity difference between the encoder and decoder increases with the number of source symbols [2]. Therefore, it would be beneficial if binarization is perfomed on m-ary data source before compression algorithms are applied on it. The process where binarization is followed by compression is most notably found in Context-based Adaptive Binary Arithmetic Coding (CABAC) [20] which is used in H.264/AVC Video Coding Standard [21], High Efficiency Video Coding (HEVC) Standard [22], dynamic 3D mesh compression [23], Audio Video Coding Standard (AVS) [24], Motion Compensated-Embedded Zeroblock Coding (MC-EZC) in scalable video coder [25], multiview video coding [26], motion vector encoding [27][28], and 4D lossless medical image compression [29].
There are many binary conversion techniques which are, or can be used for the binarization process. The most common among all is binary search tree [30][31][32]. In this, Huffman codeword is used to design an optimal tree [18]. However, there are two limitations to it. First, the probability of all the symbols should be known prior to encoding that may not be possible in all the applications. Although there are methods to overcome the above problem, they come at an additional cost of complexity. For example, binary search tree is updated with the change in incoming symbol probabilities. Second, as with the Huffman coding, the optimality is achieved only when the probability distribution of symbols are in the powers of two. Apart from binary search tree, there are other binarization schemes like unary binarization scheme [20], truncated unary binarization scheme [20], fixed length binarization scheme [20], Golomb binarization scheme [9,20,[33][34], among many [5][6][7][8][9][10][11][12][13]20,[30][31][32][33][34]. All of them are optimal for only certain type of symbol probability distributions, and hence, can only conserve the entropy of the data for that probability distribution of the source symbols. Currently, there is no binarization scheme that is optimal for all probability distributions of the source symbols which would result in achieving overall optimal data compression. This paper presents a generalized optimal binarization algorithm. The novel binarization scheme conserves the entropy of the data while converting the m-ary source data into m − 1 binary strings. Moreover, the binarization technique is independent of the data type and can be used in any field for storing and compressing data. Furthermore, it can efficiently represent data in the fields which require data to be easily written and read in binary form.
The paper is organized as follows. Section 2 describes the binarization and de-binarization process that will be carried out at the encoder and decoder, respectively. The optimality proof of the binarization scheme is provided in section 3. In section 4, the complexity associated with the binarization process is discussed. Lastly, section 5 concludes by stating the advantages of the presented binarization scheme over others, and its applications.
Binarization and De-binarization
The binarization of the source symbols is carried out at the encoder using the following two steps:
A symbol is chosen, and a binary data stream is created by assign-
ing '1' where the chosen symbol is present and '0' otherwise, in the uncompressed data. 2. The uncompressed data is rearranged by removing the symbol chosen in step 1.
The two steps are iteratively applied for m−1 symbols. It needs to be explicitly emphasized that the binarization of symbol occurs on the previously rearranged uncompressed data and not on the original uncompressed data. Here, the algorithm reduces the uncompressed data size with the removal of binarized symbols from the data, leading to the conservation of entropy. After binarizing every symbol, there are m − 1 binary data streams corresponding to m source symbols. It is because the m − 1 th binary string would represent m−1 th and m th symbols as '1' and '0', respectively. The binarization scheme demostrated here is optimal i.e., the overall entropy of binarized data streams is equal to the entropy of original data containing m-ary source. The proof of optimality is provided in section 3. After binarization, the binarized data streams can be optimally compressed using any universal compression algorithm, including the algorithms that optimally compress only binary data (for example: binary arithmetic coding). Table 1 shows the binarization process through an example. A sample input data 'AABCBACBBACCABACB' is considered for the process and as can be seen, it contains three source symbols 'A', 'B', and 'C'. In Table 1, 'Binarization order' states the sequence in which the symbols are binarized. For instance, in 'ABC' binarization order, 'A' is binarized first, followed by 'B', and then finally by 'C'. The row 'Data' shows the uncompressed data available to be binarized after each iteration. Below the 'Data' row is the binarized value of each symbol. As can be seen in each first iteration, the symbol that has be binarized is marked '1', while others are marked '0'. In the next iteration, the symbol that was binarized in the current step is removed from the uncompressed data. Although shown in Table 1, the binarization process does not require to binarize last symbol, because the resultant string contains all '1's that provide no additional information and is redundant. It also needs to be noted that each binarization order results in different sets of binary strings. At the decoder, the decoding of the compressed data is followed by debinarization of m-ary source symbol. The order of decoding follows the order of encoding for perfect reconstruction at minimum complexity. With the encoding order information, the de-binarization can be perfectly reconstructed in multiple ways other than the encoding order, but the reordering of sequence after every de-binarization will increase the time as well as the decoder complexity. The de-binarization of the source symbols is also carried out in two steps shown below, and these steps are recursively applied to all the binary data streams representing m-ary source symbols:
1. Replace '1' with the source symbol in the reconstructed data stream. 2. Assign the values of the next binary data stream in sequence to the '0's in the reconstructed data stream.
An example of de-binarization process is shown in Table 2. The debinarization order follows the same order as of binarization process. In Table 2, the row 'Data' represents the reconstructed data at each iteration. The value '1' is replaced by the symbol to be de-binarized in the respective iteration, while '0's are replaced by the binary string of the next symbol to be de-binarized. Finally, after the last iteration, the original input data 'AABCBACBBACCABACB' is losslessly recovered for all binarization and de-binarization order.
Optimality of Binarization Scheme
Let the data source be Y ∈ {Y 1 , Y 2 , . . . , Y m }, and X ∈ {0, 1} be the binary source for each source symbol. The entropy of a m-ary source Y is defined as,
H(Y ) = − m i=1 p(Y i ) log p(Y i )(1)
where p(Y i ) is the probability of Y th i source symbol. Subsequently, the entropy of m-ary data source Y with length N is H(Y N ). Similarly, H(X N ) is the entropy of binary source with data length N. As explained in the binarization algorithm, the uncompressed data is rearranged after the binarization of the previous symbol/s to N(1 − i−1 j=1 p(Y j )) data length i.e., the length of the original data subtracted by the length of all the previously binarized source symbols. Hence, the overall entropy of the m binarized
strings is m i=1 H(X N (1− i−1 j=1 p(Y j )) i
). Here, m binary strings are considered for mathematical convenience.
To achieve the optimal binarization of m-ary source, the entropy of m-ary source data must equal the total entropy of binary strings. Therefore,
H(Y N ) = m i=1 H(X N (1− i−1 j=1 p(Y i )) i ) (2) N H(Y ) = m i=1 N 1 − i−1 j=1 p(Y i ) H(X i )(3)
The probability distribution of the binary source X i is the probability distribution of m-ary source Y i when the first i − 1 source symbols have already been binarized i.e., removed from the original data. Thus, H(X i ) can be rewritten in terms of Y i in the following way:
H(Y ) = m i=1 1 − i−1 j=1 p(Y j ) H p(Y i ) 1 − i−1 j=1 p(Y j )(4)H(Y ) = − m i=1 1 − i−1 j=1 p(Y j ) p(Y i ) 1 − i−1 j=1 p(Y j ) log p(Y i ) 1 − i−1 j=1 p(Y j ) − m i=1 1 − i−1 j=1 p(Y j ) 1 − i j=1 p(Y j ) 1 − i−1 j=1 p(Y j ) log 1 − i j=1 p(Y j ) 1 − i−1 j=1 p(Y j )(5)H(Y ) = − m i=1 p(Y i ) log p(Y i ) 1 − i−1 j=1 p(Y j ) − m i=1 (1 − i j=1 p(Y j )) log 1 − i j=1 p(Y j ) 1 − i−1 j=1 p(Y j )(6)H(Y ) = − m i=1 p(Y i ) log p(Y i ) m j=i p(Y j ) − m i=1 ( m j=i+1 p(Y j )) log m j=i+1 p(Y j ) m j=i p(Y j )(7)H(Y ) = − m i=1 p(Y i ) log p(Y i ) − p(Y i ) log m j=i p(Y j ) − m i=1 m j=i+1 p(Y j ) log m j=i+1 p(Y j ) + m i=1 m j=i+1 p(Y j ) log m j=i p(Y j ) (8) H(Y ) = − m i=1 p(Y i ) log p(Y i ) − m i=1 m j=i+1 p(Y j ) log m j=i+1 p(Y j ) + m i=1 m j=i p(Y j ) log m j=i p(Y j ) (9) H(Y ) = − m i=1 p(Y i ) log p(Y i ) + m j=1 p(Y j ) log m j=1 p(Y j ) (10) H(Y ) = − m i=1 p(Y i ) log p(Y i ) + 1 log 1 (11) H(Y ) = − m i=1 p(Y i ) log p(Y i )(12)
The reduction of equation 2 to equation 12 (also equation 1) proves that the binarization scheme preserves entropy for any m-ary data source.
Computational Complexity of Binarization Scheme
The computational complexity of the presented method is the linear function of the input data length. The binarization and de-binarization process only acts as a filter, assigning or replacing 0's and 1's, respectively, for an occurrence of a source symbol without any additional table or calculation, that is created or performed for the other binarization techniques. Suppose, the length of input data is N, m is the number of source symbols, and Y is the source. For the first symbol, the length of the binary string would be N. The length of binary string for the second symbol would be the length of all the symbols, except the first symbol (see Table 1). Likewise, the length of i th binary string would be the length all symbols yet to be binarized. Mathematically, the length can be written as N
1 − i−1 j=1 p(Y i ) , where p(Y i ) is the probability of i th symbol. The total number of binary assignment would be m i=1 N 1 − i−1 j=1 p(Y i ) .
As can be seen, the computational complexity of the binarization and de-binarization process is linear in terms of the input data length.
Conclusion: Advantages and Applications
The proposed binarization scheme has the following advantages over others. Firstly, it is optimal for every data set. As proved and shown in this paper, the binarization scheme conserves entropy of m-ary data source. Secondly, the proposed method eliminates the need for knowing the source symbols at all. It works optimally without the knowledge of source because the binarization of the source symbols can occur in any order as shown Table 1, and all orders conserve m-ary source entropy, which can be inferred from the derivation shown in section 3. Thirdly, adding to the previous point, the coding is independent of the occurrence of the source symbols. In other words, any source symbol can be encoded in any order subject to the constraint that decoding is performed in the same order. The optimality is independent of the source order in the data set. Fourthly, unlike variable length codes, there is no need to know the probability distribution of the source symbols beforehand. It can be updated as the symbols occur. However, even without the knowledge of probability distribution, the presented method is optimal. Lastly, it has low complexity that is feasible for practical data compression.
One of the immediate usage of the presented binarization technique is in CABAC used in video and image compression. In addition, CABAC with the proposed binarization scheme can potentially replace Context-based Adaptive Arithmetic Coding used in various image compression standards [35], including JPEG2000 [36]. Furthermore, the binarization scheme can be applied to all the universal compression algorithms that have less complexity and resource requirements for binary data, than m-ary data.
Table 1 :
1An example of binarization processBinarization Order
Iteration
First
Second
Third
Data
AABCBACBBACCABACB
BCBCBBCCBCB
CCCCC
ABC
11000100010010100
10101100101
11111
Data
AABCBACBBACCABACB
BCBCBBCCBCB
BBBBBB
ACB
11000100010010100
01010011010
111111
Data
AABCBACBBACCABACB
AACACACCAAC
CCCCC
BAC
00101001100001001
11010100110
11111
Data
AABCBACBBACCABACB
AACACACCAAC
AAAAAA
BCA
00101001100001001
00101011001
111111
Data
AABCBACBBACCABACB AABBABBAABAB BBBBBB
CAB
00010010001100010
110010011010
111111
Data
AABCBACBBACCABACB AABBABBAABAB AAAAAA
CBA
00010010001100010
001101100101
111111
Table 2 :
2An example of de-binarization process De-binarization OrderIteration
First
Second
Third
ABC
11000100010010100
AA101A011A00A1A01
AAB1BA1BBA11ABA1B
Data
AA000A000A00A0A00 AAB0BA0BBA00ABA0B AABCBACBBACCABACB
ACB
11000100010010100
AA010A100A11A0A10
AA1C1AC11ACCA1AC1
Data
AA000A000A00A0A00 AA0C0AC00ACCA0AC0 AABCBACBBACCABACB
BAC
00101001100001001
11B0B10BB1001B10B
AAB1BA1BBA11ABA1B
Data
00B0B00BB0000B00B AAB0BA0BBA00ABA0B AABCBACBBACCABACB
BCA
00101001100001001
00B1B01BB0110B01B
11BCB1CBB1CC1B1CB
Data
00B0B00BB0000B00B 00BCB0CBB0CC0B0CB AABCBACBBACCABACB
CAB
00010010001100010
110C01C001CC101C0
AA1C1AC11ACCA1AC1
Data
000C00C000CC000C0 AA0C0AC00ACCA0AC0 AABCBACBBACCABACB
CBA
00010010001100010
001C10C110CC010C1
11BCB1CBB1CC1B1CB
Data
000C00C000CC000C0
00BCB0CBB0CC0B0CB AABCBACBBACCABACB
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| [] |
[
"Privacy-Preserving Multi-Operator Contact Tracing for Early Detection of Covid19 Contagions",
"Privacy-Preserving Multi-Operator Contact Tracing for Early Detection of Covid19 Contagions"
] | [
"Davide Andreoletti \nNetworking Laboratory\nUniversity of Applied Sciences of Southern Switzerland\nMannoSwitzerland\n",
"Omran Ayoub \nDipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly\n",
"Silvia Giordano \nNetworking Laboratory\nUniversity of Applied Sciences of Southern Switzerland\nMannoSwitzerland\n",
"Massimo Tornatore \nDipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly\n",
"Giacomo Verticale \nDipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly\n"
] | [
"Networking Laboratory\nUniversity of Applied Sciences of Southern Switzerland\nMannoSwitzerland",
"Dipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly",
"Networking Laboratory\nUniversity of Applied Sciences of Southern Switzerland\nMannoSwitzerland",
"Dipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly",
"Dipartimento di Elettronica, Informazione e Bioingegneria\nPolitecnico di Milano\nMilanoItaly"
] | [] | The outbreak of coronavirus disease 2019 (covid-19) is imposing a severe worldwide lock-down. Contact tracing based on smartphones' applications (apps) has emerged as a possible solution to trace contagions and enforce a more sustainable selective quarantine. However, a massive adoption of these apps is required to reach the critical mass needed for effective contact tracing. As an alternative, geo-location technologies in next generation networks (e.g., 5G) can enable Mobile Operators (MOs) to perform passive tracing of users' mobility and contacts with a promised accuracy of down to one meter. To effectively detect contagions, the identities of positive individuals, which are known only by a Governmental Authority (GA), are also required. Note that, besides being extremely sensitive, these data might also be critical from a business perspective. Hence, MOs and the GA need to exchange and process users' geo-locations and infection status data in a privacy-preserving manner. In this work, we propose a privacy-preserving protocol that enables multiple MOs and the GA to share and process users' data to make only the final users discover the number of their contacts with positive individuals. The protocol is based on existing privacy-enhancing strategies that guarantee that users' mobility and infection status are only known to their MOs and to the GA, respectively. From extensive simulations, we observe that the cost to guarantee total privacy (evaluated in terms of data overhead introduced by the protocol) is acceptable, and can also be significantly reduced if we accept a negligible compromise in users' privacy. | 10.1109/gcwkshps50303.2020.9367403 | [
"https://arxiv.org/pdf/2007.10168v1.pdf"
] | 220,646,768 | 2007.10168 | dd873e730be3050be041442c0a31ff46151aa4a1 |
Privacy-Preserving Multi-Operator Contact Tracing for Early Detection of Covid19 Contagions
Davide Andreoletti
Networking Laboratory
University of Applied Sciences of Southern Switzerland
MannoSwitzerland
Omran Ayoub
Dipartimento di Elettronica, Informazione e Bioingegneria
Politecnico di Milano
MilanoItaly
Silvia Giordano
Networking Laboratory
University of Applied Sciences of Southern Switzerland
MannoSwitzerland
Massimo Tornatore
Dipartimento di Elettronica, Informazione e Bioingegneria
Politecnico di Milano
MilanoItaly
Giacomo Verticale
Dipartimento di Elettronica, Informazione e Bioingegneria
Politecnico di Milano
MilanoItaly
Privacy-Preserving Multi-Operator Contact Tracing for Early Detection of Covid19 Contagions
Index Terms-Mobile OperatorsPrivacyCovid19
The outbreak of coronavirus disease 2019 (covid-19) is imposing a severe worldwide lock-down. Contact tracing based on smartphones' applications (apps) has emerged as a possible solution to trace contagions and enforce a more sustainable selective quarantine. However, a massive adoption of these apps is required to reach the critical mass needed for effective contact tracing. As an alternative, geo-location technologies in next generation networks (e.g., 5G) can enable Mobile Operators (MOs) to perform passive tracing of users' mobility and contacts with a promised accuracy of down to one meter. To effectively detect contagions, the identities of positive individuals, which are known only by a Governmental Authority (GA), are also required. Note that, besides being extremely sensitive, these data might also be critical from a business perspective. Hence, MOs and the GA need to exchange and process users' geo-locations and infection status data in a privacy-preserving manner. In this work, we propose a privacy-preserving protocol that enables multiple MOs and the GA to share and process users' data to make only the final users discover the number of their contacts with positive individuals. The protocol is based on existing privacy-enhancing strategies that guarantee that users' mobility and infection status are only known to their MOs and to the GA, respectively. From extensive simulations, we observe that the cost to guarantee total privacy (evaluated in terms of data overhead introduced by the protocol) is acceptable, and can also be significantly reduced if we accept a negligible compromise in users' privacy.
I. INTRODUCTION
Following recent surge of the coronavirus disease (Covid-19) epidemic, various governmental and organizational bodies are expressing strong interest in employing mobilecommunication technologies to early detect contagions. With the term 'early detection', we refer to the identification of positive individuals before they show any symptoms. This generally happens during the incubation period of the virus (around 5 days for Covid-19 [1]), or even during the entire course of the disease. As asymptomatic people unknowingly diffuse the virus, early detection is fundamental to drastically limit virus spread [1].
Several smartphone-based apps for early detection are already available [2]- [4]. These apps allow a user to know whether she encountered positive individuals or not (e.g., by correlating her mobility with that of known positive cases). In most countries, to comply with strict local privacy regulations, these apps are developed with privacy as a primary design constraint. However, app-based approaches suffer from several drawbacks. First, it is hard to reach the critical mass needed for an effective contact tracing (a typical safe number is 60% of population, a very challenging target [3]). In addition, these apps require the continuous use of data acquisition technologies (e.g., the GPS and the Bluetooth) that extensively consume devices' batteries. We also note that people is less likely to keep such apps installed on their smartphones at the very beginning of the epidemics, when early detection is decisive to contain the diffusion of the virus.
Contact tracing exploiting users' mobility data collected by mobile telecom operators (MOs) is regarded as a promising alternative to app-based solutions [5]. In the upcoming years, once 5G will have consolidated its penetration, MOs will possess technologies to perform a continuous and accurate tracking of users devices. For instance, Ref. [6] shows that the average accuracy of device positioning in ultra-dense 5G networks will be on the sub-meter order. A great advantage of a passive and continuous positioning is the very limited involvement of the final users. Users are not required to install any application on their smartphones, but only to give an explicit consent to track their position (explicit consent that is currently commonly granted to several apps, such as [2], [4]), therefore more easily reaching the critical mass. Even though users (and governments) are becoming more concerned regarding possible violation of the privacy of positioning data by the MOs [7] (e.g., a MO might sell them to third parties), we remark that MOs already estimate subscribers' positions to improve their services [8], and that the proposed approach guarantees that MOs do not get any sensitive data beyond it.
To effectively obtain an early detection of infections through mobility tracing, in addition to users' mobility data discussed above, the identities of positive individuals, which are only known to a Governmental Authority (GA) through the nation medical institutions, are required, and this is (a possibly even more)-sensitive information that must not be exposed. Therefore, MOs and the GA are required to collect, exchange and process this mobility data (from MO) and infection data (from the GA) in a secure and a privacy-preserving manner.
In this work, we propose a privacy-preserving protocol that enables GA and MOs to securely share and process users' data, such that each user is guaranteed to be the only person who knows the number of contacts with positive individuals she had (henceforth referred to as user's score). The protocol is built on consolidated privacy-enhancing strategies (e.g., secure secret sharing and homomorphic encryption) that guarantee total privacy to users, i.e., the mobility and the infection status of a user are only known to her MO and to the GA, respectively. This privacy is achieved at an acceptable cost in terms of data overhead exchanged among MOs and GA, as shown from extensive simulations. With slight modifications, the proposed protocol can also be employed i) to make this user discover how many positive people were in her same locations, but not necessarily in her close proximity (e.g., in a pub) and ii) to make the GA know only the identities of the users with a score above a given threshold. The identification of these users would make it possible to more easily stop the diffusion of the virus, but it poses a privacy dilemma and does not comply to several privacy regulations. In this work, we only provide the technical means to realize such identification in a privacy-preserving manner. Note also that, in case the number of these users is high, such procedure requires the exchange of a significant data overhead. However, we also show that this overhead can be heavily reduced at a negligible reduction of users' privacy.
The rest of the paper is structured as follows: in Section II we briefly review some existing approaches for privacypreserving contact tracing. Section III describes the involved entities and their privacy requirements. We present the building blocks of the proposed protocol and the protocol itself in Sections IV and V, respectively. In Section VI we show some illustrative results obtained by simulation. Finally, Section VII concludes the paper.
II. RELATED WORK
Existing solutions for contact tracing are generally based on smartphone apps of two main types: i) location-based (e.g., PrivateKit from MIT [2]) in which user's locations are acquired (e.g., with the GPS technology) and correlated with the locations of positive individuals; ii) token-based (e.g., TraceTogether [3] and Immuni [4]) that exchange anonymized tokens with smartphones in the proximity of the user (i.e., by exploiting the Bluetooth), and successively match the received tokens with those of known positive persons. A user who is tested positive can deliberately share her data (either location or received tokens) with a trusted authority, who then broadcasts it to all the others. Based on this, the app returns if a user has been in contact with a positive individual [2]. As operations are done on users' devices, privacy is mostly preserved, i.e., users' location and contacts are not exposed to the authority.
However, several privacy issues are still pending. For instance, in location-based apps each user receives the location data of a positive person, whose identity might be obtained from re-identification attacks [3], [9]. In TraceTogether [3] users send their phone numbers and all the received tokens to the authority, which in turn sends a message to those users who met some positive person. As users' contacts are exposed to the authority, this solution would hardly be adopted in countries with strict privacy laws, and several solutions are proposed to solve this issue [3]. For example, users might send to the authority only their tokens, and then perform anonymized queries to know if they met some positive person. However, other malicious behaviors are possible given that users obtain the tokens of persons in their proximity. Specifically, a user can craft a query to discover if the person that she met at a given time is positive [3].
In this work, we exploit consolidated privacy-preserving techniques (e.g., as those employed in [10], [11]) to compute the number of contacts that a user had with positive people, while guaranteeing that users' contacts, locations and infection status are not disclosed to illegitimate parties (see subsection III-B for further details). Differently from tokenbased solutions that only detect users' proximity, our protocol allows to compute also the number of positive persons within a given place. Unlike existing location-based apps, however, we assume that users' locations are estimated by MOs without any involvement of the users (e.g., using techniques for accurate geo-localization from cellular signals, such as those proposed in [6]). In this respect, authors in [5], [12] argue that MOs might play a decisive role in fighting the spreading of a virus, provided that users' privacy is guaranteed.
III. MODELING OF INVOLVED ENTITIES
In this Section, we formally define the concept of users' scores, and we describe the role and privacy requirements of the entities involved in their computation. Before doing that, we introduce the concepts of contact and infection status. We say that two persons user i and user j have a contact iff the distance between them is below a given threshold th. We encode this information in the binary variable c
(t) ij = 1 if Dist(loc (t) i , loc (t) j ) < th, and 0 otherwise, where loc (t) i
and loc (t) j refer to the geo-location (e.g., latitude and longitude) of user i and user j at time t, respectively, while Dist is a measure of geographical distance. Concerning the infection status, we then introduce the binary variable s (t) i = 1 in case user i is considered positive at time t, and 0 otherwise. Score i is the number of contacts that user i has, during a given period of time, with positive individuals. In formulas:
Score i = t j:cij =1 s (t) j (1) Similarly, Score (Loc) i
is the number of positive persons that were in a certain location Loc at the same time of user i , and is computed as Score
(Loc) i = j∈Loc s (t)
j , where the considered locations are assumed to be chosen by user i herself.
A. Role of Involved Entities
The GA is an entity established by the government to monitor the infection status of individuals within a certain region and, specifically, to collect from medical institutions the identities of positive individuals willing to share this data.
MOs are instead telecom companies that provide mobile connectivity within the considered region. Without loss of generality, we assume that each user is served by only one MO, and that the whole area is covered by all the MOs. Then, we also assume that MOs estimate the locations of their users at time t, i.e.,l oc (t) i , ∀i from cellular signals received by users' devices (e.g., as done in [6]).
B. Privacy Requirements and Security Models
We assume that the GA and the MOs are honest-butcurious, i.e., they honestly execute the protocol but also try to violate other parties' privacy from the received data. Privacy requirements for each type of data are illustrated below.
1) Users' Locations: estimates of a user's locations should only be known to her MO.
2) Users' Contacts: information regarding contacts between two users should only be known to their MOs. In addition, if these users are subscribers of different MOs, each MO should not know anything neither about the identity of the other MO's user, nor about the number of contacts between its users and any other user of its competitors (e.g., how many contacts user i and user j have during a given period).
3) Users' Infection Status and Scores: The infection status of a user should only be known to the GA and to the user herself (say user i ). Score i and Score (Loc) i should only be known to user i , except when Score i is greater than a threshold χ. In this case, Score i and the identity of user i might also be known to the GA (see subsection V-C for the details).
IV. BUILDING BLOCKS OF THE PRIVACY-PRESERVING PROTOCOL
A. Existing Privacy-Preserving Building Blocks 1) Shamir Secret Sharing: A Shamir Secret Sharing (SSS) scheme [13] allows to securely distribute a secret s among a set of participants in such a way that s can only be recovered if a sufficient number of them cooperate. The piece of secret s that each participant receives is called share, and it is referred to as s . In this work, we employ a (2, 2) SSS, i.e., s is reconstructed only if 2 out of the 2 considered participants cooperate. SSS has several homomorphic properties, i.e., each participant can perform several operations on the shares that result in the same operations over the corresponding secrets (e.g., linear combinations). Then, participants can compute s 1 ·s 2 using the Mult protocol presented in [14], or they can use the EQ and Comp protocols [14] to perform the equality check and the comparison operations. In the latter, participants input their shares s 1 and s 2 and obtain the share b eq (resp., b ge ), where b eq = 1 (resp., b ge = 1) iff s 1 = s 2 (resp., s 1 ≥ s 2 ) and 0 otherwise.
2) Paillier Cryptosystem: Paillier [15] is a secure cryptosystem with the following properties: i) it is asymmetric, i.e., anyone can encrypt a message, but only the owner of the private key can decrypt it; ii) it is probabilistic, i.e., two encryptions of the same plaintext yield different ciphertexts and iii) it is homomorphic with respect to the summation of two ciphertexts (computed as Enc(m 1 + m 2 ) = Enc(m 1 ) · Enc(m 2 )) and to the product between a ciphertext and a plaintext (computed as Enc(m 1 · m 2 ) = Enc(m 1 ) m2 ).
B. New Privacy-Preserving Primitives based on SSS 1) Secure Square Distance: the Secure Square Distance module takes in input the shares of the coordinates of points i and j, i.e., x i , y i , x j , y j and returns d 2 ij , where d ij is the euclidean distance between these points. This module is based on the Mult subroutine.
2) ObliviousTransfer: the ObliviousTransfer module (OT) allows a sender to deliver some data to a receiver without knowing which data has been transmitted. OT inputs i) a set of 2N shared elements arranged into a table with N rows and two columns (namely, attribute and value) and ii) the share attribute x . This module is based on the Mult and EQ subroutines and outputs the share value i if the attribute at row i is equal to attribute x , and 0 otherwise. This value is computed as value i = N j=1 eq jx ·value j , where eq jx = 1 if attribute x = attribute j , and 0 otherwise.
V. THE PRIVACY-PRESERVING PROTOCOL
The proposed protocol works in three main phases, namely contact tracing, score computation and communication with users. We describe these phases in the following subsections. We refer to the generic users user i and user j as subscribers of M O k and M O k , respectively, but the described operations are valid for each user and MO.
A. Privacy-Preserving Contact Tracing
In this phase, M O k obtains the binary value c ij encoding the information about its generic user i 's contacts, ∀i. Firstly, M O k estimates the current location of user i , i.e., (l at
(t) i ,long (t)
i ) by analyzing cellular signals coming from her device [6]. From this data, the M O k can independently assess the contacts among its subscribers, but not with other MOs' users (since a free exchange of users' mobility data is prohibited by the considered privacy requirements). Hence, we propose to perform the privacy-preserving computation of c ij as follows.
M O k and M O k compute the projections of their users' estimated positions on an euclidean plane (e.g.,x (t) i ,ŷ (t) i ) and exchange these values among them in form of secret shares. Then, they execute the Secure Square Distance module and obtain d 2 ij , being d 2 ij the squared euclidean distance between the generic user i and user j . The Comp module is then employed to compare d 2 ij with the threshold th 2 and obtain c ij . The MOs finally exchange these shares and recover the secret c ij (that is 1 if user i and user j has a contact, and 0 otherwise). A representation of this phase is depicted in Fig. 1.
B. Secure Computation of Users' scores
Id N k = N ame N k ||P honeN umber N k Enc GA (score N k )
with each other Enc GA (s The obtained data are then arranged by M O k in a table that we represent in Table I. Such table has N k rows (one of each subscriber of M O k ) and three columns, which are Index, Identity and Score. The first refers to the index of the row at which a certain user's data is stored. Without loss of generality, we assume that user i 's data is stored at the i-th row. The second one stores the identities of the users (e.g., anything allowing to univocally identify them, such as full names and telephone numbers). The third represents the Score values of users in encrypted form.
C. Communication with users
In this phase, we show how to distribute users' scores only to the legitimate entity (i.e., either the user herself or the GA). We consider the scenarios of User-Triggered Communication and GA-Triggered Communication. In the former, scores are requested by user i herself, and are kept secret to any other entity. In the latter, the GA identifies only the users with a score greater than a given threshold χ.
1) User-Triggered Communication: user i directly asks to M O k the values Enc GA (score i ) and Enc GA (score (Loc) i ), for any location she is interested in. Then, user i exploits the homomorphic properties of the Paillier cryptosystem to compute Enc GA (Score i ·T oken i ), where T oken i is a random value known only to her. Enc GA (Score i · T oken i ) is then sent to the GA, which deciphers it and sends Score i · T oken i back to user i . Finally, user i removes the mask T oken i and obtains Score i . A similar computation is performed to obtain score (Loc) i . We represent this phase in Fig. 2. As SSS is proven information-theoretic secure [13], no information about locations is obtained from the single shares owned by each MO.
2) Users' Contacts: at each execution of the contact tracing phase, pairs of MOs distribute to each other new shares of their users' locations. This prevents a leakage of users' identities (which cannot be inferred from locations' shares), as well as from counting the number of contacts between two users. Then, during the score computation phase, M O k can homomorphically compute Enc GA (s j + 0) (which yields a different ciphertext without altering the hidden infection status), in such a way that M O k cannot count the number of contacts between user i and user j .
3) Users' Infection Status and Scores: M O k computes user i 's scores by performing homomorphic summations on values encrypted by the GA but, since it does not know the private encryption key, it does not discover any plaintext. Then, in the user-triggered communication scenario, user i sends Enc GA (s i · T oken i ) to the GA. As the latter does not know T oken i , it cannot obtain the actual values of the scores. Finally, in the GA-triggered communication scenario, GA and M O k execute the OT module. From this execution, the GA learns the identity of user i and M O k learns nothing. As the GA is considered a honest-but-curious entity, we assume that it executes the OT module only to identify users with the highest chance to be positive (i.e., if score i ≥ χ). Clearly, the GA might execute this module regardless of the value of score i and learn the identity and scores of all the users. In the next subsection, we discuss a possible extension of the protocol to cope with this malicious behaviour of the GA.
E. Extension of the protocol for dishonest participants
We now describe how the protocol can be improved to address two malicious schemes. In the first one, the GA tries to obtain the identity of user i when Score i < χ. The proposed solution works as follows: M O k selects two random variables τ 1 and τ 2 and computes Enc GA (τ 1 · score x + τ 2 ), ∀x. These values are then sent to the GA in form of secret share, i.e., Enc GA (τ 1 · score x + τ 2 ) , ∀x and given in input to the OT module. From its execution, M O k and the GA obtain Enc GA (τ 1 · score i + τ 2 ) . M O k sends its share to the GA, which can then recover the secret Enc GA (τ 1 · score i + τ 2 ) and, from it, the plaintext τ 1 · score i + τ 2 . Finally, the GA sends to M O k both τ 1 · score i + τ 2 and score i . Since the GA never obtains the values τ 1 and τ 2 , it cannot counterfeit a score i ≥ χ and a corresponding valid τ 1 · score i + τ 2 . M O k detects a cheat if score i < χ or the actual τ 1 · score x + τ 2 cannot be computed from score i . If the GA does not cheat, the OT module is executed again as previously described, and the GA obtains identity i .
In the second malicious scheme, M O k counterfeits the encryption infection status of user i . To address this issue, the GA sends to M O k the infection status of users multiplied by a constant, e.g., Enc GA (s i · T oken GA ), where T oken GA is known to the GA only. The GA detects a cheat if the ciphered score computed by M O k does not decrypt to a multiple of T oken GA (i.e., Score i · T oken GA ).
VI. ILLUSTRATIVE NUMERICAL RESULTS
A. Simulation Settings
We perform our experiments considering a population of N = 1.5 millions users, whose mobility is traced every 20 seconds within an overall period of 1 hour. The initial position of the generic user i is given by x i = R i cos(θ i ), y i = R i sin(θ i ), being R i and θ i two random variables that follow the Gaussian distribution (with zero mean and standard deviation equal to 3800m) and the uniform distribution defined over [0, 2π], respectively. Users move following the Gauss-Markov model [16] (40% of them at an average speed of 0.01m/s, 40% at 1m/s and the remaining 20% at 14m/s). The region occupied by the population is 1900km 2 large, and is covered by K ∈ [2,5] MOs, who have the same number of subscribers N K . 1% of the whole population is assumed to be currently positive.
B. Data Overhead
We now show the overhead generated in each phase of execution of the protocol, being b the bit-length of the shares exchanged by participants (in our simulations b = 25 bits). 1) Contact tracing phase: We assume that two users have a contact if their distance is below th = 2m. The overhead generated to evaluate if there is a contact is 18b 2 + 10b. The number of these evaluations depend on the number of users currently located within a given area, which in turns depends on its size. To avoid comparisons among users with a negligible probability to meet, we assume that contacts are searched within non-overlapping squares of size l. In Fig. 3, we show the average and maximum overhead generated by each MO to execute the contact tracing phase over an area of size l ∈ {10, 35, 60, ..., 285} meters. From this figure, we observe a super-linear increase of both the average and maximum overhead per area with increasing l. We also notice that the overhead is higher when decreasing the number of involved MOs K. While the average overhead is always less than 6.8 Mbytes, the maximum overhead grows significantly with l. As an example, when K = 2 the maximum overhead goes from 0.01 to 134 Mbytes when l goes from 10 to 285 meters.
2) Score Computation phase: With 4096 bit-long ciphertexts [17], the overhead at each execution of the score computation phase (every 1 hour in our simulations) is 768 Mbytes from the GA to the MOs (i.e., obtained by delivering the infection status of users), and 153 Mbytes among MOs (i.e., obtained by exchanging the infection status of their users in case of contact).
3) Communication with users phase: The overhead generated in the user-triggered communication is negligible (i.e., 1.5 Kbytes/user). On the other hand, the GA-triggered communication generates a total overhead of 14N χ N k b 2 + 2N χ N k b + N b bits, where N χ is the number of users whose score is ≥ χ. For instance, for K = 5 and χ = 10 the overhead is 4650 Mbytes. Although this value can be considered acceptable, we note that it would be much higher if longer periods and a higher number of users were considered. To reduce this overhead, the GA sends to the MO both the share index i and a range [index i −η − , index i +η + ] that indicates the rows of Table I in which the identity of user i should be searched. Since the GA-triggered communication is issued only for users with a score ≥ χ, the MO discovers that one among its users with index ∈ [index i − η − , index i + η + ] in Table I has a higher-then-average chance to be positive. Hence, there is a trade-off between overhead and user i 's privacy. We measure privacy as the probability that her MO discovers that user i has a score ≥ χ, i.e., privacy i = 1− 1 η , where η = η − +η + . In Fig. 4, we show the trade-off between overhead and privacy with varying η ∈ [1,200], for χ ∈ [10,20,30,40]. From this figure, we observe that a very high level of privacy can be reached at a remarkable reduction of the overhead. For instance, for χ = 10 the overhead drops from 4650 to 5 Mbytes if we accept 99.5% of the total privacy.
C. Computational Time
In Table II we show the average and maximum time needed to compute the contacts among users, for several values of l. We note that l should not exceed 85 meters to allow a sampling of users' mobility every 20 seconds. Then, the GA-triggered communication for a single user takes τ · η, with τ = 6ms on a Intel Core I7 computer. When K = 5, the identities of users with score ≥ 10 are obtained in 66 minutes if total privacy is considered (i.e., if η = 3 · 10 5 ). This value drops to 2.64 seconds if the 99.5% of privacy is considered sufficient.
VII. CONCLUSION
We proposed a privacy-preserving protocol that enables a GA (owning users' infection status) and several MOs (owning accurate estimations of users' positions) to compute the number of contacts that users have with positive persons, during a considered period. The protocol guarantees that such measure is only obtained by the legitimate user, and that her infection status and mobility data are known, respectively, only to her MO and to the GA. The protocol can also be employed i) to make a user know the number of positive people who stayed in her same area (even thought not in close contact with her) and ii) to make the GA discover the identities only of users with the highest chance to be positive. We evaluated the cost of privacy in terms of overhead generated by the protocol. From extensive simulations, we observed that the overhead is acceptable, and can further be reduced at a negligible reduction of users' privacy.
In this phase, M O k securely computes the score values of user i . To do so, the GA sends to M O k the infection status (in encrypted form) of user i during a considered period (e.g., in the last day), i.e., Enc GA (s(t) i ), ∀t. At each time instant of the considered period, M O k and M O k obtain c (t) ij as described in the previous subsection. If this value is 1 (i.e., there is a contact between these users at time t), M O k and M O k exchange Fig. 1: Positioning process and privacy-preserving contact tracing performed by a pair of MOs
.
Then, M O k computes Enc GA (Score i ) by homomorphically executing the summation in Eq. 1. Similarly, M O k computes Score
summing the encrypted infection status of all users within area Loc at a given time, which are asked to all the remaining MOs.
Fig. 2 :
2Representation of the User-triggered communication 2) GA-Triggered Communication: M O k sends to the GA index x and Enc GA (score x ), ∀x. Then, the GA deciphers Enc GA (score x ) and obtains (index x , score x ), ∀x. In case ∃i : score i ≥ χ, the GA and M O k jointly execute the OT subroutine described in subsection IV-B2. To do so, M O k sends to the GA index x and identity x , ∀x, while the GA sends to M O k index i . With these values in input, the OT module returns to M O k and to the GA their shares identity i . Finally, M O k sends its share identity i M O k to the GA, which combines it with identity i GA and recover the identity of user i . In the next subsection, we show how the proposed protocol fulfills the considered privacy requirements under the honest-but-curious security model. D. Fulfillment of Privacy Requirements 1) Users' Locations: during the contact tracing phase, estimated users' locations are distributed among pairs of MOs as secret shares.
Fig. 3 :
3Overhead of data exchanged by each MO, during the contact tracing phase, within areas of size l
Fig. 4 :
4Trade-off between data overhead and privacy with varying the range η, for several values of the threshold χ and K = 5
TABLE I :
IData of subscribers of M O kIndex
Identity
Score
1
Id 1 = N ame 1 ||P honeN umber 1
Enc GA (score 1 )
...
...
...
i
Id 1 = N ame i ||P honeN umber i
Enc GA (score i )
...
...
...
N k
TABLE II :
IITime needed to perform the contact tracing phase within a given subareaSize of Subareas l (meters)
10
35
60
85
110
135
160
185
210
235
260
285
Avg Timing (seconds)
2 · 10 −5
2.4 · 10 −2
1.8 · 10 −1
6.7 · 10 −1
1.7
3.8
7.2
12.4
20.2
30.8
45.7
64.1
Max Timing (seconds)
7 · 10 −2
1
5.1
15.6
37.7
78.3
143.5
244.1
388.5
599.1
892.7
1260.6
0
50
100
150
200
4.7
4.
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| [] |
[
"An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasma applications",
"An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasma applications"
] | [
"A Alvarez Laguna \nCentre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance\n\nLaboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance\n",
"T Pichard \nCentre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance\n",
"T Magin \nKarman Institute for Fluid Dynamics\nWaterloosesteenweg 721640Sint Genesius RodeBelgium\n",
"P Chabert \nLaboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance\n",
"A Bourdon \nLaboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance\n",
"M Massot \nCentre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance\n"
] | [
"Centre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance",
"Laboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance",
"Centre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance",
"Karman Institute for Fluid Dynamics\nWaterloosesteenweg 721640Sint Genesius RodeBelgium",
"Laboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance",
"Laboratoire de Physique des Plasmas\nCNRS\nSorbonne Université\nUniv. Paris Sud\nEcole Polytechnique\nF-91128PalaiseauFrance",
"Centre de Mathématiques Appliquées\nEcole Polytechnique\nRoute de Saclay91128Palaiseau CedexFrance"
] | [] | We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas. The model considers electrons and ions as separate fluids, comprising the electron inertia and charge separation. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed testcases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations. AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. Numerical methods for the ideal multi-fluid equations coupled to Maxwell's equations have been proposed by a 2 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. number of authors[39,40,41,42,43,44,45,46], for the study of plasma sheaths[47], and for the study of plasma expansion in vacuum with the isentropic electrostatic approximation[48,49,50,51,52]. The main difficulties of the low-temperature multi-fluid plasma model and the solutions proposed in the current literature are summarized in the following. First, the stability of explicit schemes for the multi-fluid equations solving for the Poisson equation requires the resolution of the electron plasma wave frequency[52]. Alvarez Laguna et al. [44,45]have proposed implicit time integration for the previous set of equations coupled to full Maxwell's equations. However, the inversion of the matrix is computationally costly, which can be improved by the use of GPUs[46]. Alternatively, a more advantageous approach is the asymptotic-preserving (AP) scheme proposed such as the one proposed by Degond et al. [50,51,52]. AP schemes [53] preserve the quasi-neutral asymptotic limit without the resolution of the Debye length nor the plasma frequency. These schemes do not need an implicit solver, which results in a significant advantage from the computational point of view. However, the AP scheme from [50, 51] is not able to tackle the asymptotic limit of the small electron-to-ion mass ratio, finding large errors in the velocity of the electrons even with high order schemes. Additionally, the AP scheme proposed in [50] demands the solution of a second order equation for the electric potential that requires the storage of the solution in two different time-steps, which increases the computational stencil.The second difficulty of the multi-fluid equations is due to the small mass of the electrons, which results in a very large electron speed of sound. However, in most of the cases, the electron fluid travels at much smaller speeds than the electron thermal speed. This corresponds to a low-Mach regime for electrons, which is known to give numerical problems due to an excessive numerical dissipation that restricts the time step of compressible solvers[54]. Different strategies are proposed in order to tackle the low-Mach regime in compressible solvers. One method are the so-called all-regime flux-splitting methods such as the AUSM + -up [55] (applied to the multi-fluid equations in[44,45,46]) or the preconditioning methods to remove the stiffness of the low-Mach regime[56]. Similarly, the operator-splitting Lagrange-projection scheme is combined to the preconditioning method by Chalons et al.[57]. Additionally, the AP schemes are also used to tackle the incompressible asymptotic behaviour in the Euler equations[58].The third difficulty is related to the regime where the temperature of the ions is much lower than the electron temperature. In low-temperature plasmas, the electric potential, in general, scales as the electron temperature (in eV). This results in a Lorentz force that is much larger than the ion pressure flux. The main problem arises when the ion convective flux is treated with an upwind scheme and the Lorentz force (which involves the gradient of the potential) with a cell-centered scheme. The upwind scheme provides accurate non-oscillatory solutions in homogeneous problems, but they can lose accuracy in the presence of stiff source terms[59]. A solution to this problem is to use well-balanced schemes that upwind the source terms in a consistent manner[60,61,62].In the present work, we propose a numerical scheme that addresses these three problems and design three dedicated discriminating test-cases in order to benchmark such a method. Regarding the stiffness introduced by the smallness of the Debye length and the electron mass, we propose a novel operator splitting method based on the all-regime Lagrange-projection [57] coupled to the Poisson equation. We retrieve a numerical scheme that has the AP property of the quasi-neutral limit with massless electrons. Therefore, neither the Debye length nor the electron plasma frequency need to be resolved to be stable and accurate, without the need of an implicit solver. As compared to previous AP schemes for the multi-fluid equations[50,51], our approach does not need to solve an equivalent second order equation for the electric potential, but it solves the standard Poisson equation. This results in a much simpler algorithm for the potential. Additionally, the main advantage is that the numerical scheme can simultaneously tackle the problem of the small mass of electrons. The solution of these two problems with a unique AP scheme is an original contribution of this work. This can be considered a step forward that could help to reduce the numerical cost of solving the electron fluid dynamics coupled to the one of ions. Finally, a well-balanced discretization of the Lorentz force in the ion momentum equation is proposed. In the results we show that an incorrect discretization of this term can lead to spurious numerical instabilities, which can be avoided with well-balanced schemes.The numerical scheme is then benchmarked against three numerical set-ups that allows us to assess our strategy in terms of accuracy and computational cost. The first one simulates a quasi-neutral periodic perturbation in a thermal plasma. This case is used to test the asymptotic preserving property of the discretization for a small Debye length and a small electron-to-ion mass ratio. The same case is reproduced in a low-temperature plasma in order to assess the well-balanced discretization of the ions. We demonstrate that the proposed numerical strategy allows for a dramatic reduction of the computational time in cases with quasi-neutral plasma, as compared to a standard discretization. Finally, a low-temperature plasma discharge between two floating walls is simulated. This realistic set-up is able to capture the physics of the electrically charged plasma sheath coexisting with a quasi-neutral bulk and encompasses most of the difficulties of more realistic configurations, while amenable to detailed analysis of the proposed methods. | null | [
"https://arxiv.org/pdf/1904.13092v1.pdf"
] | 140,222,397 | 1904.13092 | e2ef6f31ec8ab9e27e1f2495ca3b553ed38b9ea3 |
An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasma applications
A Alvarez Laguna
Centre de Mathématiques Appliquées
Ecole Polytechnique
Route de Saclay91128Palaiseau CedexFrance
Laboratoire de Physique des Plasmas
CNRS
Sorbonne Université
Univ. Paris Sud
Ecole Polytechnique
F-91128PalaiseauFrance
T Pichard
Centre de Mathématiques Appliquées
Ecole Polytechnique
Route de Saclay91128Palaiseau CedexFrance
T Magin
Karman Institute for Fluid Dynamics
Waterloosesteenweg 721640Sint Genesius RodeBelgium
P Chabert
Laboratoire de Physique des Plasmas
CNRS
Sorbonne Université
Univ. Paris Sud
Ecole Polytechnique
F-91128PalaiseauFrance
A Bourdon
Laboratoire de Physique des Plasmas
CNRS
Sorbonne Université
Univ. Paris Sud
Ecole Polytechnique
F-91128PalaiseauFrance
M Massot
Centre de Mathématiques Appliquées
Ecole Polytechnique
Route de Saclay91128Palaiseau CedexFrance
An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasma applications
Low-temperature plasmasFinite Volume MethodAsymptotic-preserving schemeWell-balanced schemeMulti-fluid model
We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas. The model considers electrons and ions as separate fluids, comprising the electron inertia and charge separation. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed testcases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations. AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. Numerical methods for the ideal multi-fluid equations coupled to Maxwell's equations have been proposed by a 2 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. number of authors[39,40,41,42,43,44,45,46], for the study of plasma sheaths[47], and for the study of plasma expansion in vacuum with the isentropic electrostatic approximation[48,49,50,51,52]. The main difficulties of the low-temperature multi-fluid plasma model and the solutions proposed in the current literature are summarized in the following. First, the stability of explicit schemes for the multi-fluid equations solving for the Poisson equation requires the resolution of the electron plasma wave frequency[52]. Alvarez Laguna et al. [44,45]have proposed implicit time integration for the previous set of equations coupled to full Maxwell's equations. However, the inversion of the matrix is computationally costly, which can be improved by the use of GPUs[46]. Alternatively, a more advantageous approach is the asymptotic-preserving (AP) scheme proposed such as the one proposed by Degond et al. [50,51,52]. AP schemes [53] preserve the quasi-neutral asymptotic limit without the resolution of the Debye length nor the plasma frequency. These schemes do not need an implicit solver, which results in a significant advantage from the computational point of view. However, the AP scheme from [50, 51] is not able to tackle the asymptotic limit of the small electron-to-ion mass ratio, finding large errors in the velocity of the electrons even with high order schemes. Additionally, the AP scheme proposed in [50] demands the solution of a second order equation for the electric potential that requires the storage of the solution in two different time-steps, which increases the computational stencil.The second difficulty of the multi-fluid equations is due to the small mass of the electrons, which results in a very large electron speed of sound. However, in most of the cases, the electron fluid travels at much smaller speeds than the electron thermal speed. This corresponds to a low-Mach regime for electrons, which is known to give numerical problems due to an excessive numerical dissipation that restricts the time step of compressible solvers[54]. Different strategies are proposed in order to tackle the low-Mach regime in compressible solvers. One method are the so-called all-regime flux-splitting methods such as the AUSM + -up [55] (applied to the multi-fluid equations in[44,45,46]) or the preconditioning methods to remove the stiffness of the low-Mach regime[56]. Similarly, the operator-splitting Lagrange-projection scheme is combined to the preconditioning method by Chalons et al.[57]. Additionally, the AP schemes are also used to tackle the incompressible asymptotic behaviour in the Euler equations[58].The third difficulty is related to the regime where the temperature of the ions is much lower than the electron temperature. In low-temperature plasmas, the electric potential, in general, scales as the electron temperature (in eV). This results in a Lorentz force that is much larger than the ion pressure flux. The main problem arises when the ion convective flux is treated with an upwind scheme and the Lorentz force (which involves the gradient of the potential) with a cell-centered scheme. The upwind scheme provides accurate non-oscillatory solutions in homogeneous problems, but they can lose accuracy in the presence of stiff source terms[59]. A solution to this problem is to use well-balanced schemes that upwind the source terms in a consistent manner[60,61,62].In the present work, we propose a numerical scheme that addresses these three problems and design three dedicated discriminating test-cases in order to benchmark such a method. Regarding the stiffness introduced by the smallness of the Debye length and the electron mass, we propose a novel operator splitting method based on the all-regime Lagrange-projection [57] coupled to the Poisson equation. We retrieve a numerical scheme that has the AP property of the quasi-neutral limit with massless electrons. Therefore, neither the Debye length nor the electron plasma frequency need to be resolved to be stable and accurate, without the need of an implicit solver. As compared to previous AP schemes for the multi-fluid equations[50,51], our approach does not need to solve an equivalent second order equation for the electric potential, but it solves the standard Poisson equation. This results in a much simpler algorithm for the potential. Additionally, the main advantage is that the numerical scheme can simultaneously tackle the problem of the small mass of electrons. The solution of these two problems with a unique AP scheme is an original contribution of this work. This can be considered a step forward that could help to reduce the numerical cost of solving the electron fluid dynamics coupled to the one of ions. Finally, a well-balanced discretization of the Lorentz force in the ion momentum equation is proposed. In the results we show that an incorrect discretization of this term can lead to spurious numerical instabilities, which can be avoided with well-balanced schemes.The numerical scheme is then benchmarked against three numerical set-ups that allows us to assess our strategy in terms of accuracy and computational cost. The first one simulates a quasi-neutral periodic perturbation in a thermal plasma. This case is used to test the asymptotic preserving property of the discretization for a small Debye length and a small electron-to-ion mass ratio. The same case is reproduced in a low-temperature plasma in order to assess the well-balanced discretization of the ions. We demonstrate that the proposed numerical strategy allows for a dramatic reduction of the computational time in cases with quasi-neutral plasma, as compared to a standard discretization. Finally, a low-temperature plasma discharge between two floating walls is simulated. This realistic set-up is able to capture the physics of the electrically charged plasma sheath coexisting with a quasi-neutral bulk and encompasses most of the difficulties of more realistic configurations, while amenable to detailed analysis of the proposed methods.
Introduction
Low-temperature plasmas are fundamental to a wide range of technological applications -from lighting to semiconductor manufacturing and electric propulsion. These plasmas are created by an electrical discharge in a gas. The physical conditions where they occur are particularly diverse: pressures varying from below a millitorr to few hundred atmospheres, different excitation methods (e.g., inductively coupled plasma (ICP), capacitively coupled plasma (CCP), dielectric barrier discharge (DBD), magnetron), diverse geometries, and power ranges (see, e.g., [1,2]). For that reason, a number of different descriptions are used for their study, from kinetic approaches, e.g., Particle-In-Cell/Monte Carlo Collision (PIC-MCC) and Direct Simulation Monte Carlo (DSMC) method, to fluid and global models (see [3,4] for a review on the modeling and numerical approaches of low-temperature plasmas).
In atmospheric pressure discharges, the drift-diffusion approximation (e.g., [5,6,7,8,9,10,11]) is a valid simplification. However, under low-pressure conditions, the momentum transfer between species is smaller since the number of collisions decreases for decreasing pressures. This results in non-equilibrium conditions that may lead to a differential motion between the heavy species and electrons. This differential motion is responsible for plasma instabilities such as the two-stream instability [12] or the electron drift instability that is observed to cause anomalous transport in Hall thrusters [13,14,15,16,17,18,19]. Even though the electron inertial term is small in the electron momentum equations, these instabilities cannot be explained without it. Similarly, electron inertia cannot be completely neglected to fully explain the physics of sheaths [20], in electron sheaths and presheaths [21,22], in plasma sheath instabilities [23], Langmuir probes [24], in magnetized plasmas [25,26,27], and Hall thrusters [28,29].
For that reason, in low-pressure discharges, alternative models to the drift-diffusion approach such as kinetic and hybrid models, the plasma moment equations or the multicomponent non-equilibrium one velocity fluid model [30,31] are needed. Kinetic simulations provide a very accurate description of the state of the plasma, but are computationally very expensive. On the other hand, hybrid models [32,33,34] are a cheaper alternative that combines the fluid and the kinetic description. However, both methods need to resolve the Debye length and the electron plasma frequency if the time integration is explicit and the quasi-neutrality hypothesis is not assumed.
In Table 1, we present characteristic conditions of a typical low-temperature low-pressure Argon plasma RF discharge (from [1]). We first note that under these pressure conditions, the Knudsen number is of order one for electrons and ten times smaller for ions. The kinetic phenomena is important and transport models are needed to provide a closure for the plasma fluid equations. The second important feature of Table 1 is related to the smallness of the electron-to-ion mass ratio and the normalized Debye length. Most of the explicit methods (PIC methods, hybrid methods or deterministic discretization of the multi-fluid equations) require to resolve these scales in order to guarantee the stability of the scheme. However, if we consider an explicit 1D simulation of the discharge of Table 1 that resolves the Debye length and the plasma frequency with ten spatial and temporal points, the numerical set-up would need around 10 4 spatial points (in 1D) and 10 6 time steps to simulate one transit time of an ion acoustic wave. Therefore, this set-up involves a significant computational cost for a 1D simulation. Implicit methods can guarantee the numerical stability with larger time-steps. However, the Newton method to solve the linear system can become also very expensive.
In the present work, we consider the isothermal plasma fluid equations under collisionless conditions. This system of equations contains the numerical difficulties associated to the low-temperature moment plasma equations in electrostatic conditions. The model considers the first two moments of Vlasov's equation for electrons and ions while assuming both temperatures to be constant. The dynamics of the two fluids are coupled through the electric potential, which satisfies a Poisson equation. These equations are widely used in the sheath theory [35,36,37] and in the study of plasma waves [12]. Although kinetic phenomena play an important role in the plasma sheath [21], a fluid model that does not assume quasi-neutrality can potentially capture the interaction between the macroscopic scales and the sheaths.
The most general form of the multi-fluid plasma equations considers the electron inertial term in the electron momentum equation and charge separation effects. This is equivalent to considering a finite electron mass and a finite Debye length. As mentioned before, this allows for representing plasma instabilities such as the two-stream or the drift waves and charged regions of the plasma such as the sheaths. Nevertheless, both the electron mass and the Debye length are two very small parameters as compared to the ion and macroscopic scales. Consequently, these small scales impose very restrictive numerical constraints related to the resolution of the Debye length, electron plasma waves, and electron acoustic waves. Consequently, due to these requirements, the time-steps and mesh sizes are not significantly more advantageous than in the kinetic or hybrid approach. Moreover, the multiscale character of the problem can lead to very large discretization errors when the small scales are not properly resolved (see [38]). Outline of the paper. First, we present the set of equations, its normalization, and the asymptotic study when both the mass of electrons and the Debye length are very small as compared to the ion scales. Second, we present a standard discretization of the system in order to illustrate the difficulties of the numerical discretization. Third, we describe the operator splitting scheme and the well-balanced treatment of the ion source-terms while proving that the acoustic step preserves the asymptotic behaviour. Fourth, we simulate a two-stream perturbation in thermal and low-temperature plasmas. Finally, we self-consistently simulate a low-temperature discharge between two floating plates. The numerical scheme proves to be accurate both in the quasi-neutral limit and when the sheaths are included in the computational domain.
Set of equations and asymptotic behaviour
We consider the isothermal plasma (electron + ion) equations. The equations are obtained by taking the first two moments of the kinetic equation for electrons and ions (see, e.g., [63] for a derivation). These moments correspond to the mass and momentum balance laws for the two charged species. The system considers ionization reactions and, in the present paper, we neglect the effect of recombination and the elastic collisions. As explained in [1], in most discharges these scales are much smaller than the ionization scale. Nevertheless, the elastic collisions in low-pressure conditions do not impose a major difficulty from the numerical point of view and for that reason are not the focus of the present work. Finally, the system is closed with the Poisson equation for the electric potential.
The equations in dimensional form read
∂ t n e + ∂ x (n e u e ) = n e ν iz , (1a) ∂ t n i + ∂ x (n i u i ) = n e ν iz , (1b) m e ∂ t (n e u e ) + ∂ x m e n e u 2 e + p e = n e e∂ x φ,(1c)m i ∂ t (n i u i ) + ∂ x m i n i u 2 i + p i = −n i e∂ x φ,(1d)∂ 2 xx φ = n e − n i 0 e,(1e)
where n e and n i stand for the electron and ion number density respectively, and u e and u i the electron and ion velocities. The electron-impact ionization rate coefficient ν iz is a function of the electron temperature ν iz = n n K ion (T e ), for instance through Arrhenius' law, K ion (T e ) = A exp[−ε ion /(k B T e )], where the quantity ε ion is the ionization energy. The neutral number density n n is assumed to be constant. The partial pressures of the electron and ion fluids are assumed to obey the perfect gas law, p e = n e k B T e and p i = n i k B T i , where k B is Boltzmann's constant, and T e and T i , the electron and ion constant temperatures, respectively. The plasma is considered to be in thermal non-equilibrium, i.e., T e = T i , at constant temperatures. Despite the isothermal assumption, the considered set of equations contains the main difficulties of the lowtemperature moment plasma models. These difficulties, as described in the introduction, are related to the mass disparity between electrons and ions, the small Debye length, and the low-temperature of the ions.
Normalized equations
The set of equations (1a)-(1d) and (1e) are normalized by introducing some reference quantities: n 0 , the characteristic number density common to electrons and ions, L 0 = l, the reference length, T e , the electron temperature, and T i , the ion temperature. The rest of the reference variables are calculated as a combination of the previous ones: the reference velocity common to electrons and ions is based on the Bohm velocity u 0 ≡ u B = k B T e /m i , the charateristic time t 0 = L 0 /u 0 is obtained from the reference velocity and reference distance, whereas the reference potential φ 0 = k B T e /e is based on the thermal energy of electrons. The normalized set of equations reads
∂tn e + ∂x(n eūe ) =n eν iz , (2a) ∂tn i + ∂x(n iūi ) =n eν iz , (2b) ∂t(n eūe ) + ∂x n e ū 2 e + ε −1 = ε −1n e ∂xφ,(2c)∂t(n iūi ) + ∂x n i ū 2 i + κ = −n i ∂xφ,(2d)∂ 2 xxφ = χ −1 (n e −n i ) ,(2e)
where the non-dimensional parameters are defined as:
Electron-ion mass ratio: ε = m e /m i , ion-to-electron temperature: Note that, for the sake of simplicity of notation, we use the square of the Debye length χ as a non-dimensional parameter. Additionally, we define the non-dimensional Debye length λ = √ χ as it will be used to describe characteristic lengths of the problem. We highlight the importance of the three nondimensional parameters ε, χ, and κ in order to build a numerical scheme. In Table 1, we present the typical values in an Argon RF discharge. As discussed in the introduction, the smallness of the ε and χ impose very restrictive numerical constraints. In the following section, we study the asymptotic behaviour when these two parameters tend to zero.
κ = T i /T e ,
Asymptotic behavior
We study the multiscale asymptotic behavior [64] with respect to ε and χ. Previous work [52,50] performed this study only with respect to the Debye length, λ. However, the inclusion of ε in the analysis is fundamental as the electron velocity is generally much smaller than the thermal speed of electrons. Since ε and χ are the smallest parameters of the system, we do not include κ orν iz in our study.
From a physics point of view, we can consider three different asymptotic behaviours that correspond to different plasma phenomena, as illustrated in Fig. 1. The complete problem χ F ε corresponds to the system of eqs. (2). This system considers finite Debye length and electron inertia. This problem resolves all the possible scales, being the fastest one corresponding to the electron plasma waves. The main problem of designing a numerical scheme for these small scales is that it might be very inefficient to represent the macroscopic scales and leading to consistency problems due to an imbalanced numerical dissipation when the two parameters are small.
The regime where the Debye length tends to zero for arbitrarily small ε corresponds to a quasineutral plasma with the electrons that can move at bulk speed closer to the electron thermal velocity. This problem is of interest specially in the presence of a magnetic field that can produce drift motions at very high speed, such as in Hall effect thrusters. This regime allows for the representation of plasma instabilities such as the two-stream instability or the electron-drift instability. We denote this problem as 0 F ε .
Alternatively, we can consider the asymptotic behaviour of ε tending to zero for a finite Debye length. In this regime, the electrons move at speeds comparable to the ion sound velocity (Bohm's velocity) and the Debye length is arbitrarily small. This regime is important, for instance, in the plasma-sheath transition. We denote this problem as χ F 0 .
Finally, we consider the asymptotic limit where the electrons travel at speeds comparable to the ion velocity in a quasi-neutral plasma, i.e., ε → 0 and χ → 0 This behaviour is present in most of the phenomena occuring at ion scales in cold and thermal plasmas. For that reason, we focus in this problem in the rest of the paper. We denote the problem as 0 F 0 . In the following, we show the set of equations corresponding to the problem 0 F 0 and we demonstrate that lim f plasma instabilities such as the two-stream instability or the 0 F " . haviour of " ! 0 for a finite Debye length. In this regime, the velocity (Bohm's velocity) and the Debye length is arbitrarily plasma-sheath transition. We denote this problem as F 0 . he electrons travel at speeds comparable to the ion velocity in ehaviour is present in most of the phenomena occuring at ion it is the subject of our study in this paper. We denote the quations corresponding to the problem 0 F 0 and we demonstrate ic boundary conditions. x e e x i i t (n iūi ) + @x · (n iū 2 i + n i ) =n i @x¯ , @x¯ = 1 n e @xn e .
χ→0 χ F 0 = lim ε→0 0 F ε ≡ 0 F 0 in! → 0 $ → 0 e.g.
of equations that depend on two small parameters " and µ @tn e + @x · (n eūe ) = 0, @tn i + @x · (n iūi ) = 0,
eūe ) + @x · ⇥n e ū 2 e + " 1 ⇤ =n e "
@x¯ ,
t (n iūi ) + @x · ⇥n i ū 2 i + ⇤ = n i @x¯ , @ 2 xx¯ = µ 1 (n e n i ) ,
propose the following expansion for the variables of the problem
x, t) = f (0,µ) + "f (1,µ) + O(" 2 ).
f equations (4a)-(4e), we find that the system of equations for the
n (0,µ) e + @x · (n (0,µ) en (0,µ) i ) = 0, n (0,µ) i + @x · (n (0,µ) iū (0,µ) i ) = 0, 1 n (0,µ) e @xn (0,µ) e = @x (0,µ) , @x · hn (0,µ) i ⇣ū (0,µ) 2 i + ⌘i = n (0,µ) i @ x (0,µ) , @ 2 xx (0,µ) = µ 1 ⇣n (0,µ) e n (0,µ) i ⌘ .
for the variables of the problem µ F " in terms of µ of the form
, t) = (",0) f + µ (",1) f + O(µ 2 ).
r to prove Proposition 1, we first propose the following expansion
f (x, t) = f (0,µ) + "f (1,µ) + O(" 2 ).
g this expansion in the system of equations (4a)-(4e), we find tha s reads
µ F 0 : 8 > > > > > > > > > > > > < > > > > > > > > > > > > : @ tn (0,µ) e + @x · (n (0,µ) en (0,µ) i ) = 0, @ tn (0,µ) i + @x · (n (0,µ) iū (0,µ) i ) = 0, 1 n (0,µ) e @xn (0,µ) e = @x @ tn (0,µ) i + @x · hn (0,µ) i ⇣ū (0,µ) 2 i + ⌘i = n ( i @ 2 xx (0,µ) = µ 1
this system as µ F 0 . tively, we propose an expansion for the variables of the problem
f (x, t) = (",0) f + µ (",1) f + O(µ 2 ).
of equations corresponding to the zero-th order terms reads
0 F " : 8 > > > > > > > > > < > > > > > > > > > : @ t (",0)n e + @x · ( (",0)n e (",0)ū e ) = @ t (",0)n i + @x · ( (",0)n i (",0)ū i ) = @ t (",0)n e + @x ·@ tn (0,µ) i + @x · hn (0,µ) i ⇣ū (0,µ) 2 i + ⌘i = n (0,µ) i @ x (0,µ) , @ 2 xx (0,µ) = µ 1 ⇣n (0,µ) e n (0,µ) i ⌘ .
expansion for the variables of the problem µ F " in terms of µ of
f (x, t) = (",0) f + µ (",1) f + O(µ 2 ).
nding to the zero-th order terms reads
@ t (",0)n e + @x · ( (",0)n e (",0)ū e ) = 0, @ t (",0)n i + @x · ( (",0)n i (",0)ū i ) = 0, @ t (",0)n e + @x · h (",0)n e ⇣ (",0)ū2 e + " 1 ⌘i = " 1 (",0)n e @ x (",0) ,0)n i (",0)ū i + @x · h (",0)n i ⇣ (",0)ū2 i + ⌘i = (",0)n i @ x (",0) , (",0)n e = (",0)n i .
eq. (7) for the problem µ F 0 in order to study the problem F (0, -th order terms is very similar to system µ F 0 , with the di ere 6 w-temperature plasma fluid equations system of equations that corresponds to lim
µ!0 µ F 0 is the same a 0 F 0 : 8 > > > > > > > > < > > > > > > > > :n e =n i , @tn i + @x · (n iūi ) = 0,
@x · (n eūe ) = @x · (n iūi ), @t(n iūi ) + @x · (n iū 2 i + n i ) =n i @x¯ , @x¯ = 1 n e @xn e . r the following system of equations that depend on two small p µ F " :
8 > > > > > > > > < > > > > > > > > :
@tn e + @x · (n eūe ) = 0, @tn i + @x · (n iūi ) = 0, @t(n eūe ) + @x · ⇥n e ū 2 e + " 1 ⇤ =n e " @x¯ , @t(n iūi ) + @x · ⇥n i ū 2 i + ⇤ = n i @x¯ , @ 2 xx¯ = µ 1 (n e n Proposition 1, we first propose the following expansion for the
f (x, t) = f (0,µ) + "f (1,µ) + O(" 2 ).
ectrostatic isothermal multi-fluid plasma equations.
) with periodic boundary conditions, which we denote as the c expansions for the problem F " : (1) In terms of the small . We define the problem F 0 as the system of equations for 5 n consider the asymptotic behaviour of " ! 0 for a finite Debye length. In this regime, the s comparable to the ion sound velocity (Bohm's velocity) and the Debye length is arbitrarily portant, for instance, in the plasma-sheath transition. We denote this problem as F 0 . the asymptotic limit where the electrons travel at speeds comparable to the ion velocity in i.e., " ! 0 and ! 0 This behaviour is present in most of the phenomena occuring at ion mal plasmas. For that reason it is the subject of our study in this paper. We denote the following, we show the set of equations corresponding to the problem 0 F 0 and we demonstrate " ⌘ 0 F 0 in the case of periodic boundary conditions. Quasi-neutral plasma with massless electrons $ → 0 > > > > > > > : @t(n iūi ) + @x · (n iū i + n i ) =n i @x , @x¯ = 1 n e @xn e .
! → 0 ! → 0 $ → 0 e.
the following system of equations that depend on two small param µ F " :
8 > > > > > > > > < > > > > > > > > :
@tn e + @x · (n eūe ) = 0, @tn i + @x · (n iūi ) = 0, @t(n eūe ) + @x · ⇥n e ū 2 e + " 1 ⇤ =n e " @x¯ ,
@t(n iūi ) + @x · ⇥n i ū 2 i + ⇤ = n i @x¯ , @ 2 xx¯ = µ 1 (n e n i ) ,
roposition 1, we first propose the following expansion for the varia
f (x, t) = f (0,µ) + "f (1,µ) + O(" 2 ).
sion in the system of equations (4a)-(4e), we find that the system
µ F 0 : 8 > > > > > > > > > > > > < > > > > > > > > > > > > : @ tn (0,µ) e + @x · (n (0,µ) en (0,µ) i ) = 0, @ tn (0,µ) i + @x · (n (0,µ) iū (0,µ) i ) = 0, 1 n (0,µ) e @xn (0,µ) e = @x (0,µ) , @ tn (0,µ) i + @x · hn (0,µ) i ⇣ū (0,µ) 2 i + ⌘i = n (0,µ) i @ x (0,µ) , @ 2 xx (0,µ) = µ 1 ⇣n (0,µ) e n (0, i as µ F 0 .
opose an expansion for the variables of the problem µ F " in terms
f (x, t) = (",0) f + µ (",1) f + O(µ 2 ). f(x, t) = f (0,µ)
By injecting this expansion in the system of equations (4 order terms reads
µ F 0 : 8 > > > > > > > > > > > > < > > > > > > > > > > > > : @ tn (0,µ) e + @x · ( @ tn (0,µ) i + @x · ( n @ tn (0,µ) i + @x · hn (0,µ) i
⇣ū We denote this system as µ F 0 . Alternatively, we propose an expansion for the variab f(x, t) = (",0) f
The system of equations corresponding to the zero-th ord
0 F " : 8 > > > > > > > > > < > > > > > > > > > : @ t (",0)n e + @ x @ t (",0)n i + @ @ t (",0)n e + @x · h (",0)n e @ t (",0)n i (",0)ū i + @x · h (",0
We denote this system as 0 F " . µ F 0 : > > > > > > > > > > > > : n (0,µ) e
x e x @ tn
(0,µ) i + @x · hn (0,µ) i ⇣ū (0,µ) 2 i + ⌘i = n (0,µ i @ 2 xx (0,µ) = µ 1 ⇣n
is system as µ F 0 . vely, we propose an expansion for the variables of the problem µ F
f (x, t) = (",0) f + µ (",1) f + O(µ 2 ).
f equations corresponding to the zero-th order terms reads
0 F " : 8 > > > > > > > > > < > > > > > > > > > : @ t (",0)n e + @x · ( (",0)n e (",0)ū e ) = @ t (",0)n i + @x · ( (",0)n i (",0)ū i ) = @ t (",0)n e + @x · h (",0)n e ⇣ (",0)ū2 e + " 1 ⌘i = @ t (",0)n i (",0)ū i + @x · h (",0)n i ⇣ (",0)ū2 i + ⌘i = (",0)n e =
is system as 0 F " . se the expansion of eq. (7) for the problem µ F 0 in order to stud uations for the zero-th order terms is very similar to system µ F 6 P scheme for the low-temperature plasma fluid equations roposition 1 The system of equations that corresponds to lim µ!0
0 F 0 : 8 > > > > > > > > < > > > > > > > > :
@tn i + @x · (n @x · (n @t(n iūi ) + @x · (n iū 2 i + roof Let us consider the following system of equations that dep µ F " :
8 > > > > > > > > < > > > > > > > > :
@tn e + @x · (n @tn i + @x · (n @t(n eūe ) + @x · ⇥n e ū 2 e + " @t(n iūi ) + @x · ⇥n i ū 2 i + @
In order to prove Proposition 1, we first propose the following
f (x, t) = f (0,µ) + "f (1,µ)
symptotic behaviour of the electrostatic isothermal multi-fluid plasma equations.
system of equations (2a)-(2e) with periodic boundary conditions, which we denote as the sider two di↵erent asymptotic expansions for the problem F " : (1) In terms of the small terms of the small parameter . We define the problem F 0 as the system of equations for n e =n i , @tn i + @x(n iūi ) = 0, @x(n eūe ) = @x(n iūi ),
n iūi ) + @x(n iū 2 i + n i ) =n i @x¯ , @x¯ = 1 @xn e . (4a) (4b) (4c) (4d) (4e)
AP scheme for the low-temperature plasma fluid equations the zero-th order terms of the expansion in terms of ". Similarly, we d the zero-th order terms of the expansion in terms of .
F 0 0 F 0 : 8 > > > > > > > > < > > > > > > > > :n e = @tn i + @x(n iūi ) = @x(n eūe ) = @t(n iūi ) + @x(n iū 2 i + n i ) = @x¯ =
Proof. In order to prove Proposition 1, we first propose the following ex
f (x, t) = f (0, ) + "f (1, ) + O(
By injecting this expansion in the system of equations (2a)-(2e), we find order terms reads 8 > > > > @ tn Let us consider the system of equations (2a)-(2e) with periodic boundary conditions, which we denote as the problem χ F ε . We consider two different asymptotic expansions for the problem χ F ε : (1) In terms of the small 5 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
(0, ) e + @x(n (0, ) en (0, ) i ) = 0 ! 0 ! 0 " ! 0 " ! 0
parameter ε and (2) in terms of the small parameter χ. We define the problem χ F 0 as the system of equations for the zero-th order terms of the expansion in terms of ε. Similarly, we define 0 F ε as the the system of equations for the zero-th order terms of the expansion in terms of χ. Proposition 1. The system of equations that corresponds to lim
n e =n i , ∂tn i + ∂x(n iūi ) =n eν iz , ∂x(n eūe ) = ∂x(n iūi ), ∂t(n iūi ) + ∂x(n iū 2 i + κn i ) =n i ∂xφ, ∂xφ = 1 n e ∂xn e . (4a) (4b) (4c) (4d) (4e)
Proof. In order to prove Proposition 1, we first propose the following expansion for the variables of the problem
χ F ε f (x, t) = f (0,χ) + εf (1,χ) + O(ε 2 ).(5)
By injecting this expansion in the system of equations (2a)-(2e), we find that the system of equations for the zero-th order terms reads
χ F 0 : ∂tn (0,χ) e + ∂x(n (0,χ) en (0,χ) i ) =n (0,χ) eν iz , ∂tn (0,χ) i + ∂x(n (0,χ) iū (0,χ) i ) =n (0,χ) eν iz , 1 n (0,χ) e ∂xn (0,χ) e = ∂xφ (0,χ) , ∂t n (0,χ) iū (0,χ) i + ∂x n (0,χ) i ū (0,χ) 2 i + κ = −n (0,χ) i ∂ x φ (0,χ) , ∂ 2 xx φ (0,χ) = χ −1 n (0,χ) e −n (0,χ) i . (6a) (6b) (6c) (6d) (6e)
We denote this system as χ F 0 . Alternatively, we propose an expansion for the variables of the problem χ F ε in terms of χ of the form
f (x, t) = (ε,0) f + χ (ε,1) f + O(χ 2 ).(7)
The system of equations corresponding to the zero-th order terms reads 0 F ε :
∂t (ε,0)n e + ∂x (ε,0)n e (ε,0)ū e = (ε,0)n eν iz , ∂t (ε,0)n i + ∂x (ε,0)n i (ε,0)ū i = (ε,0)n eν iz , ∂t (ε,0)n e (ε,0)ū e + ∂x (ε,0)n e (ε,0)ū2 e + ε −1 = ε −1 (ε,0)n e ∂ x (ε,0) φ, ∂t (ε,0)n i (ε,0)ū i + ∂x (ε,0)n i (ε,0)ū2 i + κ = − (ε,0)n i ∂ x (ε,0) φ, (ε,0)n e = (ε,0)n i . (8a) (8b) (8c) (8d) (8e)
We denote this system as 0 F ε . We can use the expansion of eq. (7) for the problem χ F 0 in order to study the problem F (0,0) ≡ lim
F (0,0) ≡ lim χ→0 χ F 0 : ∂tn (0,0) e + ∂x(n (0,0) en (0,0) i ) =n (0,0) eν iz , ∂tn (0,0) i + ∂x(n (0,0) iū (0,0) i ) =n (0,0) eν iz , 1 n (0,0) e ∂xn (0,0) e = ∂xφ (0,0) , ∂t n (0,0) iū (0,0) i + ∂x n (0,0) i ū (0,0) 2 i + κ = −n (0,0) i ∂ x φ (0,0) , n (0,0) e =n (0,0) i . (9a) (9b) (9c) (9d) (9e)
By using the condition (9e) in eqs (9a) and (9b), we obtain that the fluxes are conserved, i.e., ∂x(n
(0,0) en (0,0) i ) = ∂x(n (0,0) iū (0,0) i
). This proves that the system F (0,0) is the system proposed as 0 F 0 in eqs. (4). Alternatively, we use the expansion of eq. (5) for the problem 0 F ε . The equations for the zero-th order terms are denoted as (0,0) F ≡ lim ε→0 0 F ε . In this new system of equations, the electron momentum equation is modified as
compared to the one in 0 F ε . (0,0) F ≡ lim ε→0 0 F ε : ∂t (0,0)n e + ∂x (0,0)n e (0,0)ū e =n eν iz , ∂t (0,0)n i + ∂x (0,0)n i (0,0)ū i =n eν iz , 1 (0,0)n e ∂x (0,0)n e = ∂ x (0,0) φ, ∂t (0,0)n i (0,0)ū i + ∂x (0,0)n i (0,0)ū2 i + κ = − (0,0)n i ∂ x (0,0) φ, (0,0)n e = (0,0)n i . (10a) (10b) (10c) (10d) (10e)
This system of equations is the same as the problem F (0,0) . Consequently, in the case of periodic boundary conditions, both limits result in the same system of equations (4a)-(4e), i.e., (0,0) F = F (0,0) = 0 F 0 .
We highlight that the system 0 F ε , which corresponds to the limit of the Debye length tending to zero, is the one that was previously considered by the works [52,50]. In this system, the electric potential becomes a Lagrange multiplier that imposes the charge neutrality. In system (8), we decided to write the set of equations in this form for clarity. Nevertheless, a differential equation for the electric potential can be retrieved from the conservation of electric charge, as done in [50].
The asymptotic behaviour proposed in this paper considers massless electrons as in [65,66]. It is therefore very different to the one derived in [52,50]. By including in our analysis the electron-to-ion mass ratio, we find that the quasi-neutrality is found in eq. (4a) and the electron density follows the Boltzmann distribution in eq. (4e). Furthermore, the electric current is conserved by equation (4d) and the ion mass and momentum equations are unchanged. The main difference as compared to [50] is the limit for the zero-th order of the potential in eq. (4e) and the electron momentum equation.
Study of a bounded low-temperature plasma through the fluid plasma equations
In low-temperature plasma industrial applications, plasmas are confined. As a result, the charged particles that are produced inside the reactor through ionization, are lost through the boundaries when they strike the wall. Since the thermal motion of electrons is larger than that of ions, the surface will charge negatively with respect to the plasma (in electropositive plasmas), forming a charged boundary layer called the plasma sheath.
The analytical models for the sheath and presheath rely on the isothermal multi-fluid equations [67], while assuming the inertia of electrons and temperature of ions to be negligible. Nevertheless, important kinetic phenomena taking place are not included in this model [68].
Let us consider a 1D domain of length l filled with a plasma between two floating walls, with no secondary electron emission, the distribution function of electrons is a Maxwellian and all the electrons that touch the wall are absorbed by the wall. With these assumptions, the flux of electrons collected by the wall (see, e.g., [1]) both in dimensional and dimensionless units read:
Dimensional: n e u e | wall = n e k B T e 2πm e and dimensionless:n eūe | wall =n
e √ 2πε .(11)
A steady solution is found when the ionization inside the bulk of the plasma balances the particle loss as follows 2n e u e | wall = l 0 n e ν iz dx.
As mentioned by Riemann [37], the ionization frequency is an eigenvalue of the problem. Consequently, there is only one ionization frequency that finds a steady state solution for a given distance between plates. In this paper, we propose a numerical methodology that proves to be convergent to find this eigenvalue.
With the previous assumptions, the potential at the pre-sheath φ p and the wall φ W , in dimensional units [69], as follows
φ p = − k B T e 2e and φ W = k B T e e ln m e 2πm i 1/2 ,(13)
where φ p is the potential drop needed to accelerate the ions to Bohm's speed (neglecting the ion pressure gradient) and φ W is the potential drop in the sheath. In Fig. 2, we illustrate the steady state solution of a bounded plasma between two floating plates.
Standard upwind finite volume discretization
We present a standard discretization of the system (2) in order to illustrate the associated numerical difficulties. An example of a simulation of a low-temperature discharge with this discretization can be found in Alvarez Laguna et al. [38]. Alternatively, a similar discretization is described in [50] in order to illustrate a standard solver of the Euler-Poisson system.
We use a finite volume discretization where the domain x ∈ [0, l] is divided into N elements of equal length ∆x. We approximate the value of the unknowns as a piecewise function inside the volume Ω j . The flux at the interfaces is approximated by a numerical flux function that is in general a function of the values on the right and left of the cell interface. The source is approximated by a piecewise constant value. After making these assumptions, the first order 1D finite volume discretization for the cell j reads
n ē n ī n eūē n iūi n+1 j = n ē n ī n eūē n iūi n j − ∆t ∆x n eūē n iūī n e (ū 2 e + ε −1 ) n i (ū 2 i + κ) n j+1/2 − n eūē n iūī n e (ū 2 e + ε −1 ) n i (ū 2 i + κ) n j−1/2 +∆t (n eν iz ) n eν iz n e ε −1 ∂xφ −n i ∂xφ n j(14)
8 AP scheme for the low-temperature plasma fluid equations
A. Alvarez Laguna et al. withφ n+1 j+1 − 2φ n+1 j +φ n+1 j−1 = ∆x 2 χ n n ej −n n ij and ∂xφ n j = 1 2∆x φ n j+1 −φ n j−1 .(15)
The numerical fluxes at the interfaces can be calculated with different Riemann solvers, e.g., Roe as in [38], Lax-Friedrich as in [50] or HLL as in the results of this paper using the standard discretization. We note here that the election of the Riemann solver for this problem has a small impact in the results as the numerical dissipation is dominated by the low-Mach regime of electrons, as it will be shown in the results. TVD reconstruction can improve the results of the standard discretization as shown in [50,38]. As explained in [50,52], the stability of the time discretization is restricted by a CFL condition that takes into account the convective scales of both fluids and the characteristic time scales of the source terms. The convective CFL reads
CFL conv = ∆t|λ e,i max | ∆x with |λ e,i max | = max (|ū e,i +c e,i |, |ū e,i −c e,i |) .(16)
Wherec e,i is the dimensionless speed of sound of electrons and ions, i.e., ε −1/2 and κ 1/2 , respectively. Note that due to the mass disparity between ions and electrons, the CFL condition of the electrons is typically more restrictive than this of the ions. Therefore, the convective CFL condition of electrons uses the maximum eigenvalues of electrons that are |λ e max | = max |ū e + ε −1/2 |, |ū e − ε −1/2 | . Similarly, the source terms impose a constraint in the time step. The stability condition for the electrostatic force is related to the resolution of the electron plasma wave [51] ∆t ω pe < 1 with ω pe = n e χε .
Finally, the ionization term has the stability constraint as follows
∆t ν iz < 1.(18)
The stability condition reads max CFL conv , ∆t ω pe , ∆t ν iz i∈N < 1.
As it can be seen in Table 1, if the cell size is larger than the Debye length, the most restrictive constraint is the resolution of the electron plasma waves. If the Debye length is resolved, then the convective condition is sufficient to fulfil the condition ∆t < ω −1 pe . As shown in [38,70], when this scheme does not spatially resolve the Debye length, the simulation leads to large spurious charge separation errors that can excite plasma modes, leading to an erroneous solution. Similarly, the truncation error of the upwind discretization of the fluxes of the electrons leads to a large error in the flux of the electrons. As it will be shown in the results and previously noted in [48,50], this is due to the low-Mach regime of the electron when the bulk speed is much smaller than the thermal speed. Figure 3: Illustration of the characteristic scales for the discretization of the two-fluid electrostatic problem. We assume conditions where the ions and electrons travel at a positive speed of similar magnitude, (ū i ∼ūe) andū i κ 1/2 and ue ε −1/2 . The ionization frequency is supposed to be such thatν iz ωpe and thus is not in the figure.
j j + 1/2 j 1/2 x ū e " 1/2ū e + " 1/2 u i + 1/2 u i 1/2 ! 1 pe t x
Acoustic/transport operator splitting strategy
We present an alternative to the standard discretization of section 3. We present a novel operator splitting strategy that decouples the acoustic and transport phenomena of electrons. In 1D, this method is analogous to an explicit Lagrange-Projection [57] method. Nevertheless, the present splitting does not need a moving Lagrangian 9 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. mesh and can be naturally expressed for multi-dimensional problems (see Chalons et al. [57] for the extension of the operator splitting method to multiple dimensions).
We propose to approximate the system of equations (2a)-(2e) in the successive solution of the following systems. The first step solves for the electron acoustic system together with the Poisson equation and the second solves for the electron transport (advection) and the ion equations. It should be noted that the first system contains the small scales related to the parameters ε and λ, whereas the second system solves for the slow dynamics, of order O(1), as explained in Section 2.2.
Electron acoustic and electrostatic system. ∂tn e +n e ∂xū e = 0, (20a)
∂t(n eūe ) +n eūe ∂xū e + ∂xp e =n e ε ∂xφ,(20b)∂ 2 xxφ =n e −n i χ ,(20c)
Electron transport and ion dynamics.
∂tn e +ū e ∂xn e =n eν iz , (21a) ∂t(n eūe ) +ū e ∂x(n eūe ) = 0, (21b)
∂tn i + ∂x(n iūi ) =n eν iz , (21c) ∂t(n iūi ) + ∂x n i ū 2 i + κ = −n i ∂xφ,(21d)
where the electron pressurep e = ε −1n e , in eq. (20b).
Strategy to solve the equations Given a state at the time (n e ,n eūe ,n i ,n iūi ,φ) n j at the time t n and the cell center x j . The scheme is split into 1. By numerically solving the system (20), we update the state (n e ,n eūe ,φ) n j to the value at t * , i.e., (n e ,n eūe ,φ) * j . 2. By numerically solving the system (21), we update the state (n e ,n eūe ,n i ,n iūi ,φ) * j to the value t n+1 , i.e., (n e ,n eūe ,n i ,n iūi ,φ) n+1 j .
In the following, we present the properties and description of the two steps of the numerical scheme.
j j + 1/2 j 1/2 x ū e " 1/2ū e + " 1/2 ! 1 pe t x AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. with¯ n+1 j+1 2¯ n+1 j +¯ n+1 j 1 = x 2 2 ⇣n n ej n n ij ⌘ and @x¯ n j = 1 2 x ¯ n j+1 ¯ n j 1 .(12)
The numerical fluxes at the interfaces can be calculated with di↵erent Riemann solvers, e.g., Roe as in [35], Lax-Friedrich as in [47] or HLL as in the results of this paper using the standard discretization. We note here that the election of the Riemann solver for this problem has a small impact in the results as the numerical dissipation is dominated by the low-Mach regime of electrons, as it will be shown in the results. TVD reconstruction can improve the results of the standard discretization as shown in [47,35]. The stability of the time discretization is restricted by a CFL condition that takes into account the convective scales of both fluids and the characteristic time scales of the source terms. The convective CFL reads
CFL conv = t| e,i max | x with | e,i max | = max (|ū e,i +c e,i |, |ū e,i c e,i |) .(13)
Wherec e,i is the dimensionless speed of sound of electrons and ions, i.e., " 1/2 and 1/2 , respectively. Note that due to the mass disparity between ions and electrons, the CFL condition of the electrons is typically more restrictive than this of the ions. Therefore, the convective CFL condition of electrons uses the maximum eigenvalues of electrons that are | e max | = max |ū e + " 1/2 |, |ū e " 1/2 | . Similarly, the source terms impose a constraint in the time step. The CFL condition for the electrostatic force is related to the resolution of the electron plasma wave
CFL elec = t ! pe with ! pe = rn e 2 " .(14)
Finally, the ionization term has the CFL as follows
CFL iz = t ⌫ iz .(15)
The CFL condition reads max
⇣ CFL conv , CFL elec , CFL iz ⌘ i2N < 1.(16)
As it can be seen in Table 1, if the cell size is larger than the Debye length, the most restrictive constraint is the resolution of the electron plasma waves. If the Debye length is resolved, then the convective condition is su cient to fulfil the condition t < ! 1 pe . As shown in [35,61], when this scheme does not spatially resolve the Debye length, the simulation leads to large spurious charge separation errors that can excite plasma modes, leading to an erroneous solution. Similarly, the truncation error of the upwind discretization of the fluxes of the electrons leads to a large error in the flux of the electrons. As it will be shown in the results and previously noted in [45,47], this is due to the low-Mach regime of the electron when the bulk speed is much smaller than the thermal speed.
Acoustic/transport operator splitting strategy
We present an alternative to the standard discretization of section 3. We present a novel splitting strategy based on the work of Chalons et al. [54] to the plasma fluid equations. Analogously, we propose to approximate the system of equations (4a)-(4e) in the successive solution of the following systems. The first step solves for the electron acoustic system together with the Poisson equation and the second solves for the electron transport (advection) and the ion equations. It should be noted that the first system contains the small scales related to the parameters " and , whereas the second system solves for the slow dynamics, of order O(1), as explained in Section 2.2.
Electron acoustic and electrostatic system. @tn e +n e @x ·ū e = 0, (17a) @t(n eūe ) +n eūe @x ·ū e + @xp e =n e " @x¯ ,
@ 2 xx¯ =n e n i 2 ,(17b)8 j j + 1/2 j 1/2ū e + " 1/2 u i + 1/2 u i 1/2 t x(17c)
AP scheme for the low-temperature plasma fluid equations Electron transport and ion dynamics. @tn e +ū e · @xn e @t(n eūe ) +ū e · @x(n eūe ) @tn i + @x · (n iūi ) @t(n iūi ) + @x · ⇥n i ū 2 i + ⇤ wherep e = " 1n e , in eq. (17b).
Strategy to solve the equations Given a state at the time (n e ,n eūe ,n i ,n iūi ,¯ ) n j at the time t n 1. By numerically solving the system (17), we update the state ( 2. By numerically solving the system (18), we update the stat (n e ,n eūe ,n i ,n iūi ,¯ ) n+1 j .
In the following, we present the properties and description of 4.1. Properties and discretization of the electron acoustic and elec The system eq. (17) can be written in conservative form, i.e., t we define the variable ⌧ e ⌘ 1/n e . In this variable, the problem (17 @t⌧ e ⌧ e @x ·ū e = 0 @tū e + ⌧ e @xp e = 1 "
2 @ 2 xx¯ = 1 ⌧
We approximate ⌧ e (x, t)@x by the solution at time t n , i.e., ⌧ e (x new variable, we obtain the following system of equations for the @t⌧ e @ m ·ū e = 0 @tū e + @ mpe = 1 "
The left-hand-side of the system is conservative in this new var are acous = (c e , c e ). For that reason, the system is called acous still with the previous spacial variable x. As this term cannot be will be treated in the discretization as a source term.
Properties and discretization of the electron acoustic and electrostatic system
The system eq. (20) can be written in conservative form, i.e., the compression terms written as fluxes. To do this, we define the variable τ e ≡ 1/n e . In this variable, the problem (20) reads
∂tτ e − τ e ∂xū e = 0,(22a)
∂tū e + τ e ∂xp e = 1 ε ∂xφ,
χ∂ 2 xxφ = 1 τ e −n i .(22b)
10 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
We approximate τ e (x, t)∂x by the solution at time t n , i.e., τ e (x, t n )∂x. We define dm = n e (x, t n )dx. By using this new variable, we obtain the following system of equations for the electrons
∂tτ e − ∂ mūe = 0,(23a)∂tū e + ∂ mpe = 1 ε ∂xφ.(23b)
The left-hand-side of the system is conservative in this new variable. The eigenvalues of the homogeneous system are λ acous = (−ε −1/2 , ε −1/2 ). For that reason, the system is called acoustic. Note that the source term in equation (23b) is still with the previous space variable x. As this term cannot be written in conservative form in the variable m, it will be treated in the discretization as a source term.
Discretization of the electron acoustic and electrostatic system
The system (23) with Poisson's eq. (22c) will be discretized in time by using a semi-implicit discretization. In this approach, we want to discretize the Lorentz force in the electrons and the electron density in Poisson's equation implicitly, as follows
τ * e − τ n e ∆t − [∂ mūe ] * = 0, (24a) u * e −ū n e ∆t + [∂ mpe ] n = 1 ε ∂xφ * . (24b) χ ∂ 2 xxφ * = 1 τ * e −n n i .(24c)
The full discretization of system (24) at t * is given by numerical scheme 1.
Numerical Scheme 1. The discretization of the electron acoustic and electrostatic system (20) reads as follows.
The discretized electron density is n * ej =n n ej 1 +n
n i j ∆t 2 εχ 1 + ∆t ∆x ū n e j+1/2 −ū n e j−1/2 − ∆t 2nn ej [∂ 2 mmpe ] n j + ∆t 2nn e j εχ .(25)
with the discretization of the pressure Laplacian as (26) and the velocity at the cell interfacē
u n e j+1/2 =ū e R +ū e L 2 −n e j+1/2 ε −1/2 2 f a (M e ) (τ e R − τ e L ) withn e j+1/2 =n e R +n e L 2 ,(27)
where f a (M e ) = O(M e ). For this function, we choose the formula proposed by Liou [71], as follows
f a (M ) = (1 − M o ) 2M 2 + 4M 2 o 1 − M 2 o(28)
The cut-off Mach M co is chosen to avoid having a null numerical dissipation when the electrons have zero velocity.
11
AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
The discretized potential is solved by the following expression
φ * j+1 − 2φ * j +φ * j−1 = ∆x 2 χ n * ej −n n ij .(30)
The discretized electron velocity is computed as follows
u * ej =ū n ej − ∆t n n ej ∆x p n e j+1/2 −p n e j−1/2 + ∆t ε ∂xφ * j ,(31)
where the gradient of the electric potencial is discretized as
∂xφ * j = 1 2∆x φ * j+1 −φ * j−1 ,(32)
and the pressure flux from the expression
p e j+1/2 = ε −1 (n e R +n e L ) 2 −n e j+1/2 ε −1/2 2 f a (M e ) (ū e R −ū e L ) .(33)
The previous numerical scheme is obtained as follows. In order to build a scheme with an asymptotic-preserving property, we follow a similar approach to this of Dimarco et al. [58]. The divergence of the velocity at time t * in eq. (24a) is computed by taking the divergence of eq. (24b). This divergence of the velocity reads
[∂ mūe ] * = [∂ mūe ] n − ∆t ∂ 2 mmpe n + ∆t ε ∂ m ∂xφ * .(34)
We discretize the term due to the Lorentz force at the time t * by using Poisson's eq. (24c), as follows
∂ m ∂xφ * = 1 n * e ∂ 2 xxφ * = 1 χ (1 − n n i τ * e ) .(35)
Note that the ion density is computed at time t n in the Poisson equation. This approximation is justified by the fact that the ions move slower than the electrons in the scales for the acoustic step. The divergence of the velocity is then introduced in the equation for the conservation of mass in the acoustic step. This reads
τ * e − τ n e ∆t − [∂ mūe ] n = ∆t εχ [1 − n n i τ * e ] − ∆t ∂ 2 mmpe n .(36)
We note that the LHS of the equation is the same as in an explicit discretization, whereas the RHS are terms that appear due to the implicitation of the scheme. The first term of the RHS is a diffusion term that is due to the Lorentz force and the second is a diffusion term due to the pressure gradient. We highlight the fact that the diffusion due to the Lorentz force, which will stabilize the scheme, appears without the need of solving Poisson's equation and additionally, it is linear in τ * e . In order to discretize in space eq. (36), we follow the formalism of Chalons et al. [57].
1 +n n ij ∆t 2 εχ τ * ej = τ n ej 1 + ∆t ∆x ū n e j+1/2 −ū n e j−1/2 + ∆t 2 εχ − ∆t 2 ∂ 2 mmpe n j .(37)
The pressure diffusion term is discretized as follows. By changing from the variable m to the space variable, the pressure gradient reads ∂ mpe =n −1 e ∂xp e . By assuming the electrons to be isothermal, the pressure isp e =n e ε −1 and the pressure gradient in the mass variable ∂ mpe = ε −1 ∂x lnn e . Therefore, the Laplacian in the mass variable reads ∂ 2 mmpe = (εn e ) −1 ∂ 2 xx lnn e . The space discretization of this term is based on centered difference as follows
Similarly, the velocity at the interface is approximated by a Riemann solver that is described in Appendix A. The original velocity at the interface reads u n e j+1/2 =ū e R +ū e L 2 −n e j+1/2 ε −1/2 2 (τ e R − τ e L ) withn e j+1/2 =n e R +n e L 2 , where the subscripts R and L refer to the values on the right and left of the interface j + 1/2. As explained in Appendix B, the truncation error is proportional to ε −1/2 ∂ 2 xx τ e ∆x. As suggested in [57], in order to control this error when ε is small, we rescale the numerical error produced by an imbalanced numerical dissipation in the low Mach regime. This yieldsū n e j+1/2 =ū
e R +ū e L 2 −n e j+1/2 ε −1/2 2 f a (M e ) (τ e R − τ e L ) ,(40)
where f a (M e ) = O(M e ). For this function, we choose the formula proposed by Liou [71], as follows
f a (M ) = (1 − M o ) 2M 2 + 4M 2 o 1 − M 2 o ,(41)
The cut-off Mach M co is chosen to avoid having a null numerical dissipation when the electrons have zero velocity. With this equation, the electron density at the time t * is calculated as follows n * ej =n n ej 1 +n
n i j ∆t 2 εχ 1 + ∆t ∆x ū n e j+1/2 −ū n e j−1/2 − ∆t 2nn ej [∂ 2 mmpe ] n j + ∆t 2nn e j εχ .(43)
After solving this equation, the electric potential is computed by solving Poisson's by the linear system described in eq. (30).
The last part of the acoustic step is to solve the electron velocity equation. For this, we use an approximate Riemann solver in order to discretize the pressure flux in eq. (24b), as follows u * ej =ū n ej − ∆t n n ej ∆x p n e j+1/2 −p n e j−1/2 + ∆t ε ∂xφ * j .
The gradient of the electric potencial is discretized as
∂xφ * j = 1 2∆x φ * j+1 −φ * j−1(45)
The pressure flux from the Riemann solver in Appendix A reads p e j+1/2 = ε −1 (n e R +n e L ) 2 −n e j+1/2 ε −1/2 2 (ū e R −ū e L ) .
As done previously, the truncation error of eq. (44) is proportional to ε −1/2 ∂ 2 xxūe ∆x (see Appendix B). In this case, we choose to balance the numerical dissipation with the technique proposed by Liou [71].
p e j+1/2 = ε −1 (n e R +n e L ) 2 −n e j+1/2 ε −1/2 2 f a (M e ) (ū e R −ū e L ) .(47)
As in Chalons et al. [57], by preconditioning the numerical dissipation with the factor f a , the truncation error of the numerical system is of order O(∆x).
As the Lorentz force is treated implicitly, the scheme is shown to be unconditionally stable for the plasma wave frequency. Additionally, with the preconditioning of the low-Mach regime, the electrons can have a larger time step, not limited by the sound waves, as it will be shown in the results. The linear stability analysis to determine the CFL condition is not trivial due to the non-linear term ∂ 2 mmpe n j and hence is left for a future work.
13
AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
Asymptotic-preserving property of the acoustic and electrostatic system
In this section, we prove the consistency of the acoustic and electrostatic system when χ and ε tend to zero with respect to the asymptotic limit of system (4) explained in Section 2.2. Property 1. In the limits ε → 0 and χ → 0, the discretized electron density at t * reads lim ε→0 χ→0n * ej −n n ij = 0.
Proof. First, we may rewrite f a (M e ) at the interface j + 1/2 in the limit ε → 0 as
f a (M e ) = C 1 j+1/2 + εC 2 j+1/2 with C 1 j+1/2 = (1 − M o ) 2 (1 − M 2 o ) 2 n ejūej + n ej+1ūej+1 n ej + n ej+1 2 , C 2 j+1/2 = 4M 2 o (1 − M 2 o ) 2 n ejūej + n ej+1ūej+1 n ej + n ej+1 2 .
Then, injecting (40) and (47) into (43) leads tō n * ej =n n ej εχ +n n ij ∆t 2
χ εK 1 j − ε 1/2 K 1/2 j − K 0 j + ∆t 2nn ej (49a) K 1 j = 1 + ∆t ∆xū ej+1 −ū ej−1 2 , (49b) K 1/2 j = ∆t ∆x n e j+1/2 2 C 1 j+1/2 + εC 2 j+1/2 τ ej+1 − τ ej −n e j+1/2 2 C 1 j−1/2 + εC 2 j−1/2 τ ej − τ ej−1 ,(49c)K 0 j = ∆t 2 ∆x 2 lnn n ej+1 + lnn n ej−1 − 2 lnn n ej .(49d)
One observes thatn * ej is a function of ε and χ continuous in (0, 0), then the two limits commute and (48) holds.
We highlight that eq. (48) is the quasi-neutral limit that occurs at scales O(χ −1 ), as shown in eq. (4a).
Property 2. Assumingn n ej =n n ij = 0, then, in the limits ε → 0 and χ → 0, the discretized electric potential at t * satisfies lim ε→0 χ→0
[∂ 2 xxφ ] * j − ∂ 2 xx lnn e * j = 0,(50)
where the operator [∂ 2 xx ] j stands for the central discretization in space at the cell j. This expression is the discretization of the divergence of eq. (4e).
Proof. By injecting eq. (43) into eq. (30), we obtain the following expression for Poisson's equation discretization
[∂ 2 xxφ * ] j = εn n ej −n n ij εK 1 j − ε 1/2 K 1/2 j − K 0 j χ εK 1 j − ε 1/2 K 1/2 j − K 0 j + ∆t 2nn ej ,(51)
where the coefficients K i j are defined in (49). This is again continuous in (ε, χ) = (0, 0), thus the two limits commute and provide lim ε→0 χ→0
[∂ 2 xxφ ] * j =n n ij n n ej ∆x 2 lnn n ej+1 + lnn n ej−1 − 2 lnn n ej .
Using property 1, we may replace in this expressionn n ij byn * ej in this expression. And using the continuity of the logarithm, the hypothesisn n ej =n n ij = 0 and property 1, we may replacen n ej in this expression to obtain (50).
As seen above, the advantage of this method is that it is AP by solving a standard Poisson equation, i.e., eq. (30). However, when this equation is discretized by a standard explicit discretization, i.e., eq. (15), the electric potential tends to infinity when χ → 0 if the ∆x is not small enough, resulting in a large numerical error.
In the discrete electron momentum equation of the acoustic step in eq. (44), we can find problems of consistency due to the small Mach number (ε → 0). In the following property we explain how these problems are solved by preconditioning the numerical dissipation.
Property 3. If f a = O(ε 1/2 ), then, the truncation error of the electron momentum equation in the acoustic step in eq. (22b) is uniform with respect to ε < 1.
Proof. The proof is analogous to the one presented in [57,72]. We assume that the variables describe a smooth flow so we can expand them in Taylor series to study the truncation error (see Appendix B for the detailed derivation of the truncation error). With this development, we use the numerical pressure flux of (47), and we neglect second order terms, i.e., O ∆x 2 , O ∆t 2 . Consequently, the discretization is consistent with
∂tū e + τ e ε −1 ∂xn e − ε −1 ∂xφ = O (∆t) + O f a ε −1/2 ∆x + O (∆x) .(52)
The function f a in eq. (41) is a continuous function of ε, as follows,
f a (ε) = C 1 j+1/2 + εC 2 j+1/2 .(53)
Assuming that C 1 j+1/2 and C 2 j+1/2 are of order one, then, f a = O(ε 1/2 ) and therefore the truncation error is uniform with respect to ε < 1.
Discretization of the electron transport and ion dynamics system
In this subsection we present the discretization for the system (21). As explained before, this system represents the larger scales related to the transport of electrons and the ion scales.
Discretization of the electron transport system
The transport system for the electrons (eqs.(21a)-(21b)) is a hyperbolic system. By using the upwind scheme proposed in Chalons et al. [57], we can discretize this system by the following numerical scheme.
Numerical Scheme 2. The discretized transport system for the electrons (eqs.(21a)-(21b)) reads
where the velocity in the interface is defined in eq. (40) and
n ē n eūe L/R = n ē n eūe L ifū n e j+1/2 ≥ 0 n ē n eūe R else.(55)
As in Chalons et al. [57], the CFL condition associated to the transport system is
max j∈N ∆t ∆x (ū n e j−1/2 ) + − (ū n e j+1/2 ) − , ∆tν iz j ≤ 1,(56)
where he superscript stands for b ± = b±|b| 2 .
15
AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
Discretization of the ion dynamics system
The low-temperature plasmas have an additional difficulty in the integration of the ion dynamics. As the plasma potential scales with the electron temperature, the Lorentz force on the ions is in general much larger than the ions pressure gradient. Naive discretizations of the source term may create numerical instabilities (see, e.g., Section 5.2 for the present problem or [60,61,62,73,74] in a general framework). In order to tackle this problem, we propose an upwind discretization [60] of the source term in eqs. (21c)-(21d),as presented in the following numerical scheme.
Numerical Scheme 3. The discretization of the ion dynamics system (21c)-(21d) reads
U i n+1 j = U i n j − ∆t ∆x F i n j+1/2 − F i n j−1/2 + ∆t S i φ j * + S i Colls j n(57)
where
U i = (n i ,n iūi ) T , F i = n iūi ,n iū 2 i +n i κ T , S i φ = 0, −n i ∂xφ T , and S i Colls = (n eν iz , 0) T (58)
The discretization of the convective fluxes F i is performed with an approximate Riemann HLL scheme [75]. The discretization of the collisional source term is at the cell center, i.e.,
S i Colls j = (n eν iz ) n j , 0 T .(59)
Alternatively, as the Lorentz force is dominant in low pressure discharges, we propose a well-balanced scheme as in [60]. We decompose the source term into the contribution at the interfaces. For the sake of simplicity of notation, we do not include the subscript corresponding the time discretization. The discretization reads
S i φ j = G U ij−1 , U ij , U ij+1 ,φ j−1 ,φ j ,φ j+1 (60) = G L U ij−1 , U ij ,φ j−1 ,φ j + G R U ij , U ij+1 ,φ j ,φ j+1 .
where the functions G L/R read
G L (U ij−1 , U ij ,φ j−1 ,φ j ) = 1 2 I + A(U ij−1 , U ij ) A(U ij−1 , U ij ) −1 Ĝ (U ij−1 , U ij ,φ j−1 ,φ j ), (61a) G R (U ij , U ij+1 ,φ j ,φ j+1 ) = 1 2 I − A(U ij , U ij+1 ) A(U ij , U ij+1 ) −1 Ĝ (U ij , U ij+1 ,φ j ,φ j+1 ). (61b)
The matrix A is the Jacobian matrix of the convective flux andĜ is the source term computed in the interface. They are defined as
A U ij , U ij+1 = 0 1 −ū 2 i j+1/2 +c 2 i 2ū i j+1/2 and (62) G U ij , U ij+1 ,φ j ,φ j+1 = ĝ mass j+1/2 g mom j+1/2 = 0 n i j+1/2φ j+1−φj ∆x .(63)
whereū
i j+1/2 = n ijūij + n ij+1ūij+1 n ij + n ij+1 andn i j+1/2 =n ij +n ij+1 2 .
In the particular case of isothermal flow with no mass source term, G L/R simplify to
G L (U ij−1 , U ij ,φ j−1 ,φ j ) =ĝ mom j−1/2 (64a) 1 2c i sign(c i +ū i j−1/2 ) − 1 2c i sign(c i −ū i j−1/2 ) 1 +ū i j−1/2 2c i sign(c i +ū i j−1/2 ) − sign(ū i j−1/2 −c i ) + 1 2 sign(c i +ū i j−1/2 ) + sign(ū i j−1/2 −c i ) , G R (U ij , U ij+1 ,φ j ,φ j+1 ) =ĝ mom j+1/2 (64b) − 1 2c i sign(c i +ū i j+1/2 ) + 1 2c i sign(c i −ū i j+1/2 ) 1 −ū i j+1/2 2c i sign(c i +ū i j+1/2 ) − sign(ū i j+1/2 −c i ) − 1 2 sign(c i +ū i j+1/2 ) + sign(ū i j+1/2 −c i ) .
Note that the velocity of the ions is evaluated at the interface between the cells. As sign(δu i ) is a discontinuous function, in numerical experiments, we observe that the sign(δu i ) function introduces discontinuities in the solution that are advected by the fluids, as the model has no physical dissipation. For that reason, we use the following approximation for the sign(δu i ) function
sign(δu i ) ≈ tanh δu i u i∞ (65)
where δu i =c i + u i orc i − u i and u i∞ is a characteristic speed of the problem.
Summary of the numerical scheme
In Fig. 5, we summarize the steps followed in order to obtain the numerical results corresponding to the AP scheme. We highlight that the steps in the electron acoustic system need to be done sequentially. The order of steps 4) and 5) can be inverted obtaining the same result. 1) Solve for the electron density " * with eq. (25) by evaluating:
• " & '(/* with eq. (27) and , with eqs. (28) and (29).
Electron acoustic and electrostatic system (20)
2) Solve for the electric potential " * with the Poisson solver of eq. (30).
3) Solve for the electron velocity " * with eq. (31) by evaluating:
Electron transport and ion dynamics system (21)
5) Solve for the ion " 4'( and velocity " 4'( with eq. (57) by evaluating: • &78/9 with a Riemann solver, e.g., HLL [75]. • &78/9
;<==> with eq. (59).
Numerical results
In the present section, we present the numerical results of the proposed scheme. We first simulate a periodic two-stream perturbation in a thermal plasma (T e = T i ). In this problem we study: (1) the convergence of our numerical scheme with the time step and the mesh size, (2) the asymptotic behaviour and consistency of the scheme when χ → 0, and (3) to assess the low-Mach correction (limit when ε → 0) in the electron fluxes.
Second, we simulate the same instability in a low-temperature plasma order to study: (1) the performance of the well-balanced scheme in the ions, (2) assess the spatial convergence in a low-temperature plasma, and (3) to evaluate the computational performance as compared to a standard discretization.
Finally, we propose a numerical set-up to simulate a plasma sheath between two floating conducting walls. In this case, we simulate a physical domain that contains charge separation regions. We show that our numerical set-up captures the physics of the classical plasma sheath [37], computing the potential drop as predicted by the theory. As the numerical domain contains both a quasi-neutral and charged regions, in this test we assess the ability of the scheme to represent both limits.
Propagation of a two-stream periodic perturbation in a collisionless thermal plasma
The two-stream instability occurs in a uniform plasma of density n 0 where the electrons have a relative velocity u 0 with respect to the ions [12]. Perturbations of the equilibrium state can lead to instabilities for sufficiently long wavelengths k. In our case, we will restrict our work to study the propagation of a periodic perturbation in a stable plasma. This case has been previously used for the convergence study of the two-fluid model in a number of works [49,48,50]. Our set of equations is slightly different to these previous works as we study the isothermal case whereas in Crispel et al. [49,48,50] the fluid model assumes an isentropic law. As in [1], in our case, the dispersion relation is found to be F (k, ω) = α 4 ω 4 + α 3 ω 3 + α 2 ω 2 + α 1 ω + α 0 = 0 (66) where the coefficients α i (k; ε, κ, n 0 , v 0 , χ) read
α 4 = εχ, α 3 = −2εχku 0 , α 2 = εk 2 χ v 2 0 − κ − εn 0 − k 2 χ − n 0 (67a) α 1 = 2 k 2 κχ + n 0 , and α 0 = −εk 2 v 0 n 0 v 0 − εχ k 4 κv 2 0 + 2kω 3 v 0 + k 4 κχ + k 2 n 0 (κ + 1) . (67b)
Firstly, we consider a perturbation in a thermal plasma, i.e., where the ions have the same temperature as the electrons. We consider a mass ratio of ε = 10 −4 and a domain of length L = 1 with 10 4 Debye lengths, i.e., λ = 10 −4 .
The background plasma has a normalized density of n 0 = 1 and the electrons travel at a velocity of u 0 = 1 that corresponds to Bohm's speed at these conditions. We fix the wavelength of the perturbation to k = 2π. Under these conditions, the characteristic polynomial, eq. (66), has four real solutions, i.e., the perturbation is stable. There are two high frequency solutions, which correspond to the electron plasma waves, and two low frequency that are quasi-neutral. For our study, we use one of the low frequency solutions, i.e., ω = 8.8857268. The initial field is the analytical solution for a small perturbation at the given frequency, wavelength, and plasma conditions. In the present case, this solution is
U (x, t = 0) = n ē u ē n ī u ī φ = 1 + 2.41425 × 10 −2 sin(2πx) 1 + 10 −2 sin(2πx) 1 + 2.41425 × 10 −2 sin(2πx) 3.41425 × 10 −2 sin(2πx) 2.41421 × 10 −2 sin(2πx) .(68)
In Fig. 6, we present the solution of the Lagrange-projection AP scheme without the low-Mach correction for the electrons at t = 0.7071 which corresponds to one period of the wave. We present the results with four different mesh resolutions N = 50, 100, 200, and 400 points. We highlight that the mesh size is not adapted to the Debye length as there are 200 Debye lengths inside one cell in the case of lowest resolution. The simulations use a CFL conv = 0.7, which corresponds to a time step of ∆t = 1.4141 × 10 −4 , 0.7071 × 10 −4 , 0.3535 × 10 −4 , and 0.176775 × 10 −4 . We also notice that this time step is much larger than the period of a plasma wave, i.e., ω −1 pe = 10 −6 . A standard simulation that resolved the Debye length with 10 mesh points and the plasma frequency with 10 time-steps, would have 2000 more mesh points and a time step 10 3 times smaller.
In Fig. 7, we present the solution forn e ,n i ,ū e , and φ at t = 0.0626, i.e., after 900 iterations of the standard HLL scheme described in Sec. 3 that uses 100 mesh points and a CFL conv = 0.7. These results illustrate that indeed, using the same conditions than in the Lagrange-projection AP scheme with N = 100, the standard discretization is unstable. It is also important to note that the scheme becomes unstable very fastly due to the rapid unstable response of the electrons.
The same simulation is presented with using the Lagrange-projection AP scheme with the low-Mach correction for the electrons in Fig. 8. We show the previous four discretization resolutions with CFL conv = 0.7. With the low-Mach correction, the solution for the electron velocity converges to the analytical solution when the mesh size is decreased, showing a much smaller error.
As explained in [54], when the numerical diffusion is rescaled for the low-Mach regime, the CFL condition is not restricted by the acoustic waves. This allows for the adaptation of the CFL to the eigenvalue λ e =ū e rather than to λ e =ū e ±c e . In Fig. 9, we present the comparison of the solution for CFL conv = 0.7, 7.1, and 28.5 with the low-Mach correction and 100 cells. We highlight that the largest CFL corresponds to an ion CFL conv i = 0.28. The results show that the numerical diffusion is lower when the CFL is adapted to the ion scales. In the simulation with the largest CFL, the time step is ∆t = 1.99 × 10 4 ω −1 pe , i.e., 10 4 larger than an explicit non-AP discretization. In Fig. 10, the convergence of the L 2 error norm is presented for three different cases: the AP scheme without low-Mach correction for the electrons and CFL conv = 0.7 (left), the AP scheme with low-Mach correction for the electrons and CFL conv = 0.7 (center), and the AP scheme with low-Mach correction for the electrons and CFL conv = 28.5 (right). We clearly see that the low-Mach correction of the electrons reduces the error of the electron velocity by one order of magnitude. When the CFL conv = 0.7, the convergence of the error corresponds to a first order scheme. In the case of CFL conv = 28.5, the error norm is reduced, especially in the ion quantities. However, we observe that the error convergence is slower than a first-order scheme. This is caused by the low-Mach regime of the ions. Since the perturbation of the ion velocity is small, the Mach of the ions varies from M i = 0 to 0.03. This could be fixed with a low-Mach correction, similar to the one performed in the electrons. Nevertheless, the interest of this paper is low-temperature plasmas, which means that in general, the velocity of the ions is much larger than the ion thermal speed. For that reason, the low-Mach regime of the ions is not treated in the present paper. In the following example, in a low-temperature plasma, we will show how this convergence improves when the Mach of ions is larger.
Propagation of a two-stream periodic perturbation in a collisionless low-temperature plasma
As mentioned above, in low-temperature plasmas, the bulk velocity of ions is usually much larger than the ion thermal speed. When the ion velocities are comparable or larger than the ion thermal speed, this can produce numerical problems due to the imbalance between the cell-centered source terms and the upwinded fluxes. In order to illustrate the performance of the numerical scheme in a case of low-temperature plasmas, we simulate a two-stream periodic perturbation in a plasma with an ion-to-electron temperature ratio of κ = 10 −4 . We use the dispersion relation of eq. (66) with ε = κ = λ = 10 −4 . We choose n 0 = 1 and, in this case, v 0 = 0.7. Note that v 0 = 1 produces a resonance and therefore the perturbations would be too large to study a linear wave 21 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al. propagation. With these conditions, the initial condition reads
U (x, t = 0) = n ē u ē n ī u ī φ = 1 + 3.3328 × 10 −3 sin(2πx) 0.7 + 10 −3 sin(2πx) 1 + 3.33285 × 10 −3 sin(2πx) 3.333 × 10 −3 sin(2πx) 3.3328 × 10 −3 sin(2πx) .(69)
The period of the oscillations is T = 0.99995. We run the simulation and compare the results to the initial field after one period. In Fig. 11, we show the comparison between the scheme with the well-balanced source term in eq. (60) Under these conditions, the Mach regime of the maximum velocity of the ions is M i = 0.33 and the electrons M e = 0.007. The convergence for different mesh sizes is presented in Fig. 12. In this case, the ions are in a larger Mach regime and therefore the compressible solver is less dissipative than in the thermal plasma case (κ = 1). As a result, as shown in Fig. 13, the error convergence is indeed first-order accuracy in space for this scheme, despite the extremely low Mach regime of the electrons and the limit of λ → 0. Finally, we compare the computational performance of the present numerical scheme. We highlight that the Poisson solver with periodic boundaries is solved by adding a Lagrange multiplier to the system that imposes the total charge in the domain to be zero and by solving the linear system with a LU decomposition implemented in the library LAPACK [76]. We choose to use a linear solver instead of a spectral method as we consider it to be more representative of a realistic multidimentional case with non-periodic boundary conditions. This is due to the fact that most of the iterative methods for solving linear systems do not scale linearly with the number of points and therefore the computational performance will be largely penalized by number of points.
In Table 2, we present the computational performance for three different cases: (1) a first-order standard discretization that resolves the Debye length with one cell and the plasma frequency with ten time steps, (2) the same first order standard that a mesh cell that is ∆x = 100 and ∆t = 0.1ω −1 pe (the scheme needs to resolve ω −1 pe for stability reasons), and (3) the AP scheme with ∆x = 100 and ∆t = 2500ω −1 pe . We highlight that the result of the standard case without resolving the Debye length diverges as the numerical error in the electron velocity becomes very large and the perturbation is not linear after a time-step.
In Table 2, we can see that the AP scheme produces a dramatic improvement as compared to a standard case when the plasma is quasi-neutral and the characteristic size of the phenomena is very large as compared to the Debye length. As the AP scheme does not need to resolve the Debye length to preserve the quasi-neutrality limit, the AP discretization needs 100 times less mesh points and 25000 times less time steps than a scheme that resolves the scales related to the Debye length. Consequently, the CPU time taken for simulating a period of the wave is of the order of 10 9 times faster than a first-order standard scheme with similar accuracy.
Collisionless isothermal sheath
The previous numerical experiments simulate a quasi-neutral plasma. Nevertheless, as discussed in Section 2.3, the set of equations allows for capturing charge separation effects, as shown in Fig. 1. For length scales much greater than the Debye length, the plasma behaves in general as quasi-neutral. However, when a surface is in contact with the plasma, a charged boundary layer called the plasma sheath is formed.
We consider a plasma in a 1D domain of unitary size with an electron-to-ion mass ratio ε = 10 −5 , ion-to-electron temperature ratio κ = 0.025, and χ = 10 −4 . The electron flux is imposed on both boundaries by the number of particles crossing the plane with positive velocity component
n eūe (x = 0, t) = −n e √ 2πε andn eūe (x = 1, t) =n e √ 2πε .(70)
The electron and ion density and the ion flux have a Neumann boundary on both sides. Alternatively, the electric potential has a Dirichlet condition on both sidesφ(x = 0, t) =φ(x = 1, t) = 0. In the results, the potential is referenced to the potential in the plasma atx = 1/2. In the sheath theory [37], the ionization is an eigenvalue of the problem. This means that there is only one ionization rate that can produce a steady solution. In our simulation set-up, we find this eigenvalue by changing the ionization frequency such the balance between ionization and ion flux occurs at every time step. Consequently, the ionization frequency at t n is calculated as
ν iz (t n ) = |n iūi (x = 0, t n )| + |n iūi (x = l, t n )| l 0 n e (t n ) dx .(71)
The flow field is initialized as follows U (x, t = 0) = (n e ,n eūe ,n i ,n iūi ,φ) T = (1, 0, 1, 0, 0) T .
As the assumptions of the classical theory are not taken by our numerical model, we might expect a result that is slightly different from eq. (13). For this reason, we will use as a reference solution a high-order highly resolved simulation described in Section 2.3. In Fig. 14, we show the steady solution reference solution that uses a standard discretization, as in Section 3, with TVD third-order reconstruction and third-order TVD Runge-Kutta [77] scheme. In the reference solution, the domain is resolved with N = 10 4 mesh points and CFL=0.9. By doing this, we ensure that the Debye length is resolved with 100 points and so is the electron plasma period. Since the scheme is high 24 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
order and highly resolved, the discretization errors are expected to be much smaller than a first order scheme with lower resolution.
The solution shows two plasma sheaths beside the left and the right boundaries of around 6 -7 Debye lengths width. As shown in the densities, the plasma is quasi-neutral in the rest of the domain, whereas it is positively charged in the sheath. The plasma potential at the wall is very similar to the theory, despite the electron inertial and the ion pressure gradient effects that are neglected in the theory. The flux of ions and electrons are equal in the whole domain. The velocities are also equal in the quasi-neutral region and they differ when the ions reach |ū i | = 1, i.e., the Bohm's velocity in our units. This point is the plasma-sheath transition that agrees with the Bohm's criterion [36]. In Fig. 15, we present the converged results using a first-order standard discretization with CFL = 0.1 and N = 10 3 . The local error is calculated with the reference solution. The main difference with the reference solution can be clearly observed in the electron flux. The error of the electron flux is as large as the actual solution. This is due to the poor behaviour of the HLL scheme under low-Mach conditions, as explained in the previous examples. Even though the error in the electron flux seems to not have a large impact in the other variables in Fig. 15. This is only true for the steady state solution. However, if we analyze the evolution of the simulation to steady state, the solutions are radically different even though they have similar steady state solution. In Fig. 17, the solution at t = 75 is presented for the reference solution, the HLL first order scheme and the AP scheme. We observe that whereas the solution of the AP scheme is very similar to the reference solution, the HLL standard scheme develops a plasma instability. This plasma instability is identified as a two-stream instability that is induced by the numerical error in the electron velocity. The instability is not present in the HLL scheme when the CFL is reduced to CFL= 0.01 (see [38]. This numerical example illustrates that the AP scheme provides a more stable discretization that avoids spurious numerical instabilities in the quasi-neutral region, while still being able to accurately capture the regions with charge separation. Additionally, the scheme increases dramatically the accuracy of the computation of electron dynamics as well as increases by several orders of magnitude the computational efficiency.
Summary and conclusions
In low-temperature plasma applications, the plasma-wall interaction is fundamental to explain the conditions inside the bulk of the plasma. For this reason, the plasma sheaths need to be included in the computational domain to fully explain the characteristics of the discharge. Consequently, the numerical model needs to be able to resolve the Debye length in order to capture the potential drop that forms beside the walls of the device. Similarly, the electron inertia is seen to play a fundamental role in low-pressure conditions. However, in most parts of the domain, the plasma behaves as quasi-neutral with massless electrons. A numerical method that is able to tackle accurately and efficiently these different regimes is a long-standing problem in computational plasmas. In this paper, we have presented a numerical scheme for the isothermal plasma equations coupled to Poisson's equation that proves to be accurate and stable both in the quasi-neutral and charge regimes, and both when electrons behave as massless and when electron inertia plays a role.
In Section 2.2, we have discussed the asymptotic behaviour of the plasma equations when both the electron mass and the Debye length tend to zero. We find a different asymptotic behaviour as compared with previous work and that the zero-th order solution of the electric potential depends only on the electron dynamics when the electronto-ion mass ratio is included in the asymptotic expansion. This relation is known in plasma physics as Boltzmann density distribution. Alternatively, the electron momentum equation at O(1) is similar the low-Mach limit of the Euler equations.
28
AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
Based on this asymptotic behaviour, we have proposed a novel operator splitting strategy that relies on the Lagrange-projection scheme. Our approach solves for the electron acoustic system coupled to the Poisson equation in a first step and the electron transport together with the ion dynamics in a second step. We show that the first step preserves the asymptotic limit when both the electron mass and the Debye length tends to zero. This means that even when these scales are not resolved, the numerical scheme is able to find the correct asymptotic solution, without the need for an implicit scheme. Consequently, the scheme is able to achieve a great improvement in the accuracy of the solution as the numerical dissipation is properly scaled to the physics. More importantly, the numerical method improves dramatically the computational efficiency when these scales are not needed to be resolved.
The numerical scheme has the advantage of solving a standard Poisson equation, as compared to other AP and semi-implicit methods that solves for more complex parabolic or elliptic equations. Additionally, the acoustic and electrostatic step involves only the electron dynamics and, therefore, the proposed model can be used in hybrid approaches where the electrons are treated as a fluid and the ions with a kinetic approach. Moreover, we have proposed a well-balanced treatment of the Lorentz force in the ions that proves to be fundamental in order to avoid spurious numerical oscillations when the temperature of the ions is much lower than this of electrons.
In the numerical results, we have simulated a quasi-neutral two-stream perturbation in thermal and low-temperature plasmas. Additionally, we have simulated a plasma sheath in a low-temperature plasma. The two-stream perturbation allowed us for assessing the performance of the model when the plasma is quasi-neutral but the electron velocity differs from this of ions. We also have proved that the scheme is able to reproduce the numerical solution even when the Debye length and the electron sound speed are not resolved. This allows us for achieving an efficient numerical scheme as compared to a scheme that needs to resolve the scales related to the Debye length or the electron acoustic waves. In our numerical experiments, the AP scheme is 10 9 faster to simulate a plasma period of a quasineutral wave than the standard discretization that needs to resolve the Debye length and the electron plasma frequency. Additionally, the electron velocity is properly captured, avoiding the problems associated to the low-Mach limit of electrons.
Finally, we have proposed a numerical set-up to simulate the steady solution of a plasma between two floating plates. The set-up is able to find self-consistently a solution that agrees with the analytical plasma sheath theory. This solution contains three different regions: (i) the quasi-neutral plasma where the ions and electrons have the same density and velocity, (ii) the plasma-sheath transition where ions are accelerated to the Bohm's speed, and (iii) the sheath where the plasma is electrically charged and the electrons are accelerated to speeds comparable to their thermal speed. The proposed AP scheme is able to reproduce accurately these three regions. The standard discretization is not able to reproduce the electron flux as the standard numerical scheme is not able to reproduce the limit when the electron inertia is very small, even when the Debye length is properly resolved. Additionally, the standard scheme develops non-linear instabilities that are triggered by the numerical error in the electron velocity.
In the present paper, we have discussed the fundaments of an asymptotic preserving scheme for a fluid plasma model in low-temperature plasma conditions. We have highlighted that the ideas presented in this paper are the basis for future developments, extending the scheme to multi-dimensions, the inclusion of the energy equation, and the development of high-order schemes including this splitting strategy. Similarly, the scheme presented in the present paper would be specially advantageous for a non-uniform mesh where the quasi-neutral region of the plasma does not need to be resolved, whereas the boundaries resolve the Debye length in order to capture the plasma sheath.
AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
Appendix A: Approximate Riemann solver of the acoustic step
We can write the system of eq. (23) as
∂t τ ē u e + ∂ m −ū e ε −1 τ −1 e = 0 ε −1 ∂xφ .(73)
Note that the pressure is written as function of the conservative variable τ e , i.e., p e = ε −1n e = ε −1 τ −1 e . We define the conservative variables of the electron acoustic step, the flux term, and the source term as .
The eigenvalues of this matrix are λ = −n e ε −1/2 ,n e ε −1/2 . Consequently, the Lax-Friedrich numerical flux for the previous Riemann problem reads
where the density at the interface is computed asn e j+1/2 = (n e R +n e L ) /2.
Appendix B: Truncation error of the acoustic system
We study the truncation error of the upwind scheme for the acoustic step. We consider that the variables describe a smooth flow so that we can expand them in Taylor series the conservative variables. By using the Lax-Friedrich scheme of eq. (76), we obtain the following approximations in the mass equation
squared non-dimensional Debye length: χ = 0 k B T e n e e 2 L 2 0 , dimensionless frequency:ν iz = ν iz t 0 . 4 AP scheme for the low-temperature plasma fluid equations A. Alvarez Laguna et al.
as 0 F " .
Laguna et al. rms of ". Similarly, we define 0 F " as the the system of equations for ms of .
F
" and it reads
Figure 1 :
1Asymptotic behaviour of the electrostatic isothermal multi-fluid plasma equations.
equations for the zero-th order terms is very similar to system χ F 0 , with the difference in
Figure 2 :
2Solution of a plasma between two floating walls with a fluid model. The proposed numerical set-up is able to captures the physics as predicted by the theory[37].
Figure 4 :
4Illustration of the characteristic scales for the operator splitting.
Lū e L + √n e Rū e R √n e L + √n e R ε 1/2 and M o = min 1, max M co ,M .
lnn n ej+1 + lnn n ej−1 − 2 lnn n ej .
and M o = min 1, max M co ,M .
• ̅ & '(/* with eq. (33) and , with eqs.(28) and(29).• 3̅ " 1 with eq. (32)." * , " * , " * 4) Solve for the electron density " 4'( and velocity " 4'( with eq. (54) by evaluating:• " & '(/* with eq. (27) and , with eqs.(28) and(29).
. (60), (61), (62), and (63).
Figure 5 :
5Algorithm for the implementation of the AP scheme
Figure 6 :
6Solution of the two-stream periodic perturbation in a collisionless thermal plasma at t = 0.7071 for different mesh resolutions with the asymptotic preserving scheme without the low-Mach correction. The simulations use CFL conv = 0.7, corresponding to a ∆t = 1.4142 × 10 −4 , 0.7071 × 10 −4 , 0.3535 × 10 −4 , and 0.176775 × 10 −4 . Note that ∆t > ω −1 pe = 10 −6 . The absence of correction for the low-Mach regime of the electrons results in a large error on the electron velocity.
Figure 7 :Figure 8 :
78Solution of the two-stream periodic perturbation in a collisionless thermal plasma at t = 0.0636 with a standard HLL scheme discretization on 100 cells with CFL conv = 0.7. Under these conditions the discretization is unstable and diverges very rapidly due to the fast dynamics of the electrons. Solution of the two-stream periodic perturbation in a collisionless thermal plasma at t = 0.7071 for different mesh resolutions with the asymptotic preserving scheme with low-Mach correction. The simulations use CFL conv = 0.7, corresponding to a ∆t = 1.4142 × 10 −4 , 0.7071 × 10 −4 , 0.3535 × 10 −4 , and 0.176775 × 10 −4 . Note that ∆t > ω −1 pe = 10 −6 . The preconditioning of the numerical dissipation for the electrons results in a correct discretization of the electron velocity.
Figure 9 :Figure 10 :
910Solution of the two-stream periodic perturbation in a collisionless thermal plasma at t = 0.7071 with N = 100 points with the asymptotic preserving scheme with low-Mach correction and different time resolution. The time splitting strategy with low-Mach correction allows for a larger time step that does not need to resolve the electron sound waves without an implicit discretization. Comparison of the L 2 error norm in the two-stream periodic perturbation in a collisionless thermal plasma. We present the AP scheme without low-Mach correction (left), with low-Mach correction and CFL conv = 0.7 (center), and with low-Mach correction and CFL conv = 28.5 (right).
Figure 11 :
11and a cell-centered implementation of the source. Both simulations use CFL conv = 25 and N = 100. The approximation of the sign function used follows eq. (65) with u i∞ = v 0 . The results show that the solution without the well-balanced implementation maintain the quasi-neutrality of the solution. However, it develops spurious numerical oscillations. These oscillations are not present with the well-balanced implementation. Solution of the two-stream periodic perturbation in a collisionless low-temperature plasma at t = 0.99995 using the well-balanced scheme for the ions (solid lines) and with a standard discretization of the Lorentz force (dashed line). The standard discretization develops oscillations that are not present in the well-balanced scheme. The simulations use N = 100 points and CFL conv = 25.
Figure 12 :
12Solution of two-stream periodic perturbation in a collisionless low-temperature plasma at t = 0.99995 for different mesh resolutions with the asymptotic preserving well-balanced scheme. The simulations use CFL conv = 25.
Figure 13 :
13L 2 error norm as function of the mesh size in the simulation of a two-stream periodic perturbation in a collisionless low-temperature plasma.
Figure 14 :
14Collisionless isothermal sheath: Solution of a plasma between two floating walls with a third order TVD scheme using N = 10 4 mesh points. The proposed numerical set-up is able to captures the physics as predicted by the theory[37].
Figure 15 :Figure 16 :
1516Collisionless isothermal sheath: Converged first-order solution with a HLL scheme and CFL = 0.1 and 10 3 mesh points. The simulation shows a large error in the electron flux due to the large numerical error at low-Mach speeds. Collisionless isothermal sheath: Converged first-order solution with the asymptotic preserving scheme with low-Mach correction with CFL = 0.4 and 10 3 mesh points. The simulation reduces dramatically the error in the electron flux.
Figure 17 :
17Collisionless isothermal sheath: Comparison of the HLL first-order standard discretization and the asymptotic preserving scheme with low-Mach correction discretization at t = 75. The first-order standard discretization develops a numerical stability that is not present in the reference solution.
U
ac e = (τ e ,ū e ) T , F ac e = −ū e , ε −1 τ −1 e T , and S ac e = 0, ε −1 ∂xφ T .(74)The Jacobian matrix of the flux term in the LHS of eq.
e R − τ e L ) ε −1 (ne R +ne L )
≈≈
∂tτ e (t n , x j ) + ∂ 2 tt τ e (t n , x j )∆t + O(∆t 2 )* [∂ mūe ] n j ≈ 1 ne ∂xū e (t n , x j ) + ε −1/2 ∂ 2 xx τ e (t n , x j )∆x + O(∆x 2 ) + O(ε xxne (t n , x j ) + O(ε −1 ∆x 2 )Therefore, the truncation error of the discretization is[Trunc. Error] τe = ε −1/2 − CFL ∂ 2 xx τ e xxne + O(ε −1 ∆x 2 ∆t) + O(∆x 2 ) + O(ε −1/2 ∆x 3 )(77)Similarly, we obtain the following approximations for the equation for the velocity, expanded in ∂tū e (t n , x j ) + ∂ 2 ttū e (t n , x j )∆t + O(∆t 2 )* [∂ mpe ] n j ≈ ε −1 ne ∂xn e (t n , x j ) + ε −1/2 ∂ 2 xxūe (t n , x j )∆x + O(∆x 2 ) + O(ε −1/2 ∆x 3 ) * [∂ xφ ] * j ≈ 1 ε ∂xφ(t n , x j ) + O(ε −1 ∆x 2 ) Consequently,the truncation error of the velocity equations reads [Trunc. Error]ū e = ε −1/2 − CFL ∂ 2 xxūe ∆x + O(ε −1 ∆x 2 )
Table 1 :
1Characteristic values of an Argon RF discharge at 1 Pa[1].Dimensional quantities
Dimensionless quantities
Neutral density
n n
1.25 × 10 20
m −3
Electron-to-ion mass ratio
ε = m e /m i
1.36 × 10 −5
Electron density
n e,i
10 16
m −3
Ion-to-electron temperature ration
κ = T i /T e
0.025
Neutral and ion temperature
T n,i
0.05
eV
Normalized Debye length
λ = λ D /l
3.5 × 10 −3
Electron temperature
T e
2
eV
Ionization level
n e,i /n n
8 × 10 −5
Distance between plates
l
3
cm
Electron Knudsen number
Kn e
1.7
Ion-neutral collisional cross section
σ in
10 −18
m 2
Ion Knudsen number
Kn i
0.17
Electron-neutral collisional cross section
σ en
10 −19
m 2
Normalized ionization rateν iz
0.0139
Ionization constant
K ion 8.16 × 10 −18 m 3 s −1
Normalized electron collision rateν e
153.8
Ionization potential
ε ion
17.44
eV
Normalized ion collision rateν i
0.94
Electron plasma period
ω −1
pe
1.77 × 10 −10
s
Normalized plasma periodω −1
pe = ω −1
pe u B /l 1.29 × 10 −5
the case of periodic boundary conditions.e.g., electron plasma waves,
sheaths
Complete problem
e.g., plasma-sheath transition
Massless electrons with
finite Debye length
e.g., electrostatic waves
(two-stream, drift-wave instability)
Quasi-neutral plasma with
finite electron inertia
e.g., most of the plasma
phenomena at ion scales
Quasi-neutral plasma with
massless electrons
, plasma-sheath transitionMassless electrons with
finite Debye length
e.g., most of the plasma
phenomena at ion scales
Quasi-neutral plasma with
massless electrons
$ → 0
Proposition 1. The system of equations that corresponds to lim!0
Table 2 :
2Solution of the two-stream periodic perturbation in a collisionless low-temperature plasma: Comparison of the computational performance with the standard and the AP Lagrange-projection scheme.Numerical scheme and resolution
Cells Iters. for one period CPU time per iter. [s] CPU time for one period [s]
Standard ∆x = λ, ∆t = 0.1ω −1
pe
10000
10 7
41.7175369
4.2 × 10 8
Standard ∆x = 100λ, ∆t = 0.1ω −1
pe
100
10 7
1.1 × 10 −3
1.1 × 10 4
AP ∆x = 100λ, ∆t = 2500ω −1
pe
100
400
1.1 × 10 −3
0.44
The results using the AP Lagrange-projection scheme with CFL conv = 0.4 and N = 10 3 are presented inFig. 16. We note that, as we want to resolve the plasma sheath, in this case we resolve the Debye length in our discretization. The results show that the plasma sheaths are properly captured with the scheme and the simulation agrees with the reference solution. Additionally, the electron flux is correctly captured, reducing by two orders of magnitude the numerical error. The rest of magnitudes have similar values to the reference solution, with a first-order discretization and ten times lower resolution than the reference solution.
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| [] |
[
"Some Properties of Balanced Hyperbolic Compact Complex Manifolds",
"Some Properties of Balanced Hyperbolic Compact Complex Manifolds"
] | [
"Samir Marouani ",
"Dan Popovici "
] | [] | [] | We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the L 2 harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain L 1 currents on the universal covering space of a balanced hyperbolic manifold. | 10.1142/s0129167x22500197 | [
"https://arxiv.org/pdf/2107.09522v2.pdf"
] | 236,134,093 | 2107.09522 | 59707a19e84ab5f3c73b0382fb3cee3eddbfe371 |
Some Properties of Balanced Hyperbolic Compact Complex Manifolds
Samir Marouani
Dan Popovici
Some Properties of Balanced Hyperbolic Compact Complex Manifolds
We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the L 2 harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain L 1 currents on the universal covering space of a balanced hyperbolic manifold.
Introduction
In this paper, we continue the study of compact complex balanced hyperbolic manifolds that we introduced very recently in [MP21] as generalisations in the possibly non-projective and even non-Kähler context of the classical notion of Kähler hyperbolic (in the sense of Gromov) manifolds. Recall that every Kähler hyperbolic manifold is also Kobayashi/Brody hyperbolic.
On the other hand, recall that a Hermitian metric on a complex manifold X identifies with a C ∞ positive definite (1, 1)-form ω on X. If we put dim C X = n ≥ 2, a Hermitian metric ω is said to be balanced (see [Gau77] where these metrics were introduced under the name of semi-Kähler and [Mic83] where they were given this name) if dω n−1 = 0. Moreover, ω is said to be degenerate balanced (see [Pop15] for the name) if ω n−1 is d-exact. Unlike in the Kähler setting, where no d-exact Hermitian metric ω can exist on a compact complex manifold, compact complex manifolds carrying degenerate balanced metrics do exist. These manifolds include:
(i) the connected sums X = k (S 3 × S 3 ), with k ≥ 2, endowed with the Friedman-Lu-Tian complex structure constructed via conifold transitions ( [Fri89], [LT93], [FLY12]);
(ii) the quotients X = G/Γ of any semi-simple complex Lie group G by a lattice Γ ⊂ G ( [Yac98]).
In [MP21], we generalised the notion of degenerate balanced compact complex manifolds starting from the observation that this is a kind of hyperbolicity property.
Throughout the text, π X : X −→ X will stand for the universal cover of X. If ω is a Hermitian metric on X, we let ω = π X ω be the Hermitian metric on X that is the lift of ω. According to [Gro91], a C ∞ k-form α on X is said to be d(bounded) with respect to ω if π X α = dβ on X for some C ∞ (k − 1)-form β on X that is bounded w.r.t. ω. Now, recall that a compact complex manifold X is said to be Kähler hyperbolic in the sense of Gromov (see [Gro91]) if there exists a Kähler metric ω on X (i.e. a Hermitian metric ω with dω = 0) such that ω is d(bounded) with respect to itself. In [MP21, Definition 2.1], we introduced the following 1-codimensional analogue of this:
Any such metric ω is called a balanced hyperbolic metric.
The implications among these notions are summed up in the following diagram. (See [MP21] for relations with other hyperbolicity notions.)
X is Kähler hyperbolic =⇒ X is balanced hyperbolic =⇒ X is degenerate balanced
We now outline the specific properties of these classes of manifolds that we prove in this paper.
(I) Case of balanced and degenerate balanced manifolds
In the first part of the paper, we obtain some general results on compact complex manifolds carrying balanced metrics (and, in some cases, results on Gauduchon metrics) and then we use them to infer vanishing results for degenerate balanced manifolds. See §.2.1 for a reminder of the terminology used in what follows.
(a) Our first main result, obtained as a consequence of the computation in Lemma 2.1, is a Hard Lefschetz Isomorphism between the De Rham cohomologies of degrees 1 and 2n-1 that holds on any compact complex balanced manifold satisfying a mild ∂∂-type condition.
For any Hermitian metric ω on an n-dimensional complex manifold, we will use throughout the paper the notation ω p := ω p /p! for any integer p between 2 and n.
Theorem 1.1. Let X be a compact complex manifold with dim C X = n.
is well defined and depends only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, C). (ii) If, moreover, X has the following additional property: for every form v ∈ C ∞ 1, 1 (X, C) such that dv = 0, the following implication holds:
v ∈ Im ∂ =⇒ v ∈ Im (∂∂),(2)
the map (1) is an isomorphism.
As a consequence of this discussion, we obtain the following vanishing properties for the cohomology of degenerate balanced manifolds.
Proposition 1.2. Let X be a compact degenerate balanced manifold.
(i) The Bott-Chern cohomology groups of types (1, 0) and (0, 1) of X vanish: H 1, 0 BC (X, C) = 0 and H 0, 1 BC (X, C) = 0. (ii) If, moreover, X satisfies hypothesis (2), its De Rham cohomology group of degree 1 vanishes: H 1 DR (X, C) = 0.
Note that degenerate balanced manifolds that satisfy hypothesis (2) do exist. Indeed, Friedman showed in [Fri19] that the manifolds X = k (S 3 × S 3 ), with k ≥ 2, endowed with the Friedman-Lu-Tian complex structure constructed via conifold transitions ( [Fri89], [LT93], [FLY12]) are even ∂∂-manifolds.
(b) Our study of the cohomology of degree 2 in this setting centres on seeking out possible positivity properties of balanced hyperbolic manifolds. As an alternative to Question 1.4. in [MP21], wondering about possible positivity properties, in the senses defined therein, of the canonical bundle K X of any balanced hyperbolic manifold X, we concentrate this time on whether there are "many" (in a sense to be determined) closed positive currents T of bidegree (1, 1) on such a manifold.
The starting point of this investigation is Proposition 5.4 in [Pop15], reproduced as Proposition 2.10. in [MP21]: a compact complex manifold X is degenerate balanced if and only if there exists no non-zero d-closed (1, 1)-current T ≥ 0 on X. In other words, the compact degenerate balanced manifolds X are characterised by their pseudo-effective cone E X (namely, the set of Bott-Chern cohomology classes of d-closed positive (1, 1)-currents on X) being reduced to the zero class.
This prompts one to ask the following Question 1.3. Let X be a compact complex manifold. Is it true that X is balanced hyperbolic if and only if its pseudo-effective cone E X is small (in a sense to be determined)?
In §.2.3 and §.2.4 we provide some evidence for this by first showing that both the balanced hypothesis on a given Hermitian metric ω (see Lemma and Definition 2.2) and the Gauduchon hypothesis (see Lemma and Definition 2.12) enable one to define a notion of ω-primitive De Rham cohomology classes of degree 2 (resp. ω-primitive Bott-Chern cohomology classes of bidegree (1, 1)). For example, if ω is balanced, we set
H 2 DR (X, C) ω-prim := ker {ω n−1 } DR ∧ · ⊂ H 2 DR (X, C),
after we have showed that the linear map:
{ω n−1 } DR ∧ · : H 2 DR (X, C) −→ H 2n DR (X, C) C, {α} DR −→ {ω n−1 ∧ α} DR ,
is well defined and depends only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, C). We go on to show that a class c ∈ H 2 DR (X, C) is ω-primitive if and only if it can be represented by an ω-primitive form (cf. Lemma 2.3), a fact that does not seem to hold in the Gauduchon context of §.2.4. We then show that, when the balanced metric ω is not degenerate balanced, the ω-primitive classes form a complex hyperplane H 2 DR (X, C) ω-prim in H 2 DR (X, C) that depends only on the balanced class {ω n−1 } DR . (See Corollary 2.5.) Finally, we are able to define a positive side H 2 DR (X, R) + ω and a negative side
H 2 DR (X, R) − ω of the hyperplane H 2 DR (X, R) ω-prim := H 2 DR (X, C) ω-prim ∩ H 2 DR (X, R) in H 2 DR (X, R) and get a partition of H 2 DR (X, R): H 2 DR (X, R) = H 2 DR (X, R) + ω ∪ H 2 DR (X, R) ω-prim ∪ H 2 DR (X, R) − ω .
A similar study of the case where ω is only a Gauduchon metric in §.2.4 leads to the characterisation of the pseudo-effective cone as the intersection of the non-negative sides of all the hyperplanes H 1, 1 BC (X, R) ω-prim determined by Aeppli cohomology classes [ω n−1 ] A of Gauduchon metrics ω on X:
E X = [ω n−1 ] A ∈G X H 1, 1 BC (X, R) ≥0 ω ,(3)
where G X is the Gauduchon cone of X (introduced in [Pop15] as the set of all such Aeppli cohomology classes, see §.2.1 for a reminder of the terminology).
In §.3.1, we answer a version of Question 1.3 on the universal covering space of a balanced hyperbolic manifold in the following form. Throughtout the paper, L p ω , L p ω , resp. L p g will stand for the space of objects that are L p with respect to the metric ω, ω, resp. g.
Proposition 1.4. Let (X, ω) be a balanced hyperbolic manifold and let π : X −→ X be the universal cover of X. There exists no non-zero d-closed positive (1, 1)-current T ≥ 0 on X such that T is L 1 ω , where ω := π ω is the lift of ω to X. This result provides the link with the second part of the paper that we now briefly outline.
(II) Case of balanced hyperbolic manifolds
The results in the second part of the paper mirror, to some extent, those in the first part. The main difference is that the stage changes from X to its universal covering space X. Specifically, when X is supposed to be balanced hyperbolic, we obtain vanishing theorems for the L 2 harmonic cohomology of X.
(a) In this setting, our main result in degree 1 and its dual degree 2n − 1 is the following Theorem 1.5. Let X be a compact complex balanced hyperbolic manifold with dim C X = n. Let π : X −→ X be the universal cover of X and ω := π ω the lift to X of a balanced hyperbolic metric ω on X.
There are no non-zero ∆ ω -harmonic L 2 ω -forms of pure types and of degrees 1 and 2n − 1 on X:
H 1, 0 ∆ ω ( X, C) = H 0, 1 ∆ ω ( X, C) = 0 and H n, n−1 ∆ ω ( X, C) = H n−1, n ∆ ω ( X, C) = 0,
where ∆ ω := dd ω + d ω d is the d-Laplacian induced by the metric ω.
The differential operators d, d ω , ∆ ω and all the similar ones are considered as closed and densely defined unbounded operators on the spaces L 2 k ( X, C) of L 2 ω -forms of degree k on the complete complex manifold ( X, ω). (See reminder of some basic results on complete Riemannian manifolds and unbounded operators in §.3.1.) (b) To introduce our results in degree 2, we start by reminding the reader of the following facts of [Dem84] (see also [Dem97, VII, §.1]). For any Hermitian metric ω on a complex manifold X with dim C X = n, one defines the torsion operator τ = τ ω := [Λ ω , ∂ω ∧ · ] of order 0 and of type (1, 0) acting on the differential forms of X, where Λ ω is the adjoint of the multiplication operator ω ∧ · w.r.t. the pointwise inner product , ω defined by ω. The Kähler commutation relations generalise to the arbitrary Hermitian context as
i[Λ ω ,∂] = ∂ + τ(4)∆ τ = ∆ τ + ∆ .(5)
When the metric ω is Kähler, one has τ = 0 and one recovers the classical formula ∆ = ∆ + ∆ . However, we will deal with a more general, possibly non-Kähler, case.
In the context of balanced hyperbolic manifolds, our main result in degree 2 is the following Theorem 1.6. Let X be a compact complex balanced hyperbolic manifold with dim C X = n. Let π : X −→ X be the universal cover of X and ω := π ω the lift to X of a balanced hyperbolic metric ω on X.
There are no non-zero semi-positive ∆ τ -harmonic L 2 ω -forms of pure type (1, 1) on X:
α 1, 1 ∈ H 1, 1 ∆ τ ( X, C) | α 1, 1 ≥ 0 = {0}, where τ = τ ω := [Λ ω , ∂ ω ∧ ·].
As a piece of notation that will be used throughout the text, whenever u is a k-form and (p, q) is a bidegree with p + q = k, u p, q will stand for the component of u of bidegree (p, q). of the second-named author. The former wishes to express his gratitude to the latter for his constant guidance while this work was carried out, as well as to his Tunisian supervisor, Fathi Haggui, for constant support. Both authors are very grateful to the referee for their careful reading of the manuscript and their useful remarks and suggestions.
Properties of degenerate balanced manifolds
In this section, we investigate the effect of the balanced condition on the cohomology of degrees 1 and 2, while pointing out the peculiarities of the degenerate balanced case.
Background
Given a complex manifold X with dim C X = n ≥ 2 and a Hermitian metric ω (identified with its underlying C ∞ positive definite (1, 1)-form ω) on X, we will put ω r := ω r /r! for r = 1, . . . , n. Moreover, we denote by C r, s (X) = C r, s (X, C) the space of smooth C-valued (r, s)-forms on X for r, s = 1, . . . , n. If X is compact, recall the classical definitions of the Bott-Chern and Aeppli cohomology groups of X of any bidegree (p, q): H p, q BC (X, C) = ker(∂ : C p, q (X) → C p+1, q (X)) ∩ ker(∂ : C p, q (X) → C p, q+1 (X)) Im (∂∂ : C p−1, q−1 (X) → C p, q (X)) H p, q A (X, C) = ker(∂∂ : C p, q (X) → C p+1, q+1 (X)) Im (∂ : C p−1, q (X) → C p, q (X)) + Im (∂ : C p, q−1 (X) → C p, q (X)) .
We will use the Serre-type duality (see e.g. [Sch07]):
H 1, 1 BC (X, C) × H n−1, n−1 A (X, C) −→ C, ({u} BC , {v} A ) → {u} BC .{v} A := X u ∧ v,(6)
as well as the pseudo-effective cone of X introduced in [Dem92, Definition 1.3] as the set
E X := [T ] BC ∈ H 1, 1 BC (X, R) / T ≥ 0 d-closed (1, 1)-current on X .
Recall that a Hermitian metric ω on X is said to be a Gauduchon metric (cf. [Gau77]) if ∂∂ω n−1 = 0. For any such metric ω, ω n−1 defines an Aeppli cohomology class and the set of all these cohomology classes is called the Gauduchon cone of X (cf. [Pop15]):
G X := {ω n−1 } A ∈ H n−1, n−1 A (X, R) | ω is a Gauduchon metric on X ⊂ H n−1, n−1 A (X, R).
The main link between the cones G X and E X on a compact n-dimensional X is provided by the following reformulation observed in [Pop16] of a result of Lamari's from [Lam99]. The pseudoeffective cone E X ⊂ H 1, 1 BC (X, R) and the closure of the Gauduchon cone G X ⊂ H n−1, n−1 A (X, R) are dual to each other under the duality (6). This means that the following two statements hold.
(1) Given any class c 1, 1 BC ∈ H 1, 1 BC (X, R), the following equivalence holds:
c 1, 1 BC ∈ E X ⇐⇒ c 1, 1 BC .c n−1, n−1 A ≥ 0 for every class c n−1, n−1 A ∈ G X .
(2) Given any class c n−1, n−1 A ∈ H n−1, n−1 A (X, R), the following equivalence holds:
c n−1, n−1 A ∈ G X ⇐⇒ c 1, 1 BC .c n−1, n−1 A ≥ 0 for every class c 1, 1 BC ∈ E X .
Finally, recall that a compact complex manifold X is said to be a ∂∂-manifold (see [DGMS75] for the notion, [Pop14] for the name) if for any d-closed pure-type form u on X, the following exactness properties are equivalent:
u is d-exact ⇐⇒ u is ∂-exact ⇐⇒ u is∂-exact ⇐⇒ u is ∂∂-exact.
On a complex manifold X with dim C X = n, we will often use the following standard formula (cf. e.g. [Voi02, Proposition 6.29, p. 150]) for the Hodge star operator = ω of any Hermitian metric ω applied to ω-primitive forms v of arbitrary bidegree (p, q):
v = (−1) k(k+1)/2 i p−q ω n−p−q ∧ v, where k := p + q.(7)
Recall that, for any k = 0, 1, . . . , n, a k-form v is said to be (ω)-primitive if ω n−k+1 ∧ v = 0 and that this condition is equivalent to Λ ω v = 0, where Λ ω is the adjoint of the operator ω∧· (of multiplication by ω) w.r.t. the pointwise inner product , ω defined by ω.
We will also often deal with C ∞ (1, 1)-forms α. If α = α prim + f ω is the Lefschetz decomposition, where α prim is primitive and f is a smooth function on X, we get Λ ω α = nf , hence
α = α prim + 1 n (Λ ω α) ω.(8)
We will often write (1, 1)-forms in this form.
On the other hand, we will often indicate the metric with respect to which certain operators are calculated. For example, d ω and ∆ ω := dd ω + d ω d are the adjoint of d, resp. the d-Laplacian, induced by the metric ω.
Case of degree 1
The starting point is the following Lemma 2.1. Let ω be a Hermitian metric on a complex manifold X with dim C X = n. Fix a form u = u 1, 0 + u 0, 1 ∈ C ∞ 1 (X, C). (i) The following formula holds:
d (ω n−1 ∧ u) = i(∂u 1, 0 −∂u 0, 1 ) ∧ ω n−2 + i (∂u 0, 1 ) prim − (∂u 1, 0 ) prim ∧ ω n−2 + i n Λ ω (∂u 1, 0 ) − Λ ω (∂u 0, 1 ) ω n−1 ,(9)
where d = d ω is the formal adjoint of d w.r.t. the L 2 ω inner product, while the subscript prim indicates the ω-primitive part in the Lefschetz decomposition of the form to which it is applied.
In particular, if du 1, 0 = 0 and du 0, 1 = 0, we get
d (ω n−1 ∧ u) = 0.
(ii) If ω is balanced and du 1, 0 = 0 and du 0, 1 = 0, then
∆(ω n−1 ∧ u) = 0,
where ∆ = ∆ ω = dd + d d is the d-Laplacian induced by ω.
(iii) If X is compact, ω is degenerate balanced and du 1, 0 = du 0, 1 = 0, then u = 0.
Proof. (i) All 1-forms are primitive, so from the standard formula (7) we get: u 1, 0 = −iω n−1 ∧ u 1, 0 , hence (ω n−1 ∧ u 1, 0 ) = −iu 1, 0 . Meanwhile, d = − d , so applying − d to the previous identity and writing the (1, 1)-form∂u 1, 0 in the form (8), we get the first line below:
d (ω n−1 ∧ u 1, 0 ) = i ∂u 1, 0 + i (∂u 1, 0 ) prim + i n (Λ ω∂ u 1, 0 ) ω = i∂u 1, 0 ∧ ω n−2 − i(∂u 1, 0 ) prim ∧ ω n−2 + i n (Λ ω∂ u 1, 0 ) ω n−1 ,
where the second line follows from the standard formula (7). Running the analogous computations for u 0, 1 or taking conjugates, we get:
d (ω n−1 ∧ u 0, 1 ) = −i∂u 0, 1 ∧ ω n−2 + i(∂u 0, 1 ) prim ∧ ω n−2 − i n (Λ ω ∂u 0, 1 ) ω n−1 .
Formula (9) follows by adding up the above expressions for d (ω n−1 ∧ u 1, 0 ) and d (ω n−1 ∧ u 0, 1 ).
(ii) If ω is balanced, we get d(ω n−1 ∧ u) = ω n−1 ∧ du = 0 since du = 0 under the assumptions. Since we also have d (ω n−1 ∧ u) = 0 by (i), the contention follows.
(iii) If ω is degenerate balanced, there exists a smooth (2n − 3)-form Γ such that ω n−1 = dΓ. Hence, ω n−1 ∧ u = d(Γ ∧ u) ∈ Im d because we also have du = 0 by our assumptions. However, ω n−1 ∧ u ∈ ker ∆ by (ii) and ker ∆ ⊥ Im d by the compactness assumption on X. Thus, the form ω n−1 ∧ u ∈ ker ∆ ∩ Im d = {0} must vanish.
On the other hand, the pointwise map ω n−1 ∧ · : Λ 1 T X −→ Λ 2n−1 T X is bijective, so we get u = 0 from ω n−1 ∧ u = 0.
We now use Lemma 2.1 to infer its consequences announced in the introduction.
• Proof of (i) of Proposition 1.2. This follows at once from (iii) of Lemma 2.1.
Another consequence of Lemma 2.1 is that the balanced condition, combined with the mild ∂∂type condition in (ii) of Theorem 1.1, enables one to get a Hard Lefschetz Isomorphism between the De Rham cohomology spaces of degrees 1 and 2n − 1.
• Proof of Theorem 1.1. (i) Lemma 2.1 is not needed here. Let u be a smooth 1-form. Since dω n−1 = 0, d(ω n−1 ∧ u) = 0 whenever du = 0, while ω n−1 ∧ u = d(f ω n−1 ) whenever u = df for some smooth function f on X. This proves the well-definedness of the map (1).
Similarly, if ω n−1 = γ n−1 +dΓ for some smooth (2n−2)-form γ n−1 and some smooth (
2n−3)-form Γ, then ω n−1 ∧u = γ n−1 ∧u+d(Γ∧u) for every d-closed 1-form u. Hence, {ω n−1 ∧u} DR = {γ n−1 ∧u} DR whenever {ω n−1 } DR = {γ n−1 } DR , so the map (1) depends only on {ω n−1 } DR . (ii) Since H 1 DR (X, C)
and H 2n−1 DR (X, C) have equal dimensions, by Poincaré duality, it suffices to prove that the map (1) is injective.
Let u be an arbitrary smooth d-closed 1-form on X. We start by showing that there exists a smooth function f : X → C such that ∂u 0, 1 = ∂∂f on X. To see this, notice that the property du = 0 translates to the following three relations holding:
(a) ∂u 1, 0 = 0; (b) ∂u 0, 1 +∂u 1, 0 = 0; (c)∂u 0, 1 = 0.
(10)
Thus, the (1, 1)-form ∂u 0, 1 is d-closed (since it is∂-closed by (c) of (10)) and ∂-exact. Thanks to hypothesis (2), we infer that ∂u 0, 1 is ∂∂-exact. Thus, there exists a smooth function f as stated. Using (b) of (10), we further infer that∂u 1, 0 = −∂u 0, 1 = −∂∂f , so∂(u 1, 0 − ∂f ) = 0. From the identities ∂(u 0, 1 −∂f ) = 0 and∂(u 1, 0 − ∂f ) = 0 and from (a) and (c) of (10), we get:
d(u 1, 0 − ∂f ) = 0 and d(u 0, 1 −∂f ) = 0. This means that (u − df ) 1, 0 ∈ ker d and (u − df ) 0, 1 ∈ ker d.
From this and from (i) of Lemma 2.1, we deduce that
ω n−1 ∧ (u − df ) ∈ ker d .(11)
On the other hand, if ω is balanced and if {ω n−1 ∧u} DR = 0 ∈ H 2n−1 DR (X, C) (i.e. ω n−1 ∧u ∈ Im d),
then ω n−1 ∧ (u − df ) ∈ Im d.(12)
From (11), (12) and ker d ⊥ Im d, we infer that ω n−1 ∧ (u − df ) = 0. Since u − df is a smooth 1-form on X and the pointwise-defined linear map:
ω n−1 ∧ · : C ∞ 1 (X, C) −→ C ∞ 2n−1 (X, C), α → ω n−1 ∧ α,
is bijective, we finally get u − df = 0, so {u} DR = 0 ∈ H 1 DR (X, C). This proves the injectivity of the map (1) whenever ω is balanced and X satisfies hypothesis (2).
In the degenerate balanced case, we get the vanishing of the first Betti number of the manifold.
• Proof of (ii) of Proposition 1.2. If ω is degenerate balanced, the map (1) vanishes identically. Meanwhile, by Theorem 1.1, the map (1) is an isomorphism. We get H 1 DR (X, C) = 0.
Case of degree 2: De Rham cohomology
The balanced property of a metric enables one to define a notion of primitivity for 2-forms.
Lemma and Definition 2.2. Let ω be a balanced metric on a compact complex manifold X with dim C X = n. The linear map:
{ω n−1 } DR ∧ · : H 2 DR (X, C) −→ H 2n DR (X, C) C, {α} DR −→ {ω n−1 ∧ α} DR ,(13)
is well defined and depends only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, C). We set:
H 2 DR (X, C) ω-prim := ker {ω n−1 } DR ∧ · ⊂ H 2 DR (X, C)
and we call its elements (ω-)primitive De Rham 2-classes.
Proof. Since dω n−1 = 0, for every d-closed (resp. d-exact) 2-form α, ω n−1 ∧ α is d-closed (resp. d-exact). This proves the well-definedness of the map. Meanwhile, if Ω ∈ C ∞ n−1, n−1 (X, C) is such that Ω = ω n−1 + dΓ for some smooth (2n − 3)-form Γ, then, for every d-closed 2-form α,
Ω ∧ α = ω n−1 ∧ α + d(Γ ∧ α). Hence, {Ω ∧ α} DR = {ω n−1 ∧ α} DR whenever {Ω} DR = {ω n−1 } DR .
This proves the independence of the map {ω n−1 } DR ∧ · of the choice of representative of the class {ω n−1 } DR .
We now observe a link between primitive 2-classes and primitive 2-forms.
Lemma 2.3. Let ω be a balanced metric on a compact complex manifold X with dim C X = n. For any class c ∈ H 2 DR (X, C), the following equivalence holds:
c is ω-primitive ⇐⇒ ∃ α ∈ c such that α is ω-primitive.
Proof. "⇐=" Suppose α ∈ C ∞ 2 (X, C) such that dα = 0, α ∈ c and α is ω-primitive. Then, ω n−1 ∧ α = 0, hence {ω n−1 ∧ α} DR = 0. This means that the class c = {α} DR is ω-primitive.
" =⇒ " Suppose the class c is ω-primitive. Pick an arbitrary representative β ∈ c. The ωprimitivity of c = {β} DR translates to {ω n−1 ∧ β} DR = 0 ∈ H 2n DR (X, C). This, in turn, is equivalent to the existence of a form Γ ∈ C ∞ 2n−1 (X, C) such that ω n−1 ∧ β = dΓ.
Meanwhile, we know from the general theory that the map
ω n−1 ∧ · : C ∞ 1 (X, C) −→ C ∞ 2n−1 (X, C) is an isomorphism. Hence, there exists a unique u ∈ C ∞ 1 (X, C) such that Γ = ω n−1 ∧ u. We get: ω n−1 ∧ β = dΓ = ω n−1 ∧ du,
where the last identity follows from the balanced property of ω. Consequently,
ω n−1 ∧ (β − du) = 0, proving that α := β − du is a primitive representative of the class c = {β} DR .
Finally, we can characterise the degenerate balanced property of a given balanced metric in terms of primitivity for 2-classes.
Lemma 2.4. Let ω be a balanced metric on a compact complex manifold X with dim C X = n. The following equivalence holds:
H 2 DR (X, C) ω-prim = H 2 DR (X, C) ⇐⇒ ω is degenerate balanced. Proof. "⇐=" Suppose that ω is degenerate balanced. Then ω n−1 is d-exact, hence ω n−1 ∧ α is d-exact (or equivalently {ω n−1 ∧ α} DR = 0 ∈ H 2n DR (X, C)) for every d-closed 2-form α. This means that the map {ω n−1 } DR ∧ · : H 2 DR (X, C) −→ H 2n DR (X, C) vanishes identically, so H 2 DR (X, C) ω-prim = H 2 DR (X, C). " =⇒ " Suppose that H 2 DR (X, C) ω-prim = H 2 DR (X, C). This translates to ω n−1 ∧ α ∈ Im d, ∀ α ∈ C ∞ 2 (X, C) ∩ ker d.(14)
Since both ω n−1 and α are d-closed, they both have unique L 2 ω -orthogonal decompositions:
ω n−1 = (ω n−1 ) h + dΓ and α = α h + du,
where (ω n−1 ) h and α h are ∆ ω -harmonic, while Γ and u are smooth forms of respective degrees 2n − 3 and 1. We get:
ω n−1 ∧ α = (ω n−1 ) h ∧ α h + d (ω n−1 ) h ∧ u + Γ ∧ α h + Γ ∧ du , ∀ α ∈ C ∞ 2 (X, C) ∩ ker d.
Together with (14), this implies that
(ω n−1 ) h ∧ α h ∈ Im d, ∀ α h ∈ ker ∆ ω ∩ C ∞ 2 (X, C).(15)
Meanwhile, since (ω n−1 ) h is ∆ ω -harmonic (and real), ω (ω n−1 ) h is again ∆ ω -harmonic (and real).
Hence, Im d (ω n−1 ) h ∧ ω (ω n−1 ) h = |(ω n−1 ) h | 2 ω dV ω ≥ 0,
where the first relation follows from (15) by choosing α h = ω (ω n−1 ) h . Consequently, from Stokes's Theorem we get:
X |(ω n−1 ) h | 2 ω dV ω = 0,
hence (ω n−1 ) h = 0. This implies that ω n−1 is d-exact, which means that ω is degenerate balanced.
Corollary 2.5. Let ω be a balanced metric on a compact complex manifold X with dim C X = n.
The following dichotomy holds:
(a) if ω is not degenerate balanced, H 2 DR (X, C) ω-prim is a complex hyperplane in H 2 DR (X, C) depending only on the balanced class {ω n−1 } DR ; (b) if ω is degenerate balanced, H 2 DR (X, C) ω-prim = H 2 DR (X, C).
Proof. This follows immediately from Lemma and Definition 2.2, from Lemma 2.4 and from H 2n DR (X, C) C.
We shall now get a Lefschetz-type decomposition of H 2 DR (X, C), induced by an arbitrary balanced metric ω, with H 2 DR (X, C) ω-prim as a direct factor. Recall that the balanced condition dω n−1 = 0 is equivalent to d ω ω = 0.
Thanks to the orthogonal 3-space decompositions:
C ∞ k (X, C) = ker ∆ ω ⊕ Im d ⊕ Im d ω , k ∈ {0, . . . , 2n},
where ker ∆ ω ⊕ Im d = ker d and ker ∆ ω ⊕ Im d ω = ker d ω , applied with k = 2 and k = 2n − 2, we get unique decompositions of ω, resp. ω n−1 :
ker d ω ω = ω h + d ω η ω and ker d ω n−1 = (ω n−1 ) h + dΓ ω ,(16)
where ω h ∈ ker ∆ ω as a 2-form, (ω n−1 ) h ∈ ker ∆ ω as a (2n − 2)-form, while η ω and Γ ω are smooth forms of respective degrees 3 and 2n−3. Since ω and ω n−1 are real, so are their harmonic components ω h and (ω n−1 ) h . Moreover, it is well known that ω ω = ω n−1 and that the Hodge star operator ω maps d-exact forms to d ω -exact forms and vice-versa. Hence, we get:
ω ω h = (ω n−1 ) h and ω (d ω η ω ) = dΓ ω .(17)
Thus, ω h is uniquely determined by ω and is d-closed (because it is even ∆ ω -harmonic). Therefore, it represents a class in H 2 DR (X, R). Definition 2.6. For any balanced metric ω on a compact complex manifold X, the De Rham co-
homlogy class {ω h } DR ∈ H 2 DR (X, R) is called the cohomology class of ω. Of course, if ω is Kähler, ω h = ω, so {ω h } DR is the usual Kähler class {ω} DR .
Lemma 2.7. Suppose there exists a balanced metric ω on a compact complex manifold X. Then, for every α ∈ C ∞ 2 (X, C) such that dα = 0 and {α} DR ∈ H 2 DR (X, C) ω-prim , we have:
ω h , α ω = 0,
where , ω is the L 2 inner product induced by ω.
Proof. Since {α} DR ∈ H 2 DR (X, C) ω-prim , there exists a form Ω ∈ C ∞ 2n−1 such that ω n−1 ∧ α = du. We get: α, ω h ω = X α ∧ ω ω h (a) = X α ∧ (ω n−1 ) h (b) = X α ∧ (ω n−1 − dΓ ω ) = X α ∧ ω n−1 = X du = 0,
where Stokes implies two of the last three equalities (note that α ∧ dΓ ω = d(α ∧ Γ ω )), while (a) follows from (17) and (b) follows from (16).
Conclusion 2.8. Let X be a compact complex manifold with dim C X = n. Suppose there exists a non-degenerate balanced metric ω on X. Then, the De Rham cohomology space of degree 2 has a Lefschetz-type L 2 ω -orthogonal decomposition:
H 2 DR (X, C) = H 2 DR (X, C) ω-prim ⊕ C · {ω h } DR ,(18)
where the ω-primitive subspace H 2 DR (X, C) ω-prim is a complex hyperplane of H 2 DR (X, C) depending only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, C), while ω h is the ∆ ω -harmonic component of ω and the complex line C · {ω h } DR depends on the choice of the balanced metric ω.
If ω is Kähler, the Lefschetz-type decomposition (18) depends only on the Kähler class
{ω} DR ∈ H 2 DR (X, C) since ω h = ω in that case.
Lemma 2.9. The assumptions are the same as in Conclusion 2.8.
For every α ∈ C ∞ 2 (X, C) ∩ ker d, the coefficient of {ω h } DR in the Lefschetz-type decomposition of {α} DR ∈ H 2 DR (X, C) according to (18), namely in {α} DR = {α} DR, prim + λ {ω h } DR ,(19)
is given by
λ = λ ω ({α} DR ) = {ω n−1 } DR .{α} DR ||ω h || 2 ω = 1 ||ω h || 2 ω X α ∧ ω n−1 . (20) Proof. Since {α} DR, prim ∈ H 2 DR (X, C) ω-prim , we have {ω n−1 } DR .{α} DR, prim = 0, so {ω n−1 } DR .{α} DR = λ ω n−1 ∧ ω h = λ (ω n−1 ) h ∧ ω h = λ (ω n−1 ) h ∧ ω (ω n−1 ) h = λ ||(ω n−1 ) h || 2 ω = λ ||ω h || 2 ω .
This gives (20).
Formula (20) implies that λ ω ({α} DR ) is real if the class {α} DR ∈ H 2 DR (X, R)
is real. This enables one to define a positive side and a negative side of the hyperplane H 2
DR (X, R) ω-prim := H 2 DR (X, C) ω-prim ∩ H 2 DR (X, R) in H 2 DR (X, R) by H 2 DR (X, R) + ω := {α} DR ∈ H 2 DR (X, R) | λ ω ({α} DR ) > 0 , H 2 DR (X, R) − ω := {α} DR ∈ H 2 DR (X, R) | λ ω ({α} DR ) < 0 .
These open subsets of H 2 DR (X, R) depend only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, R). Since {α} DR is ω-primitive if and only if λ ω ({α} DR ) = 0, we get a partition of H 2 DR (X, R):
H 2 DR (X, R) = H 2 DR (X, R) + ω ∪ H 2 DR (X, R) ω-prim ∪ H 2 DR (X, R) − ω
depending only on the cohomology class {ω n−1 } DR ∈ H 2n−2 DR (X, R).
The next (trivial) observation is that the ω-primitive hyperplane H 2 DR (X, C) ω-prim ⊂ H 2 DR (X, C) depends only on the ray R >0 · {ω n−1 } DR generated by the De Rham cohomology class of ω n−1 in the De Rham version of the balanced cone B X, DR ⊂ H 2n−2 DR (X, R) of X. (We denote by B X, DR the set of De Rham cohomology classes {ω n−1 } DR induced by balanced metrics ω.) Lemma 2.10. Let X be a compact complex non-degenerate balanced manifold with dim C X = n. Let ω and γ be balanced metrics on X. The following equivalence holds:
H 2 DR (X, C) ω-prim = H 2 DR (X, C) γ-prim ⇐⇒ ∃ c > 0 such that {ω n−1 } DR = c {γ n−1 } DR . Proof. "
⇐=" This implication follows from proportional linear maps having the same kernel.
"=⇒" This implication follows from the following elementary fact. Suppose T, S :
E −→ C are C-linear maps on a C-vector space E such that ker T = ker S ⊂ E is of C-codimension 1 in E. Then, there exists c ∈ C \ {0} such that T = cS.
To see this, let {e j | j ∈ J} be a C-basis of ker T = ker S and let e ∈ E such that {e} ∪ {e j | j ∈ J} is a C-basis of E. Then, T (e) and S(e) are non-zero complex numbers, so there exists a unique c ∈ C \ {0} such that T (e) = c S(e). Now, fix an arbitrary u ∈ E. We will show that T (u) = c S(u). There is a unique choice of λ ∈ C and v ∈ ker T = ker S such that u = λ e + v. Hence, T (u) = λ T (e) = c (λ S(e)) = c S(u).
In our case, the assumption H 2 We will now see that not only do proportional balanced classes {ω n−1 } DR and {γ n−1 } DR induce the same hyperplane of primitive classes in H 2 DR (X, C), but they can be made to also induce the same Lefschetz-type decomposition (18). This is fortunate since, in general, the complex line C · {ω h } DR depends on the choice of the balanced metric ω, unlike H 2 DR (X, C) ω-prim which depends only on the balanced class {ω n−1 } DR ∈ H 2n−2 DR (X, C). Lemma 2.11. Let X be a compact complex non-degenerate balanced manifold with dim C X = n.
DR (X, C) ω-prim = H 2 DR (X, C) γ-
(i) If ω and γ are balanced metrics on X such that ω n−1 = c γ n−1 for some constant c > 0, there exists a constant a > 0 such that ω h = a γ h .
(ii) For every ray R >0 · {ω n−1 } DR in the De Rham version of the balanced cone B X, DR ⊂ H 2n−2 DR (X, R) of X, the balanced metrics representing the classes on this ray can be chosen such that they induce the same Lefschetz-type decomposition (18).
Proof. (i) Since ω = c 1 n−1 γ, we get ω = const · γ and d ω = const · d γ . The latter identity implies ∆ ω = const · ∆ γ , hence ker ∆ ω = ker ∆ γ . In particular, (ω n−1 ) h = c (γ n−1 ) h and thus
ω h = ω (ω n−1 ) h = const · γ (γ n−1 ) h = const · γ h ,
where in all the above identities const stands for a positive constant that may change from one occurrence to another.
(ii) Fix a balanced De Rham class {γ n−1 } DR ∈ B X, DR ⊂ H 2n−2 DR (X, R) and fix a balanced metric γ (whose choice is arbitrary) such that γ n−1 represents the class {γ n−1 } DR . For every constant c > 0, the balanced class c {γ n−1 } DR can be represented by the form ω n−1 := c γ n−1 which is induced by the balanced metric ω := c 1 n−1 γ. From (i), we get C {ω h } DR = C {γ h } DR . Since we also have H 2 DR (X, C) ω-prim = H 2 DR (X, C) γ-prim by Lemma 2.10, the contention follows.
The proof of (ii) of the above Lemma 2.11 shows that the line C {ω h } DR in the Lefschetz-type decomposition (18) induced by a given ray R >0 · {ω n−1 } DR in the De Rham version of the balanced cone B X, DR ⊂ H 2n−2 DR (X, R) of X still depends on the arbitrary choice of a balanced metric γ such that γ n−1 represents a given class {γ n−1 } DR on this ray. To tame this dependence, we can fix an arbitrary Hermitian (not necessarily balanced) metric ρ on X and make all the choices of harmonic representatives and projections be induced by ρ. Thus, we get L 2 ρ -orthogonal decompositions:
ω = ω h, ρ + d ρ η ω, ρ and ω n−1 = (ω n−1 ) h, ρ + d Γ ω, ρ ,
where ω h, ρ ∈ ker ∆ ρ as a 2-form, (ω n−1 ) h, ρ ∈ ker ∆ ρ as a (2n − 2)-form, while η ω, ρ and Γ ω, ρ are smooth forms of respective degrees 3 and 2n−3. Since ω and ω n−1 are real, so are their ∆ ρ -harmonic components ω h, ρ and (ω n−1 ) h, ρ . In this way, every non-zero balanced class {ω n−1 } DR induces a Lefschetz-type decomposition analogous to (18) that depends only on the class {ω n−1 } DR and on the background metric ρ:
H 2 DR (X, C) = H 2 DR (X, C) ω-prim ⊕ C · {ω h, ρ } DR ,
where the hyperplane H 2 DR (X, C) ω-prim depends only on the class {ω n−1 } DR . In other words, we remove the dependence of the line C {ω h } DR in the Lefschetz-type decomposition (18) on a representative of the class {ω n−1 } DR and replace it with the dependence on a fixed background metric ρ.
Case of degree 2: Bott-Chern and Aeppli cohomologies
Let us finally point out that the theory developed in §.2.3 in the context of the Poincaré duality for the De Rham cohomology spaces of degrees 2 and 2n − 2 can be rerun in the context of the duality (6) between the Bott-Chern and Aeppli cohomology spaces of bidegrees (1, 1), resp. (n − 1, n − 1).
Since all the results and constructions of §.2.3, except for Lemma 2.3, have analogues in the new context with very similar proofs, we will leave most of these proofs to the reader.
In fact, the new cohomological setting allows for the theory of §.2.3 to be repeated in the more general context of Gauduchon (not necessarily balanced) metrics and the Aeppli cohomology classes they define in H n−1, n−1 A (X, R). We start with the following analogue of Lemma and Definition 2.2.
Lemma and Definition 2.12. Let ω be a Gauduchon metric on a compact complex manifold X with dim C X = n. The linear map:
[ω n−1 ] A ∧ · : H 1, 1 BC (X, C) −→ H n, n A (X, C) C, [α] BC −→ [ω n−1 ∧ α] A ,
is well defined and depends only on the cohomology class [ω n−1 ] A ∈ H n−1, n−1 A (X, C). We set:
H 1, 1 BC (X, C) ω-prim := ker [ω n−1 ] A ∧ · ⊂ H 1, 1 BC (X, C)
and we call its elements (ω-)primitive Bott-Chern (1, 1)-classes.
Proof. The well-definedness follows at once from the identities:
∂∂(ω n−1 ∧ α) = ∂∂ω n−1 ∧ α = 0, α ∈ C ∞ 1, 1 (X, C) ∩ ker d, ω n−1 ∧ ∂∂ϕ = ∂(ω n−1 ∧∂ϕ) +∂(ϕ ∂ω n−1 ) ∈ Im ∂ + Im∂, ϕ ∈ C ∞ 0, 0 (X, C),
where the latter takes into account the fact that ∂∂ω n−1 = 0. That the map [ω n−1 ] A ∧ · depends only on the Aeppli cohomology class [ω n−1 ] A follows from:
(ω n−1 + ∂Γ +∂Γ) ∧ α − ω n−1 ∧ α = ∂(Γ ∧ α) +∂(Γ ∧ α) ∈ Im ∂ + Im∂, α ∈ C ∞ 1, 1 (X, C) ∩ ker d.
The following result is the analogue of Lemma 2.4.
Lemma 2.13. Let ω be a Gauduchon metric on a compact complex manifold X with dim C X = n.
The following equivalence holds:
H 1, 1 BC (X, C) ω-prim = H 1, 1 BC (X, C) ⇐⇒ ω n−1 ∈ Im ∂ + Im∂ (i.e. ω n−1 is Aeppli-exact).
Proof. "⇐=" If ω n−1 ∈ Im ∂ + Im∂, [ω n−1 ] A = 0, so the map [ω n−1 ] A ∧ · vanishes identically.
" =⇒ " Suppose that H 1, 1 BC (X, C) ω-prim = H 1, 1 BC (X, C). This translates to ω n−1 ∧ α ∈ Im ∂ + Im∂, ∀ α ∈ C ∞ 1, 1 (X, C) ∩ ker d.
Since ω n−1 is (∂∂)-closed, it has a unique L 2 ω -orthogonal decomposition:
ω n−1 = (ω n−1 ) h + (∂Γ ω +∂Γ ω ),
with an (n − 1, n − 1)-form (ω n−1 ) h ∈ ker ∆ A, ω and an (n − 1, n − 2)-form Γ ω . (See (23) below.) On the other hand, α is d-closed, so it has a unique L 2 ω -orthogonal decomposition:
α = α h + ∂∂ϕ,
where α h is ∆ BC, ω -harmonic and ϕ is a smooth function on X. (See again (23) below.) Thus, for every α ∈ C ∞ 1, 1 (X, C) ∩ ker d, we get:
ω n−1 ∧ α = (ω n−1 ) h ∧ α + ∂(Γ ω ∧ α) +∂(Γ ω ∧ α) = (ω n−1 ) h ∧ α h + ∂ (ω n−1 ) h ∧∂ϕ +∂ ϕ ∂(ω n−1 ) h + ∂(Γ ω ∧ α) +∂(Γ ω ∧ α),
where for the last identity we used the fact that∂∂(ω n−1 ) h = 0.
Thanks to assumption (21), the last identity implies that
(ω n−1 ) h ∧ α h ∈ Im ∂ + Im∂, ∀ α h ∈ C ∞ 1, 1 (X, C) ∩ ker ∆ BC, ω .(22)
Meanwhile, since (ω n−1 ) h is ∆ A, ω -harmonic (and real), ω (ω n−1 ) h is ∆ BC, ω -harmonic (and real). Hence, Im ∂ + Im∂ (ω n−1 ) h ∧ ω (ω n−1 ) h = |(ω n−1 ) h | 2 ω dV ω ≥ 0, where the first relation follows from (22) by choosing α h = ω (ω n−1 ) h . Consequently, from Stokes's Theorem we get:
X |(ω n−1 ) h | 2 ω dV ω = 0,
hence (ω n−1 ) h = 0. This implies that ω n−1 ∈ Im ∂ + Im∂ and we are done.
The analogue in this context of Corollary 2.5 is the following Corollary 2.14. Let ω be a Gauduchon metric on a compact complex manifold X with dim C X = n.
The following dichotomy holds:
(a) if ω n−1 is not Aeppli exact, H 1, 1 BC (X, C) ω-prim is a complex hyperplane of H 1, 1 BC (X, C) depending only on the Aeppli-Gauduchon class {ω n−1 } DR ∈ G;
(b) if ω n−1 is Aeppli exact, H 1, 1 BC (X, C) ω-prim = H 1, 1 BC (X, C).
To get a Lefschetz-type decomposition of H 1, 1 BC (X, C) induced by an arbitrary Gauduchon metric ω, we use the orthogonal 3-space decompositions featuring the Aeppli-, resp. Bott-Chern-Laplacians induced by the metric ω: C ∞ n−1, n−1 (X, C) = ker ∆ A, ω ⊕ (Im ∂ + Im∂) ⊕ Im (∂∂) , C ∞ 1, 1 (X, C) = ker ∆ BC, ω ⊕ Im (∂∂) ⊕ (Im ∂ + Im∂ ),
where ker ∆ A, ω ⊕ (Im ∂ + Im∂) = ker(∂∂) and ker ∆ BC, ω ⊕ (Im ∂ + Im∂ ) = ker(∂∂) . Thus, we get unique decompositions of ω, resp. ω n−1 : ker(∂∂) ω = ω h + (∂ ωū ω +∂ ω u ω ) and ker(∂∂) ω n−1 = (ω n−1 ) h + (∂Γ ω +∂Γ ω ),
where ω h ∈ ker ∆ BC, ω as a (1, 1)-form, (ω n−1 ) h ∈ ker ∆ A, ω as an (n − 1, n − 1)-form, while u ω and Γ ω are smooth forms of respective bidegrees (1, 2) and (n − 1, n − 2). Since ω and ω n−1 are real, so are their harmonic components ω h and (ω n−1 ) h . Since ω ω = ω n−1 and since the Hodge star operator ω maps Aeppli-harmonic forms to Bott-Chern-harmonic forms and vice-versa, we get:
ω ω h = (ω n−1 ) h and ω (∂ ωū ω +∂ ω u ω ) = ∂Γ ω +∂Γ ω .(25)
Thus, ω h is uniquely determined by ω and is d-closed (because it is even ∆ BC, ω -harmonic). Therefore, it represents a class in H 1, 1 BC (X, R). Definition 2.15. For any Gauduchon metric ω on a compact complex manifold X, the Bott-Chern cohomlogy class [ω h ] BC ∈ H 1, 1 BC (X, R) is called the cohomology class of ω.
Of course, if ω is Kähler, ω h = ω, so {ω h } BC is the usual Bott-Chern Kähler class {ω} BC .
The analogue of Lemma 2.7 is the following Lemma 2.16. Suppose there exists a Gauduchon metric ω on a compact complex manifold X. Then, for every α ∈ C ∞ 1, 1 (X, C) such that dα = 0 and [α] BC ∈ H 1, 1 BC (X, C) ω-prim , we have:
ω h , α ω = 0,
where , ω is the L 2 inner product induced by ω.
The analogue in this context of Conclusion 2.8 is the following Conclusion 2.17. Let X be a compact complex manifold with dim C X = n. Let ω be a Gauduchon metric on X such that ω n−1 is not Aeppli-exact. Then, the Bott-Chern cohomology space of bidegree (1, 1) has a Lefschetz-type L 2 ω -orthogonal decomposition:
H 1, 1 BC (X, C) = H 1, 1 BC (X, C) ω-prim ⊕ C · [ω h ] BC ,(26)
where the ω-primitive subspace H 1, 1 BC (X, C) ω-prim is a complex hyperplane of H 1, 1 BC (X, C) depending only on the cohomology class [ω n−1 ] A ∈ H n−1, n−1 A (X, C), while ω h is the ∆ BC, ω -harmonic component of ω and the complex line C · [ω h ] BC depends on the choice of the Gauduchon metric ω.
We also have the following analogue of Lemma 2.9.
α ∈ C ∞ 1, 1 (X, C) ∩ ker d, the coefficient of [ω h ] BC in the Lefschetz-type decomposition of [α] BC ∈ H 1, 1 BC (X, C) according to (26), namely in [α] BC = [α] BC, prim + λ [ω h ] BC ,(27)
is given by
λ = λ ω ([α] BC ) = [ω n−1 ] A .[α] BC ||ω h || 2 ω = 1 ||ω h || 2 ω X α ∧ ω n−1 .(28)
As in §.2.3, formula (28) implies that λ ω ([α] BC ) is real if the class [α] BC ∈ H 1, 1 BC (X, C) is real. Thus, we can define a positive side and a negative side of the hyperplane H 1, 1 BC (X, R) ω-prim := H 1, 1 BC (X, C) ω-prim ∩ H 1, 1 BC (X, R) in H 1, 1 BC (X, R) by
H 1, 1 BC (X, R) + ω := [α] BC ∈ H 1, 1 BC (X, R) | λ ω ([α] BC ) > 0 , H 1, 1 BC (X, R) − ω := [α] BC ∈ H 1, 1 BC (X, R) | λ ω ([α] BC ) < 0 .
These are open subsets of H 1, 1 BC (X, R) that depend only on the cohomology class [ω n−1 ] A ∈ H n−1, n−1 A (X, R). Since [α] BC is ω-primitive if and only if λ ω ([α] BC ) = 0, we get a partition of H 1, 1 BC (X, R):
H 1, 1 BC (X, R) = H 1, 1 BC (X, R) + ω ∪ H 1, 1 BC (X, R) ω-prim ∪ H 1, 1 BC (X, R) − ω
depending only on the cohomology class [ω n−1 ] A ∈ H n−1, n−1 A (X, R).
As a consequence of these considerations, we get Proposition 2.19. Let X be a compact complex manifold with dim C X = n. The pseudo-effective cone E X ⊂ H 1, 1 BC (X, R) of X is the intersection of all the non-negative sides
H 1, 1 BC (X, R) ≥0 ω := H 1, 1 BC (X, R) + ω ∪ H 1, 1 BC (X, R) ω-prim
of hyperplanes H 1, 1 BC (X, R) ω-prim determined by Aeppli-Gauduchon classes [ω n−1 ] A ∈ G X :
E X = [ω n−1 ] A ∈G X H 1, 1 BC (X, R) ≥0 ω ,(29)
Proof. By the duality between the pseudo-effective cone E X and the closure G X of the Gauduchon cone (see §.2.1), we know that a given class [T ] BC ∈ H 1, 1 BC (X, R) lies in E X (i.e.
[T ] BC can be represented by a closed semi-positive (1, 1)-current) if and only if
X T ∧ ω n−1 ≥ 0 for all [ω n−1 ] A ∈ G X .
The last condition is equivalent to λ ω ([T ] BC ) ≥ 0, hence to [T ] BC ∈ H 1, 1 BC (X, R) ≥0 ω , for all [ω n−1 ] A ∈ G X , so the contention follows.
Based on these considerations, we propose Question 1.3 as a problem for further study.
Properties of balanced hyperbolic manifolds
The discussion of balanced hyperbolic manifolds featured in this section will mirror that of degenerate balanced manifolds of the previous section.
Background and L 1 currents on the universal cover
It is a classical fact due to Gaffney [Gaf54] that certain basic facts in the Hodge Theory of compact Riemannian manifolds remain valid on complete such manifolds. The main ingredient in the proof of this fact is the following cut-off trick of Gaffney's that played a key role in [Gro91,§.1]. It also appears in [Dem97, VIII, Lemma 2.4].
K ν ⊂K ν+1 for all ν ∈ N and X = ν∈N K ν ,
and a sequence (ψ ν ) ν∈N of C ∞ functions ψ ν : X −→ [0, 1] satisfying, for every ν ∈ N, the conditions:
ψ ν = 1 in a neighbourhood of K ν ,
Supp ψ ν ⊂K ν+1 and
||dψ ν || L ∞ g := sup x∈X |(dψ ν )(x)| g ≤ ε ν ,
for some constants ε ν > 0 such that ε ν ↓ 0 as ν tends to +∞.
In particular, the cut-off functions ψ ν are compactly supported. One can choose ε ν = 2 −ν for each ν (see e.g. [Dem97, VIII, Lemma 2.4]), but this will play no role here.
An immediate consequence of Gaffney's cut-off trick is the following classical generalisation of Stokes's Theorem to possibly non-compact, but complete Riemannian manifolds when the forms involved are L 1 . Lemma 1.1.A.]) Let (X, g) be a complete Riemannian manifold of real dimension m. Let η be an L 1 g -form on X of degree m − 1 such that dη is again L 1 g . Then X dη = 0.
Lemma 3.2. ([Gro91,
By the form η being L 1 with respect to the Riemannian metric g (L 1 g for short) we mean that its L 1 -norm is finite:
||η|| L 1 g := X |η(x)| g dV g (x) < +∞,
where dV g is the volume form induced by g.
Proof of Lemma 3.2. Let (ψ ν ) ν∈N be a sequence of cut-off functions as in Lemma 3.1 whose existence is guaranteed by the completeness of (X, g). The (m − 1)-form ψ ν η is compactly supported for every ν ∈ N , so the usual Stokes's Theorem yields:
X d(ψ ν η) = 0, ν ∈ N .
Meanwhile, d(ψ ν η) = dψ ν ∧ η + ψ ν dη, so we get:
X ψ ν dη = X dψ ν ∧ η ≤ ||dψ ν || L ∞ g ||η|| L 1 g ≤ ε ν ||η|| L 1 g , ν ∈ N,(30)
for some sequence of constants ε ν ↓ 0. Since η is L 1 g , ε ν ||η|| L 1 g ↓ 0 as ν → +∞. On the other hand, since dη is L 1 g , the properties of the functions ψ ν imply that
lim ν→+∞ X ψ ν dη = X dη.
Together with (30), these arguments yield X dη = 0, as desired.
We now apply this standard cut-off function technique to prove Proposition 1.4 stated in the introduction. It is an analogue in our more general context of balanced hyperbolic manifolds of Proposition 5.4 in [Pop15] according to which compact degenerate balanced manifolds are characterised by the absence of non-zero d-closed positive (1, 1)-currents.
Note that, due to X being compact, any pair of Hermitian metrics ω 1 and ω 2 on X are comparable in the sense that there exists a constant C > 0 such that (1/C) ω 1 ≤ ω 2 ≤ C ω 1 . Thus, their lifts ω 1 := π ω 1 and ω 2 := π ω 2 are again comparable on X by means of the same constant:
(1/C) ω 1 ≤ ω 2 ≤ C ω 1 . Therefore, the L 1 ω -assumption on T is independent of the choice of Hermitian metric on X if this metric is obtained by lifting a metric on X. However, the L 1 -condition changes for metrics on X that are not lifts of metrics on X. But we will not deal with the latter type of metrics.
Proof of Proposition 1.4. Let n = dim C X. The balanced hyperbolic assumption on X means that π ω n−1 = d Γ on X for some smooth L ∞ ω -form Γ of degree (2n − 3) on X. If a current T as in the statement existed on X, we would have
0 < X T ∧ π ω n−1 = X d( T ∧ Γ) = 0,(31)
which is contradictory. The last identity in (31) follows from Lemma 3.2 applied on the complete manifold ( X, ω) to the L 1 ω -current η := T ∧ Γ of degree 2n − 1 whose differential dη = T ∧ π ω n−1 is again L 1 ω . That η is L 1 ω follows from T being L 1 ω (by hypothesis) and from Γ being L ∞ ω , while dη being L 1 ω follows from T being L 1 ω and from π ω n−1 being L ∞ ω (as a lift of the smooth, hence bounded, form ω n−1 on the compact manifold X).
We now recall the following standard result saying that some further key facts in the Hodge Theory of compact Riemannian manifolds remain valid on complete such manifolds X when the differential operators involved (e.g. d, d , ∆) are considered as closed and densely defined unbounded operators on the spaces L 2 k (X, C) of L 2 -forms of degree k on X. The only major property that is lost in passing to complete manifolds is the closedness of the images of these operators. As usual, any differential operator P originally defined on C ∞ • (X, C) is extended to an unbounded operator on L 2
• (X, C) by defining its domain Dom P as the space of L 2 -forms u such that P u, computed in the sense of distributions, is again L 2 . u → ||u|| + ||du||, u → ||u|| + ||d u||, u → ||u|| + ||du|| + ||d u||. (c) The d-Laplacian ∆ = ∆ g := dd + d d has the following property :
∆u, u = ||du|| 2 + ||d u|| 2(32)
for every form u ∈ Dom ∆. In particular, Dom ∆ ⊂ Dom d ∩ Dom d and ker ∆ = ker d ∩ ker d .
(d) There are L 2 -orthogonal decompositions in every degree (indicated by a •):
L 2 • (X, C) = H • ∆ (X, C) ⊕ Im d ⊕ Im d ker d = H • ∆ (X, C) ⊕ Im d and ker d = H • ∆ (X, C) ⊕ Im d ,(33)
where H • ∆ (X, C) := {u ∈ L 2 • (X, C) | ∆u = 0} is the space of ∆-harmonic L 2 -forms, while Im d := L 2
• (X, C) ∩ d(L 2 •−1 (X, C)) and Im d : C)).
= L 2 • (X, C) ∩ d (L 2 •+1 (X,
An immediate consequence of (32) applied in degree 0 is that on a connected complete Riemannian manifold (X, g), every ∆-harmonic L 2 -function is constant:
H 0 ∆ (X, C) ⊂ C.(34)
3.2 Harmonic L 2 -forms of degree 1 on the universal cover of a balanced hyperbolic manifold Let X be a possibly non-compact complex manifold with dim C X = n, supposed to carry a complete balanced metric ω. In subsequent applications, the roles of X and ω will be played by X, the universal cover π : X −→ X of a compact balanced hyperbolic manifold (X, ω), resp. ω := π ω.
A well-known consequence of the Kähler commutation relations is the fact that, if ω is Kähler, the induced d-Laplacian ∆ = ∆ ω commutes with the multiplication operator ω l ∧ · acting on differential forms of any degree on X, for every l.
We will see that, when ω is merely balanced, the commutation of ∆ with the multiplication operator ω n−1 ∧ · acting on differential forms no longer holds. However, we will now compute this commutation defect on 1-forms.
The computation will continue that of (i) in Lemma 2.1. For the sake of generality and for a reason that will become apparent later on, we will work with the more general operators
d h := h∂ +∂, h ∈ C ,
acting on C-valued differential forms on X and the associated Laplacians
∆ h := d h d h + d h d h .
The first stages of the computation lead to the following result in which no completeness assumption is necessary.
Lemma 3.4. Let X be a complex manifold with dim C X = n. Suppose there exists a balanced metric ω on X. Then, for any h ∈ C and any 1-form ϕ on X, the following identity holds:
[∆ h , L ω n−1 ]ϕ = |h| 2 d − 1 h d − 1 h − d h d h ϕ ∧ ω n−1 − ih d − 1 h ϕ ∧ d h ω n−2 − i(|h| 2 + 1) ∂∂ϕ ∧ ω n−2 . (35)
Proof. • The Jacobi identity yields:
[[d h , d h ], L ω n−1 ] − [[d h , L ω n−1 ], d h ] + [[L ω n−1 , d h ], d h ] = 0. Since ω is balanced, [L ω n−1 , d h ] = 0. Writing ∆ h = [d h , d h ]
, the above equality reduces to
[∆ h , L ω n−1 ] = [d h , L ω n−1 ] d h + d h [d h , L ω n−1 ].(36)
• Note also the following formula for the formal adjoint of d h involving the Hodge star operator:
d h = −h d 1 h .(37)Indeed, d h = (h∂ +∂) =h (− ∂ ) + (− ∂ ) = −h ( 1 h ∂ +∂) = −h d 1 h .
No assumption on ω is needed here.
• As an application of (37), we observe the following formula for every (1, 1)-form α:
d h (ω n−1 ∧ α) = −ih d − 1 h (Λ ω α) ∧ ω n−1 .(38)
Again, no assumption on ω is needed. To see this, we first multiply the Lefschetz decomposition (8) of α by ω n−1 and we get: ω n−1 ∧α = (Λ ω α) ω n . Hence, (ω n−1 ∧ α) = Λ ω α, so we get the first equality below:
−h d 1 h (ω n−1 ∧ α) = −h d 1 h (Λ ω α) = −h 1 h ∂(Λ ω α) −h ∂ (Λ ω α).
Applying (37) to the l.h.s. term above and the standard formula (7) to the r.h.s. term, we get:
d h (ω n−1 ∧ α) = i∂(Λ ω α) ∧ ω n−1 − ih∂(Λ ω α) ∧ ω n−1 . Since i∂ − ih∂ = −ih d − 1 h
, the above equality is nothing but (38).
• Computation of the first term on the r.h.s. of (36) on 1-forms ϕ = ϕ 1, 0 + ϕ 0, 1 .
Using formula (38) with α := h∂ϕ 0, 1 +∂ϕ 1, 0 , we get the second equality below:
[d h , L ω n−1 ] d h ϕ = d h (ω n−1 ∧ (h∂ϕ 0, 1 +∂ϕ 1, 0 )) − ω n−1 ∧ d h d h ϕ = −ih d − 1 h hΛ ω (∂ϕ 0, 1 ) + Λ ω (∂ϕ 1, 0 ) ∧ ω n−1 − d h d h ϕ ∧ ω n−1 .(39)
On the other hand, the standard formula (7) yields:
ϕ = i(ϕ 0, 1 − ϕ 1, 0 ) ∧ ω n−1 .
Since ω is balanced, this implies the first equality on each of the two rows below:
∂ ϕ = i∂(ϕ 0, 1 − ϕ 1, 0 ) ∧ ω n−1 = i∂ϕ 0, 1 ∧ ω n−1 = iΛ ω (∂ϕ 0, 1 ) ω n ∂ ϕ = i∂(ϕ 0, 1 − ϕ 1, 0 ) ∧ ω n−1 = −i∂ϕ 1, 0 ∧ ω n−1 = −iΛ ω (∂ϕ 1, 0 ) ω n .
Taking − in each of the above two equalities and using the standard identities − ∂ =∂ , − ∂ = ∂ , we get:∂ ϕ = −iΛ ω (∂ϕ 0, 1 ) and ∂ ϕ = iΛ ω (∂ϕ 1, 0 ).
Putting together (39) and (40), we get:
[d h , L ω n−1 ] d h ϕ = −ih d − 1 h (ih∂ ϕ − i∂ ϕ) ∧ ω n−1 − d h d h ϕ ∧ ω n−1 = hh d − 1 h d − 1 h ϕ ∧ ω n−1 − d h d h ϕ ∧ ω n−1 .
We have thus obtained the following formula:
[d h , L ω n−1 ] d h ϕ = |h| 2 d − 1 h d − 1 h − d h d h ϕ ∧ ω n−1(41)
for every smooth 1-form ϕ whenever the metric ω is balanced.
• Computation of the second term on the r.h.s. of (36) on 1-forms ϕ = ϕ 1, 0 + ϕ 0, 1 .
We start by computing
[d h , L ω n−1 ] ϕ = d h (ω n−1 ∧ ϕ) − (d h ϕ) ω n−1 .(42)
Since ω n−1 ∧ ϕ 1, 0 = i ϕ 1, 0 and ω n−1 ∧ ϕ 0, 1 = −i ϕ 0, 1 , formula (37) yields the first line below:
d h (ω n−1 ∧ ϕ) = ih d 1 h (ϕ 1, 0 − ϕ 0, 1 ) = i (∂ϕ 1, 0 +h∂ϕ 1, 0 − ∂ϕ 0, 1 −h∂ϕ 0, 1 ) = i n Λ ω (h∂ϕ 1, 0 − ∂ϕ 0, 1 ) ω n−1 − i(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim ∧ ω n−2 + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) ∧ ω n−2 ,(43)
where we used the Lefschetz decomposition (8) of the (1, 1)-formh∂ϕ 1, 0 − ∂ϕ 0, 1 and then the standard formula (7) to express the value of on the primitive forms ∂ϕ 1, 0 (of type (2, 0)),∂ϕ 0, 1 (of type (0, 2)) and (h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim (of type (1, 1)) and got (∂ϕ 1, 0 ) = ∂ϕ 1, 0 ∧ ω n−2 and
(∂ϕ 0, 1 ) =∂ϕ 0, 1 ∧ ω n−2 , (h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim = −(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim ∧ ω n−2 .
On the other hand, we get
d h ϕ = −h d 1 h ( ϕ 1, 0 + ϕ 0, 1 ) = −h ( 1 h ∂ +∂)(−iϕ 1, 0 ∧ ω n−1 + iϕ 0, 1 ∧ ω n−1 ) (i) = −h − ī h ∂ϕ 1, 0 ∧ ω n−1 + ī h ∂ϕ 0, 1 ∧ ω n−1 − i∂ϕ 1, 0 ∧ ω n−1 + i∂ϕ 0, 1 ∧ ω n−1 (ii) = −h i 1 h ∂ϕ 0, 1 −∂ϕ 1, 0 ∧ ω n−1 = −h iΛ ω 1 h ∂ϕ 0, 1 −∂ϕ 1, 0 ω n = iΛ ω (h∂ϕ 1, 0 − ∂ϕ 0, 1 ),(44)
where the balanced assumption on ω was used to get (i) and the equalities ∂ϕ 1, 0 ∧ ω n−1 =∂ϕ 0, 1 ∧ ω n−1 = 0, that hold for bidegree reasons, were used to get (ii).
Noticing that the last term in (44) also features within the first term on the second line in (43), the conclusion of (43) can be re-written as
d h (ω n−1 ∧ ϕ) = 1 n (d h ϕ) ω n−1 − i(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim ∧ ω n−2 + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) ∧ ω n−2 .(45)
From this and from (42), we get:
[d h , L ω n−1 ] ϕ = 1 n − 1 (d h ϕ) ω n−1 − i(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim ∧ ω n−2 + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) ∧ ω n−2 .
Hence, using the balanced hypothesis d h ω = 0, we get the first two lines below:
d h [d h , L ω n−1 ] ϕ = 1 n − 1 dd h ϕ ∧ ω n−1 − id h (h∂ϕ 1, 0 − ∂ϕ 0, 1 ) ∧ ω n−2 + n − 1 n i d h Λ ω (h∂ϕ 1, 0 − ∂ϕ 0, 1 ) ∧ ω n−1 + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) ∧ d h ω n−2 − i(|h| 2 ∂∂ϕ 0, 1 + ∂∂ϕ 1, 0 ) ∧ ω n−2 .
Now, formula (44) shows that the term on the second line above equals minus the first term on the r.h.s. of the first line. Hence, the sum of these two terms vanishes and we get:
d h [d h , L ω n−1 ] ϕ = i ∂ϕ 1, 0 + (∂ϕ 0, 1 −h∂ϕ 1, 0 ) −h∂ϕ 0, 1 ∧ d h ω n−2 − i(|h| 2 + 1) ∂∂ϕ ∧ ω n−2 = −ih d − 1 h ϕ ∧ d h ω n−2 − i(|h| 2 + 1) ∂∂ϕ ∧ ω n−2 .(46)
• Conclusion.
Putting together (36), (41) and (46), we get (35). The proof of Lemma 3.4 is complete.
Recall that for any Hermitian metric ω on an n-dimensional complex manifold X, the pointwise Lefschetz map:
L ω n−1 : Λ 1 T X −→ Λ 2n−1 T X, ϕ −→ ψ := ω n−1 ∧ ϕ,
is bijective and a quasi-isometry (in the sense of Lemma 4.3).
We will now integrate the result of Lemma 3.4 expressing the commutation defect between ∆ h and L ω n−1 on 1-forms. We need to assume our balanced metric ω to be complete to ensure that the two meanings of d h coincide and the L 2 ω -inner products can be handled as in the compact case (see (b) and (c) of Theorem 3.3).
Proposition 3.5. Let X be a complex manifold with dim C X = n. Suppose there exists a complete balanced metric ω on X.
Then, for any h ∈ C and any 1-form ϕ ∈ Dom (∆ − 1 h ) on X, the following identity holds:
∆ h (ω n−1 ∧ ϕ), ω n−1 ∧ ϕ = |h| 2 ∆ − 1 h ϕ, ϕ .(47)
Proof. Throughout the proof, ϕ will stand for an arbitrary smooth 1-form on X.
• We first notice that
d h d − 1 h ϕ = ((|h| 2 + 1)/h) ∂∂ϕ, hence d h − ih d − 1 h ϕ ∧ ω n−2 = −ih d − 1 h ϕ ∧ d h ω n−2 − i(|h| 2 + 1) ∂∂ϕ ∧ ω n−2 .
These are the last two terms of formula (35).
Putting ψ := ω n−1 ∧ ϕ and using (35) with its last two terms transformed as above, we get:
∆ h ψ, ψ = ∆ h ϕ ∧ ω n−1 , ϕ ∧ ω n−1 + (|h| 2 d − 1 h d − 1 h − d h d h ) ϕ ∧ ω n−1 , ϕ ∧ ω n−1 −ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) = d h d h ϕ ∧ ω n−1 , ϕ ∧ ω n−1 + |h| 2 d − 1 h d − 1 h ϕ ∧ ω n−1 , ϕ ∧ ω n−1 −ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) (i) = d h d h ϕ, ϕ + |h| 2 d − 1 h d − 1 h ϕ, ϕ − ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) = ||d h ϕ|| 2 + |h| 2 ||d − 1 h ϕ|| 2 − ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) ,(48)
where (i) followed from Lemma 4.1 applied to the (necessarily primitive) 1-forms ϕ, d h d h ϕ and
d − 1 h d − 1 h ϕ.
• We now transform the last term in (48), namely T (ϕ) :
= −ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) .
Since the multiplication map ω n−2 ∧ · : Λ 2 T X −→ Λ 2n−2 T X is bijective, there exists a unique 2-form β such that d h (ω n−1 ∧ ϕ) = ω n−2 ∧ β. Thus, using (45) for the second equality below, we get:
ω n−2 ∧ β = d h (ω n−1 ∧ ϕ) = ω n−2 ∧ 1 n(n − 1) (d h ϕ) ω − i(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) .
The uniqueness of β implies that
β = − i(h∂ϕ 1, 0 − ∂ϕ 0, 1 ) prim + i(∂ϕ 1, 0 −h∂ϕ 0, 1 ) + 1 n(n − 1) (d h ϕ) ω.(49)
In particular, the primitive part β prim of β in the Lefschetz decomposition is the form inside the large parenthesis and Λ ω β = 1 n−1 d h ϕ. On the other hand, we have
d − 1 h ϕ = − 1 h ∂ϕ 1, 0 + (− 1 h ∂ϕ 0, 1 +∂ϕ 1, 0 ) prim +∂ϕ 0, 1 + 1 nih (d h ϕ) ω,(50)
where the value of the last term follows from formula (44). This implies that β and −ih d − 1 h ϕ have the same primitive part:
β prim = −ih (d − 1 h ϕ) prim .(51)
We get:
d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) = d − 1 h ϕ ∧ ω n−2 , β ∧ ω n−2 = (d − 1 h ϕ) prim , β prim + (n − 1) 2 n 1 nih d h ϕ, 1 n(n − 1) d h ϕ ,
where the last equality follows from formula (74) in the Appendix. From this and from (49)-(51), we get:
T (ϕ) = −ih d − 1 h ϕ ∧ ω n−2 , d h (ω n−1 ∧ ϕ) = ||∂ϕ 1, 0 || 2 + ||(∂ϕ 0, 1 −h∂ϕ 1, 0 ) prim || 2 + |h| 2 ||∂ϕ 0, 1 || 2 − (1 − 1 n ) ||d h ϕ|| 2 .(52)
• Putting (48) and (52) together and writing 1 n ||d h ϕ|| 2 = |h| 2 1 n || 1 ih d h ϕ|| 2 , we get:
∆ h ψ, ψ = |h| 2 ||d − 1 h ϕ|| 2 + |h| 2 1 n || 1 ih d h ϕ|| 2 + |h| 2 || − 1 h ∂ϕ 1, 0 || 2 + ||(− 1 h ∂ϕ 0, 1 +∂ϕ 1, 0 ) prim || 2 + ||∂ϕ 0, 1 || 2 .
Thanks to the expression of d − 1 h ϕ obtained in (50), this translates to
∆ h ψ, ψ = |h| 2 ||d − 1 h ϕ|| 2 + ||d − 1 h ϕ|| 2 = |h| 2 ∆ − 1 h ϕ, ϕ ,
which is (47). Proposition 3.5 is proved.
An immediate consequence of Proposition 3.5 is the following Hard Lefschetz-type result for spaces of harmonic L 2 ω -forms induced by a given complete balanced metric ω and different operators ∆ − 1 h and ∆ h . Note that h = − 1 h for all h ∈ C . This is the price we have to pay in the non-Kähler balanced context to get this kind of results.
Corollary 3.6. Let X be a complex manifold with dim C X = n. Suppose there exists a complete balanced metric ω on X. Then, for any h ∈ C , the map
ω n−1 ∧ · : H 1 ∆ − 1 h (X, C) −→ H 2n−1 ∆ h (X, C), ϕ −→ ω n−1 ∧ ϕ,
is well-defined and an isomorphism.
Proof. The well-definedness, namely the fact that this map takes ∆ − 1 h -harmonic L 2 ω -forms to ∆ hharmonic L 2 ω -forms, follows at once from Proposition 3.5 and from the form ω n−1 being ω-bounded. The fact that this map is an isomorphism follows from the standard fact that the corresponding pointwise map is bijective.
Corollary 3.7. Let X be a complex manifold with dim C X = n. Suppose there exists a complete balanced metric ω on X such that ω n−1 = dΓ for an ω-bounded smooth (2n − 3)-form Γ. Then
∆ψ, ψ ≥ 1 4||Γ|| 2 L ∞ ω ||ψ|| 2(53)
for every pure-type form ψ ∈ Dom(∆) of degree 2n − 1.
Proof. Taking h = 1 in Proposition 3.5, (47) gives:
∆ψ, ψ = ∆ −1 ϕ, ϕ = ||(∂ −∂)ϕ|| 2 + ||(∂ −∂) ϕ|| 2 ≥ ||(∂ −∂)ϕ|| 2 ,
for every (2n − 1)-form ψ, where ϕ is the unique 1-form such that ψ = ω n−1 ∧ ϕ. (See isomorphism (69) for r = 1.) Meanwhile, ψ is of pure type (either (n, n − 1) or (n − 1, n)) if and only if ϕ is of pure type (respectively, either (1, 0) or (0, 1)). In this case, ∂ϕ and∂ϕ are of different pure types, hence orthogonal to each other, hence ||(∂ −∂)ϕ|| 2 = ||(∂ +∂)ϕ|| 2 . Thus, we get:
∆ψ, ψ ≥ ||dϕ|| 2 ,(54)
for every pure-type (2n − 1)-form ψ ∈ Dom(∆).
To complete the proof, we adapt the proof of Theorem 1.4.A. in [Gro91] to our context.
Since any 1-form ϕ is primitive, Lemma 4.1 gives: |ψ| 2 = |ω n−1 ∧ ϕ| 2 = |ϕ| 2 . In particular,
||ψ|| = ||ϕ||.(55)
Meanwhile, we have: ψ = ω n−1 ∧ ϕ = dΓ ∧ ϕ = d(Γ ∧ ϕ) + Γ ∧ dϕ. In other words,
ψ = dθ + ψ ,
where θ := Γ ∧ ϕ and ψ := Γ ∧ dϕ.
To estimate θ, we write:
||θ|| ≤ ||Γ|| L ∞ ω ||ϕ|| = ||Γ|| L ∞ ω ||ψ||,(57)
where (55) was used to get the last equality.
To estimate ψ , we write:
||ψ || ≤ ||Γ|| L ∞ ω ||dϕ|| ≤ ||Γ|| L ∞ ω ∆ψ, ψ 1 2 ,(58)
where (54) and the fact that ϕ is of pure type were used to get the last inequality.
To find an upper bound for ||ψ||, we write:
||ψ|| 2 = ψ, dθ + ψ ≤ | ψ, dθ | + | ψ, ψ |,(59)
where (56) was used to get the first equality. For the first term on the r.h.s. of (59), we get:
| ψ, dθ | = | d ψ, θ | ≤ ||d ψ|| ||θ|| ≤ ∆ψ, ψ 1 2 ||Γ|| L ∞ ω ||ψ||,(60)
where (57) was used to get the last inequality. For the second term on the r.h.s. of (59), we get:
| ψ, ψ | ≤ ||ψ || ||ψ|| ≤ ||Γ|| L ∞ ω ∆ψ, ψ 1 2 ||ψ||,(61)
where (58) was used to get the last inequality. Adding up (60) and (61) and using (59), we get
||ψ|| ≤ 2 ||Γ|| L ∞ ω ∆ψ, ψ 1 2 ,
which is (53). The proof is complete.
For the record, if we do not assume ψ to be of pure type and use the full force of (47) rather than (54), we can run the argument in the proof of Corollary 3.7 with minor modifications starting from the observation that
ω n−1 = d − 1 h Γ − 1 h
, where Γ h := h Γ n, n−3 + Γ n−1, n−2 + (1/h) Γ n−2, n−1 + (1/h 2 ) Γ n−3, n for every h ∈ C and the Γ p, q 's are the pure-type components of Γ. Then, we get the following analogue of (53):
||ψ|| ≤ C h ||Γ − 1 h || ∆ h ψ, ψ 1 2 + ∆ − 1 h ψ, ψ 1 2(62)
for every form ψ ∈ Dom(∆ h ) ∩ Dom(∆ − 1 h ) (not necessarily of pure type) of degree 2n − 1, where C h := max(1, 1/|h|).
The occurrence of two different Laplacians on the r.h.s. of (62) (recall that h = − 1 h for every h ∈ C ) is the downside of that estimate that we avoided in Corollary 3.7 by restricting attention to pure-type forms. The advantage of dealing with a single Laplacian is demonstrated by Theorem 1.5 in the introduction that we now prove as a consequence of the above discussion.
Proof of Theorem 1.5. The pair ( X, ω) satisfies the hypotheses of Corollary 3.7 (playing the role of the pair (X, ω) therein). When applied to (n, n − 1)-forms and to (n − 1, n)-forms ψ ∈ Dom(∆ ω ), inequality (53) gives the following implication:
∆ ω ψ = 0 =⇒ ψ = 0.
This proves the vanishing of H n, n−1 ∆ ω ( X, C) and H n−1, n ∆ ω ( X, C). Meanwhile, the Hodge star operator = ω commutes with ∆ ω , so it induces isomorphisms
ω : H 1, 0 ∆ ω ( X, C) −→ H n, n−1 ∆ ω ( X, C) and ω : H 0, 1 ∆ ω ( X, C) −→ H n−1, n ∆ ω ( X, C).
Therefore, the spaces H 1, 0 ∆ ω ( X, C) and H 0, 1 ∆ ω ( X, C) must vanish as well.
3.3 Harmonic L 2 -forms of degree 2 on the universal cover of a balanced hyperbolic manifold
We will discuss 2-forms in a way analogous to the discussion of 1-forms we had in §.3.2. The context and the notation are the same. The analogue of Lemma 3.4 is Lemma 3.8. Let X be a complex manifold with dim C X = n. Suppose there exists a balanced metric ω on X. Then, for any h ∈ C and any 2-form α on X, the following identity holds:
[∆ h , L ω n−1 ]α = −(|h| 2 + 1) i∂∂(Λ ω α) ∧ ω n−1 − ω n−1 ∧ ∆ h α.(63)
Proof. We compute separately the two terms applied to α on the r.h.s. of the consequence (36) of the Jacobi identity and the balanced hypothesis on ω.
The first term is
[d h , L ω n−1 ] d h α = −ω n−1 ∧ d h d h α,(64)
since d h (ω n−1 ∧ d h α) = 0 owing to the vanishing of ω n−1 ∧ d h α for degree reasons.
To compute d h [d h , L ω n−1 ]α, we notice that [d h , L ω n−1 ]α = d h (ω n−1 ∧ α) − ω n−1 ∧ d h α = −ih d − 1 h (Λ ω α) ∧ ω n−1 − ω n−1 ∧ d h α,
where the last identity follows from (38). Thus, using the balanced hypothesis on ω, we get:
d h [d h , L ω n−1 ]α = −ihd h d − 1 h (Λ ω α) ∧ ω n−1 − ω n−1 ∧ d h d h α.(65)
Finally, d h d − 1 h = ((|h| 2 + 1)/h) ∂∂, so (63) follows from (64) and (65).
We now deduce the following analogue of Proposition 3.5.
Proposition 3.9. Let (X, ω) be a complete balanced manifold, dim C X = n ≥ 2.
For any h ∈ C and any 2-form ϕ ∈ Dom (∆ h ) on X, the following identity holds:
∆ h (ω n−1 ∧ α), ω n−1 ∧ α = (|h| 2 + 1) ||∂(Λ ω α)|| 2 .(66)
Proof. An immediate consequence of (63) is the identity
∆ h (ω n−1 ∧ α) = −(|h| 2 + 1) i∂∂(Λ ω α) ∧ ω n−1 .
Taking the pointwise inner product (w.r.t. ω) against ω n−1 ∧α and using the Lefschetz decomposition α 1, 1 = α 1, 1 prim + (1/n) (Λ ω α 1, 1 ) ω of the (1, 1)-type component of α, its analogue for the (1, 1)-form i∂∂(Λ ω α) and the fact that the product of any primitive 2-form with ω n−1 vanishes, we get:
∆ h (ω n−1 ∧ α), ω n−1 ∧ α = −(|h| 2 + 1) ∆ ω (Λ ω α) ω n , (Λ ω α) ω n = −(|h| 2 + 1) ∆ ω (Λ ω α), Λ ω α ,(67)
where ∆ ω f := Λ ω (i∂∂f ) for any function f on X. It is standard that the Laplacian ∆ ω is a nonpositive operator on functions. Identity (71) in Lemma 4.1 with k = 0 and r = n was used to get the last equality in (67). Now, we need the following simple observation.
Lemma 3.10. Let (X, ω) be a complete balanced manifold, dim C X = n ≥ 2. For any function f ∈ Dom ( ∆ ω ), we have: ∆ ω f, f = −||∂f || 2 .
Proof of Lemma 3.10. The formula ∂ = − ∂ gives the third equality below:
∆ ω f, f = Λ ω (i∂∂f ), f = i∂f, ∂ (f ω) = −i ∂ f, ∂ (f ω n−1 ) = −i ∂ f, (∂f ∧ ω n−1 ) , where we used the balanced hypothesis on ω to get the last equality. Now,∂f is a (0, 1)-form, hence primitive, so the standard formula (7) yields:
(i∂f ) = −∂f ∧ ω n−1 , or equivalently (∂f ∧ ω n−1 ) = i∂f, since = −Id on forms of odd degree. The contention follows.
End of proof of Proposition 3.9. Integrating (67) and applying Lemma 3.10 with f = Λ ω α, we get (66). The next consequence of the above discussion can be conveniently worded in terms of Demailly's torsion operator τ = τ ω := [Λ ω , ∂ω ∧ ·] and the induced Laplacian ∆ τ := [d + τ, d + τ ] mentioned in the introduction.
Corollary 3.11. Let (X, ω) be a connected complete balanced manifold, dim C X = n ≥ 2. For any (1, 1)-form α 1, 1 ∈ Dom (∆ τ ), the following implication holds: ∆ τ α 1, 1 = 0 =⇒ Λ ω α 1, 1 is constant.
Appendix
A key classical fact used by Gromov in [Gro91] is that some of the Lefschetz maps at the level of differential forms are quasi-isometries w.r.t. the L 2 -inner product. We spell out the equalities involving pointwise inner products that lead to more precise statements that were used in earlier parts of our text.
Let ω be an arbitrary Hermitian metric on an arbitrary complex manifold X with dim C X = n. As usual, for any r = 1, . . . , n, we put ω r := ω r /r!. Recall the following standard fact.
For every k ≤ n and every r ≤ n − k, the pointwise Lefschetz operator:
L r ω : Λ k T X −→ Λ k+2r T X, L r ω (ϕ) = ω r ∧ ϕ,(69)
is injective. When r = n − k, L n−k ω is even bijective.
We will compare the pointwise inner products ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω and ϕ 1 , ϕ 2 ω for arbitrary k-forms ϕ 1 , ϕ 2 ∈ Λ k T X. We will use the following standard formula (cf. e.g. [Voi02]):
[L r ω , Λ ω ] = r(k − n + r − 1) L r−1 ω on k-forms,(70)
for any integer r ≥ 1, where Λ = Λ ω = (ω ∧ ·) is the adjoint of the Lefschetz operator L ω w.r.t. the pointwise inner product , ω induced by ω.
(1) Case of primitive forms
Recall that for any non-negative integer k ≤ n, a k-form ϕ is said to be primitive w.r.t. ω (or ω-primitive, or simply primitive when no confusion is likely) if it satisfies any of the following equivalent two conditions: ω n−k+1 ∧ ϕ = 0 ⇐⇒ Λ ω ϕ = 0.
Lemma 4.1. For every k ≤ n, every r ≤ n−k and any k-forms ϕ 1 , ϕ 2 one of which is ω-primitive, the following identity holds:
ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω = (r!) 2 n − k r ϕ 1 , ϕ 2 ω .
In particular, the analogous equality holds for the L 2 ω -inner product , ω .
Proof. To make a choice, let us suppose that ϕ 1 is primitive. We get:
ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω = Λ ω (ω r ∧ ϕ 1 ), ω r−1 ∧ ϕ 2 ω (i) = [Λ ω , L r ω ] ϕ 1 , ω r−1 ∧ ϕ 2 ω (ii)
= r(n − k − r + 1) ω r−1 ∧ ϕ 1 , ω r−1 ∧ ϕ 2 ω . . .
= r(r − 1) . . . 1 (n − k − r + 1) (n − k − r + 2) . . . (n − k) ϕ 1 , ϕ 2 ω = r! (n − k)! (n − k − r)! ϕ 1 , ϕ 2 ω , where (i) follows from ϕ 1 being primitive, (ii) follows from the standard formula (70), the remaining equalities except for the last one follow from analogues of (i) and (ii), while the last equality proves (71).
Let us also notice that, when the powers of ω are distinct, the products involved in the analogue of (71) are actually orthogonal to each other.
Lemma 4.2. Let r, s, k ∈ N with s > 0 and k ≤ n. For any (k − 2s)-form u and any ω-primitive k-form v, the following identity holds: ω r+s ∧ u, ω r ∧ v ω = 0.
(72)
In particular, the analogous equality holds for the L 2 ω -inner product , ω . Proof. We have:
ω r+s ∧ u, ω r ∧ v ω = ω r+s−1 ∧ u, Λ ω (ω r ∧ v) ω (i) = ω r+s−1 ∧ u, [Λ ω , L r ω ] v ω (ii)
= c 1 ω r+s−1 ∧ u, ω r−1 ∧ v ω = · · · = c 1 . . . c r ω s ∧ u, v ω = c 1 . . . c r ω s−1 ∧ u, Λ ω v ω = 0, where (i) follows from v being primitive, (ii) follows from the standard formula (70) with the appropriate constant c 1 (whose actual value is irrelevant here), the remaining equalities except for the last one follow from analogues of (i) and (ii) with the appropriate constants c 2 , . . . , c r , while the last equality follows again from v being primitive and proves (72).
(2) Case of arbitrary forms
Let ϕ 1 , ϕ 2 be arbitrary k-forms and let ϕ 1 = ϕ 1, prim + ω ∧ ϕ 1, 1 + · · · + ω l ∧ ϕ 1, l and ϕ 2 = ϕ 2, prim + ω ∧ ϕ 2, 1 + · · · + ω l ∧ ϕ 2, l
be their respective Lefschetz decompositions, where l is the non-negative integer defined by requiring 2l = k if k is even and 2l = k − 1 if k is odd, while the forms ϕ j, prim , ϕ j, 1 , . . . , ϕ j, l are primitive of respective degrees k, k − 2, . . . , k − 2l for every j ∈ {1, 2}.
The sense in which the Lefschetz operator (69) is a quasi-isometry for the pointwise inner product (hence also the L 2 -inner product) induced by ω is made explicit in the following Lemma 4.3. Fix integers 0 ≤ k ≤ n, 0 ≤ r ≤ n − k and arbitrary k-forms ϕ 1 , ϕ 2 .
(i) The following identity holds: ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω = (r!) 2 n − k r ϕ 1, prim , ϕ 2, prim ω (74) + ((r + 1)!) 2 n − k + 2 r + 1 ϕ 1, 1 , ϕ 2, 1 ω + · · · + ((r + l)!) 2 n − k + 2l r + l ϕ 1, l , ϕ 2, l ω .
(ii) Putting C n, k, r, s := ((r + s)!(n − k + s)!)/(s!(n − k − r + s)!) and
A n, k, r := min s=0,...,l C n, k, r, s , B n, k, r := max s=0,...,l C n, k, r, s , the following inequalities hold:
A n, k, r |ϕ| 2 ω ≤ |ω r ∧ ϕ| 2 ω ≤ B n, k, r |ϕ| 2 ω .
(iii) With the notation of (ii), if ϕ 1, s , ϕ 2, s ω ≥ 0 for every s ∈ {0, 1, . . . , l}, the following inequalities hold:
A n, k, r ϕ 1 , ϕ 2 ω ≤ ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω ≤ B n, k, r ϕ 1 , ϕ 2 ω .
Proof. (i) Using the Lefschetz decompositions (73) and Lemma 4.2, we get: ω r ∧ ϕ 1 , ω r ∧ ϕ 2 ω = ω r ∧ ϕ 1, prim , ω r ∧ ϕ 2, prim ω + l s=1 ω r+s ∧ ϕ 1, s , ω r+s ∧ ϕ 2, s ω .
Identity (74) follows from this and from Lemma 4.1.
(ii) and (iii) follow at once from (i) applied twice, with a given 1 ≤ r ≤ n − k and with r = 0.
( i )
iIf ω is a balanced metric on X, the linear map:{ω n−1 } DR ∧ · : H 1 DR (X, C) −→ H 2n−1 DR (X, C), {u} DR −→ {ω n−1 ∧ u} DR ,
prim amounts to ker({ω n−1 } DR ∧·) = ker({γ n−1 } DR ∧ ·). Hence, by the above elementary fact, it amounts to the exietence of a constantc ∈ C \ {0} such that {ω n−1 } DR ∧ · = c {γ n−1 } DR ∧ · as C-linear maps on H 2 DR (X, C). By the non-degeneracy of the Poincaré duality H 2 DR (X, C) × H 2n−2 DR (X, C) −→ C, this further amounts to the existence of a constant c ∈ C \ {0} such that {ω n−1 } DR = c {γ n−1 } DR .Now, since the forms ω n−1 and γ n−1 are real, the constant c can be chosen real. (Replace c with (c +c)/2 if necessary.) Since the balanced metric ω is non-degenerate, c = 0. If c < 0, then ω n−1 − c γ n−1 > 0 would be the d-exact (n − 1)-st power of a balanced metric. This balanced metric would then be degenerate balanced, contradicting the assumption on X. Thus, c must be positive.
Lemma 2 . 18 .
218The assumptions are the same as in Conclusion 2.17. For every
Lemma 3. 1 .
1([Gaf54]) Let (X, g) be a Riemannian manifold. Then, (X, g) is complete if and only if there exists an exhaustive sequence (K ν ) ν∈N of compact subsets of X:
Theorem 3.3. (see e.g. [Dem97, VIII, Theorem 3.2.]) Let (X, g) be a complete Riemannian manifold of real dimension m. Then: (a) The space D • (X, C) of compactly supported C ∞ forms of any degree (indicated by a •) on X is dense in the domains Dom d, Dom d and in Dom d ∩ Dom d for the respective graph norms:
(b) The extension d of the formal adjoint of d to the L 2 -space coincides with the Hilbert space adjoint of the extension of d.
Acknowledgments. This work is part of the first-named author's PhD thesis under the supervisionProof. Thanks to (5) and to ∆ τ ≥ 0 and ∆ ≥ 0, the hypothesis ∆ τ α 1, 1 = 0 translates to ∆ τ α 1, 1 = 0 and ∆ α 1, 1 = 0. Since ω is complete, these conditions are further equivalent to (i) (∂ + τ )α 1, 1 = 0, (iii)∂α 1, 1 = 0 (ii) (∂ + τ )α 1, 1 = 0, (iv)∂ α 1, 1 = 0.Thus, we get:∂where the first equality follows from (iii) of (68), the second equality follows from Demailly's Hermitian commutation relation (4) and the third equality follows from (ii) of (68). We conclude that the hypothesis ∆ τ α 1, 1 = 0 implies∂(Λ ω α 1, 1 ) = 0. This implies, thanks to Proposition 3.9 applied with h = 1, that ∆(ω n−1 ∧ α 1, 1 ) = 0, where ∆ = ∆ ω = dd + d d is the d-Laplacian induced by ω. Since ω n−1 ∧ α 1, 1 = (Λ ω α 1, 1 ) ω n = (Λ ω α 1, 1 ) and since ∆ commutes with , we get ∆(Λ ω α 1, 1 ) = 0. By completeness of ω, this means that d(Λ ω α 1, 1 ) = 0 on X, hence Λ ω α 1, 1 must be constant since X is connected.An immediate consequence of Corollary 3.11 is that the following linear map is well defined:under those assumptions, where H 1, 1 ∆τ (X, C) is the space of ∆ τ -harmonic L 2 ω -forms of type (1, 1).Proof of Theorem 1.6. The pair ( X, ω) satisfies the hypotheses of Corollary 3.11 (playing the role of the pair (X, ω) therein). By the balanced hyperbolic hypothesis on (X, ω), there exists an ω-bounded smooth (2n − 3)-form Γ on X such that ω n−1 = d Γ.Let α 1, 1 ∈ H 1, 1 ∆ τ ( X, C) such that α 1, 1 ≥ 0. Then,∂α 1, 1 = 0 (by (iii) of (68)) and real, hence we also have ∂α 1, 1 = 0. Thus, α 1, 1 is d-closed, so ω n−1 ∧ α 1, 1 = d( Γ ∧ α 1, 1 ) ∈ Im d because Γ ∧ α 1, 1 is L 2 ω and d( Γ ∧ α 1, 1 ) is again L 2 ω . On the other hand,because Λ ω α 1, 1 is constant by Corollary 3.11. Since the subspaces H 2n ∆ ω ( X, C) and Im d of the space of L 2 ω -forms of degree 2n on X are orthogonal (see (d) of Theorem 3.3), we deduce that ω n−1 ∧ α 1, 1 = 0. Equivalently, Λ ω α 1, 1 = 0. This implies that α 1, 1 = 0 since α 1, 1 ≥ 0 by hypothesis.
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Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France Email: [email protected] AND [email protected] For the first-named author only. A Yachou, Sur les variétés semi-kählériennes. University of Lille. Institut de Mathématiques de Toulouse ; Université de Monastir, Faculté des Sciences de Monastir Laboratoire de recherche AnalysePhD ThesisA. Yachou -Sur les variétés semi-kählériennes -PhD Thesis (1998), University of Lille. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France Email: [email protected] AND [email protected] For the first-named author only, also: Université de Monastir, Faculté des Sciences de Monastir Laboratoire de recherche Analyse, Géométrie et Applications LR/18/ES/16
Avenue de l'environnement 5019. Monastir, TunisieAvenue de l'environnement 5019, Monastir, Tunisie
| [] |
[
"Supplementary Material: Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells",
"Supplementary Material: Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells"
] | [
"Yunfei Huang \nTheoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany\n",
"Christoph Schell \nInstitut für Klinische Pathologie\nUniversitätsklinikum Freiburg\nD-79002FreiburgGermany\n",
"Tobias B Huber \nDepartment of Medicine IV\nFaculty of Medicine\nMedical Center\nUniversity of Freiburg\nGermany\n\nBIOSS Center for Biological Signalling Studies\nAlbert-Ludwigs-University Freiburg\nGermany\n\nIII. Department of Medicine\nUniversity Medical Center Hamburg-Eppendorf\nHamburgGermany\n",
"Ahmet Nihat ",
"Ş Imş Ek \nTheoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany\n",
"Nils Hersch \nBiomechanics\nInstitute of Complex Systems-7\nForschungszentrum Juelich\nD-52425JuelichGermany\n",
"Rudolf Merkel \nBiomechanics\nInstitute of Complex Systems-7\nForschungszentrum Juelich\nD-52425JuelichGermany\n",
"Gerhard Gompper \nTheoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany\n",
"Benedikt Sabass *[email protected] \nTheoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany\n"
] | [
"Theoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany",
"Institut für Klinische Pathologie\nUniversitätsklinikum Freiburg\nD-79002FreiburgGermany",
"Department of Medicine IV\nFaculty of Medicine\nMedical Center\nUniversity of Freiburg\nGermany",
"BIOSS Center for Biological Signalling Studies\nAlbert-Ludwigs-University Freiburg\nGermany",
"III. Department of Medicine\nUniversity Medical Center Hamburg-Eppendorf\nHamburgGermany",
"Theoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany",
"Biomechanics\nInstitute of Complex Systems-7\nForschungszentrum Juelich\nD-52425JuelichGermany",
"Biomechanics\nInstitute of Complex Systems-7\nForschungszentrum Juelich\nD-52425JuelichGermany",
"Theoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany",
"Theoretical Soft Matter and Biophysics\nInstitute of Complex Systems-2 and Institute for Advanced Simulation\nForschungszentrum Juelich\nD-52425JuelichGermany"
] | [] | This supplementary document provides details on the generation of test data, on the construction and solution of the optimization problems, on the choice of regularization parameters, and code. Figures contain further results on the performance of different TFM routines.2/17 | 10.1038/s41598-018-36896-x | null | 53,360,442 | 1810.05848 | 19da3aa188e014097f14f523cfc5dd7db9a55bc3 |
Supplementary Material: Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells
Yunfei Huang
Theoretical Soft Matter and Biophysics
Institute of Complex Systems-2 and Institute for Advanced Simulation
Forschungszentrum Juelich
D-52425JuelichGermany
Christoph Schell
Institut für Klinische Pathologie
Universitätsklinikum Freiburg
D-79002FreiburgGermany
Tobias B Huber
Department of Medicine IV
Faculty of Medicine
Medical Center
University of Freiburg
Germany
BIOSS Center for Biological Signalling Studies
Albert-Ludwigs-University Freiburg
Germany
III. Department of Medicine
University Medical Center Hamburg-Eppendorf
HamburgGermany
Ahmet Nihat
Ş Imş Ek
Theoretical Soft Matter and Biophysics
Institute of Complex Systems-2 and Institute for Advanced Simulation
Forschungszentrum Juelich
D-52425JuelichGermany
Nils Hersch
Biomechanics
Institute of Complex Systems-7
Forschungszentrum Juelich
D-52425JuelichGermany
Rudolf Merkel
Biomechanics
Institute of Complex Systems-7
Forschungszentrum Juelich
D-52425JuelichGermany
Gerhard Gompper
Theoretical Soft Matter and Biophysics
Institute of Complex Systems-2 and Institute for Advanced Simulation
Forschungszentrum Juelich
D-52425JuelichGermany
Benedikt Sabass *[email protected]
Theoretical Soft Matter and Biophysics
Institute of Complex Systems-2 and Institute for Advanced Simulation
Forschungszentrum Juelich
D-52425JuelichGermany
Supplementary Material: Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells
This supplementary document provides details on the generation of test data, on the construction and solution of the optimization problems, on the choice of regularization parameters, and code. Figures contain further results on the performance of different TFM routines.2/17
Fast calculation of the matrix M using the shift theorem in Fourier space
Recently, it was suggested by Danuser and co-workers 1 that the computational effort for calculation of M can be significantly reduced by using the convolution theorem in Fourier space. In this section, we combine Fourier-space convolution with the shift theorem to calculate traction on a regular grid directly from irregularly spaced displacements. The starting point is a general linear relation between a continuous displacement field U i (x) and the traction field F j (x ). Both are two-dimensional vector fields with (i, j) ∈ {1, 2} and extend over the whole surface of the substrate Ω. They are related as 2
U i (x) = 2 ∑ j=1 Ω G i j (x − x )F j (x ) d 2 x .(1)
This equation is to be approximated as a matrix product. To discretize the traction field, we introduce a rectangular, regular mesh with meshsize w. The position of the mesh nodes is denoted by y k . The distance between any point in the traction field at x and the mesh nodes at y l is abbreviated for simplicity as (d 1 , d 2 ) = x − y l . The traction at x is assumed be a linear combination of the traction values at the four surrounding nodes. Hence, we introduce a pyramidal shape function, located at every node y l , which scales the traction magnitude at x depending on the distances (d 1 , d 2 ) as
h(d 1 , d 2 ) =θ (d 1 )θ (d 2 )(1 − d 1 /w)(1 − d 2 /w)θ (w − d 1 )θ (w − d 2 ) + θ (−d 1 )θ (d 2 )(1 + d 1 /w)(1 − d 2 /w)θ (w + d 1 )θ (w − d 2 ) + θ (d 1 )θ (−d 2 )(1 − d 1 /w)(1 + d 2 /w)θ (w − d 1 )θ (w + d 2 ) + θ (−d 1 )θ (−d 2 )(1 + d 1 /w)(1 + d 2 /w)θ (w + d 1 )θ (w + d 2 ), where θ (x) := 1, x ≥ 0 0, x < 0 .
With the help of the shape function, the continuous traction field is linearly approximated as
F j (x ) = ∑ l h(x − y l ) f j,l . Thus, Eq. (1) becomes U i (x) = ∑ j,l Ω G i j (x − x )h(x − y l ) d 2 x f j,l = ∑ j,l M j,l (x) f j,l .(2)
In order to simplify the convolution in Eq. (2) we employ a Fourier transformation with wave vector k = (k 1 , k 2 ). The shape function becomes in Fourier spaceh(k 1 , k 2 ) = sinc 2 (k 1 )sinc 2 (k 2 ) and the Green's function reads in Fourier spacẽ G i j (k 1 , k 2 ) = 2(1 + ν)/(E(k 2 1 + k 2 2 ) 3/2 ) (k 2 1 + k 2 2 )δ i j − k i k j ν . On using the shift theorem, we find for Eq. (2)
U i (x) = ∑ j,l Ft −1 (G i j (k)h(k)e −iky l ) f j,l .(3)
The coefficient matrix M used in the main part of the paper is constructed from the M j,l (x) by inserting discrete measurement positions for x and arranging into matrix form.
Proximal gradient methods for L1 regularization (PGL) and EN regularization (PGEN)
Proximal gradient methods provide a way to robustly solve optimization problems involving locally non-differentiable, convex penalty functions. For our tests with TFM, we follow well-established approaches 3,4 . The target function to be considered for PGL and the PGEN reads
1 2 MW * W f − u 2 2 + λ 1 W f 1 + λ 2 2 W f 2 2 ,(4)
where W denotes an unitary wavelet transform. For wavelet transformation, we employ the lifting transform function provided with Ref. 5 . The non-differentiable penalty functions for L1 and EN regularization are denoted by g(f) = λ 1 ||W f|| 1 or g(f) = λ 1 ||W f|| 1 + λ 2 /2||W f|| 2 2 , respectively. The optimization procedure is iterative and based on a gradient decent for the differentiable term ||Mf − u|| 2 2 /2. If we were to employ a gradient decent for this term only, the traction at the iteration number k + 1 would be given by
f k+1 = h k with h k = f k − τM T (Mf k − u)
where τ is the step size. However, since the solution must also obey the regularization constraints, the iteration is modified as follows
f k+1 = argmin f {τg(f) + 1 2 ||h k − f|| 2 } =: prox τg(f) (h k ).(5)
Hence, the proximal gradient scheme produces incremental changes that balance a gradient decent to minimize the solution residual with a minimization of the penalty. The right-hand side of Eq. (5) is a definition of the so-called proximity operator. For the L1 norm, g(x) = λ 1 |x|, the proximity operator can be given in closed form as a threshold function to be applied to each element prox τλ 1 x 1 (x) = (|x| − τλ 1 ) + sgn(x) =: S(x, τλ 1 ). Since the variables x are for our target function (4) the wavelet coefficients, result must be transformed back to real space after every iteration. The result is, see e.g. 3,4,6 ,
f k+1 = W * S W f k − τM T (Mf k − u) ; τλ 1 .(6)
The iteration for the EN regularization in PGEN has the same structure, but an additional factor that shrinks the solution magnitude 4
f k+1 = 1 1 + τλ 2 W * S W f k − τM T (Mf k − u) ; τλ 1 .(7)
In order to obtain a stable iteration, the step size needs to obey τ < 2/ M T M 2 . Iterations are stopped when the change of the solution norm from timestep to timestep is smaller than 10 −8 %.
Bayesian compressive sensing using Laplace prior (BCSL)
To test an example of a more elaborate Bayesian hierarchical model for TFM, we employ a software representing the network shown in Fig. S14(a). The system is described in detail in Ref. 7 . A main difference between this algorithm and ABL2 is the assumption of sparsity through a Laplace prior that is employed along with further hyperpriors for the noise. The model is solved with a variational technique. As shown in the Supplementary figures, the algorithm effectively produces a sparse image, however, with strongly overestimated traction hotspots. Addition of noise to the displacement field leads to a very visible degradation of the reconstruction quality with many false traction hotspots appearing.
Bayesian Lasso (BL) and Bayesian elastic net (BEN)
As an alternative to the variational Bayes approach, hierarchical models can be solved by Markov chain Monte Carlo methods.
To test this approach in conjunction with complex models, we employ two models provided as Matlab packages together with Ref. 8 . The BL is based on the same network structure as the BCSL shown in Fig. S14(a). Furthermore, we also tested a Bayesian version of the elastic net (BEN) where the network is shown in Fig. S14(b). Both algorithms perform similarly to the BCSL for very low noise. Here, the assumption of sparsity helps to produce a clear background and allows to distinguish traction sites clearly. However, traction magnitude estimates are strongly exaggerated. As expected, the full solution of the models via Monte Carlo sampling makes the performance of BL and BEN slightly more robust than BCSL in the presence of noise. Overall, the performance of these methods in the context of TFM is unsatisfactory.
Analytical calculation of displacements around a circular traction patch
For convenience of the reader, we provide here explicit formulas for the displacement field around a single circular traction patch. Test images containing multiple patches can be assembled by adding the displacement vectors resulting from the individual patches. The analytical solution is calculated for a patch with radius R located at the origin. We employ polar coordinates, r = (r cos θ , r sin θ ), with r and θ being the radial and angular coordinate. The traction patch is described as
f (r) = f 0 , |r| < R, 0, |r| ≥ R.(8)
The traction vector is given by f x = f (r) cos γ and f y = f (r) sin γ, where γ is the angle between the x axis and the direction of the traction. Fourier transformation yields
f (ρ) = 2π 0 ∞ 0 f (r)e iρr cos(φ −θ ) rdrdθ = 2πR f 0 J 1 (ρR)/ρ,(9)
where J 1 (ρR) is a Bessel function, ρ is a radial wave vector, and φ is an angle. The traction vector in Fourier space is thus f x =f (ρ) cos γ andf y =f (ρ) sin γ. The Fourier space Green's function in polar coordinates reads
G i j (ρ, φ ) = 2(1 + ν) Eρ (1 − ν) + ν sin 2 φ −ν sin φ cos φ −ν sin φ cos φ (1 − ν) + ν cos 2 φ .(10)
3/17
According to the convolution theorem, the displacement field in Fourier space becomes
ũ x u y = 2(1 + ν) Eρ (1 − ν) + ν sin 2 φ −ν sin φ cos φ −ν sin φ cos φ (1 − ν) + ν cos 2 φ f x f y .(11)
Thus, displacements in real space can be calculated through inverse Fourier transformation
u x u y = 1 2π 2 2π 0 ∞ 0 ũ x u y e −iρr cos(φ −θ ) ρdρdφ .(12)
The result can be simplified as
u x (r, θ ) = R(1 + ν) πE (1 − ν)N 1 + νN 2 f 0 cos γ − νN 3 f 0 sin γ , u y (r, θ ) = R(1 + ν) πE − νN 3 f 0 cos γ + (1 − ν)N 1 + νN 4 f 0 sin γ .(13)
For the functions N 1 to N 4 we have for the inner region where r < R and ξ 1 = r 2 /R 2
N 1 = 4E 0 (ξ 1 ) (14a) N 2 = 4 cos(2θ ) (r 2 + R 2 )E 0 (ξ 1 ) + (r 2 − R 2 )K 0 (ξ ) 3r 2 + 4 sin 2 θ E 0 (ξ 1 ) (14b) N 3 = 2 sin(2θ ) (r 2 − 2R 2 )E 0 (ξ 1 ) + 2(R 2 − r 2 )K 0 (ξ 1 ) 3r 2 (14c) N 4 = 4 cos 2 θ E 0 (ξ 1 ) − 4 cos(2θ ) (r 2 + R 2 )E 0 (ξ 1 ) + (r 2 − R 2 )K 0 (ξ 1 ) 3r 2 .(14d)
Here, E 0 is complete elliptic integral of the first kind and K 0 is complete elliptic integral of the second kind. For the outer region where r > R and ξ 2 = R 2 /r 2 we have
N 1 = 4 r 2 E 0 (ξ 2 ) + (R 2 − r 2 )K 0 (ξ 2 ) rR (15a) N 2 = 6r 2 − 2(r 2 − 2R 2 ) cos(2θ ) E 0 (ξ 2 ) + 2(r 2 − R 2 )(cos(2θ ) − 3)K 0 (ξ 2 ) 3rR (15b) N 3 = 2 sin(2θ ) (r 2 − 2R 2 )E 0 (ξ 2 ) + (R 2 − r 2 )K 0 (ξ 2 ) 3rR (15c) N 4 = 6r 2 + 2(r 2 − 2R 2 ) cos(2θ ) E 0 (ξ 2 ) − 2(r 2 − R 2 )(cos(2θ ) + 3)K 0 (ξ 2 ) 3rR . (15d)
Implementation of the regularization routines
For L2 regularization in real space we employ a singular value composition and the routines "tikhonov" and "l_curve" provided by the Matlab package "Regularization Tools". 9 The use of this package for TFM has been described earlier. 10,11 To perform L1-and EN regularization we minimize well-established formulas 12 using the convex optimization package CVX. 13,14 The target functions for L1 regularization and EN regularization are given bŷ
f = argmin f f T M T Mf − 2u T Mf + f T f + λ 1 f 1 ,(16)
f = argmin
f f T M T M + λ 2 I 1 + λ 2 f − 2u T Mf + λ 1 f 1 .(17)
4/17
Comparison of the two formulas shows that EN regularization is a stabilized version of the L1 regularization. Below, we provide a short Matlab code for L1-and EN regularization. c v x _ e n d ; end
L1 regularization using Iterative Reweighted Least Squares (IRLS)
The L1 regularization problemf = argmin f Mf − u 2 + λ 1 f 1 can be solved with different approaches. As an alternative to the popular convex optimization, we tested the iteratively reweighted least squares algorithm (IRLS). [15][16][17] This algorithm approximates the result at every iteration i as
f i+1 = argmin f i+1 2n ∑ k=1 2m ∑ l=1 M lk f i+1 k − u l 2 + λ 1 2 2n ∑ k=1 | f i+1 k | 2 | f i k | .(18)
As initial condition for this scheme we chose f i=0 = 1. The IRLS can be seen as a reweighted, iterative L2 scheme. In our tests, this method exhibited favorable properties with less pronounced over-estimation of local traction.
Implementation of Bayesian TFM routines
The code for Bayesian TFM includes standardization of the data, finding the optimal regularization parameters, solving the regularized problem, and finally undoing the standardization. For regularization in BL2 and ABL2 we employ the Matlab package "Regularization Tools" that requires a singular value decomposition of the problem. 9 Below, we provide an exemplary code for BL2. (c) Same data as before, but regularization done with IRLS. For all samples, the regularization parameter should be chosen well above the turning point of the L-curve to avoid partial suppression of traction patterns. In our experience, the IRLS produces more accurate results than the L1 regularization with CVX.
e ( f ' * ( X' * X+ lambda2 * R ) / ( 1 + lambda2 ) * f −2 * u ' * X * f + lambda1 * ( norm ( f , 1 ) ) ) ;
f u n c t i o n F = r e c o n s t r u c t i o n _ B L 2 (X, u , n o i s e _ u ) % T h i s c o d e r e q u i r e s t h e M a t l a b p a c k a g e ' ' R e g u l a r i z a t i o n T o o lFigure S6 .
2S6t a = 1 / v a r ( n o i s e _ u ) ; % s t a n d a r d i z e t h e i n p u t d a t a s d = s t d (X ) ; X= (X−r e p m a t ( mean (X, 1 ) , s i z e (X , 1 ) , 1 ) ) . / r e p m a t ( s t d (X) , s i z e (X , 1 ) , 1 ) ; u = u−mean ( u ) ; p , beta , U, s , V, u , C , X, aa ,XX ) ;middle_down = m i d d l e − 0 . 5 * s t e p _ s i z e ; e v i d e n c e _ d o w n = l o g e v i d e n c e ( middle_down , beta , U, s , V, u , C , X, aa ,XX ) ; ] = t i k h o n o v (U, s , V, fu , f l a m b d a ) ; % c a l c u l a t e l o g ( d e t ( A ) ) A = f a l p h a * fC+ f b e t a * XX; L = c h o l (A ) ; Comparison of results of L1 regularization using CVX and IRLS for different λ 1 . See Fig. 3 of the main text. (a) Artificial data with traction spots having all the same magnitude. Displacements are corrupted with 2% noise. (a-i) The L-curve exhibits a turning point indicating a transition from a data-dominated to a regularization-dominated regime. (a-iii)-(a-v) Traction fields obtained for λ A 1 , λ B 1 and λ C 1 . (b) Artificial data with traction spots having different magnitudes and 2% displacement noise. (b-iii)-(b-v) Traction reconstruction with CVX.
Figure S14 .
S14Directed acyclic graphs representing the complex Bayesian models (BCSL, BL and BEN) that have been tested with this work. Some test results using these methods for TFM are shown in the supplementaryFig. S8.
Figure S1. Classical methods for selecting the regularization parameter λ 2 with the L-curve and GCV for strong noise σ n /σū 0.85 The L2 regularization parameters suggested by the L-curve criterion and the GCV differ considerably, about by a factor of ten. Data is artificial and consists of 15 circulars traction spots.
Traction microscopy to identify force modulation in subresolution adhesions. S J Han, Y Oak, A Groisman, G Danuser, Nat. Methods. 12Han, S. J., Oak, Y., Groisman, A. & Danuser, G. Traction microscopy to identify force modulation in subresolution adhesions. Nat. Methods 12, 653-656 (2015).
. L D Landau, E Lifshitz, Theory of elasticity. 7109Course Theor. Phys.Landau, L. D. & Lifshitz, E. Theory of elasticity, vol. 7. Course Theor. Phys. 3, 109 (1986).
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. P C Hansen, 46Regularization tools version 4.0 for matlab 7.3. Numer. algorithmsHansen, P. C. Regularization tools version 4.0 for matlab 7.3. Numer. algorithms 46, 189-194 (2007).
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Regularization and variable selection via the elastic net. H Zou, T Hastie, J. R. Stat. Soc. Ser. B Stat. Methodol. 67Zou, H. & Hastie, T. Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B Stat. Methodol. 67, 301-320 (2005).
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"Chemically non-perturbing SERS detection of catalytic reaction with black silicon"
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"E Mitsai \nInstitute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia\n",
"A Kuchmizhak \nInstitute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia\n\nSchool of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia\n",
"E Pustovalov \nSchool of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia\n",
"A Sergeev \nInstitute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia\n\nSchool of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia\n",
"A Mironenko \nInstitute of Chemistry\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia\n",
"S Bratskaya \nSchool of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia\n\nInstitute of Chemistry\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia\n",
"D P Linklater \nJohn st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia\n\nMelbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia\n",
"A Balčytis \nJohn st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia\n\nMelbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia\n",
"E Ivanova \nJohn st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia\n",
"S Juodkazis \nJohn st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia\n\nMelbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia\n"
] | [
"Institute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia",
"Institute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia",
"School of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia",
"School of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia",
"Institute of Automation and Control Processes\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia",
"School of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia",
"Institute of Chemistry\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia",
"School of Natural Sciences\nFar Eastern Federal University\nVladivostokRussia",
"Institute of Chemistry\nFar Eastern Branch\nRussian Academy of Sciences\n690041VladivostokRussia",
"John st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia",
"Melbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia",
"John st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia",
"Melbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia",
"John st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia",
"John st\nSwinburne University of Technology\n3122Hawthorn, VictoriaAustralia",
"Melbourne Centre for Nanofabrication\nANFF\n151 Wellington Road3168ClaytonVICAustralia"
] | [] | All-dielectric resonant micro-and nano-structures made of the high-index dielectrics recently emerge as a promising surface-enhanced Raman scattering (SERS) platform which can complement or potentially replace the metal-based counterparts in routine sensing measurements. These unique structures combine the highly-tunable optical response and high field enhancement with the noninvasiveness, i.e., chemically non-perturbing the analyte, simple chemical modification and recyclability. Meanwhile, the commercially competitive fabrication technologies for mass production of such structures are still missing. Here, we attest a chemically inert black silicon (b-Si) substrate consisting of randomly-arranged spiky Mie resonators for a true non-invasive (chemically non-perturbing) SERS identification of the molecular fingerprints at low concentrations. Based on comparative insitu SERS tracking of the para-aminothiophenol (PATP)-to-4,4' dimercaptoazobenzene (DMAB) catalytic conversion on the bare and metal-coated b-Si, we justify applicability of the metal-free b-Si for the ultra-sensitive non-invasive SERS detection at concentration level as low as 10 −6 M. We perform supporting finite-difference time-domain (FDTD) calculations to reveal the electromagnetic enhancement provided by an isolated spiky Si resonators in the visible spectral range. Additional comparative SERS studies of the PATP-to-DMAB conversion performed with a chemically active bare black copper oxide (b-CuO) substrate as well as SERS detection of the slow daylight-driven PATP-to-DAMP catalytic conversion in the aqueous methanol solution loaded with colloidal silver nanoparticles (Ag NPs) confirm the non-invasive SERS performance of the all-dielectric crystalline b-Si substrate. Proposed SERS substrate can be fabricated using easy-to-implement scalable technology of plasma etching amenable on substrate areas over 10 × 10 cm 2 making such inexpensive all-dielectric substrates promising for routine SERS applications, where the non-invasiveness is of mandatory importance. | 10.1039/c8nr02123f | [
"https://arxiv.org/pdf/1802.09783v1.pdf"
] | 21,699,495 | 1802.09783 | e2883f4a260b13194c57cd0ea49ee88d6efb0945 |
Chemically non-perturbing SERS detection of catalytic reaction with black silicon
E Mitsai
Institute of Automation and Control Processes
Far Eastern Branch
Russian Academy of Sciences
690041VladivostokRussia
A Kuchmizhak
Institute of Automation and Control Processes
Far Eastern Branch
Russian Academy of Sciences
690041VladivostokRussia
School of Natural Sciences
Far Eastern Federal University
VladivostokRussia
E Pustovalov
School of Natural Sciences
Far Eastern Federal University
VladivostokRussia
A Sergeev
Institute of Automation and Control Processes
Far Eastern Branch
Russian Academy of Sciences
690041VladivostokRussia
School of Natural Sciences
Far Eastern Federal University
VladivostokRussia
A Mironenko
Institute of Chemistry
Far Eastern Branch
Russian Academy of Sciences
690041VladivostokRussia
S Bratskaya
School of Natural Sciences
Far Eastern Federal University
VladivostokRussia
Institute of Chemistry
Far Eastern Branch
Russian Academy of Sciences
690041VladivostokRussia
D P Linklater
John st
Swinburne University of Technology
3122Hawthorn, VictoriaAustralia
Melbourne Centre for Nanofabrication
ANFF
151 Wellington Road3168ClaytonVICAustralia
A Balčytis
John st
Swinburne University of Technology
3122Hawthorn, VictoriaAustralia
Melbourne Centre for Nanofabrication
ANFF
151 Wellington Road3168ClaytonVICAustralia
E Ivanova
John st
Swinburne University of Technology
3122Hawthorn, VictoriaAustralia
S Juodkazis
John st
Swinburne University of Technology
3122Hawthorn, VictoriaAustralia
Melbourne Centre for Nanofabrication
ANFF
151 Wellington Road3168ClaytonVICAustralia
Chemically non-perturbing SERS detection of catalytic reaction with black silicon
(Dated: February 28, 2018)PACS numbers:
All-dielectric resonant micro-and nano-structures made of the high-index dielectrics recently emerge as a promising surface-enhanced Raman scattering (SERS) platform which can complement or potentially replace the metal-based counterparts in routine sensing measurements. These unique structures combine the highly-tunable optical response and high field enhancement with the noninvasiveness, i.e., chemically non-perturbing the analyte, simple chemical modification and recyclability. Meanwhile, the commercially competitive fabrication technologies for mass production of such structures are still missing. Here, we attest a chemically inert black silicon (b-Si) substrate consisting of randomly-arranged spiky Mie resonators for a true non-invasive (chemically non-perturbing) SERS identification of the molecular fingerprints at low concentrations. Based on comparative insitu SERS tracking of the para-aminothiophenol (PATP)-to-4,4' dimercaptoazobenzene (DMAB) catalytic conversion on the bare and metal-coated b-Si, we justify applicability of the metal-free b-Si for the ultra-sensitive non-invasive SERS detection at concentration level as low as 10 −6 M. We perform supporting finite-difference time-domain (FDTD) calculations to reveal the electromagnetic enhancement provided by an isolated spiky Si resonators in the visible spectral range. Additional comparative SERS studies of the PATP-to-DMAB conversion performed with a chemically active bare black copper oxide (b-CuO) substrate as well as SERS detection of the slow daylight-driven PATP-to-DAMP catalytic conversion in the aqueous methanol solution loaded with colloidal silver nanoparticles (Ag NPs) confirm the non-invasive SERS performance of the all-dielectric crystalline b-Si substrate. Proposed SERS substrate can be fabricated using easy-to-implement scalable technology of plasma etching amenable on substrate areas over 10 × 10 cm 2 making such inexpensive all-dielectric substrates promising for routine SERS applications, where the non-invasiveness is of mandatory importance.
I. INTRODUCTION
For decades, the surface-enhanced Raman scattering (SERS) effect mainly based on the augmentation of the inelastic scattering of the probed molecules placed in the vicinity of the electromagnetic "hot spots" is considered as a non-invasive characterization technique allowing label-free quantitative identification of the vibrational molecular fingerprints, even at single-molecule level [1][2][3][4][5]. To create such hot spots, the plasmon-active nanostructured substrates are widely used, allowing multi-fold enhancement of the Raman yield via electromagnetic effect as well as additional enhancement via the modification of the molecule electronic structure (a charge transfer).
Fast development in design of the SERS-active substrates have led to the growing interest in their applications for in-situ studies of the catalytic reactions allowing identification and quantification of the intermediates, transition states and final products [6]. Aside from such challenges as the chemical and thermal sta- * [email protected] bility of the plasmon-active SERS substrates, their inert (non-invasive) character has been recently put into question after the demonstration of the spontaneous conversion of para-aminothiophenol (PATP) to 4,4'dimercaptoazobenzene (DMAB) on the silver-and goldcoated substrates during SERS measurements [7][8][9][10]. Multiple efforts were undertaken to study the PATP, particularly owing to its affinity to the most-common plasmonic-active metals as well as appearance of the several characteristic Raman bands in its SERS spectra [8,[11][12][13][14][15][16]. These bands were initially assigned to the b 2 -type non-totally symmetric vibrations of the PATP molecule emerging in the detected spectra via the charge transfer (or chemical enhancement) mechanism [11,14,15]. However, more recent studies assigned those bands to the dimercaptoazobenzene (DMAB) molecule produced from PATP via a plasmon-driven catalytic oxidation [8]. This shows that for certain molecules, SERS is not always a non-invasive technique veiling the exact information on the vibrational fingerprints of the initial molecules deposited onto the plasmon-active SERS substrate. In this respect, the search for the chemically inert SERS substrates which can overcome this limitation allowing for a true non-invasive bio-identification with good sensitivity is of mandatory importance.
Recently, all-dielectric resonant micro-and nanostructures made of the high-index dielectrics (Si, Ge, GaAs, etc.) emerge as a promising platform which can complement or potentially replace the metal-based counterparts in routine SERS measurements [17][18][19][20]. The main advantages of such structures are their low invasiveness, reproducibility, tunability of optical response, recyclability, as well as wide feasibilities for surface functionalization via covalent binding of selective receptors to provide hostguest molecular recognition [21,22]. Generally, the isolated or ordered arrays of the high-index resonant structures supporting various types of Mie resonances were suggested for SERS measurements [23][24][25][26]. Noteworthy, fabrication of such structures is challenging as it involves costly, non-scalable and time-consuming electron-or ionbeam milling techniques making all-dielectric resonators too expensive for routine biosensing experiments. Laserinduced forward transfer technique allows rapid fabrication of all-dielectric resonant structures via pulse-laser melting of the donor substrate resulting in ejection of the spherical-shape drops which can be transfered onto a receiving substrate [27,28]. Meanwhile, such laserassisted fabrication of large-scale commercially competitive substrate still requires rather long fabrication cycles and complicated alignment (adjustment) procedures, while the shape of the produced resonant structures can be controllably tuned only in a rather narrow range of parameters.
In this study, we demonstrate an application of a chemically inert black-Si (b-Si) substrate consisting of randomly-arranged spiky Mie resonators for a true noninvasive SERS identification of the molecular fingerprints at low concentrations. Based on comparative in-situ SERS tracking of the PATP-to-DMAB catalytic conversion on the bare and metal-coated b-Si, we justify applicability of the metal-free b-Si for the ultra-sensitive non-invasive SERS detection at concentration level as low as 10 −6 . Supporting finite-difference time-domain (FDTD) calculations were undertaken to show the origin of the electromagnetic enhancement of the isolated spiky Si resonators in the visible spectral range. By detecting the PATP-to-DMAB conversion on a chemically active bare black copper oxide (b-CuO) substrate as well as a slow-rate conversion photo-catalyzed by colloidal silver in an aqueous methanol solution on b-Si we further confirm the chemically non-perturbing character of SERS performance of the all-dielectric crystalline substrate.
II. METHODS
Substrate fabrication. The b-Si substrates were fabricated following a simple reactive ion etching procedure in O 2 and SF 6 gas mixture [29][30][31]. The b-CuO substrates are made by chemical oxidation of industrial grade copper foil (Flewsolutions Pty) using earlier developed processing steps [32].
Finally, gold and silver films of variable thicknesses ranging from 20 to 200 nm are coated onto the b-Si and b-CuO substrates using e-beam evaporation procedure. All films are deposited at a constant deposition rate of 1 nm/s while rotating the sample holder to ensure uniform deposition. SERS measurements on the silvercoated substrates were performed within 24 hours after evaporation to avoid silver oxidation.
Substrate characterization. The nano-topography of the resulting bare and metal-coated blackend substrates are carefully inspected by the scanning electron microscopy (SEM, Carl Zeiss Ultra55+). Additionally, cross-section cuts of the bare and metal-capped blackend surfaces of both types are prepared to check the vertical size of the surface structures. To do this, the focused ionbeam milling (FIB, Carl Zeiss CrossBeam 1540) is utilized with a Ga + -ion beam at 30 kV and current ranging from 50 to 200 pA. Higher ion-beam currents are applied to produce the initial cut, while a high-frequency beam scanning at low current was utilized to minimize the curtain effect on the cross sectional cut.
Reflection coefficient of both, the bare blackend substrates in the visible and near IR spectral ranges is measured using an optical integrating sphere spectrometer (Varian, Cary 5000).
FDTD calculations. Enhancement and local structure of the electromagnetic fields near the isolated tips of variable size on the b-Si surface are accessed using finitedifference time-domain modeling (FDTD, Lumerical Solutions package). The modeled geometry is extracted from the corresponding SEM images of the produced FIB cuts. The dielectric function of Si is modeled using the experimental library data [33]. The size of the square unit cell is 0.5 × 0.5 × 0.5 nm 3 , while the computational volume is limited by perfectly matched boundaries. The linearly-polarized Gaussian light source with the lateral size fitting the size of the experimental optical spot is used to excite the structure. The total-field scattered field source of the same size is applied for calculations of the scattering spectra (for details, see supporting material; Fig. S2).
Raman spectroscopy. A commercial Raman apparatus (Alpha, WiTec) equipped with a 532-nm CW semiconductor laser is used for in-situ Raman detection of the PATP-to-DMAB conversion. The linearly-polarized radiation is focused by an objective lens (100x, Carl Zeiss) with a numerical aperture (N A) of 0.9, providing the excitation of the local section (≈0.41 µm 2 ) of the surface. The focal depth of the lens, which can be approximately estimated as a doubled Rayleigh length in air ≈λ(NA −2 )=1.3 µm ensures the uniform irradiation of the high-aspect-ratio textures of the used blackend surfaces. The incident power is tuned using a built-in attenuator. The Raman signal from the adsorbed molecules within the focal volume is collected with the same lens and analyzed using a grating-type spectrometer (1800 lines/mm) equipped with an electrically-cooled CCD camera.
In this study, the PATP is used as a model adsorbate owing to its strong SERS response, simple structure and affinity to the most common plasmonic materials as silver and gold used in this study. Moreover, these molecules represent the perfect probe to study the enhancement mechanisms in SERS. The drops of the alcoholic solution of the PATP at its molar concentration of 10 −6 M are deposited onto the surface of the SERS substrates using the home-build drop deposition system and Raman measurements are conducted after complete evaporation of the drop. Additionally, the bare or silver-capped b-Si substrates are attested for in situ detection of the PATP-to-DMAB conversion in water/methanol solution (1/1, v/v) at the PATP molar concentration of 5·10 −2 M. In this experiment, SERS substrates are immersed into a quartz cell filled with the above mentioned solution, where slow PATP-to-DMAB conversion is stimulated via surface plasmon resonance (SPR)-assisted catalysis by colloidal silver nanoparticles (Ag NPs) at PATP/Ag molar ratio 500/1. Spherical-shape Ag NPs having the diameter ranging from 5 to 10 nm are synthesized via citrate reduction method and purified by dialysis (see details in the Supporting information). For SERS studies, the laser radiation is focused onto the substrate surface through the solution with long-working-distance lens of N A = 0.25 exciting the local section of ≈1.3 µm 2 ) of the b-Si surface, while similar lens and spectrometer are used to analyze the Raman response. The position of the focal spot is calibrated by maximizing yield from the Raman band of the crystalline Si. The sealed quartz cells with the substrate and solutions are maintained under daylight exposure between the SERS measurements.
III. RESULTS AND DISCUSSIONS
As mentioned, the surface of the b-Si contains randomly-arranged vertically oriented spikes with the height of about 600±150 nm ( Fig.1(a)), tip curvature radius smaller than 10 nm and an average density of ≈60 spikes/µm 2 . Non-uniform coverage of the b-Si with the deposited Ag (or Au) film produces even more textured surface with strong plasmonic response and high SERS yield ( Fig. 1(b)). In a sharp contrast, b-CuO surface has substantially more chaotic orientation of the textures with respect to the surface normal, while their average height exceeds 1 µm with the broad deviation of sizes in a lateral dimension (Fig. 1(c)). The deposition of the metal film results in formation of the metal flakes on the textured surface ( Fig. 1(d)). Noteworthy, the density of the spikes is similar to those for b-Si, while the SERS yield is also comparable, as shown in previous studies [32,34,35]. Meanwhile, the density of the surface textures both substrate ensure that several (≈10-20) randomly arranged spiky nano-features will be always within the laser irradiated spot (see Methods for details). Such dense surface structure allows both substrates to almost perfectly absorb the radiation in the visible spectral range as it is illustrated by the measurements of the reflection coefficient ( Fig. 1(e)). Specifically, for the 532-nm laser wavelength used further for Raman spectroscopy, the reflection coefficient as small as 2.1% is found for the bare b-CuO, dropping down to 1.3% -for the black Si. The nano-texture is acting as a gradient refractive index region and acts as an anti-reflection coating.
For the probed spectral region defined by the spectrometer grating, only two distinct Raman bands at 1587 and 1078 cm −1 (marked in blue; Fig.2(a, left)) which are the νCS(a1) and νCC(a1) vibration modes of the PATP molecules [11,36] capping the surface of the bare b-Si are distinguishable. Time-resolved control SERS measurements show that no substantial changes of the relative intensity of the both identified modes occur within several minutes of measurements ( Fig. 2(a,left)). Noteworthy, even a fifty-fold increase of the irradiation intensity or longer accumulation time do not allow detection of the PATP on the surface of the smooth crystalline Si substrate (the upper-most curve in Fig. 2(a, left)), indicating the pronounced SERS effect for the bare b-Si. For the both PATP Raman bands, the non-metallic b-Si substrate provides an averaged enhancement factor of ≈10 3 as compared to the smooth crystalline Si surface. This is larger by more than an order of magnitude considering the effect of an increased surface area.
Particularly, this enhancement can be attributed to the more efficient absorption of laser radiation (decreased reflectivity) as well as to the enhanced electromagnetic fields near the cone-shaped Mie resonators randomly distributed across the b-Si surface. Supporting FDTD calculations are performed to access the enhancement of the electromagnetic (EM) field near the isolated b-Si spiky resonators of variable sizes (see Methods for details). The calculated enhancement of the normalized squared field amplitude (E) 2 /(E 0 ) 2 as high as 25 is found on the surface of metal-free spiky Si resonators, sufficiently high to boost the SERS yield, which is proportional to the forth power of the EM amplitude, via the field enhancement mechanism (see Fig. 2(b)). Indeed, the position of the EM hot spots on the surface of the spiky tip as well as the maximal amplitude are size-dependent ( Fig. 2(b)). Meanwhile, for the produced b-Si with an averaged height of 550±150 nm, the squared EM-field amplitude never drops below 15. Noteworthy, according to our calculations the field enhancement inside the tip reaches (E) 2 /(E 0 ) 2 ≈12 for the certain tipped structures, indicating that each isolated structure can act as a typical all-dielectric resonator [19,37,38], which can support various types of modes (see the Fig. S2 in the Supporting information). In this respect, the b-Si substrate can be considered as a disordered array of the densely packed spiky Mie resonators. As the distance between the neighboring structures is rather small (see Fig. 1(a,b)), the enhancement of the EM field in their gaps can also boost the SERS yield. High density of the structures poten- tially provide broadband excitation in the visible spectral range ensuring that for the fixed size deviation of the Si resonators on the surface, at least several of them will be resonantly excited by the visible laser radiation, even under tight focusing conditions.
In addition to the substantially enhanced EM field near the surface of such dielectric resonators, use of crystalline materials gives an additional modality to the chemical SERS measurements allowing in situ detection of the local temperature distribution via tracking the spectral shape and position of the temperature-dependent c-Si Raman band [39,40], which can be resonantly excited within the Si resonator [26,41]. In the range of applied intensities used for Raman measurements, we did not observe any substantial variations of the shape and position of the c-Si band (see Fig. S3 in the Supporting information) indicating potentially weak contribution of the temperature effect to the obtained SERS data. This is consistent with a recently reported laser trapping/pinning of micro-beads on bare b-Si at high light intensities which usually creates strong temperature gradients and convection flows on plasmonic metal coated substrates [42].
In a sharp contrast, for b-Si coated with a 100-nm thick Ag film (Ag/b-Si) several evident Raman bands at 1140, 1185, 1385 and 1432 cm −1 appear in the measured spectra (green-color areas in Fig. 2(a, right)). Time-resolved SERS measurements show that after first 10 seconds of laser irradiation of molecules on the substrate, only the low-intensity bands assigned to PATP can be identified. The appearance and continuous increase in intensity of all the bands attributed to the DMAB can be seen for longer irradiation times presumably indicating the monotonous conversion of the PATP to DMAB via a plasmon-driven catalytic reaction. It should be stressed that the bare b-Si substrate expectedly provides about 2 orders of magnitude lower SERS yield for the PATP bands comparing to the metal-coated one. This presum-ably is due to the absence of the chemical SERS enhancement mechanisms proceeding via the chemical interactions at the substrate-analyte interface. Meanwhile, for such metal-free chemically inert substrate, SERS identification of the adsorbed analyte molecules has true noninvasive character.
The relative intensity ratio of the Raman bands at 1432 and 1587 cm −1 is typically used to access the PATPto-DMAB conversion rate [14,16]. However, both the monomer and dimer molecules contribute to the total intensity of the Raman band at 1587 cm −1 complicating the analysis of the conversion kinetics and total yield. In this study, without loss of generality, we use the time evolution of the normalized intensity of the DMAB Raman band at 1432 cm −1 to preliminary access the PATPto-DMAB conversion, assuming that the concentration of DMAB increases to a certain maximum value within the laser-exposed volume for the used substrate. Figure 2(c) shows the average conversion plotted against the laser irradiation time of the PATP molecules deposited onto the Ag/b-Si substrate for various thicknesses of metal coating and irradiation intensities of 13.1 and 120 µW/µm 2 . At low intensity level, the monotonous growth of the DMAB signal is observed for all Ag/b-Si substrates within the first 50-90 seconds of the irradiation cycle (colored solid curves in Fig. 2(c)). Upon reaching maximum, the DMAB Raman band intensity decreases due to photo-degradation of the analyte and a growing contribution of luminescence background from newly formed complexes. This process is illustrated as the grey-color areas in Fig. 2(c). The transition between two regimes indicates the end of the PATP-to-DMAB conversion within the exposed volume having a fixed number of the initial PATP molecules. Meanwhile, almost 10-fold higher intensity of 120 µW/µm 2 of the laser radiation results in complete conversion of the PATP-to-DMAB within the first 10 seconds of the laser exposure (dashed curves in Fig.2(c)).Noteworthy, the higher Raman yield for the characteristic PATP (DMAB) bands, as well as faster reaction and degradation rates correspond to thicker Ag films, indicating the key role of metal in the SERS enhancement, PATP-to-DMAB conversion and photo-degradation processes [43].
The obtained results are further analyzed by fitting the initial part of the intensity evolution of the DMAB Raman band at 1432 cm −1 (see Fig. 2(c)) and normalizing the obtained line slope on the incident laser intensity. In this way, we estimated the average PATP-to-DMAB conversion rate for various types of the applied film thickness, summarizing the obtained trends in Fig. 3(a). Noteworthy, SERS enhancement of PATP (DMAB) bands measured on the Au-coated b-Si is almost order of magnitude weaker, which can be explained in terms of more favorable EM enhancement under 532-nm excitation of the Ag-coated b-Si. Additionally, the chemically active Ag coating provides faster PATP-to-DMAB conversion compared to the Au coating of the same thickness (see also Fig. S4 in Supporting information) [44]. higher reactivity of the silver toward the hydrogen sulfide [45] and its derivatives, responsible for easier formation of a charge transfer complex and chemical amplification, it have been reported that the formation of DMAB on the Ag surfaces is much easier under the same experimental condition than that on Au ones, considering an easier formation and a higher activity of triplet oxygen molecules ( 3 O 2 ) on the Ag surfaces [46,47]. Furthermore, with the activation of 3 O 2 species driven by plasmon resonances, the 514.5 nm excitation was shown to be more efficient for DMAB formation on the Ag surfaces, whereas, the 632.8 nm excitation was preferred on the Au ones [47].
In a sharp contrast to the chemically stable bare b-Si, the b-CuO which surface can contain the chemically active metallic sites, as it was confirmed by previous XPS analysis [32]. This would allow the PATP molecules to be converted to the DMAB under laser exposure even without any metal coating. It should be stressed, the copper oxide have been shown to be a catalyst for the amine oxidation [48]. In our experiments, for bare b-CuO we found rather slow PATP-to-DMAB conversion rate (see three bottom curves in Fig. 3(b)), while the coverage of the b-CuO with the 100-nm thick Ag film provides the pronounced SERS yield with an average conversion rate of 750 (s×W) −1 (four upper spectra in Fig. 3(b)).
Although, the bare b-CuO demonstrates slightly better SERS performance being compared to the bare b-Si substrate, the latter provides the way for non-invasive SERS registration with temperature feedback modality and moderate SERS yield allowing non-perturbing identification of the molecule fingerprints at relatively low concentration. To further illustrate this remarkable feature of the bare b-Si, we studied its applicability for noninvasive detection of the slow daylight-driven PATP-to-DAMP catalytic conversion in the aqueous methanol solution loaded with the Ag NPs (see Methods for details).
For pure solution of the PATP (without Ag NPs), corresponding time-resolved SERS spectra detected near the b-Si surface demonstrate only two Raman bands attributed to the monomer molecules even within rather long period of 4 days under daylight exposure ( Fig. 4(a)). It should be noted, for the same solution silver-coated (100-nm thick) b-Si converts the PATP-to-DMAB within the first minute of irradiation maintaining the conversion rate similar to those obtained for the dried layer (see Fig. S5 in the Supporting information). This makes the metal-coated substrate non-applicable for SERS detection of the considered model reaction in solutions, as the sensing surface is covered by the PATP layer owing to strong affinity of the PATP to the plasmon-active metals. Such SERS measurements obtained from this nearsurface PATP layer veil the exact information on the slow conversion kinetics proceeding in the solution.
Finally, to stimulate the slow daylight-driven PATPto-DMAB catalytic conversion, the Ag catalyst was added to the PATP solution and the same time-resolved measurements were performed on the bare b-Si substrate immersed into the solution (see Methods for details). These measurements show that only after 4 days of daylight exposure a clear fingerprint of DMAB molecules appeared in the measured SERS spectra (Fig. 4(b)). Hence, the bare b-Si can be considered as a substrate allowing chemically non-perturbing study of the catalytic kinetics in the solution.
IV. CONCLUSIONS AND OUTLOOK
To conclude, we successfully demonstrate the noninvasive SERS identification of the PATP molecules with non-metallic b-Si surface at concentration as low as 10 −6 M and an average SERS enhancement of 10 3 . The enhancement can be attributed to the efficient absorption of the laser radiation by anti-reflective b-Si as well as enhancement of the electromagnetic field near the spiky nano-features acting as Mie resonators, as it was confirmed by the supporting FDTD calculations. Thermal effects can be monitored by observation of 521 cm −1 crystalline Si vibration mode and adds additional value for precise monitoring of (photo-)catalytic reactions on b-Si substrates. More importantly, comparative in situ SERS tracking of the PATP-to-DMAB catalytic conversion on the Ag-and Au-capped b-Si as well as bare and metal-coated b-CuO confirms the chemically nonperturbing character of the non-metallic all-dielectric crystalline substrate. This remarkable feature allows to use such substrates for non-invasive SERS tracing of the catalytic processes in the solution, which was demonstrated by detecting slow PATP-to-DMAB conversion catalyzed by the colloidal silver nanoparticles under daylight exposure in the aqueous methanol solution.
Mie resonances explored here for the chemically nonperturbing SERS detection can be also exploited in agglomerated networks of nanoparticles which provided detection of atmospheric oxygen and nitrogen as well as polymer on the laser ablated surfaces (without metal coating) [49]. Here demonstrated, a single-point SERS measurement technique on dielectric surfaces can be combined with a surface mapping which provides statistical distribution of the hot-spots over the surface and can be directly linked to the analyte concentration [50]. Quantitative detection of analytes in solution is a required key technology of bio-medical and environmental sensing.
ACKNOWLEDGMENTS
The work was partially supported by the FEBRAS Program for Basic Research "Far East" (projects Nos. 18-3-002, 18-3-012). A.K. acknowledges the partial support from Russian Foundation for Basic Research (17-02-00571-a) and RF Ministry of Science and Education (Contract No. MK-3287.2017.2) through the Grant of RF President. We are grateful to M. Larkins for b-CuO SERS test samples and discussions.
FIG. 1 .
1B-Si and b-CuO substrates and their optical properties. (a-d) False-color normal-view SEM images of the bare (a) and the Ag-coated b-Si (b), as well as the bare (c) and Ag-coated black b-CuO (d). Scale bar corresponds to 500 nm. Side-view SEM images (in the right) show cross-sectional FIB cuts of the corresponding substrates. (e) Reflection coefficient of the bare b-Si (purple) and b-CuO substrates (orange).
FIG. 2 .
2SERS detection of the PATP-to-DMAB conversion with bare and silver-coated b-Si. (a) Series of Raman SERS spectra of the PATP layer adsorbed on the bare (left) and Ag-coated (right) b-Si substrates. The total irradiation time of the PATP molecules is indicated near each spectrum, while the accumulation time for each spectrum is 10 s. The characteristic bands of the PATP at 1078 and 1587 cm −1 are highlighted by the blue-color areas, while four specific bands attributed to DMAB are highlighted by green. The Raman band at 1432 cm −1 used to access the conversion are marked with the red circle. Each spectrum is shifted on the vertical axis for better displaying. (b) Squared electric-field amplitude inside and near the isolated spiky Si resonators of variable height ranging from 400 to 600 nm calculated at 532 nm. (c) PATP-to-DMAB conversion versus laser irradiation time. The conversion is defined as a normalized intensity of the DMAB characteristic Raman band at 1432 cm −1 measured for the b-Si coated with 20-(green), 100-(green) and 200-nm thick (purple) Ag film (blue) at 13.1 µW/µm 2 . Similar conversion measured at 120 µW/µm 2 for b-Si coated with 20-nm-thick Ag film is shown by the dashed curve. Grey-color area indicate the photo-degradation of the analyte.
PATP-to-DMAB conversion rate for Ag-and Au-coated b-Si and b-CuO substrates. (b) Series of SERS spectra of the PATP layer adsorbed on the bare and Ag-coated b-CuO substrates.
FIG. 4 .
4Non-invasive SERS detection of the PATP-to-DMAB daylight-driven catalytic conversion in the aqueous methanol solution. Time-resolved SERS spectra of the PATP at molar concentration of 5×10 −2 dissolved in pure (a,b) and Ag NP-loaded aqueous methanol solution (c). For these measurements, bare (a,c) and Ag-coated (b) b-Si substrates were used. Detection procedure is schematically illustrated in the insets. The accumulation times for each spectrum are 30 (a,c) and 5 s (b). For measurements with bare b-Si substrates, both solutions were maintained under daylight irradiation, while the total incubation time is indicated near each spectrum.
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| [] |
[
"Can a Dove prism change the past of a single photon?",
"Can a Dove prism change the past of a single photon?"
] | [
"Miguel A Alonso \nThe Institute of Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA\n\nCenter for Coherence and Quantum Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA\n",
"Andrew N Jordan \nCenter for Coherence and Quantum Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA\n\nDepartment of Physics and Astronomy\nUniversity of Rochester\n14627RochesterNew YorkUSA\n\nInstitute for Quantum Studies\nChapman University\n1 University Drive92866OrangeCAUSA\n"
] | [
"The Institute of Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA",
"Center for Coherence and Quantum Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA",
"Center for Coherence and Quantum Optics\nUniversity of Rochester\n14627RochesterNew YorkUSA",
"Department of Physics and Astronomy\nUniversity of Rochester\n14627RochesterNew YorkUSA",
"Institute for Quantum Studies\nChapman University\n1 University Drive92866OrangeCAUSA"
] | [] | We reexamine the thought experiment and real experiment of Vaidman et al. [1,2], by placing Dove prisms in the nested Mach-Zehnder interferometer arms. In those previous works, the criterion of whether a single photon was present, or not, was the presence of a "weak trace", indicating the presence of a nonzero weak value. This was verified by slightly varying the mirror angle at a given frequency, which was then detected on a position sensitive detector at the oscillation frequency. We show the presence of the Dove prisms gives identical weak values everywhere to the previous configuration because the prisms change neither the path difference, nor the mode profile in the aligned case. Nevertheless, the same slight variations of the interferometer mirrors now give a signal at the first mirror of the nested interferometer. We can interpret this result as a misaligned optical interferometer, whose detailed response depends on the stability of the elements, or as the detector coupling to a nonzero effective weak value. | 10.1007/s40509-015-0044-8 | [
"https://arxiv.org/pdf/1501.01287v1.pdf"
] | 118,808,130 | 1501.01287 | e094a534135b3cc0b0da75a85279ee2270c16547 |
Can a Dove prism change the past of a single photon?
Miguel A Alonso
The Institute of Optics
University of Rochester
14627RochesterNew YorkUSA
Center for Coherence and Quantum Optics
University of Rochester
14627RochesterNew YorkUSA
Andrew N Jordan
Center for Coherence and Quantum Optics
University of Rochester
14627RochesterNew YorkUSA
Department of Physics and Astronomy
University of Rochester
14627RochesterNew YorkUSA
Institute for Quantum Studies
Chapman University
1 University Drive92866OrangeCAUSA
Can a Dove prism change the past of a single photon?
(Dated: January 6, 2015)
We reexamine the thought experiment and real experiment of Vaidman et al. [1,2], by placing Dove prisms in the nested Mach-Zehnder interferometer arms. In those previous works, the criterion of whether a single photon was present, or not, was the presence of a "weak trace", indicating the presence of a nonzero weak value. This was verified by slightly varying the mirror angle at a given frequency, which was then detected on a position sensitive detector at the oscillation frequency. We show the presence of the Dove prisms gives identical weak values everywhere to the previous configuration because the prisms change neither the path difference, nor the mode profile in the aligned case. Nevertheless, the same slight variations of the interferometer mirrors now give a signal at the first mirror of the nested interferometer. We can interpret this result as a misaligned optical interferometer, whose detailed response depends on the stability of the elements, or as the detector coupling to a nonzero effective weak value.
Recent papers by Lev Vaidman and collaborators have explored the possibility of inferring where a photon was in the past, based on a present detector outcome [1,2]. This continues the tradition of John Wheeler who proposed and carried out experiments to infer a photon's past trajectory, concluding from a detection event that it came by both paths or by a single path in a Mach-Zehnder interferometer, depending on whether a second beam splitter is placed in the interferometer or not. This decision can be made after the photon passes the first beam-splitter, leading to the photon's "delayed choice" of behaving like a particle or a wave [3]. Related interesting results, such as the "interaction free measurement" (the ability to detect an object's presence with a single photon by detecting an event that would have been impossible had the object been there [4]), involve "counterfactual" reasoning: If an event could have happened in a given situation, but did not (or its reverse), what can we infer about it?
Vaidman and collaborators have analyzed a Mach-Zehnder interferometer within a Mach-Zehnder interferometer. They argue that conditioned on certain detector clicks, one can infer the photon's past. This question does not fall within usual quantum mechanics, and different interpretations give different answers: Bohr says the question is not well posed -so don't ask it [5]; Wheeler says the photons can retro-actively change their past reality [3]; Bohm gives a definite trajectory [6], but it can be "surrealistic" [7], etc.
Vaidman has analyzed this situation within the Aharonov-Bergmann-Lebowitz two-state vector formalism [8], where there is a forward evolving wavefunction from the prepared single photon state, and a backward evolving wavefunction from the detection event. This analysis indicates something striking: the regions of overlap between these forward and backward evolving wavefunctions are nonzero in two regions -one is the outer arm of the interferometer, and the other is inside the in-ner interferometer (see Fig. 1). This result is interpreted as the photon's past (conditioned on both the pre-and post-selection) being in both the outer arm of the interferometer, as well as the inner arm of the interferometer -despite there being no connection between them! This interpretation has similarities to the transactional interpretation [9].
In order to test this idea, Vaidman has provided a criterion as to how to decide this question: the photon's past is determined by it leaving a "weak trace" behind. This can be checked experimentally by weakly measuring a projection operator on that arm of the interferometer to check the photon's presence or absence. This criterion also relies on ascribing reality to the weak valuethe presence of a non-zero weak value is interpreted as evidence that the photon was present. This is done in experiment [1], by slightly tilting in an oscillating fashion every mirror in the interferometer with a different frequency. By detecting the light on a split detector, the signal is then Fourier analyzed, and a peak at that frequency is defined as a weak trace. Figs. 1(a,b) show the setup used by Vaidman and colleagues, and illustrate the effect of slightly tilting some of the mirrors.
There have been several comments concerning Ref.
[1]. Refs. [10] points out that alignment is critical: for example, if the mirrors A and B are tilted together, their effect will cancel out, leaving no trace of their (combined) tilts on the detector. A related comments by Svensson criticises associating a (postselected) weak value in a misaligned interferometer to the particle position in an aligned one [11]. These interpretational questions have led to a dispute with Zubairy and colleagues, about the actual counterfactuality of a counterfactual communication proposal. In that dispute, Zubairy claims there can be a secure communication channel between two parties by having both parties select on events that can be interpreted as having no photon inside the communication channel -this also relies on making a claim about where arXiv:1501.01287v1 [quant-ph] 6 Jan 2015 When mirror E is tilted instead, the two legs going through mirrors A and B bend, but interfere without relative misalignment and cancel exactly before reaching mirror F. (c) When Dove prisms with different orientations are added to all legs, the robustness of the inner Mach-Zehnder interferometer to tilts of mirror E is broken. The blue lines indicate half-wave plates whose fast axes are at 45 • with respect to the interferometer plane (coming out of the plane on the side indicated by a circle), introduced for polarization correction. The inset in (c) shows that a Dove prism acts as a parity operator in one transverse direction but not in the other. the photon was in the past, given certain detection results [12]. Vaidman criticized that proposal because it is a more complicated realization of the nested interferometer [13]. Just as he claimed the photon was in the nested interferometer in the past, despite there being no way to enter it, so too, in the Zubairy proposal, he claims that photons are actually in the public communication channel, and the communication is therefore not counterfactual. This resulted in further debate [14,15].
The purpose of this paper is to further explore the concept of Vaidman's "weak trace" criteria in the modified set-up of Fig. 1(c). We show the experimental procedure of varying the mirrors to reveal the weak trace is not as simple as it appears. By adding Dove prisms to the inner interferometer, the weak values remain exactly the same, but the process of wiggling the interferometer mirrors reveals a new situation. There will now be a weak trace from mirror E, despite the fact the mirror E's weak value is 0. This fact motivates the title of the paper: If we follow the principle of the weak trace, we must conclude that the photon's presence or absence at mirror E depends on the presence or absence of the (later) Dove prisms.
Setup.-We begin our analysis with a review of the two-state vector formalism. The system is prepared in a state |Ψ and post-selected in state Φ|. The selected state is indicated by a detector click in the simplest realization. The pre-selected state is propagated forward in time, and the post-selected state is propagated backward in time to meet at a point in between -for us, this is in the past. The weak value [16,17] A w of any operator A at the intermediate point may be formally calculated to give,
A w = Φ|A|Ψ Φ|Ψ .
(1)
We now apply this formula to projection operators Π j , where j indicates the various points inside the interferometer. Using our conventions for the phase shifts acquired by the optical elements, the two-state vector is
Φ| |Ψ = 1 √ 3 ( A|+i B|− C|) 1 √ 3 (|A +i|B −|C ),(2)
which give the weak values,
Π A,w = 1, Π B,w = −1, Π C,w = 1, Π E,w = 0, Π F,w = 0.(3)
It is important to stress that these results are calculated for an aligned interferometer. In particular, all discussion of the transverse mode structure is suppressed because it is irrelevant -the weak value is a system quantity only.
Adding the Dove prisms in Fig. 1(c) does nothing to the above calculations. Let us see why: If the interferometer has a laser injected into it, prepared in a mode that is symmetric about the optical axis, the beam will be refracted when it enters the prism, bounce off the lower surface, and refract again when it leaves the prism. The resulting exiting mode is essentially identical to the one that entered, leaving no effect on the light beam. Consequently, propagating the system state either forward or backward through the Dove prism will be equivalent to free space propagation over a similar optical path length.
Suppose now that the beam is misaligned with the optical axis. Such misalignment can be introduced by slightly tilting one of the interferometer mirrors. The paths resulting from such misalignments are represented as red lines in Fig. 1. Let the transverse coordinates with respect to each aligned, unfolded path, be written as x (within the plane of the interferometer) and y (out of this plane). As can be seen in Fig. 1(c) and its inset, the effect of a Dove prism along the path that includes mirror A is to reflect the mode in x around the optical axis, as well as to reverse the transverse momentum in that direction, k x , while leaving y and k y unchanged. Notice that the longer path length and extra internal reflection introduced by this prism will give an extra phase to the photon. We compensate for this phase by putting other Dove prisms in the other two arms of the interferometer to rebalance the path lengths, but we orient these at right angles to the first, so that they act as parity operators in the y direction (i.e., out of the interferometer plane). This way, the paths resulting from mirror deflections in the interferometer plane will be essentially unchanged by these prisms, whereas out-of-plane deflections would be reflected about the optical axis. As we will now see, the insertion of the prisms changes the stability of the interferometer to mirror tilts.
Results.-The effect of the Dove prisms on the interferometer's sensitivity to tilts in the mirrors can be understood easily by thinking of the three unfolded paths that join the source with the detector: through mirrors E, A, and F (path EAF), through mirrors E, B, and F (path EBF), and through mirror C (path C). Tilting mirror j by a small angle α j /2 (defined as positive if clockwise for mirrors A, E, and F, which face down, and counterclockwise for mirrors B and C, which face up) tilts the reflected beam by α j and hence gives it a momentum kick by an amount k sin α j ≈ kα j , so that the effect can be modeled by a phase factor U j = exp(ikα j x). Free propagation has the effect of converting momentum kicks into spatial translations due to beam walk-off. The three field contributions at the detector plane, due to each of the three paths, are
ϕ EAF (x) ≈ 1 √ 3 ϕ(x − z E α E − z A α A − z F α F ) × exp[ik(α E + α A + α F )x],(4)ϕ EBF (x) ≈ − 1 √ 3 ϕ(x − z E α E − z B α B − z F α F ) × exp[ik(α E + α B + α F )x],(5)ϕ C (x) ≈ 1 √ 3 ϕ(x − z C α C ) exp(ikα C x),(6)
where ϕ(x) is the field profile for a beam traveling in free space over the same optical distance, and z j is the optical distance between mirror j and the detector. Note that we neglect phase factors common to all three paths, as well as the dependence in y. When the Dove prism is inserted, though, the first of these contribution suffers a change in the sign of α E given the parity flip:
ϕ (Dove) EAF (x) ≈ 1 √ 3 ϕ(x + z E α E − z A α A − z F α F ) × exp[ik(−α E + α A + α F )x].(7)
Note that polarization is not considered in the previous analysis, despite its important effect on the phase shifts each element imposes. Recall that, as in the original experiment [1], the initial and final beam splitters are polarizing ones, and that suitably-oriented polarizers (shown as black lines in Fig. 1) are used at the entrance and exit of the system to guarantee equal weighting of the three paths. The effect of polarization can be addressed by inserting six half-wave plates (indicated as blue lines in Fig. 1(c)), two along each leg, to guarantee that all equivalent optical elements see the same polarization and therefore all legs accummulate the same phase and interfere appropriately. This way, the previous analysis remains valid.
We now calculate the effect of slight tilts in the mirrors on the signal at the detector plane. Without the Dove prisms, the state at the detector is a sum of (4), (5), and (6). We assume only a symmetric initial meter mode. The expected shift of the photon position (the centroid of the beam) when the Dove prisms are not inserted is, in the small angle approximation, independent of tilts of mirrors E and F:
x = z A α A − z B α B + z C α C .(8)
When the Dove prisms are inserted, the contribution in (4) is replaced with that in (7), so that the centroid now depends on tilts on E, but still not on F:
x (Dove) = −2z E α E + z A α A − z B α B + z C α C .(9)
That is, the stronger dependence is now on α E , not only because of the largest numerical factor but because z E is larger than the distances from any other mirror to the detector. Note that the asymmetry between E and F is not due to the fact that the Dove prisms were inserted before mirrors A, B, and C; if they were to be placed right after these mirrors, the previous result would hold with only a change in the sign of α A . The signals detected by the split detector are proportional to the above centroids. We observe that the meter shift at the detector from mirror E is controlled not by the standard weak value Π E but by an effective weak value,
Π E,w = −2,(10)
originating from the reflection of the meter profile about the optical axis along path EAF.
Discussion.-When a mirror within the system is slightly tilted, it introduces a factor of exp(ikα j x) ≈ 1 + ikα j x to the spatial profile (the meter vector), assumed to be initially an even function (e.g., a Gaussian mode). The operator x within the second term of this factor produces a different, odd, spatial transverse mode (e.g., a HG01 mode) orthogonal to the first. This odd mode is needed at the split detector to produce a signal. If this odd mode were to follow the same path as the even one from the tilted mirror to the detector, the signal would be proportional to the weak value of the projection operator for the corresponding mirror. This is the case for all mirrors if the Dove prisms are not in place, due to the alignment properties of this system. When the Dove prisms are inserted, though, the even and odd modes traveling from mirror E follow very different paths, since the Dove prism's action as a parity operator produces a sign change for the odd mode only within one of the legs of the inner Mach-Zehnder interferometer. Therefore, the odd mode generated by tilting mirror E is directed in its entirety to the detector, while the even mode leaving this mirror does not reach the detector. The resulting "weak trace" is then not proportional to the weak value for the corresponding mirror. Notice that one could also design other systems where the converse is true: a given mirror has a nonzero weak value, but tilting it would not produce a detectable signal on a split detector. One such system would result, for example, from shifting to the left the final beam splitter in the current system, so that it captures instead the other output port of the inner interferometer.
Conclusions.-We have shown that by adding the Dove prisms to the nested interferometer geometry of Vaidman et al., we change the stability of the system. This results in a misalignment of mirror E now having a detectable effect, whereas previously it did not appear. The weak values of the various projection operators are identical to those used by those authors in the presence of the Dove prisms, because in the aligned case, these prisms act as identity operators, other than adding optical path length. The perfect phase balance of the inteferometer is not affected at all. We showed that the presence of the parity operation indicates that any misalignment of mirror E will leave a weak trace.
What, then, do we conclude from our results? On one hand, following the arguments of Vaidman, we can say that the presence of the Dove prisms causes the photon to be present (in the past) on mirror E because it now leaves a weak trace -altering its past, depending on the presence of this optical element or not. Indeed, more radical things were said by Wheeler, who said the presence or absence of a beam splitter retroactively changed the photon path to one or both paths. Following this line of reasoning, we could further make a "delayed choice" experiment, by choosing to either insert the prisms or not after the photon has passed mirror E -and cause the photon to either leave a trace or not ("it was there or it was not") on the split detector! However, the comparison with the 3 box problem [18], where one infers that (-1) particle is in box B because its weak value is -1, shows a danger with this interpretation: Does one conclude that the Dove prisms cause (-2) particles to be present on mirror E? Or that if we put the Doves before or after mirror A, that we can change it from +1 particle to -1 particle? On the other hand, we can simply interpret this result in terms of the stability of the interferometer. Adding the Dove prisms allows us to break the interferometer stability in the presence of a tilt on mirror E.
We stress that because the weak values themselves are unchanged, it is possible to (at the same time) make a seperate measurement of the weak value of E using other methods, i.e. a quantum nondemolition measurement with a separate meter, which will then couple to the (zero) weak value. So, one conclusion is that the question of leaving a trace on a detector is not as simple as calculating a weak value. The misalignment of an interferometer can bring in other effects that must be accounted for.
How, then, does this connect to the Vaidman/Zubairy debate? In order to make any claim about the past of the quantum particle, one must have a principle to invoke. Vaidman's criterion of the weak trace indicating that the photon was at mirrors A and B, but not E and F, has been shown to be sensitive to the details of the interferometer stability. In response to that, one could adopt the weak values themselves as a new criterionalthough to measure them in practice means to slightly break the interference [14]. If one adopts instead the criterion of perfect destructive interference as Zubairy does, then it is indeed true that if the interferometer is perfectly aligned, the sum of paths EAF and EBF cancel exactly not only at the detector, but at all points past the second nested beamsplitter. Nonetheless, the fact that a slight deviation from perfect alignment in the inner interferometer interferes with path C, making the a photon in the public channel appear without destructive interference, shows the practical fragility of the communication proposal.
FIG. 1 .
1Optical setup used by Vaidman et al. [1, 2] (a,b) and its modification (c) with Dove prisms inserted on each path. The beam splitters shown in black are 1:1 and polarizationinsensitive, while those in blue (top-left and bottom-right) are polarizing ones and act as 2:1 through a suitable choice of the input and output polarizations, selected by polarizers (thick black lines). (a) The red line illustrates the misalignment introduced by tilting mirror A. (b)
Acknowledgements.-We give our thanks for lengthy and animated discussions with Lev Vaidman and M. Suhail Zubairy, as well as for their comments on the manuscript. We both especially wish to thank José Javier Sánchez Mondragón, for his lavish hospitality at the LAOP workshop and conference in Cancún, Mexico, where this work was begun. ANJ thanks R. Kastner for discussions. MAA acknowledges funding from the National Science Foundation under grant PHY-1068325. ANJ acknowledges support from the US Army Research office Grant No. W911NF-09-0-01417.
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| [] |
[
"Local Correlation Effects on the s ± -and s ++ -wave Superconductivities Mediated by Magnetic and Orbital Fluctuations in the five-orbital Hubbard Model for Iron Pnictides",
"Local Correlation Effects on the s ± -and s ++ -wave Superconductivities Mediated by Magnetic and Orbital Fluctuations in the five-orbital Hubbard Model for Iron Pnictides"
] | [
"Jun Ishizuka \nDepartment of Physics\nNiigata University\n950-2181IkarashiNiigataJapan\n",
"Takemi Yamada \nDepartment of Physics\nNiigata University\n950-2181IkarashiNiigataJapan\n",
"Yuki Yanagi \nDepartment of Physics\nNiigata University\n950-2181IkarashiNiigataJapan\n\nDepartment of Physics\nFaculty of Science and Technology\nTokyo University of Science\n278-8510NodaJapan\n",
"Yoshiakiōno "
] | [
"Department of Physics\nNiigata University\n950-2181IkarashiNiigataJapan",
"Department of Physics\nNiigata University\n950-2181IkarashiNiigataJapan",
"Department of Physics\nNiigata University\n950-2181IkarashiNiigataJapan",
"Department of Physics\nFaculty of Science and Technology\nTokyo University of Science\n278-8510NodaJapan"
] | [
"Journal of the Physical Society of Japan LETTERS"
] | We investigate the electronic state and the superconductivity in the 5-orbital Hubbard model for iron pnictides by using the dynamical mean-field theory in conjunction with the Eliashberg equation. The renormalization factor exhibits significant orbital dependence resulting in the large change in the band dispersion as observed in recent ARPES experiments. The critical interactions towards the magnetic, orbital and superconducting instabilities are suppressed as compared with those from the random phase approximation (RPA) due to local correlation effects. Remarkably, the s++-pairing phase due to the orbital fluctuation is largely expanded relative to the RPA result, while the s±-pairing phase due to the magnetic fluctuation is reduced. | 10.7566/jpsj.82.123712 | [
"https://arxiv.org/pdf/1310.7325v2.pdf"
] | 118,821,450 | 1310.7325 | 095652140ed1ea15f0a9b1564bf9ce87578aaf5e |
Local Correlation Effects on the s ± -and s ++ -wave Superconductivities Mediated by Magnetic and Orbital Fluctuations in the five-orbital Hubbard Model for Iron Pnictides
25 Nov 2013
Jun Ishizuka
Department of Physics
Niigata University
950-2181IkarashiNiigataJapan
Takemi Yamada
Department of Physics
Niigata University
950-2181IkarashiNiigataJapan
Yuki Yanagi
Department of Physics
Niigata University
950-2181IkarashiNiigataJapan
Department of Physics
Faculty of Science and Technology
Tokyo University of Science
278-8510NodaJapan
Yoshiakiōno
Local Correlation Effects on the s ± -and s ++ -wave Superconductivities Mediated by Magnetic and Orbital Fluctuations in the five-orbital Hubbard Model for Iron Pnictides
Journal of the Physical Society of Japan LETTERS
25 Nov 2013
We investigate the electronic state and the superconductivity in the 5-orbital Hubbard model for iron pnictides by using the dynamical mean-field theory in conjunction with the Eliashberg equation. The renormalization factor exhibits significant orbital dependence resulting in the large change in the band dispersion as observed in recent ARPES experiments. The critical interactions towards the magnetic, orbital and superconducting instabilities are suppressed as compared with those from the random phase approximation (RPA) due to local correlation effects. Remarkably, the s++-pairing phase due to the orbital fluctuation is largely expanded relative to the RPA result, while the s±-pairing phase due to the magnetic fluctuation is reduced.
The discovery of the iron-based superconductors 1 has triggered an intense research effort to investigate their electronic state and superconducting mechanism. Most of the phase diagrams exhibit the tetragonal-orthorhombic structural transition and the stripe-type antiferromagnetic (AFM) transition. 1, 2 The AFM fluctuation is enhanced towards the AFM transition, 3 while the ferro-orbital (FO) fluctuation responsible for the softening of the elastic constant C 66 4, 5 is enhanced towards the structural transition. Correspondingly, two distinct s-wave pairings: the s ± -wave with sign change of the order parameter between the hole and the electron Fermi surfaces (FSs) mediated by the AFM fluctuation 6,7 and the s ++ -wave without the sign change mediated by the FO fluctuation [8][9][10] and by the antiferro-orbital (AFO) fluctuation 11 which is also responsible for the softening of C 66 through the two-orbiton process 12 were proposed. Despite the numerous efforts, the pairing state together with the mechanism of the superconductivity is still controversial.
As the details of the electronic band structure are crucial for the pairing state and mechanism, the theoretical studies have employed the realistic multi-orbital models [7][8][9][10][11][12] where the tight-binding parameters are determined so as to reproduce the first-principles band structures which had been found to agree with the angle-resolved photoemission spectroscopy (ARPES) by reducing the band width by a factor of 2 ∼ 3. 13 However, recent high-resolution ARPES measurements for Ba 0.6 K 0.4 Fe 2 As 2 14 revealed significant band (or orbital) dependence of the mass enhancement from 1.3 to 9.0. More recently, some evidences for an orbital-selective Mott transition (OSMT) in K x Fe 2−y Se 2 , 15 where the renormalization factor Z for d xy orbital becomes zero while Z for the other orbitals are finite, and for the heavy fermion behavior in KFe 2 As 2 , 16 where the system is near the OSMT, were observed. In these cases, we need to investigate the superconductivity on the basis of the strongly correlated electronic states in the presence of the large orbital dependence of Z.
In this letter, we investigate the 5-orbital Hubbard model 7 for iron pnictides by using the dynamical mean-field theory
(DMFT) 17 which is exact in infinite dimensions (d = ∞)
where the self-energy becomes local and enables us to sufficiently take into account the local correlation effects including the strong correlation regime where Z largely depends on the orbital 18 and the OSMT is realized. 19 To examine the superconductivity, we solve the Eliashberg equation in which the effective pairing interaction and the renormalized singleparticle Green's function are calculated on the basis of the DMFT. In particular, we focus our attention on the local correlation effects on the possible pairing states, the magnetic fluctuation mediated s ± -wave and the orbital fluctuation mediated s ++ -wave, beyond the random phase approximation (RPA) which was extensively developed for iron pnictides in the previous works. [7][8][9][10][11] The 5-orbital Hubbard model consists of the Fe 3d-orbitals and is given by the Hamiltonian, 7Ĥ =Ĥ 0 +Ĥ int , where the kinetic partĤ 0 is determined so as to reproduce the firstprinciples band structure for LaFeAsO and the Coulomb interaction partĤ int includes the multi-orbital interaction on a Fe site: the intra-and inter-orbital direct terms U and U ′ , Hund's rule coupling J and the pair transfer J ′ . In this paper, we set the x-y axes parallel to the nearest Fe-Fe bonds.
To solve the model, we use the DMFT 17 in which the lattice model is mapped onto an impurity Anderson model embedded in an effective medium which is determined so as to satisfy the self-consistency condition:Ĝ(iε m ) = (1/N ) kĜ (k, iε m ) with the wave vector k and the Matsubara frequency ε m = (2m + 1)πT , whereĜ(iε m ) and G(k, iε m ) are the 5×5 matrix representations of the local (impurity) Green's function and the lattice Green's function, respectively, which are explicitly given bŷ
G(iε m ) = Ĝ −1 (iε m ) −Σ(iε m ) −1 ,(1)G(k, iε m ) = (iε m + µ) −Ĥ 0 (k) −Σ(iε m ) −1 ,(2)
whereΣ(iε m ) is the 5 × 5 matrix representation of the impurity (local) self-energy andĜ(iε m ) is that of the bare impurity Green's function describing the effective medium. Within the DMFT, the spin (charge-orbital) susceptibility is given in the 25 × 25 matrix representation aŝ
χ s(c) (q) = 1 − (+)χ 0 (q)Γ s(c) (iω n ) −1χ 0 (q)(3)
withχ 0 (q) = −(T /N ) kĜ (k + q)Ĝ(k), where k = (k, iε m ), q = (q, iω n ) and ω n = 2nπT . In eq. (3), Γ s(c) (iω n ) is the local irreducible spin (charge-orbital) vertex in which only the external frequency (ω n ) dependence is considered as a simplified approximation 20 and is explicitly given bŷ
Γ s(c) (iω n ) = −(+) χ −1 s(c) (iω n ) −χ −1 0 (iω n )(4)withχ 0 (iω n ) = −T εmĜ (iε m + iω n )Ĝ(iε m ), wherê χ s(c) (iω n )
is the local spin (charge-orbital) susceptibility. When the largest eigenvalue α s (α c ) of (−)χ 0 (q)Γ s(c) (iω n ) in eq. (3) for a wave vector q with iω n = 0 reaches unity, the instability towards the magnetic (charge-orbital) order with the corresponding q takes place.
To examine the superconductivity mediated by the magnetic and charge-orbital fluctuations which are extremely enhanced towards the corresponding orders mentioned above, we write the effective pairing interaction for the spin-singlet state using the spin (charge-orbital) susceptibility and vertex given in eqs. (3) and (4) obtained within the DMFT in the 25 × 25 matrix representation aŝ
V (q) = 3 2Γ s (iω n )χ s (q)Γ s (iω n ) − 1 2Γ c (iω n )χ c (q)Γ c (iω n ) + 1 2 Γ (0) s +Γ (0) c(5)
with the bare spin (charge-orbital) vertex:
[Γ (0) s(c) ] ℓℓℓℓ = U (U ), [Γ (0) s(c) ] ℓℓ ′ ℓℓ ′ = U ′ (−U ′ + 2J), [Γ (0) s(c) ] ℓℓℓ ′ ℓ ′ = J(2U ′ − J) and [Γ (0) s(c) ] ℓℓ ′ ℓ ′ ℓ = J ′ (J ′ )
, where ℓ ′ = ℓ and the other matrix elements are 0. Substituting the effective pairing interaction eq. (5) and the lattice Green's function eq. (2) into the linearized Eliashberg equation:
λ∆ ll ′ (k) = − T N k ′ l1l2l3l4 V ll1,l2l ′ (k − k ′ ) × G l3l1 (−k ′ )∆ l3l4 (k ′ )G l4l2 (k ′ ),(6)
we obtain the gap function ∆ ll ′ (k) with the eigenvalue λ which becomes unity at the superconducting transition temperature T = T c . In eq. (6), ∆ ll ′ (k) includes the 1/d corrections yielding the k dependence of the gap function responsible for the anisotropic superconductivity which is not obtained within the zeroth order of 1/d. 17 If we replaceΓ s(c) withΓ (0) s(c) and neglectΣ, eq. (5) yields the RPA result of V (q). 7-11 Therefore, eq. (6) with eqs. (2) and (5) is a straightforward extension of the RPA to include the vertex and the self-energy corrections within the DMFT without any double counting.
In the actual calculations with the DMFT, we solve the effective 5-orbital impurity Anderson model, where the Coulomb interaction at the impurity site is given by the same form asĤ int with a site i and the kinetic energy responsi-consistency condition as possible, by using the exact diagonalization (ED) method for a finite-size cluster to obtain the local quantities such asΣ andχ s(c) . Since the multiorbital system requires rather CPU-time and memory consuming calculations, we employ the clusters with the site number N s = 4 within a restricted Hilbert space. 21 We have also performed preliminary calculations with N s = 2 22 and have confirmed that the results with N s = 4 are qualitatively consistent with those with N s = 2 and quantitatively improved especially for the intermediate interaction regime as previously observed in the DMFT+ED approaches for the multi-band and multi-orbital models. [23][24][25][26] In fact, the DMFT results from the ED with N s = 4 are quantitatively in good agreement with the precise results from the numerical renormalization group 27 for the 2-orbital Hubbard model and those from the continuous-time quantum Monte Carlo 28 for the 3orbital Hubbard model. 26 As for the 5-orbital Hubbard model, the ED results with N s = 3 are found to agree with those with N s = 2. 26 Therefore, we expect that the ED calculations with N s = 4 yield quantitatively reliable results also for the present 5-orbital Hubbard model. All calculations are performed at T = 0.02eV for the electron number n = 6.0 corresponding to the non-doped case. We use 32 × 32 k-point meshes and 1024 Matsubara frequencies in the numerical calculations with the fast Fourier transformation. Here and hereafter, we measure the energy in units of eV.
In the previous RPA study, 8 it was found that the s ±pairing is mediated by the magnetic fluctuation near the AFM order for U > U ′ , while the s ++ -pairing is mediated by the orbital fluctuation near the FO order for U < U ′ , where the superconductivity is investigated in the wide parameter space by treating U , U ′ , J and J ′ as independent parameters apart from the condition satisfied in the isolated atom: U = U ′ +2J and J = J ′ . Correspondingly, we consider the two specific cases with U > U ′ and U < U ′ to elucidate the correlation effects beyond the RPA on the magnetic and orbital orders and the those fluctuations mediated superconductivity.
First, we consider the case with U > U ′ , where the magnetic fluctuation dominates over the orbital fluctuation. In Fig. 1, several physical quantities are plotted as functions of U with U = U ′ + 2J, J/U = 0.1 and J = J ′ . Fig. 1 (a) shows the renormalization factor de-
fined by: Z ℓ = 1 − dΣ ℓ (ε) d(ε) ε→0 −1 with orbital ℓ = d x 2 −y 2 , d 3z 2 −r 2 , d zx , d yz and d xy .
When U increases, all of Z ℓ monotonically decrease with increasing the variance of Z ℓ . We find that Z ℓ for ℓ = d xy is the smallest for all U and finally becomes zero at U c ∼ 5 while Z ℓ for ℓ = d xy are finite revealing the OSMT, 21 as recently discussed in K x Fe 2−y Se 2 15 and KFe 2 As 2 16 where the ARPES experiments are well accounted for by the slave-spin mean-field 15,29 and the slaveboson mean-field (Gutzwiller) 16 approximations yielding the OSMT with Z dxy → 0. We note that, even in the intermediate correlation regime away from the OSMT, the large orbital dependence of Z ℓ results in the significant change in the band dispersion 21 which is consistent with the recent highresolution ARPES measurements for Ba 0.6 K 0.4 Fe 2 As 2 . 14 Figs. 1 (b) and (c) show the U dependence of the largest eigenvalues α s and α c for several wave vectors q, where α s(c) shows a maximum at q = q max . When U increases, both at U AFM c ∼ 2.40 where the magnetic susceptibility with q ∼ (π, 0) corresponding to the stripe-type AFM diverges. The largest eigenvalue λ of the Eliashberg equation (6) is also plotted in Fig. 1 (b) and is found to increase with increasing α s and finally reaches unity at U SC c ∼ 2.34 where the superconducting instability occurs. For comparison, we also plot the RPA results of α s for q max and λ in Fig. 1 (b) and find that the critical interactions U AFM c and U SC c from the DMFT are about twice larger than those from the RPA 7 due to the correlation effects beyond the RPA and are consistent with the values of the effective Coulomb interactions derived from the downfolding scheme based on first-principles calculations. 30 In Figs. 2 (a)-(f), we show the d xy intra-orbital components of the spin (charge-orbital) susceptibility χ s (χ c ) and the pairing interaction V , together with the band-diagonal components of the gap functions ∆ with the lowest Matsubara frequency iε m = iπT for U = 2.28, U ′ = 1.824 and J = J ′ = 0.228. In this case, the enhanced spin susceptibility for q ∼ (π, 0), i. e., the stripe-type AFM fluctuation yields the large positive value of the effective pairing interaction V for q ∼ (π, 0) resulting in the gap function with sign change between the electron and hole FSs, i. e., the s ± -wave state. Figs. 2 (g)-(l), we also show the corresponding RPA results for U = 1.15, U ′ = 0.92 and J = J ′ = 0.115. As the q dependence of χ s and V from the DMFT becomes weak as compared to the RPA results due to the local correlation π 0 π 0 π 0 DMFT s ± -wave result as shown in Fig. 1 (b).
(a) χ s xy (q) (b) χ c xy (q) (c) V xy (q) (d) ∆ 2 (k) (e) ∆ 3 (k) (f) ∆ 4 (k)q x q x q x q y q y q y k y k y k y k x π 0 π 0 q x q y q x q y k x k x k x π 0 q x q y RPA s ± -wave (g) χ s xy (q) (h) χ c xy (q) (i) V xy (q) (j) ∆ 2 (k) (k) ∆ 3 (k) (l) ∆ 4 (k)
Next, we consider the case with U < U ′ , where the orbital fluctuation dominates over the magnetic fluctuation. Figs. 3 (a)-(c) show the renormalization factor Z ℓ and the largest eigenvalues α s , α c and λ as functions of U ′ with U = 0.25U ′ + 2J, J/U = 0.1 and J = J ′ . When U ′ increases, Z ℓ for all ℓ monotonically decrease with keeping the smallest value for ℓ = d xy , similar to the case of Fig. 1 (a). When U ′ increases, both α s and α c increase with α s < α c and α c becomes unity at U FO c ∼ 2.28 where the orbital susceptibility with q ∼ (0, 0) corresponding to the FO diverges. We note that q max = (0, π/4) just below U FO c with α c = 0.98 and q max = (0, 0) just above U FO c with α c = 1.03, while it is difficult to determine q max precisely at U FO c with α c = 1 within the present numerical resolution as χ c diverges almost simultaneously for q ∼ (0, 0) and then we call the FO in a broad sense. With increasing α c , λ increases and finally reaches unity at U SC c ∼ 1.54 where the superconducting instability occurs. For comparison, we also plot the RPA results of α c for q max and λ in Fig. 3 (c), and find that U largely expanded as compared to the RPA result, in contrast to the case with the s ± -pairing phase which is reduced (see Fig. 1 (b)).
In Figs. 4 (a)-(f), we show the same physical quantities as in Figs. 2 (a)-(f) for U = 0.4, U ′ = 1.28 and J = J ′ = 0.04. In this case, the enhanced orbital susceptibility in the whole q space yields the negative value of the effective pairing interaction V for all q resulting in the gap function without sign change, i. e., the s ++ -wave state. In Figs. 4 (g)-(l), we also show the corresponding RPA results for U = 0.25, U ′ = 0.8 and J = J ′ = 0.025. As the q dependence of χ c from the DMFT becomes weak as compared to the RPA result due to the local correlation effects, the local (q-averaged) component of the pairing attraction |V | becomes considerably larger than the RPA result for the same value of α c for q max resulting in the remarkable enhancement of the s ++ -pairing phase which is expanded far away from the FO critical interaction U FO c (α c = 0.82 for U SC c ) in contrast to the RPA result (α c = 0.95 for U SC c ) as shown in Fig. 3 (c). In summary, we have investigated the electronic state and the superconductivity in the 5-orbital Hubbard model for iron pnictides by using the DMFT+ED method in conjunction with the linearized Eliashberg equation. All of the critical interactions towards the magnetic, orbital and superconducting instabilities have been found to be suppressed as compared to the RPA results. Remarkably, the s ++ -pairing phase due to the orbital fluctuation is largely expanded as compared to the
(q) (b) χ c xy (q) (c) V xy (q) (d) ∆ 2 (k) (e) ∆ 3 (k) (f) ∆ 4 (k) q x q x q y q y q y k x k x k x k y k y k y π 0
π 0 π 0 RPA s ++ -wave (j) ∆ 2 (k) (k) ∆ 3 (k) (l) ∆ 4 (k) q x q x q x q y q y q y k x k x k x k y k y k y RPA result, while the s ± -pairing phase due to the magnetic fluctuation is reduced. This is caused by the local correlation effects which enhance the local, i. e., the q-independent magnetic (orbital) fluctuation resulting in the local component of the repulsive (attractive) pairing interaction responsible for the suppression (enhancement) of the s ± (s ++ )-pairing. Although the case with U < U ′ is not realistic and the FO fluctuation enhanced there (d xy intra-orbital component) is not corresponding to the softening of C 66 , the same effects due to the local correlation are expected to occur in the s ++pairing in the realistic cases with the electron-phonon interaction 9-11 and/or the mode-coupling effects of the Coulomb interaction 31 and will be discussed in subsequent papers.
Fig. 1 .
1(Color online) (a) The renormalization factor Z ℓ with ℓ = d x 2 −y 2 , d 3z 2 −r 2 , dzx, dyz and dxy, (b) and (c) the largest eigenvalues αs and αc for several q and λ which reach unity towards the magnetic, chargeorbital and superconducting instabilities, respectively, as functions of U with U = U ′ + 2J, J/U = 0.1 and J = J ′ for n = 6.0 and T = 0.02. The RPA results of αs for qmax and λ are also plotted by thin lines in (b).
Fig. 2 .
2(Color online) DMFT results for the dxy intra-orbital components of the spin susceptibility χ s (a), the charge-orbital susceptibility χ c (b) and the pairing interaction V (c), and those for the band-diagonal components of the gap function ∆ with the lowest Matsubara frequency iεm = iπT for band 2 (d) and band 3 (e) (band 4 (f)) with the hole (electron) FSs (solid lines) for U = 2.28, U ′ = 1.824 and J = J ′ = 0.228, where αs = 0.964 for qmax. (g)-(l) The corresponding RPA results for U = 1.15, U ′ = 0.92 and J = J ′ = 0.115, where αs = 0.964 for qmax.
Fig. 3 .
3DMFT are larger than those from the RPA due to the correlation effects beyond the RPA. Remarkably, the DMFT (Color online) (a) The renormalization factor Z ℓ with ℓ = d x 2 −y 2 , d 3z 2 −r 2 , dzx, dyz and dxy, (b) and (c) the largest eigenvalues αs and αc for several q and λ which reach unity towards the magnetic, chargeorbital and superconducting instabilities, respectively, as functions of U ′ with U = 0.25U ′ + 2J, J/U = 0.1 and J = J ′ for n = 6.0 and T = 0.02. The RPA results of αc for qmax and λ are also plotted by thin lines in (c).
Fig. 4 .
4(Color online) DMFT results for the dxy intra-orbital components of the spin susceptibility χ s (a), the charge-orbital susceptibility χ c (b) and the pairing interaction V (c), and those for the band-diagonal components of the gap function ∆ with iεm = iπT for band 2 (d) and band 3 (e) (band 4 (f)) with the hole (electron) FSs (solid lines) for U = 0.4, U ′ = 1.28 and J = J ′ = 0.04, where αc = 0.76 for qmax. (g)-(l) The corresponding RPA results for U = 0.25, U ′ = 0.8 and J = J ′ = 0.025, where αc = 0.76 for qmax.
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AcknowledgmentsThis work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology.
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| [] |
[
"Interaction induced local moments in parallel quantum dots within the functional renormalization group approach",
"Interaction induced local moments in parallel quantum dots within the functional renormalization group approach"
] | [
"V S Protsenko \nM. N. Mikheev Institute of Metal Physics\nUral Federal University\n620990, 620002Ekaterinburg, EkaterinburgRussia, Russia\n",
"A A Katanin \nM. N. Mikheev Institute of Metal Physics\nUral Federal University\n620990, 620002Ekaterinburg, EkaterinburgRussia, Russia\n"
] | [
"M. N. Mikheev Institute of Metal Physics\nUral Federal University\n620990, 620002Ekaterinburg, EkaterinburgRussia, Russia",
"M. N. Mikheev Institute of Metal Physics\nUral Federal University\n620990, 620002Ekaterinburg, EkaterinburgRussia, Russia"
] | [] | We propose a version of functional renormalization-group (fRG) approach, which is, due to including Litim-type cutoff and switching off (or reducing) the magnetic field during fRG flow, capable describing singular Fermi liquid (SFL) phase, formed due to presence of local moments in quantum dot structures. The proposed scheme allows to describe the first-order quantum phase transition from "singular" to the "regular" paramagnetic phase with applied gate voltage to parallel quantum dots, symmetrically coupled to leads, and shows sizable spin splitting of electronic states in the SFL phase in the limit of vanishing magnetic field H → 0; the calculated conductance shows good agreement with the results of the numerical renormalization group. Using the proposed fRG approach with the counterterm, we also show that for asymmetric coupling of the leads to the dots the SFL behavior similar to that for the symmetric case persists, but with occupation numbers, effective energy levels and conductance changing continuously through the quantum phase transition into SFL phase. arXiv:1607.08544v2 [cond-mat.str-el] | 10.1103/physrevb.94.195148 | [
"https://arxiv.org/pdf/1607.08544v2.pdf"
] | 119,106,708 | 1607.08544 | ddf0da22555f9ff5baf2693e2799304f515e6963 |
Interaction induced local moments in parallel quantum dots within the functional renormalization group approach
V S Protsenko
M. N. Mikheev Institute of Metal Physics
Ural Federal University
620990, 620002Ekaterinburg, EkaterinburgRussia, Russia
A A Katanin
M. N. Mikheev Institute of Metal Physics
Ural Federal University
620990, 620002Ekaterinburg, EkaterinburgRussia, Russia
Interaction induced local moments in parallel quantum dots within the functional renormalization group approach
(Dated: June 14, 2018)
We propose a version of functional renormalization-group (fRG) approach, which is, due to including Litim-type cutoff and switching off (or reducing) the magnetic field during fRG flow, capable describing singular Fermi liquid (SFL) phase, formed due to presence of local moments in quantum dot structures. The proposed scheme allows to describe the first-order quantum phase transition from "singular" to the "regular" paramagnetic phase with applied gate voltage to parallel quantum dots, symmetrically coupled to leads, and shows sizable spin splitting of electronic states in the SFL phase in the limit of vanishing magnetic field H → 0; the calculated conductance shows good agreement with the results of the numerical renormalization group. Using the proposed fRG approach with the counterterm, we also show that for asymmetric coupling of the leads to the dots the SFL behavior similar to that for the symmetric case persists, but with occupation numbers, effective energy levels and conductance changing continuously through the quantum phase transition into SFL phase. arXiv:1607.08544v2 [cond-mat.str-el]
I. INTRODUCTION
During last decades electronic and transport properties of quantum dots have attracted considerable attention due to possibility of realizing non-trivial quantum phenomena and promising applications in different nanodevices [1,2], in particular quantum computation systems [3][4][5][6][7][8]. The electron-electron interaction may have significant effect on electronic properties and transport in quantum dot systems and is responsible for many interesting physical phenomena.
While relatively strong Coulomb interaction leads to Kondo effect [9][10][11][12], for some special geometries of quantum dot systems even relatively weak interaction plays important role. This concerns in particular a system of parallel quantum dots, connected symmetrically to common leads, where the formation of the so called singular Fermi liquid (SFL) state [13,14], which possesses a local moment, decoupled from the leads, and almost unitary conductance, is possible near half filling.
To date, a wide variety of analytical and numerical methods have been developed to describe the effect of the Coulomb interaction in quantum dot structures. The numerical renormalization group (NRG) method provides a strightforward approach to describe spectral functions and conductivity of quantum dots in the presence of an interaction [15][16][17][18]. However, being very successful, this method requires significant computational effort, which grows exponentially with the number of interacting degrees of freedom. This circumstance does not allow one to directly apply these methods to fairly large systems.
The recently prposed nanoscopic dynamic vertex approximation (nano-DΓA) [19,20], allows one to treat effects of interaction in nanostructures, and has a potential of describing non-local correlations beyond nano-DMFT [21][22][23][24][25]. In the most complete, parquete, formulation nano-DΓA approach however also requires substantial computational resources.
In this regard, the recently proposed functional renormalization-group (fRG) approach [26][27][28] is very promising because it allows to reformulate a many particle problem in terms of coupled differential equations for irreducible vertex functions, which, after several approximations, can be easily integrated even for complex systems. This method has already been successfully applied to systems consisting of a small number of quantum dots arranged in different geometries [29][30][31][32]. In these studies it was shown that the results obtained with the sharp frequency cutoff are in good agreement with the Bethe ansatz and NRG data [30]. At the same time, the fRG approach with the sharp cutoff scheme is not applicable to describe the SFL state, since, e.g., for parallel quantum dots it yields an artificially low conductance at low magnetic fields for gate voltages V g less than some critical value |V g | V c g [30], which contradicts the NRG results [13].
To describe physical properties in the local moment SFL state, we propose in the present paper an fRG scheme with continously switching off (or decreasing) the external magnetic field with decreasing cutoff. This scheme is similar to a counterterm technique, used earlier to treat the first order quantum phase transitions in lattice fermionic systems [33,34] within fRG approach. This technique was not however applied to description of quantum dot structures. We show that within this approach it is possible to describe correctly the dependence of the conductance on the magnetic field and the gate voltage in a good agreement with NRG data. We also found that the influence of an asymmetrical dot-lead coupling can be straightforwardly investigated using the same formalism. To demonstrate this, as an example, we consider the particular case of a parallel quantum dot system with different coupling of the leads to the dots. An especially interesting observation is the continuous behavior of the calculated observables at the quantum phase transition to the SFL state in this case, leading to the resonance form of the conductance curve in the vicinity of the phase transition point.
The plan of the paper is the following. In Sect. II we formulate the model of parallel quantum dots with arbitrary couplings to the leads. In Sect. III we introduce the fRG approach with the counterterm, compare its results to the standard fRG approach and numerical renormalization group data. In Sect. IV we analyze presence of the local moments and the origin of spin splitting of electronic spectrum in the SFL regime, as well as the conductance of quantum dots, symmetrically and asymmetrically coupled to the leads. Finally, in Sect. V we present the conclusions.
II. MODEL
The parallel quantum dots connected to two conducting leads (see Fig. 1) can be modelled by the Hamiltonian [30]
H = H dot + H leads + H coup ,(1)
where
H dot = jσ 0 σ n j,σ + U 2 n j,σ − 1 2 n j,σ − 1 2 (2)
describes quantum dots with equal on-site Coulomb repulsion U and single energy level 0 σ = V g − σH/2 (σ = +1/ − 1 for the spin up/down electrons), controlled by the gate voltage V g and magnetic field H, d † j,σ (d j,σ ) -denote creation (annihilation) operators for an electron with spin σ localized on the j-th quantum dot and n j,σ = d † j,σ d j,σ . The next term, H leads , takes into account the two equivalent non-interacting semi-infinite leads, which are modeled by the following tight-binding Hamiltonian
H leads = −τ α=L,R ∞ j=0 σ (c † α,j+1,σ c α,j,σ + H.c.),(3)
here c † α,j,σ (c α,j,σ ) are the leads creation (annihilation) operators for an electron with spin direction σ on the j lattice site of the left α = L or right α = R lead and τ is the hopping matrix element in the leads. Finally, the last term H coup introduces coupling between each quantum dot i and leads α, and is given by
H coup = − α=L,R i,σ (t α i c † α,0,σ d i,σ + H.c.),(4)
where t α i are the corresponding hopping parameters (i = 1, 2, α = L, R).
For the symmetric case t α i = t one can apply canonical transformation to the even d e,σ and the odd d o,σ orbitals
d e(o),σ = (d 1,σ ± d 2,σ )/ √ 2,(5)L R QD1 QD2 t R 2 t R 2 t R 1 t L 2 t L 1 L R t QD1 QD2t
FIG. 1. Left: Schematic representation of parallel double quantum dots (DQ1 and DQ2) connected to two left (L) and right (R) leads by equal tunneling matrix elements t α i . Right: Schematic representation of a configuration of parallel quantum dots, symmetrically coupled to the leads, obtained as a result of the transformation (5). The vertical up-down arrow indicates the presence of two-particle interactions between quantum dots (see Eq. 17 in the Section IV A of the paper).
such that the coupling part H coup takes the form
H coup = −t α=L,R σ (c † α,0,σ d e,σ + H.c.),(6)
therefore only the even orbitals are directly connected to the leads by the hopping amplitudet = √ 2t and the parallel double dot system considered in the present study can be equivalently mapped onto the system is shown on the right side of the Fig. 1.
After projection of the leads and taking the wide-band limit (see Refs. [30,35,36]), the inverse of the noninteracting propagator for electrons with spin σ represents the 2×2 matrix in the dot-space, which has the following structure
G −1 0,σ (iω) i,j = iω − 0 σ + i Γ L i + Γ R i sign(ω) δ ij + i α=L,R Γ α i Γ α j sign(ω)(1 − δ ij ),(7)
with the energy independent hybridization strength Γ α i = π|t α i | 2 ρ lead (0), where ρ lead is the local density of states at the last site of the left or right lead (the leads are equivalent).
III. FRG APPROACH
A. Formulation of the method
To describe the correlation effects in quantum dots at zero temperature T = 0, we, following to Ref. [30], project out the leads, introduce some cutoff of the bare single-particle Green function of the dots G 0 → G Λ 0 , and apply the one-particle irreducible fRG scheme [27,28], yielding an infinite hierarchy of differential flow equations for the self-energy Σ and the n-particle vertices Γ (2n) , n ≥ 2. Truncating the fRG flow equations by neglecting of the flow of the vertex functions with n ≥ 3 leads to a closed system of the flow equations for the Σ and the two-particle vertex Γ (4) ≡ Γ, which has the standard form [29,30]
∂ Λ Σ Λ α α = − dω 2π e iω0 + S Λ ββ (iω) Γ Λ α β αβ ,(8)
and
∂ Λ Γ Λ α β αβ = dω 2π S Λ γγ (iω) G Λ δδ (−iω)Γ Λ α β δγ Γ Λ δ γ αβ − dω 2π S Λ γγ (iω) G Λ δδ (iω) (9) × Γ Λ α γ αδ Γ Λ δ β γβ + (δ γ, δ γ ) − Γ Λ β γ αδ Γ Λ δ α γβ + (δ γ, δ γ ) , where G Λ σ (iω) = G Λ 0,σ (iω) −1 − Σ Λ σ −1
, the Greek multi-indices collect the dot and spin indexes and repeated indices imply summation. The S Λ denotes the single-scale propagator (from now, we omit dots indexes for brevity)
S Λ σ = G Λ σ ∂ Λ G Λ 0,σ −1 G Λ σ .(10)
As in Refs. [29,30] we neglect frequency dependence of the self-energy and two-particle vertices, which was previously shown to describe very accurately conductivity of the single-impurity Anderson model in both, weakand strong coupling regimes; taking into account frequency dependence is not expected to improve results, because of the neglect of the three-particle vertices, see Refs. [35,37].
The interacting part of the energy energy can be obtained from the flow equation [28,35]
∂ Λ E Λ int = dω 2π e iω0 + α G Λ 0 − G Λ ∂ Λ G Λ 0 −1 αα .(11)
The corresponding conductance of a double quantum dot system is given by the relation (see, e.g., Ref. [30])
G = 2G 0 σ | i,j (Γ L i Γ R j ) 1/2 G Λ→0 ji,σ (0)| 2 ,(12)
where G 0 = 2e 2 h is the conductance quantum [38].
B. fRG approaches without counterterm
The standard fRG scheme to study a correlated quantum dots uses sharp cutoff in frequency space [29][30][31][32]
G Λ 0 = Θ(|ω| − Λ)G 0 ,(13)
where Heaviside theta function Θ cuts out infrared modes with Matsubara frequency |ω| < Λ. Fig. 2 shows the zero temperature linear conductance G as a function of a magnetic field H at the half-filling (V g = 0) in the particular case of full coupling symmetry t α i = t. One can see that there is a clear qualitative difference between the magnetic field H dependence of the conductance G(H) at the half-filling (V g = 0) within the NRG approach, applied following Refs. [39][40][41], and the fRG approach based on the sharp cutoff scheme. The conductance obtained within the latter approach first smoothly increases with decreasing magnetic field and then suddenly drops when the magnetic field becomes sufficiently small, in contrast to the NRG result. This drop of the conductance originates from the unphysical behavior of the vertex functions in the fRG flow, namely, with decreasing of the magnetic field they first converge to finite, but unphysically large values, and, for even smaller fields, diverge at Λ → 0.
The results of the approach with sharp cutoff can be somewhat improved using Litim-type [42] Λ-dependence of the bare propagator [43]
G Λ 0,σ −1 = [G 0,σ ] −1 + iI (Λ − |ω|) × Θ (Λ − |ω|) sign(ω),(14)
where I is the 2×2 identity matrix, with respect to the indexes of the dots. Using this cutoff, we find that over the whole range of studied magnetic fields the conductance is larger than that obtained in the fRG approach with the sharp momentum cutoff (see Fig. 2), which leads to some improvement of agreement with NRG data for not too low magnetic fields and allows us to continue the renormalization group flow toward a weaker magnetic fields.
Despite some improvement of the results, the smooth cutoff itself does not allow to describe correctly conductivity at small magnetic fields. Furthermore, we found that at low magnetic field these fRG schemes break down not only for the half-filled symmetric case (V g = 0, t α i = t), as considered above, but also in both, symmetric and asymmetric cases, when |V g | < V c g , where V c g is a critical value of gate voltage, which is discussed below. We consider this situation as rather general, and related to the formation of the SFL state in zero magnetic field, that according to the phase diagram obtained in Ref. [14] occurs near half filling.
C. fRG apprach with the counterterm
To describe the SFL state we decrease the magnetic field continuously during the flow, similarly to a counterterm extension of the fRG [33,34]. To this end, we introduce additional term σχ Λ /2 in the propagator of the quantum dots
G Λ 0,σ = G Λ 0,σ −1 + σ 2 Iχ Λ −1 ,(15)
corresponding to additional Λ-dependent external magnetic field χ Λ , which is switched off in the end of the flow, if we set χ Λ→0 = 0, and plays a role of infrared regulator in the bosonic spin sector. In this study, we use two different counterterms with linear and exponential cutoff dependence
χ Λ 1 =H min(1, Λ/Λ c ), χ Λ 2 =H 1 + exp [(Λ c − Λ) /Λ 0 ] ,(16)
that allow to begin the fRG flow with state at the field H ini = H + χ Λ→∞ = H +H, while at the end of the fRG flow one obtains renormalized vertices describing the system in the physical field H fin = H; Λ c ,H and Λ 0 are independent parameters, (Λ c Λ 0 ), which determine the scale and sharpness of switching off the H.
In the present study, the parameter Λ 0 was fixed to be approximately equal to the fRG scale, where the splitting occurs, but can be changed in a rather broad range. The value of the parameter Λ c is chosen according to the counterterm fieldH: it should provide a slow switching off the counterterm, when the additional magnetic field is chosen to be small and at the same time the counterterm should not be switched off too slowly for large values ofH, otherwise significant errors might occur. In other respects these parameters are arbitrary.
One can see from Fig. 2 that use of the counterterm technique eliminates the unphysical behaviour of the conductance at low magnetic fields. In the presence of counterterm, Σ Λ→0 σ converges to a finite value, which entails the nonzero value of the conductance and significantly improves agreement with the NRG data. Although the calculated conductance deviates slightly from the corresponding NRG results, the conductance per spin almost reaches the unitary limit value at zero magnetic field, G σ (H = 0)| Vg=0 ≈ 0.98e 2 /h.
We have verified that the proposed fRG scheme is stable with respect to a choice of the actual form of the counterterm, as well as parameters Λ c ,H, and Λ 0 . In particular, in Fig. 3 we plot the dependence of the effective energy levels of the quantum dots Λ j,σ = 0 σ + Σ Λ jj,σ on the cutoff parameter Λ in different schemes. One can see, that although the flow of the levels is scheme dependent, the final result does not depend on the scheme and initial magnetic field; the latter can be varied in a rather broad range.
n j,σ = dω 2π e iω0 + G Λ=0 0,σ (iω) −1 − Σ Λ→0 σ −1 jj at T = 0
are shown in Fig. 4. One can see that the self-energies and occupation numbers are strongly split (with respect to spin projection) in some range |V g | < V c g near the halffilling V g = 0 even when H → 0. As we discuss below, this reflects the formation of local moment in this range of gate voltages. There is small splitting of self-energies also in the vicinity of the transition point (in the shaded area) in the paramagnetic phase |V g | > V c g , which is an artifact of the present approach. We also note, that in the vicinity of the transition point, the proposed fRG approach may overestimate the renormalization of the vertices and for this reason may become less accurate for V g ≈ V c g (in the shaded areas in the Fig. 4a). However, we have found that even for these gate voltages all the observables are described correctly in comparison with the NRG calculations.
The local moment appears due to the specific charge redistribution, which in the symmetric case t α i = t is related to the even and odd orbitals d e(o),σ , yielding one electron in the odd orbital, disconnected from leads, and changing the character of the spin-spin correlations [14]. To show this explicitly, we plot the total occupation numbers n e(o) = σ n e(o),σ = σ d † e(o),σ d e(o),σ in Fig. 5(a). In agreement with Ref. [14] in finite range of gate voltages |V g | < V c g the odd orbital is occupied by one electron n o = 1, forming a S = 1/2 lo- cal moment, which is aligned along the direction of the magnetic field. The occupation numbers n e,σ for the even orbital behave smoothly due to the level broadening caused by the hybridization of the even orbital with leads. In Fig. 5(b) we also plot the gate voltage dependence of the square of the spin in the even(odd) orbitals S 2 e(o) = (1/4) σ,σ (d † e(o),σ σd e(o),σ ) 2 (σ are the Pauli matrices). The square of the spin on the odd orbital, S 2 o is equal to S(S + 1) = 3/4 in the region |V g | < V c g and becomes zero for the |V g | > V c g . In contrast, the square of the spin in the even orbital behaves smoothly and does not shows a significant change at the V g = ±V c g . The average of the square of the total spin S 2 t (see Fig. 5(b)) takes an almost constant value, S 2 t ≈ 1.2, for |V g | < V c g and has a weakly pronounced maximum at V g = 0. In this region of gate voltage S 2 t > S 2 e + S 2 o = S 2 1 + S 2 2 , which indicates the presence of ferromagnetic correlations.
The dependence of the interaction energy E int = E − E 0 , i.e. a difference between the energy E of interacting and E 0 of non-interacting systems, on the gate voltage (shown in Fig. 5(c)) indicates that the jumps of occupation numbers correspond to the cusp of the E(V g ) dependence, confirming first-order phase transition at |V g | = V c g . In order to understand the mechanism that leads to the existence of spin splitting of energy levels in the SFL state, we rewrite the Hamiltonian (2) for the symmetric case t α i = t in terms of even and odd orbitals (up to a constant contribution) as
H dot = k,σ 0 σ − U 2 n k,σ − U S e S o + U 2 n e,↑ n e,↓ + n o,↑ n o,↓ + 1 2 n o n e + d † e,↑ d o,↑ d † e,↓ d o,↓ + H.c. .(17)
The Hamiltonian (17) has a form of the two-orbital Anderson model with intra-orbital, inter-orbital, Hund exchange interaction, and pair electron hopping equal to U/2. In the SFL (local moment) phase even for the infinitesimally small magnetic field, the Hund exchange leads to strong splitting of itinerant electronic states
−U S e S o ∼ − U 4 σ σn e,σ .
As a result the Zeeman energy splitting of even states becomes ∆ ↑↓ ∼ U/2, instead of ∆ ↑↓ = H in the absence of interaction U , and therefore it is strongly enhanced. Due to half filling of the odd orbital, the pair hopping term is not active in the SFL phase.
The results showing spin splitting in SFL phase refer to H → 0 limit of Green functions at T = 0. Presence of the local moment makes however non-commutative limits T → 0 and H → 0. In the opposite limit H = 0, T → 0 our NRG calculations confirm the validity of Logan et al. result [44], expressing the Green function through the arithmetic average of spin-split Green functions G(H = 0) = (G ↑ + G ↓ )/2, calculated at small finite magnetic field (which can be therefore extracted from fRG calculation).
The resulting dependence of the conductance G(V g ), obtained at T = 0 and H → 0 is represented in Fig. 6. We can see that while the conductance within the both sharp and smooth cutoffs drops to zero in the SFL state, the fRG approach with the cutoff procedure (15) leads to a finite conductance in the whole range |V g | < V c g . The conductance in fRG almost reaches unitary value G 0 = 2e 2 /h at V g = 0 and in the symmetric case t α i = t shows discontinuity at the gate voltage V c g , corresponding to a quantum phase transition from SFL to a regular FL ground state, which agrees well with our NRG results and previous NRG analysis for strong on-site Coulomb repulsion [13]. It can be seen that the conductance in the singular Fermi liquid regime is well described by the proposed approach; as expected, the agreement with NRG becomes better as U/Γ is decreased. At the same time, for U/Γ α i = 6, the disagreement with NRG is still not too large close to half-filling. Furthermore, we have verified that this holds for U/Γ α i 10. The conductance in the opposite limit T → 0 at H = 0 may differ from the above result due to additional phase shift [44]. Neglecting vertex corrections, the conductance in this limit is given by
G = 4G 0 | i,j (Γ L i Γ R j ) 1/2 G Λ→0
ji (H = 0)| 2 , which yields somewhat smaller value, than at T = 0, H → 0.
B. Asymmetric coupling to the leads
In order to consider influence of asymmetric coupling to the leads on the SFL phase and behaviour of the conductance we also carried out the fRG calculation for the cases, when the coupling part of the double quantum dot Hamiltonian have the more general form with different hopping parameters t α i . It turns out that for rather different sets of the hopping parameters we face again with the problem of the unphysical behaviour of the vertex functions in the standard fRG schemes at low magnetic field; as for the isotropic case the fRG approach with the counterterm fixes this problem and physical behavior near the half-filling can be described.
As an example, in Fig. 7 we plot the conductance (a), effective energy levels (b) and occupation numbers (c) of the parallel quantum dot system with up-down coupling asymmetry Γ L 1 = Γ R 1 = U/3 and Γ L 2 = Γ R 2 = U/6. From Fig. 7b) one can see that as in the perfectly symmetric case there exists a critical gate voltage V c g below which energy levels acquire again finite spin splitting in the presence of infinitesimal magnetic field. As expected, in the case when the quantum dots are asymmetrically connected to leads the value of the splitting become different for both quantum dots. For |V g | < V c g the energy levels of quantum dot, weakers hybridized with the leads, are strongly split, which leads to a significant difference between the average occupation numbers of up-and downspin states for this dot (see Fig. 7c). In contrast, energy levels of the quantum dot 1,σ , stronger hybridized with the leads, do not exhibit such a strong splitting and the corresponding occupation numbers n 1,σ are more sensitive to a change of the gate voltage. The overall behavior of the renormalized energy levels and occupation numbers is nevertheless similar to that obtained for the symmetric coupling to the leads, showing that SFL state is not restricted to the symmetric coupling of quantum dots to the leads. The main difference with the symmetric case is that the transition to SFL state is continuous in the presence of the asymmetry. The absence of the discontinuities in the asymmetric case is a consequence of the interaction induced non-zero hopping between even and odd orbitals. The corresponding transition to SFL phase can be therefore viewed as a (second-order) quantum phase transition.
The conductance of parallel quantum dot system with up-down coupling asymmetry shown in Fig. 7a) has also no discontinuities and shows a sharp asymmetric Fanolike resonance at a gate voltage V c g , indicating the SFL behavior. The NRG calculations for asymmetric cases become progressively more complicated (especially when all hopping parameters are different), and they are not therefore presented here. Due to rich phase diagram of these asymmetric cases, with several independent hopping parameters and gate voltage, a more detail analysis of these cases will be given in the upcoming publication [45].
V. CONCLUSION
In summary, we have proposed the version of the functional renormalization-group approach, which is able to describe interaction-induced local moments in quantum dot systems, which occur due to peculiarities of geometric structure of these systems. We applied the proposed scheme to obtain occupation numbers, square of the spin, and conductance as a function of gate voltage in parallel quantum dots, symmetrically coupled to leads, and found good agreement with NRG calculations. Investigation of the gate voltage dependence of the interaction energy shows confirms first order quantum phase transition at the gate voltage V c g where abovementioned quantities show a jump. For parallel quantum dots, differently coupled to the leads, the conductance shows Fano-like resonance.
The proposed approach can be further used for describing equilibrium and non-equilibrium processes in complex quantum dot systems or organic molecules, which are not accessible for NRG approach. The presence of local moments in some geometries of these systems can be in particular utilized in quantum computation devices.
FIG. 2 .
2(Color online). Linear conductance 2G/G0 (G0 = 2e 2 /h) at T = 0 as a function of magnetic field H for the symmetric case Γ α i = U/4 and Vg = 0. Dashed green and dashed-dotted blue lines correspond to fRG approximation in the sharp-cutoff and Litim-type cutoff scheme, respectively. Solid black upper line: NRG calculation. Solid red line: fRG approach with the counterterm χ Λ 1 , Eq. (16) withH = 0.1U and Λc = 0.05U .
FIGFIG. 4 .
4. 3. (Color online). Renormalization of the energy levels Λ σ = Λ 1(2),σ with spin up σ =↑ (lower curves) and down σ =↓ (upper curves) in the fRG approximation with different counterterms for Γ α i = U/4 (note that in this caseΛ 1,σ = Λ 2,σ ), and H = Vg = 0. Solid (black) line and dasheddotted-dotted (red) lines: the linear counterterm (Λc =H/2) withH = 0.2U andH = 0.02U , respectively. Dashed (green) and dashed-dotted (blue) lines: exponential counterterm (Λc = 10Λ0 = 0.05U ) withH = 0.2U andH = 0.1U , respectively. IV. SPIN SPLITTING, LOCAL MOMENTS IN THE SFL PHASE, AND THE CONDUCTANCE A. Symmetric coupling to the leadsThe results of the calculation of spin-dependent selfenergies Σ Λ→0 ij,σ (in the following the renormalized energy levels Λ→0 j,σ are used as a measure of the diagonal diagonal self-energies Σ Λ→0ii,σ ) and occupation numbers Upper panel: Effective energy levels Λ→0 1(2),σ (thick solid (red) and dashed (black) line for σ =↑, ↓, respectively) and off-diagonal self-energies (effective hopping between the levels) ΣΛ→0 12,σ (thin solid (green) and dashed (blue) line for σ =↑, ↓) in units of U as a function of the gate voltage for Γ α i = U/4 and H → 0. Lower panel: the average occupation numbers n 1(2),σ (solid (red) line and dashed (black) line for σ =↑, ↓) as a function of the gate voltage for the same parameters. The calculations were performed within the fRG approach with the linear counterterm (H/U = 0.1, Λc/U = 0.05).
FIG. 5 .
5(Color online). a) and b) The occupation numbers n e(0) and the average square of magnetic moment S 2 e(0)in the even (dashed (black) lines) and odd (solid (red) lines) orbitals, as well as the average of the square of the total spin S 2 t = (S1 + S2) 2 (dashed-dotted (blue) line) as a function of the gate voltage for Γ α i = U/4; (c) The interaction energy Eint of the double quantum dot system as a function of Vg/U .
FIG. 6 .
6(Color online). The dependence of linear conductance G on gate voltage Vg for Γ α i = U/4 (upper plot), Γ α i = U/2 (middle plot) and Γ α i = U/6 (lower plot), H → 0 (at temperature T = 0) within the fRG approach with the linear counterterm (H/U = 0.1, Λc/U = 0.05, solid black line) and NRG calculation (dashed red line). Dashed-dotted green and dashed-dashed dotted blue lines on upper plot are fRG approaches in the sharp-cutoff and Litim-type cutoff scheme without counterterm. dashed (red) line for σ =↑ and thick solid (black) line for σ =↓) and Λ→0 2,σ (thin dashed (blue) line for σ =↑ and thin solid (green) line for σ =↓), as well as the spin-resolved average occupation numbers (c) of each quantum dot n 1(2),σ (with the same notation of the curves as on (b)) in the asymmetric case with Γ L 1 = Γ R 1 = U/3 and Γ L 2 = Γ R 2 = U/6 obtained by the fRG approach with the linear counterterm (H/U = 0.1, Λc/U = 0.05).
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| [] |
[
"Electric-Field Control of Magnetic Anisotropies: Applications to Kitaev Spin Liquids and Topological Spin Textures",
"Electric-Field Control of Magnetic Anisotropies: Applications to Kitaev Spin Liquids and Topological Spin Textures"
] | [
"Shunsuke C Furuya \nDepartment of Physics\nIbaraki University\n310-8512MitoIbarakiJapan\n",
"Masahiro Sato \nDepartment of Physics\nIbaraki University\n310-8512MitoIbarakiJapan\n"
] | [
"Department of Physics\nIbaraki University\n310-8512MitoIbarakiJapan",
"Department of Physics\nIbaraki University\n310-8512MitoIbarakiJapan"
] | [] | Magnetic anisotropies originate from the spin-orbit coupling and determine the phase that magnetic systems fall into. We develop a microscopic theory for DC electric-field controls of magnetic anisotropies in magnetic Mott insulators and discuss its applications to Kitaev materials and topological spin textures. First, we derive a Kitaev-Heisenberg model based on a microscopic approach.In the presence of a strong intra-atomic spin-orbit coupling, DC electric fields add non-Kitaev interactions such as a Dzyaloshinskii-Moriya interaction and an off-diagonal Γ interaction to the Kitaev-Heisenberg model and can induce a topological quantum phase transition between Majorana Chern insulating phases. We also investigate the inter-atomic Rashba spin-orbit coupling and its effects on topological spin textures. DC electric fields turn out to create and annihilate magnetic skyrmions, hedgehogs, and chiral solitons. arXiv:2110.06503v2 [cond-mat.str-el] 21 Apr 2022Here, U d,p , J H , and ∆ dp = E d − E p are microscopic parameters of H U and denote the on-site Coulomb repulsion of d, p orbitals, the p-orbital Coulomb exchange, and the d-orbital eigenenergy (E d ) relative to the p-orbital one (E p ), respectively[55]. Note that the Kitaev coupling K is independent of the SOC because we discarded the J eff = 3/2 in the derivation of Eq. (3) (see Fig. S3 of the Supplementary Material). Inclusion of the J eff = 3/2 levels makes K depend on the SOC [70, 71].We can build the Kitaev-Heisenberg model on the honeycomb lattice from our model (3)[55,69]. Unlike Ref.[31], our Hamiltonian contains the Heisenberg interaction. This discrepancy is attributed to details of intrinsic hoppings (SeeFig. S3of Supplementary Material and Ref.[69]). The ferromagnetic superexchange J F < 0 accords with a universal law, the Goodenough-Kanamori rule[72][73][74]. By contrast, the antiferromagnetic Kitaev coupling K > 0 is less universal, and arises from a specificity of our model. Including direct hoppings of J eff = 1/2 and J eff = 3/2 orbitals at M 1 and M 2 turns the Kitaev coupling K ferromagnetic[31,55,62,69].We now give our attention to E-driven interactions absent in the model (3) on the xy plane. Inplane and our-of-plane electric fields yield different hopping processes. The in-plane electric field yields hoppings, δH t (E) = −I σ=± {E y (p † z,σ d 1,yz,σ +p † x,σ d 1,xy,σ + H.c.) + E x (p † z,σ d 2,zx,σ + p † y,σ d 2,xy,σ + H.c.} with an over- | null | [
"https://arxiv.org/pdf/2110.06503v2.pdf"
] | 238,744,223 | 2110.06503 | 357e979471d994ac360470a76fe005599a0a96c3 |
Electric-Field Control of Magnetic Anisotropies: Applications to Kitaev Spin Liquids and Topological Spin Textures
Shunsuke C Furuya
Department of Physics
Ibaraki University
310-8512MitoIbarakiJapan
Masahiro Sato
Department of Physics
Ibaraki University
310-8512MitoIbarakiJapan
Electric-Field Control of Magnetic Anisotropies: Applications to Kitaev Spin Liquids and Topological Spin Textures
(Dated: April 22, 2022)
Magnetic anisotropies originate from the spin-orbit coupling and determine the phase that magnetic systems fall into. We develop a microscopic theory for DC electric-field controls of magnetic anisotropies in magnetic Mott insulators and discuss its applications to Kitaev materials and topological spin textures. First, we derive a Kitaev-Heisenberg model based on a microscopic approach.In the presence of a strong intra-atomic spin-orbit coupling, DC electric fields add non-Kitaev interactions such as a Dzyaloshinskii-Moriya interaction and an off-diagonal Γ interaction to the Kitaev-Heisenberg model and can induce a topological quantum phase transition between Majorana Chern insulating phases. We also investigate the inter-atomic Rashba spin-orbit coupling and its effects on topological spin textures. DC electric fields turn out to create and annihilate magnetic skyrmions, hedgehogs, and chiral solitons. arXiv:2110.06503v2 [cond-mat.str-el] 21 Apr 2022Here, U d,p , J H , and ∆ dp = E d − E p are microscopic parameters of H U and denote the on-site Coulomb repulsion of d, p orbitals, the p-orbital Coulomb exchange, and the d-orbital eigenenergy (E d ) relative to the p-orbital one (E p ), respectively[55]. Note that the Kitaev coupling K is independent of the SOC because we discarded the J eff = 3/2 in the derivation of Eq. (3) (see Fig. S3 of the Supplementary Material). Inclusion of the J eff = 3/2 levels makes K depend on the SOC [70, 71].We can build the Kitaev-Heisenberg model on the honeycomb lattice from our model (3)[55,69]. Unlike Ref.[31], our Hamiltonian contains the Heisenberg interaction. This discrepancy is attributed to details of intrinsic hoppings (SeeFig. S3of Supplementary Material and Ref.[69]). The ferromagnetic superexchange J F < 0 accords with a universal law, the Goodenough-Kanamori rule[72][73][74]. By contrast, the antiferromagnetic Kitaev coupling K > 0 is less universal, and arises from a specificity of our model. Including direct hoppings of J eff = 1/2 and J eff = 3/2 orbitals at M 1 and M 2 turns the Kitaev coupling K ferromagnetic[31,55,62,69].We now give our attention to E-driven interactions absent in the model (3) on the xy plane. Inplane and our-of-plane electric fields yield different hopping processes. The in-plane electric field yields hoppings, δH t (E) = −I σ=± {E y (p † z,σ d 1,yz,σ +p † x,σ d 1,xy,σ + H.c.) + E x (p † z,σ d 2,zx,σ + p † y,σ d 2,xy,σ + H.c.} with an over-
Introduction.-The spin-orbit coupling (SOC) is of significance to topological electronic states of matters [1][2][3][4][5][6]. Recently, magnetic anisotropies, a direct descendent of SOC, has enjoyed renewed theoretical and experimental interests for their essential roles in topological states of magnetic materials such as topological spin textures (TST) [7,8] and Kitaev spin liquids [9][10][11][12][13].
The SOC yields antisymmetric magnetic anisotropies also known as the Dzyaloshinskii-Moriya interaction (DMI) [14,15]. Competitions between the DMI and ferromagnetic exchange interactions yield TST such as magnetic skyrmions [Figs. 1 (a) and (b)] [16][17][18][19][20], magnetic hedgehogs [21][22][23][24], and chiral solitons [ Fig. 1 (c) and (d)] [25][26][27]. Carrying a nonzero topological index, TST is robust against disturbances and thus promising for device applications [7,8,28].
The SOC also gives inversion-symmetric magnetic anisotropies that render the Kitaev material topological [9][10][11][12][13][29][30][31][32][33][34][35]]. The Kitaev model and its derivatives [9,13,[29][30][31][32][33][34][35] are significant to fundamental physics and simultaneously relevant for quantum computation. In light of scientific interests and engineering applications, controlling methods of these SOC-driven topological states of magnetic materials are currently one of the central issues in condensed-matter physics, quantum physics, and applied physics [36][37][38][39][40][41][42].
The DC electric field holds promise for external controls of topological states of magnetic materials. Noncollinear magnetic orders of TSTs can drive the electric polarization [45][46][47][48][49][50][51]. For device applications, however, we need to create the TST even when we cannot expect the noncollinear magnetic order in the first place [e.g., the ferromagnetic state of Figs. 1 (b),(d)]. Controlling magnetic anisotropies will enable such a creation of the TST. The situation becomes even more nontrivial in Kitaev materials with no magnetic orders. On the other hand, recent technological advances such as electric double-layer transistors [52,53] and intefacial engineerings [38,54,55] make strong DC electric fields of ∼ 1-10 MV/cm available. Scanning tunneling microscopes (STMs) can also yield DC electric fields of ∼ 10 MV/cm (6), including helical (HL), Skyrmioncrystal (SkX), and ferromagnetic (FM) phases [16-18, 43, 44]. (c) Chiral soliton lattice (CSL) in spin chain. (d) Schematic phase diagram of classical Heisenberg ferromagnetic chain with DMI, including HL, CSL, and FM phases [25][26][27]. (e) Kitaev model on honeycomb lattice.
locally [55][56][57]. To design materials and experiments, we need detailed information of how the external DC electric field changes microscopic parameters qualitatively and quantitatively [58,59]. However, such a microscopic theory is not available yet for DC electric-field controls of magnetic anisotropies.
This Letter develops a microscopic quantum theory for DC electric-field controls of magnetic anisotropies in magnetic Mott insulators. We show how the DC electric field E controls TSTs and Kitaev quantum spin liquids (QSLs). We emphasize that our theory is generic and applicable to various magnetic systems beyond these specific examples. (1) with zeroth-order term omitted. The black solid, blue solid, and red dashed arrows denote the total hoppings Ht(E), the intrinsic hoppings Ht(0), and the E-driven hoppings δHt, respectively.
Framework.-The spin Hamiltonian determining magnetic properties of Mott insulators originates from virtual hopping processes of electrons under strong interactions. These hopping processes are well described by Hubbardlike tight-binding models [15,31,[59][60][61][62][63]. Considering the locality of virtual hopping processes in Mott insulating phase, we deal with models with three sites, two magnetic-ion sites (M 1 and M 2 ) with half-occupied d orbitals and a ligand site (L) with fully occupied p orbitals. We limit ourselves to S = 1/2 quantum spin systems with superexchange interactions mediated by p orbitals for simplicity. Note that we can treat the S = 1/2 spin and a J eff = 1/2 pseudospin equally, as we see later.
The SOC and a crystal-electric field (CEF) lift the orbital degeneracy and define orbitals relevant to lowenergy quantum spin systems. The SOC can generate spin-flipping electron hoppings intrinsically or otherwise extrinsically in collaboration with the DC electric field E. Spin-flipping hoppings lead to magnetic anisotropies [15]. This Letter deals with microscopic Hubbard-like models with a generic Hamiltonian H = H U (E) + H t (E), where H U (E) and H t (E) are intra-atomic and inter-atomic interactions under electric fields E, respectively. H t (0) denotes the intrinsic electron hoppings and the difference δH t (E) := H t (E) − H t (0) contains extrinsic E-driven hoppings.
We regard hoppings H t (E) as a perturbation to H U (E). Performing the fourth-order perturbation expansion, we obtain the effective spin Hamiltonian [59],
H spin = P H U P + P H t 1 E g − H U QH t 3 P,(1)
where P is the projection operator to the Mott-insulating ground states of H U (E) with the eigenenergy E g and Q = 1 − P is to its complementary space. The zerothorder term P H U (E)P is mostly irrelevant but gives the uniform Zeeman energy, P H U P = − a=x,y,z j h a S a with h = gB for the magnetic field B = µ 0 H [59], where g is the electron's g tensor. Hereafter, we call h the magnetic field for simplicity. The fourth-order process gives the ligand-mediated superexchange interaction. Figure 2 shows a diagrammatic representation of the fourth-order process in Eq. (1). The black, blue, and red arrows represent the total hoppings H t (E), the intrinsic ones H t (0), and the E-driven ones, δH t (E), respectively. The intrinsic O(|E| 0 ) diagram typically contains the Heisenberg superexchange interaction [59,60]. Hopping amplitudes of the E-driven hoppings are O(|E|) because they arise from a distortion of electron orbitals' wave functions [55]. Importantly, the electric field can break a reflection symmetry. We may rephrase the reflection-symmetry-breaking distortion as a locally generated E-driven electric polarization. The atomic-scale electric polarization triggers intrinsically forbidden hopping processes δH t (E). We obtain the O(|E|) correction to the spin Hamiltonian following the diagram. This |E|linear approximation works even under (currently feasible) strong electric fields regardless of whether the field is extrinsic or comes from interfacial electric fields.
j J eff = 1/2 p x p y p z e g J eff = 3/2 (a) (b) M 1(2) L M 1 L M 2 L 1 L 2 x y z x y z (e) M 1 L M 2 E D F x =D F y (c) M 1 L M 2 (d) E D F z M 2 M 1 FIG. 3. (a)
Kitaev-Heisenberg model.-Based on the generic framework, we discuss Kitaev materials. We consider a low-spin d 5 electron configuration under the octahedral CEF and the strong SOC [31], where the J eff = 1/2 doublet hosts the J eff = 1/2 (pseudo)spin [ Fig. 3 (a)]. The J z eff = 1/2 state is a spin-orbit-entangled superposition of t 2g states, |↑, l z = 0 and |↓, l z = 1 [31,[64][65][66][67][68][69]. Here, we assume an isosceles right triangle M 1 -L-M 2 formed on the xy plane [ Fig. 3 (b)]. In terms of the creation and annihilation operators of the J eff = 1/2 doublet, the intrinsic hoppings are given by [55]:
H t (0) = t √ 3 σ=± [(p † y,σ + ip † z,−σ )d 1,Γ7+,σ + (p † x,σ + σp † y,−σ )d 2,Γ7+,σ + H.c.]. (2)
Here, Γ 7+ labels the irreducible representation of the J eff = 1/2 doublet [64,66]. The intrinsic hoppings (2) lead to the following spin Hamiltonian, 46,55], keeping a reflection symmetry about the xy plane, (x, y, z) → (x, y, −z). The reflection symmetry about the xy plane permits the DMI with the nonzero z component [ Fig. 3 (c)]. Explicit calculations of Eq. (1) show that the in-plane electric field adds
H spin (0) = J F S 1 · S 2 + KS z 1 S z 2 − a=x,y,z j=1,2 h a S a j (3) with J F = − 8 3 t 4 ( lap integral I [D F · S 1 × S 2 = D z F (S 1 × S 2 ) z with D z F JF = − 4I t (E x +E y )(U d −Up+∆ dp )+(E x −E y )JH 2(U d −Up)+JH
to the Kitaev-Heisenberg Hamiltonian (3) in accordance with the reflection symmetry [55].
The out-of-plane electric field E = E z e z violates the reflection symmetry about the xy plane but keeps another reflection symmetry, (x, y, z) → (y, x, z), about a plane M := {(x, y, z)|x = y} perpendicular to the M 1 -M 2 bond. This reflection symmetry confines the DMI vector D F to the intersection of the M plane and the xy plane. Straightforward calculations give
H spin = J F S 1 · S 2 + KS z 1 S z 2 − a=x,y,z j=1,2 h a S a j + D F · (S 1 × S 2 ) + Γ [(S x 1 + S y 1 )S z 2 + S z 1 (S x 2 + S y 2 )],(4)
with [55]. The out-of-plane electric field also drives the symmetric off-diagonal magnetic anisotropy, Γ (S y 1 S z 2 + S z 1 S y 2 ), known as the Γ interaction [75,76] in the context of Kitaev materials. Both the DMI and the Γ interaction accord with the reflection symmetry about M.
D F = D x F (e x + e y ), D x F = 16t 3 3 E z I( 1 U d −Up+∆ dp ) 2 JH 4(U d −Up+∆ dp ) 2 −JH 2 , and Γ = 32t 3 9 E z I( 1 U d −Up+∆ dp ) 2 U d −Up+∆ dp 4(U d −Up+∆ dp ) 2 −JH 2
Gapped QSL.-Much effort is being made to investigate effects of non-Kitaev interactions on QSL states of the pure Kitaev model in connection with Kitaev materials such as α-RuCl 3 [77][78][79][80][81][82][83][84].
The Kitaev model can have a gapped QSL phase with the Chern number C = 1 when h x h y h z = 0 [29]. The Majorana mass gap ∝ |h x h y h z | is the third order of the external magnetic field. Interestingly, the Γ interaction generates the mass ∝ |h x Γ | of the Majorana fermion together with the magnetic field [75,76]. If Γ = 0 for E = 0, the Majorana mass ∝ |h x E z | is the second order about the external electromagnetic fields. The Edriven Γ interaction can induce topological phase transitions between gapped QSLs with different Chern numbers [76]. A recent study [85] pointed out that the DMI can also drive toplogical quantum phase transitions between gapped QSLs under magnetic fields.
We can build the Kitaev-Heisenberg-Γ model without the DMI when L 1 and L 2 are crystallographically equivalent [Figs. 1 (e) and 3 (e)] under electric fields along the [111] direction [55]. If L 1 and L 2 are nonequivalent, the electric field also adds the DMI to the Kitaev-Heisengerg-Γ model. We may expect topological phase transitions due to the E-driven DMI and Γ interaction in a Kitaev material close enough a topological quantum transition line [76].
For α-RuCl 3 , we can use parameters U d = 2.5 eV, U p = 1.5 eV, ∆ dp = 5.5 eV, J H = 0.7 eV, and t = 0.88 eV [84]. These parameters give J F = −3 meV. As we mentioned, our model gives antiferromagnetic K > 0 but can make it ferromagnetic K < 0 with slight modifications of the model. If we ignore the E dependence of J F as we did thus far in this Letter, we find that a DC electric field 1-10 MV/cm change |D x F | and |Γ | only by 10 −4 -10 −3 % of J F , which is too much underestimated. Note that the 1-10 MV/cm DC electric field can reduce J F by 1-10 % [59]. This reduction enhances the change in the ratios |D x F /J F | and |Γ /J F |. We can further enhance the DC electric-field effect, by including Rashba-SOC-driven hoppings besides the hitherto considered intra-atomic SOC.
Rashba spin-orbit coupling.-We can effectively regard that the single-electron HamiltonianĤ 1 contains the Rashba SOC, namely, −α R (σ × k) · e(r), where k is the wavevector of the electron in the crystal and e(r) = E(r)/|E(r)| is the unit vector parallel to the electric field E(r). We can deem E(r) the external electric field, a surface electric field for thin-film materials [86,87], or an interfacial electric field for field-effect transistors [52,53]. The coefficient α R is proportional to |E(r)|. The Rashba SOC affects the single-electron Hamiltonian and turns into spin-flipping hoppings and eventually into the DMI.
For demonstration, we again consider the isosceles right triangle of Fig. 3 (b). We also inherit the electron configuration, the low-spin d 5 configuration under the octahedral CEF. The only difference from the previous J eff = 1/2 model lies in SOC mechanisms. This time, we consider Rashba-SOC-driven hoppings but ignore the d-orbital splitting due to the SOC [55]. Given that the d xy orbital carries the S = 1/2 spin, we find H t (0) = t σ (p † y,σ d 1,xy,σ +p † x,σ d 2,xy,σ +H.c.). The superexchange interaction is again ferromagnetic:
H spin (0) = J R S 1 · S 2 with J R = −8t 2 JH 4(U d −Up+∆ dp ) 2 −JH 2 ( 1 U d −Up+∆ dp ) 2 < 0 [59].
The Rashba SOC due to the out-of-plane electric field E = E z e z gives [1,2]
δH t (E) = s,s [iλp † y,s (σ ss × d 1 ) z d 1,xy,s + iλp † x,σ (σ ss × d 2 ) z d 2,xy,s + H.c.],(5)
with a unit vector d j pointing toward the ligand site from M j . The hopping amplitude λ equals to α R |k|. The E-driven hoppings (5) yield the in-plane [55]. Note that the DMI violates the inversion symmetry about the M plane because the Rashba SOC −α R (σ × k) z does.
DMI D R · S 1 × S 2 with D R = D x R e x + D y R e y and D x R = 32λt 3 ( 1 U d −Up+∆ dp ) 2 JH 4(U d −Up+∆ dp ) 2 −JH 2 and D y R = −32λt 3 ( 1 U d −Up+∆ dp ) 2 U d −Up+∆ dp 4(U d −Up+∆ dp ) 2 −JH 2
For a parameter set U d = 3 eV, U p = 2 eV, ∆ dp = 5 eV, J H = 1 eV, t = 0.1 eV, and λ = 0.05 eV, we obtain |D x R /J R | ≈ 0.46. We used a value λ = 0.05 eV, much smaller than λ = α R |k| ∝ |E(r)| estimated in some materials [88,89]. Because α R is well controllable with the electric field [89], we may expect that the external DC electric field can control the DMI substantially both for Kitaev spin liquids and TSTs.
Magnetic skyrmion lattice.-We can directly apply our Rashba SOC argument to controls of magnetic skyrmion lattice [16-18, 43, 44]. We exemplify this application by considering a two-dimensional (2D) array of edgesharing octahedra whose centers have magnetic ions and vertices have ligand ions [ Fig. 4 (a)]. The magnetic ions form the square lattice on the xy plane. We can build this many-body model with the small square plaquette of Fig. 4 (b) made of two isosceles right triangles M 1 -L 1 -M 2 and M 1 -L 2 -M 2 . Assuming the Rashba SOC (5) on the M j -L j bonds, we obtain the spin Hamiltonian for the building block of Fig. 4
(b): (a) (c) (d) E q 1 q 2 q 3 (b) M 1 M 2 E xH spin = J R S 1 · S 2 + D ⊥ e ⊥ · (S 1 × S 2 ) − h z j S z j ,
where we applied both the electric and magnetic fields parallel to the z axis. The vector
D ⊥ e ⊥ with D ⊥ = √ 2(D x R +D y R ) isH SkX = −|J R | r (S r · S r+ex + S r · S r+ey ) − h z r S z r + D ⊥ (E z ) r (S r × S r+ex · e y − S r × S r+ey · e x ).(6)
The model (6) realizes the Néel-type magnetic skyrmion lattice because a 90 (6) into a model that exhibits the Bloch-type skyrmion lattice [43]. The out-of-plane electric field thus controls the ratio Fig. 1 Fig. 4 (c)] with the same square plaquette, we can construct a one-dimensional version of the model (6). This ferromagnetic chain exhibits the CSL phase. The ML 2 chain is typically realized for (M, L) = (Cu, O). The electric field controls the ratio D ⊥ (E z )/J R (E z ) and turns the ferromagnetic state into the CSL [ Fig. 1 (d)] [25][26][27].
• rotation (S x r , S y r , S z r ) → (−S y r , S x r , S z r ) turns the modelD ⊥ (E z )/J R (E z ) [
Magnetic hedgehog lattice.-The Rashba SOC is appropriate to drive the chiral structure in a certain direction by generating a uniform DMI. This feature of the Rashba SOC will also be useful for multiple-q states.
Provided that a magnetic material shows a double-q state with the two q vectors lying on the q x -q y plane [q 1 and q 2 of Fig. 4 (e)] and the electric field E adds another q vector (q 3 ). This feature of the electric field will drive the magnetic hedgehog lattice where the triple-or quadrupleq spin texture is required [8,24,90].
Discussions.-This Letter provides the general theoretical foundation to DC electric-field controls of magnetism [36][37][38][91][92][93][94][95][96][97], which takes on a growing importance with ongoing advances in strong DC electric-field source [36,37,52,53], including a single-cycle terahertz laser pulse [98][99][100][101][102][103]. For theoretical reference, we discuss a simple antiferromagnetic model with a d 9 electron configuration and weak intra-atomic SOC within p orbitals in the supplementary material [55].
For experiments, we propose to use (quasi-)2D systems. We can generally apply strong enough electric fields to quasi-2D materials with state-of-the-art techniques such as double-layer transistors [53] and STM [56]. The above-mentioned α-RuCl 3 [104] and other Kitaevcandidate materials [31] have a quasi-2D layered honeycomb structure. Quasi-2D materials are compatible with the strong DC electric-field source [36,37,52,53]. Be-sides, the quasi-2D structure allows us to use the strong electric field on the interface to another material. Generation of strong surface electric fields were already experimentally available [52,53]. Ref. [38] gives an experimental controlling method of the DMI with the interface electric field in the SrRuO 3 -SrIrO 3 bilayer system. To conclude, we point out that 2D van der Waals magnets exhibit large electric-field effects. Electric-field switching between ferromagnetic and antiferromagnetic states were already experimentally reported [105,106]. Our manuscript is thereby expected to stimulate electric-field controls of magnetic aniostropies in those quasi-2D magnets. In this Section, we derive the spin Hamiltonian of the J eff = 1/2 model [Figs. S1 (a) and (b)]. The J eff = 1/2 doublet at M j are a spin-orbit-entangled superposition of t 2g orbitals. The up state (|+ j ) and the down state (|− j ) state of the (pseudo)spin at M j are given by [46,65,68,69]
|+ j = 1 √ 3 (|d j,xy,↑ + |d j,yz,↓ + i |d j,zx,↓ ), (S1.1) |− j = 1 √ 3 (|d j,xy,↓ − |d j,yz,↑ + i |d j,zx,↑ ), (S1.2)
where |d j,a,σ = d † j,a,σ |0 is a spin-σ state of the d aorbital electron at M j and |0 is the vacuum of the creation operator d † j,a,σ . This doublet is labeled as Γ 7+ using the Bethe's notation of the double group [66]. Hereafter, we denote σ =↑, ↓ as σ = +, −, respectively. Note that under the strong spin-orbit coupling (SOC), we should adopt the so-called JJ coupling scheme instead of the LS coupling one [68,69]. Here, J is a superposition of the angular momentum L and the spin S [see Eq. (S1.5)]. Whereas the SOC enters into wave functions of hybridized orbitals [see Eq. (S4.69)] in the LS-coupling scheme, it does not in the strongly spin-orbit-entangled states (S1.1) and (S1.2) in the JJ-coupling scheme. This difference comes out of a fact that the former deals with the SOC perturbatively but the latter does nonperturbatively. In Eqs. (S1.1) and (S1.2) formulated within theJJ-coupling scheme, the coefficients are fully determined by the crystalline symmetry.
It is straightforward to write the intrinsic hoppings H t (0) in terms of t 2g -orbital operators. As Figs. S1 (c) and (d) show, the electron in the p y orbital can hop directly to the d xy orbital at M 1 but cannot to that at M 2 because the d xy orbital has the odd parity about a reflection (x, y, z) → (x, −y, z) but the p y orbital has the even parity. Likewise, the electron in the p x orbital can hop directly to the d xy orbital at M 2 but cannot to that at M 1 because of the difference in the parity about another reflection (x, y, z) → (−x, y, z). Such crystalline symme- tries permits the following intrinsic hoppings at E = 0:
M 1 J eff = 1/2 p x p y p z J eff = 3/2 (a) L L M 2 x y z (b) (c) (d)H t (0) = t σ=± (p † y,σ d 1,xy,σ + p † z,σ d 1,zx,σ + p † x,σ d 2,xy,σ + p † z,σ d 2,zx,σ + H.c.). (S1.3)
The creation and annihilation operators of the t 2g -orbital electrons depend on each other in the J eff = 1/2 model because they are spin-orbit entangled in the J eff = 1/2 doublet. Here, we note that the orbital angular momentum L = −l d and the spin,
S d := 2 a=xy,yz,zx s,s =± d † j,a,s σ ss d j,a,s , (S1.4)
form the effective total angular momentum J eff as follows [67].
J eff = −l d + S d . (S1.5)
Hereafter, we set = 1 for simplicity. S d and L of the Γ 7+ doublet are antiparallel to each other because electrons in the d 5 configuration feel the strong SOC ξL · S d with large positive ξ [67]. We can easily confirm
J z eff |± j = ± 1 2 |± j . (S1.6)
Since there are few possibilities of confusions, we simply represent this pseudospin J eff as S j and call it a spin, as we did in the main text. The spin operator S j is thus defined as
S z j = 1 2 (|+ jj +| − |− jj −|), (S1.7) S ± j = |± jj ∓| . (S1.8)
Our purpose in this subsection is to write the spin Hamiltonian in terms of this spin operator S j . For this purpose, we represent creation and annihilation operators of the t 2g orbitals in terms of the J eff = 1/2 doublet, d † j,Γ7+,σ and d j,Γ7+,σ . They are defined as
|+ j =: d † j,Γ7+,+ |0 , |− j =: d † j,Γ7+,− |0 . (S1.9)
An operator, n j,Γ7+,σ = d † j,Γ7+,σ d j,Γ7+,σ , counts the number of electrons with the σ spin in the Γ 7+ doublet. We can relate t 2g -orbital operators d j,a,σ with a = xy, yz, zx to the Γ 7+ -orbital operator d j,Γ7+,σ as follows. For example, the operator d j,xy,σ satisfies
d j,xy,+ |σ j = 1 √ 3 δ σ,+ |0 , d j,xy,− |σ j = 1 √ 3 δ σ,− |0 , (S1.10)
where δ a,b is Kronecker's delta. These relations indicate d j,xy,σ P = Qd j,xy,σ P = Q 1 √ 3 d j,σ P, (S1.11) at low energies, where P = j=1,2 |+ jj +|+|− jj −| is the projection operator to the Hilbert subspace spanned by the Γ 7+ doublets and Q = 1 − P is the projection to its complementary. Note a simple relation P d j,a,σ P = 0 for a = xy, yz, zx. Likewise, we obtain d j,yz,σ P = Qd j,yz,σ P = Q − 1 √ 3 σd j,−σ P, (S1.12)
d j,zx,σ P = Qd j,zx,σ P = Q i 1 √ 3 d j,−σ P.
(S1.13)
These relations lead to
H t (0)P = t √ 3 σ=± [(p † y,σ + ip † z,−σ )d 1,Γ7+,σ + (p † x,σ + σp † y,−σ )d 2,Γ7+,σ + H.
c.] P, (S1.14)
P H t (0) = P t √ 3 σ=± [(p † y,σ + ip † z,−σ )d 1,Γ7+,σ + (p † x,σ + σp † y,−σ )d 2,Γ7+,σ + H.c.] .
(S1. 15) At low energies, we can abbreviate these relations as Eq.
(2) in the main text. We are now ready to write down the many-body Hamiltonian
H = H U + H t (0), (S1.16)
of the the J eff = 1/2 model at E = 0. The intrinsic hoppings and the intra-atomic interactions H U are given by Eq. (2) and
H U = U d j=1,2
n j,Γ7+,+ n j,Γ7+,− + U p µ=x,y,z n pµ,+ n pµ,−
+ j=1,2 V j σ=± n j,σ + V p σ=± µ=x,y,z n pµ,σ − J H µ =µ s µ · s µ − j=1,2 a=x,y,z
h a S a j , (S1. 17) respectively. Here, n pµ,σ := p † µ,σ p µ,σ is the number operator of the p µ -orbital electron. U d and U p denote the intra-band Coulomb repulsions for the d and p orbitals, V j and V p are the on-site potentials at M j and L, and J H > 0 is the ferromagnetic direct exchange between spins s µ in the p µ orbitals.
B. Spin Hamiltonian at zero electric fields
Using the intrinsic hoppings (2) as a perturbation to Eq. (S1.17), we can derive the spin Hamiltonian (1),
H spin = P − 2 9 t 4 1 U d − U p + V 1 − V p + 1 U d − U p + V 2 − V p 2 1 2(U d − U p ) + V 1 + V 2 − 2V p − J H (S 1 · S 2 − 2S z 1 S z 2 ) − a=x,y,z j=1,2
h a S a j P + const. (S1.18)
Provided that V 1 = V 2 , the spin Hamiltonian becomes the Kitaev-Heisenberg-like one (3) with
J F = − 8 9 t 4 1 U d − U p + ∆ dp 2 1 2(U d − U p + ∆ dp ) − J H , (S1.19) K = −2J F , (S1.20)
where ∆ dp = V 1 −V p = V 2 −V p corresponds to an eigenenergy difference of the J eff = 1/2 doublet and the p orbitals when no electron occupies them.
C. Building Kitaev-Heisenberg model
We can build the Kitaev-Heisenberg model x y plane and derive the effective spin Hamiltonian,
H KH = a=x,y,z i,j a (KS a i S a j + J H S i · S j ) − j h · S j ,(H spin = P KS z 3 S z 3 + J F S 3 · S 4 − j=3,4
h · S j P.
(S1.22)
To combine the octahedra of Figs. S2 (a) and (b) so as to make them share an edge and form the green M 2 -M 3 bond, the xyz and x y z coordinates should satisfy (x , y , z ) = (y, z, x), (S1.23)
turning Eq. (S1.22) into
H spin = P KS x 3 S x 3 + J F S 3 · S 4 − j=3,4
h · S j P.
(S1. 24) In fact, we can easily confirm the relation (S1.23) by overlapping the xy and x y planes with a single octahedron [ Fig. S2 (c)]. Repeating the same procedure, we obtain the Kitaev-Heisenberg model (S1.21) on the honeycomb lattice.
Let us comment on the difference of our result (S1.21) from that by Jackeli and Khalliulin [31]. They built the Kitaev model based on electron hoppings between J eff = 1/2 and J eff = 3/2 orbitals [ Fig. S3 (a)]. These hopping processes yield the Ising-type interaction S γ i S γ j whose component γ is perpendicular to the plane where the triangle M 1 -L-M 2 is put. It is the S z i S z j interaction when the triangle is put on the xy plane. By contrast, we did not include the J eff = 3/2 orbitals. In other words, we assume the strong splitting between the J eff = 1/2 and J eff = 3/2 orbitals due to the CEF and the SOC. Instead, we explicitly take the p orbitals into the perturbation expansion [ Fig. 2]. As our straightforward calculations show, the S + i S − j + S − i S + j interaction can emerge in the process (b) whereas it cannot in the process (a) when the triangle is put on the xy plane.
D. In-plane electric fields
Now, we apply the in-plane electric field E = E x e x + E y e y to the J eff = 1/2 model. Recall that we put the three-site model on the xy plane. The in-plane electric field breaks the C 2 rotation symmetry and the σ v reflection symmetry about the y = x plane. The remaining symmetry is the σ v reflection symmetry about the xy plane. The in-plane electric field permits hoppings that were intrinsically forbidden. The in-plane DC electric field induces hoppings between p orbitals and t 2g orbitals, + E x (p † z,σ d 2,yz,σ + p † y,σ d 2,xy,σ ) + H.c.], (S1. 25) where −IE y is the matrix element of the −E y P y term in the single-electron Hamiltonian:
−IE y = p x,σ |(−E y P y )|d 1,xy,σ = −eE y dr p x,σ |r y r|d j,xy,σ .
(S1.26)
The matrix element (S1.26) represents a reflectionsymmetry-breaking distortion of the wave functions of d 1,xy,σ and p x,σ orbitals caused by the electric field. The perturbation V = −E · P gives rise to the following perturbation correction to |d 1,xy,σ :
|d 1,xy,σ → |d 1,xy,σ − p x,σ |V |d 1,xy,σ E p − E d |p x,σ + · · · (S1.27)
The similar expansion holds for |p x,σ . The matrix element p x,σ |V |d 1,xy,σ = −IE y gives a measure of how much the proximate d xy and p x orbitals mix with each other. Their mixture results in the distortion of orbital probability clouds along the direction of E (Fig. S4). The distortion of the orbitals affects the spin Hamiltonian as the addition of the E-driven hoppings δH t (E).
To evaluate the constant I, we adopt the wave functions r|d j,xy,σ and r|p xy,σ of the hydrogenlike atom in terms of the the polar coordinate r = r(sin θ cos φ, sin θ sin φ, cos θ):
r|d j,xy,σ ≈ R nd (r)Y xy (θ, φ), (S1.28) r|p x,σ ≈ R 2p (r)Y x (θ, φ), (S1.29) R 3d (r) = 4 81 √ 30 Z M a 0 3/2 Z M a 0 r 2 exp − Z M 3a 0 r , (S1.30) R 4d (r) = 1 768 √ 5 Z M a 0 3/2 12 − Z M r a 0 Z M r a 0 2 × exp − Z M r 4a 0 , (S1.31) R 2p (r) = 1 2 √ 6 Z L a 0 3/2 Z L a 0 r exp − Z L 2a 0 r , (S1.32) Y xy (θ, φ) = 15 16π sin 2 θ sin(2φ), (S1.33) Y x (θ, φ) = 3 4π sin θ cos φ. (S1.34)
Here, a 0 , Z M , and Z L are the Bohr radius and the atomic numbers of M j and L, respectively [46,64]. n = 3, 4, · · · represent the principal quantum number. For n = 3, these wave functions lead to
I ≈ ea 0 16 27 Z 7/2 M Z 5/2 L Z M 3 + Z L 2 −7 .
(S1.35)
For n = 4, it becomes
I ≈ ea 0 √ 3 16 Z 7/2 M Z 5/2 L Z M 4 + Z L 2 −8 .
(S1.36)
Note the following symmetry, p a,σ |P b |d 1,ab,σ = I, (S1.37)
for a, b = x, y, z and a = b. In analogy with the intrinsic hoppings H t (0), we replace the t 2g -orbital operators with the Γ 7+ ones:
δH t (E) ∼ − I √ 3 σ=± [E y (σp † z,−σ + p † x,σ )d 1,Γ7+,σ + E x (σp † z,−σ + p † y,σ )d 2,Γ7+,σ + H.c.),
(S1. 38) where ∼ denotes the approximate identity relation at low energies. The E-driven hopping adds the following interactions to the spin Hamiltonian H spin :
P δH t (E) 1 E g − H U QH t (0) 3 P + P H t (0) 1 E g − H U QδH t (E) 1 E g − H U QH t (0) 2 P + P H t (0) 1 E g − H U QH t (0) 1 E g − H U QδH t (E) 1 E g − H U QH t (0)P + P H t (0) 1 E g − H U QH t (0) 2 1 E g − H U
QδH t (E)P (S1.39) Equation (S1.39) denotes the O(|E| 1 ) terms of diagrams in Fig. 2. Straightforward calculations for a triangle M 1 -L-M 2 lead to
H spin = J F S 1 · S 2 + KS z 1 S z 2 + D F · S 1 × S 2 , (S1.40) with D F = D z F e z and D z F J F = − 4I t (E x + E y )(U d − U p + ∆ dp ) + (E x − E y )J H 2(U d − U p ) + J H .
(S1.41)
E. Out-of-plane electric field
In contrast to the in-plane electric field, the out-ofplane electric field E = E y e z keeps the σ v reflection symmetry about the y = x plane but breaks the σ v one about the z = 0 plane. It induces hoppings,
δH t (E) = −IE z σ=± j=1,2 [(p † y,σ + p † x,σ )d 1,yz,σ + H.c.] ∼ − IE z √ 3 σ,j [(σp † y,−σ + ip † x,−σ )d j,Γ7+,σ + H.c.],
(S1. 42) leading to
H spin = J F S 1 · S 2 + KS z 1 S z 2 + D F · S 1 × S 2 + Γ [(S x 1 + S y 1 )S z 2 + S z 1 (S x 2 + S y 2 )] (S1.43) with D F = D x F (e x + e y ), (S1.44) D x F = 16 3 E z I 1 U d − U p + ∆ dp 2 · J H 4(U d − U p + ∆ dp ) 2 − J H 2 , (S1.45) Γ = 32t 3 9 E z I 1 U d − U p + ∆ dp 2 · U d − U p + ∆ dp 4(U d − U p + ∆ dp ) 2 − J H 2 .
(S1.46) lattice is put on the (111) plane. Now we apply the electric field E [111] = E [111] (e x + e y + e)/ √ 3 so that the electric field is perpendicular to the honeycomb lattice. Then, the resultant spin Hamiltonian is given by
H KHΓ = a=x,y,z i,j a (KS a i S a j + J H S i · S j ) − j h · S j + Γ (E [111] ) a=x,y,z b,c =a i,j a (S b i S c j + S c i S b j ),
(S1. 47) where Γ (E [111] ) is proportional to E [111] :
Γ = 32t 3 9 2E [111] √ 3 I 1 U d − U p + ∆ dp 2 · U d − U p + ∆ dp 4(U d − U p + ∆ dp ) 2 − J H 2 .
(S1. 48) We assumed that the ligand sites are all crystallographically equivalent, where the DMI is canceled in the manybody Hamiltonian (S1.47).
S2. RASHBA SPIN-ORBIT INTERACTION IN TIGHT-BINDING MODELS
Here, we remove the intra-atomic SOC but add the inter-atomic SOC, the Rashba SOC. The Rashba SOC enters into the single-electron HamiltonianĤ 1 :
H 1 = p 2 2m + V (r) − α R e(r) · σ × p, (S2.49)
where p and m are the momentum and the mass of the electron and V (r) is the potential that the electron feels. e(r) = E(r)/|E(r)| is the unit vector parallel to the electric field as defined in the main text. We normalized e(r) to make α R ∝ |E(r)| following the convention. The single-electron Hamiltonian keeps the C 2v symmetry [64] in the absence of the electric and magnetic fields (i.e., E(r) = h = 0). The last term ofĤ 1 is nothing but the Rashba SOC. The Rashba SOC enters into the second-quantized many-electron model via the hopping amplitude, a matrix element of the single-electron Hamiltonian. The hopping amplitude between the p x orbital at L and the d a orbital at M j is given by
p x,s |Ĥ 1 |d j,p x,s |(−α R e(r) · σ × p)|d j,a,s = −α R |k|e z (σ ss × (−id j )) z δ j,2 , (S2.52)
where we replaced the momentum p = k by −i |k|d j with the unit vector d j parallel to the M j -L bond. The Kronecker's delta δ j,2 appears because the vector d j is proportional to e x for j = 1 and to e y for j = 2. Hence, we arrived at the E-driven hoppings (5) in the main text.
We consider the d 5 low-spin configuration under the octahedral CEF [ Fig. S5 (a)] and the isosceles right triangle M 1 -L-M 2 [ Fig. S5 (b)]. In this setup, the intrinsic and E-driven hoppings are given by
H t (0) = t σ (p † y,σ d 1,xy,σ + p † x,σ d 2,xy,σ + H.c.), (S2.53) δH t (E) = s,s [iλp † y,s (σ ss × d 1 ) z d 1,xy,s + iλp † x,σ (σ ss × d 2 ) z d 2,xy,s + H.c.]. (S2.54)
The spin Hamiltonian (1) following the diagram of Fig. 2.
We obtain the O(|E|) correction to the spin Hamiltonian by replacing one of the four H t (E) by δH t (E) and the other by H t (0). Namely, we find
P H t (E) 1 E g − H U QH t (E) 3 P ≈ P H t (0) 1 E g − H U QH t (0) 3 P + P δH t (E) 1 E g − H U QH t (0) 3 P + H.c. + P H t (0) 1 E g − H U QδH t (E) 1 E g − H U QH t (0) 2 P + H.c. (S2.55)
The spin Hamiltonian for E = 0 contains neither magnetically anisotropic terms nor inversion-asymmetric terms because the Hamiltonian H t (0) + H U for E = 0 is magnetically isotropic and inversion symmetric. The Rashba interaction (S2.54) breaks those spin and spatial symmetries at the same time. To show that the Rashba SOC leads to the DMI, we consider the second terms, FIG. 18 in Ref. [59]. The process (a) gives
P δH t (E) 1 E g − H U QH t (0) 3 P + H.c.,(S2.P δH t (E) 1 E g − H U QH t (0) 3 P + H.c. process (a) = P iλt U d − U p + ∆ dp t 2 2(U d − U p ) + 2∆ dp − J H 2 U d − U p + ∆ dp · σ1,σ2,··· ,σ5=± 1 2 δ σ1,−σ2 (−δ σ4,σ2 δ σ3,σ1 − δ σ4,−σ2 δ σ3,−σ1 )(σ σ4σ5 × d 2 ) z d 2,σ5 d 1,σ3 d † 2,σ2 d † 1,σ1 + σ1,σ2,··· ,σ5 1 2 δ σ1,−σ2 (δ σ4,σ2 δ σ3,σ1 + δ σ4,−σ2 δ σ3,−σ1 )(σ σ4σ5 × d 1 ) z d 1,σ5 d 2,σ3 d † 2,σ2 d † 1,σ1 + iλt U d − U p + ∆ dp t 2 2(U d − U p ) + 2∆ dp + J H 2 U d − U p + ∆ dp · σ1,σ2,··· ,σ5=± 1 2 δ σ1,−σ2 (−δ σ4,σ2 δ σ3,σ1 + δ σ4,−σ2 δ σ3,−σ1 )(σ σ4σ5 × d 2 ) z d 2,σ5 d 1,σ3 d † 2,σ2 d † 1,σ1 + σ1,σ2,··· ,σ5=± 1 2 δ σ1,−σ2 (δ σ4,σ2 δ σ3,σ1 − δ σ4,−σ2 δ σ3,−σ1 )(σ σ4σ5 × d 1 ) z d 1,σ5 d 2,σ3 d † 2,σ2 d † 1,σ1 P, (S2.57)
where we already removed the p-orbital operators by using the projection P [see, for example, Eq. (A3) of Ref. [59]].
Rewriting the d-orbital creation and annihilation operators in terms of the spin S j (j = 1, 2), we obtain
P δH t (E) 1 E g − H U QH t (0) 3 P + H.c. process (a) = P −4λt 3 2(U d − U p ) + 2∆ dp − J H 1 U d − U p + ∆ dp 2 · [d y 2 (−S z 1 S y 2 − S y 1 S z 2 ) + d x 2 (−S z 1 S x 2 − S x 1 S z 2 ) + d y 1 (−S z 1 S y 2 − S y 1 S z 2 ) + d x 1 (−S z 1 S x 2 − S x 1 S z 2 )] + −4λt 3 U d − U p + 2∆ dp + J H 1 U d − U p + ∆ dp 2 · [d y 2 (−S z 1 S y 2 + S y 1 S z 2 ) + d x 2 (−S z 1 S x 2 + S x 1 S z 2 ) + d y 1 (−S z 1 S y 2 + S y 1 S z 2 ) + d x 1 (−S z 1 S x 2 + S x 1 S z 2 )] P. (S2.58)
The process (b) of FIG. 18 in Ref. [59] gives
P δH t (E) 1 E g − H U QH t (0) 3 P + H.c. process (b) = P iλt U d − U p + ∆ dp t 2 2(U d − U p ) + 2∆ dp − J H 2 U d − U p + ∆ dp · σ1,··· ,σ5 δ σ1,σ2 (−δ σ4,σ2 δ σ3,σ1 )(σ σ4σ5 × d 2 ) z d 2,σ5 d 1,σ3 d † 2,σ2 d † 1,σ1 + iλt U d − U p + ∆ dp t 2 2(U d − U p ) + 2∆ dp − J H 2 U d − U p + ∆ dp · σ1,··· ,σ5 δ σ1,σ2 (δ σ4,σ2 δ σ3,σ1 )(σ σ4σ5 × d 1 ) z d 1,σ5 d 2,σ3 d † 2,σ2 dd † 1,σ1 + H.c. P = P −8λt 3 2(U d − U p ) + 2∆ dp − J H 1 U d − U p + ∆ dp 2 (d y 2 S z 1 S y 2 + d x 2 S z 1 S x 2 ) + −8λt 3 2(U d − U p ) + 2∆ dp − J H 1 U d − U p + ∆ dp 2 (d y 1 S y 1 S z 2 + d x 1 S x 1 S z 2 ) P. (S2.59)
Combining these contributions of the two processes together, we obtain
P δH t (E) 1 E g − H U QH t (0) 3 P + H.c. = P −4λt 3 2(U d − U p ) + 2∆ dp − J H 1 U d − U p + ∆ dp [(d y 1 − d y 2 )(S 1 × S 2 ) x − (d x 1 − d x 2 )(S 1 × S 2 ) y ] + −4λt 3 2(U d − U p ) + 2∆ dp + J H 1 U d − U p + ∆ dp [(d y 1 + d y 2 )(S 1 × S 2 ) x − (d x 1 + d x 2 )(S 1 × S 2 ) y ] P, (S2.60)
which is nothing but the DMI. The other terms,
P H t (0) 1 E g − H U QδH t (E) 1 E g − H U QH t (0) 2 P + H.c. (S2.61)
lead to the same result after repeating a similar procedure to the above one. Collecting all these contributions, we reach the final result:
H spin = JS 0 · S 1 + D x (S 1 × S 2 ) x + D y (S 1 × S 2 ) y , (S2.62) D x = − (d y 1 − d y 2 ) 16λt 3 2(U d − U p ) + 2∆ dp − J H + (d y 1 + d y 2 ) 16λt 3 2(U d − U p ) + 2∆ dp + J H 1 U d − U p + ∆ dp 2 , (S2.63) D y = (d x 1 − d x 2 ) 16λt 3 2(U d − U p ) + 2∆ dp − J H + (d x 1 + d x 2 ) 16λt 3 2(U d − U p ) + 2∆ dp + J H 1 U d − U p + ∆ dp 2 .
(S2.64)
Note that d 1 = e x and d 2 = e y for the spatial configuration of Fig. S5 (b).
S3. STRENGTH OF EXTERNAL AND INTERNAL ELECTRIC FIELDS
This section gives brief supplemental information to the estimation of required electric-field strength. Mainly, there are two resources of the DC electric field. One is to apply it externally using, for example, field-effect transistors. The other is generated internally by crystal structures. We call the former external electric fields and the latter internal ones.
Currently, we can realize the external DC field of the strength ∼ 10 MV/cm using, for example, double-layer transistors [52,53]. On the other hand, the internal DC electric field can be even stronger. Let us consider the octahedral CEF [See Figs. S2 (a), (b), and S6]. The strength of the CEF is typically ∼ 1 eV [84]. If the ligand is ∼ 0.1-1 nm away from the magnetic ion, the internal CEF is ∼ 10-100 MV/cm. This internal CEF is responsible for the Rashba SOC generated on the interface of different materials. This is the reason why the Rashba SOC can be strong, as we briefly saw in the main text. We can find another interesting situation in scanning tunneling microscopes (STM). A tip of the STM induces a DC electric field strong enough to induce the tunneling electron current on the surface. If the STM tip is ∼ 1 nm distant from the surface and the ∼ 1 V voltage is applied, the surface feels the ∼ 10 MV/cm electric field [57].
S4. FIELD-DRIVEN DZYALOSHINSKII-MORIYA INTERACTION IN MINIMAL MODEL
The main text discusses the intra-atomic SOC in the d orbital and the inter-atomic SOC. By contrast, this section discusses a simple model with a weak intra-atomic SOC within the ligand site to supplement the main text. Also, we suppose a d 9 electron configuration unlike the models discussed in the main text. Let us name this model a minimal model after its simplicity.
A. Hybridization of p orbitals
To define the minimal model Hamiltonian, we briefly review the hybridization of p orbitals. Let us denote the eigenenergy of the s and p orbitals in the absence of SOC as ε s and ε p , respectively. The Hamiltonian to describe the hybridization is given by
H sp = 8 n=1 c † n H sp c n , (S4.65)
where c † n denotes the creation operators of the s-orbital electron (s † ± ) and the p-orbital ones (p † µ,± for µ = x, y, z). We list them as follows.
(c † 1 , c † 2 , · · · , c † 8 ) = (s † + , p † x,+ , p † y,+ , p † z,− , s † − , p † x,− , p † y,− , p † z,+ ). (S4.66)
Note that the spin of the p z orbitals in Eq. (S4.66) are inverted in advance to simplify H sp . H sp contains the eigenenergies, ε s and ε p , of the s and p orbitals, the uniaxial anisotropy ∆, and SOC λL p ·S p , where the eigenenergies ε s and ε p are supposed to be those in the absence of the anisotropy or SOC. The 8 × 8 matrix H sp is then block-diagonalized.
H sp = ε s 0 0 0 0 0 0 0 0 ε p −i λ 2 λ 2 0 0 0 0 0 i λ 2 ε p −i λ 2 0 0 0 0 0 λ 2 i λ 2 ε p + ∆ 0 0 0 0 0 0 0 0 ε s 0 0 0 0 0 0 0 0 ε p i λ 2 − λ 2 0 0 0 0 0 −i λ 2 ε p −i λ 2 0 0 0 0 0 − λ 2 i λ 2 ε p + ∆ = ε s 0 0 0 0 H p− 0 0 0 0 ε s 0 0 0 0 H p+ . (S4.67)
Here, we did not include the sp hybridization for simplicity though it is straightforward to include the sp hybridization as evident from the matrix representation (S4.67). Ignoring the s orbital is consistent with an approximation performed later in this Section.
H p+ = ε p i λ 2 − λ 2 −i λ 2 ε p −i λ 2 − λ 2 i λ 2 ε p + ∆ . (S4.68)
H p− is obtained from H p+ by replacing λ → −λ and i → −i. Performing the first-order perturbation with respect to λ, we obtain the hybridization of p orbitals. where |p µ,σ =:p † µ,σ |0 denotes the hybridized p µ -orbital state. The hybridizations of |p x,+ , |p y,+ , and |p z,− are derived likewise. The minimal model has the following electron manybody Hamiltonian. where n j,a1g,σ := d † j,a1g,σ d j,a1g,σ and n pz,σ = p † z,σ p z,σ number operator of the a 1g -orbital electron at M j and the non-hybridized p z orbital at the ligand site, respectively. The eigenenergies of the a 1g orbital and the p z orbital are encoded in the on-site potentials V j and V p [58,59]. U d and U p represent the on-site Coulomb interaction. Generally, DC electric fields also affect the intra-atomic interaction (S4.76), as we mentioned in the main text. However, we do not consider it for the following reason. The E-driven hopping amplitude in Eq. (S4.78) is proportional to E a . Hence, as long as we are focused on additional spin-spin interactions generated by E with the O(|E| 1 ) precision, we can keep H U (E) ≈ H U (0).
H = H U + H t (0) + δH t (E),(S4.
We consider the d 9 configuration at M j and regard the a 1g orbital among them is half filled (Fig. S6). In other words, we assume a tetragonal CEF at each M j . The tetrahedral CEF lifts the d-orbital degeneracy. Note that an octahedral CEF is a special case of the tetragonal one. The octahedral CEF lifts the d orbitals into e g and t 2g . The tetragonal CEF with the lower symmetry than the octahedral one further splits e g into a 1g and b 1g and t 2g into e g and b 2g [64]. The a 1g orbital has the d 3z 2 −r 2 symmetry and thus the highest eigenenergy under the tetragonal CEF.
We incorporate the DC electric field into the E-driven hoppings (S4.78). The Hamiltonian (S4.75) does not include the p-orbital hybridization. It is straightforward to take it into account. Recall that all the p-orbital levels are below the Fermi surface and the p z one is closest to the Fermi surface. At low energies, we keep the hybridized p z orbital only. Hence, we simply replace p z,± in Eqs. (S4.76) and (S4.77) withp z,± . On the other hand, the E-driven hoppings (S4.78) are involved with the operators p x,σ and p y,σ other than the low-energy relevant orbital. Since it costs too large energy to excitẽ p x -andp y -orbital states, the term (S4.78) would seem irrelevant to low-energy physics. However, this is not the case becausep x,σ andp y,σ can remove an electron from thep z orbital with a small probability. The probability is given by (λ/2∆) 2 at low temperatures. Hereafter, we drop the tilde in the creation and annihilation operators (i.e.,p z,σ → p z,σ ) to simplify the notation.
C. Symmetries
When E = 0, the minimal model has the D 4h symmetry due to the tetragonal CEF at the M j sites. The linear M 1 -L-M 2 alignment as well as the uniaxial CEF at the ligand site respect the D 4h symmetry. The D 4h group contains D 2h as its subgroup made of C 2a rotations (a = x, y, z), σ a reflections (a = xy, yz, zx), and an inversion at the ligand site. The D 2h symmetry is evident from the geometrical structure of the model such as the equidistant linear alignment of the three sites and the tetragonal CEF.
SOC, λl p ·s p , transcribes the D 4h crystalline symmetry into a spin symmetry, where l p and s p are the orbital angular momentum and the spin due to the p orbitals, respectively. The spin operator s p is defined as s p := 1 2 µ=x,y,z s,s =± p † µ,s σ ss p µ,s , (S4. 83) where σ = (σ x , σ y , σ z ) represents the Pauli matrices and σ ss denotes their (s, s ) components. If SOC was absent, the minimal model has the D 4h × SU(2) symmetry, a direct product of the D 4h crystalline symmetry and the SU(2) spin-rotation symmetry. SOC violates the SU(2) symmetry down to D 4h in the following sense. For example, SOC is antisymmetric under the inversion r → −r because it turns (l p , s p ) → (−l p , s p ), but symmetry under the simultaneous inversion (l p , s p ) → (−l p , −s p ) of the real space and the spin space. SOC lowers the symmetry D 4h × SU(2) → D 4h . We denote the latter D 4h asD 4h to distinguish it from the pure crystalline symmetry. An arbitrary element M ∈D 4h acts on l p and s p equivalently. In other words, M treats s p as if it was an angular momentum. Then, the following invariance is obvious.
M (λl p · s p )M −1 = λl p · s p . (S4.84)
We can translate theD 4h -group operation in terms of p † µ,σ operators. For example, the C 2z rotation ofD 4h turns them into C 2z (p † x,σ , p † y,σ , p † z,σ )C −1 2z = iσ(−p † x,σ , −p † y,σ , p † z,σ ). (S4. 85) Operations of theD 4h group on p † µ,σ are defined likewise. Hence, we findD 4h keeps the whole intra-atomic interactions at the ligand site invariant.
TheD 4h symmetry acts on S j carried by the a 1g orbital similarly to s p . To represent it in terms of d † j,a1g,σ , we need to reflect the a 1g spatial symmetry. For example, C 2z acts d † j,a1g,σ as C 2z d † j,a1g,σ C −1 2z = iσd j,a1g,σ . (S4. 86) In terms of the spin S j := 1 2 s,s d † j,a1g,s σ ss d j,a1g,s , the relation (S4.86) can read as
C 2z S j C −1 2z = (−S x j , −S y j , S z j ) , (S4.87)
where denotes the transpose. The transformation (S4.87) accords with the aforementioned requirement that S j transforms similarly to the orbital angular momentum l p . Now we can confirm that the full Hamiltonian (S4.75) satisfies C 2z HC −1 2z = H.
We can see that D 4hnoninvariant hoppings such as p † y,σ d j,a1g,σ are absent in the minimal model because of C 2z p † y,σ d j,a1g,σ C −1 2z = −p † y,σ d j,a1g,σ . Similar arguments hold for the other members of theD 4h group.
D. Without electric fields
Note that both the CEF at M j and the ligand site respect the D 4h symmetry other than the inversion. The inversion symmetry exists because the ligand site is precisely located at the inversion center of the M 1 -M 2 bond. Indeed, the resultant spin Hamiltonian fully respects thẽ D 4h symmetry. In particular, the spin Hamiltonian cannot have the DMI owing to the inversion symmetry. The spin Hamiltonian H spin is given by Eq. (1) of the main text. When E = 0, all the H t in Eq. (1) are supposed to be H t (0): 88) in the absence of the magnetic field. Carrying out the fourth-order expansion, we find H spin = J min S 1 · S 2 , (S4.89)
H spin = P H t (0) 1 E g − H U QH t (0) 3 P, (S4.J min = 4t 4 1 (U d − U p + ∆ dp ) 2 2 2∆ dp + U d − U p + 1 U d ,(S4.90)
where the parameters ∆ dp , U d , U p are defined similarly to the other models discussed in the main text.
E. With electric fields
The electric field adds the hopping (S4.82) into the perturbation expansion. Collecting all the O(|E| 1 ) terms in the expansion of Fig. 2 of the main text, we find that the DC electric field E = E x e x + E y e y modifies the spin Hamiltonian from Eq. (S4.89) to H spin = J min S 1 · S 2 + D min · S 1 × S 2 , (S4.91) D min = α min n, (S4.92) n = 1 |E| (E y e x − E x e y ), (S4.93)
α min = 8|E|It 3 λ 2∆ 1 (∆ dp + U d − U p ) 2 1 2∆ dp + 2U d − U p .
(S4.94)
F. DMI and symmetries
The electric field on the xy plane lowers the symmetry of the minimal model from D 2h to C 2v = {I, C 2e , σ xy , σ ez }, where I, C 2e , and σ ez denote the identity, the C 2 rotation around E, and the reflection along n (i.e., n → −n), respectively. The spin Hamiltonian (S4.91) respects the C 2v symmetry. To see this, we look into C 2v operations to DMI. To discuss those point-group operations, we can assume E = Ee x and n = −e y without loss of generality. The C 2e = C 2x rotation, the σ xy and σ ez = σ zx reflections act on the spin operator S j as
C 2x S j C −1 2x = (S x j , −S y j , −S z j ) ,(
FIG
. 1. (a) Néel-type magnetic skyrmion. (b)Schematic phase diagram of square-lattice classical Heisenberg ferromagnetic model with DMI
FIG. 2 .
2Diagramatic representation of spin Hamiltonian
FIG. 4 .
4(a) Edge-sharing octahedra ML4 projected to xy plane. Magnetic ions form the square lattice (black solid lines). Intrinsic and E-driven hoppings occur on dashed gray bonds. (b) Four-site model for nearest-neighbor superexchange interaction of magnetic ions. (c) One-dimensional version of square-lattice model (a), ferromagnetic spin chain with uniform DMI. (d) q vectors of 3q-hedgehog state[8,24,90].
depicted as the red-fringe arrow in Figs. 4 (a), (b), and (c). With many square plaquettes ofFig. 4 (b), we can build a square-lattice ferromagnet [Fig. 4 (a)],
ACKNOWLEDGMENTS
The authors are grateful to Tetsuo Hanaguri for fruitful discussions. S.C.F. and M.S. are supported by a Grant-in-Aid for Scientific Research on Innovative Areas "Quantum Liquid Crystals" (Grant No. JP19H05825). M.S. is also supported by JSPS KAKENHI (Grant No. 20H01830) Supplementary Material for "Electric-Field Control of Magnetic Anisotropies: Applications to Kitaev Spin Liquids and Topological Spin Textures" Shunsuke C. Furuya and Masahiro Sato Department of Physics, Ibaraki University, Mito, Ibaraki 310-8512, Japan S1. FIELD-DRIVEN MAGNETIC ANISOTROPIES IN KITAEV-HEISENBERG MODEL A. Model at zero electric fields
FIG. S1. (a) J eff = 1/2 model's electron configurations in Γ7+ orbitals and px,y,z orbitals at Mj and L sites. (b) Geometrical configuration of three sites, M1,2 and L.M1-L-M2 forms a right-angle isosceles triangle. (c) Electron hopping between dxy orbital at M1 and py orbital at L. (d) Electron hopping between dxy orbital at M2 and px orbital at L. The hopping between dxy at M1 (M2) and px (py) orbital at L is forbidden by symmetries.
FIG
. S2. (a) zz bond (red line) of Kitaev-Heisenberg model on xy plane (light blue plane). (b) xx bond (blue line) of Kitaev-Heisenberg model on yz plane (light yellow plane). (c) Orthogonality of two planes.
S1.21) on the honeycomb lattice from the spin Hamiltonian(3) in the main text. Note that we put two triangles M 1 -L 1 -M 2 and M 1 -L 1 -M 2 on the xy plane to derive the Hamiltonian (3) of the main text. The M 1 -M 2 bond in this spin Hamiltonian corresponds to the zz bond (the red line) ofFig. S2 (a). The other bonds a = x, y are derived similarly. Let us consider the xx bond ofFig. S2 (b). We consider a local x y z coordinate system and put trianglesM 3 -L 3 -M 4 and M 3 -L 4 -M 4 on the FIG.S3. (a)Second-order process to yield Kitaev model discussed in Ref.[31]. (b) Fourth-order process to yield Kitaev-Heisenberg model discussed in this paper. The arrows depict electron hoppings between proximate orbitals.
E
δH t (E) = −I σ=± [E y (p † z,σ d 1,yz,σ + p † x,σ d 1,xy,σ )FIG. S4. (a) Proximate dxy and px orbitals with zero overalp integral. (b) dxy orbital distorted by electric field E along the vertical direction. This distortion makes the overlap of the two proximate orbitals nonzero, yielding the E-driven hopping δHt(E).
F
. Kitaev-Heisenberg-Γ model under [111] electric field When we build the Kitaev honeycomb model from our three-site models [Figs. S2 (a) and (b)], the honeycomb . S5. (a) d 5 electron configuration of d orbitals under octahedral CEF and p orbitals. (b) Spatial configuration of agnetic ions M1,2 and ligand ion L. We emphasize that the intra-atomic SOC is not considered here in contrast to the setup of Fig. S1. We deal with the inter-atomic SOC, that is, the Rashba SOC of Eq. (S2.54).
a,s . (S2.50) When E(r) = 0, the hopping amplitude in our three-site model on the isosceles right triangle is reduced to p x,s |Ĥ 1 |d j,a,s = tδ a,xy δ j,2 δ s,s , (S2.51) with a real constant t [see Fig. S1 (d)]. The Rashba SOC adds another term to the right hand side of Eq. (S2.51). For example, E(r) = E z e z gives
. (S2.55) The second line of Eq. (S2.55) is similarly calculated. The fourth-order perturbation processes are divided into two classes, the processes (a) and (b) of
FIG
. S6. Splittings of d orbitals by octahedral and tetragonal CEF[64]. The tetragonal CEF derives from the octahedral one in the AB6 unit of a crystal by replacing four of B atoms with C atoms or distorting AB6 uniaxially. The 3 × 3 matrix H p± describes the hybridization of |p x,∓ , |p y,∓ , and |p z,± .
|p z,+ = |p z,+ − λ 2∆ (|p x,− + i |p y,− ), (S4.69) |p x,− + i |p y,− = |p x,− + i |p y,− + λ ∆ |p z,+ , (S4.70) |p x,− − i |p y,− = |p x,− − i |p y,− , (S4.71)
|p z,− = |p z,− + λ 2∆ (|p x,+ − i |p y,− ), (S4.72) |p x,+ − i |p y,+ = |p x,+ − i |p y,+ − λ ∆ |p z,+ , (S4.73) |p x,+ + i |p y,+ = |p x,+ + i |p y,+ . (S4.74) B. Model
= j + 1 mod 2. The C 2x symmetry (S4.95) imposes a constraint, (D min ) x = 0, and the σ xy symmetry (S4.96) imposes (D min ) z = 0. The σ zx symmetry (S4.97) imposes both the constraints, (D min ) x = (D min ) z = 0.
n j,a1g,+ n j,a1g,− + U p n pz,+ n pz,−[(E x p † x,σ + E y p † y,σ )d j,a1g,σ + H.c.],75)
H U = U d
j=1,2
+
j=1,2
V j
σ=±
n j,a1g,σ + V p
σ=±
n pz,σ ,
(S4.76)
H t (0) = t
j=1,2 σ=±
(−1) j (p †
z,σ d j,a1g,σ + H.c.), (S4.77)
δH t (E) = −I
j=1,2 σ=±
(S4.78)
according to Eqs. (S4.69) and (S4.72). Eqs. (S4.69) ans (S4.72) givesbecause the projection operator P acts on the ligand site as P = |p z,+ p z,+ | + |p z,− p z,− |. Therefore, we may at low energies and rewrite the E-driven hoppings (S4.78) asP p †
x,σ P = 0,
Qp †
x,σ P = Q σ
λ
2∆p
†
z,−σ P, (S4.79)
P p †
y,σ P = 0,
Qp †
y,σ P = Q −i
λ
2∆p
†
z,−σ P, (S4.80)
approximate
p †
x,σ ≈ σ
λ
2∆p
†
z,−σ , p †
y,σ ≈ −i
λ
2∆p
†
z,−σ ,
(S4.81)
δH t (E) ≈
Iλ
2∆
j=1,2 σ=±
[(E x σp †
z,−σ + iE yp †
z,−σ )d j,a1g,σ
+ H.c.],
(S4.82)
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See Supplemental Material for details about technical calculations of microscopic parameters. See Supplemental Material for details about technical calculations of microscopic parameters.
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| [] |
[
"Standard Model Explanation of the Ultra-high Energy Neutrino Events at IceCube",
"Standard Model Explanation of the Ultra-high Energy Neutrino Events at IceCube"
] | [
"Chien-Yi Chen \nDepartment of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n",
"P S Bhupal Dev \nConsortium for Fundamental Physics\nSchool of Physics and Astronomy\nUniversity of Manchester\nM13 9PLManchesterUnited Kingdom\n",
"Amarjit Soni \nDepartment of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n"
] | [
"Department of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA",
"Consortium for Fundamental Physics\nSchool of Physics and Astronomy\nUniversity of Manchester\nM13 9PLManchesterUnited Kingdom",
"Department of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA"
] | [] | The recent observation of two PeV events at IceCube, followed by an additional 26 events between 30 -300 TeV, has generated considerable speculations on its origin, and many exotic New Physics explanations have been invoked. For a reliable interpretation, it is however important to first scrutinize the Standard Model (SM) expectations carefully, including the theoretical uncertainties, mainly due to the parton distribution functions. Assuming a new isotropic cosmic neutrino flux with a simple unbroken power-law spectrum, Φ ∝ E −s for the entire energy range of interest, we find that with s = 1.5 -2, the SM neutrino-nucleon interactions are sufficient to explain all the observed events so far, without the need for any beyond the SM explanation. With more statistics, this powerful detector could provide a unique test of the SM up to the PeV scale, and lead to important clues of New Physics. | 10.1103/physrevd.89.033012 | [
"https://arxiv.org/pdf/1309.1764v2.pdf"
] | 118,604,357 | 1309.1764 | 0ad2f65b0e354a31a963135af791ea40ba8d4a7c |
Standard Model Explanation of the Ultra-high Energy Neutrino Events at IceCube
Chien-Yi Chen
Department of Physics
Brookhaven National Laboratory
11973UptonNew YorkUSA
P S Bhupal Dev
Consortium for Fundamental Physics
School of Physics and Astronomy
University of Manchester
M13 9PLManchesterUnited Kingdom
Amarjit Soni
Department of Physics
Brookhaven National Laboratory
11973UptonNew YorkUSA
Standard Model Explanation of the Ultra-high Energy Neutrino Events at IceCube
MAN/HEP/2013/22
The recent observation of two PeV events at IceCube, followed by an additional 26 events between 30 -300 TeV, has generated considerable speculations on its origin, and many exotic New Physics explanations have been invoked. For a reliable interpretation, it is however important to first scrutinize the Standard Model (SM) expectations carefully, including the theoretical uncertainties, mainly due to the parton distribution functions. Assuming a new isotropic cosmic neutrino flux with a simple unbroken power-law spectrum, Φ ∝ E −s for the entire energy range of interest, we find that with s = 1.5 -2, the SM neutrino-nucleon interactions are sufficient to explain all the observed events so far, without the need for any beyond the SM explanation. With more statistics, this powerful detector could provide a unique test of the SM up to the PeV scale, and lead to important clues of New Physics.
I. INTRODUCTION
Recently the IceCube collaboration has reported the first observation of extremely intriguing two events with neutrino energies above 1 PeV [1]. These are by far the highest neutrino energies that have ever been observed. As such they could potentially represent the first detection of a non-atmospheric, astrophysical high energy neutrino flux, thus opening an avenue for a number of astrophysical objects and mechanisms to provide information complementary to that obtained from electromagnetic or hadronic observations [2]. In a follow-up search with improved sensitivity and extended energy coverage, IceCube has reported additional 26 events with deposited energies ranging from 30 to 300 TeV [3]. These 28 events are the result of the first 662 days of data-taking, and give a preliminary significance of 4.1σ with respect to the reference atmospheric neutrino background model. These observations may well hold the key to understanding neutrino masses, the nature of neutrino mass-hierarchy, their relevance to Dark Matter, or in general, to physics beyond the Standard Model (SM). For these reasons, it is extremely important to understand every possible aspect of these IceCube events.
This realization has spurred a lot of interest on the origin of these ultra-high energy (UHE) neutrino events and their underlying spectral shape. Various extra-terrestrial sources (e.g., gamma-ray bursts, active galactic nuclei, early supernovae, baby neutron stars, starburst galaxies, cosmogenic) [4] with different power-law fluxes have been analyzed. From a particle physics point of view, several beyond SM phenomena, e.g., decaying heavy Dark Matter [5], lepto-quark resonance [6], decay of massive neutrinos to light ones over cosmological distances [7], and pseudo-Dirac neutrinos oscillating to their sterile counterparts in a mirror world [8], have been proposed. Most of these proposals are motivated by some specific features in the IceCube data such as a (slight) paucity of muon tracks, a (possible) apparent energy gap between 300 TeV and 1 PeV, and possibly a bit higher than expected event rate above PeV.
Our primary aim in this paper is to carefully scrutinize the IceCube observations with respect to the SM expectations, taking into account the theoretical uncertainties, mainly due to the parton distribution functions (PDFs). Following the IceCube analysis [3], which did not find significant clustering of the events in time or space compared to randomized sky maps, we assume a simple isotropic astrophysical power-law spectrum for the UHE neutrino flux Φ ∝ E −s , and show that for s = 1.5 -2, the current data is consistent with the SM, within the theoretical and experimental uncertainties. Thus there is no significant feature of the current data requiring an exotic particle physics explanation, other than of course a very interesting new cosmic neutrino flux. However, we want to stress that there are mild indications of slight potential deficit of muons in comparison to other flavors and perhaps a little excess of above PeV events. If these features attain clear statistical significance as more data is accumulated, then some new physics interactions may well become necessary; but at this juncture, these considerations appear somewhat premature.
The rest of the paper is organized as follows: In Section II, we calculate various neutrino-induced scattering cross sections in the SM, along with their differential distributions, for a reference PDF set, and compare the predictions for the central value as well as the 90% confidence level (CL) range of the PDFs at Leading Order (LO), Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO). In Section III, we use the IceCube experimental parameters to carefully estimate the corresponding number of events predicted by the SM, along with its theoretical uncertainty, and compare our results with the IceCube observed events for a simple power-law cosmic neutrino flux. We also perform a χ 2 -analysis to find the best-fit spectral index and normalization for this new flux. Finally, our conclusions are given in Section IV.
II. STANDARD MODEL CROSS SECTION
The expected neutrino-induced event rate at IceCube can be schematically written as
dN dE EM = T · Ω · N eff (E ν ) · σ(E ν ) · Φ(E ν )(1)
where E ν is the incoming neutrino energy in the laboratory frame, E EM is the electromagnetic (EM)-equivalent deposited energy, T is the time of exposure, Ω is the solid angle of coverage, N eff is the effective number of target nucleons, σ is the neutrino-induced interaction cross section, and Φ is the incident neutrino flux.
The main particle physics ingredient in Eq. (1) is the neutrino-induced interaction cross section [9]. Within the SM, neutrinos interact with matter only through the exchange of weak gauge bosons. The dominant processes (in most of the energy range of interest) are the chargedcurrent (CC) and neutral-current (NC) neutrino-nucleon deep inelastic scattering (DIS) mediated by t-channel W and Z respectively:
ν + N → + X (CC), (2) ν + N → ν + X (NC),(3)
where = e, µ, τ represents the SU (2) L lepton-flavor, N = (n + p)/2 is an isoscalar nucleon (n and p being the neutron and proton, respectively) in the renormalization group-improved parton model, and X is the hadronic final state. The neutrino interactions with the electrons in the target material can generally be neglected with respect to the neutrino-nucleon interactions due to the electron's small mass (m e M N ). There is however an important exception for theν e e − interaction when the incoming neutrino energy is between about 4 -10 PeV. In this case, the resonant production of the SM W -boson gives a significant enhancement in theν e e cross section, which peaks at E res ν = M 2 W /2m e = 6.3 PeV, and is commonly known as the Glashow resonance [10].
A. Differential Cross Sections
The neutrino-nucleon differential scattering cross sections at leading order are given by [11]
d 2 σ CC νN dxdy = 2G 2 F M N E ν π M 2 W Q 2 + M 2 W 2 × xq(x, Q 2 ) + xq(x, Q 2 )(1 − y) 2 ,(4)d 2 σ NC νN dxdy = G 2 F M N E ν 2π M 2 Z Q 2 + M 2 Z 2 × xq 0 (x, Q 2 ) + xq 0 (x, Q 2 )(1 − y) 2 ,(5)
where −Q 2 is the invariant momentum-square transfer to the exchanged vector boson, M N and M W (Z) are the nucleon and intermediate W (Z)-boson masses respectively, and G F is the Fermi coupling constant. The differential distributions in Eqs. (4) and (5) are with respect to the Bjorken scaling variable x and the inelasticity parameter y, where
x = Q 2 2M N yE ν and y = E ν − E E ν ,(6)
E being the energy carried away by the outgoing lepton in the laboratory frame, and x is the fraction of the initial nucleon momentum taken by the struck quark. Here q,q (q 0 ,q 0 ) are respectively the quark and anti-quark density distributions in a proton, summed over valence and sea quarks of all flavors relevant for CC (NC) interactions [11]:
q = u + d 2 + s + b,(7)
q =ū +d 2 + c + t,
q 0 = u + d 2 (L 2 u + L 2 d ) +ū +d 2 (R 2 u + R 2 d ) +(s + b)(L 2 d + R 2 d ) + (c + t)(L 2 u + R 2 u ),(8)q 0 = u + d 2 (R 2 u + R 2 d ) +ū +d 2 (L 2 u + L 2 d ) +(s + b)(L 2 d + R 2 d ) + (c + t)(L 2 u + R 2 u ),(9)with L u = 1 − (4/3)x W , L d = −1 + (2/3)x W , R u = −(4/3)x W and R d = (2/3)x W where x W = sin 2 θ W ,(10)
and θ W is the weak mixing angle. For theνN cross sections, Eqs. (4) and (5) are the same but with each quark distribution function replaced by the corresponding anti-quark distribution function, and vice-versa, i.e., q ↔q, q 0 ↔q 0 .
The differential cross sections for the dominant neutrino-electron scattering processes are given by [12]
dσν e e→νee dy = G 2 F m e E ν 2π R 2 e + L 2 e (1 − y) 2 (1 + 2m e E ν y/M 2 Z ) 2 + 4(1 − y) 2 × 1 + Le(1−2meEν /M 2 W ) 1+2meEν y/M 2 Z (1 − 2m e E ν /M 2 W ) 2 + Γ 2 W /M 2 W ,(11)dσν ee→νµµ dy = G 2 F m e E ν 2π 4 1 − m 2 µ /2m e E ν 2 (1 − 2m e E ν /M 2 W ) 2 + Γ 2 W /M 2 W ,(12)dσν ee→ντ τ dy = G 2 F m e E ν 2π 4 1 − m 2 τ /2m e E ν 2 (1 − 2m e E ν /M 2 W ) 2 + Γ 2 W /M 2 W ,(13)dσν e e→had dy = dσν ee→νµµ dy Γ(W → hadrons) Γ(W → µν µ ) ,(14)
where L e = 2x W − 1 and R e = 2x W are the chiral couplings of Z to electron, and Γ W GeV is the total width of the W -boson.
The main source of theoretical uncertainties in the neutrino-nucleon differential cross sections given by Eqs. (4) and (5) comes from the PDFs. The size of the PDF uncertainty with respect to the x and y variables defined in Eq. (6) can be seen from the distributions given in Fig. 1. For concreteness, we have shown the results for a fixed incoming neutrino energy E ν = 1 PeV, and have used the NNPDF2.3 PDF sets [13] based on a global data set including the recent LHC data. We use the PDF sets (central values and 90% CL error) given for α s (M Z ) = 0.118, and compare the cross section results evaluated with these PDFs at LO, NLO and NNLO, as shown in Fig. 1 by the solid, dashed and dot-dashed lines respectively, along with their corresponding 90% CL bands shown by the (dark to light) shaded regions. We also show the y-distribution of the electron antineutrino-electron cross section (ν e e → anything) given by Eqs. (11)-(14), which is of course independent of the PDF uncertainties. The distributions for the antineutrino-nucleon cross sections are similar to those for the neutrino-nucleon cross sections, just with slightly smaller values, and are not shown here.
From Fig. 1, we find that the PDF uncertainties in the y-distributions are constant in the low-y region, while they grow for smaller values of x. This is due to the uncertainties in the shape of light-quark and gluon distributions in the small-x and high Q 2 region. The lowest x and highest Q 2 scales accessed to date are by the DIS fixed target experiments at HERA [14]. Including these DIS data in their global PDF analysis, NNPDF2.3 could go down to x min = 10 −9 in the x-grid, and up to Q 2 max = 10 8 GeV 2 in the Q-grid [13]. The cross sections calculated using these PDFs have significantly reduced errors at low x, as compared to previous analyses (see e.g., [11,15]). Similar improved results in the low-x regime were obtained in Ref. [16] using the HER-APDF1.5 [14,17] PDF sets.
B. Total Cross Section
The total neutrino-nucleon cross section is obtained by integrating the differential cross sections in Eqs. (4) and (5) over the x and y variables:
σ(E ν ) ≡ 1 0 1 0 dxdy d 2 σ dxdy ,(15)
For completeness, we show in Fig. 2 the integrated SM cross sections for the CC and NC neutrino-nucleon (νN ) and antineutrino-nucleon (νN ) interactions, computed using the NNPDF2.3 PDF sets at NNLO. The numerical integrations in Eq. (15) were carried out using an adaptive Monte Carlo routine. For our numerical purposes, we take the lower limit of the x-integration to be 10 −6 in order to avoid large uncertainties in the low-x grids. The shaded regions in Fig. 2 reflect the 90% CL PDF uncertainties in the total cross section, and also to some extent, the uncertainties from the precision of the numerical integration technique used. Our results for the total cross section agree well with those calculated using other PDF sets; for a comparison, see e.g., [15,16].
At very high neutrino energies, the cross sections are dominated by contributions from very small x, which currently have a large uncertainty directly associated with the underlying QCD dynamics at high energies [9]. In this regime, one might have to go beyond the DGLAP formalism [18] used by conventional PDF fits, and to consider the non-linear evolution of the parton distribution arising due to the physical process of recombination of partons in the parton cascade. This can be characterized by the saturation scale of the growth of the parton distribution, forming a Color Glass Condensate [19]. Such saturation effects lead to slightly higher values of the neutrino-nucleon cross section and a steeper energy dependence at very high energies (E ν > ∼ 100 PeV) [20]. However, since the current IceCube events are observed at PeV scale and below, these non-linear effects are of less importance, and hence, we do not include them in our analysis.
At higher orders in QCD, the expressions (4) and (5) must be convoluted with appropriate quark and antiquark density distributions. The heavy quark masses should be taken into account at higher energies in the calculation of the structure functions [21], but the LO cross sections still give us a good estimate of the dominant contributions up to PeV energies. In fact, the numerical values of the cross sections at LO as shown in Fig. 2 agree with the NLO results given in Ref. [16] up to 5% or so for the current IceCube energy range of interest.
In Fig. 2 we also show the totalν e e − scattering cross section which has the 'Glashow resonance' at 6.3 PeV, and gives the dominant contribution in the vicinity of the resonance energy. Other neutrino-electron cross sections are not shown here since they are many orders of magnitude smaller than the νN cross sections [9]. The 'Glashow resonance' option has been considered earlier [22] to explain the two PeV events at IceCube. However, this possibility was disfavored by a dedicated followup analysis [1]. The average incoming neutrino energies required to explain the two PeV events are found to lie below the Glashow resonance region, and hence, the contributions fromν e e − scattering to the total number of events turn out to be negligible (see Section III). This effect could however be important if an excess of events is observed in the 3 -6 PeV deposited energy range.
III. SM PREDICTION FOR THE NUMBER OF EVENTS
To the best of our knowledge, no attempt has been made so far to quantify the PDF uncertainty effects on the number of events expected in each of the deposited energy bins at IceCube. It is important to include these effects for a better comparison of the observed IceCube events with any particle physics explanation. In order to obtain a reliable estimate of the number of IceCube events expected due to the SM interactions, we determine the values of various parameters in Eq. (1) as follows:
• T =662 days for the IceCube data collected between May 2010 and May 2012 [3].
• N eff (E ν ) = N A V eff (E ν )
where N A = 6.022 × 10 23 mol −1 is the Avogadro number which is equal to 6.022 × 10 23 cm −3 water equivalent for interactions with the ice nuclei. For interactions with electrons, N A should be replaced with (10/18)N A for the number of electrons in a mole of H 2 O. Note that a natural ice nucleus with 10 protons and 8 neutrons is close to being isoscalar, and hence, we use the generic reference PDF sets, without treating the protons and neutrons separately.
• V eff (E ν ) = M eff /ρ ice is the effective neutrino target volume, where ρ ice = 0.9167 g · cm −3 is the density of ice, and M eff is the effective target mass which includes the background rejection cuts and event containment criteria. It depends on the incoming neutrino energy and attains its maximum value M max eff 400 Mton (corresponding to V max eff 0.44 km 3 water-equivalent) above 100 TeV for ν e CC events [3], and above 1 PeV for other CC and NC events. There is some flavor bias at low energies caused by the deposited energy threshold due to missing energy in escaping particles from ν µ and ν τ CC events as well as all flavor NC events, which decreases M eff for these events as compared to the ν e CC events.
• For the incoming neutrino flux, we assume a Fermishock astrophysical flux falling as an unbroken power-law spectrum:
Φ(E ν ) = CE −s ν(16)
for the entire energy range of interest. The exact energy dependence governed by the spectral index s largely depends on the extra-terrestrial source evolution models. For a given value of s, the flux normalization C should be chosen to be consistent with the observational upper bound on the fluxes. Following the previous IceCube analyses [1,3], we first show our results for s = 2 with an all-flavor normalization
C = 3.6 × 10 −8 GeV · sr −1 · cm −2 · s −1 ,(17)
which is the integral upper limit on the UHE cosmic neutrino flux obtained in a previous IceCube search [23]. This normalization includes equal strength of neutrinos and antineutrinos summed over all neutrino flavors, and assumes an equal flavor ratio of ν e : ν µ : ν τ = 1 : 1 : 1 (same for antineutrinos), since neutrino oscillations over astronomical distances tend to equalize the neutrino flavors reaching the Earth, regardless of the initial flux composition [24]. We will also perform a χ 2analysis with the existing IceCube data to find the best-fit value of the flux normalization for different spectral indices.
• The solid angle of coverage Ω = 2π sr for an isotropic neutrino flux in the southern hemisphere (downward events at IceCube), while for those coming from the northern hemisphere (upward events) we must take into account the attenuation effects due to scattering within the Earth which can be represented by multiplying Ω with an energydependent shadow factor [11,25] S
(E ν ) = 0 −1 d(cos θ) exp − z(θ) L int (E ν ) ,(18)
where θ is the angle of incidence of the incoming neutrinos above nadir, z(θ) is the effective column depth which represents the amount of material encountered by an upgoing neutrino in its passage through the Earth, and L int (E ν ) = 1/σN A is the interaction length. The Earth attenuation effects are relevant at energies above 100 TeV. For the upgoingν e 's, the interaction length is very small near the Glashow resonance (due to its enhanced cross section), and hence, theseν e 's do not survive their passage through the Earth to the detector. For the upgoing ν τ 's, there is significant energy loss due to regeneration effects inside the Earth, which leads to fast τ -decays producing secondary neutrinos (of all flavor) with lesser energy than the original incident one [26], thereby shifting the energy of the upgoing ν τ 's downward when they reach the detector.
• The visible energy relevant for detection is the EMequivalent deposited energy E EM in Eq. (1), which is always smaller than the incoming neutrino energy E ν by a factor which depends on the interaction channel. For NC events given by Eq. (3), the cross section is identical for all flavors, and the fraction of energy imparted to the outgoing hadrons X is determined by the inelasticity parameter y. The resulting hadronic shower yields fewer number of photo-electrons than an equivalent-energy electromagnetic shower by a factor F X [27] which is a function of the hadronic final state energy E X = yE ν . We parametrize this energy dependence as [28]
F X = 1 − E X E 0 −m (1 − f 0 ),(19)
where E 0 = 0.399 GeV, m = 0.130 and f 0 = 0.467 are the best-fit values from the simulations of a hadronic vertex cascade [27].
Thus for NC events, the total deposited EMequivalent energy is given by E EM,had = F X yE ν . On the other hand, for ν e N CC events given by Eq. (4), the final state electron deposits its entire energy, E EM,e = (1 − y)E ν into an electromagnetic shower, and there is also an accompanying hadronic shower with deposited energy E EM,had . The factor F X reduces the deposited energy for a hadronic shower to about 80 -90% of an equivalent-energy EM shower.
The ν µ N CC events are similar in properties to those due to ν e N CC, assuming the final state muon energy to be completely measurable. We have not included in our analysis the effects of muon energy loss during its propagation in rock outside the detector since the IceCube search only considered the interaction vertices well contained within the detector volume, and discarded the events with through-going muon tracks originated outside the detector in order to remove the cosmic ray muon background.
The ν τ N CC events are however more complicated, with properties somewhat between NC and ν e N CC events. At the relevant energies (50 TeV < ∼ E ν < ∼ 2 PeV), tau leptons will travel only about 10 -50 m before decaying, so we do not expect them to produce the characteristic "double bang" signature [29] at IceCube as it has a string separation of 125 m [30]. These distinct signatures may only be visible at IceCube for τ -energies exceeding about 5 PeV when they travel far enough that the cascades from their production and decay are well-separated. The "double bang" could still be observed with less energetic τ 's in the proposed DeepCore experiment with string spacings as small as 42 m [31]. About 80% of the τ -decays in the current sample result in a shower, with decays to electrons in an EM shower, and hadronic decays involving multiple mesons in a hadronic shower. The rest ∼ 20% of the taus will produce muons which will give distinct muon tracks. The hadronic showers due to τ -decay will deposit an energy of roughly (1/2)F X (1−y)E ν (the other half being carried away by the associated ν τ 's), whereas the leptonic final states will deposit roughly (1/3)(1 − y)E ν , the rest being carried away by the final-state neutrinos.
Our final results for the expected number of events due to the SM interactions in all the 11 energy bins analyzed at IceCube are shown in Fig. 3 for an E −2 flux. The expected background due to atmospheric neutrinos and atmospheric muons were taken from Ref. [3], along with the combined statistical and systematic uncertainties (black shaded) which total to 10.6 +5.0 −3.6 events. The SM predictions for the number of signal events were obtained using the NNPDF2.3 [13] PDF sets, and to check the robustness of these results, we compared the results computed using the MSTW2008 PDFs [32]. We also compare the results obtained with the two PDF sets at LO, NLO and NNLO, including in our analysis the 90% CL PDF uncertainties on the predicted signal+background events for each bin. We find that our signal+background fit is consistent with the IceCube observed data points within their current uncertainties for all the energy bins, except the very first one where we predict an excess of about 2 events over that observed. This is slightly different from the IceCube signal+background best-fit given in Ref. [3]. This mild disagreement may be due to one or several of the following reasons: (i) The Ice-Cube best-fit was derived from a global fit of the deposited energy and zenith distribution of the data to a combination of the atmospheric neutrino background and an isotropic astrophysical flux in the range of 60 TeV -2 PeV, which does not include the lowest energy bin shown in Fig. 3. (ii) The PDF uncertainties in the cross section which are not shown in the IceCube bestfit. (iii) The uncertainty in the flux normalization. The IceCube fit has taken the per-flavor normalization to be E 2 ν Φ(E ν ) = (1.2 ± 0.4) × 10 −8 cm 2 · s −1 · sr −1 , whereas our results are derived assuming the central value of 1.2 × 10 −8 cm 2 · s −1 · sr −1 which is the current upper limit for an equal-flavor composition. (iv) some additional experimental effects relevant at lower energies to reduce the atmospheric background (e.g., hit probability) not captured in our simple analysis. It is also important to note here that we have directly used the true value of the inelasticity parameter y for a given PDF set in our numerical analysis, and not the the average inelasticity parameter as used in some of the previous analyses. The total number of SM signal events in each channel over the entire energy range of interest shown in Fig. 3 are summarized in Table I background events, we obtain a total of 31.46 +7.13 −4.85 sig-nal+background events, which is consistent with the 28 events observed by IceCube. From the energy distribution shown in Fig. 3, we find that although the SM expectations for the number of events in the highest energy bin observed so far is slightly below the current experimental central value, it is still consistent within the theoretical and experimental uncertainties. Note however that the SM fit shown in Fig. 3 was obtained with the current upper limit on the normalization of an astrophysical E −2 flux; hence, any additional excess in the future data and/or improvement in the flux limit would make it extremely difficult to explain by the SM alone, and could give us an important clue to some new physics.
In Fig. 4, we show the distribution of the declination angles of the events shown in Fig. 3, and find that out of the 19.58 +1.77 −0.61 signal events, 12.64 +0.26 −0.29 are downgoing and 8.21 +1.87 −0.96 upgoing. Combining this information with the distribution of the atmospheric background events, our signal+background fit seems to be in good agreement with the IceCube data obtained so far. Apart from the deposited energy and declination angle distributions, we can also understand several other features of the IceCube data with our simple SM interpretation:
• There are more downgoing events (about 60%) than upgoing due to the Earth attenuation effects. • The number of muon tracks (2.89 +0.04 −0.06 downgoing and 2.20 +0. 31 −0.53 upgoing) predicted in the SM seems to be consistent with the 7 track-like events (1 upgoing, 6 downgoing) observed by IceCube, with 4 of the downgoing events in the lower energy bins consistent with the expected 6.0 ± 3.4 background atmospheric muons. So at the moment, there does not seem to be a statistically significant paucity of muons. We must however emphasize that if there is indeed a persistent paucity of muons in future as more data is accumulated, and results in a significant statistical discrepancy with the SM expectations shown here, one will have to seriously consider some beyond SM explanation. For instance, one possible solution in such a (currently hypothetical) scenario could be due to a lepton-flavor violating (LFV) gauge interaction in a warped extradimensional setup [33]. In these models, one may have ν µ N → ν τ X CC interactions mediated by a TeV-scale W in the t-channel which could cause the paucity of muon events, while being consistent with the current experimental limits on LFV processes. Another alternative solution to the 'muon problem' could be due to the presence of an Rparity violating interaction in a supersymmetric scenario. However, as we have emphasized earlier, it is premature to consider such exotic possibilities without a clear statistical deviations of the IceCube data from the SM expectations.
• For the duration of the current data-taking by Ice-Cube, the lack of events between 2 -10 PeV, as would be expected by an unbroken E −2 flux considered here, indicates that there might be a break or cut-off in the spectrum close to 2 PeV, or the spectrum could be softer (such as E −2.2 or E −2.3 , but with a higher value of the flux normalization). However, it is difficult to explain all the observed events with a softer spectrum, as we will see below.
In the analysis presented above, we have assumed a simple unbroken power-law flux given by Eq. (16) with s = 2 and the flux normalization given by Eq. (17). To ascertain the range of the spectral index s compatible with the existing IceCube data, we perform a χ 2 -analysis, with the χ 2 -value is defined as
χ 2 = i (f SM i − f IC i ) 2 δf 2 i ,(20)
s χ 2 min C (GeV · sr −1 · cm −2 · s −1 ) where f SM i and f IC i are the number of events in the i-th energy bin, as predicted by the SM signal+background and as observed by IceCube respectively, and δf i is the corresponding experimental uncertainty in the i-th bin as reported by IceCube. The results are summarized in Table II and also in Fig. 5 for some typical values of s. For a given value of s, we fix the overall flux normalization C by minimizing the χ 2 -value over the 7 energy bins with non-zero observed number of events. The resulting energy distribution is shown in Fig. 5. Here we have chosen the central values of the NNPDF2.3 NNLO PDF sets. The corresponding PDF uncertainties are similar to those shown in Fig. 3, and hence, not shown here for purposes of clarity. We find that a softer spectrum (s > 2) does not give a good fit to the existing data, and the best-fit range of the spectral index is s = 1.8 -2, though the current data does not exclude a harder spectrum up to s = 1.5, provided there is a cut-off in the spectrum close to 2 PeV. The corresponding flux normalization range given in Table II is consistent with the current upper bounds from IceCube [23,34,35], and could be tested with more data in future.
IV. CONCLUSION
In summary, for a reliable search for signals of New Physics by the powerful IceCube detector, it is desirable to have a very good understanding of all aspects of the observed UHE neutrino events. Here we have shown that from a particle physics point of view, the current data seems to be consistent with the SM explanation for a simple astrophysical power-law flux, Φ = CE −s ν with C =(0.2 -3)×10 −8 GeV · sr −1 · cm −2 · s −1 and s = 1.5 -2, and so far does not require any New Physics. However, it is extremely important to bear in mind that as the statistics solidifies with the accumulation of more Ice-Cube data, it would provide us with a unique test of the SM with the highest neutrino energies ever observed in Nature, and if any significant deviation from the current energy spectrum emerges, it will call for a beyond SM scenario.
FIG. 1 .
1The x-and y-distributions for the neutrino-nucleon CC and NC cross sections for a fixed energy Eν = 1 PeV. The solid, dashed and dot-dashed lines correspond to the distributions with the PDFs at LO, NLO and NNLO respectively, and their 90% CL uncertainties are shown by the dark to light shaded regions. The y-distribution of the electron antineutrino-electron cross section is also shown (lower solid line on the right panel).
FIG. 2 .
2The neutrino-induced scattering cross sections for the dominant SM processes as a function of the incoming neutrino energy. The νN andνN cross sections were computed using NNPDF2.3 at NNLO, and their 90% CL ranges are shown by the shaded regions.
FIG. 3 .FIG. 4 .
34The SM signal+background events, along with their 90% CL PDF uncertainties (green shaded), for the IceCube deposited energy bins between 16 TeV -2 PeV. The IceCube data points (with error bars) and the atmospheric background (black shaded) were taken from Ref.[3]. The SM signal events were computed for an E −2 flux using the procedure outlined in the text, and using two different PDF sets (NNPDF2.3 and MSTW2008) at LO, NLO and NNLO for a better comparison. The zenith angle distribution of the SM signal+background events, along with the 90% CL PDF uncertainties (green shaded). The IceCube data points (with error bars) and the atmospheric background (black shaded) were taken from Ref.[3].
FIG. 5 .
5The energy distribution of the SM signal+background events (similar to Fig. 3) for a power-law flux with different spectral indices. The flux normalization is fixed by minimizing the χ 2 -value. For PDFs, we have taken the central values of the NNPDF2.3 NNLO PDF sets.
TABLE I .
ITotal number of SM signal events expected from
different final states in the deposited energy range 16 TeV -
2 PeV. The theoretical errors are derived using the 90% CL
PDF uncertainties.
. The central values and the theoretical errors shown here are derived using the NNPDF2.3 NNLO and its 90 % CL uncertainties. The corresponding numbers for LO and NLO and also for MSTW PDFs are of similar magnitude and are not shown here. Note that at the moment, the IceCube detector can not distinguish between electromagnetic and hadronic shower events, and hence, collectively calls them the 'cascade' events, whereas the muons appear as distinct 'track' events. Thus we find that in the energy range of interest, the SM predicts 15.76 +1.78−0.66 cascade events and
5.09 +0.36
−0.59 muon tracks. Combining this with the 10.6 +5.0
−3.6
TABLE II .
IIThe best-fit flux normalization values for different spectral indices of the incoming UHE neutrino flux.Events per 662 Days
Deposited EM-Equivalent Energy (TeV)
s=1.5
s=1.8
s=2
s=2.2
s=2.5
10 -1
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| [] |
[
"Kazakhstan engineering",
"Kazakhstan engineering"
] | [
"N Baibekov S ",
"A Dossayeva A ",
"Center "
] | [] | [] | This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration, we propose a number of lemmas and theorems that, together with the Dirichlet and Euler theorems, make it possible to prove the infinity of prime twins. | 10.20431/2347-3142.0504002 | [
"https://arxiv.org/pdf/1805.00346v1.pdf"
] | 119,313,127 | 1805.00346 | c27e0aaef9de77dd2daf1c6da6697f415f8f0026 |
Kazakhstan engineering
N Baibekov S
A Dossayeva A
Center
Kazakhstan engineering
Prime numbersprime-twinscomposite numbersintegersalgorithmarithmetical progressionmatrix of prime numbersspecial factorial
This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration, we propose a number of lemmas and theorems that, together with the Dirichlet and Euler theorems, make it possible to prove the infinity of prime twins.
Introduction
In 1912, during 5-th International Mathematical Congress, which was held at the University of Cambridge, it was noted that there is a number of open problems in the field of prime numbers. One of them is a problem of proving the infinity of number of prime twin numbers. This problem, which has not been solved for more than 2000 years as of today, is now referred as the second Landau problem.
The aim of this work is to prove the infinity of prime twins.
In [1] and [2] we proposed two versions of the analytic proof of the infinity of prime twin numbers, which, we believe, give a correct result about the infinity of twins. In this paper we propose another new proof of the infinity of twins, which, if possible, turned out to be more "elementary" and more explicit. We believe that the successful use of the fundamental works of world famous scientists (Eratosthenes, Euler, Dirichlet and others) allowed us to do this.
First, for convenience, as in [1] and [2], we introduce the following notation. As known, a consecutive multiplication of natural numbers is called a factorial: ∏ =1 = ! In the future, sequential multiplication of primes will occur quite often, so for such cases we use the notation: 2 * 3 * 5 * 7 * 11 * … * = ∏ =1 = ! ′. Here is a prime number with a serial number . A combination of symbols ! ′ represent consistent multiplication of prime numbers from 2 to . It will be called a special factorial (or primorial) of prime number . For example, 4 ! ′is a special factorial of fourth prime number 4 = 7 or 4 ! ′ = 7! ′ = 2 * 3 * 5 * 7 = 210.
To achieve the goal set for the proposed work, we develop matrices of prime numbers.
Matrix of prime numbers
Let us represent the set of natural numbers (except for number 1) in a form of a matrix family with elements of ( , , ), where is a serial number of rows, is a serial number of columns and kis a serial number of matrix .
( , , ) = ( + 1) + ! ′( − 1) = + ( − 1), (1) where = 1, 2, … , ∞; = 1, 2, … , ! ′. = + 1 -is the first number of plurality of numbers that are in row. Note, index k of the parameters and denotes affiliation of these parameters to matrix .
It appears from the equation (1) that the sequence of numbers in any row of the matrix , is an arithmetic progression with first term equal to = + 1. The difference in this progression is = ! ′. In this case, the maximum number of rows of matrix should be equal to a special factorial ! ′, i.e.: , = ! ′ The number of columns can be arbitrarily large up to infinity (Fig. 1). As known, the number 1 is not a prime number, otherwise all numbers that are multiples of 1 would be composite numbers. The number 1 is also not a composite number, since it is not divisible by other numbers. That is why the number 1 is located separately in the upper left corner of the matrices. In Fig. 1, only four matrices 1 , 2 , 3 and 4 are shown for clarity. The number of rows of other matrices exceeds the paper size, so they are not shown in Fig 1. It follows from (1) that each matrix from the family of matrices , where = 1, 2, 3, … , ∞, differs from each other, primarily by the number of rows and the difference in arithmetic progression. For example, in case of matrix −1 , the arithmetic progression constant will be equal to −1 = −1 ! ′, with the maximum number of rows being −1, = −1 ! ′. In papers [2], and also in [1], to determine the properties of the developed matrices, we proposed a number of lemmas and theorems, as well as their detailed proofs. Therefore, for convenience, we present them without proof. In particular: Lemma 1. To determine whether an arbitrarily chosen number can occupy the same place in matrices and +1 , the following lemma was proved:
Any integer Z > 0 occupies only one specific place in the matrix
Lemma 2.
If the randomly selected number takes a place located in the first column of the matrix , then it occupies the same place in the matrix + , but in other cases it takes different places in matrices
and + . Considering the set of numbers in an arbitrary row of matrix , and to determine the distribution of these numbers in case of matrix +1 , we also proved the following lemma: From Dirichlet's theorem for primes in an arithmetic progression it follows that if the first term and the difference of the progression are not mutually prime, then in this progression there will not be one prime number or there will be only one prime number. Moreover, this prime number is the first term of the arithmetic progression in question. It also follows from Dirichlet's theorem that if the first term and the difference of the arithmetic progression are mutually prime, then in this progression there is an infinite set of primes [3], [4] and [5].
In Fig.1 all of the rows in which there are only composite numbers (and possibly with one prime number), i.e. the first number and the difference in the arithmetic progression of which are not mutually prime, are, for clarity, repainted in a dark color and such rows will be called ordinary rows. The number of ordinary (repainted) rows will be denoted as ,, where the index k indicates belonging of these rows to the matrix .
In this figure, all the rows in which there is an infinite number of prime numbers, i.e. the first number and the difference in the arithmetic progression of which are mutually prime, have been left uncolored for clarity. The total number of uncolored rows is denoted as ,, where the index k also shows the belonging of these rows to the matrix .
It follows from (1) that if > 2, then prime twin numbers can not be located in a one particular uncolored row. Twins can only be in two adjacent uncolored rows, the first term and the difference in the arithmetic progression of which are mutually prime, and on the other hand the difference of the ordinal numbers of these two adjacent uncolored rows must be equal to 2. A pair of such uncolored rows is called a pair of twin rows or twin-rows. For two numbers located in different rows, but in one column of pairs of twin-rows, the equality | ( , , ) − ( , ± 2, )| = 2 is always valid. A pair of such prime numbers are twins. For example, from Fig. 1 it can be seen that prime twin numbers (5 and 7), (29 and 31), (41 and 43), and others are located in different rows, but in one column. The total number of pairs of twin rows is denoted by , where the index k also shows the belonning of these rows to the matrix .
If a row has an infinite set of prime numbers, but it is located from the same row at a distance greater than 2, then such row is called a single row. Therefore, in single rows there can not be any prime twin numbers. The total number of single rows is denoted as , where the index k also shows the belonging of these rows to the matrix . Then the total number of uncolored rows is = + 2 (2) If a total number of all rows to be denoted as , then Ω = + = + 2 + (3) On the other hand,:
Ω = , = = ! ′ (4)
The aim of this work is to determine the total number of primes-twins. Therefore, in the future we will be focusing on the pairs of twin-rows.
3.
An infinite amount of prime twin numbers As it was mentioned above, all twin prime numbers are mainly located in paired of twin-rows. If in this case at some point, for example, when considering matrix, all pairs of twin-rows disappear (i.e. they for some reason have become ordinary or single rows), then, it is obvious that they will not appear in any of the next matrices. In this case, it means that the number of twins should be limited.
Consider the number of rows of matrix in which prime numbers are being distributed.
( −1 ) = ( − 1)! ′ ( −1 − 1)! ′ = − 1 (6)
Theorem 2 is proved.
A number of corollaries follow from this theorem.
Corollary 1 from Theorem 2.
It follows from Lemma 3 that an infinite set of numbers in a chosen row (for example, in an uncolored) of the matrix −1 , are redistributed inside rows of the matrix . On the other hand, from Theorem 2 we can state that an infinite set of prime numbers in the same uncolored row of matrix −1 , are redistributed inside − 1 uncolored rows of matrix .
2 It follows from (1) and Fig. 1 that for our arithmetic progressions the condition 2 ≤ ≤ + 1 is satisfied. In this case, if the number 1 is transferred to the matrix , then all the members of the last row go to the 1st row and the penultimate row becomes the last row. Then the condition is 1 ≤ ≤ fulfilled automatically. Therefore, all the conclusions obtained for the conditions 1 ≤ ≤ are applicable and are used for our cases. We notice that in this case the first number of this last row will be equal = = ! ′. Thus it is a row of composite numbers, which is a colored row.
Consequently, the infinite set of some part of the composite numbers that were in an uncolored row of the matrix −1 in the case of the matrix are separately redistributed as a separate ordinary (colored) row for which the first term and the difference of the arithmetic progression are not mutually prime.
Corollary 2 of Theorem 2.
Now consider the general case. Suppose that there are −1 uncolored rows in a matrix −1 . Then, continuing the conclusions of Lemma 3, we find that the infinite set of numbers in these rows are redistributed inside −1 rows of the matrix . On the other hand, it also follows rom Theorem 2 that the infinite set of prime numbers in these −1 uncolored rows of the matrix −1 are redistributed in −1 ( − 1) uncolored rows of matrix .
Consequently, the infinite set of some of the composite numbers that were in all of the uncolored −1 rows of the matrix −1 , in the case of the matrix are separately redistributed in −1 separate ordinary (colored) rows, for each of which the first term and the difference of the arithmetic progression are not mutually prime.
Corollary 3 of Theorem 2.
Now we determine exactly which composite numbers are separated from an infinite set of prime numbers and redistributed as separate colored rows in the matrix .
As it was shown above, the first term and the difference of the progression of each of these colored rows are not mutually prime; from (1) we obtain that:
( , ) ≢ 1 and ( , ! ′) ≢ 1 This happens when the value of the first member of the arithmetic progression of the observed row is equal to the value (or a multiple of that value) of one of the following primes 1 , 2 , 3 , … , . We will show this more clearly and in detail. For example, in the case of matrix 1 , only one prime number 1 = 2 is being considered. Consequently, an infinite set of composite numbers, multiples of 2, which were in an infinite row of natural numbers, i.e. in the matrix 0 , in the case of the matrix 1 are now separated from the prime numbers and arranged as a separate colored row (Fig. 1, 1 ). Note that the first member of this row is 2, i.e. 1 = 1 = 2. Thus, during the formation of the matrix 1 , an infinite set of composite numbers that can be divided by 2 are fully identified and output in a separate row, which for clarity is recolored into a dark color (Fig. 1, 1 ). Note that the prime number 1 = 2 is also separated from the other prime numbers and being the only prime number is located in the given colored row along with other identified composite numbers.
In the case of matrix 2 , the next prime number is 2 = 3. In this case, all of the numbers that in the case of the matrix 1 were in its only uncolored row are redistributed along the three rows of the matrix 2 . In this case, one of these rows, in which the first term and the progression constant are not mutually prime, becomes colored. That is, for this row the following condition must be fulfilled:
( 2 , 1 2 ) ≢ 1. From this condition we obtain that the fulfillment of the requirement ( 2 , 1 ) ≢ 1 has already been considered in the case of matrix A 1 . Therefore, for this row, the following condition must be fulfilled:
( 2 , 2 ) ≢ 1. This automatically implies that the first number of this colored row should be 2 = 2 = 3. The remaining numbers of this row must be equal to the multiple of the prime number 2 = 3. Thus, it should be noted that during the formation of the 2 matrix an infinite set of composite numbers that are divisible by 3 are fully revealed and are also output in a separate row, which for clarity is recolored into a dark color (Fig. 1, 2 ). Note that in this case the number 2 = 3 is also separated from other prime numbers and being the only prime number is located in the given colored row along with other identified composite numbers.
As a result, we get that in matrix 2 an infinite set of composite numbers, which are divided by 1 = 2 and 2 = 3, are fully revealed and distributed in separate colored rows. Moreover, we also get that the prime numbers 1 and 2 will not participate in the distribution of other prime numbers in the uncolored rows of the matrix 2 .
This pattern of distribution of prime and composite numbers is observed and occurs in all subsequent matrices. For example, continuing the above algorithm up to the matrix , we get that an infinite set of composite numbers that are not mutually prime with the difference of the arithmetic progression = ! ′ = 1 2 3 … , are distributed over the individual colored rows of the matrix . It should be noted that these separated composite numbers can be conditionally divided into k groups:
The 1st group is an infinite set of composite numbers that are divisible by the number 1 = 2, The 2nd group is an infinite set of composite numbers that are divisible by the number 2 = 3, The 3rd group is an infinite set of composite numbers, which are divisible by the number 3 = 5, … The -th group is an infinite set of composite numbers, which are divisible by the number . Summarizing these groups, we can say that in the colored rows of the matrix an infinite set of those composite numbers that are not coprime with the numbers 1 , 2 , 3 , … , will be distributed. It should be said that in this case, i.e. in the case of matrix , the following prime numbers 1 , 2 , 3 , … , are also separated from other prime numbers and will not participate in the redistribution of an infinite set of other prime numbers in ( − 1)! ′ rows of the matrix . Each of them will be located in the corresponding colored rows of the matrix , as the first number of this row and the corresponding arithmetic progression.
Theorem 3. The number of pairs of twin rows in the matrix increases monotonically with increasing order number k of the given matrix, and in each row of these pairs of twin rows there are an infinite number of primes.
Proof of Theorem 3.
It follows from Lemma 3 that all the numbers in one uncolored row of the matrix −1 (including in one of the rows of an arbitrary pair of twin rows) are redistributed over rows of the matrix . Consequently, all the numbers in an arbitrary pair of twin rows of the matrix −1 are redistributed in pairs of rows of the matrix .
Consider the case, suppose there are −1 pairs of twin rows in the matrix −1 .We choose one pair from them and will observe its first row. It follows from Theorem 2 that all prime numbers in this row are redistributed in − 1 rows of the matrix . This means that, from the analyzed pairs of rows of the matrix , one pair ceases to be a pair of twin rows.
If we consider the redistribution of the numbers in the second row of the given pair of twin rows of the matrix −1 , we similarly get that all the numbers in this row are redistributed along the other rows of the matrix . Moreover, the set of prime numbers will be redistributed along − 1 rows of the matrix. This means that from the analyzed pairs of rows of the matrix , one more pair ceases to be a pair of twin rows.
As a result, we obtain that all prime numbers in an arbitrary pair of twin rows of the matrix −1 , in the case of the matrix , are redistributed along its − 2 pairs of twin rows.
From the above, we find that from the analyzed pairs of rows of the matrix , two pairs of rows cease to be a pair of twin rows. Thus, in this matrix there are 2 pairs, each of which consists of one colored (ordinary) and one uncolored row.
If we take into account that there are −1 pairs of rows-twins in the matrix −1 , then the total number of pairs of twin rows in the matrix will be equal to: = −1 ( − 2) (7) As it was shown above, there is only one pair of twin rows in the matrix 2 (Fig. 1). With this in mind, it is easy to establish that the number of pairs of twin rows in matrix is determined by the following expression: = ( 2 − 2)( 3 − 2) * … * ( − 2) = ( − 2)! ′ , where ≥ 2 (8) It follows from (8) that as the number of the matrix increases, the number of pairs of twin rows in it increases monotonically. On the other hand, the sequence of numbers in each row of pairs of twin rows is an arithmetic progression. Moreover, as it was shown above, the first term and the difference of each of these progressions are mutually prime. Thus, it follows from Dirichlet's theorem that every row of any pair of twin rows has an infinite quantity of prime numbers.
Theorem 3 is proved.
It follows from (8) that lim →∞ = ∞, i.e. as k → ∞, the number of pairs of twin rows of the matrix tends towards infinity. If we assume that in each pair of twin rows there is at least one pair of prime twins, then we can definitely say that the number of twins is infinite. And this is possible when the prime numbers in each uncolored row, including rows of any pair of twin rows of the matrix , are located closer and denser than in case of the matrix −1 .
In this case, from an infinite set of prime numbers in each of two rows of any pair of twin rows, at least 2 prime numbers will necessarily be in the same column and thus they will be twins. Yet, it really so?
To see this, we first analyze the following general case. Suppose that in the matrix −1 there are −1 single (uncolored) rows, −1 ordinary (colored) rows and −1 pairs of twin rows. Obviously, the total amount of rows is: Ω −1 = −1 + 2 −1 + −1 = −1 ! ′ (9) Let's now define how many uncolored (single, as well as in pairs of twin rows) and colored (regular) rows will be in the matrix .
It follows from Theorem 2 that all prime numbers in −1 single (noncolored) rows of the matrix −1 are redistributed along −1 ( − 1) single rows of matrix , i.e.:
= −1 ( − 1)(10)
In addition, from this equality and Lemma 3 we also obtain that −1 rows become ordinary colored rows. The number of these newly formed colored rows is equal to: = −1 .
(11) Here, the superscript s of the parameter shows that these colored (usual) rows of the matrix appear from single (uncolored) rows of matrix −1 . From Lemma 3 we can also obtain that all numbers located in −1 colored rows of the matrix −1 are redistributed over −1 rows of the matrix , i.e.: = −1 (12) Here, the superscript b of the parameter shows that these colored (ordinary) rows of the matrix appear from the ordinary (colored) rows of matrix −1 . Let's now consider −1 pairs of twin rows of the matrix −1 . From Theorem 3, and also from (7) and (8), we will find that the total number of pairs of twin rows in matrix is equal to:
= −1 ( − 2) = ( − 2)! ′, where ≥ 2.
At the same time, as shown above, some pairs of rows (total in 2 −1 ) in the matrix , cease to be pairs of twin rows: one row of each of these pairs of rows will appear to be single (uncolored) rows, and the other will appear to be ordinary (colored) rows. Hence, there appear new colored and uncolored rows in matrix , and their total quantity is equal to: = 2 −1 (13) = 2 −1 (14) Here, the superscript t of the parameters and shows that these analyzed rows appear from pairs of twin rows of matrix −1 .
As a result of (8) -(14) we get that the total number of colored and uncolored rows in the matrix will be equal to: (15) and (16) we can find the total number of rows in the matrix : Ω = + = −1 + 2 −1 + −1 (17) Comparing (9) and (17), we get: Ω = Ω −1 = ! ′ The last equality identically coincides with Lemma 4, which confirms the correctness of the above conclusions. Now let's consider the number of prime and composite numbers located in uncolored rows of the matrix and not exceeding such that: ≫ = ! ′ .
= + + = −1 + 2 −1 + −1 (15) = + + 2 = −1 ( − 1) + 2 −1 − 2 −1 (16) Using
(18) Let 0 , 0 and 0 be a quantity of respectively prime, composite numbers, as well as all numbers contained in the arithmetic progression, consisting of a series of natural numbers 3 from 1 to x: 0 = 0 + 0 . Let , , and , be a quantity of respectively prime, composite numbers, as well as all numbers 4 not exceeding and located in one uncolored row with the serial number i: , = + , Then , and are total numbers of respectively prime, composite, as well as all available numbers not exceeding and located in all non-colored rows of the matrix , are determined by simple summation:
= ∑ ( ) =1 = ∑ ( ) =1
3 An arithmetic progression consisting of a series of natural numbers from 1 to x is not part of the matrix considered (Fig. 1). Therefore, in order to describe elements of this progression, the index "0" is used. 4 Numerous scientific works are devoted to the problem of prime numbers [4]. For example, in 1935 Siegel obtained that for any fixed A> 1 for 1 ≤ ≤ log , for an arithmetic progression with parameters ( , ) ≡ 1 и 1 ≤ ≤ , it is true that:
( , , ) = ( ) ( ) + ( exp(− √ln )).
where D is the difference of the progression, l is the first term of the progression. This expression in the scientific literature is known as the Siegel-Walphlisch formula. In this formula, ( ) is the Euler function, A is a positive constant, = ( ) > 0 is a constant, where Li( ) = ∫ ln 2 . At the present time, many functions are known regarding the quantity of prime numbers not exceeding x in an arithmetic progression. Moreover, if the generalized Riemann hypothesis is considered correct, then, as known, the following formula holds:
( , , ) = ( ) ( ) + (√ ln )
It should be said that the last function is the closest to the real values of ( , , ). Nevertheless, many problems related to even more exact approximations remain unresolved [4], [5].
= + = ∑ ( + ) ( ) =1
Here the summation is carried out over uncolored rows. Let's introduce several new notation terms. Let , , , and , be average numbers of prime, composite and all numbers not exceeding and located in one uncolored row of matrix . Then,
On the other hand, it is obvious that the total number of all numbers (prime and composite) not exceeding and located in any row of the matrix are equal to each other and defined by the relation: It follows from Theorem 1 that the prime numbers that are distributed along ( −1 ) rows of the matrix −1 , in the case of matrix , are being redistributed along its ( ) rows. On the other hand, from the corollary to Theorem 2, we obtain that this redistribution occurs without any participation of a prime number . Therefore, the following equalities hold:
. Moreover, the serial numbers (column number) and (row number), in whicha given number is located, are determined in a uniquely way:where -residue obtained by dividing the ( − ) on ! ′, i.e.:
Lemma 3 .
3The set of numbers located in one selected row of the matrix , are redistributed in + rows of the matrix + . At the same time, sequence of numbers contained in each of these + rows, is an arithmetic progression with a constant + = + ! ′. This lemma automatically implies the proof of the following lemma: Lemma 4. The set of numbers located in all the rows of the matrix are redistributed over + ! ′ rows of the matrix + .
Theorem 1 .
1The number of unpainted rows of matrix, in which prime numbers are being distributed, is determined by the Euler's function ( ), where = ! ′a constant in arithmetic progressions of numbers that are in each of these rows. Proof of theorem 1. By definition the Euler's function ( ) determines the number of positive integers , not exceeding and which are mutually prime 2 with , i.e. 1 ≤ ≤ and ( , ) ≡ 1. On the other hand, from Dirichlet's theorem it follows that if the first term and the difference of arithmetic progression are mutually prime, then in this progression there are infinitely many prime numbers. Consequently, the number of rows of the matrix , on which an infinite number of prime numbers is being distributed is equal to the value of Euler's function ( ): = ( ) = ( ! ′) Prime numbers, which are located in one unpainted row of the matrix − , are redistributed in − unpainted rows of matrix . Proof of theorem 2. From Theorem 1 it follows that in the case of the matrix −1 prime numbers are redistributed according to its ( −1 ) = ( −1 ! ′) = ( −1 − 1)! ′ rows. In case of matrix redistribution of prime numbers occurs in ( ) = ( ! ′) = ( − 1)! ′ rows. Consequently, if we consider the set of prime numbers that are in one of the unpainted rows of −1 matrix, then their redistribution in the case of the matrix occurs over uncolored rows, the number of which is equal to ( )
Here and later on we shall consider the cases when condition (18) is satisfied, i.e. ≫ = ! ′ . Let's determine the average density of prime numbers not exceeding and located in one uncolored row of the matrix : The density of prime numbers in uncolored rows of the matrix grows monotonically with the increase in the sequence number k of the observed matrix. And the density of prime numbers in each uncolored row in limit tends towards the limit value: →∞ = Proof of theorem 4.
The number of columns, as in the case of matrix , can be arbitrarily large up to infinity.39
69
99 129 159 …
24
25 235 445 655 …
w 9
10
40
70 100 130 160 …
25
26 236 446 656 …
10
11
41
71 101 131 161 …
26
27 237 447 657 …
n 11
12
42
72 102 132 162 …
27
28 238 448 658 …
u 12
13
43
73 103 133 163 …
28
29 239 449 659 …
m 13
14
44
74 104 134 164 …
29
30 240 450 660 …
b 14
15
45
75 105 135 165 …
30
31 241 451 661 …
e 15
16
46
76 106 136 166 …
31
32 242 452 662 …
r 16
17
47
77 107 137 167 …
32
33 243 453 663 …
17
18
48
78 108 138 168 …
33
34 244 454 664 …
18
19
49
79 109 139 169 …
34
35 245 455 665 …
19
20
50
80 110 140 170 …
35
36 246 456 666 …
20
21
51
81 111 141 171 …
36
37 247 457 667 …
21
22
52
82 112 142 172 …
37
38 248 458 668 …
22
23
53
83 113 143 173 …
38
39 249 459 669 …
23
24
54
84 114 144 174 …
39
40 250 460 670 …
24
25
55
85 115 145 175 …
40
41 251 461 671 …
25
26
56
86 116 146 176 …
41
42 252 462 672 …
26
27
57
87 117 147 177 …
…
…
…
…
…
…
27
28
58
88 118 148 178 …
207
208 418 628 838 …
28
29
59
89 119 149 179 …
208
209 419 629 839 …
29
30
60
90 120 150 180 …
209
210 420 630 840 …
30
31
61
91 121 151 181 …
210
211 421 631 841 …
Matrix A 3
Matrix A 4
Fig.1. Matrices of prime numbers 1
The number of rows of matrix 4 exceeds the paper size, so here only a fragment is shown.Also, for this reason, other matrices are not shown in this figure.
( 1 ) = 0 ( 0 ) − 1 2,( 2 ) = 1,( 1 )− 1 3,( 3 ) = 2,( 2 )− 1 …………………..…………..…….. , ( ) = −1. ( −1 ) − 1 From the last equality we get that: , = −1, ( −1 ) ( ) − 1 = −1, − 1 − 1 ,(21)
where the small value 1 = 1( ). From (20) and (21) we obtain that:where the small value 2 = 1 , = 1 ( ) ,.From (22) it follows that: prime numbers in any uncolored row of matrix are more densely distributed than in case of the previous matrix −1 , with an increase in order number k of the matrix , the density of prime numbers in any of its uncolored rows increases continuously. On the other hand, as k → ∞, it becomes obvious that:Consequently, taking this into account, we obtain from (22) that:i.e. as → ∞ in the limit, we get that lim Equation(26)means that as the order number of the matrix increases, the density of prime numbers in its uncolored rows grows monotonically and tends to the limiting value.Theorem 4 is proved.It should be said that the fulfillment of Theorem 4, in particular (26), is possible when prime numbers in any uncolored row of matrix are more densely distributed than in case of uncolored rows of previous matrices. This means that in each pair of twin rows of the matrix there will undoubtedly be twins. Now let's go back to the main problem of this work.Theorem 5. The hypothesis of an infinite quantity of prime twins is true.Proof of theorem 5.Consider the matrix for a truly large → ∞. It follows from Dirichlet's theorem that every row of any pair of twin rows of contains an infinite set of primes.On the other hand, it follows from Theorem 3 that as the order number of the matrix grows, the number of pairs of twin rows in it grows monotonically, such that limIt also follows from Theorem 4 and (26) that as the order number of the matrix grows, the density of prime numbers in each uncolored row grows monotonically. This means that in each pair of twin rows there are twin numbers. This, in turn, implies that the number of twins is infinite.Theorem 5 is proved.Conclusion.In this paper, in order to study the properties of prime twin numbers, we first introduce the notion of a matrix of prime numbers. Then we consider the density of prime numbers in separate rows of a matrix to be considered. In this paper we prove a number of lemmas and theorems that together with the Dirichlet and Euler theorems make it possible to prove the infinity of prime twins.AUTHOR'S BIOGRAPHYSeidikassym N. Baibekov Nationality -Republic of Kazakhstan Education -Physical faculty of Leningrad State University Academic degree -Doctor of technical science Academic title-Professor Author of more than 130 scientific articles and about 20 monographs, textbooks and manuals Academic interests -mathematics (number theory), physics, ITtechnology and others.
Infinite Number of Twin Primes. S N Baibekov, A A Durmagambetov, Advances in Pure Mathematics. 6Baibekov S.N., Durmagambetov A.A., Infinite Number of Twin Primes. Advances in Pure Mathematics, 2016, 6, 954-971.
Proof of an Infinite Number of Primes-Twins. S N Baibekov, A A Durmagambetov, International Journal of Scientific and Innovative Mathematical Research. 5Baibekov S.N., Durmagambetov A.A., Proof of an Infinite Number of Primes- Twins. International Journal of Scientific and Innovative Mathematical Research (IJSIMR), 2017, 5 (19), 6-17
. K Prachar, Primzahl Verteilung, Springer-VerlagBerlinPrachar K., Primzahl Verteilung. Springer-Verlag, Berlin, 1957.
A L Karatsuba, Fundamentals of analytic number theory. M., Nauka. 240Karatsuba A.L., Fundamentals of analytic number theory. M., Nauka, 1983, 240 p.
The Introduction to the theory of algebraic numbers. М М Postnikov, Science. Postnikov М.М. The Introduction to the theory of algebraic numbers. М.: Science, 1986.
The authors would be sincerely grateful to those who will send their comments and suggestions to: baibekovsn@mail. P , ruP.S. The authors would be sincerely grateful to those who will send their comments and suggestions to: [email protected]
| [] |
[
"ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE",
"ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE",
"ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE",
"ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE"
] | [
"Philip Greengard ",
"Jeremy G Hoskins ",
"ANDNicholas F Marshall ",
"Amit Singer ",
"Philip Greengard ",
"Jeremy G Hoskins ",
"ANDNicholas F Marshall ",
"Amit Singer "
] | [] | [] | This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W 2 . In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = −∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented. Ω |∇ϕ| 2 dµ. | null | [
"https://arxiv.org/pdf/2201.13386v2.pdf"
] | 246,430,256 | 2201.13386 | c475eda46328217b54d2df641c4e9f4bbb019106 |
ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE
Philip Greengard
Jeremy G Hoskins
ANDNicholas F Marshall
Amit Singer
ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE
This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W 2 . In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = −∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented. Ω |∇ϕ| 2 dµ.
1. Introduction 1.1. Introduction. Let µ and ν be probability measures supported on a bounded convex set Ω ⊂ R N . The quadratic Wasserstein distance W 2 (µ, ν) is defined by (1) W 2 (µ, ν) 2 := inf π Ω×Ω |x − y| 2 dπ(x, y),
where the infimum is taken over all transference plans π (probability measures on Ω × Ω such that π[A × Ω] = µ[A] and π[Ω × A] = ν[A] for all measurable sets A). Computing the quadratic Wasserstein distance is a nonlinear problem. In this paper, we consider the case where µ and ν have smooth positive densities f and g with respect to Lebesgue measure: dµ = f dx and dν = g dx. In this case, there is a classic local linearization of W 2 based on a weighted negative homogeneous Sobolev norm, which is derived from linearizing a general Monge-Ampére equation, see §2.
In particular, the weighted negative homogeneous Sobolev norm · Ḣ−1 (dµ) is defined for functions u such that Ω udµ = 0 by
(2) u Ḣ−1 (dµ) := sup Ω u ϕ dµ : ϕ Ḣ1 (dµ) = 1 , where (3) ϕ 2Ḣ 1 (dµ) :=
That is, the spaceḢ −1 (dµ) is the dual space ofḢ 1 (dµ). Under fairly general conditions, if δµ denotes a perturbation of µ, then (informally speaking) we have W 2 (µ, µ + δµ) = δµ Ḣ−1 (dµ) + o( δµ ), see for example [Theorem 7.2.6 [29]] for a precise statement. Under stronger assumptions, if δµ = ε, then the error term can be shown to be O(ε 2 ), see §3.5 for details. In this paper, we study how to leverage the connection between thė H −1 (dµ)-norm and the W 2 metric for computational purposes. In particular, our computational approach is based on a connection between theḢ −1 (dµ)-norm and the Witten Laplacian, which is a Schrödinger operator of the form
H = −∆ + V,
where V is a potential that depends on f , see Figure 1. We show that this connection provides a method of computation whose computational cost can be controlled by the amount of regularization used when defining the potential, see §4 for details. For the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays, we present a code for computing this linearization of W 2 based on the Witten Laplacian, and present a number of numerical examples, see §5. This Witten Laplacian perspective leads to several potential applications; in particular, a method of defining an embedding discussed and methods of smoothing discussed in §6.
1.2.
Background. The quadratic Wasserstein distance W 2 is an instance of Monge -Kantorovich optimal transport whose study was initiated by Monge [18] in 1781 and generalized by Kantorovich [12] in 1942. The theory of optimal transport has been developed by many authors; for a summary see the book by Villani [29]. Recently, due to new applications in data science and machine learning, developing methods to compute and approximate optimal transport distances has become an important area of research in applied mathematics, see the surveys by Peyré and Cuturi [22] and Santambrogio [25].
In this paper, we focus on the connection between the W 2 metric and thė H −1 (dµ)-norm, which can be used to approximate W 2 , see §2. 3 and §3.5. The connection betweenḢ −1 (dµ) and W 2 follows from the work of Brenier [3] in 1987 who discovered that under appropriate conditions the solution to Monge-Kantorovich optimal transport with a quadratic cost can be expressed using a map that pushes forward one measure to the other; moreover, this map is the gradient of a convex function, see §2.1. Brenier's theorem reduces the problem of computing Wasserstein distance to solving a generalized Monge-Ampére equation. Linearizing this general Monge-Ampére equation gives rise to an elliptic equation, which corresponds to a weighted negative homogeneous Sobolev norm, see §2.3 below or see [ §4.1.2, §7.6 of [29]]. The connection between Wasserstein distance and negative homogeneous Sobolev norms has been considered by several different authors as is discussed in the following section.
1.3. Related work. Several authors have considered the connection between negative homogeneous Sobolev norms and the W 2 metric in several different contexts. In analysis, this connection has been used to establish estimates by many authors, see for example [13,15,23,26,27]. Moreover, this connection also arises in the study of how measures change under heat diffusion, see [5,21,30]. The weighted negative homogeneous Sobolev norm has also been considered in connection to maximum mean discrepancy (MMD) which is a technique that can be used to compute a distance between point clouds, and has many applications in machine learning, see [1,19,20]. Authors have also considered this connection in papers focused on computing the W 2 metric, see [4,9,14]. Moreover, the negative homogeneous Sobolev norm has been considered in several applications to seismic image and image processing, see [8,10,31].
We emphasize three related works. First, Peyre [23] establishes estimates for the W 2 metric in terms of the unweighted negative homogeneous Sobolev norm · Ḣ−1 (dx) . Let µ and ν be probability measures on Ω with densities f and g with respect to Lebesgue measure. The result of Peyre says that if 0 < a < f, g < b < +∞, then
b −1/2 f − g Ḣ−1 (dx) ≤ W 2 (µ, ν) ≤ a −1/2 f − g Ḣ−1 (dx) ;
a discussion and concise proof of this result can be found in [ §5.5.2 of [25]]. Informally, this result says that if µ and ν have their mass spread out over Ω, then the Wasserstein distance is equivalent to the negative unweighted homogeneous Sobolev norm of f − g; as a consequence, the weighted Sobolev norm is most interesting for probability distributions that have regions of high and low density.
Second, Engquist, Ren, and Yang [9] study the application of Wasserstein distance to inverse data matching. In particular, they compare the effectiveness of several different Sobolev norms for their applications; we note that the definitions of the norms they consider differ from the norm (2) that we consider, see the discussion in §3.6 below, but their results do indicate that negative homogeneous Sobolev norms may not be an appropriate substitute for the W 2 metric for some applications.
Third, Yang, Hu, and Lou [31] consider the implicit regularization effects of Sobolev norms in image processing. In §6 we mention a similar potential applications to image smoothing; our perspective is slightly different, but the underlying idea of this potential application is the same as [31].
Preliminaries and motivation
In this section, we briefly summarize material from Chapters 0.1, 0.2, 2.1, 2.3, 4.1, 4.2, and 7.6 of the book by Villani [29]. In order to make these preliminaries as concise as possible we state all definitions and theorems for our special case of interest; in particular, we assume that all probability measures have smooth densities with respect to Lebesgue measure, and restrict our attention to transport with respect to a quadratic cost function. This section is organized as follows: we consider the Monge-Kantorovich transport problem and Brenier's theorem in §2.1, the Monge-Ampére equation in §2.2, and then discuss linearization of the W 2 metric in §2.3. The main purpose of these preliminaries is to provide background for (13), stated at the end of §2.3, which clarifies the statement that theḢ −1 (dµ)-norm is a linearization of the W 2 metric.
2.1. Monge-Kantorovich optimal transport and Brenier's theorem. Let Ω ⊂ R n be a bounded convex set, and µ and ν be probability measures on Ω that have positive smooth densities f and g, respectively, with respect to the Lebesgue measure. A transference plan π is a probability measure on Ω × Ω such that
π[A × Ω] = µ[A], and π[Ω × A] = ν[A],
for all measurable subsets A of Ω. We denote the set of all transference plans by Π(µ, ν), and define the transportation cost I(π) with respect to the quadratic cost function |x − y| 2 by I(π) = Ω×Ω |x − y| 2 dπ(x, y).
In this case, the Monge-Kantorovich optimal transport cost T (µ, ν) is defined by (4) T (µ, ν) = inf π∈Π(µ,ν) I(π).
Since we are considering a quadratic transportation cost, the Monge-Kantorovich optimal transport cost T (µ, ν) is the square of the W 2 metric:
(5) W 2 (µ, ν) 2 := T (µ, ν).
Under the above conditions, Brenier's theorem states that the Monge-Kantorovich optimization problem (4) has a unique solution π satisfying
(6) dπ(x, y) = dµ(x) δ(y = ∇ϕ(x)),
where ϕ is a convex function such that ∇ϕ#µ = ν. Here # denotes the pushforward, and δ denotes a Dirac distribution. To be clear, given T : Ω → Ω, the push-forward ν := T # µ is defined by the relation ν(A) = µ(T −1 (A)) for all measurable sets A, or equivalently, by the relation
Ω ζ(x)dν(x) = Ω ζ(T (x))dµ(x),
for all continuous functions ζ on Ω. Combining (4), (5), and (6) gives
(7) W 2 (µ, ν) 2 = Ω |x − ∇ϕ(x)| 2 dµ(x).
Monge-Ampére equation.
Recall that by assumption µ and ν have densities f and g, respectively, with respect to the Lebesgue measure. We can express (6) as
Ω ζ(∇ϕ(x))f (x)dx = Ω ζ(y)g(y)dy,
for all bounded continuous functions ζ on Ω. Changing variables y = ∇ϕ(x) on the right hand side and using the fact that ζ is arbitrary gives
(8) f (x) = g(∇ϕ(x)) det D 2 ϕ(x).
Upon rearranging this equation as
det D 2 ϕ(x) = f (x) g(∇ϕ(x)) ,
it is clear that is an instance of the general Monge-Ampére equation
det D 2 ϕ(x) = F (x, ϕ, ∇ϕ),
which has been studied by many authors, see for example [16,17,28].
Linearization of general Monge-Ampére equation.
To linearize this general Monge-Ampére equation we assume that f > 0 is positive, ∇ϕ ≈ Id is close to the identity, and f ≈ g. More precisely, assume
(9) ϕ(x) = |x| 2 2 + εψ(x), and (10) g(x) = 1 + εu(x) f (x),
for some ε > 0. Substituting (9) into (7) gives
(11) W 2 (µ, ν) 2 = ε 2 Ω |∇ψ| 2 dµ,
which expresses W 2 (µ, ν) 2 in terms of ψ. Substituting (9) and (10) into (8) gives
f = (1 + εu + ε 2 R 1 )(f + ε∇f · ∇ψ + ε 2 R 2 )(1 + ε∆ψ + ε 2 R 3 ),
where R 1 , R 2 , R 3 are remainder functions depending on f, ψ, u. It follows from rearranging terms that
(12) Lψ = u + εR, where Lψ = −∆ψ − ∇(log f ) · ∇ψ,
and R denotes some remainder function depending on f , ψ, and u. Thus, if ψ satisfies Lψ = u, then by (11) and (12) we expect that
(13) W 2 (µ, ν) = ε Ω |∇ϕ| 2 dµ + O(ε 2 ),
which, roughly speaking, says that Lψ = u is a linearization of the quadratic Wasserstein optimal transport problem, see §3.5 for a more precise version of (13).
Characterizing the operator L
So far we have presented background material that motivates why the quadratic Wasserstein distance W 2 is related to the operator L defined by
Lψ = −∆ψ − ∇(log f ) · ∇ψ.
In this section, we discuss the connection between the operator L, the negative weighted homogeneous Sobolev norm · Ḣ−1 (dµ) , and the quadratic Wasserstein distance W 2 in detail. The section is organized as follows: We start, in §3.1, by stating and proving a version of Green's first identity that L satisfies. Second, in §3.2 we state a result connecting theḢ −1 (dµ)-norm to the solution of an elliptic boundary value problem involving L. Third, in §3.3, we give a characterization of theḢ −1 (dµ)-norm in terms of a divergence optimization problem. Fourth, in §3.4 we consider a characterization of theḢ −1 (dµ)-norm involving the Witten Laplacian, which can be derived from L by a change of variables. Fifth, in §3.5 we provide a more precise version of the statement that theḢ −1 (dµ)-norm is a linearization of the W 2 metric. Finally, in §3.6, we discuss weighted Sobolev norms in relation to the Fourier transform.
3.1. Green's first identity analog for L. Let Ω be a bounded convex domain in R N , ϕ be a once differentiable function, and ψ be a twice differentiable function. Green's first identity states that
(14) Ω ϕ(−∆ψ)dx = Ω (∇ϕ) · (∇ψ)dx − ∂Ω ϕ(∂ n ψ)ds,
where ∂ n ψ = n · ∇ψ, and n is an exterior unit normal to the surface element ds.
Proposition 3.1. Suppose that µ is a measure on Ω which has a density f with respect to Lebesgue measure: dµ = f dx. Further assume that f is once differentiable, f > 0 on Ω, and L :
= −∆ − ∇(log f ) · ∇. Then,(15)Ω ϕ (Lψ)dµ = Ω (∇ϕ) · (∇ψ)dµ, whenever ∂ n ψ = 0 on ∂Ω or ϕ = 0 on ∂Ω.
Proof. By the definition of L and the fact that dµ = f dx we have
Ω ϕ(Lψ)dµ = Ω ϕf (−∆ψ) − ϕ (∇f ) · (∇ψ)dx.
Since we assumed ∂ n ψ or ϕ vanishes on ∂Ω it follows from (14) that
Ω ϕf (−∆ψ) − ϕ (∇f ) · (∇ψ)dx = Ω (∇(ϕf )) · (∇ψ) − ϕ (∇f ) · (∇ψ)dx.
Finally, observe that expanding ∇(ϕf ) with the product rule and canceling terms gives:
Ω f (∇ϕ) · (∇ψ) + ϕ(∇f ) · (∇ψ) − ϕ (∇f ) · (∇ψ)dx = Ω (∇ϕ) · (∇ψ)dµ,
which establishes (15).
Elliptic boundary value problem.
Let Ω be a bounded convex domain in R N , and suppose that µ is a measure on Ω which has a density f with respect to Lebesgue measure: dµ = f dx. Assume that f is once differentiable and f > 0 on Ω. For functions u such that Ω udµ = 0 we define theḢ −1 (dµ)-norm by
u Ḣ−1 (dµ) := sup Ω u ϕ dµ : ϕ Ḣ1 (dµ) = 1 , where ϕ 2Ḣ 1 (dµ) := Ω |∇ϕ| 2 dµ.
The following result characterizes theḢ −1 (dµ)-norm in terms of an elliptic boundary value problem involving the operator L.
Proposition 3.2. We have (16) u 2Ḣ −1 = Ω |∇ψ| 2 dµ = Ω ψ(Lψ)dµ = Ω ψ u dµ,
where ψ is a solution to the elliptic boundary value problem
(17) Lψ = u in Ω ∂ n ψ = 0 on ∂Ω, where L := −∆ + ∇(− log f ) · ∇.
Proof of Proposition 3.2. The second and third equalities in (16) are a direct consequence of the definition of L and Proposition 3.1, so we only need to show that u 2Ḣ −1 = Ω |∇ψ| 2 dµ . Assume that ψ is a solution to (17) substituting Lψ = u. It follows from (15) that
(18) Ω u ϕ dµ = Ω (Lψ) ϕ dµ = Ω (∇ψ) · (∇ϕ) dµ.
Using the Cauchy-Schwarz inequality gives
Ω (∇ψ) · (∇ϕ)dµ ≤ Ω |∇ψ| 2 dµ 1/2 Ω |∇ϕ| 2 dµ 1/2 , which implies that sup Ω u ϕ dµ : Ω |∇ϕ| 2 dµ = 1 ≤ Ω |∇ψ| 2 dµ 1/2 .
On the other hand, from (18) we have
Ω u ϕ dµ = Ω |∇ψ| 2 dµ 1/2 , when ϕ = ψ Ω |∇ψ| 2 dµ −1/2
, so we conclude that u 2Ḣ −1 = Ω |∇ψ| 2 dµ as was to be shown.
3.3. Divergence formulation. TheḢ −1 (dµ)-norm can also be formulated as an optimization problem over vector fields satisfying a divergence condition. We note that our computational approach does not directly use this divergence formulation, and that our purpose of stating the following result is for completeness and its connection to other methods.
Proposition 3.3. We have u 2Ḣ −1 (dµ) = min F Ω F 2 dµ,
where the minimum is taken over vector fields F with continuous first-order partial derivatives that satisfy
(19) − 1 f div(f F ) = u in Ω n · F = 0 on ∂Ω.
Note that it will become clear from the proof that it would suffice to assume that (19) holds in a weak sense.
Proof. First, observe that if ψ is a solution to the elliptic boundary value problem (17), then F = ∇ψ is admissible to the minimization since
− 1 f div(f ∇ψ) = −∆ψ − ∇(log f ) · ∇ψ = Lψ = u
and n · ∇ψ = ∂ n ψ = 0. And from Proposition 3.2 we have
u 2Ḣ −1 (dµ) = Ω |∇ψ| 2 dµ ≥ min F Ω F 2 dµ.
To complete the proof it suffices to show that
Ω |∇ψ| 2 dµ ≤ min F Ω F 2 dµ.
If we define F = ∇ψ + G, then G satisfies
− 1 f div(f G) = 0 in Ω n · G = 0 on ∂Ω. Expanding F = ∇ψ + G gives Ω F 2 dµ = Ω (|∇ψ| 2 + 2∇ψ · G + |G| 2 ) dµ.
To complete the proof we will show that Ω ∇ψ · G dµ = 0. Observe that
Ω ∇ψ · G dµ = − Ω ψ 1 f div(f G) dµ + Ω 1 f div(ψf G) dµ.
The first integral on the right hand side is zero since 1 f div(f G) = 0 in Ω. The second integral on the right hand side is zero since by the Gauss divergence theorem
Ω 1 f div(ψf G)dµ = Ω div(ψf G) dx = ∂Ω ψf G · n ds,
and n · G = 0 by assumption. This completes the proof.
3.4. Witten Laplacian formulation. TheḢ −1 (dµ)-norm can also be defined in terms of a boundary value problem involving the Witten Laplacian H, which is a Schrödinger operator of the form
(20) H := −∆ + V,
where V is a potential depending on f defined by
V := − 1 4 f −2 |∇f | 2 + 1 2 f −1 ∆f = f 1/2 ∆f −1/2 .
Alternatively, the Witten Laplacian H can be defined by the similarity transform
(21) Hψ = f 1/2 L(f −1/2 ψ),
which symmetrizes L in the sense that the resulting operator H is self-adjoint with respect to L 2 (dx); for a discussion of the Witten Laplacian and some spectral estimates see [6]. In the following Proposition, we give an elliptic equation involving H that can be used to compute theḢ −1 (dµ)-norm; this proposition is an immediate consequence of the fact that the Schrödinger operator definition (20) is consistent with the similarity transform definition (21).
Proposition 3.4. Using the notationũ = f 1/2 u andψ = f 1/2 ψ we have u Ḣ−1 (dµ) := Ωũ (x)ψ(x)dx 1/2 ,
whereψ is the solution to the elliptic boundary value problem:
Hψ =ũ in Ω ψ = 0 on ∂Ω.
Proof. This formulation is an immediate consequence of the identity
Hψ = f 1/2 L(f −1/2 ψ) = −∆ψ + − 1 4 f −2 |∇f | 2 + 1 2 f −1 ∆f ψ,
which is straightforward to verify: expanding
f 1/2 L(f −1/2 ψ)
gives
Hψ = −f 1/2 (div(f −1/2 (∇ψ)− 1 2 f −3/2 (∇f )ψ)− ∇f f 1/2 ·(f −1/2 (∇ψ)− 1 2 f −3/2 (∇f )ψ), = 1 2 f −1 (∇f ) · (∇ψ) − ∆ψ + 1 2 f −1 (∇f ) · (∇ψ) − 3 4 f −2 ∇f 2 ψ + 1 2 f −1 (∆f )ψ − f −1 (∇f ) · (∇ψ) + 1 2 f −2 |∇f | 2 ψ),
and after canceling terms we have
Hψ = −∆ψ − 1 4 f −2 |∇f | 2 ψ + 1 2 f −1 (∆f )ψ.
From the above calculation, it is also clear that
V = f 1/2 ∆f −1/2
since all terms involving ∇f cancel.
Remark 3.1 (Alternate form of the potential). Using the fact that
∆(− log f ) = −f −1 ∆f + f 2 |∇f | 2 ,
we can write V as
V = 1 4 |∇F | 2 − 1 2 ∆F,
where F := − log f .
3.5.
Linearization remainder estimate. Recall that previously in §2.3 we gave the informal estimate
W 2 (µ, ν) = ε Ω |∇Ψ| 2 dµ + O(ε 2 ),
where Ψ satisfies
LΨ = u, where L := −∆ − ∇(log f ) · ∇.
The purpose of this section, is to make this informal statement more precise; we emphasize that the result proved in this section is for illustrative purposes: results involving weaker assumptions and weaker regularity conditions are possible. Let µ and ν be probability measures with densities f and g with respect to the Lebesgue measure: dµ = f dx and dν = gdx. Assume that
ϕ(x) = |x| 2 2 + εψ(x),
where ψ is a smooth function satisfying ∂ n ψ = 0 on ∂Ω. Further, assume that
g(x) = (1 + εu(x))f (x),
where u is a smooth function. Assume that ϕ satisfies the nonlinear equation
(22) f (x) = g(∇ϕ(x)) det D 2 ϕ(x).
Let Ψ be the solution to the elliptic boundary value problem
LΨ = u in Ω ∂ n Ψ = 0 on ∂Ω, where L := −∆ − ∇(log f ) · ∇.
We have the following result.
|W 2 (µ, ν) − ε u Ḣ−1 (dµ) | ≤ C Ω,f,ψ,u ε 2 ,
where C f,ψ,u can be chosen in terms of almost everywhere upper bounds on:
|∇f |, |H f |, |∇ψ|, |H ψ |, |u|, |∇u|, |Ω|, and λ 1 (L) −1 ,
where |∇f | denotes the magnitude of the gradient of f , |H f | denotes the operator norm of the Hessian of f , |Ω| denotes the measure of Ω, and λ 1 (L) denotes the smallest positive eigenvalue of L.
We demonstrate this result numerically in §5.3.
Proof of Proposition 3.5. By the Lagrange remainder formulation of Taylor's Theorem, and (22) we have
(23) f = (1 + εu + ε 2 R 1 ))(f + ε∇f · ∇ψ + ε 2 R 2 )(1 + ε∆ψ + ε 2 det H ψ ),
where the remainder functions R 1 , R 2 can be expressed by
R 1 (x) = ∇u(ξ 1 ) · ∇ψ(x), and R 2 (x) = (∇ψ(x)) H f (ξ 2 )∇ψ(x),
where ξ 1 , ξ 2 are points on the line segment between x and x + ε∇ψ(x). It follows that
(24) ε(−∆ψ − ∇(log f ) · ∇ψ − u)f = ε 2 R,
where the remainder function ε 2 R consists of all terms in the expansion of the right hand side of (23) that include ε to power at least 2. By the definition of L and Ψ we can rewrite (24) as (Lψ − LΨ)f = εR.
Multiplying both sides of this equation by ψ + Ψ and integrating over Ω gives
Ω (ψ + Ψ)(Lψ − LΨ)dµ = ε Ω (ψ + Ψ)R dx.
Using (15) to rewrite the left hand side gives
Ω |∇ψ| 2 dµ − Ω |∇Ψ| 2 dµ = ε Ω (ψ + Ψ)R dx.
By (9) and (16) it follows that
(25) W 2 (µ, ν) 2 − ε 2 u 2Ḣ −1 (dµ) = ε 3 Ω ψRdx + Ω ΨRdx .
We can bound the L 2 norm of Ψ by the L 2 norm of u and the inverse of the smallest positive eigenvalue of L; therefore, we can complete the proof by using Cauchy-Schwarz and almost everywhere bounds on all other quantities.
Remark 3.2. We note that it is possible to obtain various estimates on |W 2 (µ, ν) − ε u Ḣ−1 (dµ) | from (25) in terms of L p norms of the quantities ∇f, H f , ∇ϕ, H ψ , u, and ∇u instead of almost everywhere bounds. Moreover, results that guarantee bounds on the solution of general Monge-Ampére equations could be used to provide bounds for ∇ψ and H ψ in terms of f and u, for example see [17].
3.6. Fourier transform and weighted Sobolev norms. In this section, we discuss a family of weighted Sobolev norms defined by Engquist, Ren, and Yang [9] using the Fourier transformf
(ξ) = R N f (x)e −2πix·ξ dx.
In [9] the authors compare the W 2 metric to a family of weighted Sobolev norms defined in Fourier domain for the purpose of inverse data matching; in particular, in [Eq. 8 and Remark 2.1 of [9]] they define
(26) u 2H s (w) = R N w * u s 2 dξ where u s (ξ) := (2π|ξ|) s u(ξ),
where w and u denote the Fourier transform of w and u, respectively, and * denotes convolution
(f * g)(x) = R N f (x − y)g(y)dy.
Here we refer to the norm defined in (26) as theH s (w)-norm to avoid confusion with theḢ 1 (dµ)-norm, which we defined in (3) by also see [ §7.6 of [29]]. First, observe that if s = 1 and w = 1 is the constant function, then w = δ is the Dirac delta distribution and
u 2H s (w) = R N (2π|ξ|) u(ξ) 2 dξ = R N (2πiξ) u(ξ) 2 dξ = R N |∇u| 2 dx,
where the final inequality follows from the Plancherel theorem. However, if s = 1 and w is arbitrary, then in general
u 2H s (w) = R N | w * u 1 | 2 dξ = R N |u 1 | 2 w 2 dx = R N |∇u| 2 w 2 dx = u 2Ḣ 1 (dµ) ,
where u 1 denotes the inverse Fourier transform of u 1 (ξ) = 2π|ξ| u(ξ), and µ is assumed to have density w 2 with respect to Lebesgue measure. It does follow from the Plancherel theorem that
R N |u 1 | 2 dx = R N |∇u| 2 dx.
However, in general, the functions |u 1 | 2 and |∇u| 2 are not equal, and thus in general their integrals against w 2 are not equal; in particular, their integrals against w 2 can be very different when w 2 is localized in space. Roughly speaking, the issue is that taking the absolute value of ξ does not commute with taking the convolution. It is possible to define theḢ 1 (dµ)-norm in Fourier domain by defining u(ξ) := 2πiξ u; however, this does not seem to lead to a viable way to compute the dualḢ −1 (dµ)norm except in dimension N = 1; we note that quadratic Wasserstein distance also has a simple characterization in 1-dimension, see Remark 3.3. If F −1 and G −1 are the pseudo-inverse of F and G defined by
F −1 (t) = min{x ∈ R : F (x) ≥ t} and G −1 (t) = min{x ∈ R : G(x) ≥ t}, then W 2 (µ, ν) 2 = 1 0 (F −1 (t) − G −1 (t)) 2 dt,
see for example [Remark 2.30 of [22]].
Computation and regularization
In this section, we consider the connection between theḢ −1 (dµ)-norm and the Witten Laplacian, see §3.4, from a computational point of view. Recall that the Witten Laplacian is a Schrödinger operator of the form
H = −∆ + V,
where V is a potential. Roughly speaking, the advantage of considering this formulation is that all the complexity of the problem has been distilled into the potential V , which can be regularized to manage the computational cost. This section is organized as follows. First, in §4.1 we consider a spectral decomposition of −∆ by its Neumann eigenfunctions and define the fractional Laplacian (−∆) γ . Second, in §4.2, we change variables using fractional Laplacians to precondition our elliptic equation involving H. Third, in §4.3, we observe how using the heat equation to define a smoothed version of V can control the condition number of our problem. Finally, in §4.4 we discuss the computational cost of the described method.
4.1. Spectral decomposition of the Laplacian. Suppose that Ω is a bounded convex domain. Recall that λ is a Neumann eigenvalue of the Laplacian −∆ on Ω if there is a corresponding eigenfunction ϕ such that
−∆ϕ = λϕ in Ω ∂ n ϕ = 0 on ∂Ω,
where ∂Ω denotes the boundary of Ω, and n denotes an exterior unit normal to the boundary. The Neumann eigenvalues of the Laplacian are nonnegative real numbers that satisfy 0 = λ 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · +∞, and the corresponding eigenfunctions {ϕ k } ∞ k=0 form an orthogonal basis of square integrable functions on Ω. We can use this basis of Neumann eigenfunctions to define the fractional Laplacian (−∆) γ for γ ∈ R and ψ = k α k ϕ k by
(−∆) γ ψ = α 0 ϕ 0 + k>0 λ γ k α k ϕ k ,
where we include the constant term α 0 ϕ 0 , independent of γ, so that the operator is well-defined and invertible for both positive and negative γ. With this definition, the operator (−∆) γ is invertible, which will become relevant in the following section; in particular, we will use the invertible operator (−∆) γ to precondition the elliptic equation Hψ = u.
Preconditioning the elliptic equation.
Recall that our goal is to solve the elliptic equation
(27) Hψ = (−∆ + V )ψ = u,
with Neumann boundary conditions; note that to simplify notation we dispense with the tilde notationψ andũ from §3.4 and just write ψ and u. In order to precondition (27) we define U and Ψ by U := (−∆) 1/2 u and Ψ := (−∆) 1/2 ψ.
It follows that
(28) AΨ = U where A := Id −P 1 + (−∆) −1/2 V (−∆) −1/2 ,
where P 1 denotes the projection onto the space of constant functions,
(P 1 Ψ)(x) = 1 |Ω| Ω Ψ(y)dy,
and Id denotes the identity operator. We remark that the projection P 1 is necessary in the definition of A since we have defined (−∆) −1/2 to preserve constant functions, while the Laplacian −∆ destroys constant functions.
Since (−∆) −1/2 is invertible the dimension of the null space of A is the same as the dimension of the null space of H, which is 1-dimensional. In particular, we have Hf −1/2 = 0, and A(−∆) 1/2 f −1/2 = 0.
Let λ 1 (H) and λ 1 (A) denote the smallest positive eigenvalue of H and A, respectively. If ψ 1 (H) is a normalized eigenvector associated with λ 1 (H), then it follows from the Courant-Fisher Theorem that
λ 1 (A) ≥ cλ 1 (H),
where c = ( Ω ∇ψ 1 (H) 2 dx) −1 . In the following, we treat c as a fixed constant, which empirically we find is the case. Under this assumption, the condition number of A on the space of functions orthogonal to (−∆) 1/2 f −1/2 is bounded by the operator norm of A, which satisfies
A 2 ≤ 1 + V L ∞ ,
If f is an arbitrary smooth positive function, then V could still take very large values, which could make our problem ill-conditioned. In the following section, we introduce a definition of V that includes smoothing which can be used to control its L ∞ -norm.
e −∆t ψ = k e −τ λ k α k ϕ k .
Next, we define the smoothed potential V τ by
(29) V = f −1/2 τ ∆f 1/2 τ , where f τ := (e −τ ∆ f 1/2 ) 2 ,
and define the corresponding operator H τ by
H τ := −∆ + V τ .
By §3.4, the operator H τ defines aḢ −1 (dµ τ )-norm, where µ τ is the measure with density f τ . Observe that if τ = 0 then u Ḣ−1 (dµτ ) = u Ḣ−1 (dµ) , while when τ → ∞ then u Ḣ−1 (dµτ ) → u Ḣ−1 (dx) . In particular, we have
V τ L ∞ → 0, as τ → ∞,
so the parameter τ can be used to control the condition number of H, and hence can be used to control the computational cost as is discussed in the following section.
Computational cost.
Computing theḢ −1 (dµ τ )-norm using the operator
H τ = −∆ + V,
involves solving an elliptic equation involving H τ . By §4.2 this equation can be preconditioned by a change of variables resulting in a linear system
A τ Ψ = U,
where A τ is an operator with condition number O( 1 + V τ L ∞ ). Since A τ is positive definite on the space orthogonal to its null space, and since U is contained in this space, we can use Conjugate Gradient to solve this linear system to a fixed precision ε > 0 with computational cost
C solve = O C A 1 + V τ L ∞ ,
where C A is the cost to apply A. The operator A can be applied quickly if we can efficiently change between the standard basis and the basis of Neumann Laplacian eigenfunctions. In the following remark, we discuss the case Ω = [0, 1] 2 , where this transformation can be performed by a Discrete Cosine Transform (DCT).
Numerical examples
In this section we describe a numerical algorithm for using the Witten Laplacian to compute a local linear approximation of W 2 distance via theḢ −1 (dµ τ )-norm. We use the analytical tools of the previous sections, and demonstrate the method on several numerical examples. In particular, this section is organized as follows: First, in §5.1 we describe the implementation of the algorithm and provide a link to code. In §5.2 we include analytical results about Wasserstein distance for Gaussian distributions and translations that we will use to interpret the numerical results. Third, in §5.3, we provide an initial numerical example for Gaussian distributions that illustrates the result of Proposition 3.5. Next, in §5.4 we provide illustrations of how the linearization approximates Wasserstein distance for translations. Fifth, in §5.5 we include visualizations of how the linearization approximates Wasserstein distance for changes in variance. Finally, in §5.7 we present an example of computing an embedding of theḢ −1 (dµ)-norm into L 2 .
Implementation.
Algorithm 5.1 (Linearized W 2 via Witten Laplacian). We first compute the Witten potential, V , and then solve the resulting partial differential equation by converting it to a symmetric linear system which is solved using conjugate gradient. 1) Compute the potential V τ using the smoothing procedure of section 4.3.
2) Solve the linear system AΨ = U using conjugate gradient where A is defined by (28).
a) The discretized operator A of (28) can be applied to a function, f , tabulated on an equispaced grid by first approximating f as a 2-dimensional cosine expansion of the form
f (x 1 , x 2 ) ≈ n−1 k1,k2=0 α k1,k2 cos(πk 1 x 1 ) cos(πk 2 x 2 )
where n is the number of function tabulations in each spatial dimension. The coefficients α k1,k2 are computed with a Discrete Cosine Transform (DCT), which requires O(n 2 log n) operations. b) The operator ∆ −1/2 of A is applied to a cosine expansion via pointwise multiplication of the coefficients. For example, ∆ −1/2 cos(mx 1 ) = − 1 m cos(mx 1 ). c) Pointwise multiplication by V τ in spatial domain is then performed with an inverse DCT, followed by pointwise multiplication in the spatial domain. d) Conjugate gradient is iterated until convergence up to some desired error tolerance.
We implemented the preceding algorithm in Python, and have provided publicly available codes with the implementation accessible at https://github.com/ nmarshallf/witten_lw2.
Analytic formulas for W 2 for Gaussian distributions and translations.
Let µ and ν be measures on R N with densities f and g with respect to Lebesgue measure: dµ = f dx and dν = gdx. Assume that f is a Gaussian function with mean m f and diagonal covariance
Σ f = diag(σ 2 f ), where σ 2 f = (σ 2 f,1 , . . . , σ 2 f,N ) f (x) = 1 (2π) d/2 (det Σ f ) 1/2 exp − 1 2 (x − m 1 ) Σ −1 f (x − m f ) .
Similarly, assume that g is a Gaussian function with mean m g and covariance Σ g = diag(σ 2 g ). Then,
(30) W 2 (µ, ν) 2 = |m f − m g | 2 2 + |σ f − σ g | 2 2 .
That is, the square of the quadradic Wasserstein distance between Gaussian distributions with diagonal covariance matrices is equal to the square of the distance between their means plus the square of the distance between their standard deviations, see [Remark 2.31 of [22]] for a more general result.
The dependence of quadratic Wasserstein distance on the distance between means for Gaussian distributions is a special case of a general translation property. Let µ and ν be two measures on R N that have the same mean
R N xdν = R N xdµ. Let T v : R N → R N denote the translation operator T v : x → x + v. Suppose that ν v denote a translation of ν by v; more formally, ν v := T v # ν
where # denotes the push forward. Then quadratic Wasserstein distance satisfies the following relation:
(31) W 2 (µ, ν v ) 2 = W 2 (µ, ν) 2 + |v| 2 2
, That is, if two measures have the same mean and one measure is translated distance |v|, then the square of the quadratic Wasserstein distance between the measures increases by |v| 2 , see [Remark 2.19 of [22]] for a slightly more general statement of this translation result.
Numerical example: linearization of W 2 for Gaussian distributions.
In this section, we demonstrate that our code satisfies the result of Proposition 3.5 using Gaussian distributions and (30). Let µ and ν be measures on R N with densities f and g with respect to Lebesgue measure: dµ = f dx and dν = gdx. We define f :
[0, 1] 2 → R by (32) f (x) = 1 2π(det Σ f ) 1/2 exp − 1 2 (x − µ f ) Σ −1 f (x − µ f ) ,g(x) = 1 2π(det Σ g ) 1/2 exp − 1 2 (x − µ g ) Σ −1 g (x − µ g ) ,
with µ g = µ f + (0.001, 0.002) and Σ g = Σ f + 0.001 0 0 0.003 .
The covariances Σ f and Σ g are chosen such that, for numerical purposes up to precision 10 −16 , the functions f and g, are essentially supported on [0, 1] 2 and thus both are probability densities that integrate to 1. For this numerical example, we use the above Python implementation of Algorithm 5.1 for the functions f and g tabulated on a 513 × 513 equispaced grid on [0, 1] 2 , we define u = (f − g)/f , see §4.3. Recall that Proposition 3.5 says that if u Ḣ−1 (dµ) = ε, then
|W 2 (µ, ν) − u Ḣ−1 (dµ) | = O(ε 2 ).
The implementation gives
(33) u Ḣ−1 (dµ) ≈ 1.2397 × 10 −3 ;
using (30) we find that (34) |W 2 (µ, ν) − u Ḣ−1 (dµ) | ≈ 6.8598 × 10 −6 .
Thus, (33) and (34) provide a numerical demonstration of Proposition (3.5).
5.4.
Numerical example: visualizing the linearization for translations. In this section, we visualize how different metrics compare to W 2 by considering a subset of the sphere {ν : W 2 (µ, ν) = ε}; in particular, we consider the subset of this sphere that consists of translated versions of µ. Fix ε > 0, let S denote the unit circle S := {v ∈ R 2 : v 2 = 1}, and observe that
(35) W 2 (µ, µ εv )v ∈ R 2 : v ∈ S = εS,
where µ εv is the translation of µ by εv; the fact that this set is equal to εS follows from (31). In the following, we define analogs of the set defined in the left hand side of (35), where the W 2 metric is replaced by our linearization, the unweighted Sobolev norm, and the Euclidean norm, respectively. By plotting these sets, we can understand how these metrics distort slices of small spheres with respect to the W 2 metric. Let µ be a measure with density f with respect to the Lebesgue measure: dµ = f dx. Suppose that µ εv is the translation of µ by εv, which is the measure with density f εv (x) := f (x + εv). First, we use the weighted negative homogeneous Sobolev norm based on the regularized Witten Laplacian formulation described in §4 to define
T witten := (f − f εv )/f τ Ḣ−1 (dµτ ) v ∈ R 2 : v ∈ S ,
second, we use the unweighted Sobolev norm to define
T sobolev := f − f εv Ḣ−1 (dx) v ∈ R 2 : v ∈ S ,
and third, we use the Euclidean norm to define
T euclid := f − f εv L 2 (dx) v ∈ R 2 : v ∈ S .
For this numerical example, we use the function f plotted in Figure 1, see §1. 1.
This function f : [0, 1] 2 → R is defined by f (x) = 1 c exp(9x 1 )(cos(16πx) + 1)ζ(x),
where ζ is a bump function supported in [.1, .9] 2 that is equal to 1 on [.2, .8] 2 , and c is a constant that normalizes f so that it is a probability density; given f we define the potential V τ , see Figure 1. We plot the sets T witten , T sobolev , and T euclid in Figure 2. Note that the set εS is included for reference and is plotted using a dotted line in the plots of Figure 2. First, consider the plot of T euclid in Figure 2. Since the probability measures f and f εv are probability densities, they can be thought of as being normalized to have L 1 -norm equal to 1, which is the reason that the scale of T euclid is much larger than εS which appears as a dot. The shape of T euclid can be interpreted as follows: if the image f is translated up, then the vertical stripes will mostly overlap, see Figure 1, resulting in a small change in the Euclidean distance. In contrast, if the image is shifted left, then the strips will become misaligned resulting in a large change in the Euclidean distance; this explains the barbell shape of the set T euclid . Next, consider the plot of T sobolev in Figure 2 corresponding to the unweighted Sobolev norm, which partially corrects the scaling. Finally, the plot of T wittin which is the linear approximation of Wasserstein distance computed using the method described in this paper nearly recovers the circle with only a small deformation. 5.5. Numerical example: visualizing effect of changing variance. In this section, we again visualize how different metrics compare to W 2 by considering a subset of the sphere {ν : W 2 (µ, ν) = ε}. By assuming that the density f of µ is a Gaussian function with a diagonal covariance matrix we can consider the subset of {ν : W 2 (µ, ν) = ε} consisting of Gaussian distributions with the same mean, but whose diagonal covariance matrix is different. Let f be a Gaussian function centered at (1/2, 1/2) with diagonal covariance matrix Σ
f = diag(σ 2 f ) where σ 2 f = (σ 2 f,1 , σ 2 f,2 ) f (x) := 1 2π(det Σ f ) 1/2 exp − 1 2 (x − (1/2, 1/2) ) Σ −1 f (x − (1/2, 1/2) )
.
We plot f and its regularized potential V τ in Figure 3. Figure 3. The function f (left) and its regularized potential V τ (right).
If the density g of ν is the Gaussian function centered at (1/2, 1/2) with diagonal covariance matrix Σ g = diag(σ 2 g ), then recall that by (30) we have (36)
W 2 (µ, ν) = |σ f − σ g |.
Fix ε > 0, let S denote the unit circle S := {v ∈ R 2 : v 2 = 1}, and observe that
(37) W 2 (µ, µ εv )v ∈ R 2 : v ∈ S = εS,
where here µ εv is the measure with density f εv , where f εv is a Gaussian function centered at the (1/2, 1/2) with diagonal covariance
Σ fεv = diag((σ f + εv)) 2 .
That is, f εv changes the standard deviations σ f of the Gaussian f by εv. The fact that (37) holds follows from (36). As in the previous section, we study analogs of the set defined in the left hand side of (37), where the W 2 metric is replaced by our linearization, the unweighted Sobolev norm, and the Euclidean norm, respectively. In particular, we define
V witten := (f − f εv )/f τ Ḣ−1 (dµτ ) v ∈ R 2 : v ∈ S , V sobolev := f − f εv Ḣ−1 (dx) v ∈ R 2 : v ∈ S , and V euclid := f − f εv L 2 (dx) v ∈ R 2 : v ∈ S .
We plot the sets V witten , V sobolev , and V euclid in Figure 2. Note that the set εS is included for reference and is plotted using a dotted line in the plots of Figure 2. First, observe that the plots of V euclid and V sobolev in Figure 4 appear stretched in the (1, 1) and (−1, −1) directions: when the vector εv changing the standard deviations is in the positive quadrant this corresponds to increasing the standard deviation of both variables. Similarly, the negative quadrant (where both components of v are negative) corresponds to decreasing the standard deviation of both variables. Both of these deformations result in a similarly large change. In contrast, the other quadrants (where the components of v have different signs) correspond to increasing one standard deviation in one direction while decreasing the standard deviation in the other direction; this explains the asymmetry of V euclid and V sobolev . In contrast, the plot of V witten in Figure 4 roughly preserves the circle with only a small deformation. 5.6. Numerical example: managing noise and computational cost. In this section, we remark how the method can be used in more practical situations involving images. In particular, we consider a 129 × 129 image containing a biomolecule. In order to manage both the computational cost and noise, we define the potential V τ with τ = 0.01, see Figure 5. Observe that the maximum value of the potential in Figure 5 is about 150. Therefore, we expect the computational cost to be proportional to the square root of 150. Using Algorithm 5.1 to compute distances based on this image has an average time of about 0.035 (seconds) on a laptop, where the average is taken over 128 computations. In this section, we discuss an immediate extension of the described method to defining an embedding of the negative weighted homogeneous Sobolev norm. In particular, we can define a map respectively, where the Gaussian function is defined by (32). We compute Φ f (g) and Φ f (h) and plot the result in Figure 6. We find that φ f (g) − φ f (h) L 2 = 4.949 × 10 −3 , and using the analytic formula for the W 2 metric between these Gaussian distributions gives | φ f (g) − φ f (h) L 2 − W 2 (ν, η)| ≈ 5.080 × 10 −5
g → Φ f (g) such that (38) Φ f (g) − Φ f (h) L 2 = g − h Ḣ−1 (dµ) .
where µ and η are the measures associated with g and h, respectively, which verifies the effectiveness of this embedding as a local approximation of the W 2 metric.
Discussion
In this paper we have studied a classic linearization of Wasserstein distance. In particular, we focused on the connection between W 2 and the Witten Laplacian which is a Schrödinger operator of the form
H = −∆ + V.
From a computational point of view, the principle advantage of this formulation is that the computational cost of solving H = −∆ + V can be roughly bounded by the square root of the maximum value of the potential (since this influences the condition number of solving Hψ = u after an appropriate transformation). The potential V can be smoothed until an acceptable computational cost is achieved. For example, if the maximum value of V is 100, then the number of iterations for Conjugate Gradient will be ∼ 10, where each iteration has the same cost of computing the unweighted Sobolev norm. The numerical experiments indicate how this Witten Laplacian distance will, roughly speaking, preserve the Wasserstein distance for small balls around measures. This perspective opens the possibility of many interesting applications. For example, the operator H can be used to smooth images via the diffusion f → exp(−τ H)f, where exp denotes the operator exponential, which is interesting since the infinitesimal generator H of the diffusion has a connection to Wasserstein distance. More generally, the fractional diffusion
f → exp(−τ H α )f,
for α > 0 can be considered where H α is the operator to power α, which is well defined since H is positive semi definite. There are other interesting applications about defining embeddings into Euclidean space, and potential applications to graphs. Another potential application is related to maximum mean discrepancy (MMD), which has been consider by other authors, see the discussion in §1.3, but our perspective may offer some new ideas.
Figure 1 .
1Function f (left) and its regularized potential V (right), see §5.4 for details about this example.
Proposition 3. 5 .
5Under the assumptions of §3.5 we have
theḢ −1 (dµ)-norm defined in (2) by u Ḣ−1 (dµ) := sup Ω u ϕ dµ : ϕ Ḣ1 (dµ) = 1 ,
Remark 3. 3 (F
3W 2 in 1-dimension). Let µ and ν be measures on R with densities f and g with respect to Lebesgue measure: dµ = f dx and dν = g dx. Let F, G : R → [0, 1] denote the cumulative distribution functions: (
4. 3 .
3Smoothing when defining the potential. Recall that the potential V in the definition of H can be defined byV := f −1/2 ∆f 1/2 .The basic idea is to run the heat equation on f 1/2 and use the resulting smoothed function to define the potential V . Given a function f , we define a 1-parameter family of norms parameterized by τ > 0 as follows. Let {λ k } ∞ k=0 and {ϕ k } ∞ k=0 denote the Neumann Laplacian eigenvalues and eigenfunctions, see §4.1. We define the Neumann heat kernel e −τ ∆ for a function ψ = k α k ϕ k by
Remark 4. 1 (
1Spectral decomposition of Laplacian on unit square). In the case Ω = [0, 1] 2 , the Neumann Laplacian eigenvalues and eigenfunctions can be indexed by k = (k 1 , k 2 ) ∈ Z 2 >0 and are of the formλ k = k 2 1 + k 2 2 and ϕ k (x) = c k1 c k2 cos(πk 1 x 1 ) cos(πk 2 x 2 ),where x = (x 1 , x 2 ) and c k1 and c k2 are constants to normalize ϕ k to have unit L 2 norm: c k1 = 1/ √ 2 if k 1 > 0 and c k1 = 1 if k 1 = 0. Thus, expanding a function on the unit square in these Neumann eigenfunctions is equivalent to expanding a function in the double cosine series, which can be efficiently achieved by the Discrete Cosine Transform (DCT). In particular, if a function on the unit square [0, 1] 2 is represented by an n×n array, then the computational cost of expanding in a double cosine series using the DCT is O(n 2 log n) operations.
Figure 2 .
2T euclid (left), T sobolev (middle), and T witten (right).
Figure 4 .
4V euclid (left), V sobolev (middle), and V witten (right).
Figure 5 .
5Function f (left) and its regularized potential V (right) 5.7. Numerical example: local embedding.
Indeed, it follows from §3.4 that if Φ f (g) = H −1/2 ((f − g)/ f ), where H = −∆+V and the potential V depends on f , then (38) holds. The operator H −1/2 with Neumann boundary conditions is well defined since H is positive definite on a subspace that contains (f − g)/ √ f . Computationally, H −1/2 can be computed using a version of H with a regularized potential via an iterative method based on approximating √ x by Chebyshev polynomials. To illustrate this embedding we define Gaussian functions f, g, h with means µ f = (1/2, 1/2), µ g = µ f + (0.001, 0.002), µ h = µ f + 0.003, −0.002) and standard deviations σ f = (1/16, 1/14), σ g = σ f + (0.001, 0.003), σ h = σ f + (−0.001, 0.002),
Figure 6 .
6Φ f (g) (left) and Φ f (h) (right).
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| [] |
[
"Kinematics of Arbitrary Spin Matter Fields in Loop Quantum Gravity",
"Kinematics of Arbitrary Spin Matter Fields in Loop Quantum Gravity"
] | [
"Refik Mansuroglu \nInstitute for Quantum Gravity\nFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)\nStaudtstraße 791058ErlangenGermany\n",
"Hanno Sahlmann \nInstitute for Quantum Gravity\nFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)\nStaudtstraße 791058ErlangenGermany\n"
] | [
"Institute for Quantum Gravity\nFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)\nStaudtstraße 791058ErlangenGermany",
"Institute for Quantum Gravity\nFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)\nStaudtstraße 791058ErlangenGermany"
] | [] | Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin 1 2 fields and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling.We will study implications of the Gauß and diffeomorphism constraint for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin 1 2 and higher. In the case of spin 1 2 , we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss in how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields. arXiv:2011.13848v2 [gr-qc] 10 May 2021 1 Note that we are using the convention that the first index of he transforms at s(e) and the second one at t(e). At the same time, we are using the convention for e • f , which is used in[15], such that he · h f = h e•f holds. | 10.1103/physrevd.103.106010 | [
"https://arxiv.org/pdf/2011.13848v2.pdf"
] | 227,209,542 | 2011.13848 | 963717c446a6fd077003c00f67763c965e2c6f62 |
Kinematics of Arbitrary Spin Matter Fields in Loop Quantum Gravity
Refik Mansuroglu
Institute for Quantum Gravity
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Staudtstraße 791058ErlangenGermany
Hanno Sahlmann
Institute for Quantum Gravity
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Staudtstraße 791058ErlangenGermany
Kinematics of Arbitrary Spin Matter Fields in Loop Quantum Gravity
(Dated: May 12, 2021)
Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin 1 2 fields and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling.We will study implications of the Gauß and diffeomorphism constraint for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin 1 2 and higher. In the case of spin 1 2 , we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss in how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields. arXiv:2011.13848v2 [gr-qc] 10 May 2021 1 Note that we are using the convention that the first index of he transforms at s(e) and the second one at t(e). At the same time, we are using the convention for e • f , which is used in[15], such that he · h f = h e•f holds.
I. INTRODUCTION
The understanding of quantum matter fields combined with a theory of quantum gravity is an important step towards a grand unified theory. On the one hand, matter fields yield access to verifying the theory of quantum gravity. On the other hand, quantum gravity can act as a natural regulator of quantum matter which solves conceptual problems in quantum field theory.
Loop quantum gravity uses a formulation of general relativity as a constrained gauge theory, with a Gauß constraint encoding SU(2) gauge invariance (invariance under spatial frame rotations). It was realized early on [1-3] that, to solve the Gauss constraint, gravity and fermionic excitations have to be coupled. A very compelling solution was first suggested in [1, 2] and later expanded on in [3]: the fermions sit at the open ends of gravitational spin networks. It was then realized that to consistently deal with adjointness relations, the density weight between the fermionic canonical variables has to be balanced [4]. Detailed derivations from classical actions have been considered [5][6][7]. More recently, the coupling of fermion states to gravity has been investigated also from the perspective of spin foam models for loop quantum gravity [8,9]. The coupled gravity-fermion states we are considering here are precisely the boundary states in the spinfoam formalism.
In the present work we expand on this in two ways. On the one hand, the picture of spin 1 2 matter immediately suggests a generalization to point excitations of spin other than 1 2 . We will use this general picture in most of the work, and also begin a discussion of how it could be derived from classical actions for fields of higher spin. On the other hand, we investigate the consequences of Gauß * [email protected] † [email protected] and diffeomorphism constraint, by studying various examples of quantum states and by general considerations.
In section II we generalize the matter Hilbert space of [4] to excitations with arbitrary spin quantum numbers and use well-known methods to implement gauge invariance [1][2][3]. In particular, we combine the idea of the gauge invariant path observables of [3] with the quantum theory of [4] to obtain simpler gauge invariant observables. In section III we discuss the implementation [10][11][12] of symmetry under spatial diffeomorphisms in detail. For the case of spin 1 2 we make sure that we can remove gauge transformations from the diffeomorphism constraint locally, to obtain a constraint that generates exclusively local spatial diffeomorphisms.
We then discuss symmetry properties of the quantum states imposed by the diffeomorphism constraint and the statistics of the matter fields in examples. It turns out that assuming the spin statistics connection from quantum field theory, simple rules can be formulated for certain symmetric states to vanish with the implementation of the diffeomorphism constraint. In section IV we suggest candidates for the classical actions describing the semiclassical limits of the considered quantum theory. A Hamiltonian formulation yields contributions to Gauß, diffeomorphism and Hamilton constraint constraints from the matter action, but also new constraints. We make some simple observations about this constrained Hamiltonian formulation, but also point out thorny issues that makes those classical theories quite complicated. In section V, an embedding of a U(1) gauge symmetry into the theory for Dirac fermions is considered. The theory of electromagnetism is already well understood within the context of vacuum loop quantum gravity [13], and the coupling to fermions is contained in [5]. Our discussion leads to a formulation of positive and negative electromagnetic charges or particles and antiparticles, respectively.
Throughout the paper, we use the signature (− + ++) for the metric. The spatial slice the canonical theory will be based on is denoted by Σ. Four-dimensional spacetime indices are denoted by lower case Greek letters µ, ν, ρ, ... ∈ {0, ..., 3}. Spatial indices are denoted by lower case letters a, b, c, ... ∈ {1, 2, 3}.
Indices which correspond to a spin 1 2 representation of SU(2) are denoted by capital letters A, B, C ∈ {1, 2}. Spin 1 representations are denoted by lower case letters starting with i, j, k, ... ∈ {1, 2, 3} and four-dimensional spin 1 2 (Dirac) representations are denoted by capital letters starting with I, J, K, ... ∈ {0, 1, 2, 3}. Higher spin representations are built by the symmetrized direct sum of Dirac representations and are indexed by the multiindices denoted by capital script letters A, B, C, ... ∈ {A, i, (A 1 A 2 A 3 ), ...}.
II. LOOP QUANTUM GRAVITY WITH MATTER FIELDS
In this section, we will sketch the construction of an unconstrained Hilbert space for gravity and matter of arbitrary spin, and the implementation of the Gauss constraint. This is a natural generalization of the construction of [4]. We also consider the generalization of natural observables first suggested in [1, 2] and later studied in [3]. Using creation and annihilation operators, which both act pointwise, enables us to reduce an ambiguity of these path observables [3].
Let us start with matter-free loop quantum gravity. The gravitational observables act on cylindrical functions which form the Ashtekar-Lewandowski Hilbert space, H AL = L 2 (A, dµ AL ), (II.1) via multiplication of holonomies and the action of the derivation X S , respectively [14] π
j (h) e Ψ[A] = π j (h) e [A]Ψ[A], (II.2) S E i Ψ[A] = i(X i S Ψ)[A]. (II.3)
The matter degrees of freedom are described by a Fock space based on point-like excitations. The total unconstrained Hilbert space can hence be described by the tensor product
H = H AL ⊗ F ± h (j) , (II.4)
where F ± denotes the (anti)symmetric Fock space over the one particle Hilbert space
h (j) = x∈Σ C 2j+1 . (II.5)
This can be equivalently written as h (j) = {f : Σ −→ C 2j+1 |f (x) = 0 for finitely many x} (II. 6)
f | f = x∈Σ f (x)f (x). (II.7)
The matter Fock space comes with creation and annihilation operators satisfying the canonical (anti)commutation relations, θ(x) A , θ † (y) B ± = δ A B δ x,y (II. 8) θ(x) A , θ(y) B ± = 0 (II.9) θ † (x) A , θ † (y) B ± = 0, (II. 10) where we denoted the indices ranging over the spin j representation space by the script letters A, B. These can be constructed by 2j many symmetrized Weyl spinor indices.
x, y are points in the spatial slice Σ. Note that the creation and annihilation operators can be constructed to both act as a spacetime scalar and hence pointwise [4].
On the way towards a physical Hilbert space, we need to implement the Gauss constraint. For this, we only regard quantum states lying in the kernel of the quantum Gauss constraint operator, or equivalently states that are invariant with respect to the unitary action U g of gauge transformations generated by the Gauss constraint operator. The action of U g reads 1 U g π j (h e ) U −1 g = g(s(e)) · π j (h e ) · g −1 (t(e)), (II.11)
U g θ(x)U −1 g = g(x) · θ(x), (II.12) U g θ † (x)U −1 g = θ † (x) · g −1 (x). (II.13)
When implementing the Gauss constraint, we arrive at the Hilbert space of SU(2) invariant states. One suitable basis is given by a generalization of spin network states, which also admit spin representations of matter fields at the vertices of the underlying spin network graph. We want to characterize these in the following subsection.
A. Generalized Spin Network States
The kinematic Hilbert space is spanned by the tensor products of spin network states with Fock states. To implement the Gauss constraint, one can follow either refined algebraic quantization [11,16] or a reduced phase space quantization both yielding the same result.
We have to filter for all states which are invariant under gauge transformations [1,5] leaving only certain combinations of holonomies, intertwiners and matter fields where the gauge transforamtion cancels. Given a graph γ and a set of matter fields θ p1 , ..., θ p N , together with (II.11 -II.13) we can now deduce the characteristics of the quantum states in the Hilbert space H G of gauge invariant states with the following characteristics:
The matter field θ has to be attached to a vertex of the underlying graph γ. This might also be a 2-valent vertex, although we do not consider them in the vacuum theory.
For an n-valent vertex with a single matter field θ of spin j attached, only intertwiners of the form
ι : j 1 ⊗ ... ⊗ j n → j (II.14)
can be gauge-invariantly coupled with θ. In particular, the set of spin quantum numbers (j 1 , ..., j n ) is restricted by the Clebsch-Gordan rules for spin coupling.
For an n-valent vertex with an arbitrary number N of particles, only intertwiners of the form
ι : j 1 ⊗ ... ⊗ j n → k (II.15)
can be gauge-invariantly coupled. Again, the set of spin quantum numbers (j 1 , ..., j n ) is restricted by the theory of spin coupling. Here k denotes a spin quantum number the N particles can couple to. k is restricted by Clebsch-Gordan theory to be in the set k ∈ {0, 1, ..., N · j} (II. 16) or k ∈ 1 2 , 3 2 , ..., N · j .
(II.17)
The second set is valid only for N odd and j halfintegral. If we assume anti-commutation relations, as known as fermionic quantization, then the total spin is further bounded by a total number of 2j + 1 particles or equivalently k ≤ j + 1 2 − N 2 . This encodes the finiteness of the antisymmetrized Fock space at any point.
In particular, N = 0 reduces to the vacuum spin network case. Furthermore, we can generalize this for mixing different types (spins) of particles at one and the same point. Then, we would get the tensor product of coupled spin variables k 1 , ..., k l each corresponding to one specific type. The intertwiner then has the form Figure 1. Exemplary generalized spin network state. Vertices with a spin 1 2 particle are depicted by stars and vertices without matter by dots. The spin 1 2 particles couple to the star intertwiner gauge invariantly. For the sake of clarity, arrows depicting the direction of the edges are omitted.
ι : j 1 ⊗ ... ⊗ j n → k 1 ⊗ ... ⊗ k l .(
attached. As the corresponding intertwiner has to couple to 0, only a spin k representation of the said holonomy qualifies for a gauge invariant spin network state.
The total Hilbert space of gauge invariant states is then spanned by the states described above. Another way to describe the gauge invariant states is that they are obtained by a pairing (summation over the free indices) of a generalized spin network as defined in [14] with a suitable matter state with particle excitations at the nongauge invariant vertices.
We complete the discussion on the implementation of the Gauss constraint by discussing an example of such a generalized spin network state shown in Figure 1. There, we can see a graph with spin 1 2 particles denoted by stars, which lie at the vertices. The black dots on the other hand denote vertices without matter. We might have also sketched a pair of spin 1 2 particles, but they will have no effect to the state. Single particles do influence the spin quantum numbers of the adjacent holonomies as they have to couple to j = 1 2 in order to yield a gauge invariant state in total. The generalized spin networks can be formulated with arbitrary particle types with an adequate pictorial notation. To keep the example simple, we leave it by inserting spin 1 2 particles only. Furthermore note that we did not depict the directions of the edges in Figure 1 for the sake of clarity. The direction can indicate index positions of holonomies and intertwiners. However, using the respective (pseudo-)metric, we can arbitrarily lower and raise the indices and therefore change the directions of the edges, anyway.
B. Path Observables
After we have introduced the Hilbert space H G of gauge invariant generalized spin network states, we will now face the natural question of how to create and annihilate particles. Obviously, it is not possible to create or annihilate a single spin 1 2 particle without leaving H G . Instead, we can introduce operators which are coupled gauge invariantly.
[1, 2] suggests to couple creators and annihilators by holonomies. The idea was continued by [3]. However there, the annihilation operator is a density of weight one. Therefore, every appearing annihilation operator has to be smeared gauge covariantly, i.e. with a holonomy with variable endpoint lying within an open subset R ⊂ Σ whose closure is compact. To do this in a welldefined manner, an arbitrary but fixed rule of how to construct the edge e pp and therefore the holonomy h e pp with fixed starting point p and variable endpoint p has to be applied. This is necessary since there are a priori infinitely many different edges with specific startingand endpoints when integrating over the endpoint of the holonomy h e pp .
In our case, using the scalar creation and annihilation operators of [4], this complication is bypassed. A gauge invariant creation operator of two different chiral components may take the form
θ A p AC π 1 2 (h e ) C Bθ B q = j= 1 2 θ p θ q , (II.19)
where p, q ∈ Σ and e(0) = p, e(1) = q we depicted this so called path observable also by a graph with the stars indicating creation of, in this case, two Weyl spinors. The direction of e is again implicitly encoded in the starting and endpoint p, q. The order of the action of the operators follows the convention that the symbol on the right acts first (just as in the algebraic formulae). The operator (II.19) acts on a generalized spin network state by generating a matter field at the point p, a spin 1 2 holonomy along the edge e and a matter field at the point q. The resulting state can be written again as a linear combination of spin network states now corresponding to a potentially larger graph.
If we wanted to define an operator which annihilates a particle at one or both endpoints of the holonomy, we have to be careful whether one uses already the smeared version of the annihilation operator or the version of [1] of the annihilation operator. With the smeared versionθ † A , we can define, for instancê
θ † p A π 1 2 (h e ) A Bθ B q = j= 1 2 θ † p θ q , (II.20)
where we depicted the annihilation operator by an empty circle. The operator (II.20) now annihilates a particle at the point p, creates a holonomy along e and creates a particle at the endpoint q of the holonomy. Since the particles θ are indistinguishable, this operator effectively transports a particle lying at p along e to q.
For the sake of completion, we also show the last variant of the path observables including two spin 1 2 fermions and one holonomŷ
θ † p A π 1 2 (h e ) A B BCθ † qC = j= 1 2 θ † p θ † q .
(II.21)
Given a particle of higher spin, we can define the creation two matter fields, for instancê
θ A p AB π j (h e ) B Cθ C q , (II.22)
which creates two spin j particles at the points p, q ∈ Σ and a spin j holonomy inbetween.
The last operator we want to draw attention to can be obtained as a special case of (II.20) by choosing a trivial edge and p = q and consequently also a trivial holonomy π 1 2 (h e ) = 1. We end up with the operator which creates and afterwards annihilates a particle θ at the point p. This operator is related to the well known number operatorN in quantum field theory. It can be also shown [17] that the (anti)commutator of the path observables (II.19 -II.21) and (II.23) form again path observables if the holonomies meet at endpoints. This statement still holds when using the scalar creation and annihilation operators. For two edges e(0) = p, e(1) = f (0) = q, f (1) = r, for example, we find
e θ p θ q , f θ † q θ r ± = e • f θ p θ r , (II.24)
which is a direct consequence of the canonical (anti)commutation relations of θ, θ † and h e . One can read (II.24) such that stars and open circles can be linked to a longer holonomy. Indeed [3,17], this rule applies to all possible combinations of path observables. If there are multiple possibilities to link stars and circles, we will be left with a linear combination of those links. On the other hand, if there is no such possibility, then the (anti)commutator vanishes. (Anti)Commutators involving the number operator also behave in the same way, for instance
θ † q θ , f θ † q θ r ± = f θ q θ r .
(II. 25) With the operators introduced in this section we have found an intuitive formulation of creation and annihilation of parts of generalized spin network states including matter fields. As a next step, we want to take a look at the diffeomorphism constraint and its effect on the quantum states.
III. DIFFEOMORPHISM SYMMETRY
On the way to a quantum theoretical formulation of matter fields, we need to understand the symmetries of the theory. The generators of these symmetries are besides the Gauss constraint -on a kinematical level -also the diffeomorphism constraint. In this section, we will discuss the action of the diffeomorphism constraint in the quantum theory, and the symmetry properties it imposes upon states. We refer to Appendix A for a review of the standard derivation of the diffeomorphism constraint for vacuum loop quantum gravity [10,15]. There we extend it by the contribution from Dirac theory of spin 1 2 particles [5,18] and discuss details of the separation of gauge and diffeomorphism symmetry.
A. Diffeomorphism Invariant Spin Network States with Matter Fields
In generally covariant theories, the diffeomorphism symmetry ensures that physical information may only be extracted from the equivalence classes of diffeomorphism invariant states. In particular, the absolute point inside the spatial hypersurface p ∈ Σ has no physical relevance, rather we can deform Σ by semi-analytic diffeomorphisms and do not change any physical observable. In matter-free loop quantum gravity, states fulfilling the diffeomorphism constraint are obtained via an averaging method [10,12,20] yielding equivalence classes of deformed states.
It will make a significant difference what conditions are imposed on the diffeomorphisms which constitute the diffeomorphism symmetry. This has been discussed already in previous work [10,25]. We will work in the semianalytic category [26].
In the following, we will be particularly interested in the interplay between the (anti)symmetrization imposed on quantum states that are based on graphs with symmetries by the diffeomorphism constraint on the one hand, and the (anti)symmetrization of the state due to the statistics of the matter field on the other hand. The fact that (anti)symmetry is imposed in some cases due to the diffeomorphism constraint is a novel feature in loop quantum gravity.
The spin network decomposition of the Hilbert space H G contains all the spin network states that admit matter fields at the vertices of γ such that they are lying in the kernel of the Gauss constraint. Let Diff (Σ) denote the group of semi-analytic diffeomorphisms. We will consider the following subgroups
Diff γ = {φ ∈ Diff(Σ) φ(γ) = γ} (III.1) TDiff γ = φ ∈ Diff γ φ(e) = e and φ(v) = v ∀e ∈ E(γ), v ∈ V (γ) (III.2) GS γ = Diff γ TDiff γ . (III.3)
The group Diff γ consists of those diffeomorphisms which map the graph γ onto itself, while TDiff γ maps the graph trivially onto itself, i.e. it maps every edge e ∈ E(γ) and every vertex v ∈ V (γ) onto itself. Note that this also ensures that all fermions are mapped along with the vertex they are attached to in the first place, as the diffeomorphism constraint can be reduced to spatial diffeomorphisms. The quotient of Diff γ and TDiff γ again forms a group whose elements we call graph symmetries φ ∈ GS γ . These describe the permutations of edges and vertices within the graph γ, which can be achieved by a semi-analytic diffeomorphism. In particular, the number of graph symmetries # GS γ is finite. We define the diffeomorphism invariant states by averaging over all the diffeomorphisms φ ∈ Diff(Σ) taking two steps. First we define the action of a projection op-eratorP γ on a spin network state Ψ γ corresponding to the graph γP
γ Ψ γ := 1 # GS γ φ∈GSγÛ φ Ψ γ . (III.4)
Here,Û φ acts on the spin network state by mapping edges of holonomies and vertices of intertwiner as well as the vertices where a matter field is attached:
U φ h e = h φ(e)Ûφ ι = ιÛ φ θ p = θ φ(p) . (III.5)
In the second step, we average over the rest of the diffeomorphism group, namely the diffeomorphisms φ ∈ Diff (Σ) Diff γ , which move the graph γ. This group, however, has infinite cardinality, such that we have to define the state ( η(Ψ γ ) | in the dual space H * G with the following action on a spin network state Φ α ∈ H
( η(Ψ γ ) | Φ α = φ∈Tγ Û φPγ Ψ γ | Φ α .
(III. 6) In the literature, η is called the rigging map [10,20]. Note that as Φ α has a convergent norm, also the sum in (III.6) will be convergent such that ( η(Ψ γ ) | is well-defined.
We want to understand the behavior of spin network states under diffeomorphism symmetry. We want to discuss a condition under which a spin network state will be mapped to 0. This way, it is possible to identify states that do not appear in nature as they are annihilated by diffeomorphism symmetry. We want to discuss one sufficient condition such that given a spin network state Ψ γ the group averaged state ( η(Ψ γ ) | vanishes 2 .
Lemma III.1. Let γ be a spin network graph and Ψ γ a spin network state. If there exists a graph symmetry ψ ∈ GS γ such thatÛ ψ Ψ γ = −Ψ γ . Then the diffeomorphism averaged state vanishes, in other words
( η(Ψ γ ) | = 0. (III.7)
Proof. We prove the hypothesis by proving that the pro-jectionP γ Ψ γ vanishes. As GS γ is a group with finitely many elements, we can rearrange the averaging sum in the following waŷ
P γΨ γ = 1 # GS γ φ∈GSγÛ φ Ψ γ = 1 2# GS γ φ∈GSγÛ φ Ψ γ + φ∈GSγÛ φ•ψ Ψ γ = 1 2# GS γ φ∈GSγÛ φ Ψ γ + φ∈GSγÛ φÛψ Ψ γ = 1 2# GS γ φ∈GSγÛ φ Ψ γ − φ∈GSγÛ φ Ψ γ = 0, (III.8)
where we used the group homomorphism propertŷ U ψ•φ =Û ψÛφ , which becomes apparent from the definition ofÛ . In the first step, we permuted the finitely many addends of the second sum adequately. It follows that also ( η(Ψ γ ) | = 0.
In order to identify states which get annihilated due to the diffeomorphism symmetry, it suffices to find one graph symmetry which maps the spin network state Ψ γ onto its own negative. As we require physical states to be invariant under diffeomorphism symmetry, we may call the states which satisfy the condition of Lemma III.1 unphysical. Note, however, that this condition might not be necessary for having an unphysical state, as we can also imagine multiple addends canceling only in the ensemble but not two terms alone. In the following, we will focus on the condition characterized by Lemma III.1.
B. A Specific Spin Network Graph
If we consider the spin-statistics theorem known from quantum field theory on curved spacetime [28] as a guiding principle, we can study the behavior of spin network states under exchange of fermions or bosons by permuting the respective vertices via a graph symmetry. The exchange of the particles as well as the permutation of edges on the graph will yield signs which may lead to the condition needed for Lemma III.1. As spin network graphs can, in general, be very asymmetric, i.e. there might be only few or no nontrivial graph symmetries, we can hardly make statements about the physicality of general spin network states. Because of this, we start with the simplest spin network state admitting two fermions. In Appendix B, we consider a more general class of spin network states. Let us consider a state which consists of an edge e with a vertex at the starting-and endpoint each
Ψ e =θ A p AB π 1 2 (h e ) B Cθ C q |0 = C· j= 1 2 p q . (III.9)
Here, θ A creates a spin 1 2 fermion in the Weyl representation. The state Ψ e contains two fermions at two distinct points p, q ∈ Σ. Up to a constant C, which is determined by normalization of the intertwiner, we can depict the algebraic formula of the spin network state by a spin network graph. The fermions sitting at the intertwiners at p and q are depicted by a star. Although, this graphical notation is very similar to the notation in [3], there is a subtle difference between (III.9), which is a state, and (II. 19), which is an operator, having the same graphical representation. When being applied to the vacuum state 3 , the operator (II.19) yields the state (III.9). Thus, the similarity in the notation is justified.
The state (III.9) can be generalized to a system of two spin j ∈ N0 2 particles with a suitable gravitational interaction inbetween
Ψ e = θ A p (ι p ) AB π j (h e ) B C θ C q = C· j p q , (III.10) where now π j (h e ) A B
is a spin j representation of SU(2), ι : j ⊗j → 0 is the intertwining operator and θ is a vector in the spin j representation space of SU(2). The state Ψ e has one graph symmetry φ : Σ → Σ, which maps
φ : p → q φ : q → p φ : e → e −1 . (III.11)
The diffeomorphism φ exchanges the two particles and hence is a good candidate for fulfilling the conditions of Lemma III.1. Also here, the state (III.10) can be read in the two different ways, as a creation operator or as the state per se. In the following, we will think about it as creation operators, albeit it will not make a difference taking the opposite perspective. IfΨ creates the state Ψ, then the action of the Rigging map η onΨ can be expressed via the action on Ψ in the following way
η Ψ |0 = η (|Ψ ) , (III.12)
where |0 denotes the vacuum state. Let us apply the diffeomorphism φ to (III.10) and get
U φ Ψ e = θ A φ(p) (ι φ(p) ) AB π j h φ(e) B C θ C φ(q) = θ A q (ι q ) AB π j h −1 e B C θ C p . (III.13)
In order to convert this expression into something comparable to (III.10), we have to better understand the intertwiner ι AB which couples two general spin j holonomies.
To do this, we can use the fundamental representation of SU(2) to build up any other irreducible representation.
In particular, we find the following
Lemma III.2. Let ι : j ⊗ j → 0, j ∈ N0 2 be a gauge invariant intertwiner of SU(2). It holds ι AB = (−1) 2j ι BA , (III.14)
i.e. the intertwiner is symmetric for integral spin and anti-symmetric for half-integral spin.
Proof. We will proof this by giving an explicit construction of ι. At first, note that the subspace of gauge invariant intertwiner, which couple
j ⊗ j ∼ = 0 ⊕ 1 ⊕ ... ⊕ 2j (III.15)
is one-dimensional. If we therefore find one gaugeinvariant intertwiner ι as described above, it will be unique up to normalization. We rewrite the indices µν as a number of 2j symmetrized spin 1 2 indices and make the educated guess
ι (A1...A2j )(B1...B2j ) = (A1|(B1| ... |A2j )|B2j ) , (III.16)
where we denote the symmetrization grouping by a vertical line |. Hence, (III.16) is symmetric in A 1 , ..., A 2j as well as in B 1 , ..., B 2j . To prove the intertwining property, we will successively use the intertwining property of
π 1 2 (g) A C π 1 2 (g) B D AB = CD , (III.17)
with an arbitrary element g ∈ SU(2). If we now build π j (g) from π 1 2 (g) analogously, we end up with the desired intertwining property
ι (A1...A2j )(B1...B2j ) π 1 2 (g) A1 C1 π 1 2 (g) B1 D1 · · · × × · · · π 1 2 (g) A2j C2j π 1 2 (g) B2j D2j = ι (C1...C2j )(D1...D2j ) ⇐⇒ ι AB π j (g) A C π j (g) B D = ι CD , (III.18)
which proves (III. 16). Finally, we can read off
ι (A1...A2j )(B1...B2j ) = (−1) 2j ι (B1...B2j )(A1...A2j ) ι AB = (−1) 2j ι BA , (III.19)
yielding a sign factor −1 for each of the in (III.16).
We are now ready to take a closer look at (III.13) and compare it to (III.10)
U φ Ψ e = (ι q ) AB π j h −1 e B C θ A q θ C p = π j (h e ) B A (ι q ) BC (−1) 2j θ C p θ A q = (−1) 4j π j (h e ) B A (ι q ) CB θ C p θ A q = Ψ e , (III.20)
where we used the (anti)commutation relations of θ in the first step, the intertwining property in the second step, and Lemma III.2 in the last step. From (III. 20) we deduce that the simple spin network graph survives the diffeomorphism group averaging. However, if we were to choose the opposite statistics, the state would lie inside the kernel of the rigging map η. Note that the above considerations also hold true for any disjoint union of an arbitrary spin network state with a pair of particles (III.10).
C. Behavior of General Spin Network States
It quickly becomes apparent that we cannot ensure that there always exists a diffeomorphism which performs the desired exchange of particles. A counter example can be constructed from (III.10) just by gauge invariantly coupling a spin network on one of the star vertices but not on the other one. This state can be written as
Ψ asymm = F D θ A p (ι p ) DAB π j (h e ) B C θ C q = j j± 1 2 , (III.21)
where F D denotes a spin network graph which couples to the rest gauge invariantly. As we can see, there is no way to exchange the two particles without changing the topology of the graph. Still, we can study the behavior on more general spin network graphs, which do admit a graph symmetry exchanging two particles. In Appendix B, we challenge this idea and construct the most general spin network states admitting two particles and a graph symmetry which exchanges the two. There, we found rules for the detailed construction of states which survive the diffeomorphism constraint. Given a generic spin network state with graph symmetry, we can decompose the intertwiner on the symmetry axis into the Wigner basis and deduce from the statistics of the particles which components will survive or vanish after group averaging. Specifically, fermions will let those components survive which gather a total minus sign and bosons will let those components survive which are invariant under the diffeomorphism φ general .
IV. CLASSICAL ACTIONS
As we have seen, quantum matter fields can be naturally coupled to holonomies which represent the fundamental building blocks of quantum geometry and are built from representations of SU(2). To embed the discussions of the previous sections into a more complete picture, we want to discuss the corresponding classical theory from where we would start a canonical quantization in the first place.
For the classical theory of spin 1 2 fields, we will review the work [4,5,18]. Subsequently, we generalize the idea to spin 0 as well as higher spin fields. For this, we follow the same steps of the canonical quantization program as in the vacuum theory but now for an action
S = S Gravity + S Matter . (IV.1)
For spin 1 2 there is the well known Dirac action and the action for higher spin fields was already investigated early on [33,34].
Note that the experimentally confirmed theories of integral spin particles are actually Yang-Mills theories of connection 1-forms. This puts us into a dilemma of choosing between a Fock quantization of integral spin creators and annihilators or a loop quantization of the holonomies defined by path ordered exponentials of connections (analogous to gravity degrees of freedom), which would be a natural choice within the backgroundindependent loop quantization of gravity. While a Fock quantization would fit better into the particle picture of the previous sections, the loop quantization of a Yang-Mills action with an underlying gauge symmetry group G is already well understood (see for instance [35] and [36]). However, the corresponding G-holonomies decouple from the gravitational holonomies such that the spin interaction character of the particles is lost and the classical derivation would be inconsistent with the quantum theory discussed above. Note that not only on a quantum level the two perspectives yield different theories, but also the classical theories turn out to be inequivalent (see Appendix C for details).
A. Dirac Spinors
In a gravity theory of fermionic matter, we can achieve a Hamiltonian formulation. The Dirac action for a spin 1 2 particle Ψ and its conjugate momentum Π := det(g)Ψ † reads
S Dirac = i 2 M d 4 x − det(g) Ψγ α e µ α ∇ µ Ψ − ∇ µ Ψγ α e µ α Ψ , (IV.2)
with Ψ = Ψ † γ 0 , e µ α being the tetrad field and ∇ the covariant derivative which annihilates e. The spatial part of the corresponding connection can later be identified with the Ashtekar connection A [5]. If we now introduce a foliation described by lapse function N and shift vector field N a via n µ = 1 N (T µ − N µ ) and n α = −δ 0 α . We can then write the tetrad as
e µ α = i * (e) µ α − n α n µ , (IV.3)
with the triad e a i := i * (e) a i being the pullback of the tetrad onto the spatial hypersurface Σ. We also decompose the Dirac spinor into two chiral components Ψ = (ψ, η), which are both Weyl spinors. Plugging in all these quantities and splitting up the derivatives into the ones along the time vector field T µ and the spatial derivatives D a , the Dirac action takes the form of a constrained system with Dirac contributions to the Gauss, diffeomorphism and Hamilton constraint [5].
The canonical variables are now given by the chiral components (ψ, η) and their conjugate momenta (π ψ , π η ) = i det(g)(ψ † , η † ) satisfying the anti-Poisson relations
ψ A (x), ψ B (y) ± = 0 {π ψ A (x), π ψ B (y)} ± = 0 (IV.4) ψ A (x), π ψ B (y) ± = δ A B δ x,y , (IV.5)
and similar for η. Also the anti-Poisson relations which mix ψ and η vanish. Note that the momenta π ψ go with a relative factor det(q) in comparison to the matter fields ψ. It turns out that the discussion is being simplified by transforming the (spacetime scalar) Weyl spinors to halfdensities [4] ξ = 4 det(q)ψ and ρ = 4 det(q)η. (IV.6)
As a consequence, the conjugate momenta π ξ , π ρ are also half-densities. They satisfy the simple reality conditions
π ξ = iξ † π ρ = iρ † . (IV.7)
With both contributions, Holst and Dirac, the constraints can be written in the following form
G i = 1 κ D a E a i + i(ξ † τ i ξ + ρ † τ i ρ) (IV.8) V a = 1 κ F i ab E b i + i 2 ξ † D a ξ + ρ † D a ρ − c.c. (IV.9) H = 1 2κ √ q (2[K a , K b ] i − F i ab )[E a , E b ] i + + E a i 2 det(q) D a (ξ † σ i ξ + ρ † σ i ρ)+ + i(ξ † σ i D a ξ − ρ † σ i D a ρ − c.c.)+ − K i a (ξ † ξ − ρ † ρ) , (IV.10)
where the constant κ has to be taken into account since S Holst and S Dirac carry different units. The constraints can then be interpreted as the generators of gauge transformations as in matter-free loop quantum gravity. The kinematical Hilbert space can then be finally set up as a tensor product space of cylindrical functions of holonomies h e together with the antisymmetric Fock space F − (h (j) ) as discussed in section II. For a way to write the Fock space in which the states are (wave) functions, in keeping with the gravitational Hilbert space, see [4].
B. Integral Spin Quantum Fields
The Spin 0 Field
Before we consider higher spin quantum fields, we want to shortly discuss the classical theory of a spin 0 and a spin 1 field explicitly. As a first step, let us take a look at the real, massless Klein-Gordon field described by the Klein-Gordon action
S KG = − 1 2 M d 4 x − det(g)g µν D µ φD ν φ. (IV.11)
Since the matter fields φ are spacetime and SU(2) scalars, the covariant derivatives can also be replaced by partial derivatives.
We can perform a Legendre transformation, going over to Arnowitt-Deser-Misner (ADM) variables [37] for the metric and to φ and its conjugate momentum
π = ∂L ∂(∂ 0 φ) = det(q) N (−∂ 0 φ + N a ∂ a φ) (IV.12)
The term πφ can be manipulated in such a way that the canonical variables are both half-densities ξ = 4 det(q)φ and π ξ = π 4 det(q)
.
(IV. 13) In this case, we will have to keep the covariant derivatives in order to absorb the half-density factor inside the matter field φ. However, the transformation to the halfdensity ξ changes the symplectic structure. We can go back to canonical variables by redefining the Ashtekar connection. In the symplectic structure, we gather the following excess term due to the product rule:
πφ = π ξξ − π ξ ξ L t 4 det(q) 4 det(q) .
(IV.14)
The excess term can be reformulated in terms of the flux variableĖ a i . In order to arrive there, we write We can use both expressions of (IV.15) in the following linear combination
L t 4 det(q) 4 det(q) = 3 2 − 1 2 L t 4 det(q) 4 det(q) = 3 2 · 1 4 q abq ab + 1 2 · 1 2ė a i e i a = e i a 4 det(q)Ė a i . (IV.16)
We can combine the term (IV.16) with the symplectic term −Ė a i A i a coming from the gravity action. Finally, the new canonical variable, which is conjugate to E, reads
A i a →Ã i a = A i a + π ξ ξ e i a 4 det(q) . (IV.17)
This also means that every appearance of the Ashtekar connection A in the Holst contributions of the constraints has to be replaced by the new connectionà minus the excess term. We can finally write down the constraints in terms of the new canonical variables
V a = π ξ D a ξ (IV.18) H = 1 2 q ab D a ξD b ξ + π 2 ξ . (IV.19)
In these equations, D a is the covariant derivative using the Levi-Civita connection, which can be expressed in terms of E a i . One can see that -as expected for a spin 0 field -there is no contribution to the Gauss constraint, hence the gauge transformation of φ is trivial. The diffeomorphism constraint takes a similar form as in the Dirac case and can be proven to generate spatial diffeomorphisms. From this point on, we can follow the same steps as in the Dirac case and describe the quantum theory by a tensor product of the vacuum spin network states with a Fock space of spin 0 particles. Finally, we end up with a rather boring theory, since there is no constraint on the entanglement of the spin 0 particles with the gravitational holonomies but rather the particles may be placed at any point in Σ, also displaced from the spin network graph γ. As the spin 0 particle is gauge invariant, we would not expect anything different.
The Spin 1 Field
A more interesting theory emerges from an action of the spin 1 particle. For the reasons discussed in the beginning of this section, we will refrain from starting with a Yang-Mills theory, although most of the experimentally confirmed matter theories are Yang-Mills theories of connection 1-forms in classical field theory of integral spin matter fields. Rather we choose a minimal action for a spin 1 particle inspired by the Klein-Gordon action. A massive spin 1 particle can be described by a rank 1 tensor φ I , I ∈ {0, 1, 2, 3} and the action [38]
S = M d 4 x N 2 det(q) − g µν η IJ ∇ µ φ I ∇ ν φ J + − m 2 φ I φ I + e µ I ∇ µ φ I e ν J ∇ ν φ J , (IV.20)
where we lower and raise internal indices I, J, ... with the Minkowski metric η and ∇ is the covariant derivative with the spin connection ω I µ J , which defines a parallel transport of the spin 1 representations and is compatible with the tetrad e I µ . Note that it is important for the following analysis that we consider matter fields with internal structure φ I rather than spacetime tensors A µ . We show in Appendix C that the two theories are inequivalent in the presence of gravity.
The action of the spin 1 field (IV.20) yields the equations of motion
g µν ∇ µ ∇ ν φ I − m 2 φ I − e µ I e ν J ∇ µ ∇ ν φ J = 0. (IV.21)
If we contract (IV.21) with ∇ ρ e ρ K η KI =: ∇ I , then we get the so called Lorentz condition
m 2 e µ I η IJ ∇ µ φ J =: m 2 ∇ J φ J = 0. (IV.22)
The Lorentz condition reduces the degrees of freedom by 1, such that we end up with 3 degrees of freedom as expected for a spin 1 particle. Note, however, that this formulation is only valid if we impose m 2 = 0. If we insert the Lorentz condition (IV.22) into the equations of motion (IV.21), we can derive the spin 1 equivalent of the Klein Gordon equation
g µν ∇ µ ∇ ν φ I − m 2 φ I = 0. (IV.23)
We can perform a Legendre transformation analogous to the spin 0 case before. The action can be written again in terms of three constraints
S = M d 4 x π Iφ I − ω J 0 I π I φ J + N a π I ∇ a φ I + − N 2 det(q) q ab η IJ ∇ a φ I ∇ b φ J + π I π I det(q) + + m 2 φ I φ I − e a i ∇ a φ i ∇ I φ I − π 0 ∇ I φ I , (IV.24)
where we again used the metric in terms of the shift vector field N a and the lapse function N being the Lagrange multipliers for the diffeomorphism and Hamilton constraint and denoted the conjugate momentum to φ I by π I . As in the spin 0 case, we can then rearrange the halfdensity weights 4 det(q) to define the new half-density matter field ξ := 4 det(q)φ and its conjugate momentum
π ξ = π 4 √ det(q)
by extending the Ashtekar connection by an adequate excess term analogue to (IV.17) to make the symplectic structure invariant under this transformation. If we take a closer look at the variables φ I and π I , it turns out that one of the four degrees of freedom is fixed by a constraint. These couple with the constraints arising for the spin connection ω.
In Appendix D, we list these technical hurdles in detail. Although a satisfying solution to these problems is not known yet, reminiscent structures from previously discussed theories appear. This lets us conjecture the spin 1 field contributions to the Gauss, diffeomorphism and Hamilton constraint to take a similar form compared to the previous matter theories, namely
G i = k ij π j ξ ξ k (IV.25) V a = π i ξ D a ξ i (IV.26) H = 1 2 q ab D a ξ i D b ξ i + π i ξ π ξ i + m 2 ξ i ξ i , (IV.27)
where the covariant derivative D contains the Ashtekar connection A and is compatible with the triad e. In order to formulate the constraints in terms of the extended Ashtekar connectionÃ, one would have to choose another covariant derivative and collect correction terms. Note that we already left out the terms in the Hamilton constraint arising from the additional term (∇φ) 2 of the action (IV.20). The Gauss constraint G i can now be interpreted as the generator of gauge transformations of the spin 1 particle ξ. To see this, we calculate the Poisson bracket of the canonical variables (π ξ , ξ) with the smeared Gauss constraint G(Λ)
{G(Λ), φ i } = −Λ j k ij φ k (IV.28) {G(Λ), π i } = Λ k i kj π j . (IV.29)
The vector constraint V a and Hamilton constraint H on the other hand, take a similar form as in the previous theories.
Higher Integral Spin Fields
Although there is still work to be done in the case of spin 1, this gives rise to the conjecture that similar matter theories with higher spin can be treated in a similar way. However, in general the introduction of auxiliary fields is needed to derive a generalized Klein-Gordon equation and the Lorentz conditions
g µν ∇ µ ∇ ν φ I1...Ij = 0 (IV.30) η I1J e µ J ∇ µ φ I1...Ij = 0, (IV.31)
with φ being a symmetric tensor field, which describe an integral spin j field when the Lorentz conditions (IV.31) are imposed. The auxiliary fields vanish on shell as long as we do not add potential terms. In addition to that, the minimal Lagrangian for an arbitrary integral spin j field contains the terms (IV.20) in addition to terms which are proportional to the auxiliary fields. According to [38] the Lagrangian for particles with integral spin reads
L = − det(g) 1 2 φ (j) I1...Ij g µν ∇ µ ∇ ν − m 2 φ (j) I1...Ij + + − det(g) j 2 (∇φ (j) ) I2...Ij (∇φ (j) ) I2...Ij + + O(φ (j−1) ) (IV.32)
where we pull internal indices with the Minkowski metric η and defined (∇φ (j) ) I1... −1) ) collects all the terms which are at least linear in the introduced auxiliary fields φ (j−q) with j ≥ q > 0. These auxiliary fields vanish on shell. Moreover, the additional term (∇φ (j) ) 2 reduces the number of degrees of freedom of the matter field φ (j) to 2j +1. Note that this method only works for massive fields with m 2 = 0. We see from the first term of (IV.32) that we recover the same terms as before in the Gauss, diffeomorphism and Hamilton constraint when doing a constraint analysis. The remaining terms all contain at least one auxiliary field. Since we can reformulate the classical equations of motion in a way that the auxiliary fields vanish
Ij−1 = η IJ e µ I ∇ µ φ (j) JI1...Ij−1 . The Landau symbol O(φ (jφ (j−q) = 0 ∀j ≥ q > 0, (IV.33)
we will end up with the exact same form of the constraints (IV.25 -IV.27) after we implemented the conditions (IV.33). We conjecture that the most general form of the constraints will read where we denoted any function of the auxiliary fields φ j−q which vanishes when implementing the on shell conditions (IV.33) by O(φ (j−q) ) and τ i being the basis elements of the Lie algebra su(2) in the spin j representation. Note that the order of implementing the constraints becomes important now. Since we would like to implement the Gauss constraint as the generator of SU(2) gauge transformations, we can simplify this discussion by first projecting onto the subspace where (IV.33) holds. Otherwise, we would have to deal with the definition of a Fock space, in particular creation and annihilation operators for each of the auxiliary fieldsφ (j−q) and the action of the Gauss constraint would be more complicated.
G i = π
The covariant phase space formalism [39] seems particularly useful for this endeavor, as it clearly brings out the points in the construction of the phase space in which the equations of motion can be used.
C. Half-Integral Spin Quantum Fields
We finally suggest a general theory of loop quantum gravity with particles of spin j ∈ 2N0+1 2 . We are looking for a theory of gravity minimally interacting with matter fields, i.e. we want to find an action which yields the same equations of motion as in the free theory with covariant derivatives. Even on classical level, the interacting theories of higher spin particles is a current topic of interest [40][41][42][43]. There are several unsolved problems including No-Go theorems for certain theories of higher spin particles. The very idea of a free theory of a halfintegral spin particle can be traced back to [34], which is in turn based on [33]. The systematic approach based on this work needs the introduction of auxiliary matter fields, which vanish on shell in the free theory but not necessarily in an interacting theory.
There, a half-integer spin n + 1 2 ∈ N 0 + 1 2 particle is described by a 1 2 (n + 1), 1 2 n ⊕ 1 2 n, 1 2 (n + 1) representation of the Lorentz group ψ I1...In with the Weyl index being suppressed. ψ is a symmetric tensor and satisfies the spinor trace condition where we carefully intertwine 4-dimensional representation space indices I, J, ... with 4-dimensional spacetime indices µ, ν, ... nontrivially via the tetrad e µ I (cf. (IV.2)). Unfortunately, (IV.39) is not an Euler-Lagrange equation, i.e. cannot be derived from an action by a variational principle unless we introduce auxiliary matter fields [44]. For a spin 3 2 particle, for instance, the Lagrangian density which yields the equations (IV.38) and (IV.39) for the massless case reads
L = β 2 η IJ ψ I iγ K e µ K D µ − m ψ J − 2 3 βχ e µ I η IJ D µ ψ J − 1 3 βχ iγ K e µ K D µ + 2m χ, (IV.40)
where we introduced an auxiliary Dirac spinor field χ. The corresponding Euler-Lagrange equations can be written in the following form [44] (
−iγ K e ν K D ν + m)ψ I + i 2 γ I e µ K D µ ψ K = = 2 3 e µ J η IJ D µ + 1 4 γ I γ K e µ K D µ χ (IV.41) e µ I D µ ψ I = −(iγ K e µ K D µ + 2m)χ. (IV.42)
The coefficients in front of each of the terms in (IV. 40) are chosen such that the auxiliary spinor field vanishes on shell. The equation χ = 0 can be obtained by contracting (IV.41) with e µ I D µ and substituting (IV.42) therein. This reproduces the equations (IV.38) and (IV.39) for the special case of a spin 3 2 particle. For higher spins, the Lagrangian yielding the equations of motion (IV.38, IV.39) can be constructed in a similar way [44]. For the spin 3 2 particle, however there is a way to write the Lagrangian without the necessity of introducing auxiliary matter fields. The Rarita-Schwinger Lagrangian [34] takes the compact form
L RS =ψ I γ IJK e µ J D µ − imσ IK ψ K with γ IJK = 1 3! γ [I γ J γ K] and σ IK = [γ I , γ K ].
(IV. 43) In the canonical theory, the Rarita Schwinger field is described by Ψ i and the corresponding canonical momentum, which is linear in Ψ 0 [45][46][47]. Again it turns out to be advantageous to go over to density weight 1 2 fields. On the kinematical level, the four-dimensional Rarita Schwinger field can thus be quantized like a triple of Dirac fields. One can perform the analogue constraint analysis as in the Dirac case and again arrives at contributions to Gauss, diffeomorphism and Hamilton constraint, as well as new constraints coming from the fact that Ψ 0 turns out to be non-dynamical.
The classical theory of higher half-integral spin quantum fields can also be systematically constructed similar to (IV.32). Again, we refrain from writing out the whole Lagrangian, but indicate the dependence of L on the auxiliary fields [44]
L = β 2 − det(g)ψ (j) I1...In ie µ I γ I ∇ µ − m ψ (j) I1...In + + O(ψ (j−1) ) (IV.44)
with n = j − 1 2 and the matter degrees of freedom described by the symmetric tensor field ψ (j) , which also carries a Dirac spinor representation. Also here, the important term yielding the desired constraints is the first one in (IV.44), while the other terms vanish if we implement the on shell conditions. The Gauss and diffeomorphism constraints finally have the same form as (IV.34) and (IV.35). The Hamilton constraint on the other hand will have the form H = e a I φ I1...In γ I D a φ I1...In + m 2 φ I1...In φ I1...In + + O(ψ (j−q) ).
(IV.45)
The Hamilton constraint differs from the integral case by its linear dispersion relation. In the case of half-integral spin, we conjecture that it will be possible to introduce half-densities ξ describing the matter field without changing the symplectic structure. This is harder in the case of integral spin, but an adequate redefinition of the connection also yielded a formulation in terms of half-densities in the spin 0 case.
V. ELECTROMAGNETIC CHARGE AND ANTIPARTICLES
In this last section, we want to challenge the extension of loop quantum gravity with matter fields by a U(1) gauge field. This is used to describe the electromagnetic charge and interaction among particles as well as between particles and spacetime. Therefore, it yields a better suiting description of most particles of the standard model of particle physics.
On the level of loop quantum gravity without matter fields, there have already been investigations on the loop quantization of a U(1) theory [13,35] and also [4] discussed that case implicitly in the context of spin 1 2 particles. We want to review these results and combine them with our previous results.
We will start with a formulation of the constraint algebra of the U(1) and SU(2) Gauss constraint as well as the diffeomorphism and Hamilton constraint. Subsequently, we can define creation and annihilation operators for particles and antiparticles analogously to the path observables of section II.
A. A Kinematical Hilbert Space for Charged Fermions
We will start with the Yang-Mills action for a U(1) connection one-form A µ on curved spacetime
S YM = − 1 4 dt Σ d 3 x − det(g)F µν F ρσ g µρ g νσ , (V.1) with the curvature F µν = ∂ µ A ν − ∂ ν A µ .
In F µν , we can replace the partial derivatives by covariant derivatives D corresponding to the Levi-Civita connection Γ for the transport of spacetime structures, the Ashtekar connection for SU(2) structures and the U(1) connection A µ for U(1) structures. Performing a Legendre transformation of (V.1) with respect to A µ and the conjugate momentum
E µ = ∂L ∂(∂ 0 A µ ) = − det(g)F µ0 , (V.2)
we end up with a Hamiltonian consisting of the following constraints only [5]
G i = D a E a i + i ξ † τ i ξ + ρ † τ i ρ (V.3) G = D a E a + ξ † ξ + ρ † ρ (V.4) V a = F i ab E b i + F ab E b + i 2 det(q) ξ † D a ξ + ρ † D a ρ − c.c. (V.5) H = 1 2 det(q) 2[K a , K b ] i − F i ab [E a , E b ] i + + E a j D a ξ † σ j ξ + ρ † σ j ρ + − iE a j ξ † σ j D a ξ − ρ † σ j D a ρ − c.c. + + E a j K j a ξ † ξ − ρ † ρ + q ab E a E b + B a B b . (V.6)
where we defined B a := 1 2 bc a F bc . The U(1) Gauss constraint corresponds to the Lagrange multiplier A 0 . Also the diffeomorphism and Hamilton constraint arise from the action in an analogous manner. This is not surprising, since we can express gravity theory in terms of Ashtekar's variables [23] as an SU(2) Yang-Mills theory as well. A more elegant, geometric approach to the constraint analysis of the U(1) Yang-Mills theory is derived by [48,49].
The U(1) Gauss constraint generates U(1) gauge transformations. Note that both, ξ and ρ, have the same sign in the U(1) gauge constraint, which corresponds to the same sign of electromagnetic charge. Importantly, the two Gauss constraints decouple, which can be seen by calculating the Poisson brackets between the smeared SU(2) and U(1) Gauss constraints
{G(Λ), G(Λ)} = 0. (V.7)
Also, since Λ is abelian, one can show
{G(Λ), G(Λ )} = 0. (V.8)
As for the vector constraint V a , we can reformulate it in the same way as before yielding the constraint W ( N ), which generates spatial diffeomorphisms. This constraint behaves similarly as an element of the Poisson algebra. The commutator with the U(1) Gauss constraint takes the form
G(Λ), W ( N ) = Σ d 3 x ΛD a E a , sp L N A b E b + Λ(ξ † ξ + ρ † ρ), iN a (ξ † ∂ a ξ + ρ † ∂ a ρ − c.c.) = G N (Λ) = G sp L N Λ , (V.9)
which is analogue to {G(Λ), W ( N )} = G( sp L N Λ) [15]. The Poisson relations excluding the U(1) Gauss constraint G(Λ) remain untouched. This ensures that the kinematical Hilbert space with the two Gauss constraints and the diffeomorphism constraint being implemented is stable under the action of the constraint operators, since the constraint algebra is closed under the Poisson bracket. Let us also review the unconstrained Hilbert space of the theory including U(1) representations [13]. We build the Hilbert space additionally with U(1) holonomies h e (A) := P exp e A ∈ U(1), (V. 10) which satisfies the same properties as the SU(2) holonomies in addition to the fact that they are abelian. These will build up our Hilbert space, as we can formulate a * algebra with them together with the electric fields E and E [13]. Hence, we define the Hilbert space of charged matter fields on loop quantum gravity as the Cauchy completion of the span of smooth cylindrical functions with respect to both connections A and A
H := γ H γ H γ := span Ψ γ [A, A] ∈ Cyl ∞ γ , (V.11)
where we now call a function of A and A cylindrical if it can be written as a function of holonomies
Ψ[A, A] = f (h e1 , ..., h e N , h e1 , ..., h e N ) ∈ U(1). (V.12)
The prime in H γ denotes that we only consider the orthogonal components of each of the H γ , i.e. H γ does not include cylindrical functions which are also cylindrical with respect to a graphγ γ which is strictly included in γ.
The inner product can be constructed just like the Ashtekar-Lewandowski inner product but using the Haar measure of SU(2)×U(1) instead of SU(2). The orthogonal components are further decomposed into the irreducible representations of SU(2) labelled by spin quantum numbers j ∈ N0 2 and U(1) labelled by charge quantum numbers n ∈ Z. This yields the decomposition
H γ ∼ = j, l, n H γ, j, l, n , (V.13)
where the spin vectors j and l encode the spin representations of the SU(2) holonomies and n encodes the representation of the U(1) holonomies and the H γ, j, l, n only contain cylindrical functions admitting the respective spin or charge representations of the SU(2) and U(1) holonomies.
In order to solve the U(1) Gauss constraint, note that the n representation of h e transforms like a holonomy π n (h e ) → π n (g(e(0))) π n (h e ) π n g −1 (e(1)) = e −inφ(e(0)) π n (h e ) e inφ(e(1)) , (V.14)
with φ : Σ → u(1). Taking into account the matter fields, we find the transformation
θ p → e −in θ φ(p) θ p . (V.15)
If we consider gauge invariant cylindrical functions only, all the exponents in the gauge transformation of (V.14 -V. 15) have to cancel. The U(1) invariance can hence be translated into the simple condition
e∈E(v) e(1)=v n e − e∈E(v) e(0)=v n e − θ∈Θ(v) n θ = 0 ∀v ∈ V (γ), (V.16)
where E(v) denotes the set of edges which start or end at the vertex v ∈ V (γ) and Θ(v) the set of matter fields which are attached to v. Note that n θ is fixed for a fixed particle type θ. In particular for spin 1 2 fermions, it holds that n θ = n ω as pointed out before. All the quantum states which satisfy (V.16) in addition to being SU(2) invariant define the Hilbert space H G of gauge invariant quantum states. However, for U(1) invariance we need the notion of antiparticles, which is not yet present in our discussion. Let us consider the following U(1) variant example Example V.1. Consider two charged fermions of spin 1 2 and charge n = 1 connected by an edge ê
θ A p AB π 1 2 (h e ) B C π 1 (h e )θ C q |0 = j = 1 2 n = 1 n = 2 j = 0 p q . (V.17)
Since the two particles carry the same charge, a total electromagnetic charge of 2 quanta is flowing out of the two fermion system.
The particles θ carry an intrinsic charge and therefore can be seen as sources (or sinks for the opposite sign, respectively) of electromagnetic charge. The U(1) holonomies on the other hand, do not provide a source of charge because the same amount of electromagnetic charge flows through every point along the holonomy.
For U(1) invariant states, the U(1) Gauss constraint states that the overall electromagnetic flux of the quantum state vanishes. Therefore, there has to be either vacuum or both, positively and negatively charged particles in order to balance each other. This raises the question of how to describe antiparticles (i.e. particles of the same type with an electromagnetic charge of opposite sign) within our theory.
B. Antiparticles
From the classical action, the concept of antiparticles is not manifest. Indeed, also in flat spacetimes, antiparticles appear first in the Fourier transform of the quantum fields. In loop quantum gravity, however, we cannot access this tool, but may define an analogue version of the particle-antiparticle pair inspired by flat quantum field
theoryθ = 1 √ 2 θ + +π θ− (V.18) π θ = 1 √ 2 θ − +π θ+ , (V.19)
and similar for ω. Here, we defined the creationθ ± and annihilation operators θ † = −iπ θ± for particles (+) and antiparticles (−). The chiral components of the Dirac spinor are both, creating a particle, but also annihilating an antiparticle if we are serious about the analogy in flat quantum field theory. The momenta π θ and π ω on the other hand should also create an antiparticle in addition to annihilating a particle. The definitions (V.18) and (V.19) are moreover compatible with the reality conditions π θ = iθ † and impose the new reality conditions
π θ± = iθ † ± . (V.20)
Another way of recovering particle and antiparticle as Weyl spinors in the theory is achieved by switching the terms θ and θ † for one of the two Weyl components [50]. This way, also the electromagnetic charge of one of the chiral components is flipped. This does not change the anticommutation relations and is therefore a freedom of choice in the quantum theory. Let us finally look at the U(1) invariant version of example V.1.
Example V.2. Consider the particle-antiparticle pair connected by an SU(2) and a U(1) holonomŷ
θ † p A π 1 2 (h e ) A B π 1 (h e )θ B q |0 =θ p − A π 1 2 (h e ) A B π 1 (h e )θ B + q |0 = j = 1 2 n = 1 p q . (V.21)
The antiparticle is depicted by an empty circle. Note that the former annihilation operatorθ † does not annihilate the vacuum anymore, but the operators θ † ± do. Because of that, only the creation operatorsθ ± survive when acting on the vacuum state. Moreover, since we have both, a source and a sink of electromagnetic charge, the U(1) Gauss constraint is satisfied.
Example V.2 shows that we can define the path observables creating elements of gauge invariant quantum states also for SU(2)×U(1) gauge theory. The path observable in example V.1 carries a total charge of 2 quanta and thus cannot be made U(1) invariant without introducing extra structure. Instead, we would have to add a second pair of antiparticles, which absorbs the excess charge. This might look as follows Example V.3. Consider a pair of particles and a pair of antiparticles each connected by an edge with spin 1 2 and charge 1. Furthermore, the two pairs are connected by another edge, which does not carry any spin but neutralizes the charge of the two pairs. The pictorial representation reads j = 1 2 n = 1 j = 1 2 n = 1
j = 0 n = 2 p q r s . (V.22)
Example V.3 can be further generalized using an arbitrary gauge invariant generalized spin network state the same way as in the uncharged case. The fact that we need as many particles as antiparticles (or equivalently positive and negative charges) is also in agreement with flat quantum field theory.
VI. SUMMARY AND OUTLOOK
In this work, we have studied various kinematical aspects of the coupling of quantum matter to loop quantum gravity.
Following [4,5], but using the language of Fock spaces, we defined the Hilbert space H and basic operators for a quantum theory of spin 1 2 fields, and we generalized it to higher spin quantum fields in a straightforward way. An orthonormal basis for the gauge invariant states in H can be found by the generalization of the notion of spin network states known from matter-free loop quantum gravity. This generalization contains matter fields which can be located at the vertices of the underlying spin network graph γ. Inspired by the gauge invariant observables defined by [1, 3], we were able to introduce a closed (anti)commutator algebra of creation and annihilation operators for gauge invariant generalized spin network states.
While the Gauss constraint could be directly understood to ensure the gauge invariance of the states, one has to add a phase space dependent generator of a gauge transformation to the diffeomorphism constraint before it generates spatial deformations without gauge transformations (Appendix A). We have checked that this is possible at least locally, which is enough for the further considerations in the present work. It also fixes the order of implementation of the constraints, since the action of the diffeomorphism constraint would generate terms proportional to the Gauss constraint otherwise.
The implementation of the diffeomorphism constraint on the space of gauge invariant generalized spin network states was performed with the group averaging technique in analogy to the matter-free case [10,20]. We paid special attention to the interplay between the (anti)symmetrization imposed on quantum states that are based on graphs with symmetries by the diffeomorphism constraint on the one hand, and the (anti)symmetrization of the state due to the statistics of the matter field on the other. The fact that (anti)symmetry is imposed in some cases due to the diffeomorphism constraint is a novel feature in loop quan-tum gravity. In special symmetric cases and by consideration of particle exchange generated by a graph symmetry, we have found simple rules which characterize whether a state is annihilated by the diffeomorphism averaging or not. For these considerations, we assumed a spin-statistics connection motivated by the fact that a spin-statistics theorem exists for quantum field theories on curved spacetime [28].
While we found that it is straightforward to generalize the quantum kinematics of [4] and [18] to matter theories with higher spin, it is an important question whether this quantum theory can be derived from a classical higher spin action. These actions contain auxiliary matter fields which have no physical significance whatsoever as they vanish on shell. Using this technique, the kinematical constraints on the classical phase space could be elaborated and indeed take a similar form as in the lower spin case.
Although we completely discussed the case of spin 0 and 1 2 only, we can conjecture a general behavior by solving the on shell conditions for the auxiliary fields in the higher spin case. However, it is not clear how to consistently solve the second class constraint with the Lagrange multiplier − A a in order to get from the SO(3, 1) connection ω to an SU(2) connection A. In addition, in the four-dimensional representation of the spin 1 particle we encountered further second class constraints, since the 0-component of the momentum π 0 seems not to be an independent variable. For the simplified situation of flat spacetime, solving the additional constraints yields nonlocal dependencies, and hence a complicated theory.
In any case, this should be further investigated before tackling even higher spins.
Finally, we briefly considered charged Dirac fermions coupled to gravity and U(1) Yang-Mills theory, and rederived earlier results [5,35]. We found a way to represent both, particle and anti-particle on the Fock space. Due to the requirement of gauge invariance, creation of particles goes hand in hand with that of anti-particles. We have sketched the consequences for the gauge invariant observables introduced earlier. We note that it would not be easy to tackle the classical theory of electromagnetically charged higher spin fields. The problem here is that it is not clear whether the vanishing conditions for the auxiliary fields still hold. There is already existing work on a Lagrangian formulation of higher spin theories including an electromagnetic field [33,42], which might be used to extend the available theory to gravitationally and electromagnetically interacting theories.
Altogether we found that there is a very simple and elegant kinematics for coupling quantum matter of arbitrary spin to loop quantum gravity. Implementation of the Gauss constraint leads to a tight coupling between matter and gravity already at the kinematical level, and the diffeomorphism constraint imposes interesting symmetry properties on the joint quantum states in certain situations. But obviously there are still many open problems on this path to a unified theory of matter and grav-ity.
ACKNOWLEDGMENTS
We thank Kristina Giesel and Thomas Thiemann for interesting discussions during the completion of this work. Moreover, we thank Thomas Thiemann for very detailed and valuable feedback.
We thank the anonymous referee for a suggestion that improved the presentation of the material substantially.
RM thanks the Elite Graduate Programme of the Friedrich-Alexander-Universität Erlangen-Nürnberg for its support and for encouraging discussions among its members and faculty.
H.S. would like to acknowledge the contribution of the COST Action CA18108.
Appendix A: The Generator of Purely Spatial Transformations
From the combined action of Holst gravity [19] with Dirac fermions we can read off the smeared diffeomorphism constraint [5],
V ( N ) = Σ dx 3 N a V a = Σ dx 3 N a 1 κ E b i F i ab + + i 2 ξ † D a ξ + ρ † D a ρ − c.c. , (A.1)
where ξ and ρ are left-and right-handed half-density spinors in the Weyl representation as defined in [4] and D the SU(2) covariant derivative. However, following the standard discussions of the vacuum theory [15,20], we do not stick to (A.1) as a diffeomorphism constraint. We will rather add a term proportional to the Gauss constraint, with which we might define a new constraint within the same constraint algebra. Using integration by parts and vanishing boundary conditions, we find the following form of the new constraint:
W ( N ) = Σ dx 3 N a V a − N a A i a G i = Σ dx 3 E a i ∂ a N b A i b + N b ∂ b A i a + + i 2 N a ξ † ∂ a ξ + ρ † ∂ a ρ − c.c. . (A.2)
The constraint W ( N ) is what is often called the diffeomorphism constraint in the literature [15,21,22]. The reason for this name becomes apparent when calculating the action of (A.2) on the phase space variables. Furthermore, it is important to note that the constraint W ( N ) is dependent on a section in the principal fiber bundle, which is in general only defined locally. Hence, we cannot subtract the terms encoding the gauge transformations for any diffeomorphism but only on those with a support in an open neighborhood around a given point 5 .
The Poisson brackets of the constraint with respect to the phase space coordinates E a i , ξ, ρ, A i a , iξ † , iρ † read
W ( N ), E a i = N b ∂ b E a i − ∂ b N a E b i + ∂ b N b E a i = sp L N E a i (A.3) W ( N ), A i a = ∂ a N b A i b + N b ∂ b A i a = sp L N A i a (A.4) W ( N ), ξ = N a ∂ a ξ + 1 2 ∂ a N a ξ = sp L N ξ (A.5) W ( N ), iξ † = N a ∂ a iξ † + 1 2 ∂ a N a iξ † = sp L N iξ (A.6)
and analogously for ρ and ρ † . The right hand side of (A.3 -A.6) describes the infinitesimal action of a finite spatial diffeomorphism, which is in particular the flow of the vector field N . This is why we call the right hand side of (A.3 -A.6) the Lie derivative [15,18]. The Lie derivative has been chosen to act trivially in the local trivialization chosen for the connection and the fields like E in associated bundles. In the chosen trivializations, the constraint W ( N ) thus "ignores" the internal structure of the phase space variables completely. For the original vector constraint V ( N ) we find the following Poisson relations
V ( N ), E a i = L sp N E a i + k ij N b A j b E a k − N a G i = L N E a i − N a G i (A.7) V ( N ), A i a = L sp N A i a + D a N b A i b = L N A i a (A.8) V ( N ), ξ = L sp N ξ + N b A j b τ j ξ = L N ξ (A.9) V ( N ), iξ † = L sp N iξ † + N b A j b τ j iξ † = L N iξ † . (A.10)
We see that V ( N ) gives the desired Poisson relations which are gauge covariant Lie derivatives of the canonical variables except in (A.7). Here, we have a term left which is proportional to the Gauss constraint. From a classical point of view, at least all the phase space variables which lie on the constraint hypersurface are transformed by V ( N ) as expected for a gauge covariant diffeomorphism. If we consider the complete phase space a priori, then we will see that E a i is transformed like a spacetime vector density and a gauge covector plus a term which is proportional to its covariant divergence.
We conclude by pointing out the difference of the actions of the spatial diffeomorphism constraint W ( N ) and the gauge covariant diffeomorphism constraint V ( N ). While W ( N ) generates the action (via pullback) of diffeomorphisms of Σ on phase space functions (see (A.3 -A.6)), the action of V ( N ) is more subtle and involves changes in the internal space. In particular, since it is gauge invariant in itself, its action on holonomies can not change the position of their endpoints. Rather it will generate diffeomorphisms that leave these endpoints fixed. Since the constraints V ( N ) and W ( N ) only differ by a term proportional to G i , they generate the same closed constraint algebra. This becomes apparent when looking at the Poisson relations involving G(Λ) and W ( N ), for instance
{G(Λ), G(Λ )} = G([Λ, Λ ]) (A.11) {G(Λ), W ( N )} = −G( L sp N Λ) (A.12) {W ( N ), W ( M ))} = W ([ N , M ]). (A.13)
The Poisson brackets involving G(Λ) and V ( N ) are also closed 6 . These take the same form as in matter-free loop quantum gravity [23,24]. Therefore, in the quantum theory we will implement the symmetry group Dif f (M ) consisting of spatial diffeomorphisms only.
Appendix B: Diffeomorphism Symmetric General Spin Network States with Graph Symmetry
In this appendix, we want to study the behavior of generic spin network states which admit two particles and a graph symmetry which exchanges the particles. The example of section III B can be generalized by coupling the two spin 1 2 holonomies to spin 1 and by closing the graph gauge invariantly. We get
Ψ symm,1 = θ A p AB π 1 2 (h e1 ) B C (σ r ) CD i × × π 1 2 (h e2 ) E D EF θ F q F i = 1 2 1 2 p q , 1
(B.1) 6 We also stay with a closed Poisson algebra when taking into consideration the Hamilton constraint H(N ).
where (σ r ) i are the Pauli matrices, which couple 1 2 ⊗ 1 2 ⊗1 at r ∈ Σ, and F i denotes an arbitrary spin network graph, which couples the rest gauge invariantly, and is invariant with respect to a diffeomorphism φ 1 , which rotates p to q and h e1 to h e2 and vice versa. This state is the loop quantum gravity analogue of the triplet state in flat quantum field theory [30]. If we act with the graph symmetry φ, we get
U φ Ψ symm,1 = θ A q AB π 1 2 (h e2 ) B C (σ r ) CD i × × π 1 2 (h e1 ) E D EF θ F p F i = −Ψ symm,1 ,
where we used the antisymmetry of the two Levi-Civita symbols AB and the symmetry of σ AB and anticommuted the two fermions. We deduce that this state will not survive the group averaging procedure, if we stick to the spin-statistics connection. If there is a triplet state in loop quantum gravity, it better not have a graph symmetry as described above.
From the previous calculations, it becomes clear that given a spin network graph which has a suitable graph symmetry the resulting sign is determined by the edge or vertex, respectively, which intersects the symmetry axis. Let us sketch the most general such spin network graph including two particles
Ψ general = ι1 ... ιn−1 ... ... e1
... en .
(B.2)
The graph symmetry φ general will exchange the two particles and map the edge e i to e n−i+1 and vice versa as well as the points where the intertwiner ι i lies to ι n−i and vice versa. In principle, there is also the possibility that e i is mapped to e −1 n−i (and vice versa) for some i. This case is present when the direction of these edges is flipped by φ general , which can, however, be transformed to the previous case by inserting two intertwiners ι
π j (h e ) A B = (−1) 2j ι BC π j h −1 e C D ι DA . (B.3)
The sign of (B.3) appears twice as long as e n−i = e i and similar for the contribution of the intertwiner. Therefore, the only contribution to the sign may arise from the object attached to the center of the graph. If n is odd, e n+1 2 determines whether the state will survive the group averaging, whereas for n even, ι n 2 determines whether the state will survive the group averaging.
Let us discuss the case for odd n first. With (B.3), we can prepare the state such that we only have to consider how to resolve e n+1 2 → e −1 n+1 2
. By preparing the direction of the edges symmetrically around e n+1 2 , there has to be one nontrivial 2-valent intertwiner ι AB , which takes care of the direction of the edges, at the beginning-or at the endpoint, but not at both points of the holonomy which needs an intertwiner ι AB at the left vertex but none at the right one. Alternatively, one can invert the directions of e n−1 2 and e n+3 2 and get an intertwiner ι AB at the right vertex. The same argument holds for every other edge, too. We can use (B.3) again, to reformulate
ι AC π j h e n+1 2 C B → ι AC π j h −1 e n+1 2 C B = (−1) 2j ι BC π j h e n+1 2 C A . (B.5)
Unfortunately, we cannot control what spin couples to the many intertwiners inbetween the two particles. Hence, the spin j of the holonomy lying in the center and the corresponding arising sign is not determined by the spin of the particles θ. That way, the spin of the holonomy intersecting the symmetry axis of φ general has to be of the same type (half-integral or integral) as the spin of the particles θ.
If we have an intertwiner intersecting the symmetry axis, the result is even more subtle. Let us sketch the most general such intertwiner where we depicted the rotation symmetry axis by a dashed blue line. As we can see, holonomies which lie on the opposite of the symmetry axis must carry the same spin. Apart from that, holonomies might also lie on the symmetry axis. Their spin is arbitrary and does not underlie symmetry constraints, since they are invariant under φ general . Let the intertwiner (B.6) be of the form ι n 2 : j 1 ⊗ j 1 ⊗ ... ⊗ j n ⊗ j n ⊗ j n+1 ⊗ j n+2 → 0, (B.7) where the spins j 1 , ..., j n appear double for they are corresponding to the holonomies with reflections, whereas j n+1 and j n+2 represent the holonomies which lie on the symmetry axis and hence only appear once. Without loss of generality, we can ignore j n+1 and j n+2 in the following considerations, as they do not contribute to a possibly resulting sign. We can now expand the intertwiner ι n 2 into a linear combination of intertwiners of 3-valent vertices, where we can freely choose the coupling scheme [31]. Consider the following expansion jn jn j1 j1
∈ span
... , (B.8)
where the black dashed lines depict SU(2) representations without simultaneously describing a parallel transport in Σ, i.e. its beginning-and endpoint coincide. Note that the spin quantum numbers l 1 , ..., l n as well as the k 1 , ..., k n−1 are always integers. The intertwiners of the 3valent vertices represent the building blocks of (B.8) and are described by the Wigner 3j-symbols [31,32], which satisfy a number of symmetry identities, in particular This means that under the exchange of two identical spins under φ general , we gather a sign from the i-th branch if 2j i and l i are not both odd or both even. Consequently, all the contributions are multiplied. Obviously, this is again not a very strong statement, as we can produce arbitrary signs within the span of (B.8).
This indefiniteness can be illustrated with the following This transformation does not describe a symplectomorphism, since the symplectic structure is not conserved E µȦ µ = e µ I π I ė J µ φ J + e J µφJ = π Iφ I + e µ Iė J µ π I φ J , (C.2)
where π I denotes the conjugate momentum to φ I and E µ the conjugate momentum to A µ . The first term in (C.2) is the desired one but the second one is odd. Note that in flat spacetime where we require e I µ = δ I µ , the transformation is indeed a symplectomorphism. In (C.2) we used the identity
E µ = ∂L ∂Ȧ µ = ∂L ∂φ I ∂φ I ∂Ȧ µ = π I e µ I . (C.3)
This brings us to the conclusion that the two theories to describe a spin 1 field are not equivalent. Since the Yang-Mills connection A µ does not admit an SU(2) structure, there is no interaction term including A and ω, albeit it is there for the action for the spin 1 field φ. The matter field A hence decouples from the SU(2) holonomies, which appear in matter-free loop quantum gravity.
Appendix D: Constraints of the Spin 1 Action
The action (IV.20) resembles very much the Klein-Gordon action but now including a nontrivial SU(2) interaction. We want to mention some technical details and hurdles which arise when studying the classical constraints. As in the matter-free theory, ω 0 still has to be reformulated to yield the Lagrange multiplier for the Gauss constraint. The conjugate momentum π reads π I = det(q) 1 N ∇ 0 φ I − N a N ∇ a φ I + η 0I ∇ K φ K .
(D.1)
We can read off that π 0 is independent of the time derivative of φ 0 since the first and part of the third term in (D.1) cancel. We get π 0 = − det(q)e a i ∇ a φ i . which we have to add to the action (IV.24). X here acts as the Lagrange multiplier. As we can see, the term ∇ I φ I · f already appears in the Hamilton constraint and can therefore be left out when doing the constraint analysis. For this, we would have to calculate the Poisson brackets of f (X) with the other constraints to identify possible secondary constraints. 8 At last, we express the spin connection ω by the Ashtekar connection A. In the matter-free theory, the spin connection can be written in terms of two variables
ω 0i a = 1 2γ A i a − − A i a (D.4) i jk ω jk a = 1 2 A i a + − A i a , (D.5)
where A a is the Ashtekar connection and − A a is nondynamical [19]. Next to the lapse function and the shift vector field, we hence also have the Lagrange multipliers ω 0i 0 , ω jk 0 and − A a . In the vacuum theory, hence, the following constraints ∂L ∂ ω 0i 0 = 0 ∂L ∂ ω jk 0 = 0 ∂L ∂ − A i a = 0 (D. 6) hold and yield the well-known Gauss constraint next to defining relations for the Lagrange multiplier ω 0i 0 = ω 0i 0 (A, Γ, N, N a ) and the connection Γ i a = Γ i a (E) such that
− A i a = A i a − 2 γ Γ i a . (D.7)
If we include the spin 1 field, however, we get the constraints It becomes apparent that the split of the indices of the spin connection ω a into 0 and i makes it troublesome to recover the desired information from (D.8 -D.10). When including the spin 1 field, we only get a contribution of the j component of the matter field. The 0 component on the other hand does not appear in the Gauss constraint but rather in the defining relations for Γ and ω 0i 0 if we follow the same steps as in the matter-free case.
∂L ∂ ω 0i 0 = −∂ a E a i − k li E a k γ 2 + 1 2γ − A l a − γ 2 − 1 2γ A l a + 2φ [0 π i] = 0 (D.8) ∂L ∂ ω jk 0 = γ 2 i jk ∂ a E a i − E a[j A k] a + 2φ [j π k] = 0 (D.9) ∂L ∂ − A i a = − γ 2 + 1 2γ k im E a k ω 0m 0 + γ 2 + 1 2γ abc e b[i| e d|j] N d A j c + − A j c + − γ 2 + 1 2γ abc ∂ b (N e ci ) + γ 2 + 1 2γ 2 abc ijk N e j b A k c − − A k c + + N a 2 jk i φ j π k − 2φ [0 π i] − q ab N jk i φ j ∇ b φ k − φ [0 ∇ b φ i] + − e a
[1] H. A. Morales-Tecotl and C. Rovelli, Fermions in quantum gravity, Phys. Rev. Lett. 72, 3642 (1994), arXiv:grqc/9401011. [2] H. Morales-Tecotl and C. Rovelli, Loop space represen- 8 The constraint analysis for a spin 1 particle in flat spacetime yields a secondary constraint g(Y ) = Σ d 3 xY ∂ i π i − (∂a∂ a − m 2 )φ 0 . Its Poisson bracket with f (X) is constant, so f (X) and g(Y ) form a second class pair. The constraint algebra is closed, but the variables π 0 and φ 0 are determined by the solution of f and g, namely π 0 = −∂ i φ i and φ 0 = −(∂a∂ a − m 2 ) −1 ∂ i π i . φ 0 hence is nonlocal.
i a .(IV.15)
..Ij ξ J1...Jj + O(φ (j−q) ) (IV.34) V a = π I1...Ij ξ D a ξ I1...Ij + O(φ (j−q) ) (IV.35) H = q ab D a ξ I1...Ij D b ξ I1...Ij + π ξ I1...Ij π I1...Ij ξ + + m 2 ξ I1...Ij ξ I1...Ij − e a i D a ξ iI2...Ij D I ξ II2...Ij + − π 0I2...Ij ξ D I ξ II2...Ij + O(φ (j−q) ), (IV.36)
γ
J ψ JI2...In = 0 (IV.37) with the Dirac matrices γ I . (IV.37) reduces the total number of degrees of freedom in the chiral components of ψ to 2 n + 1 2 + 1 as expected for a spin n + 1 2 representation 4 . To run the program of loop quantum gravity for half-integral spin fields, we would need a Lagrangian from which we can derive the generalization of the Dirac equation (−ie µ I γ I D µ + m)ψ I1...In = 0 (IV.38) η IJ e µ I D µ ψ JI2...In = 0, (IV.39)
π j h e n+1 2 .
2For instance, the edges e n
= l 1 , l i ∈ {0, ..., 2j i } k i ∈ {|l i − k i−1 |, ..., l i + k i−1 } k n−1 = l n
1 m 2 m 3 = (−1) 2j+l j j l m 2 m 1 m 3 . (B.9)
|k] ∇ l φ l + δ j i φ |0] ∇ l φ
This was pointed out to one of the authors by Lewandowski[27] in the context of loop quantum gravity without matter.
The vacuum state is the Ashtekar-Lewandowski vacuum in the gravitational sector and the Fock vacuum in the matter sector. The former is uniquely fixed by spatial diffeomorphism invariance, see[26,29].
The number of possible independent values of ψ is given by the number of partitions of n identical blocks into four parts.
As a matter of fact, this might yield nontrivial effects depending on the topology of Σ.
The theory is, however, not U(1) invariant, since we include a mass term.
with n = 3 and j 1 = j 2 = j 3 = 1 2 , which is the case of a 6-valent vertex with six spin 1 2 holonomies. It holds that 2j i = 1 for all i = 1, 2, 3. Hence, the sign depends on the choice of the l i . However, we can choose l 1 = l 2 = l 3 = 1 on the one hand, and l 1 = l 2 = 1, l 3 = 0 on the other hand. The first intertwiner will not gather a phase when applyingÛ φ general , whereas the second intertwiner gathers a sign −1 when doing so.Given a generic spin network state like (B.2) and an intertwiner intersecting the symmetry axis together with its expansion into the basis (B.8), we can tell from the statistics of the particles θ which components will survive or vanish after group averaging. Specifically, fermions will let those components survive which gather a total minus sign and bosons will let those components survive which are invariant under the diffeomorphism φ general .If we would extend the group of diffeomorphisms to include also diffeomorphisms which are smooth except at a finite number of points suggested by[25], the group of graph symmetries of the spin network state (B.10) is also significantly extended. It is possible to exchange any two of the edges at the vertex (B.10), for instance. However, this would generate both signs for l 1 = l 2 = 1 and l 3 = 0, i.e. the state does not survive the group averaging irrespective of the statistics of the matter fields. Apparently, this extended diffeomorphism group defines another theory and makes clear that the initial choice of the symmetry group is crucial for the analysis of the kinematical Hilbert space.Appendix C: An Alternative Spin 1 ActionIn section IV B, we discuss a spin 1 particle described by a tensor with an SU(2) structure φ I . If we were to consider a U(1) Yang-Mills theory 7 instead, we might identify the connection 1-form A µ with φ in the following way A µ = e I µ φ I . (C.1)
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| [] |
[
"Machine Learning for Better Models for Predicting Bond Prices",
"Machine Learning for Better Models for Predicting Bond Prices"
] | [
"Swetava Ganguli [email protected] ",
"Jared Dunnmon [email protected] "
] | [] | [] | Bond prices are a reflection of extremely complex market interactions and policies, making prediction of future prices difficult. This task becomes even more challenging due to the dearth of relevant information, and accuracy is not the only consideration-in trading situations, time is of the essence. Thus, machine learning in the context of bond price predictions should be both fast and accurate. In this course project, we use a dataset describing the previous 10 trades of a large number of bonds among other relevant descriptive metrics to predict future bond prices. Each of 762,678 bonds in the dataset is described by a total of 61 attributes, including a ground truth trade price. We evaluate the performance of various supervised learning algorithms for regression followed by ensemble methods, with feature and model selection considerations being treated in detail. We further evaluate all methods on both accuracy and speed. Finally, we propose a novel hybrid time-series aided machine learning method that could be applied to such datasets in future work. | null | [
"https://arxiv.org/pdf/1705.01142v1.pdf"
] | 20,579,021 | 1705.01142 | d035beee817c4ace81a040d5918f00068fccc4ee |
Machine Learning for Better Models for Predicting Bond Prices
Swetava Ganguli [email protected]
Jared Dunnmon [email protected]
Machine Learning for Better Models for Predicting Bond Prices
Bond prices are a reflection of extremely complex market interactions and policies, making prediction of future prices difficult. This task becomes even more challenging due to the dearth of relevant information, and accuracy is not the only consideration-in trading situations, time is of the essence. Thus, machine learning in the context of bond price predictions should be both fast and accurate. In this course project, we use a dataset describing the previous 10 trades of a large number of bonds among other relevant descriptive metrics to predict future bond prices. Each of 762,678 bonds in the dataset is described by a total of 61 attributes, including a ground truth trade price. We evaluate the performance of various supervised learning algorithms for regression followed by ensemble methods, with feature and model selection considerations being treated in detail. We further evaluate all methods on both accuracy and speed. Finally, we propose a novel hybrid time-series aided machine learning method that could be applied to such datasets in future work.
I. Introduction
Key Problem:
Bond markets are generally characterized by a substantial dearth of trading information with respect to the amount of information available to equity traders. While equity traders can access stock bids, offers, and trades within 15 minutes of these activities, analogous information on bonds is only available to those who engage a fee-for-data contractor, and even then only in relatively small subsets compared to the overall volume of bond trades. The asymmetry in required versus available information leads to the current state wherein many bond prices are in fact days old and do not accurately represent recent market developments [1].
Our Goal:
The goal of this project is to use the techniques and algorithms of machine learning and a set of data describing trade histories, intermediate calculations, and historical prices made available (on Kaggle) by Benchmark Solutions, a bond trading firm, in order to more accurately predict up-to-date bond prices using data that would be viable to obtain at a particular moment in time [1]. The high volume of data characteristic of this problem is common in such financial modeling endeavors, and hinders the formation of fully descriptive a priori theoretical models. In this report, we develop strategies to effectively utilize the data provided for bond price prediction via thorough investigation of the space of available machine learning models and combination with methods from time-series analysis. 1
Strategy and Methods:
Feature Selection: An important aspect of this task is creating class-balanced training and test data sets while identifying appropriate metrics for assessment of prediction success. Critical features are analyzed and extracted using low order modeling techniques like Principal Component Analysis (PCA) and correlation analysis.
Supervised Learning Methods:
We first investigate computationally inexpensive techniques such as Generalized Linear Models (GLMs) and regression trees. We also assess the viability of methods like Principal Component Regression (PCR) and Support Vector Regression (SVR).
Ensemble Methods: Since we have a regression problem at hand, regression trees are combined as weak learners in ensemble methods like Bagging, LS-Boosting and Random Forests to reduce overfitting and to potentially take advantage of the large size of the dataset.
Hybrid Time-Series Methods Because each bond includes historical data on five different quantities for the last ten trading periods, we investigate the possibility of feature space augmentation or reduction using Time-Series (TS) analysis. Ideally, predictions from TS methods would either provide new features with additional explanatory power or enable reduction of the feature set size while retaining explanatory power.
Neural Networks: We experiment with applying neural networks to this problem, as they are known to fit even highly nonlinear data well given sufficient neurons.
II. Exploratory Data Analysis
The data used for this project contains 61 attributes observed for each of 762,678 bonds: 3 Nominal, 12 Discrete Ordinal, 1 Observation Weight and 45 Continuous (Ratio) Attributes, including a ground truth trade price. To predict the bond price (often called the "trade price"), the data delineates a unique ID of the bond (nominal discrete attribute), a categorical ID of the bond (nominal discrete attribute), a weight/importance of each bond (continuous ratio attribute), the bond coupon (continuous ratio attribute), years to maturity (continuous ratio attribute), whether the bond is callable or not (nominal discrete binary variable), seconds after the trade occurred that it was reported (continuous ratio attribute), notional amount of the trade (quantitative discrete attribute), the type of trade that occurred (2 = customer sell, 3 = customer buy, 4 = trade between dealers), and a fair price estimate based on implied hazard and funding curves of the bond issuer (continuous ratio attribute). This last attribute is referred to from hence forth as the "curve-based price." In addition, the dataset also has information about the last 10 trades that occurred on each bond considered, including the time difference between a trade and the previous trade (continuous ratio attribute), the trade price (continuous ratio attribute), the notional trade amount (continuous ratio attribute), the trade type (binary discrete nominal attribute), and the curve-based price (continuous ratio attribute).
Correlated Attributes:
We observe from the correlation matrices that attributes The key problems encountered in the process of feature selection and in creating training and test data sets are:
1. The time series length is either on the borderline or below the minimum number of points required for a statistically consistent time-series prediction 2. Categorical attributes have non-uniform distributions 3. The amount of data characterizing the various categories of bond importance is distinctly non-uniform Points 1 and 2 are direct manifestations of the dearth of data for bond price prediction. It is also important to correctly predict the highly weighted bonds well since they are usually of higher priority in a portfolio. Due to the problems mentioned above, it is difficult to conduct typical k-fold cross-validation wherein the training sets would be class-balanced. Instead, in order to utilize all the data given, we utilize a 70-30 hold-out cross-validation. We therefore create weight balanced training and test sets using the following algorithm.
Algorithm for Cross-Validation:
Step I Randomly create 5 instances of weight balanced training and test sets.
Step II Run Machine Learning Algorithm on each of these 5 training and test sets.
Step III Report the appropriate metric (discussed below) from each of the 5 independent runs.
Step IV The final value of the evaluation metric is the average of these 5 values.
To demonstrate that our sets are indeed weight balanced, we plot the PDF of the bond weights for one instance of the training and test sets in Figure 1.
Model Evaluation Metric:
Given that errors in bond pricing are equally detrimental in upward and downward directions [2], we evaluate our predictions based on the simple weighted L 1 -norm of the difference between the actual price and our predictions per sample (i.e. per bond). Thus, the model evaluation metric that we choose is the Weighted Error in Prediction per Sample (WEPS) which is defined as:
WEPS = ∑ m i=1 w i ( y true − y predict ) ∑ m i=1 w i(1)
Note that all prediction errors are calculated using our crossvalidation algorithm.
Statistical Significance:
Let e 1 and e 2 be the errors obtained from two different models M 1 and M 2 . Since the number of records in the training datasets and test datasets for all models is the same, say n, we can write the observed difference in the error as:
d = e 1 − e 2 .
The variance of d can be computed as
σ 2 d ≈σ 2 d = 1 n (e 1 (1 − e 1 ) + e 2 (1 − e 2 )) ∼ O(10 −6 ) (2)
The 95% confidence interval in our case is then given by
(d t = True Difference) d t = d ± 1.96σ d(3)
Importantly, this implies that any improvements in the WEPS metric out to the fifth decimal place are indeed statistically significant.
Feature Generation and Selection:
Feature selection and generation is handled as follows:
Correlation Analysis: No attributes supplied are strongly correlated. Mild correlations exist in only 2 sets of attributes. Thus, this method is not particularly informative.
PCA in Supervised
Learning: PCA is run on the full dataset with the goal of determining if there exists a reduced feature set that retains the majority of the explanatory power of the full feature set.
Scoring Function for Ensemble Methods: Random Forests (RF) are used for feature ranking. RF will select features randomly with replacement and group every subset in a separate subspace (called the random subspace). We use a scoring function with the following methodology.
If feature X 2 appears in 25% of the trees, then score it. Otherwise, we do not consider ranking the feature because we do not have sufficient information about its performance. We then assign the performance score of every tree in which X 2 appears to X 2 and average the score. Our search method is recursive: For example, if we drop the worst 20% in the first round, we do so in all following rounds until the desired number of features is attained. The 20% parameter has been determined via numerical experiment.
IV. Models from Supervised Learning
We now proceed to explaining implementation and performance of the various models utilized here. All algorithms were implemented in Matlab for ease of workflow, and all results referenced in the text can be found in Figure 4.
Generalized Linear Models:
Several models from supervised learning were investigated.
First, an unweighted generalized linear model was implemented using two different link functions and the full feature set in order to investigate the underlying distribution of the data. While financial data often has an underlying normal variation, it is important to ensure that this assumption is valid before proceeding. We report the results of Ordinary Least Squares (OLS) regression using link functions for the normal and gamma distributions. Evaluating the training and test errors for these different cases illustrates that normal variation appears to best characterize the data. To improve on these results, Weighted Least Squares (WLS) was performed using the evaluation weights to appropriately govern which points are treated with highest importance in the regression. WLS gives noticeable 3.1 ¢ improvement over OLS (in the context of errors on the order of $1).
Principal Component Regression:
We next proceed to implementing PCR using the reduced feature set. We reduce the size of the feature set to 23 features using this procedure, as the PCA routine reports that all principal components higher than 23 are nearly linearly dependent. In order to make a prediction based on PCA, we extract the transform utilized in the PCA algorithm and apply this directly to the test data. Once the data is transformed in this way, we can run GLM models as usual. It was explicitly confirmed that transforming the regression coefficients back to the original covariate space gives the same predictions for OLS, validating the prediction procedure we use. Interestingly enough, despite the fact that the first few principal components tend to explain variance in the input best, it is in fact the independent principal component in our data with the lowest eigenvalue (i.e. the last one in the reduced feature set) that provides the vast majority of the explanatory power with respect to the bond price. This is illustrated explicitly in Figure 2. Investigation of this feature's constituents reveal that it exclusively contains all of the historical curve and trade prices, implying that these variables have substantial impact on correct prediction. In fact, the most explanatory PCA variables are so potent that while using 23-feature WLS gives an error of $0.9191 in 12 seconds, a 3-feature WLS using only the three most explanatory PCA components gives a WEPS of only $1.2637 in just 3 seconds. This 75% reduction in speed could certainly become important in dealing with massive datasets often encountered in this area of finance.
Support Vector Regression:
We briefly investigate the possibility of using Support Vector Regression (SVR) to predict bond prices, but found the model estimation process to be so time-intensive that it precluded effective parameter tuning. Specifically, the LibSVM package did not report an SVR result in 5 days of computation time while the LibLinear package took 3 days just to estimate a single model. It is possible that this has to do with the size of the memory cache allocated to storage of the support vectors. Regardless, given these model estimation times, performing a parameter sweep over the critical SVR model parameters proved impractical given the time constraints of the project, and results for this method are therefore not reported.
Regression Trees:
Regression trees are known to overfit the data, thereby forming highly biased predictors. This behavior is observed in our analysis as well. The WEPS on the training set is extremely low (0.5¢ to 1¢) while the WEPS on the test set is $2.5 to $3. To mitigate overfitting, we experiment by (i) changing the number of data samples required at each node to make a decision, (ii) implementing different metrics for growing and pruning the trees (e.g. entropy, misclassification errors, etc.), (iii) varying the number of predictors randomly sampled at each node to make a decision, and (iv) controlling the depth of the trees. However, the test WEPS does not decrease in any of these cases. A full list of tested conditions and representative results can be found in Figure 4.
V. Models from Ensemble Methods
We use Bagging of data samples, LS-Boosting (Sequence of Decision Trees), and Random Forests (Bagged Trees) as our ensemble regression methods with regression trees as the weak learner. Again, note that all prediction errors are calculated according to our cross-validation algorithm.
Random Forests with Regression Trees:
Random forests can be a useful method for feature selection as outlined above. However, they are computationally very expensive. Furthermore, they generally produce a WEPS of slightly above $1, which is inferior to even simple models like GLMs. Various methods have been experimented with to reduce the overfitting in each weak learner and to accelerate the convergence of the random forest algorithm, including: (i) changing the number of data samples required at each node to make a decision, (ii) different metrics for growing and pruning the trees (e.g. entropy, misclassification errors, etc.), and (iii) number of trees grown for the majority vote. However, none of these methods improves the performance of the random forest. The training and test errors along with the execution time shown in Figure 4 are characteristic of the experiments we conducted.
LS -Boost with RT as Weak Learner:
We also experimented with the LS Boost algorithm with regression trees as the weak learner. Boosting is known to perform well due to exponential penalizations of observations proportional to the error in their prediction. J, the number of terminal nodes in trees, is the critical parameter that can be optimized for a given dataset. Hastie et al. [3] comment that typically 4 ≤ J ≤ 8 works well for boosting and results are fairly insensitive to the choice of J in this range. We therefore choose J = 6. This algorithm performs better than random forests in that WEPS stagnates at around 80¢ and the computation time is 3 times lower. Characteristic values of WEPS on the training and test sets after performing cross-validation are shown in the summary table of Figure 4.
VI. Models from Hybrid Time-Series Methods
A relatively uncommon idea that was explored during this project was the possibility of using time series analysis to supplement the given feature set. Ideally, using a time-series method to make a prediction for the new price could allow for much of the historical data contained in the five time series for trade price, curve price, trade type, trade size, and time delay to be incorporated in a concise fashion. The goal of our work was to implement a time-series forecasting method that would allow us to create a set of time-series predictions that could be used to either augment our feature set or even replace all of the historical features in a concise fashion.
One issue that is immediately apparent with the dataset utilized here is that each bond contains time-series data for the last ten instances of the five variables mentioned above. Ten points is a very small number for normal time-series predictions, to the point that inbuilt Matlab packages for many standard time-series analyses cannot estimate model parameters with any degree of certainty from such a small dataset. Knowing this, we proceeded to investigate several potential options for time-series analysis in order to find even a very simplistic model that could at least output a reasonable prediction in the majority of cases with the goal of using it for feature-set augmentation as discussed above. For the sake of brevity, these complex time series models will be concisely described below, with provided references detailing full model specification.
Cointegration Models
One method of time-series prediction involves a procedure known as cointegration analysis. Briefly, a set of univariate time series can be considered cointegrated if some linear combination of these series and their lags is statistically stationary in time. A linear combination of time series (and lags) that is cointegrated is known as a cointegration relationship. If a set of cointegration relationships exists amongst a group of time series, these relationships can be used to forecast values for a subset of the time series at later points using historical data. The canonical statistical test for determining whether or not a set of time series data contains a cointegration relationship is the Engle-Granger test [4].
The fundamental assumption of the Engle-Granger test is that a time-stationary linear combination of two time series, y t and z t , is required for cointegration, such that,
y t − βz t = u t ,(4)
with u t stationary. If u t were known a priori, one could use an established statistical method such as the Dickey-Fuller test to evaluate stationarity [4]. However, in this case, we estimate u t using ordinary least squares and analyze the stationarity of the estimated series. A second iteration of this procedure is performed on the first differences of the each time series with the lagged residuals included. The combined output of the stationarity tests is presented as a single test statistic that can be used to evaluate the existence of a cointegration relationship. We performed the Engle-Granger (E-G) test in Matlab on the time-series data from each of the nearly one million bonds in our dataset. Given that we only have ten points for each quantity for each bond, it is not unexpected that the majority of the E-G tests reported a p-value substantially above any reasonable significance boundary. In other words, this result means that it is generally not possible to reject the null hypothesis of there being no cointegration relationship amongst the different time series variables describing each bond. This makes it difficult to specify a cointegration-based model such as a Vector Auto-Regressive (VAR) or Vector Error-Correction (VEC) model to forecast bond price based on cointegration relationships amongst previous time-series quantities. Notably, however, there was a nontrivial proportion of the data (around 10 %) for which the E-G test did allow for confident rejection of the null, implying the existence of at least one cointegration relationship. This result suggests that with access to additional historical data, it might well be possible to form a viable prediction for each bond that would allow for reduction of said historical data to a single time-series prediction that could then be input into the machine learning model discussed here [4].
Auto-Regressive Moving Average (ARMA) Models
Another, slightly simpler method for approaching time-series forecasting is to predict future values of a variable based on its historical behavior. While there exist a wide variety of methods for accomplishing this task, ARMA models are quite common due to their simplicity and intuitiveness. In particular, the ARMA model incorporates two separate modes of capturing time-series behavior. The first can be summarized by considering that recent values of a time-series variable can be very good predictors of the present value. This is captured by the AR (Auto-Regressive) parameter in the model. The second line of thought, the Moving Average (MA) portion of the model, captures the fact that a large shock at a previous period would not only affect that period, but periods in the near future as well [4]. The combination of these two models can be written as the following in terms of a time-series y t , AR coefficients φ i , MA coefficients θ j , and deviations from the AR model t ,
y t = p ∑ i=1 φ i y t−i + t + q ∑ j=1 θ j t−j ,(5)
where p and q are the specification parameters of an ARMA(p, q) model. Specification of p and q can be reliably achieved using the Box-Jenkins methodology, a welldocumented procedure using easily computed autocorrelation functions [4]. The form of several visualized autocorrelation and partial correlation functions implies that an ARMA(1,1) model would potentially be appropriate for this data. Given the results of the cointegration analysis and the fact that the majority of observations allow statistically viable fitting to an ARMA(1,1) model, we pursue this approach in our time-series modeling. In terms of results, the ARMA model was utilized in the following fashion. One of the data series provided, but not originally used in the regression is a categorical variable that is a relatively non-informative bond ID number. This variable identifies bonds that are of the same class (issuer, period, etc.). Because fitting an ARMA(1,1) model to each observation independently would take substantial computational resources and thus might not be the most helpful from a predictive standpoint, we used knowledge gained from our PCA analysis to define a method for efficiently integrating a time-series analysis into our models via this bond ID number. Specifically, our previous analyses indicated that historical trade and curve prices were responsible for most of the explanatory power of the model. We therefore aimed to use time-series estimates of the difference between the trade price and the curve price to extract additional information that would improve our results. Our algorithm is as follows:
(i) Estimate an ARMA(1,1) model for 10 samples of each bond type in the training set using a variable defined as the difference in the trade and curve prices at each time point (ii) Average ARMA parameters to create an average TS model for the difference in trade and curve price for that bond type (iii) Forecast one period forward from the historical data, which gives a prediction of the difference between the trade and curve price for each bond type for the prediction period (iv) Use this forecast variable as a new feature in GLM models
The ultimate goal of this procedure was to integrate given data on the type of the bond in a manner more consistent with fundamental economic behavior as opposed to a simple categorical label. Theoretically, this series should have greater explanatory power than the bond identification number series alone because it incorporates time-series data. We illustrate this in practice by performing a reduced-feature set GLM 2 using only data from the current time period, the curve and trade price from the first historical period, and either the bond identification number or the ARMA(1,1) variable. As shown in Figure 4, including the bond identification number changes training and test error only by 0.04 ¢ and 0.01 ¢, respectively, for this reduced-feature case while inclusion of the ARMA(1,1) variable instead lowers training and test error by 2 ¢ and 2.3 ¢, respectively. Inclusion of the ARMA variable in the full WLS model similarly yields a respective reduction in training and test error by 0.5 ¢ and 0.6 ¢. The fact that inclusion of the time-series variable enhances the performance of the GLM in both training and test errors suggests that this variable does adds new information to the model instead of simply causing overfitting. Importantly, if the time series model for each bond type is precomputed, it is a simple matter to use this feature to provide supplementary information about the bond price evolution in a simple GLM.
VII. Neural Networks
Neural networks (NN) are very well-suited for function fitting problems. A neural network with enough neurons can fit any data with arbitrary accuracy. They are particularly well suited for addressing non-linear problems. We therefore experiment with two-layer (one hidden layer, one output layer) neural networks trained with the Levenberg-Marquardt optimization algorithm and simple back-propagation. The training and testing errors along with the execution time from this exercise are shown in Figure 4. Two-Layer NNs perform quite well on our dataset, reducing test error to 73 ¢ in only 2 hours. WEPS reduction with network size beyond 20 neurons is quite gradual. Function for Ensemble Methods: Random rests (RF) are used for feature ranking. RF will lect features randomly with replacement and oup every subset in a separate subspace (called ndom subspace). We use a scoring function with e following methodology. If feature X2 appears 25% of the trees, then, score it. Otherwise, we o not consider ranking the feature because we o not have sufficient information about its perrmance. We then assign the performance score every tree in which X2 appears to X2 and avere the score. For example:If Performance(Tree1) 1-WEPS = 0.85, Performance(Tree2) = 0.70, Perrmance(Tree3) = 0.30. Then, the importance of ature X2 = (0.85+0.70+0.30)/3 = 0.6167. Our search ethod is recursive: For example, let's say in the st round we drop the worst 20%, second too and on until we get the desired number of features. is number has been experimented with.
Models from Supervised Learning at all prediction errors are calculated according to ss-validation algorithm.
lized Linear Models models from supervised learning were imple-. First, an unweighted generalized linear model plemented using various link functions and the ture set in order to investigate the underlying tion of the data. While financial data often has an ing normal variation, it is important to ensure that umption is valid before proceeding. We report the of Ordinary Least Squares (OLS) regression using ctions for the normal, gamma, Poisson, and binostributions. Evaluating the training and test errors e different cases illustrates that normal assumption to best characterize the data. Further, the normal tion of the residuals reinforces this conclusion. To e on these results in terms of minimizing our error Weighted Least Squares (WLS) was performed he evaluation weights to appropriately govern oints are treated with highest importance in the ion. WLS gives noticeable improvement over OLS.
Methods
Best WEPS Train
VIII. Conclusions and Future Work
At this point, we can make several definitive conclusions regarding the relative performance of the tested models in predicting bond price: (i) GLM models perform well with low computational cost (order of seconds), (ii) Feature set augmentation with TS models improves results, (iii) Ensemble methods do not substantially improve results, and require much more computational investment, (iv) Neural networks give very accurate results without overfitting in reasonable amounts of time (order of hours), (v) NNs and GLMs give best results in terms of combined speed and accuracy.
There exist several fruitful directions in which to take future work. First, obtaining a dataset with longer time histories would allow for statistically significant specification of more detailed time-series models for improvement of feature augmentation. Investigating the performance of different classes of time-series models as machine learning feature generation mechanisms would be useful. Second, the success of neural networks on this dataset implies that investigating the application of multilayer networks and deep learning methods to this problem may yield better bond price predictions. Finally, exploiting parallel implementation of these algorithms for model tuning would greatly enhance our ability to make the most accurate predictions possible. All of these routes could yield improvements to our current results, and we intend to investigate several of these in the coming months.
IX. Summary of Algorithm Performance
Figure 1 :
1Demonstration of weight-balanced training and test datasets.
Figure 2 :
2WLS WEPS versus PCA Feature Number.
Figure 3 :
3Test Error versus Training Error and Training Time.
Figure 4 :
4Summary of Results.
Price of the Last Trade and Curve-Based Price of the Last Trade are strongly correlated at all time points. This is intuitively expected. Thus, this information can be used to inform dimensionality reduction. The fact that the remainder of the variables are minimally correlated implies that each of those attributes should supply new information for our prediction.A similar conclusion can be observed when autocorrelations
are computed for these different time series. Specifically, the
mean autocorrelations for each variable are very low (ρ < 0.3)
beyond the first lagged period, indicating that each variable
contributes unique information at every time period.
Treatment of Categorical Attributes:
1 Computing time on Stanford Corn, Barley and Rye clusters is gratefully acknowledged
Empirical PDFs of the nominal attributes have been ana-
lyzed. From the PDF of the attribute that denotes whether
the bond is callable, we see that 89 % of the bonds are not
callable whereas 11 % of the bonds are callable. From the
empirical PDF of the attribute trade type of the current trade,
it is seen that the current trade has 20 % of the type 2 trade,
36 % of the type 3 trade and 43 % of the type 4 trade. Thus,
there is a relatively uniform sampling of the three trade types.
While preparing the cross-validation datasets, this will be
taken into account such that they are class-balanced. Further-
more, in the ensemble methods used (which are regression
tree-based since we are predicting a a continuous output),
these categorical variables are handled appropriately in cases
where they are nominal or ordinal.
III. Cross-Validation, Feature Selection and
Model Evaluation Metric
Cross-Validation Strategy:
Specify!an!AutoLRegressive!Moving!Average!model:!ARMA(1,1)! ! Fit!ARMA!to!4me!series!describing!difference!in!curve!and!trade!price ! Average!model!es4mated!from!10!historical!samples!per!bond!class! used!to!predict!difference!between!current!trade!and!curve!prices! ! ARMA!forecast!for!each!bond!class!used!as!addi4onal!GLM!feature! ! No4ceable!decrease!in!WEPS!compared!to!using!bond!ID!series!or%Predic/on%of%Bond%Prices%Using%Machine%Learning%
rth!
res!
! is!
Stanford!University!
n!
Future%Work%
Conclusions%
CS!229!Machine!Learning!!
ing!
ate!
ark!
re!
at!
e.!
us)!
)!
Methodology%
CrossHValida/on%and%Feature%Selec/on%
Results%
ods!
r! a!
!
BOND%PRICE%PREDICTION%!
SUPERVISED%
LEARNING!
ENSEMBLE%
METHODS!
HYBRID%TIME%
SERIES!
NEURAL%
NETWORKS%(NN)!
! GLMs!
! PCR!
! SVR!
! Bootstrapping!
! Random!Forests!
! LS!L!Boos4ng!
! ARMA!
! VAR/VEC!
! TwoLLayer!
! Mul4layer!
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
Training Error, $
Test Error, $
Test Error vs. Training Error
10
−2
10
0
10
2
0
0.5
1
1.5
2
2.5
3
Execution Time, Hours
Test Error, $
Test Error vs. Execution Time
10
0
g Error, $
. Training Error
Generalized Linear Models
Principal Component Regression
Regression Trees
Random Forests
LS-Boost
Neural Networks
ARMA
GLM w/ ARMA
10
0
0
0.5
1
1.5
2
2.5
3
Training Error, $
Test Error, $
Test Error vs. Training Error
Generalized Linear Models
Principal Component Regression
Regression Trees
Random Forests
LS-Boost
Neural Networks
ARMA
GLM w/ ARMA
! Create!5!instances!of!weightLbalanced!training!and!test!sets!
! Run!ML!Algorithms!on!each!of!these!5!training!and!test!sets!
! Report!WEPS!from!each!of!the!5!independent!runs!
! Final!WEPS!is!the!average!of!these!5!values!
CrossHValida/on%Approach:%!
PCA%Feature%Selec/on%for%GLMs%:%!
! DickeyLFuller!and!EngleLGranger!tests!used!to!assess!sta4onarity!and
coLintegra4on!of!4me!series,!respec4vely!
! Time%Series%(TS)%for%Feature%Genera/on:%!
! Only!23!out!of!58!PCA!features!effec4vely!linearly!independent!
! Feature!containing!exclusively!all!previous!prices!and!all!previous!
intermediate!curveLpredicted!prices!holds!most!explanatory!power!
Feature%Selec/on%for%Ensemble%Methods:%!
! GLM!models!generally!perform!well!with!low!cost!
! Augmen4ng!feature!set!with!TS!models!improves!performance!
! Ensemble!methods!do!not!seem!to!substan4ally!improve!results!
! Neural!networks!give!most!accurate!results!without!overLfijng!!
! Neural%networks%and%GLMs%give%best%speedHaccuracy%profiles%
! Random!Forests!(RF)!are!used!for!feature!ranking!
! Scoring!func4on!scores!each!feature!if!it!appears!on!25%!of!trees!
! Recursively!assign!average!WEPS!of!every!tree!in!which!a!feature!
appears!to!that!feature!as!its!score!!
! Improve!4me!series!models!(model!type!and!specifica4on)!
! Explore!deep!learning!(encouraged!by!performance!of!NNs)!
! Exploit!parallel!implementa4on!of!algorithms!for!model!tuning!
! If!possible,!add!more!data!to!our!4me!series!for!each!bond!!
References%
1. Has4e,!T.,!et!al,!'Elements!of!Sta4s4cal!Learning',!Springer.!
2. Hull,!J.C.,!'Op4ons,!Futures!and!Other!Deriva4ves',!Pren4ceLHall.!
3. Shumway,!R.H.,!et!al,!'Time!Series!Analysis',!Springer.!
The performances of the various methods are summarized below. The compute time is evaluated by running the code on one node on the Stanford Corn cluster.CS 229 Machine Learning • Final Project Report • Other methods tried with RT also overfit the data Random Forests (Ensemble Method) with RTMethods
WEPS
Train
WEPS
Test
Training
Time
Generalized Linear Models
OLS
0.8043
0.8455
23 seconds
WLS
0.7722
0.8147
20 seconds
Gamma
1.8681
1.9499
28 seconds
Generalized Linear Models with PCA (PCR)
23-Feature WLS
0.8626
0.9191
12 seconds
3-Feature WLS
1.1945
1.2637
3 seconds
Hybrid Time-Series Methods
ARMA(1,1) Model
0.9822
0.9862
⇠ 6 hours
WLS w/ARMA
0.7676
0.8091
12 seconds
9-Feature WLS
0.8252
0.8711
4 seconds
9-Feature WLS w/Bond
ID
0. 8256 0.8710
4 seconds
9-Feature WLS w/ARMA 0. 8054 0.8477
4 seconds
Regression Trees (RT)
All predictors
0.0055
2.4369
⇠ 83 hours
5 predictors per node
0.0117
2.5527
⇠ 23 hours
10 predictors per node
0.0066
2.6624
⇠ 28 hours
15 predictors per node
0.0058
2.9765
⇠ 32 hours
20 predictors per node
0.0056
2.9186
⇠ 37 hours
50 Regression Trees
0.9588
1.1259
⇠ 22 hours
100 Regression Trees
0.9011
1.0876
⇠ 42 hours
200 Regression Trees
0.8876
1.0735
⇠ 63 hours
300 Regression Trees
0.8354
1.0623
⇠ 78 hours
LS-Boost (Ensemble Method), Weak Learner: RT
100 Weak Learners
0.9613
1.0091
⇠ 5 hours
250 Weak Learners
0.9223
0.9897
⇠ 12 hours
400 Weak Learners
0.8668
0.9045
⇠ 19 hours
500 Weak Learners
0.8012
0.8236
⇠ 23 hours
Neural Networks (Feed-Forward)
Two-Layer 5 Neurons
0.7095
0.7344
⇠ 2 hours
Two-Layer 10 Neurons
0.6817
0.7139
⇠ 8 hours
Two-Layer 20 Neurons
0.6767
0.7108
⇠ 14 hours
Two-Layer 30 Neurons
0.6668
0.7012
⇠ 32 hours
Table 1 :
1Summary of Results
Referred to as "9-Feature WLS" inFigure 4
Benchmark Solutions, Benchmark Bond Trade Price Challenge. Benchmark Solutions, Benchmark Bond Trade Price Challenge, www.kaggle.com, 2014.
Options, Futures and Other Derivatives. J C Hull, Prentice Hall8th EdHull, J. C., Options, Futures and Other Derivatives, Prentice Hall, 8th Ed, 2011.
T Hastie, R Tibshirani, J H Friedman, The Elements of Statistical Learning. Springer2nd EdHastie, T., Tibshirani, R., Friedman, J. H., The Elements of Statistical Learning, Springer, 2nd Ed, 2009.
. R H Shumway, Time Series Anaysis. Springer3rd EdShumway, R.H., Time Series Anaysis, Springer, 3rd Ed, 2010.
| [] |
[
"Detecting Multiple Seller Collusive Shill Bidding",
"Detecting Multiple Seller Collusive Shill Bidding"
] | [
"Jarrod Trevathan es:[email protected] \nSchool of ICT\nGriffith University\nAustralia\n",
"Claire Aitkenhead \nSchool of ICT\nGriffith University\nAustralia\n",
"Nazia Majadi \nSchool of ICT\nGriffith University\nAustralia\n",
"Wayne Read \nSchool of Mathematical and Physical Sciences\nJames Cook University\nAustralia\n",
"Jarrod Trevathan "
] | [
"School of ICT\nGriffith University\nAustralia",
"School of ICT\nGriffith University\nAustralia",
"School of ICT\nGriffith University\nAustralia",
"School of Mathematical and Physical Sciences\nJames Cook University\nAustralia"
] | [] | Shill bidding occurs when fake bids are introduced into an auction on the seller's behalf in order to artificially inflate the final price. This is typically achieved by experimental results are presented. | 10.1016/j.elerap.2021.101066 | [
"https://arxiv.org/pdf/1812.10868v1.pdf"
] | 57,189,199 | 1812.10868 | a52b9360815db9e3303aa752c2d17b938c9c8556 |
Detecting Multiple Seller Collusive Shill Bidding
28 Dec 2018
Jarrod Trevathan es:[email protected]
School of ICT
Griffith University
Australia
Claire Aitkenhead
School of ICT
Griffith University
Australia
Nazia Majadi
School of ICT
Griffith University
Australia
Wayne Read
School of Mathematical and Physical Sciences
James Cook University
Australia
Jarrod Trevathan
Detecting Multiple Seller Collusive Shill Bidding
28 Dec 2018* Corresponding authorShill biddingonline auctionsShill Scorecollusionauction simulation
Shill bidding occurs when fake bids are introduced into an auction on the seller's behalf in order to artificially inflate the final price. This is typically achieved by experimental results are presented.
the seller having friends bid in her auctions, or the seller controls multiple fake bidder accounts that are used for the sole purpose of shill bidding. We previously proposed a reputation system referred to as the Shill Score that indicates how likely a bidder is to be engaging in price inflating behaviour with regard to a specific seller's auctions. A potential bidder can observe the other bidders' Shill Scores, and if they are high, the bidder can elect not to participate as there is some evidence that shill bidding occurs in the seller's auctions. However, if a seller is in collusion with other sellers, or controls multiple seller accounts, she can spread the risk between the various sellers and can reduce suspicion on the shill bidder. Collusive seller behaviour impacts one of the characteristics of shill bidding the Shill Score is examining, therefore collusive behaviour can reduce a bidder's Shill Score. This paper extends the Shill Score to detect shill bidding where multiple sellers are working in collusion with each other. We propose an algorithm that provides evidence of whether groups of sellers are colluding. Based on how tight the association is between the sellers and the level of apparent shill bidding is occurring in the auctions, each participating bidder's Shill Score is adjusted appropriately to remove any advantages from seller collusion. Performance has been tested using simulated auction data and
Introduction
Online auction fraud can take various forms, including (but not limited to) misrepresenting an item for sale, failing to pay for or deliver goods, selling black market items, and bid shielding [8,17,22]. Shill bidding is a fraudulent activity whereby a seemingly innocent bidder (i.e., a shill bidder) uses fake bids to drive up the auction's price for the seller's benefit. The seller can have her friends operate as shill bidders, and/or can register multiple bidder accounts for the sole intention to submit shill bids. Such behaviour disadvantages legitimate bidders as they are forced to pay more for an item in order to win the auction. Shill bidding is not permitted by commercial online auctioneers, and severe penalties can be incurred by those caught engaging in shill bidding [17,29].
In March 2001, a U.S. court charged three men for their participation in a ring of fraudulent bidding in hundreds of art auctions on eBay [17]. The men created more than 40 eBay user accounts using false registration information.
The fraudsters gave themselves away when it was discovered that the items were misrepresented as being of greater value. Furthermore, suspicion was raised when unrealistically high shill bids were placed. These two factors provided a firm case for prosecutors. A more recent case occurred during 2010 in the UK [29]. A man used two eBay accounts. The first account was used to list a minibus for sale. He then used the second account to submit fake bids in the auction to inflate the price. The man also misrepresented the minibus by illegally reducing its mileage. He was fined £5,000 under newly introduced laws designed to combat shill bidding. These two cases highlight that shill bidders can get caught and prosecuted. However, if it were not for the misrepresentation and excessive prices, would the perpetrators have been detected?
Commercial online auctioneers claim to monitor their auctions for shill bid-ding activity, but are reluctant to disclose their techniques. eBay's feedback reputation system does not extend to shill bidding. The only means of recourse for a bidder that suspects she is a victim of shill bidding is to contact the auctioneer. Shill bidding detection is a relatively new area in academic literature (see [1][2][3][4][5][18][19][20]). We proposed a solution that observes bidding patterns over a series of auctions for a particular seller, looking for typical shill bidding behavior [18]. The goal is to obtain statistics regarding a bidder's conduct, and deduce a measure called a Shill Score. The Shill Score indicates the likelihood that a bidder is engaging in shill behaviour. A bidder is given a value between 0 and 10. The closer the Shill Score is to 10, the more likely that the bidder has engaged in shill-like price inflating behaviour. The Shill Score targets core strategies that a shill bidder follows. A shill bidder that deviates too far from these characteristics is less effective, and will not significantly alter the auction outcome for the seller. The Shill Score's reputation-based approach acts as both a detection mechanism and a deterrent to shill bidders. To avoid detection, a shill must behave like a normal bidder, which in effect stops her from shilling.
We also extended the Shill Score in later work to detect collusive shill bidding (i.e., where a seller has multiple shill bidders in an auction) [19,20]. Colluding shill bidders can engage in more sophisticated strategies in an attempt to reduce suspicion.
In the first of the aforementioned real world shill bidding cases, three sellers were in collusion. In other situations a single seller might be operating under several aliases (i.e., multiple seller accounts). The purpose for taking such an approach is to reduce the suspicion on any particular seller (or shill bidder) by distributing the risk of shill bidding and making it less noticeable. Collusive seller behaviour can influence elements of the Shill Score. None of the shill detection techniques in the literature specifically target using multiple seller accounts to engage in shill bidding. In this paper, we examine data from multiple sellers for signs of seller collusion, and determine which sellers are hosting auctions with suspicious shill behaviour (referred to as multiple seller collusive shill bidding). The algorithm firstly identifies potential groups of colluding sellers.
The degree of the association between the each member in the suspect colluding group is then determined to provide evidence regarding how suspicious the group is. A modified Shill Score is calculated for a bidder across each suspect seller's auctions. If the collective Shill Scores for the bidder across all sellers are sufficiently high, the original Shill Score is recalculated with an adjustment to remove the advantage of engaging in collusive seller behaviour. Performance has been tested using simulated auction data and experimental results are presented. This paper is organised as follows: Section 2 discusses the online auction format and provides background on shill behaviour, the Shill Score, and ways to detect collusive shill bidding. Section 3 defines the behaviour and strategies multiple colluding sellers can use to engage in shill bidding, and presents an algorithm to extend the Shill Score to account for colluding seller behaviour. Section 4 shows how the proposed algorithm performs with simulated auction data. Section 5 provides some concluding remarks and avenues for future work. The specific online auction format being investigated in this paper resembles that of an eBay auction. The auction has a predetermined start and finish time.
Bidders submit bids at any stage between the start and finish time. Each newly submitted bid must be higher than the previously submitted bid (i.e., greater than or equal to a minimum bid increment specified by the auctioneer). When two bids are received for the same value, the bid that arrived first is accepted.
The winner is the bidder with the highest bid once the auction terminates. The winner must pay the seller the amount corresponding to the winning bid.
An auction's bid history typically contains the following information for each bid received:
< bid #, bidder id, time, bid amount > Bid # is the number of the bid in the order of when it was received, bidder id is the identity of the bidder submitting the bid, time is the time at which the bid was received (typically down to the exact second), and bid amount is the monetary value of the bid.
Note that there are some differences in how online auctioneers display their bid histories. For example, eBay masks the bidder ids for all auctions over $200.
A bidders id is replaced by two random characters from her name and padded out with *s (e.g., j*****y). There are also nuances related to automated bidding (e.g., eBays Proxy Bidding, uBids Bid Butler). However, this paper assumes that all bid history information is available (i.e., no bid masking) and that the format is standardised (i.e., no automated bidding).
Shill Bidder Characteristics and Strategies
We refer to the most extreme competitive shill bidding strategy as aggressive shilling [18]. An aggressive shill continually outbids everyone, thereby driving up the price as much as possible. This strategy often results in the shill bidder entering many bids.
In contrast, a shill bidder might only introduce an initial bid into an auction where there have been no prior bids with the intent to stimulate bidding. This behaviour is a common practice in traditional and online auctions. However, most people typically do not consider it fraudulent. Nevertheless it is still shill bidding, as it is an attempt to influence the price by introducing spurious bids.
We refer to this as benign shilling in the sense that the shill bidder does not continue to further inflate the price throughout the remainder of the auction [18]. A benign shill bidder will typically make a "one-off" bid at or near the beginning of the auction.
To be effective, a shill bidder must comply with a particular strategy that attempts to maximise the pay-off for the seller. We define aggressive shill bidders to have the following characteristics [18]:
1. Bid exclusively in auctions only held by one particular seller. However, this alone is not sufficient to incriminate a bidder. It may be the case that the seller is the only supplier of an item the bidder is after, or that the bidder really trusts the seller (usually based on the reputation of previous dealings). 2. High bid frequency. An aggressive shill will continually outbid legitimate bids to inflate the final price. Bids are typically placed until the seller's expected payoff for shilling has been reached, or until the shill risks winning the auction (e.g., near the termination time or during slow bidding).
3. Few or no winnings for the auctions participated in (as the shill's goal is to lose).
4.
Bid within a small time period after a legitimate bid. Generally a shill wants to give legitimate bidders as much time as possible to submit a new bid before the auction's closing time.
5. Bid the minimum amount required to outbid a legitimate bidder. If the shill bids an amount that is much higher than the current highest bid, it is unlikely that a legitimate bidder will submit any more bids and the shill will win the auction. 6. Bid more near the beginning of the auction. A shill's goal is to try and stimulate bidding, by bidding early a shill can influence the entire auction process compared to a subset of it. Furthermore, bidding towards the end of an auction is risky as the shill could accidentally win.
Regardless of the strategy employed (i.e., aggressive or benign), a shill will still be a bidder that often trades with a specific seller but has not won any auctions. Another factor that affects a shill bidder's strategy is the value of the current bid in relation to the reserve price (i.e., a price specified by the seller as being the minimum value for which s/he will accept for the sale to proceed).
For example, once bidding has reached the reserve price, it becomes more risky to continue shilling. However, this is conditional on whether the reserve price is a realistic valuation of the item that all bidders share. Table 1 illustrates an example auction with three bidders. Each bidder is denoted as b 1 , b 2 , and b 3 respectively. Bidders b 1 and b 3 are legitimate, whereas b 2 is a shill bidder. b 2 engages in aggressive shill behaviour by outbidding a legitimate bid by the minimal amount required to stay ahead and within a small time period of the last bid. b 2 's bids force the other bidders to enter higher bids in order to win. If b 2 was not participating in this auction, b 1 would have only needed to pay $21 in order to win. Instead, b 2 caused b 1 to pay $33, thus the shill has inflated the price by $12. Another approach might be to check bidders feedback records as accounts created for the purpose of shill bidding might have limited or no feedback. However, there seems to be numerous problems with these approaches.
Shill Bidding Example
Firstly, checking someone's IP address over time could be considered a breach of privacy. Furthermore, a shill bidder may fake her IP address to avoid detection or frame an innocent person. In addition, it may be the case that the computer is located in an Internet cafe or a place where the computer is shared.
One person may bid then log off, and then a subsequent person comes along and uses the same computer to bid. Additionally, computers are dynamically assigned IP addresses on networks. A shill bidder can get around this method of detection by disconnecting and then reconnecting his/her computer to the network. Each time the shill bidder does this, she will be assigned a different IP address.
With regard to checking feedback, there are generally two instances where auction participants could be considered suspect:
1. User IDs with zero feedback; and 2. A pattern of the same group of users bidding on different auctions by one seller.
However, there are legitimate users with zero feedback (i.e., newly joined bidders, or those who have not yet won or participated in any auctions). There are also legitimate reasons that the same group of users might bid on different auctions by the same seller. If a seller has a good reputation, loyal bidders may choose to continually deal with the seller. Additionally, an abundance of good feedback can be misleading as some sellers can engage in a practice referred to as reputation stacking [8,17]. Reputation stacking is where the seller creates multiple bidder accounts and holds a large number of auctions for low-valued items just for the purpose of generating positive feedback.
One of the first academic approaches to dealing with shill bidding is by Wang et al. [26,27]. They suggest that listing fees (referred to as a shill proof fee)
could be used to deter reserve price shilling. Reserve price shilling is a situation that occurs when a seller is charged a fee based on what the stated reserve price is. Therefore, a seller will list a lower reserve price to avoid higher fees and then use shill bids to push the price up. Wang et al.'s proposal charges a seller an increasing fee based on how far the winning bid is from the reserve price. The idea is to compel the seller into stating her true reserve price, thereby eliminating the economic benefits of reserve price shilling. However, this method is untested and does not apply to auctions without reserve prices.
Most previous work on shill detection in online auctions is based on analysing large volumes of historical auction data to search for shill patterns. Kauffman and Wood [9] used a statistical approach to detecting shill bidding behaviours and showed how the statistical data of a market would look if opportunistic behaviours do exist. They also showed how to use an empirical model to test for questionable behaviours. However, one limitation of the approach is the need to review multiple auctions over a long period of time. Furthermore, since the statistical approach was based on analyzing a large amount of historical auction data, it was not applicable to directly analysing a particular auction where shilling behaviours might be involved.
Chau et al [30] proposed a shill bidding detection method called 2-Level
Fraud Spotting, which can be used to detect fraudsters in online auctions using data mining techniques by investigating historical data from eBay auctions.
Xu and Cheng [29] propose an approach to detect shill suspects in concurrent online auctions (where multiple auctions for identical items are simultaneously taking place). Their auction model can be formally verified using a model checker according to a set of behavioural properties specified in pattern-based linear temporal logic. Dong et al. [31,32] extend on this work by verifying shill suspects using Dempster-Shafer theory of evidence. They use eBay auction data to validate whether using Dempster-Shafer theory to combine multiple sources of evidence of shilling behaviour can reduce the number of false positive results that would be generated from a single source of evidence. Later, Dong et al. [33] study the relationship between final prices of online auctions and shill activities in eBay auctions. They train a neural network using features extracted from item descriptions, listings and other auction properties. The likelihood of shill bidding is determined by the aforementioned Dempster-Shafer shill certification technique. Goel et al. [34] introduce an approach for verifying shill bidders using a multi-state Bayesian network, which supports reasoning under uncertainty.
They describe how to construct the multi-state Bayesian network and present formulas for calculating the probabilities of a bidder being a shill and being a normal bidder.
Some approaches have been proposed to detect shill bidding in real-time (i.e., while an auction is in progress) [35,36,37,38,39]. The motivation is that actions can be taken to penalise the seller or shill bidder before the auction terminates to ensure that innocent bidders do not become victims. Such actions can include suspending or cancelling an auction, economic penalties, and account suspension or cancellation. However, a problem with a purely real-time shill detection method is that there is insufficient information available from just one auction. A bidder's historical behaviour must be to some extent taken into account to provide sufficient evidence of shill bidding. The real-time proposals so are merely demonstrations of a method, but lack any sort of testing to prove their effectiveness. Furthermore, the shill behaviours outlined in these papers are arbitrary and do not share a consensus amongst the academic community about what actually constitutes shill bidding.
Beranek et al. [2] introduce a trust model based on reputation (users evaluation after performed transactions) and on examination of properties of possible fraudulent behaviour in online auctions. The evidence is expressed and combined using belief functions. Beranek and Knizek [3] extend the fraud detection approach by using contextual information whose origin is outside online auction portals. The suggested model integrates information from auctions and relevant contextual information with the aim to evaluate the behaviour of certain sellers in an online auction and determine whether it is legal or not. However, this
proposal does not focus specifically on shill bidding.
The Shill Score Reputation System
In previous work [18], we proposed a reputation system that observes a bidder's bidding patterns over a series of auctions for a particular seller. The goal is to obtain statistics regarding a bidder's conduct with the seller, and calculate a Shill Score. Potential auction participants can observe other bidders'
Shill Scores to determine the likelihood that any of them are engaging in priceinflating behaviour.
The Shill Score targets core shill bidding strategies. A shill that deviates too far from these strategies is less effective, and will not significantly alter the auction outcome. This approach acts as both a detection mechanism and a deterrent to shill bidders. To avoid detection, a shill must behave like a normal bidder, which essentially restricts her ability to engage in shill bidding.
The Shill Score basically works as follows (see [18] for specific details): A bidder b i , is examined over k auctions held by the same seller for the behaviour outlined in Section 2.2. Each characteristic of shill behaviour is assigned a rating, which is combined to form b i 's Shill Score. The Shill Score gives b i a value between 0 and 10. The closer the Shill Score is to 10, the more likely that b i is engaging in price-inflating behaviour. The algorithm's goal is to determine which bidder(s) is most inclined to be the shill out of a group of l bidders. The
Shill Score behavioural ratings are determined as follows:
• α Rating -Percentage of auctions bi has participated in.
• β Rating -Percentage of bids bi has made out of all the auctions participated in.
• γ Rating -Normalised function based on the auctions bi has won out of the auctions participated in.
• δ Rating -Normalised inter bid time for bi out of the auctions participated in.
• Rating -Normalised inter bid increment for bi out of the auctions participated in.
• ζ Rating Normalised time bi commences bidding in an auction.
Each rating is between 0 and 1, where the higher the value, the more suspicious the bidder. A bidder's Shill Score is calculated as the weighted average of these ratings:
score = ω 1 α + ω 2 β + ω 3 γ + ω 4 δ + ω 5 + ω 6 ζ ω 1 + ω 2 + ω 3 + ω 4 + ω 5 + ω 6 × 10
where ω i , 1 ≤ ω i ≤ 6, is the weight associated with each rating. Tsang et al [24] propose what they feel are the optimal selections for weight values. If a bidder wins an auction, then his/her α, β, δ, and ζ ratings are 0 for the particular auction (as the shills goal is to lose).
Shill Bidding Detection Involving Colluding Shill Bidders
The Shill Score (as outlined in [18]) considers only the basic scenario with one seller and one shill bidder. In the US case described in Section 1 [17], there were three sellers, which used 40 different aliases (40 shills in effect). The sellers understood that there was less chance that they would get caught if they used multiple bidder accounts to take alternating turns at submitting shill bids. This makes it more difficult for authorities to determine which bidders are shills, as collusive behaviour allows shill bidders to appear to be more like regular bidders. In some cases, geographical proximity can be an indication of collusion if there are several shills within a close area that participate in the auction. For example, in the shill case [17], two of the men were from California and the other was from Colorado. However, this is not a reliable indicator of shill bidding and may raise privacy concerns, as it requires examining the registration database for such relationships.
We refer to the strategies a group of shill bidders can engage in as collusive shill bidding [19,20]. We investigated approaches to shill bidding involving one seller who controls multiple shill bidders and what effect this has on the Shill Score. The main goal of shilling is to drive up the price of an item. In the situation where there is only one shill bidder, the shill's secondary goal is to attempt to do this in such a manner that it minimises her Shill Score. When there is more than one shill bidder, there are particular strategies that the group (of shill bidders) can engage in to influence some factors contributing to their individual Shill Scores. Therefore, the group's collective goal (secondary to shilling) is to minimise each member's Shill Scores.
Despite being able to use more complicated strategies, the group as a whole must still conform to certain behaviour in order to be effective as a shill. With regard to the Shill Score, all that shill bidders can do by colluding is to reduce their α and β ratings. The γ, δ, and ζ ratings are still indicative of shill bidding. For example, none of the colluding shill bidders will be inclined to win an auction. Furthermore, it is still in the group's interests to bid quickly, and by minimal amounts to influence the selling price. Therefore, inter bid times and increments will be consistent for all shill bidders. Shill bidders will also bid early in an auction and cease bidding well before the end of an auction.
There appear to be three possible strategies that can be employed by colluding shill bidders. The first strategy is referred to as the alternating bid strategy.
Two (or more) colluding shill bidders each take alternating turns at bidding, e.g., shill 1 bids, then shill 2 bids, then shill 1 bids again, etc. Table 2 presents an example of the alternating bid strategy. Here there are three bidders, denoted b 1 , b 2 and b 3 respectively. b 1 is a legitimate bidder, but b 2 and b 3 are shill The second strategy is for colluding shills to take turns at shilling for a particular auction (referred to as the alternating auction strategy). For example,
given two auctions, shill 1 will bid exclusively in auction 1 , while shill 2 bids only in auction 2 . This strategy lowers the shills' α ratings (i.e., number of auctions participated in), but does not affect their β ratings.
The third strategy is to use a combination of the alternating bid and alternating auction strategies (referred to as the hybrid strategy). The hybrid strategy can be used to alter the group's α and β ratings between the two extremes. An example of a hybrid strategy would be for shill 1 and shill 2 to alternately bid in auction 1 , shill 3 and shill 4 alternately bid in auction 2 , then shill1 and shill 3 alternately bid in auction 3 , etc. This continues until all combinations of bidders have been used, and then the process repeats. In reality, colluding shills would probably employ a hybrid strategy.
In [19,20] we describe how to extend the Shill Score to detect collusive shill bidding behaviour using a Collusion Score. While the details this approach are outside the scope of this paper, the purpose of this discussion is to highlight how differing collusive behaviours can be used in an attempt to influence the Shill Score.
A limitation of our collusive shill bidding proposal is that it only focused on one seller who controls multiple shill bidders. There is no literature that addresses the situation where multiple sellers are in collusion and what strategies they can engage in. In this paper we refer to this scenario as multiple seller collusive shill bidding.
Shill Bidding Detection Involving Multiple Seller Accounts
This section describes the behaviours and strategies multiple colluding sellers can use to engage in shill bidding and what affect this has on the Shill Score. We then present an algorithm that takes into account multiple seller collusive shill bidding strategies and adjusts the Shill Score for a specific bidder appropriately to remove any advantage that seller collusion may have.
Multiple Collusive Seller Shill Bidding Behaviour and Tactics
The goal of this paper is to address shill bidding strategies that a seller could engage in if she has control of multiple seller accounts. In order to narrow the scope of the problem, we restrict our attention to the case where there are two or more sellers in collusion, but they only control one shill bidder. That is, we are ignoring instances where the colluding sellers control multiple shill bidding accounts (this will be the focus of future work).
Let the set of all bidders in the auction dataset be
B = {b 1 , b 2 , . . . , b l }
where b i denotes the ith bidder and 1 ≤ i ≤ l.
The set of all sellers is
S = {s 1 , s 2 , . . . , s m }
where s j denotes the jth seller account and 1 ≤ j ≤ m.
The set of all auctions in the dataset is
A = {a 1 , a 2 , . . . , a n }
where n (n ≥ 0) is the total number of auctions.
The set of auctions conducted by seller s j is
A j = {a j 1 , a j 2 , . . . , a j k }
where A j ⊆ A and k (0 ≤ k ≤ n) is the total number of auctions conducted by s j . Each sellers' auctions forms a partition over A. That is,
A = A 1 ∪ A 2 ∪ · · · ∪ A j and A i ∩ A j = {}, i = j, where each A is pairwise disjoint.
Let us denote a seller engaging in collusive shill bidding behaviour as S shill .
S shill controls a series of w (0 ≤ w ≤ n) seller accounts. The challenge is to determine which subset of seller accounts in S are controlled by S shill . We will denote the colluding seller accounts as the set S where S ⊆ S.
Consider the case where S shill is using two colluding seller accounts S = {s 1 , s 2 }, and controls a single shill bidder b s (b s ∈ B). If b s were to participate entirely in auctions held by s 1 , then b s 's Shill Score will be high for s 1 , but 0 for s 2 . This is due to the α rating being high in s 1 's auctions, but 0 in s 2 's auctions as b s has not participated in any auctions held by s 2 . The same holds true if the situation is reversed and b s participates in auctions held by s 2 and does not participate in s 1 's auctions (i.e., a high α rating for s 2 , but a 0 α rating for s 1 ).
Now consider the case where S shill is using two colluding seller accounts S = {s 1 , s 2 }, and is conducting a total of four auctions (two per seller). That is, the set of auctions held by s 1 is A 1 = {a 1 1 , a 1 2 }, and the set of auctions held by s 2 is A 2 = {a 2 1 , a 2 2 }. In order to reduce the impact of b s 's α rating, the best strategy for S shill is to alternate b s evenly between the two seller's auctions.
That is, use b s in some auctions held by s 1 and s 2 , but do not use b s for all auctions held by s 1 or s 2 . The optimal sequences are as follows: Using such a sequence to evenly distribute b s 's shill bidding activity out across the auctions held by s 1 and s 2 reduces b s 's α rating for any particular seller. That is, the average number of auctions participated in per seller is lower, therefore suspicion is lower on b s for each particular seller. We refer to this approach as the alternating seller strategy. That is, S shill is alternating b s evenly across each of the auctions by s 1 and s 2 in order to avoid detection and reduce suspicion. Now let's scale up the problem to three sellers S = {s 1 , s 2 , s 3 }, who are hosting the following respective auctions
A 1 = {a 1 1 , a 1 2 , a 1 3 }, A 2 = {a 2 1 , a 2 2 , a 2 3 }, A 3 = {a 3 1 , a 3 2 , a 3 3 }.
To evenly reduce suspicion on b s for each particular seller, Sshill can use any variation of the following sequence a 1 1 , a 2 2 , a 3 3 provided that b s only participates in an even number of auctions from each seller. The less number of auctions bs participates in from each seller, the better.
However, the effect of the α rating in the Shill Score becomes more obscure if there are an uneven number of auctions conducted by each seller. Consider three sellers s 1 , s 2 and s 3 , who are hosting the following respective auctions A 1 =
{a 1 1 , a 1 2 }, A 2 = {a 2 1 , a 2 2 , a 2 3 }, A 3 = {a 3 1 , a 3 2 , a 3 3 , a 3 4 , a 3 5 }.
Assume that S shill employs the following sequence of alternating auctions a 1 1 , a 2 1 , a 3 1 . b s 's α rating will be high for s 1 , medium for s 2 , and low for s 3 . This is due to the α rating looking at the percentage of auctions for a particular seller that b s has participated in.
For each respective seller this will be s 1 − 50%, s 2 − 33.3%, and s 3 − 20%. Logic might seem to suggest to not shill in s 1 's auctions, which in this case means that the Shill Score has done its job by disrupting the usual business of shill bidding. If b s is transferred from to a 1 1 to perhaps a 3 2 as s 3 has a greater number of auctions, then s 3 's α rating will jump up to 40%, which is also an undesirable outcome.
It seems that the optimal approach for the alternating seller strategy is to host an even number of auctions for each s j ∈ S , and evenly alternate bs between each seller. The lower the number of auctions participated in, the lower the impact of the rating. As such it appears that the effect of the Shill Score in its current form on multiple seller collusive shill bidding is as follows:
1. Participate like a regular shill bidder and shill in all auctions across all seller accounts, and bs will have a high Shill Score for all sellers she has been involved with; 2. Alternately, shill in an ad hoc manner in some auctions across some/all seller accounts, and bs will have a high Shill Score for some sellers and a lower Shill Score for other sellers. But ultimately, there will be substantial evidence of shill bidding; or 3. Host an even number of auctions across all seller accounts, and alternate bs evenly across all s j ∈ S , then evenly reduce the suspicion on b s for all sellers b s has been involved with. Therefore, the Shill Score has done its job for the first two cases. It is the third case that needs addressing. Clearly the Shill Score is successful in that it forces S shill to change her behaviour by expending more effort to open multiple accounts, reducing the number of auctions with shill bidding, and going to careful lengths to avoid detection. The main advantage S shill has is to influence the Shill Score's α rating (i.e., the percentage of sellers auctions the bidder has participated in). The remaining Shill Score ratings (i. e., β, γ, δ, , ζ) are unaffected by the alternating seller strategy (provided the aforementioned assumption is in place where we are only dealing with one shill bidder being used across multiple sellers). The remainder of this section discusses an algorithm that accounts for the alternating seller strategy in an attempt to remove any remaining advantage S shill might have by engaging in such an approach to shill bidding.
The Seller Collusion Algorithm
In this section we propose the Seller Collusion Algorithm. The algorithm is designed to achieve the following goals: 6. Based on a severity measure (i.e., the extent that it is likely that shill bidding is occurring), proportionally adjust the α rating and recalculate b i 's Shill Score.
The following sections outline the specifics of each step in our proposed algorithm to combat multiple seller collusive shill bidding.
Step 1 -Identify which sellers a bidder has been involved with
Step one in the Seller Collusion Algorithm is to identify which sellers bi has been involved with, and for each identified seller, count how many of the seller's auctions bi has participated in. This refers to only whether the bidder has bid in any of the seller's auctions, not how many bids have been submitted in a particular auction. Table 3 presents an example of test data that will be used to help describe
Step 2 -Determine which sellers have an association based on their dealings with a bidder
Step two in the Seller Collusion Algorithm is to identify potential groups of suspected colluding sellers. To do this we use the Seller Association Graph
(SAG) SAG = (V, E)
where V is the set of all vertices (i.e., the sellers) and E is the set of edges (i.e., an association between sellers). One Seller Association Graph SAG i , is generated for each bidder bi (where 1 ≤ i ≤ l). Each seller is a vertex v in the graph. For two sellers s j , s k (where j, k > 0, j = k), an edge e will exist between their vertices v j and v k , if the bidder has participated in both of their auctions. Note that this is not restricted to mean concurrent participation (i.e., the auctions are running at the same time), but any historical participation. Figure 1 shows the Seller Association Graphs generated based on the data in Table 3. SAG 1 , SAG 2 and SAG 3 , correspond to b 1 , b 2 and b 3 respectively. Once each seller association graph SAG i has been defined, the respective vertices are weighted based on the number of s j 's auctions bi has participated in (Figure 2). For example, in SAG 1 , b 1 has participated in 3 of s 1 's auctions, therefore s 1 's vertex is weighted as 3, etc.
Step 3 -Identify potential groups of suspected colluding sellers
Step three in the Seller Collusion Algorithm is to determine which of the identified sellers are most likely to be in collusion with each other. The purpose of this step is to "weed out" more innocent sellers who might have had few (or significantly less) dealings with bi compared to more suspicious seller cliques.
In for sellers with a strong association, or -1 for sellers with a weak association.
SBAG = (V, E)
where V is a set of sellers and E is a set of edges between sellers.
The corresponding Shill Bidding Association Graph for SAG i is denoted as SBAG i . The non-suspect sellers' vertices and all vertex and edge weightings from SAG i are discarded in SBAG i . However, SBAG i maintains all the vertices and edges for the suspect sellers that were identified in SAG i .
Step 4 -Calculate Modified Shill Scores to produce evidence of seller collusion
Step four in the Seller Collusion Algorithm is to calculate b i 's Shill Score for each suspected colluding seller. This will indicate which sellers are likely to have shill bidding occurring in their auctions. We would expect that a bidder who was engaging in shill bidding for a seller would have a high Shill Score.
Furthermore, we would expect that the bidder would have a consistently high Shill Score across all suspect sellers in a clique.
However, given that the goal of multiple seller collusive shill bidding is to reduce the effect of the shill bidder's α rating, the Shill Score in its current form is unreliable. Instead, a Modified Shill Score (MSS) is used which removes the α rating during the calculation of the Shill Score. That is, the MSS is calculated as though b i had only participated in s j 's auctions. Each vertex in SBAG i is then weighted with b i 's MSS for the particular seller. Figure 5 presents examples of Shill Bidding Association Graphs with the vertices weighted based on the bidder's MSS for each particular seller. In the example, the sellers in SBAG 1 appear highly suspicious.
Step 5 -Classify the severity of suspected colluding sellers
Step five in the Seller Collusion Algorithm is to classify a suspected colluding group of sellers based on the severity of shill bidding that has occurred during If v i < med − 0.5 or > med + 0.5, then discard The remaining sellers in SBAG i are now those who have sufficiently high MSSs which are relatively similar. Therefore, these sellers have the most likelihood of engaging in the alternating seller strategy (Figure 7). SBAG 3 has only one vertex remaining. As a seller cannot be in collusion with herself, SBAG 3 can also be entirely discarded. However, SGAG 1 shows a strong association between s 2 and s 3 .
3.2.6.
Step 6 -Adjust the α rating of suspected shill bidders As previously mentioned, seller collusion is a way of avoiding shill bidding detection. The seller directly tries to influence the α rating of the Shill Score by increasing the number of sellers b i participates in auctions with. As such, the higher the likelihood of seller collusion, the less we can rely on b i 's α rating.
Step six in the Seller Collusion Algorithm is to adjust the α rating on b i 's Shill Score and recalculate for the remaining vertices in SBAG i .
In the original Shill Score each rating is given a weighting that influences how important the particular characteristic of shill bidding is in terms of calculating the Shill Score. The α rating is weighted by ω 1 in the Shill Score. To mitigate against the effect of the alternating seller strategy, we need to reduce the influence of the α rating. This is achieved by proportionally reducing ω 1 based on how high the M SS i is for b i in s j 's auctions. That is, the higher M SS i the lower ω 1 is for b i 's α rating.
The first operation in achieving this outcome is to alter b i 's MSS so that it is between 0 and 1:
M SS i = M SS i /10
Next, let ω 1 denote the scaling factor to reduce ω 1 by. ω 1 is calculated as follows:
ω 1 = 1 − M SS i
Finally, the original Shill Score is recalculated for b i with the influence of the α rating being reduced by ω 1 /ω 1 : score = (ω 1 /ω 1 )α + ω 2 β + ω 3 γ + ω 4 δ + ω 5 + ω 6 ζ ω 1 + ω 2 + ω 3 + ω 4 + ω 5 + ω 6 × 10
Performance
Test Setup
Testing the performance of any shill bidding detection technique is difficult.
As such, many shill bidding detection proposals overlook the testing stage, therefore their effectiveness cannot be ascertained. Using data from commercial auction sites is ultimately the best test. However, most auction sites are reluctant to supply auction data for privacy reasons and fear of lost reputation if it were discovered that shill bidding is occurring in their auctions. Even if commercial auction data can be obtained it is still unknown if shill bidding definitely occurs as shill bidders generally are not forthcoming about their behaviour.
The aforementioned testing problems are further exacerbated when looking for seller collusion in auction data. As this our work is the first to look into collusive behaviour, significantly larger datasets are required for testing. That is, potentially 1,000 of auctions involving numerous bidders and sellers.
Previously, we have undertaken two approaches to testing. One is to hold simulated auctions (on a purpose-built auction server) involving human users bidding for fake items with fake money. The users did not have any idea that there was a shill bidder in the auctions inflating the price. However, this approach is time consuming to arrange and to continually monitor the auctions.
Additionally, such an approach may not necessarily capture all of the real-world auctioning behaviours and strategies as real money and items are not involved.
If this approach were to be used to test the Seller Collusion Algorithm, the auctions could potentially involve participants in both seller and bidder roles.
During the auctions a small number of sellers would be asked to engage in seller collusion. They would not be instructed how they must do this. Once all auctions were completed the Seller Collusion Algorithm could be run against all generated auction data. However, the issues of the size of the test data set required and the amount of effort to organise the auctions really limit the viability of this approach.
The second approach to testing is to use software bidding agents to generate synthetic auction data. Previously, we have proposed a Simple Shill Bidding Agent [21] and an Adaptive Shill Bidding Agent [40] that were programed to engage in typical shill bidding strategies. These agents were pitted against a set of "Zero-Intelligence" bidding agents whose sole purpose was to randomly commence bidding up to their randomly allocated bidding limit throughout an auction. While this approach automates the testing process and can generate large amounts of data sets, the problem was that we were programming the very behaviour we were expecting. As such, the shill detection mechanism was bound to uncover the shill bidders who were perpetrating this behaviour. The simulator was also used as the front end for processing the synthetic auction data for the purposes of undertaking the tests.
Results and Analysis
The following four tests were conducted with regard to the Seller Collusion Algorithm:
1. How the proposed detection mechanism performs against a baseline of regular auctions that do not contain shill bidding (see Figure 10(a)).
How the Seller Collusion Algorithm performs on auctions involving collud-
ing groups of sellers who engage in the alternating seller strategy to avoid detection (see Figure 10(b)).
3. The impacts on Shill Scores and the operation of the Seller Collusion Algorithm when the shill bidder is not evenly alternated between the sellers' auctions (see Figure 10(c)).
Comparison between baseline Shill Scores and recalculated Shill Scores us-
ing the Seller Collusion Algorithm (see Figure 10(d)).
Furthermore, we generate a simulated auction dataset using a shill bidding agent [21]. The dataset contains 10 sellers, 51 bidders (including a shill bidder (e.g., 'Shill a')), and 30 auctions. We applied the dataset on the seller collusion algorithm and compared the algorithm with collusion score approach [19]. Figure 11 shows the comparative analysis results. Figure 11(a) presents that all bidders show normal bidding behaviour (including 'Shill a'). Figure 11(b) illustrates that all bidders show regular bidding patterns except 'Shill a' (red-filled circle). This indicates that the seller collusion algorithm performs better than collusion score approach.
We also analysed the bidding patterns of 'Shill a' and found that 'Shill a' achieved the highest MSSs for the sellers (e.g., 'seller a', 'seller b', 'seller c', and 'seller d') she participated in (see Figure 12). This indicates that 'seller a', 'seller b', 'seller c', and 'seller d' are the colluding sellers who engaged 'Shill a'
for price-inflating behaviour.
(a) Collusion Score [19] (b) Seller collusion algorithm
Conclusion
This paper presented an approach to detect shilling bidding where multiple sellers are working in collusion with each other (i.e., multiple collusive seller shill bidding). In order to rein in the scope of the problem, this paper only focused on the situation where there are two or more colluding sellers who control one shill bidder. We described the strategies colluding sellers could employ to reduce the amount of suspicion raised by shill bidding through alternating the shill bidder evenly across their auctions. We referred to the optimal approach as the alternating seller strategy. By engaging in the alternating seller strategy, the sellers can influence the rating used when calculating the shill bidders Shill Score.
The approach to detect multiple collusive seller shill bidding outlined in this paper accomplishes two goals. Firstly, we are able to potentially ascertain colluding groups of sellers employing the alternating seller strategy. This is achieved by identifying the sellers a bidder has been associated with (using the Seller Association Graph), determining potential cliques of sellers who might be engaging in shill bidding (using the Shill Bidding Association Graph), calculating a bidders Modified Shill Score for each of the sellers (i.e., removing the rating), and ranking their likelihood of collusion. Secondly, we are able to account for the alternating seller strategy, and recalculate the bidders Shill Scores for each seller. This allows the Shill Score reputation system to remain a deterrent to those attempting to engage in shill bidding. The information gathered by the Seller Collusion Algorithm could be supplied to auction houses, allowing them to monitor or take action against suspected sellers (i.e., warning auction participants, suspending/cancelling user accounts, suspending/cancelling auctions, economic penalties, legal action).
Performance has been tested using simulated auction data and experimental results are presented. The tests examined how the proposed detection mechanism performs against a baseline of regular auctions that do not contain shill bidding. We then compared how our approach performed on auctions involving colluding groups of sellers who engage in the alternating seller strategy to avoid detection. Finally, we investigated the effects of seller collusion whereby the shill bidder was not evenly alternated between the sellers auctions. This algorithm has yet to be tested against live auction data.
Future work involves investigating strategies collusive sellers can engage in whereby they control more than one shill bidder. That is, combining the work presented in this paper with the work into collusive shill bidding [19,20]. This will significantly increase the complexity of the problem and the amount of auction data required for testing, but the goal is to create a well-rounded approach to shill bidding detection that focuses on all shill bidding strategies. Finally, we plan to expand the Shill Score to operate in live auctions so that real-time actions can be taken against shill bidding perpetrators.
. b 2 and b 3 take alternating turns at outbidding b 1 . This strategy has the effect of lowering b 2 and b 3 's β ratings (i.e., the number of individual shill bids in an auction), but does not affect their α ratings.
1 .
1Identify potential colluding groups of sellers; 2. Ascertain the likelihood that shill bidding is occurring within the colluding group of sellers; and 3. Account for the influence on the rating for a specific bidders Shill Score by proportionally adjusting the ratings weight. The Seller Collusion Algorithm has six distinct steps: 1. Identify which sellers' auctions b i has participated in; 2. Determine which sellers have an association based on their dealings with b i and construct a Seller Association Graph; 3. Determine which sellers are most likely to be colluding based on b i 's dealings with the suspect sellers and construct a Shill Bidding Association Graph; 4. For each seller in a group of suspected colluding sellers, calculate b i 's Modified Shill Score (i.e., remove the α rating from consideration); 5. Check the similarities between each Modified Shill Score to determine the likelihood the sellers are operating in a group, and that they are engaging in shill bidding; and
the Seller Collusion Algorithm. There are three bidders B = {b 1 , b 2 , b 3 } and six sellers S = {s 1 , s 2 , s 3 , s 4 , s 5 , s 6 }. b 1 has participated in 3 auctions by s 1 , 40 auctions by s 2 , 30 auctions by s 3 , 2 auctions by s 4 , and no auctions held by s 5 and s 6 . Similarly, Table 1 outlines the dealings b 2 and b 3 have had with the sellers in S.
Figure 1 :
1Example Seller Association Graphs illustrating which sellers each bidder has had dealings with.
Figure 2 :
2Seller Association Graphs with weighted vertices to depict how many of each sellers auctions a bidder has participated in.
order to identify possible colluding seller groups, each edge of the Seller Association Graph is given a weighting. The weighting examines how similar (or dissimilar) each pair of sellers are in terms of the number of auctions b i has participated in with them. The first operation is to compare how far the number of auctions is from the mean number of auctions for two particular sellers s j , s k : a = (n j −n) b = (n k −n) where isn is the average of the auctions held by all the sellers in SAG i , and n j and n k are the values of the vertices for s j and s k respectively. Note that if a or b = 0 (i.e., they are equal ton), then a or b are set to 1 to avoid division by 0 in the next equation. The following equation is used to generate the edge weighting e j,k for s j and s k : to similarities in the auctions participated in, this equation also identifies groups of sellers who have constituted a large portion of a bidder's auctions. That is, sellers with a large number of similar participation quotas are more suspicious than similar sellers with whom bi has only participated in their auctions a small number of times. The equation will produce a value of 1
Figure 3 :
3Seller Association Graphs with weighted edges to gauge how strong the association is between groups of sellers based on the number of auctions a bidder has participated in with them.
Figure 3
3shows the example Seller Association Graphs highlighting suspicious seller cliques based the edge weighting metric (i.e., those edges and vertices in bold, with an edge weighting of 1). Those sellers which are identified as suspected colluding sellers are then added to a new graph we refer to as the Shill Bidding Association Graph (SBAG).
Figure 4
4presents examples of the Shill Bidding Association Graphs. SBAG 1 , SBAG 2 and SBAG 3 , correspond to b 1 , b 2 , and b 3 respectively.
Figure 4 :
4Shill Bidding Association Graphs highlighting sellers with strong associations and with vertex weightings removed.
Figure 5 :
5Shill Bidding Association Graphs with Modified Shill Score weightings added to the vertices.their auctions (i.e., those auctions for which the suspected bidder has been involved). A shill bidder is likely to have the same bidding pattern across all auctions in which they are participating. Therefore, we would expect the MSS to be similar across all auctions for the shill bidder. Sellers for which b i has a high MSS can be deemed suspect. Furthermore, sellers for which a bidder has a very similar MSS can also be considered suspicious. Sellers that fall into this category warrant further investigation.During step four, we compare the b i 's MSS for each seller. Firstly, we can remove any s j from SBAG i if b i 's MSS is below 4 for s j 's auctions as there is little evidence that shill bidding is occurring despite being identified as being part of a clique.
Figure 6 :
6Shill Bidding Association Graphs with sellers removed who have a Modified Shill Score below 5.
Figure 6
6shows that SBAG 2 can now be discarded as the b 2 has a very low Shill Score for the auctions held by s 4 , s 5 and s 6 . SBAG 3 illustrates the situation where the Shill Score for b 3 is below the threshold of 5 for s 2 , but is above 4 for s 3 . In this case s 2 's vertex is dropped from SBAG 3 .For all sellers that remain in the SBAG i we want to compare how similar their MSS scores are. To do this determine what the median MSS is for the group. The median is denoted as med. For a given seller's vertex v i the following occurs:
Figure 7 :
7Final Shill Bidding Association Graphs highlighting sellers that are highly suspicious.
Figure 8
8illustrates the final results for the Seller Collusion Algorithm with the example data. SBAG 2 and SBAG 3 have been removed from consideration as there will little evidence of seller collusion. The remaining sellers in SBAG 1 have had b 1 's α rating adjusted to account for strong evidence of collusion. As such b 1 's Shill Score with regard to these s 2 and s 3 increases as the influence of collusion on the rating is scaled down in the Shill Score.
Figure 8 :
8Final Shill Bidding Association Graphs with a recalculated Shill Score adjusted for seller collusion.
Figure 9 :
9Multiple Seller Collusive Shill Bidding Simulator.
Figure 10 :
10Simulated Test Results.
Figure 11 :
11Comparative Analysis Results.
Figure 12 :
12MSSs of 'Shill a' for the sellers she participated in.
Table 1 :
1An example auction with one shill bidder inflating the price for the seller.Bid #
bid
Price Time
15
b 1
$33
20:03
14
b 2 (Shill)
$32
12:44
13
b 1
$31
12:42
12
b 2 (Shill)
$26
5:05
11
b 1
$25
5:02
10
b 2 (Shill)
$21
2:47
9
b 3
$20
2:45
8
b 2 (Shill)
$15
1:07
7
b 1
$14
1:05
6
b 2 (Shill)
$9
0:47
5
b 3
$8
0:45
4
b 2 (Shill)
$6
0:20
3
b 3
$5
0:19
2
b 2 (Shill)
$2
0:06
1
b 1
$1
0:05
In this example, b 2 exhibits the typical shill behaviour described above. This
is evidenced by: (i) High frequency of bids (i.e., b 2 has submitted more bids than
both the other bidders); (ii) Has not won the auction despite the high number
Table 2 :
2An example auction with two colluding shill bidders alternating their bids in order to reduce suspicion.Bid #
bid
Price Time
15
b 1
$35
20:03
14
b 2 (Shill)
$32
12:44
13
b 1
$31
12:42
12
b 3 (Shill)
$26
5:05
11
b 1
$25
5:02
Table 3 :
3Example data for possible seller collusion -the number of times each bidder participated in each sellers auctions.
Recently, Tsang et al.[23] extended upon our work and created a more sophisticated tool to generate synthetic auction data. Their approach was calibrated against data collected from the New Zealand auction site, Trade Me, and can reproduce the statistical qualities of the auction data indicative to that of real data sets from Trade Me.To assist with development of the Seller Collusion Algorithm, we developed the Multiple Seller Collusive Shill Bidding Simulator. This allows a user to manually or automatically input various scenarios to automate the process of running the algorithm. The simulator undertakes all of the calculations and provides visual output of each stage of the algorithm to ensure that the calculations are being performed accurately.
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| [] |
[
"Holomorphic Frobenius actions for DQ-modules",
"Holomorphic Frobenius actions for DQ-modules"
] | [
"François Petit "
] | [] | [] | Given a complex manifold endowed with a C × -action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the C × -action is free and proper, then the category of Fequivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold. * The author has been fully supported in the frame of the OPEN scheme of the Fonds National de la Recherche (FNR) with the project QUANTMOD O13/570706given by the inclusion. Consider the subsheaf M of p * M defined by M(V ) := {s ∈ p * M(V ) | φ t (s) = s} By definition of p C × * M and M, there is a canonical map M → p C × * M. (6.3) | 10.4171/prims/58-1-5 | [
"https://arxiv.org/pdf/1803.07923v1.pdf"
] | 54,538,511 | 1803.07923 | 6e30086b9e75dac85397cc9d43eed5d6bfa4b375 |
Holomorphic Frobenius actions for DQ-modules
21 Mar 2018
François Petit
Holomorphic Frobenius actions for DQ-modules
21 Mar 2018arXiv:1803.07923v1 [math.AG]
Given a complex manifold endowed with a C × -action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the C × -action is free and proper, then the category of Fequivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold. * The author has been fully supported in the frame of the OPEN scheme of the Fonds National de la Recherche (FNR) with the project QUANTMOD O13/570706given by the inclusion. Consider the subsheaf M of p * M defined by M(V ) := {s ∈ p * M(V ) | φ t (s) = s} By definition of p C × * M and M, there is a canonical map M → p C × * M. (6.3)
Introduction
Relying on the notion of Frobenius action for Deformation quantization modules (DQ-modules) introduced in [KR08], we establish an equivalence between the category of coherent Frobenius equivariant DQ-modules and the category of modules over the sheaf of invariant sections of the DQ-algebra. This result applied to the special case of the canonical DQ-algebra W on the cotangent bundle provides an equivalence between coherent F-equivariant DQ-modules and coherent microdifferential modules on the projective cotangent bundle. This equivalence permits to deduce the codimension three conjecture for formal microdifferential modules [KV14] from the one for DQ-modules on a symplectic manifold [Pet17].
Deformation quantization algebras (DQ-algebras) are non-commutative formal deformations of the structure sheaf of a complex variety. They are used to quantize complex Poisson varieties. In the symplectic case, they are often presented as an extension of the ring of microdifferential operators to arbitrary symplectic manifolds. The ring of formal microdifferential operators E, introduced in [SKK73], is a sheaf on the cotangent bundle of a complex manifold that quantizes it as a homogeneous symplectic manifold. DQ-algebras and in particular the canonical deformation quantization of the cotangent bundle W ignore the homogeneous structure and quantize this bundle as a symplectic manifold. This allows one to produce quantizations of arbitrary complex symplectic manifolds using W (see [PS04]) and in some sense extends formal microdifferential modules to arbitrary symplectic manifolds (Note that it is always possible to quantize complex Poisson varieties as proved in [CH11,Yek05] building upon ideas of Kontsevich [Kon01]).
The ring W is an E algebra. Hence, it is natural to ask if it possible to identify those W-modules which are extension of E-modules. For that purpose, it is necessary to add an extra structure to encode the compatibility with the C × -action on the fibers of the cotangent bundle. This can be achieved by using the notion of holomorphic Frobenius action. They were introduced by Masaki Kashiwara and Raphaël Rouquier in their seminal work [KR08] which introduced an analogue of Beilinson-Bernstein's localization for rational Cherednik algebras. Objects originating from deformation quantization are defined over the ring of formal power series C[[ ]] or its localization with respects to that is the field of formal Laurent series C(( )). This makes these objects too large for many applications since what is often required is an object satisfying certain finiteness assumptions over C. To overcome this difficulty, they introduced, in [KR08], the notion of W-algebra with a holomorphic Frobenius action or F-action for short. Given a complex symplectic manifold X endowed with an action of C × and quantized by a DQ-algebra, a Faction is a compatible action of C × on the DQ-algebra, acting on the deformation parameter with a weight. This allows one to rescale W and the W-modules with respect to . These actions have been subsequently used by several authors in problems arising from the study of the representation theory of quantized conic symplectic singularities, and in particular rational Cherednik algebras (see for instance [BDMN17,BK12,BLPB12,McG12,Los12,Los15])
In this paper, we study the notion of DQ-modules endowed with a F-actions. The definition of a F-action initially provided by Kashiwara and Rouqier is a punctual definition which makes it difficult to use for problems of global nature as questions of analytic extension (i.e. extending a F-action through an analytic subset). Hence, we provide a reformulation in the spirit of G-linearization of coherent sheaves (see [MFK94, Ch.1 §3]). Given a DQ-algebra A X , on a Poisson manifold X, endowed with a F-action, and assuming that this action is free and proper, we establish an equivalence between the category of coherent DQ-modules endowed with a F-action and the category of modules over the sheaf of invariant sections on the quotient space Y = X/C × (Theorem 6.11). Here we have to work on the quotient space since C × is not simply connected and F-equivariant DQ-modules are constant along the orbits. Our result generalizes the first example of [KR08,§2.3.3] (provided without a proof) which states an equivalence of categories between good W-modules and good micro-differential modules. We extend this example to DQmodules over arbitrary Poisson manifold and relax the finiteness conditions by only requiring the DQ-modules to be coherent. To obtain this equivalence of categories, we first prove that a locally finitely generated A X -module endowed with a F-action is locally finitely generated by locally invariant sections (Theorem 5.5). This implies that if M is coherent, it locally has an equivariant presentation of length one (Corollary 5.8). We prove that the invariant sections functor and the equivariant extension functor form an adjoint pair (Proposition 6.2) and establish the coherence of the sheaf of invariant sections (Theorem 6.9). Then we can prove the equivalence announced earlier (Theorem 6.11). As an example, we construct the weight one F-action on the canonical deformation quantization W of the cotangent bundle and obtain as a corollary of Theorem 6.11 an equivalence between coherent W-modules and coherent formal microdifferential modules on the projective cotangent bundle (see Proposition 6.16 for a precise statement). Finally, we use this result to deduce the codimension three conjecture for formal microdifferential modules initially proved by Kashiwara and Vilonen (in the formal as well as in the analytic case) in [KV14] from its DQ-module version proved in [Pet17]. For that purpose, we have to extend F-action through analytic subsets, which is one of the reason, we defined F-actions in a non-punctual manner.
Preliminaries on DQ-modules
We write C for the ring of formal power series with complex coefficients in and C ,loc for the field of formal Laurent series. Let (X, O X ) be a complex manifold. We define the sheaf of C -algebras
O X := lim ← − n∈N O X ⊗ C (C / n C ). Definition 2.1. A star-product denoted ⋆ on O X is a C -bilinear associative multiplication law satisfying f ⋆ g = i≥0 P i (f, g) i for every f, g ∈ O X ,
where the P i are holomorphic bi-differential operators such that for every f, g ∈ O X , P 0 (f, g) = f g and P i (1, f ) = P i (f, 1) = 0 for i > 0. The pair (O X , ⋆) is called a star-algebra.
Definition 2.2. A DQ-algebra A X on X is a C X -algebra locally isomorphic to a star-algebra as a C X -algebra.
Notations 2.3. (i) If A X is a DQ-algebra, we set A loc X := C ,loc ⊗ C A X ,
(ii) if X and Y are two complex manifolds endowed with DQ-algebras A X and
A Y then X × Y is canonically equiped with a DQ-algebra A X×Y := A X ⊠A Y (see [KS12, §2.3]).
There is a canonical morphism of C -algebras
p ♯ 2 : p −1 2 A X → A X ⊠ A Y → A X×Y
and this morphism is flat ([KS12, lemma 2.3.2]).
(iii) We denote by Mod(A X ) the Grothendieck category of left A X -modules, by Mod coh (A X ) its full abelian subcategory whose objects consist of coherent A X -modules. We use similar notation for the left A loc X -modules.
There is a unique isomorphism A X / A X ∼ −→ O X of C X -algebra. We denote by σ 0 : A X ։ O X the epimorphism of C X -algebras defined as the following composition
A X → A X / A X ∼ −→ O X .
These data induce a Poisson bracket {·, ·} on O X defined by:
for every a, b ∈ A X , {σ 0 (a), σ 0 (b)} = σ 0 ( −1 (ab − ba)). Lemma 2.4. Let (O X , ⋆) be a star algebra and v : O X → O X be a C-linear derivation of (O X , ⋆) such that there exists v 0 ∈ Der(O X ) such that for every u ∈ O X , σ 0 • v(u) = v 0 • σ 0 (u) and v( ) = m . Then, there exists a unique sequence (v k ) k≥0 of differential operators such that for any f ∈ O X , v(f ) = i≥0 i v i (f ).
In particular, for every
u = i i u i ∈ O X , v(u) = i k i+k v k (u i ) + m i i u i (2.1) = n n i+k=n v k (u i ) + m n u n . (2.2)
Proof. This proof is an adaptation of the proof of [
f ∈ O X v(f ) = i i v i (f ).
By assumption v 0 is a differential operator. We will prove by induction that the v k are differential operators. Assume that this is true for k < l with l ∈ N. Let (P n ) n∈N be the sequence of bidifferential operators associated with the star products ⋆. By assumption v is continuous for the -adic topology, thus for every
f , g ∈ O X , v(f ⋆ g) = j≥0 v( j P j (f, g)) = n≥0 n i+j=n v i (P j (f, g)) + mn P n (f, g) and f ⋆ v(g) + v(f ) ⋆ g = n≥0 n j+k=n (P k (f, v j (g)) + P k (v j (f ), g)) .
Since
v(f ⋆ g) = f ⋆ v(g) + v(f ) ⋆ g, we obtain i+j=n v i (P j (f, g)) + mn P n (f, g) = j+k=n (P k (f, v j (g)) + P k (v i (f ), g)) .
Using the induction hypothesis, we deduce from the above expressions that
v l (f g) + Q l (f, g) = f v l (g) + v l (f )g + R l (f, g)
where Q l and R l are bidifferential operators. This implies that
[v l , g](f ) = v l (f g) − g v l (f ) = f v l (g) − Q l (f, g) + R l (f, g).
Since Q l (·, g) and R l (·, g) are differential operators, it follows from [KS12, Lemma 2.2.4] that v l is a differential operator.
The canonical deformation quantization of the cotangent bundle
Let M be a complex manifold. The cotangent bundle of M , X := T * M is equipped with the sheaf E X of formal microdifferential operators. This is a filtered, conic sheaf of C-algebras. We denote by E X (0) the subsheaf of E X formed by the operators of order m ≤ 0. These sheaves were introduced in [SKK73]. The reader can consult [Sch85] for an introduction to the theory of microdifferential modules. On X, there is DQ-algebra W X (0) which was constructed in [PS04]. Here, we review their construction.
Let C be the complex line endowed with the coordinate t and denote by (t; τ ) the associated symplectic coordinate on T * C. We set
E T * (M×C),t (0) = {P ∈ E T * M ; [P, ∂ t ] = 0}.
We consider the following open subset of T * (M × C)
T * τ =0 (M × C) = {(x, t; ξ, τ ) ∈ T * (M × C)|τ = 0}
and the morphism
ρ : T * τ =0 (M × C) → T * M, (x, t; ξ, τ ) → (x; ξ/τ ).
We obtain the C X -algebra
W X (0) : = ρ * ( E T * (M×C),t (0)| T * τ =0 (M×C) ) (2.3)
where acts as τ −1 . A section P of W X (0) can be written in a local symplectic coordinate system (x 1 , . . . , x n , u 1 , . . . , u n ) as
P = j≤0 f j (x, u i )τ j , f j ∈ O X , j ∈ Z. Setting = τ −1 , we obtain P = k≥0 f k (x, u i ) k , f k ∈ O X , k ∈ N.
We write W X for the localization of W X (0) with respect to the parameter . There is the following commutative diagram of morphisms of algebras.
E X ι / / W X E X (0) ? O O / / W X (0) ? O O
where the algebra map ι : E X → W X is given in a local symplectic coordinate system (x 1 , . . . , x n , u 1 , . . . , u n ) by
x i → x i , ∂ xi → −1 u i .
Section depending on a complex parameter
Let X be a complex manifold endowed with a DQ-algebra A X . We consider the DQ-algebras A C×X = O C ⊠A X on C × X. We denote by t the coordinate on C, by p 2 : C × X → X the projection on X and by p 1 the projection on C. Note that A C×X is a left D C -modules and in particular a left O C -module. Let t ∈ C, denote by m t the maximal ideal of O C,t and consider the morphism
i t : X → C × X, x → (t, x).
Then, we have an evaluation morphism
ev t : i −1 t A C×X → i −1 t A C×X /m t (i −1 t A C×X ) ≃ A X u → u(t). and ev t : i −1 t A loc C×X → i −1 t A loc C×X )/m t (i −1 t A loc C×X ) ≃ A loc X u → u(t).
Notations 3.1. (i) Let (f, f ♯ ) : (X, R X ) → (Y, R Y ) be a morphism of ringed spaces. As usual, we denote by f * the functor
f * : Mod(R Y ) → Mod(R X ), M → f * M : = R X ⊗ f −1 RY f −1 M.
(ii) In order to keep the number of notations to a bearable level, we will write in-
distinctly p * 2 M for A C×X ⊗ p −1 2 AX p −1 2 M and for A loc C×X ⊗ p −1 2 A loc X p −1 2 M depend-
ing of whether M is considered as an A X -module or an A loc X -module.
Definition 3.2. Let M be an A X -module (resp. a A loc X -module) and set N = p * 2 M and consider s ∈ N . The module N is a D C -module. The derivative with respect to t of a section s in N is the section ∂ t s. It is denoted s ′ and called the derivative of s. Definition 3.3. Let U be an open subset of C and let M be a coherent A Xmodule (resp. A loc X -module). Let (s(t)) t∈U be a family of section of M. We say that (s(t)) t∈U depends holomorphically on t, if locally there exists a section s ∈ p * 2 M such that ev t (s) = s(t).
Proposition 3.4. Let X be a complex manifold and F be a coherent O X -module on X and U an open subset of C × X, u ∈ p * 2 F (U ) such that for every t ∈ p 2 (U ), u(t) = 0. Then u = 0.
Proof. This question is local. So, we can assume that we are working in the vicinity of a point (t 0 , x) ∈ C × X. We identify the local ring (O X,x , m x ) with a subring of the local ring (O C×X,(t0,x) , m (t0,x) ) via the morphism of locally ringed spaces induced by the projection p 2 : C × X → X . We denote by r (t0,x) the ideal of O C×X,(t0,x) generated by m x . For every q ∈ N, we have
(p * 2 F ) (t0,x) /r q (t0,x) (p * 2 F ) (t0,x) ≃ O C,t0 ⊗ F x /m q x F x . Writing u t0 (x) for the image of u in (p * 2 F ) (t0,x) /r q (t0,x) (p * 2 F ) (t0,x) and choosing an isomorphism F x /m q x F x ≃ C r , we can identify u t0 (x) with a vector (f 1 , . . . , f r ) where the f i ∈ O C,t0 . It follows from the assumption that there exists a neigh- bourhood V of t 0 such that for every t ∈ V, f i (t) = 0. This implies that u t0 (x) = 0 that is u (t0,x) ∈ r q (x,t0) (p * 2 F ) (x,t0) . As r (x,t0) ⊂ m (x,t0) and (p * 2 F ) (t0,x)
is a finitely generated O C×X,(t0,x) -module, it follows from the Krull intersection lemma that u (t0,x) = 0. Let u 0 be the image of u via the map N → N / N . It follows from the assumptions that for every t ∈ p 1 (U ), u 0 (t) = 0 and form the isomorphism (3.1) that u 0 ∈ p * 2 (M/ M). Then by Proposition 3.4, u 0 = 0. That is u ∈ N . Let us show by recursion that u ∈ n≥0 n N . We just proved that u ∈ N . Assume that u ∈ n N and denote by u n the image of u via the map N → n N / n+1 N . By the isomorphism (3.1) we identify u n with a section of the coherent O C×X -module p * 2 ( n M/ n+1 M) such that for every t, u n (t) = 0. Thus by Proposition 3.4, u n = 0 that is u ∈ n+1 N . It follows that u ∈ n≥0 n N and
N = p * 2 M. Let U an open subset of C × X and u ∈ N (U ) such that for every t ∈ p 1 (U ), u(t) = 0. Then u = 0. Proof. Let (t, x) ∈ C × X. There exists an open neighbourhood V × U ⊂ C × X of (t, x) and finitely many u i ∈ M| U such that M| U = i A loc X u i . We consider the A U -module M ′ = i A X u i . It is a finitely generated A U -submodule of the coherent A loc U -module M. Thus, M ′ is coherent.
Shrinking V × U if necessary and multiplying u by n with n ∈ N sufficiently big, we can assume that n u ∈ A V ×U ⊗ AU M ′ . The section n u satisfies the hypothesis of the Proposition 3.5. It follows that n u = 0. But, the action of on N is invertible. It follows that u = 0.
Corollary 3.7. Let M be a coherent A X -module (resp. A loc X -module) and set N = p * 2 M. Let U an open subset of C × X and u ∈ N (U ) such that for every t ∈ p 1 (U ), u ′ (t) = 0. Then u ∈ p −1 2 M.
Proof. Since M is coherent, locally it has a presentation
0 → I → A m X → M → 0.
Since A C×X is flat over A X , the module N has the following presentation
0 → A C×X I → A m C×X → N → 0. (3.2) Let (t 0 , x 0 ) ∈ C × X. There exists an open neighbourhood V of (t 0 , x 0 ) and a section s = n i=1 a i e i ∈ A m C×X | U such that its image in N is u.
By hypothesis u ′ (t) = 0, it follows from the Proposition 3.5 (resp. the Corollary 3.6) that u ′ = 0 which implies that we can write
s ′ = j b j v j with b j ∈ A C×X and v j ∈ I. Let c j be a primitive of b j in a neighbourhood of t 0 and set w = j c j ⊗ v j . Thus (s − w) ′ = 0 in A m C×X which implies that s − w ∈ p −1 2 A m X .
Finally since s − w and s have the same image in N , it follows that u does not depend on t i.e u ∈ p −1 2 M.
Holomorphic Frobenius actions
In this section, we precise certain aspects of the definition of a F-action on a DQalgebra or on a DQ-module. This notion was introduced in [KR08, Definition 2.2 and Definition 2.4]. Let (X, {· , ·}) be a complex Poisson manifold. We assume that it comes equipped with a torus action,
C × → Aut(X), t → µ t such that µ * t {f, g} = t −m {µ * t f, µ * t g} with m ∈ Z * .
Notations 4.1.
• We denote by σ :
C × × C × → C × the group law of C × ,
• µ : C × × X → X the action of C × on X.
• µ :
C × × X → C × × X, (t, x) → (t, µ(t, x)) • for t ∈ C × , the morphism i t : X → C × × X, x → (t, x). • We write µ t (resp. µ t ) for the composition µ • i t (resp. µ • i t ).
• Consider the product of manifolds C × × X. We denote by p i the i-th projection.
• Consider the product of manifolds C × × C × × X. We denote by q i the i-th projection, and by q ij the (i, j)-th projection (e.g., q 13 is the projection from
C × × C × × X to C × × X, (t 1 , t 2 , x 3 ) → (t 1 , x 3 )).
• Recall that in all this paper, A X is DQ-algebra and we write
A C × ×X for the DQ-algebra O C × ⊠A X . Lemma 4.2. Let θ : µ −1 A C × ×X → A C × ×X be a morphism of sheaves of p −1 1 O C × - algebras such that the adjoint morphism ψ : A C × ×X → µ * A C × ×X is a continuous morphism of Fréchet C-algberas. Then the dashed arrow in the below diagram is filled by a unique morphism λ of q −1 12 O C × ×C × -algebras. If θ is an isomorphism then λ also. (id C × × µ) −1 (O C × ⊠ A C × ×X ) id × θ / / _ O C × ⊠ A C × ×X _ (id C × × µ) −1 A C × ×C × ×X λ / / ❴ ❴ ❴ ❴ ❴ A C × ×C × ×X
Proof. By adjunction, it is equivalent to show that the dashed arrow in the below diagram is filled by a unique map of q −1 12 O C × ×C × -algberas.
(O C × ⊠ A C × ×X ) id × θ / / _ (id C ×× µ) * (O C × ⊠ A C × ×X ) _ A C × ×C × ×X / / ❴ ❴ ❴ ❴ ❴ (id C ×× µ) * A C × ×C × ×X Denoting by p ⊠ the external product of presheaves, there is a morphism id p ⊠ θ : O C × p ⊠ A C × ×X → O C × p ⊠ µ * A C × ×X (4.1)
DQ-algebras, the sheaf O C × as well as µ * A C × ×X are sheaves of nuclear Fréchet C-algebras. Moreover, there exists a countable basis B of open set of C × × C × × X of the form U i × V j such that A C × X | Vj is isomorphic to a star-algebra. Evaluating the morphism (4.1), on the U i × V j ∈ B, we get the continuous morphism of topological C-algberas id Ui p ⊠ θ Vj (As the spaces we consider are nuclear, the choice of a topology on the tensor products does not matter. For instance, we endow all the tensor product of nuclear spaces with the projective tensor product topology).
(id ⊗ θ) Ui×Vj : O C × (U i ) ⊗ π A C × X (V j ) → O C × (U i ) ⊗ π µ * A C × ×X (V j )
By definition the morphisms id Ui p ⊠ θ Vj are compatible with restrictions and applying the completion functor to the above morphisms, we obtain the following diagram
O C × (U i ) ⊗ π A C × X (V j ) id ⊗ θ / / O C × (U i ) ⊗ π µ * A C × ×X (V j ) O C × (U i ) ⊗ π A C × ×X (V j ) id ⊗ θ / / O C × (U i ) ⊗ π µ * A C × ×X (V j ).
We have obtained a family of morphisms of Fréchet algebras {id ⊗ θ} Ui×Vj ∈B . We describe the completion of the topological vector spaces
O C × (U i ) ⊗ π A C × X (V j ) O C × (U i ) ⊗ π µ * A C × ×X (V j ).
Observe that, on V j , there is an isomorphism of Fréchet algebra
A C × ×X (V j ) ≃ O C × ×X (V j ).
Hence, we obtain a continuous inclusion with dense image
O C × (U i ) ⊗ π O C × X (V j ) ֒→ O C × (U i ) ⊗ π O C × ×X (V j ).
Applying the completion functor and using the fact that it commutes with products, we obtain the following isomorphisms algebras
O C × (U i ) ⊗ π O C × X (V j ) ∼ → O C × (U i ) ⊗ π O C × ×X (V j ) ≃ A C × ×C × ×X (U i × V j ).
Similarly, we have that
O C × (U i ) ⊗ π µ * A C × ×X (V j ) ≃ (id C × × µ) * A C × ×C × ×X (U i × V j ).
Hence, we have obtained a family of morphism of C-algebras
{A C × ×C × ×X (U i × V j ) → (id × µ) * A C × ×C × ×X (U i × V j )} Ui×Vj ∈B which extends to a unique morphism of sheaves on C × × C × × X, λ : A C × ×C × ×X → (id × µ) * A C × ×C × ×X .
By [KS12, Lemma 2.2.9], there is a canonical morphism
q ♯ 23 : q −1 23 A C × ×X → A C × ×C × ×X .
We obtain the morphism λ as the composition
λ : (id ×µ) −1 A C × ×X (id × µ) −1 q ♯ 23 −→ (id × µ) −1 A C × ×C × ×X λ → A C × ×C × ×X . (4.2)
We introduce the functor
Ev t : Mod(p −1 1 O C × ) → Mod(C X ) M → a −1 X (O C × ,t /m t ) ⊗ a −1 X O C × ,t i −1 t M ≃ i −1 t M/a −1 X m t i −1 t M.
In particular, Ev
t (A C × ×X ) ≃ A X and Ev t ( µ −1 A C × ×X ) ≃ µ −1 t A X .
The following definition should be compared with [KR08, Defintion 2.2] and with [BLPB12,p.15].
Definition 4.3. A F-action with exponent m on A X is the data of an isomorphism of p −1 1 O C × -algebras θ : µ −1 A C × ×X → A C × ×X such that (a) the morphism θ t := Ev t ( θ) satisifies θ 1 = id, (b) for every t ∈ C × , θ t ( n ) = t mn n , (c) the adjoint morphism of θ, ψ : A C × ×X → µ * A C × ×X is a continuous morphism of Fréchet C-algberas, (d) setting θ : µ −1 A X µ −1 p ♯ 2 −→ µ −1 A C × ×X θ → A C × ×X the below diagram commutes, (id C × ×µ) −1 µ −1 A X (id C × ×µ) −1 θ / / (id C × ×µ) −1 A C × ×X λ A C × ×C × ×X (σ × id X ) −1 µ −1 A X (σ×idX ) −1 θ / / (σ × id X ) −1 A C × ×X O O
where λ is provided by Lemma (4.2).
Definition 4.4. A F-action on
A loc X is the localization with respect to of a F-action on A X . Remark 4.5. It would be possible to define directly the notion of F-action on A loc X but the definition would be slightly more involved. Moreover, any such action would be induced by a F-action on A X . This justify the choice of our previous definition.
The pair
(i t , ev t ) : (X, A loc X ) → (C × × X, A loc C × ×X ) (4.3)
is a morphism of ringed spaces. The F-action on A X induces another morphism of ringed spaces (µ, θ) :
(C × × X, A loc C × ×X ) → (X, A loc X ). (4.4)
Remark 4.6. A word of caution about Morphism (4.4). This morphism is a morphism of C-ringed spaces but not of C -ringed spaces.
The morphism λ : (id ×µ) −1 A loc C × ×X → A loc C × ×C × ×X (4.5)
provided by Lemma 4.2 and the data of the F-action θ on A loc X allows to define a morphism of ringed space
(id ×µ, λ) : (C × × C × × X, A loc C × ×C × ×X ) → (C × × X, A loc C × ×X ).
The morphism of sheaves
σ ♯ : σ −1 O C × → O C × ×C × induces a map α : (σ × id X ) −1 A C × ×X σ ♯ ⊗ id −→ A C × ×C × ×X
which provides a morphism of ringed spaces
(σ × id X , α) : (C × × C × × X, A C × ×C × ×X ) → (C × × X, A C × ×X ).
Lemma 4.7. The morphisms of sheaves of rings θ, λ and α are flat.
Proof. The proof for θ and λ are similar. Hence, we only provide the proof for θ. Since θ is an isomorphism it is flat and
µ −1 p ♯ 2 is flat by [KS12, Lemma 2.3.2]. Thus, θ = θ • µ −1 p ♯ 2 is flat.
We now prove the flatness of α. Since σ is a submersion, for every (t 1 , t 2 ) ∈ C × × C × , there exist an open neighbourhood W of (t 1 , t 2 ) and a biholomorphism g : U × V → W such that p 1 (z 1 , z 2 ) = σ • g(z 1 , z 2 ) = z 1 . Since flatness is a local question, we can restrict α to an open neighbourhood of the form W × W ′ with W ′ an open subset of X. Hence, we obtain the following commutative diagram
(p 1 × id W ′ ) −1 A U×X (g×id W ′ ) −1 α| W ×W ′ / / ≀ A U×V ×W ′ ≀ (σ| W × id W ′ ) −1 A C × ×X α| W ×W ′ / / A C × ×C × ×X | W ×W ′ where the top morphism (g × id W ′ ) −1 α| W ×W ′ = q ♯ 13 is flat by [KS12,Definition 4.8. A F-action on a A loc X -module M is the data of an isomorphism of A loc C × ×X -modules φ : µ * M ∼ → p * 2 M (4.6) such that the diagram (id C × ×µ) * µ * M (id C × ×µ) * φ / / (id C × ×µ) * p * 2 M ∼ / / q * 23 µ * M q * 23 φ / / q * 23 p * 2 M ≀ q * 3 M (σ × id X ) * µ * M (σ×idX ) * φ / / (σ × id X ) * p * 2 M. ≀ O O (4.7)
commutes.
Following [KR08], we denote by Mod F (A loc X ) the category of (A loc X , θ)-modules whose morphisms are the morphisms of A loc X -modules compatible with the action of C × . This category is a C-linear abelian category. We write Mod F,coh (A loc X ) for the full subcategory of Mod F (A loc X ) the objects of which are coherent modules in Mod(A loc X ). Let M be an A loc X -module endowed with a F-action φ : µ * M → p * 2 M and t ∈ C × . There is the following commutative diagram defining the morphism φ t
i * t µ * M ≀ i * t φ / / i * t p * 2 M ≀ µ −1 t M φt / / M
where the vertical map are isomorphism of C X -modules. Hence, we have obtained a map of C X -module φ t : µ −1 t M → M such that (a) φ t depends holomorphically of t,
(b) φ tt ′ = φ t ′ • µ −1 t ′ φ t for t, t ′ ∈ C × , (c) φ t (am) = θ t (a)φ t (m) for a ∈ A loc X and m ∈ M.
Remark 4.9. (a) We will usually write φ tt
′ = φ t ′ •φ t instead of φ tt ′ = φ t ′ •µ −1 t ′ φ t . (b)
This implies that a F-action in our sense give rise to a F-action in the sense of [KR08]. In practice, the examples of F-action in the sense of [KR08] are also F-action in our sense.
Let M be an A loc X -module endowed with a F-action φ : µ * M → p * 2 M. The F-action provides a derivation of M. Indeed, notice that p * 2 M has a structure of left p −1 2 D C × -module. Hence, we have
µ * M φ / / p * 2 M ∂t / / p * 2 M (4.8) Let t 0 ∈ C × . Consider the morphism i t0 : X → X × C × , x → (x, t 0 ).
Applying the functor i −1 t0 to the morphism (4.8), we obtain
dφ t (·) dt | t=t0 : µ −1 t0 M / / i −1 t0 µ * M i −1 t 0 φ / / i −1 t0 p * 2 M i −1 t 0 ∂t / / i −1 t0 p * 2 M evt 0 / / M.
In particular, when t 0 = 1, we get
v : M / / i −1 1 µ * M i −1 1 φ / / i −1 1 p * 2 M i −1 1 ∂t / / i −1 1 p * 2 M ev1 / / M. In other words, v : M → M s → dφ t (s) dt | t=1 .
The morphism v is a C-linear derivation of the module M.
Invariant sections
Generalities
We start by defining the notion of locally invariant and invariant sections.
U ′ × V ⊂ C × × X of (1, x ′ ) such that for every (t, x) ∈ V × U ′ , µ(t, x) ∈ U and φ t (s µ(t,x) ) = s x .
(ii) The section s is locally invariant on U , if it is locally invariant at every
x ′ ∈ U . (iii) Assume that U ⊂ X is stable by the action of C × . A section s ∈ M(U ) is invariant if for every t ∈ C × , φ t (s) = s. Lemma 5.2. Let M ∈ Mod F (A loc X ), U ⊂ X an open subset and s ∈ M(U ) such that v(s) = 0. Then, for every x ′ ∈ U there is a neighbourhood V × U ′ of (1, x ′ ) ∈ C × × U such that for every t ′ ∈ V 1. dφt(s| t ′ U ′ ) dt | t=t ′ = 0. 2. φ t ′ (s| t ′ U ′ ) = s| U ′ Proof. (i) Since µ is continuous, there exists a neighbourhhod V ×U ′ ⊂ U of (1, x ′ ) such that µ(V × U ′ ) ⊂ U .
Hence for every t ′ ∈ V , we have the following equalities.
dφ t (s| t ′ U ′ ) dt | t=t ′ = 1 t ′ dφ tt ′ (s| t ′ U ′ ) dt | t=1 = 1 t ′ φ t ′ dφ t (s| t ′ U ′ ) dt | t=1 = 1 t ′ φ t ′ (v(s| t ′ U ′ )) = 0.
(ii) Shrinking V and U ′ if necessary Corollary 3.7 implies that locally for every Proof. The question is local and M is coherent. Hence we can assume that M is finitely generated. Thus, there exists s 1 , . . . , s n ∈ M such that M = n i=1 A loc X s i . This implies that there exists l ∈ N such for every 1 ≤ i ≤ n, there exists a ij ∈ −l A C × ×X such that
t ′ ∈ V φ t ′ (s| t ′ U ′ ) = s| U ′ .φ t (s i ) = n j=1 a ij (t)s j . Setting t = e u , this implies that v k (s) = n i=j d k du k a ij (e u ) u=0 s i . Setting N = n i=1 A X s i , it follows that for every k ∈ N, v k (s) ∈ −l N . We consider the submodules M 0 = k, i A X v k (s i ) M 0,≤p = 1≤i≤n 0≤k≤p A X v k (s i ) of −l N . It is clear that M 0 is stable by v, that M 0,≤p is a coherent A X -module,
that for every p ∈ N, M 0,≤p ⊂ M 0,≤p+1 and M 0 = p≥0 M 0,≤p . Since A X is a noetherian sheaf of algebras and −l N is a coherent A X -module, the sequence (M 0,≤p ) p∈N is locally stationary. Thus, there exists a covering (U j ) j∈J of X such that for every j ∈ J there exists p j ∈ N such that M 0 | Uj = M 0,≤pj . Hence, M 0 is coherent.
Theorem 5.5. Assume that the action µ is free and let M ∈ Mod F,coh (A loc X ). Then M is locally finitely generated by locally invariant sections.
Notations 5.6. We introduce the following notation. Let δ = (δ 1 , . . . , δ n ) (resp. η = (η 1 , . . . , η n )) with δ i > 0 for 1 ≤ i ≤ n (resp. η i > 0 for 1 ≤ i ≤ n). We denote by R(x, δ, η) the subset of C n the elements of which are the z = (z 1 , . . . , z n ) ∈ C n such that for every 1 ≤ i ≤ n
ℜ(x i ) − δ i < ℜ(z i ) < ℜ(x i ) + δ i and ℑ(x i ) − η i < ℑ(z i ) < ℑ(x i ) + η i .
We call such a subset of C n a rectangle centered in x.
Proof of Theorem 5.5. By lemma 5.4, M has a coherent A X -lattice M 0 stable by v. Let us show that this lattice is generated by locally invariant sections. This problem is local. So, it is sufficient to work in an open neighbourhood U of a point x ∈ X. Since M 0 is coherent, we can assume it is finitely generated on U by a family (e i )) 1≤i≤l . For every 1 ≤ i ≤ l there exists a ij ∈ A U with 1 ≤ j ≤ l such that It follows that v(e) = Ae. If B ∈ GL l (A U ) is such that v(Be) = 0 then, Be will provide a generating family of M 0 , formed of locally invariant sections. We have the following equalities.
v(Be) = v(B)e + Bv(e)
= v(B)e + BAe. Shrinking U if necessary, we can further assume that A U is isomorphic to a star-algebra (O U , ⋆) the star-product of which is given by
f ⋆ g = i≥0 P i (f, g)
where the P i are bidifferential operators.
Writting A = k≥0 A k k and B = j≥0 B j j with A k and B j in M l (O U ), we obtain BA = ( j≥0 j B j ) ⋆ ( k≥0 k A k ) = n≥0 n ( k+j+m=n P m (B j , A k )) and v(B) = n≥0 n i+k=n v k (B i ) + m n B n = n≥0 n i+k=n k =0 v k (B i ) + v 0 (B n ) + m n B n .
Thus, Equation (5.1) is equivalent to the recursive system of equations Notice that C n depends only of the B j with j < n.
∀n ∈ N, v 0 (B n ) + B n (mn id +A 0 ) + i+k=n i =n v k (B i ) +
Since the action of C × on X is free, v 0 does not vanish and we can find a local coordinate system (z 1 , . . . , z n ) on an open neighborhood of x (that we still denote U ) such that in this coordinate system If (z; u) is the coordinnate system on T * U associated to the coordinate system (z) = (z 1 , . . . , z n ) on U , it follows from [SK75] that the characteristic variety of the system (5.4) is {u 1 = 0}.
Since U is open, there exists a rectangle R( R(x, δ, η). Consider a function g ∈ O U (R(x, δ, η)). We have the following Cauchy problem
x, δ, η) ⊂ U . Set Y = {z ∈ U |z 1 = p 1 } and Y ′ = Y ∩ ∂f ∂z 1 + f (mn id +A 0 ) = g f | Y ′ = I l .
(5.5)
The hypersurface Y ′ is non-characteristic. . This proves that for every n ∈ N, the equation (5.2) has a solution B n with B n | Y = I l and defined on R(x, δ, η). This implies in particular that B 0 (p) = I l is invertible. Hence, B 0 is invertible in a neighbourhood V of x. This ensures that B = j≥0 j B j is invertible on V which proves the claim.
The case of free and proper actions
We now assume that the action of C × on X is free and proper. We set Y = X/C × and denote by p : X → Y the canonical projection. The morphism p is a C × -principal bundle. We say that an open subset V of X is equivariant if V = p −1 p(V ).
Lemma 5.7. Let M ∈ Mod F (A loc X ). (i) If s ∈ M is a locally invariant section, then s is locally the restriction of a globally invariant section. (ii) If M is locally finitely generated by locally invariant sections, then M is locally finitely generated by invariant sections.
Proof. (i) Let U be an open subset of X and assume that s ∈ M(U ). Let x ∈ U . Since s is locally invariant on U there exist a neigbourhoud U ′ ⊂ U of x and a neighbourhood V of 1 ∈ C × such that for every t ∈ V, x ′ ∈ U ′ , µ(t, x ′ ) ∈ U and φ t (s) = s. (a) Shrinking U ′ and V if necessary we can assume that the pair U ′ and V satisfies the following property. Let x 0 , x 1 ∈ U ′ such that there exist t ∈ C × such that x 1 = µ(t, x 0 ) then t ∈ V .
(b) Let y ∈ W = p −1 (p(U ′ )). There exists (t 0 , x 0 ) ∈ C × × U ′ such that y = µ(t 0 , x 0 ). We sets
y = φ −1 t0 (s x0 ).
Let us show that the sections is well define. Assume that there exist (t 1 , (s 1 , . . . , s n ) generating M| U . Keeping the notation and applying the construction of (i) to the family (s 1 , . . . , s n ), we obtain some open sets U ′ and V satisfying the same propreties as in (i) and a family of invariant sections (s 1 , . . . ,s n ) defined on the open subset W = p −1 (p(U ′ )) of X. It remains to prove that (s 1 , . . . ,s n ) is a generating family of M| W . Since W is stable by the action, we can assume for the sake of simplicity that X = W . The problem is now equivalent to check that the morphism of sheaves of A loc X -modules u : (A loc X ) n → M, e i →s i is an epimorphism. There is the following commutative diagram of p −1
x 1 ) ∈ C × × U ′ such that y = µ(t 1 , x 1 ). Then there exists t ∈ V such that x 1 = µ(t, x 0 ). It follows that φ −1 t1 (s x1 ) = φ −1 t0 (φ −1 t (s x1 )) = φ −1 t0 (s x0 ) =s y . (ii) Since M1 O C × -modules. µ * (A loc X ) n µ * u / / θ µ * M φ p * 2 (A loc X ) n p * 2 u / / p * 2 M.
(5.6) Let x 1 ∈ X. Then there exists x 0 ∈ U ′ and t ∈ V such that x 1 = µ(t, x 0 ). We consider the map
i (t,x0) : {pt} → C × × X, pt → (t, x 0 )
and the evalutation map
i ♯ (t,x0) : i −1 (t,x0) p −1 2 O C × ≃ O C × ,t → C.
These two maps allow us to define the morphism of ringed spaces
(i (t,x0) , i ♯ (t,x0) ) : ({pt}, C) → (C × × X, p −1 2 O C × )
Applying the functor i * (t,x0) to the diagram (5.6), we obtain
(A loc X ) n x1 ux 1 / / θ (x 0 ,t) M x1 φ (x 0 ,t) (A loc X ) n x0 ux 0 / / M x0
where the two vertical arrows are isomorphisms and the map u x0 is an epimorphism. This implies that u x1 is an epimorphism which proves the claim.
From DQ-modules to modules over the ring of invariant sections
From now on we assume that the action of C × on X is free and proper.
Equivariant extension and invariant sections functors
We define the sheaf of locally invariant sections of A X as the sheaf on X such that for every open set U
A C × X (U ) = {s ∈ A X (U )|v(s) = 0}
and we also set
A loc, C × X (U ) = {s ∈ A loc X (U )|v(s) = 0}. The sheaf of invariant sections of A X is defined as the subsheaf B Y (0) of p * A X which is given by, for every open set V ⊂ Y , B Y (0)(V ) = {s ∈ p * A X (V )|v(s) = 0} = {s ∈ p * A X (V )|θ t (s) = s}.
This is a sheaf of C-algebra. We define B Y similarly i.e.
B Y (V ) = {s ∈ p * A loc X (V )|v(s) = 0} = {s ∈ p * A loc X (V )|θ t (s) = s}.
By definition of B Y (0) and B Y , there are morphisms of algebras
p −1 B Y (0) → A C × X , (6.1) p −1 B Y → A loc, C × X .
(6.2) Lemma 6.1. The morphisms (6.1) and (6.2) are isomorphims of C-algebras.
Proof. This follows from Lemma 5.7.
We define the functor of locally invariant sections as follows
(·) C × : Mod F (A loc X ) → Mod(A loc,C × X ) M → M C × where M C × is the subsheaf of M such that for every open U ⊂ X M C × (U ) = {s ∈ M(U )|v(s) = 0}
The functor of globally invariant sections is defined by
p C × * : Mod F (A loc X ) → Mod(B Y ) M → p C × * M := p * (M C × ).
Note that by definition of p C × * , there is a natural transformation i : p C × * → p * , such that for (M, ψ) ∈ Mod F (A loc X )
Proposition 6.2. The functors
p * C × : Mod(B Y ) / / Mod F (A loc X ) : p C × * o o form the adjoint pair (p * C × , p C × * ).
Proof. (i) Let N ∈ Mod(B Y ). We start by constructing the unit of the adjunction (p * C × , p C × * ). Consider the unit η ′ : id → p * p * of the adjunction
p * : Mod(B Y ) / / Mod(A loc X ) : p * . o o
The stalk of the map η ′ N in y ∈ Y is given by
N y → lim − → y∈V A loc X (p−1(V )) ⊗ BY,y N y , n → 1 ⊗ n.
Sections of the form 1 ⊗ n are invariant section of p * C × N . Thus, the preceding map factorizes through p C × * p * C × N and we obtain the following commutative diagram.
N η ′ N / / ηN # # ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ p * p * N p C × * p * C × N i p * C × N 9 9 s s s s s s s s s (6.5)
We have obtained a natural transformation η :
id → p C × * p * C × . Let (M, ψ) ∈ Mod F (A loc X )
. Let ε ′ : p * p * → id be the counit of the adjunction
p * : Mod(B Y ) / / Mod(A loc X ) : p * . o o
We define the following natural transfromation
ε = ε ′ • i.
(6.6) By construction the folowing diagram commutes.
p * p * M ε ′ M / / M p * C × p C × * M εM : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ p * iM e e ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ (6.7)
Let x in X. The stalk at x of Morphism (6.6) is given by
A loc X,x ⊗ B Y,p(x) (p C × * M) p(x) → M x , a ⊗ m → am
This implies that the following diagram
µ * p * p C × * M φ µ * εM / / µ * M ψ p * 2 p * p C × * M p * 2 εM / / p * 2 M
where φ is provided by the functor p * C × is commutative. Hence, Morphism (6.6) is equivariant i.e. ε is a morphism of Mod F (A loc X ). In view of diagrams (6.5) and (6.7), the following diagrams are commutative
p * N p * η ′ N / / id & & p * p * p * N ε ′ p * N / / p * N p * C × N p * C × ηN / / p * C × p C × * p * C × N ε p * C × N / / p * i p * C × N O O p * C × N p * M η ′ p * M / / id & & p * p * p * M p * ε ′ M / / p * M p * p * C × p C × * M p * p * iM O O p * εM 8 8 q q q q q q q q q q p C × * M η p C × * M / / ? iM O O η ′ p C × * M 8 8 q q q q q q q q q q p C × * p * C × p C × * M p C × * εM / / i p * C × p C × * M O O p C × * M ? iM O O
which proves that (p * C × , p C × * ) is an adjoint pair.
Coherence of the sheaf of invariant sections
In this subsection, we prove the coherence of B Y . The following result is elementary.
Lemma 6.3. Let x ∈ X. Let v 0 be an holomorphic derivation of O X that does not vanishes at x. Then there exist a neighbourhood U of x such that for every
c ∈ C the map v 0 + c : O X (U ) → O X (U ) , f → v 0 (f ) + cf is surjective.
Lemma 6.4. Let s ∈ A X (X) and assume that there is a covering (U i ) i∈I of X and u i ∈ A X (U i ) such that s| Ui = u i . Then, there exist u ∈ A X (X) such that
s = u Proof. On U i ∩U j , u i | Uij = u j | Uij . As is not a zero divisor, u i | Uij = u j | Uij .
Lemma 6.5. Let x ∈ X. There exists an equivariant neighbourhood V of x and a section s ∈ A X (V ) such that v(s) = 0 and s = u with u ∈ A X (V ) an invertible section.
Proof. Let x ∈ X and m ∈ Z. Since v 0 does not vanish in x, there exists an open neighbourhood U of x and an hypersurface Y in U which is non-characterisitic for v 0 + m on U . Hence the following Cauchy problem
v 0 (f ) + mf = 0 f | Y = 1
has a solution f 0 in an open neighbourhood U ′ of x. Shrinking U ′ if necessary we can further assume that f 0 is invertible on it. We now show that there exists u = i≥0 i u i ∈ A X,x such that
u 0 = f 0 and v( u) = 0.
It follows from equation (2.1) that this is equivalent to show that there exist
u = i≥0 i u i ∈ A X,x such that for every n ∈ N i+k=n v k (u i ) + m (n + 1) u n = 0.
Thus, it remains to show that the recursive system
v 0 (u n ) + m (n + 1) u n = i+k=n k =0 v k (u i ) u 0 = f= φ −1 t0 ( u x0 ) = t −m 0 φ −1 t0 (u x0 ) and t −m 0 φ −1 t0 (u x0 )
is an invertible section. Finally, Lemma 6.4 implies the existence of u.
Assume that there exist a section s ∈ A X (X) such that v(s) = 0 and s = u with u invertible. Consider the exact sequence
0 → s n A X → A X π → A X /s n A X → 0.
(6.8) and apply the functor (·) C × to it. We obtain the left exact sequence
0 → (s n A X ) C × → A C × X → (A X /s n A X ) C × . (6.9)
Lemma 6.6. The left exact sequence (6.9) is exact.
Proof. The proof is similar to the one of Lemma 6.5.
Lemma 6.7. Assume that there exist a section s ∈ A X (X) such that v(s) = 0 and s = u with u invertible. Then
p C × * (A X /s n A X ) ≃ B Y (0)/s n B Y (0).
Proof. Notice that p * s n A X ≃ s n p * A X . Hence, p C × * (s n A X ) ≃ s n B Y (0). Applying p C × * to the sequence (6.8), we obtain the following left exact sequence
0 → s n B Y (0) → B Y (0) p C × * π −→ p C × * (A X /s n A X ). (6.10)
Let us show that the above sequence is exact. Let y ∈ Y , a ∈ p C × * (A X /s n A X ) y . Thus, there exists an open subset V ⊂ Y such that y ∈ V , a ∈ A X /s n A X (U ) where U = p −1 (V ) and a is an invariant section. Choose x ∈ U such that y = p(x). Since the sequence (6.9) is exact, there exist an open set W ⊂ U containing x and a locally invariant section u ∈ A X (W ) such that π(u) = a| W . Hence, there exists an equivariant open subset U ′ and an invariant section b ∈ A X (U ′ ) such that on a neighbourhood W ′ ⊂ W ∩ U ′ of x, b| W ′ = u| W ′ . Moreover, shrinking U ′ if necessary, we can assume that the orbit of W ′ under the action of C × is U ′ . As a and b are invariant sections, it follows that π(b) = a| U ′ . This proves that p * C × π is an epimorphism of sheaves.
The following theorem is a minor variation of [KS12, Theorem 1.2.5 (i)] and the proof is the same. (ii) The sheaf B Y is a Noetherian sheaf of C-algebras.
(iii) The sheaf A C × X is a Noetherian sheaf of C-algebras.
(iv) The sheaf A loc,C × X is a Noetherian sheaf of C-algebras.
Proof. (i) We apply Theorem 6.8. Coherency is a local property. Hence, using Lemma 6.5, we can assume that that there exist there exist a section s ∈ A X (X) such that v(s) = 0 and s = u with u invertible. Then point (i) and (ii) of Theorem 6.8 are immediately satisfied. Since p C × * is a a right adjoint it commutes with limits. Thus,
B Y (0) = p C × * A X = p C × * (lim ← − n A X /s n A X ) ≃ lim ← − n p C × * (A X /s n A X ).
Moreover, Lemma 6.7 implies that
p C × * (A X /s n A X ) ≃ B Y (0)/s n B Y (0). Hence B Y (0) ≃ lim ← − n B Y (0)/s n B Y (0).
This proves that condition (iii) of Theorem 6.8 holds as well as condition (iv) since
B Y (0)/sB Y (0) ≃ O Y . Thus, B Y (0) is Noetherian sheaf of C-algebras.
(ii) We keep the notation of (i) and consider a section s = u as above. Consider the free algebra B Y (0) T and impose the relations
∀a ∈ B Y (0), T · a = ψ u (a) · T where ψ u (a) = u −B Y ≃ B Y (0)[T ; ψ u ]/(T s − 1)B Y (0)[T ; ψ u ] is also Noetherian. (iii) & (iv) We have that A C × X ≃ p −1 B Y (0) (resp. A loc,C × X ≃ p −1 B Y )
. Hence the result follows from (i) (resp. (ii)) and [Kas03, Proposition A.14]. (ii) This follows from the isomorphism
A loc X ≃ A X ⊗ A C × X A loc,C × X .
The equivalence of categories
The aim of this subsection is to prove the following theorem.
Theorem 6.11. The adjoint pair (p * C × , p C × * ) induces a well defined adjunction
p * C × : Mod coh (B Y ) / / Mod F, coh (A loc X ) : p C × * . o o (6.11)
These functors are equivalence of categories inverse to each others.
For that purpose, we have to prove that the adjunction is well defined and that the unit and the counit of this adjunction are isomorphism.
Lemma 6.12. Let M ∈ Mod F,coh (A loc X ). The natural morphism (6.6)
ε : p * C × p C × * M −→ M
is an isomorphism.
Proof. It follows from Corollary 5.8 that the morphism (6.6) is an epimorphism. Let us prove that it is a monomorphism. Let x ∈ X and let m =
n i=1 w i ⊗m i ∈ A loc X,x ⊗ A loc,C × X,x p C × * (M) p(x)
. We can assume that m 1 , . . . , m n are invariant sections
of M defined on an equivariant open subset V of X containing x. Consider the map φ : (A loc X ) n | V → M| V , φ(e j ) = m j , 1 ≤ j ≤ n where (e i ) 1≤i≤n
is the canonical basis of (A loc X ) n . Then we get the following left exact sequence
0 → ker φ → (A loc X ) n | V φ → M| V .
The module ker φ belongs to Mod F, coh (A loc X | V ). It is locally finitely generated by invariant sections. Hence there exists an equivariant open set V ′ containing x and invariant sections s 1 , . . . , s q where s i = (s i1 , . . . , s in ) with s ij ∈ A loc,C × X (V ′ ) and generating ker φ| V ′ .
Assume that ε(m) = n j=1 w j m j = 0. Thus, w = (w 1 , . . . , w n ) ∈ ker φ x and we have
w = q i=1 α i s i with α i ∈ A loc X,x . Then n j=1 w j ⊗ m j = n j=1 ( q i=1 α i s ij ) ⊗ m j = q i=1 (α i ⊗ ( n j=1 s ij m j ) =0 ) = 0.
This proves that the map (6.6) is a monomorphism. and apply the functor p * C × p C × * to the above exact short exact sequence. Since p * C × p C × * is isomorphic to the identity functor on Mod F, coh (A loc X ), we obtain the short exact sequence
0 → p * C × p C × * M ′ → p * C × p C × * M → p * C × p C × * M ′′ → 0.
The ring A loc X is faithfully flat over A loc,C × X , this implies that the sequence
0 → p −1 p C × * M ′ → p −1 p C × * M → p −1 p C × * M ′′ → 0 (6.12)
is exact. Moreover, p : X → Y is surjective. Hence taking the stalks of the short exact sequence (6.12) in every x ∈ X, we find that for every y ∈ Y the sequence
0 → (p C × * M ′ ) y → (p C × * M) y → (p C × * M ′′ ) y → 0
is exact. This proves that p C × * is exact.
Lemma 6.14. The functors p * C × and p C × * preserve coherent modules.
Proof. (i) Since A loc X is coherent, a A loc X -modules is coherent if and only if it admits a presentation of length one by finitely generated free modules. This implies that p * C × preserves coherent modules. (ii) Let M be a coherent A loc X -module endowed with a F-action. It follows from Corollary 5.8 that there exists an equivariant open subset V of X such that M| V has a presentation of length one by free modules of finite rank in Mod F, coh (A loc X ), i.e (A loc X | V ) n → (A loc X | V ) m → M| V → 0. Applying the exact functor p C × * to the above sequence, we obtain the right exact sequence
(B Y | p(V ) ) n → (B Y | p(V ) ) m → (p C × * M)| p(V ) → 0.
As B Y is coherent, this implies that p C × * M is a coherent B Y -module.
We are now ready to prove the main result of this section.
Proof of Theorem 6.11. It follows from Proposition 6.2 and Lemma 6.14 that the adjunction (6.11) is well defined. Because of Lemma 6.12, it only remains to show that for every N ∈ Mod coh (B Y )
η N : N → p C × * p * C × N (6.13)
is an isomorphism. The B Y -module N is coherent. Hence, there is a covering (V i ) i∈I of Y such that for each i ∈ I, N | Vi has a free presentations of length one. It follows that the morphism (6.13) is an isomorphism since p * C × and p C × * are right exact functors and η BY is an isomorphism.
An example: the case of W X
In this section, we sketch the construction of the canonical weight one F-action on W X . Let M be a complex manifold and we set X = T * M . We denote by W X (0), the standard quantization of the cotangent bundle and W X := C ,loc ⊗ C W X (0) (see for instance [KS12,p.133]). We write E X for the ring of formal microdifferentials operators on X, E X (0) for the subsheaf of order zero microdifferential operators and set A C × ×X := O C × ⊠W X (0).
We endow C with a coordinate t, T * C with the coordinate system (t, τ ). Similarly, we equipped C × with the coordinate r and T * C × with the coordinate system (r, λ). We consider the map γ : T * (C × × C) → T * (C × × C), (r, t; λ, τ ) → (r, t; λ, τ /r). and the isomorphism of sheaves
γ ♯ : E T * τ =0 (C × ×C),t,∂r (0)
Let V be an open subset of C × and (U, x; u) a local symplectic coordinate system of X where U is a conical open subset of X. Then
θ V ×U : µ −1 W C × ×X (0)(V × U ) → W C × ×X (0)(V × U ) i≥0 f i (r, x; u) i → i≥0 f i (r, x; r · u)r i i .
This implies that
θ r : µ −1 r W X (0)(U ) → W X (0)(U ) i≥0 f i (x; u) i → i≥0 f i (x; r · u)r i i . A section s = i≥0 f i (x; u) i in W X (0)(U ) is invariant, if for every r ∈ C × , θ r (s) = s. That is, for every i ≥ 0, f i (x, ru) = r −i f i (x, u).
This implies that s ∈ ι( E X (0)(U )) and in particular that ( W X (0)) C × ≃ E X (0) and W C × X ≃ E X . Hence, applying Theorem 6.11, we obtain the proposition Proposition 6.16. The adjoint pair (p * C × , p C × * ) induces a well defined adjunction
p * C × : Mod coh ( E Y ) / / Mod F, coh ( W X ) : p C × * . o o
These functors are equivalence of categories inverse to each others. 7 The codimension three conjecture for formal micro-differential modules
In this subsection, we deduce the codimension-three conjecture for formal microdifferential modules from the codimension-three conjecture for DQ-modules. Let We denote by d X the dimension of X. Let l be a non-negative integer, from now on, we set X l := (C × ) l × X, similarly Z l := (C × ) l × Z and A X l := O (C × ) l ⊠ W X (0). We will need the following proposition whose proof is similar to the one of [KK81, Theorem 1.2.2].
Proposition 7.1. Let r and l be non-negative integers and M be a coherent A loc X lmodule so that Ext j A loc X l (M, A loc X l ) = 0 for any j > r. Then H j Z l (M) = 0 for any closed analytic subset Z of X and any j < codim Z − r.
Lemma 7.2. Assume X does not intersect the zero section of T * M . Let Λ be a Lagrangian subvariety of X, Z be a closed analytic subset of Λ such that codim Λ Z ≥ 2, j : X \ Z ֒→ X the inclusion and M a holonomic W X -module supported by Λ. Let (f, f ♯ ) : (X l , A X l ) → (X l ′ , A X l ′ ) a morphism of C-ringed space such that f ♯ is flat. Set V = X \ Z and denote by f V the restriction of f to V . Then
(id (C × ) l ×j) * f * V M| V ≃ f * M. Proof. Consider the following exact sequence
0 → H 0 Z l (f * M) → f * M → (id (C × ) l ×j) * (id (C × ) l ×j) −1 f * M → H 1 Z l (f * M) → 0.
Since M is holonomic and f * is exact, it follows that for any j > d X /2
Ext j A loc X l (f * M, A loc X l ) = 0.
Hence, by Proposition 7.1, H j Z l (f * M) = 0 for 0 ≤ j < 2. Then, the above exact sequence implies that
(id (C × ) l ×j) * f * V M| V ≃ (id (C × ) l ×j) * (id (C × ) l ×j) −1 f * M ≃ f * M.
Lemma 7.3. Assume X is a conical open subset of T * M and does not intersect the zero section. Let Λ be a conical Lagrangian subvariety of X, let Z be a closed conical analytic subset of Λ such that codim Λ Z ≥ 2, j : X \ Z ֒→ X the inclusion and M a holonomic W X -module supported in Λ such that M ∈ Mod F ( W X | X\Z ). Then M ∈ Mod F ( W X ).
Proof. Set V = X \ Z; On C × × V , the F-action is given by
φ ′ : µ * V M| V → p * 2,V M| V .
Applying (id C × ×j) * , we get (id C × ×j) * φ : (id C × ×j) * µ * V M| V → (id C × ×j) * p * 2,V M|V. By Lemma 7.2,
(id C × ×j) * µ * V M| V ≃ µ * M, (id C × ×j) * p * 2,V M| V ≃ p * 2 M.
Thus, we obtain a morphism φ : µ * M → p * 2 M. Applying (id (C × ) 2 ×j) * to the below diagram diagram
(id C × ×µ V ) * µ * V M| V (id C × ×µV ) * φ ′ / / (id C × ×µ V ) * p * 2,V M| V ∼ / / q * 12,V µ * V M| V q * 12,V φ / / q * 12,V p * 2,V M|V ≀ q * 3,V M (σ × id V ) * µ * V M| V (σ×idV ) * φ ′ / / (σ × id V ) * p * 2 M| V .
≀ and using the isomorphisms provided by Lemma 7.2, we obtain a commutative diagram identical to Diagram (4.7). This shows that φ is a F-action.
We recall the DQ-module version of the codimension three conjecture.
Theorem 7.4 ([Pet17, Theorem 1.5]). Let X be a complex manifold endowed with a DQ-algebra A X such that the associated Poisson structure is symplectic. Let Λ be a closed Lagrangian analytic subset of X, Z a closed analytic subset of Λ such that codim Λ Z ≥ 3 and j : X \ Z → X the open embedding. Let M be a holonomic (A loc X | X\Z )-module, whose support is contained in Λ \ Z. Assume that M has an A X | X\Z -lattice. Then j * M is a holonomic module and is the unique holonomic extension of M to X whose support is contained in Λ.
We now deduce the codimension three for formal microdifferential modules from the one for DQ-modules.
Theorem 7.5 ([KV14, Theorem 1.2]). Let M be a complex manifold, X an open subset of T * M , Λ a closed Lagrangian analytic subset of X, and Z a conical closed analytic subset of Λ such that codim Λ Z ≥ 3. Let E X the sheaf of formal microdifferential operators on X and N be a holonomic ( E X | X\Z )-module whose support is contained in Λ \ Z. Assume that N possesses an ( E X (0)| X\Z )-lattice N 0 . Then N extends uniquely to a holonomic module defined on X whose support is contained in Λ.
Proof. (i) The unicity is proved as in [KV14]. Thus we do not repeat the proof here.
(ii) Proving the coherence of an extension of N is a local problem. By the dummy variable trick, we can assume that X does not intersect the zero section of T * M . Hence, for any x ∈ X, there exists a neigbourhood V of x, a conical open subset U , of X such that V ⊂ U and a coherent E P * U -module N ′ together with a E P * U (0)lattice N 0 such that N | V ≃ (p −1 N ′ )| V and N 0 | V ≃ (p −1 N ′ 0 )| V where p U : U → P * U is the restriction to U of the canonical projection. We can further assume that Z and Λ are subset of U .
The module M := p * C × N ′ belongs to Mod F,coh ( W U | U\Z ), has a W U (0)| U\Zlattice and its support is contained in Λ \ Z. By [Pet17, Theorem 1.4], j * M is an holonomic module supported by Λ and is also endowed with a F-action by Lemma 7.3. Moreover it follows from Proposition 6.16 that p C × * j * M is a coherent E P * U -module such that
(p C × * j * M)| p(U)\p(Z) ≃ p C × * ((j * M)| U\Z ) ≃ N ′ .
This proves that p C × * j * M is a coherent extension of N ′ . This implies that j * N is a coherent E X -module.
Proposition 3. 5 .
5Let M be a coherent A X -module, set N = p * 2 M , U an open subset of C × X and let u ∈ N (U ) such that for every t ∈ p 1 (U ), u(t) = 0. Then u = 0. Proof. The O C×X -modules n N / n+1 N are coherent and n N / n+1 N ≃ p * 2 ( n M/ n+1 M). (3.1)
n≥0 n
n≥0N = (0) by [KS12, Corollary 1.2.8] which proves the claim.
Corollary 3. 6 .
6Let M be a coherent A loc X -module and let
Definition 5. 1 .
1Let (M, φ) ∈ Mod F (A loc X ), U ⊂ X and s ∈ M(U ). (i) The section s is locally invariant at x ′ if there exists an open neighbourhood
Proposition 5. 3 .
3Let (M, φ) ∈ Mod F,coh (A loc X ) and U be an open subset of X stable by the action of C × and let s ∈ M(U ). The following conditions are equivalent(1) for every t ∈ C × , φ t (s) = s, (2) v(s) = 0. Proof. (1) ⇒ (2) is clear.Let us prove that (2) ⇒ (1). It follows from Lemma 5.2 that for every t ′ ∈ C × ,dφ(s) dt | t=t ′ = 0. Then, Corollary 3.7 implies that φ(s) = s Lemma 5.4. Let M ∈ Mod F,coh (A loc X ). Locally, there exists a coherent A Xmodule M 0 ⊂ M such that M ≃ M loc 0 and v(M 0 ) ⊂ M 0 .
a
ij e j .We form the matrix A = (a ij ) 1≤i,j≤l ∈ M l (A U
Therefore, we are looking for B in GL l (A U ) such that v(B)e + BAe = 0.Hence, it is sufficient to prove the existence of B ∈ GL l (A U ) such that v(B) + BA = 0.(5.1)
0 = 0 and C n = − i+k=n i =n v k (B i ) + k+j+m=n j =n P m (B j , A k ), the system (5.2) is rewritten as ∀n ∈ N, v 0 (B n ) + B n (mn id +A 0 ) = C n (5.3)
∈ N, ∂B n ∂z 1 + B n (mn id +A 0 ) = C n . (5.4)
is locally finitely generated by locally invariant sections, there exists an open subset U of X and locally invariant sections
Corollary 5. 8 .
8Let M ∈ Mod F,coh (A loc X ). Then there exists a covering (V i ) i∈I of X by equivariant open subsets of X such that for each i ∈ I, M| Vi admits, in Mod F (A loc X | Vi ), a presentation of length one by free modules of finite rank. Proof. This follows from Theorem 5.5 and Lemma 5.7 (ii).
0 has a solution u in an open neighbourhood V ′ of x. This follows from Lemma 6.3. Finally using Lemma 5.7, we extend the section u to an invariant section s defined on an equivariant open set V .Let us check that there exists an invertible section u ∈ A X (V ) such that s = u. Let y ∈ V . By construction (see the proof of Lemma 6.5), s y
Theorem 6. 8 .
8Let k be a field of characteristic zero, R a sheaf of k-algebras and s a section of R such that(i) sR = Rs, (ii) R ·s −→ R is a monomorphism, (iii) R ≃ lim ← − n∈N R/Rs n (iv) R/Rs is a left Noetherian ring.Then R is a left Noetherian k-algebra.Theorem 6.9. (i) The sheaf B Y (0) is a Noetherian sheaf of C-algebras.
Proposition 6 .
610. (i) The sheaf A X is faithfully flat over A C × X , (ii) the sheaf A loc X is faithfully flat over A loc,C × X . Proof. (i) The proof is similar to point (i) of the proof of [KS12, Lemma 6.1.2 (a)].
M be a complex manifold, X be an open subset of T * M ,Ṫ * M = T * M \ M , P * M the projective cotangent bundle and p :Ṫ * M → P * M be the canonical projection.
Lemma 2.3.2]. This implies that α is flat. The following definition is an adpation of [KR08, Definition 2.4] along the line of [MFK94, ch.1 §3 Definition 1.6].
Proposition 6.13. The functor p C × * restricted to Mod F, coh (A loc X ) is exact. Proof. As p C × * is a right adjoint it is left exact. Let us show it is right exact on Mod F, coh (A loc X ). Consider the exact sequence 0 → M ′ → M → M ′′ → 0.
Acknowledgements. I would like to express my gratitude to Masaki Kashiwara and Pierre Schapira for their scientific insights. It is pleasure to thanks Gwyn Bellamy, Damien Calaque, Vincent Pecastaing, Mauro Porta, Marco Robalo, Yannick Voglaire for useful conversations.This is an isomorphism by Proposition 5.3. Hence, we do not make any distinction between p C × * M and M and denote both of them by p C × * M. By definition of B Y there is a morphism of sheaves of algebrasThis allows us to define the functorThere is the following adjunctionand obtain the following commutative diagramThis implies that µ * p * ≃ p * 2 p * . Hence, there is an isomorphism φ :. We obtain a functor p *
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E-mail address: francois. [email protected] address: [email protected]
| [] |
[] | [
"Hyungju Oh \nDepartment of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA\n",
"Sinisa Coh \nDepartment of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA\n",
"Marvin L Cohen \nDepartment of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA\n"
] | [
"Department of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA",
"Department of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA",
"Department of Physics\nMaterials Sciences Division\nUniversity of California at Berkeley\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUSA"
] | [] | We report on a comparative study of the electronic structure, phonon spectra, and superconducting properties for recently discovered superconducting hydrides, H3S and H3P. While the electronic structures of these two materials are similar, there are notable changes in the phonon spectra and electron-phonon coupling. The low-frequency bond-bending modes are softened in H3P and their coupling to the electrons at the Fermi surface is enhanced relative to H3S. Nevertheless, coupling to the high-frequency modes is reduced so the resulting calculated superconducting transition temperature is reduced from ∼166 K in H3S to ∼76 K in H3P. | null | [
"https://arxiv.org/pdf/1606.09477v2.pdf"
] | 118,541,899 | 1606.09477 | 7ed5dcebd505701966df7da10a6aaaa8342659d0 |
2 Jul 2016
Hyungju Oh
Department of Physics
Materials Sciences Division
University of California at Berkeley
Lawrence Berkeley National Laboratory
94720BerkeleyCaliforniaUSA
Sinisa Coh
Department of Physics
Materials Sciences Division
University of California at Berkeley
Lawrence Berkeley National Laboratory
94720BerkeleyCaliforniaUSA
Marvin L Cohen
Department of Physics
Materials Sciences Division
University of California at Berkeley
Lawrence Berkeley National Laboratory
94720BerkeleyCaliforniaUSA
2 Jul 2016(Dated: July 5, 2016)Comparative study of high-T c superconductivity in H 3 S and H 3 Pnumbers: 7115Mb7120-b7420Fg7462Bf
We report on a comparative study of the electronic structure, phonon spectra, and superconducting properties for recently discovered superconducting hydrides, H3S and H3P. While the electronic structures of these two materials are similar, there are notable changes in the phonon spectra and electron-phonon coupling. The low-frequency bond-bending modes are softened in H3P and their coupling to the electrons at the Fermi surface is enhanced relative to H3S. Nevertheless, coupling to the high-frequency modes is reduced so the resulting calculated superconducting transition temperature is reduced from ∼166 K in H3S to ∼76 K in H3P.
I. INTRODUCTION
Many materials have been proposed theoretically as conventional phonon-mediated superconductors having a high superconducting transition temperature (T c ). Based on the BCS theory 1 , materials with light masses and strong bonds are promising candidates for high-T c superconductors 2,3 because T c is scaled by the inverse square root of the atomic mass. Therefore, theoretical studies have been intensively performed focusing on the compounds consisting of the lightest hydrogen atom. In experiments, on the other hand, achieving a high-T c in hydrogen compounds has not been reported yet. Recently, it is experimentally reported that, under extreme high pressures of 100-200 GPa, sulfur hydride transforms to a metallic state and shows extremely high-T c up to ∼200K 4,5 .
To find out the crystal structure of the high-T c sulfur hydride, many ab-initio studies have been done and most of these studies have concluded that cubic H 3 S will form with a H-rich decomposition environment under high pressure [6][7][8][9][10][11][12] . Furthermore, from electron-phonon coupling (EPC) calculations [6][7][8][9][10][11][12][13][14][15] , it is revealed that strong coupling happens between high-frequency phonon modes and electrons and these strong coupling induces high-T c in the body-centered cubic H 3 S.
Here we study two types of hydrides, H 3 S and H 3 P. Following the discovery of high-T c conventional superconductivity in sulfur hydride, a hydride phosphine (H 3 P) was also reported to be a possible high-T c (T c > 100 K at pressure P> 200 GPa) superconductor via four-probe electrical measurements 16 . Hence we compare the normal and superconducting properties of these two materials. For the crystal structures of high-T c hydrides, Xray diffraction experiments 12,17 confirm that the sulfur atoms of H 3 S form a body-centered cubic structure as shown in Fig. 1. Up to now, no available experimental data for the crystal structure of H 3 P exists. Hence for comparison purposes, we assume in this study that both materials have the same crystal structure and analyze the effect of element change on material properties.
II. METHODS
The following methods are used to perform the calculations of the electronic structures, phonon properties, and superconducting properties. For the electronic structures, our calculations are based on ab-initio normconserving pseudopotentials and the Perdew-Burke-Ernzerhof 18 functional as implemented in the SIESTA 19 and Quantum-ESPRESSO 20 codes. Phonon frequencies are computed using density-functional perturbation theory 21 implemented in Quantum-ESPRESSO 20 package. Finally, EPC and Eliashberg spectral functions are obtained via the Wannier90 22 and EPW 23 packages.
For the calculation using SIESTA, electronic wavefunctions are expanded with pseudoatomic orbitals (doubleζ polarization) and a charge density cutoff of 800 Ry is used. We sample the Brillouin zone on a uniform 16×16×16 k-point mesh. For the calculation with Quantum-ESPRESSO, a plane-wave basis up to 160 Ry and a 32×32×32 k mesh size are employed.
Phonon frequencies ω qν and EPC parameters λ qν are computed on a coarse mesh (8×8×8) of reciprocal space. Next, interpolation techniques 24 based on maximally localized Wannier functions 24-26 are used to interpolate EPC parameters on a fine grid (36×36×36).
The Eliashberg spectral function α 2 F (ω) is computed by integrating the interpolated phonon frequencies ω qν and the EPC λ qν over the Brillouin zone,
α 2 F (ω) = 1 2 qν w q ω qν λ qν δ(ω − ω qν ).(1)
Here the w q is the Brillouin zone weight associated with the phonon wavevectors q. The total EPC λ is calculated as the Brillouin zone average of the mode-resolved coupling strengths λ qν :
λ = qν w q λ qν = 2 ∞ 0 dω α 2 F (ω)/ω.(2)
FIG. 1. The Im3m crystal structure assumed for H3S and H3P. The large sphere (orange) is S or P, and the small sphere (white) is H.
III. ELECTRONIC STRUCTURE
Here we discuss the electronic structure of H 3 S and H 3 P. In all of our calculations we set the conventional lattice parameter as 3Å. With this lattice parameter, the calculated pressures of both materials are 220 GPa.
The overall shapes of the band structures are similar for both materials [Figs. 2(a) and (c)]. Because phosphorus has one less valence electron than sulfur, the Fermi level (E F ) is shifted down in H 3 P. With the shift, E F of H 3 P is placed near a different peak position in the density of states (DOS). For H 3 S, the DOS at E F is calculated to be 0.45 states eV −1 f.u. −1 . A similar value (0.50 states eV −1 f.u. −1 ) for the DOS is found in the case of H 3 P. Figure 2 compares the orbital contributions to the band structure and DOS in H 3 S and H 3 P. In both materials, the DOS at E F comes dominantly from 3p orbitals of sulfur or phosphorus. The portion of 3p orbitals is twice as large as the portion of hydrogen orbitals. The Fermi surfaces originated from hydrogen orbitals are almost same in both case, forming small hole pockets centered at Γ -point.
IV. PHONON PROPERTIES
In this section we discuss the differences of the phonon properties between H 3 S and H 3 P. When the sulfur is changed to phosphorus, the characteristics of the phonon spectra differ significantly along Γ -H and H-N highsymmetry lines [Fig. 3]. The hydrogen-phosphorus bondbending modes become softer and three unstable phonon modes appear at the H high-symmetry point. Therefore we expect that in the doubled unit cell these unstable modes would be stabilized. We exclude these negative phonon modes when calculating α 2 F so that we can make an reliable comparison with H 3 S. The structural instability of body-centered cubic H 3 P is also reported by previous theoretical structural studies 27,28 .
Next we discuss the strength of the EPC for the two cases. In H 3 S, phonon modes of 150∼200 meV frequencies (which are H-S bond-stretching modes) are strongly coupled to electrons at the Fermi surface. In H 3 P, however, low-frequency modes (< 50 meV) are more relevant. These modes originate from softened H-P bond-bending motion.
To give a more quantitative discussion about the relevant energy scales of the phonons, we calculate the EPCweighted average of the phonon frequencies,
ω ln = exp 2 λ ∞ 0 dω α 2 F (ω) ω ln ω .(3)
The value of ω ln is 1580 K (136 meV) for the H 3 S and 610 K (53 meV) for the H 3 P. Therefore ω ln is more than twice as large in H 3 S relative to H 3 P.
V. SUPERCONDUCTING PROPERTIES
The total EPC λ equals 1.38 in H 3 S, whereas it reaches 1.66 in H 3 P. The Eliashberg phonon spectral functions of H 3 S and H 3 P are quite different. The EPC in H 3 S is dominated by the phonon modes at the zone center Γ point. In H 3 P, however, we observed an overall contribution of different modes to λ along Γ -H-N directions as shown in Fig. 3.
Here we discuss why there is a large difference in the EPC between H 3 S and H 3 P. First, we consider the difference in DOS. Since λ is roughly proportional to the DOS at E F , the EPC could be enhanced by the large DOS. However, in our case, there is no sufficient change in DOS to reproduce the large enhancement in EPC for H 3 P. Another point is the coupling strength between the electrons and the low-frequency hydrogen vibration. There is no significant enhancement in the electron-phonon matrix elements which is proportional to ω qν λ qν [Fig. 3]. But, the dominant modes to EPC appear at low frequencies in H 3 P [Fig. 4]. This change causes the enhancement of the EPC λ value.
Finally, we estimate the superconducting transition
T c = ω ln 1.20 exp − 1.04 (1 + λ) λ − µ * (1 + 0.62 λ) .(4)
Here µ * is the Coulomb repulsion parameter. For commonly used µ * = 0.1 we estimate T c = 166 K for H 3 S and 76 K for H 3 P. The exact value of µ * here is not that important since even with µ * = 0 we get very similar T c (219 and 96 K).
The value of λ we obtained for H 3 S and H 3 P is near the limit of applicability of the McMillan equation. However, we find that the Kresin-Barbee-Cohen model 30,31 , which is applicable for large λ, gives similar estimates for T c .
Although H 3 P has a higher λ value than H 3 S, the estimated T c is about half of that in H 3 S. This agrees well with the experimentally obtained T c of ∼ 200 K in H 3 S and ∼ 100 K in H 3 P. We expect that the deviation here from experiment might occur because we ignored unstable phonon modes in our calculation, so softening might be overestimated for H 3 P in the low-frequency regime.
VI. CONCLUSION
With the assumption of the same body-centered cubic structure and lattice parameter, we compare the electronic, phonon, and superconducting properties of H 3 S and H 3 P. The results of electronic structures show no significant difference, except for a slight change in the Fermi level due to the different number of valence electrons. However, there are notable changes in phonon spectrum and electron-phonon coupling properties. First, there exists phonon softening in low-frequency bond-bending modes, and the coupling of these modes to electrons near the Fermi surface is enhanced. As the dominant frequency regime changes from high to low frequency, the superconducting transition temperature is reduced from ∼166 K in H 3 S to ∼76 K in H 3 P.
FIG. 2 .
2Electronic band structures and density of states (DOS) per three-hydrogen formula unit (f.u.) of (a), (b) H3S and (c), (d) H3P. Dominant orbital characters are represented in blue (H orbitals), red (S or P s orbitals), and green (S or P p orbitals) color.
FIG. 3 .
3Phonon spectrum and phonon density of states (PHDOS) of (a) H3S and (b) H3P. The radius of the red circle is proportional to ωqνλqν .
FIG. 4 .
4Eliashberg spectral function α 2 F (red) and cumulative contribution to the electron-phonon coupling strength λ (blue) of (a) H3S and (b) H3P. The cumulative EPC is calculated as λ(ω) = 2ω 0 dω ′ α 2 F (ω ′ )/ω ′ .temperature T c using the McMillan equation29
ACKNOWLEDGEMENTS
. * Email, [email protected]* Email: [email protected]
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| [] |
[
"Frequent Knot Discovery",
"Frequent Knot Discovery"
] | [
"Floris Geerts [email protected] \nFoundations of Computer Science School of Informatics\nLaboratory for\nUniversity of Edinburgh\nUK\n"
] | [
"Foundations of Computer Science School of Informatics\nLaboratory for\nUniversity of Edinburgh\nUK"
] | [] | We explore the possibility of applying the framework of frequent pattern mining to a class of continuous objects appearing in nature, namely knots. We introduce the frequent knot mining problem and present a solution. The key observation is that a database consisting of knots can be transformed into a transactional database. This observation is based on the Prime Decomposition Theorem of knots. | null | [
"https://arxiv.org/pdf/cs/0410038v1.pdf"
] | 10,217,120 | cs/0410038 | 05fceecc3cb03f72c3d6741f8d9b3b317e2f67c8 |
Frequent Knot Discovery
16 Oct 2004 February 7, 2008
Floris Geerts [email protected]
Foundations of Computer Science School of Informatics
Laboratory for
University of Edinburgh
UK
Frequent Knot Discovery
16 Oct 2004 February 7, 2008
We explore the possibility of applying the framework of frequent pattern mining to a class of continuous objects appearing in nature, namely knots. We introduce the frequent knot mining problem and present a solution. The key observation is that a database consisting of knots can be transformed into a transactional database. This observation is based on the Prime Decomposition Theorem of knots.
Introduction
Many algorithms have recently been developed for mining frequent patterns. Traditionally, these patterns consist of subsets of attributes in a relational database [1]. Recently, other patterns have been mined, such as trees [27] and graphs [13,14,25,10]. However, most objects appearing in nature lack the discrete character of graph and trees. In this paper we explore the possibility of applying the framework of frequent pattern mining to a class of continuous objects appearing in nature, namely knots. A knot can be thought of as a piece of rope (where the rope has zero thicknes) which forms a loop in three-dimensional Euclidean space R 3 . Figure 1 shows an example of a knot known as the Trefoil knot.
The history of knots dates back to the late 1800's when Lord Kelvin suggested that atoms where knots in an invisible and frictionless fluid. Since then, theoretical properties of knots are extensively studied in mathematics [3]. In physics, knot invariants (e.g., the Jones polynomial) are used in statistical physics [12] and knots also appear in the context of quantum gravity [2]. Recently, knots showed up as building blocks for future quantum computers [15].
In biology, knots are used to characterize topoisomerase enzymes [22] and in polymer science, physical properties of long ring polymers, such as DNA, gels and rubbers are related to properties of knots [4,5,24]. It is shown that knots are present in such polymers with probability one when the polymers are long enough [21]. There is also much interest in developing artificial knotted biopolymers as building blocks for DNA-based computing [18].
The study of knotted polymers is done both by using experimentally obtained knots and by using knots obtained by numerical approaches based on self-avoiding random-walk simulations. Examples of questions one would like to answer in these studies are what is the probability of having a certain knot in polymers of a certain length [19], and whether the knots appear tight or loose in the knotted polymers [11].
In this article we consider the frequent knot mining problem which can be stated as follows: Given a collection of knots, find all subknots which appear frequently in this collection.
We believe that finding frequent subknots in a large collection of real or simulated knotted polymers, will contribute to a deeper understanding of the statistical properties of knotted polymers in R 3 . This article reports a first attempt for solving the frequent knot mining problem.
The solution presented in this article consists of three steps:
1. Encoding of knots in transactions;
2. Mining these transactions; and finally, 3. Decoding of the frequent itemsets into knots.
The article is organized as follows: In Section 2, definitions are given and the frequent knot mining problem is stated formally. The encoding (decoding) of knots (transactions) into transactions (knots) is described in Section 3 . In Section 4, we present the KnotMiner algorithm for mining frequent knots. Finally, conclusions are drawn in Section 5.
Preliminaries
A knot K can be thought of as a piece of rope (where the rope has zero thickness) which forms a loop in three-dimensional Euclidean space R 3 . Two knots K and K ′ are equivalent, or in symbols K ≡ K ′ , if they can be transformed into each other without cutting and pasting the ropes. We will only consider so-called tame knots. These are knots which are equivalent to piecewise linear knots, i.e., knots consisting of a finite number of straight lines. A knot is trivial if it is equivalent to a rope which forms a circle in a plane in R 3 .
A knot can be finitely represented by a knot diagram. The knot diagram of a knot K is a connected undirected planar graph, which correspond to a (generic) projection of K onto a plane. Vertices in a knot diagram correspond to places where the projection of the knot intersects, and each edge adjacent to a vertex is labelled as an undercrossing or overcrossing, whichever is the case. Given a knot diagram consisting of n vertices one can find in time polynomial in n a piecewise linear knot such that its z-projection gives the original diagram [8]. Two knot diagrams can be transformed into each other using the so-called Reidemeister moves if and only if they represent equivalent knots [3].
The connected sum of two knots K 1 and K 2 is formed by removing a small piece of rope from both knots and then connecting the four endpoints by two new pieces of rope in such a way that no new crossings are introduced, the result being a a single knot, which is denoted by K = K 1 #K 2 . This operation is illustrated in Figure 2. The connected sum K 1 #K 2 is equivalent to K 2 #K 1 and (K 1 #K 2 )#K 3 is equivalent to K 1 #(K 2 #K 3 ). The connected sum of a knot K and the trivial knot is equivalent to K [3].
A knot is called prime if for any decomposition as a connected sum, one of the factors is the trivial knot. There are infinitely many prime knots.
Theorem 1 (Prime decomposition Theorem [3,17]) Every knot K can be decomposed as a connected sum of nontrivial prime knots. If K ≡ K 1 #K 2 # · · · #K m and K ≡ L 1 #L 2 # · · · #L n , where K i and L i are nontrivial prime knots, then m = n, and after reordering each K i is equivalent to L i .
This theorem motivates the following definition. Let
K = K 1 #K 2 # · · · #K p and let L = L 1 #L 2 # · · · #L q .
Then, K is a subknot of L, or K L, if for any i = 1, . . . , p we have that
|{j | K j ≡ K i }| ≤ |{j | L j ≡ K i }| .
A knot datatabase D is a finite collection of knots. The support of a knot K in D is defined as supp(K) = |{L ∈ D | K L}|.
The frequent knot mining problem can be stated as follows: Given a knot database D and a threshold value σ ∈ N, find all knots K in D such that supp(K) > σ.
For completeness, we also state the frequent itemset mining problem. A transaction database T is a finite collection of k-tuples in N k . The support of an itemset I in T is defined as
supp(I) = |{J ∈ T | ∀ℓ : (I) ℓ ≤ (J) ℓ }|,
where (I) ℓ (resp. (J) ℓ ) denotes the ℓth component of I (resp. J). The frequent itemset mining problem is then: Given a transaction database T and a threshold value σ ∈ N, find all itemsets I in T such that supp(I) > σ.
From Knot Databases to Transaction Databases
In this section we show how to transform a knot database D into a transactional database. We assume that the knots in D are represented by knot diagrams.
We start by computing for each knot K in D its prime decomposition. Schubert [17] gives an algorithm computing this decomposition. The running time is at worst exponential in the number of vertices in the knot diagram. In this way, we obtain a set primes(D) consisting of knot diagrams for all prime knots occurring in D. Two different knot diagrams in primes(D) can represent the same prime knot, so we have to eliminate duplicates. There exists an algorithm for testing whether two knot diagrams represent equivalent knots [9,23]. However, at present, the complexity of this algorithm is not known. From here on, we assume that primes(D) does not contain duplicates and order it arbitrarily.
We now define a mapping, denoted by encode, from knots in a knot database to elements in a transaction database. Let D be a knot database, and K a knot in D. Then, encode(K) = (n 1 , . . . , n p ), with p = |primes(D)| and n i is the number of times the prime knot corresponding to the ith knot diagram in primes(D) appears in the prime decomposition of K. Clearly, encode(D) is a transaction database consisting of |primes(D)| attributes. Given a set of knots K = {K 1 , . . . , K p }, we now define the mapping, denoted by decode, from itemsets of a transaction database T ⊂ N p to knots in R 3 . Let t ∈ T and let (t) i1,...,i k = (m 1 , . . . , m k ) be a k-itemset. Then,
decode(m 1 , . . . , m k ) = K i1 # · · · #K i1 m1 times # · · · # K i k # · · · #K i k m k times .
For any knot K, we have that K ≡ decode(encode(K)).
Algorithm
We now present The KnotMiner algorithm for computing the frequent knots in a knot database.
Algorithm 1: KnotMiner Input: knot database D, σ Output: All knots K such that supp(K) > σ.
1: Compute T := encode(D) 2: Compute the set F of frequent itemsets in T 3: Output decode(F ).
The first and last step in KnotMiner are already fully explained in Section III. For the second step one can either transform T into a binary transaction database and use a standard mining algorithm like Apriori [1], Eclat [26] or FPgrowth [7]. Alternatively one can mine T directly using algorithms presented in [20] and [16]. The following result is immediate.
Theorem 2
The KnotMiner algorithm works correctly.
Concluding Remarks and Future Work
In this article we introduced the frequent knot mining problem and proposed the KnotMiner algorithm to solve it. Currently, there exists no implementation of KnotMiner. This is mainly due to the complex algorithms needed for the encoding of a knot database into a transactional databases.
However, recent research indicates that the knot decomposition of knotted polymers can be obtained by "Coulomb decomposition", which is a technique where polymers are brought into an equilibrium state using Coulomb interactions [6]. We hope to apply this technique on simulated knotted polymers and hence obtain an implementation of KnotMiner, specifically aimed for mining knotted polymer databases.
Figure 1 :
1Trefoil knot.
Figure 2 :
2Sum of two knots.
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| [] |
[
"Entanglement requirements for implementing bipartite unitary operations",
"Entanglement requirements for implementing bipartite unitary operations"
] | [
"Dan Stahlke \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA\n",
"Robert B Griffiths \nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA\n"
] | [
"Department of Physics\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA",
"Department of Physics\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA"
] | [] | We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerical examples of Schmidt rank 2 unitaries which can be deterministically implemented, by either SEP or LOCC, using an entangled resource of two qutrits with less than one ebit of entanglement. | 10.1103/physreva.84.032316 | [
"https://arxiv.org/pdf/1104.1652v3.pdf"
] | 118,567,084 | 1104.1652 | 0706f0a346bd08f67a5cd5bd642bd81336530d61 |
Entanglement requirements for implementing bipartite unitary operations
12 Jul 2011
Dan Stahlke
Department of Physics
Carnegie Mellon University
15213PittsburghPennsylvaniaUSA
Robert B Griffiths
Department of Physics
Carnegie Mellon University
15213PittsburghPennsylvaniaUSA
Entanglement requirements for implementing bipartite unitary operations
12 Jul 2011(Dated: July 12, 2011)PACS numbers: 0367Ac
We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerical examples of Schmidt rank 2 unitaries which can be deterministically implemented, by either SEP or LOCC, using an entangled resource of two qutrits with less than one ebit of entanglement.
I. INTRODUCTION
It is possible to carry out nonlocal quantum operations on multipartite systems using only local quantum operations and classical communications (LOCC) provided that the parties involved have access to a suitable entangled state, referred to as a resource. Given a large enough resource it is always possible to use teleportation to send all inputs to one party, who performs the operation and then distributes the results to the other parties using teleportation. In some cases it is possible to perform a nonlocal operation with less entanglement than is required by teleportation [1][2][3][4][5][6]. The question then arises as to how much entanglement is really necessary in order to implement a given nonlocal operation.
Our first result, that the Schmidt rank of the resource must be at least as great as that of the unitary [Theorem 1(a)], follows rather immediately from the fact that it is a separable (SEP) operation. This is analogous to the result given in [7] in which probabilistic (i.e. SLOCC) implementations are considered. Since SEP is contained in SLOCC, our Theorem 1(a) can be seen as a consequence of the result in [7], however we provide an independent proof along the way to our main result.
In contrast to the probabilistic case, the deterministic implementation of a unitary is only possible if the state meets certain entanglement requirements. For one thing, the entanglement of the resource must be at least as great as the entangling power of the unitary since entanglement cannot increase under SEP [8]. It has been shown that any deterministic controlled-unitary operator on two qubits implemented with bipartite LOCC using a resource * Electronic address:[email protected] † Electronic address:[email protected] of two entangled qubits necessarily requires a maximally entangled resource [9]. Our paper takes a different approach to the problem, using SEP, and provides a proof applicable to general unitaries of arbitrary dimension. We show that if the resource has Schmidt rank equal to that of the unitary, the resource must be uniformly entangled in the sense that all its nonzero Schmidt coefficients are the same [Theorem 1(b)]. These same restrictions apply to LOCC, as it is a particular case of SEP. It is not hard to see that if the Schmidt rank of the resource is greater than the Schmidt rank of the unitary, then the resource need not be uniformly entangled (e.g. a larger rank resource that is majorized by a smaller rank maximally entangled state). We have found that it is in fact possible for such a larger rank resource to have less entanglement than would be required for a resource of Schmidt rank equal to that of the unitary. We have found examples of protocols in both SEP and LOCC which deterministically implement a controlled phase operation using less than one ebit of entanglement. In this case the unitary has Schmidt rank two and the resource has Schmidt rank three. Although the nonlocal unitary protocol given in [10] can with certain probability consume less than one ebit of entanglement 1 , we believe that ours is the first example of carrying out such a protocol deterministically using less than one ebit of entanglement.
The remainder of this article is organized as follows. Section II sets up the problem of bipartite deterministic implementations of unitary operators using SEP. Section III provides the requisite background regarding map-state duality [11,12] and atemporal diagrams [13][14][15]. Our main result is proved in Sec. IV using what we believe to be a new method based on the use of map-state duality. In Sec. V we consider the case of a resource of larger Schmidt rank. There is a brief conclusion in Sec. VI. An appendix details the implementation of a controlled unitary using a qutrit resource state of less than one ebit of entanglement.
d A d B = dĀdB, but we do not require that d A = dĀ or d A = d B .
The separable operation must satisfy the usual closure condition [16] k
(E k ⊗ F k ) † (E k ⊗ F k ) = I A ⊗ I a ⊗ I b ⊗ I B (1)
which is depicted in Fig. 1(a).
In addition, for |Φ any pure input state on H A ⊗ H B , the outcome of the operation will be a pure state
U |Φ Φ| U † = k E k ⊗ F k |Φ Φ| ⊗ |ψ ψ| E k ⊗ F k †(2)
on HĀ ⊗ HB. Since the protocol is assumed to be deterministic, every term on the right side is proportional to the same pure state and it must be the case that
(E k ⊗ F k ) |ψ = α k U,(3)
with α k some complex number. Note that both sides of (3) are operators acting on H A ⊗ H B ; Fig. 2(a) will help interpreting it correctly. The resource |ψ is assumed to have a Schmidt rank of D ψ , which means it can be written in the form
|ψ = D ψ i=1 λ i |a i ⊗ |b i .(4)
for suitable orthonormal bases
{|a i } and {|b i } of H a and H b , with Schmidt coefficients λ i > 0 for i ≤ D ψ .
Similarly, the bipartite operator U is assumed to have a Schmidt rank of D U , meaning that it can be written in the form [17]
U = DU i=1 µ i A i ⊗ B i ,(5)
where {A i } and {B i } are bases of the operator spaces L(H A , HĀ,) and L(H B , HB,), orthonormal under the Frobenius (Hilbert-Schmidt) inner product, and µ i > 0 for i ≤ D U . Equivalently, D U is the minimum number of terms needed in order to write U in the form C i ⊗ D i , without requiring C i or D i to be from an orthonormal basis.
III. MAP-STATE DUALITY AND DIAGRAMS
Map-state duality [11,12] plays a central role in the proof that will follow. This is a general concept that is sometimes referred to as reshaping or a partial transpose [11] and in a specific manifestation is known as the Jamio lkowski or sometimes the Choi-Jamio lkowski isomorphism. States and maps are considered to both be tensors, and when a choice of orthonormal basis is fixed there is a natural linear relation between bras and kets (i.e. |i ↔ i| for all basis vectors |i ) 2 .
With this identification between bras and kets in place, the bipartite state |ψ on the Hilbert space H a ⊗ H b can be identified with the linear map ψ ′ : H b → H a obtained by turning kets into bras on the H b space:
|ψ = ij ψ ij |a i ⊗ |b j → ψ ′ = ij ψ ij |a i b j | (6)
Similarly, the operators U , E k , and F k , give rise to
U ′ = ijmn Ā j ,B n |U |A i , B m A i ,Ā j B m ,B n , E ′ k = ijm Ā j |E k |A i , a m A i ,Ā j a m | , F ′T k = ijm B j |F k |B i , b m |b m B i ,B j .(7)
In the case of these three operators, map-map duality may be a more precise term, however we will use map-state duality to refer to any such partial transpose. The primed operator for F k is denoted as F ′T k in order to draw attention to the fact that its domain and range are swapped in comparison to E ′ k .
The equations introduced so far make use of six distinct Hilbert spaces and tensors of various rank. In such situations the underlying structure of equations can be somewhat hidden when expressed using Dirac notation. Abstract index notation is more transparent but can become unwieldy. For this reason we provide atemporal diagrams, similar to those found in [13], which should aid the reader in following the arguments in the text.
Operators are designated by squares or rectangular boxes. As a matter of style, the state |ψ and its corresponding operator ψ ′ will be represented as a circle instead of a square. Lines between these boxes represent tensor contraction, and these lines are labeled by the Hilbert spaces which they correspond to. Open lines on the left of a diagram represent the input to the total linear operator defined by the diagram, and open lines on the right represent outputs. Putting the inputs on the left means that operators are to be applied in a left-to-right manner, opposite to how algebraic equations are interpreted. As has been so far described, our diagrams are to be interpreted in exactly the same way as traditional quantum circuits as used for example in Nielsen and Chuang [18]. The primary difference between our diagrams and traditional circuits is that in the latter the horizontal direction is understood to represent the passage of time whereas our diagrams make no reference to time. The presence of a summation symbol has the obvious meaning: the linear operator depicted in the diagram denotes the terms of a series. The trace or partial trace operation is just a special case of tensor contraction and is denoted by joining the relevant spaces with a line. The identity operator is represented by a line. With minor changes in style our diagrams are equivalent to the atemporal diagrams of [13], and resemble other such schemes [14,15].
a) k A a b B E k F k E † k F † k A a b B Ā B = A a b B A a b B b) k α * k A a b B E k F k U † A B Ā B = A a b B A B ψ † c) k α * k A a b B E k F kĀ B = A a b B UĀ B ψ † d) k α * k a E ′ k Ā A b F ′T k B B = a ψ ′ † b B B U ′ Ā A e) k α * kâÊ k b F T k B =âψ †b BÛ f) k α * kâÊ kÛ −1B b F T k B =âψ †b BB g) k α *
IV. ENTANGLEMENT REQUIREMENTS
Our main result is the following:
a) A B ψ E k F kĀ B a b = α k A B UĀ B b) B B F ′T k ψ ′ E ′ k Ā A b a = α k B B U ′ Ā A c)BF T kψ E k bâ = α kBÛÂ
FIG. 2. (a) Deterministic unitary operation, (3). (b)
Apply map-state duality to get (8). (c) Restrict spaces to supports and ranges of operators to get (14).
|ψ must be a uniformly (maximally) entangled state: all the nonzero Schmidt coefficients are the same.
Proof of (a). Making use of map-state duality and the operators defined in (6) and (7), equation (3) [ Fig. 2(a)] can be rewritten as [ Fig. 2
(b)] E ′ k ψ ′ F ′T k = α k U ′ .(8)
Since the rank of a product of linear operators is at most the smallest of the ranks of the individual operators, it follows that rank(ψ ′ ) ≥ rank(U ′ ). The rank of an operator is equal to the number of its nonzero singular values. Since the Schmidt decompositions (4) and (5) are essentially singular value decompositions of ψ ′ and U ′ , it is apparent that rank(ψ ′ ) = D ψ and rank(U ′ ) = D U and the inequality becomes D ψ ≥ D U . Part (a) is proved.
Proof of (b). Apply the closure condition (1) to ψ| and use the adjoint of (3) to obtain
k α * k U † (E k ⊗ F k ) = ψ| ⊗ I AB .(9)
as shown in Fig. 1(b). Next, multiply both sides on the left by U to arrive at
k α * k (E k ⊗ F k ) = ψ| ⊗ U,(10)
as shown in Fig. 1(c). Making use of map-state duality gives [ Fig. 1(d)]
k α * k (E ′ k ⊗ F ′T k ) = ψ ′ † ⊗ U ′ .(11)
The map U ′ may in general have rank less than the dimension of H B ⊗ HB or H A ⊗ HĀ (which need not be equal to each other). In this case it will be useful to denote by HB the subspace of H B ⊗ HB which forms the support (or co-image or row space) of U ′ , the orthogonal complement of its kernel (null space), and by HÂ the subspace of H A ⊗ HĀ that forms the range (or image) of U ′ . Each of these subspaces has a dimension equal to D U , and U ′ is a nonsingular (invertible) linear map of HB onto HÂ, which we hereafter denote byÛ . In the same way one can introduce subspaces Hb and Hâ of H b and H a which form the support and range of ψ ′ , and defineψ to be the corresponding nonsingular map of rank D ψ from Hb to Hâ. Next,Ê k is E ′ k with its domain restricted to Hâ, which can be strictly smaller than the support of E ′ k , and with its range restricted to HÂ, which could be smaller than the image of E ′ k . Finally,F T k is F ′T k regarded as a map from H B ⊗ HB to H b , but with domain and range restricted to HB and Hb, respectively 3 .
The result of restricting (11) to the subspaces just defined is
k α * k (Ê k ⊗F T k ) =ψ † ⊗Û ,(12)
corresponding to Fig. 1(e). Multiplying on the left byÛ −1 and tracing over HB gives
k α * kF T kÛ −1Ê k = D Uψ † ,(13)
see Fig. 1(f) and (g). Restricting (8) to subspaces results inÊ
kψF T k = α kÛ ,(14)
see Fig. 2(c). Here we have restricted the spaces over which matrix multiplications are being performed (Hb and Hâ instead of H b and H a ), however equality is still maintained because the dimensions which have been eliminated correspond to the zero Schmidt coefficients of |ψ , which is to say the zero singular values of ψ ′ . To complete the proof, make use of the assumption D ψ = D U . Then D ψ is also the rank ofÊ k and F T k : all four operators in (14) are full rank. Taking the inverse of both sides and inserting the result for U −1 in (13) leads to the result
k |α k | 2ψ−1 =ψ −1 = D ψψ † ,(15)
where k |α k | 2 = 1 follows from (1), (3) and the normalization of |ψ . With |ψ in Schmidt form,ψ is 3 It is significant that we defineF T k as F ′T k restricted to subspaces. In general it is not the case thatF k is equal to F ′ k restricted to subspaces.
diagonal, soψ = I/ D ψ . Therefore all the nonzero Schmidt coefficients of |ψ are equal to 1/ D ψ .
V. LARGER RANK RESOURCE
We have proved that a resource that is of the smallest viable Schmidt rank must be maximally entangled, but it is also possible to use a resource that is of higher Schmidt rank that is not maximally entangled. For one thing, if such a state meets an appropriate majorization criterion it can be deterministically transformed into a maximally entangled state [19]. In this case the larger rank initial resource would have greater entanglement than would be required if the smaller maximally entangled state had been used in the first place. There is however the possibility that some protocol could be devised to use a resource of larger Schmidt rank that has less entanglement than the maximally entangled state of smaller rank.
In fact, we have numerically found examples of such constructions in both SEP and LOCC. One solution in SEP uses a resource state |ψ = √ 0.81 |00 + √ 0.095(|11 + |22 ) on two qutrits to implement the two qubit controlled unitary operator U = diag{1, 1, 1, e iφ } with φ = 2cos −1 (35/36). We have verified this to be an exact solution using a computer algebra system. This resource constitutes less than one ebit of entanglement: the Von Neumann entropy is approximately 0.89 ebits. Since entropy cannot increase under SEP [8] it is necessary for the resource that is consumed to have greater entanglement than the entangling capacity of the unitary being implemented. The entangling capacity of this unitary is shown in [20] to be approximately 0.23 ebits. Since this is much less than the 0.89 ebits that we use, there remains the possibility that a different construction or an even larger rank resource could potentially lower the entanglement cost further.
We also found an LOCC protocol which, though less efficient than the SEP construction just described, allows one to carry out a bipartite unitary deterministically using a resource with less than one ebit of entanglement. The resource in this case is |ψ = √ 0.8 |00 + √ 0.1(|11 + |22 ) and the unitary implemented is U = diag{1, 1, 1, e iφ } with φ = 0.08π. The Von Neumann entropy of this resource is approximately 0.92 ebits, and this is a four round protocol (Alice, Bob, Alice, Bob).
The constructions described above are instances of a more general continuous family of solutions that we have found, covering a range of controlled phase operations. As should be expected, a larger phase φ requires a larger entanglement resource. In both the SEP and the LOCC case only certain classes of solutions were searched for, so it is possible that a more thorough search would provide more efficient protocols. The details of our SEP construction are presented in Appendix A. Our LOCC construction consists of a long list of Kraus operators in numerical form, which is available upon request.
VI. CONCLUSION
We have shown that a unitary operator of Schmidt rank D implemented as a bipartite separable operation requires an entanglement resource of Schmidt rank at least D. If the Schmidt rank of the resource is exactly equal to D, the resource must be uniformly (maximally) entangled with equal nonzero Schmidt coefficients. These restrictions apply also to LOCC, which is a subset of SEP. The proof uses map-state duality in a way which has not (so far as we know) been previously applied to problems of this type, so might have other interesting applications.
Numerical results show that the amount of entanglement required for the resource can be lowered by using a resource of Schmidt rank larger than D. A four round LOCC protocol has been found which uses a two-qutrit resource state with less than one ebit of entanglement to implement a bipartite controlled phase gate (albeit with a small phase).
Although some large classes of unitaries are known to have implementations in LOCC using resources having the minimal Schmidt rank required by Theorem 1(a) [1][2][3][4]6], it is not known whether such minimal-rank implementations are possible for all unitaries. Given a unitary of Schmidt rank D U it is always possible to find a collection of operators {E k ⊗ F k } such that (9) and (3) are satisfied with a resource of Schmidt rank D ψ = D U . But it is not known if there is a separable operation satisfying both (1) and (3). Consequently, it is possible that some unitaries may require a resource of greater rank than the lower bound given in Theorem 1(a). Even if such a minimal rank solution is always possible in SEP, it still might not be possible in LOCC. This stands in contrast to the case of SLOCC where it is known that any unitary can be implemented using a state of Schmidt rank equal to that of the unitary [7].
FIG. 1 .
1Atemporal diagrams, explained in Sec. III. (a) Closure condition, (1). (b) Apply ψ| and simplify using the adjoint ofFig. 2(a) to get(9). (c) Multiply on the right by U to get (10). (d) Apply map-state duality to get(11). (e) Restrict spaces to supports and ranges of operators to get(12). (f) Multiply byÛ −1 . (g) Trace over HB to get(13).
Theorem 1 .
1Suppose that a unitary operator U is implemented deterministically by a separable operation that makes use of the pure state entanglement resource |ψ [i.e. suppose that (1) and (3) hold]. Then (a) The Schmidt rank D ψ of |ψ is greater than or equal to the Schmidt rank D U of U . (b) If the Schmidt ranks are equal, D U = D ψ , then
U : H A ⊗ H B → HĀ ⊗ HB, using as a resource an entangled state |ψ on two ancillary systems H a and H b , by means of a separable operation {E k ⊗ F k }, k = 1, 2, . . .. Here E k : H A ⊗ H a → HĀ and F k : H B ⊗ H b → HB together form a product Kraus operator. For U to be unitary it is necessary that the dimensions of the Hilbert spaces satisfyII. NONLOCAL UNITARIES VIA
SEPARABLE OPERATIONS
We are interested in carrying out a bipartite uni-
tary map
U ′ : H B ⊗ HB → H A ⊗ HĀ (by turning bras into kets on H A and kets into bras on HB), E ′ k : H a → H A ⊗ HĀ (by turning bras into kets on H A ), and F ′T k : H B ⊗ HB → H b (by turning bras into kets on H b and kets into bras on HB),
Although the protocol given in[10] is deterministic in the sense that it always succeeds in a finite number of steps, it is probabilistic in the amount of entanglement required. For any nontrivial unitary there is a chance that the protocol requires usage of the |ψα 2 state, which has one ebit of entanglement. Thus, if the protocol only has access to a state with less than one ebit of entanglement there is a nonzero probability that the protocol cannot be carried out successfully.
It is also possible to formulate map-state duality in a basis independent manner[17], however this is not necessary for the present work.
ACKNOWLEDGMENTSWe thank Vlad Gheorghiu for his comments. The research reported here was supported in part by the National Science Foundation through Grant No. 0757251.Appendix A: Less than one ebit in SEPWe performed a numerical search for solutions to(1)and(3)with the resource and unitary taking the forms |ψ = √ c 0 |00 + (1 − c 0 )/2(|11 + |22 ), (A1)In this case the unitary U is Schmidt rank 2 and the resource is Schmidt rank 3, so the spaces H a and H b are each 3 dimensional. In order to reduce the search space we looked for operators {E k } and {F k } of the formIt is possible to take advantage of the symmetry of the resource |ψ by searching for operator sets of the formwhere l, m, n ∈ {0, 1} and L, M , and N are defined byThere is no loss of generality in this assumption, since if(1)and(3)then so. This decreases the number of independent operators (indexed by k) that need to be solved for, and in fact it turns out to be sufficient to consider only two values of k.Initially we searched for solutions with θ = π/4 and c 0 = 0.6 which, although representing more than one ebit of entanglement, is not majorized by a fully entangled resource of Schmidt rank 2. Once a solution was found, the parameters were variated until a value of c 0 was reached which represented a resource of less than one ebit of entanglement. Further constraints were added and variations made to simplify the solution and identify relations between the parameters. A family of solutions was found of the form (A6) withwhere the parameters x, y, c 0 , and θ must be solved for numerically. The asterisks in T 0 and T 1 represent parameters that can be found using the relation S k ψ ′ T T k = I/4. A sequence of solutions for x, y, c 0 , and θ were fed into an inverse symbolic calculator of our own design which uses a lookup table to convert floating point numbers into algebraic expressions. One of these solutions produced particularly simple algebraic expressions:With this algebraic solution in hand, we used the computer algebra package Sage[21]to verify that this indeed represented an exact (not just approximate to within floating point precision) solution to(1)and(3).
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The Sage Development Team. W Stein, Sage Mathematics Software (Version 4.6.1). W. Stein et al., Sage Mathematics Software (Ver- sion 4.6.1), The Sage Development Team (2010), http://www.sagemath.org.
| [] |
[
"Two Dimensional Incommensurate and Three Dimensional Commensurate Magnetic Order and Fluctuations in La",
"Two Dimensional Incommensurate and Three Dimensional Commensurate Magnetic Order and Fluctuations in La"
] | [
"− X Ba \nDepartment of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMarylandUSA\n",
"Cuo \nNRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada\n",
"J J Wagman \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n",
"G Van Gastel \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n",
"K A Ross \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n\nDepartment of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMarylandUSA\n\nNational Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA\n",
"Z Yamani \nNRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada\n",
"Y Zhao \nNational Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA\n\nDepartment of Materials Sciences and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA\n",
"Y Qiu \nNational Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA\n\nDepartment of Materials Sciences and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA\n",
"J R D Copley \nNational Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA\n",
"A B Kallin \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n\nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n",
"E Mazurek \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n",
"J P Carlo \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n\nNRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada\n",
"H A Dabkowska \nBrockhouse Institute for Materials Research\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n",
"B D Gaulin \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n\nBrockhouse Institute for Materials Research\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n\nCanadian Institute for Advanced Research\n180 Dundas St. WM5G 1Z8TorontoOntarioCanada\n"
] | [
"Department of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMarylandUSA",
"NRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMarylandUSA",
"National Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA",
"NRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada",
"National Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA",
"Department of Materials Sciences and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA",
"National Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA",
"Department of Materials Sciences and Engineering\nUniversity of Maryland\n20742College ParkMarylandUSA",
"National Institute of Standards and Technology\n20899-6102GaithersburgMarylandUSA",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"NRC, Chalk River Laboratories\nCanadian Neutron Beam Centre\nChalk RiverK0J 1J0OntarioCanada",
"Brockhouse Institute for Materials Research\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Brockhouse Institute for Materials Research\nMcMaster University\nL8S 4M1HamiltonOntarioCanada",
"Canadian Institute for Advanced Research\n180 Dundas St. WM5G 1Z8TorontoOntarioCanada"
] | [] | We present neutron scattering measurements on single crystals of lightly doped La2−xBaxCuO4, with 0 ≤ x ≤ 0.035. These reveal the evolution of the magnetism in this prototypical doped Mott insulator from a three dimensional (3D) commensurate (C) antiferromagnetic ground state, which orders at a relatively high TN , to a two dimensional (2D) incommensurate (IC) ground state with finite ranged static correlations, which appear below a relatively low effective TN . At low temperatures, the 2D IC magnetism co-exists with the 3D C magnetism for doping concentrations as low as ∼ 0.0125. We find no signal of a 3D C magnetic ground state by x ∼ 0.025, consistent with the upper limit of x ∼ 0.02 observed in the sister family of doped Mott insulators, La2−xSrxCuO4. The 2D IC ground states observed for 0.0125 ≤ x ≤ 0.035 are diagonal, and are rotated by 45 degrees within the orthorhombic basal plane compared with those previously reported for samples with superconducting ground states: La2−xBaxCuO4, with 0.05 ≤ x ≤ 0.095. We construct a phase diagram based solely on magnetic order parameter measurements, which displays much of the complexity of standard high temperature superconductivity phase diagrams discussed in the literature. Analysis of high energy-resolution inelastic neutron scattering at moderately low temperatures shows a progressive depletion of the very low energy dynamic magnetic susceptibility as x increases from 0.0125 to 0.035. This low energy, dynamic susceptibility falls off with increasing temperature on a scale much higher than the effective 2D IC TN appropriate to these materials. Appreciable dynamic 2D IC magnetic fluctuations inhabit much of the "pseudogap" regime of the phase diagram. arXiv:1304.0362v1 [cond-mat.supr-con] | 10.1103/physrevb.88.014412 | [
"https://arxiv.org/pdf/1304.0362v1.pdf"
] | 119,289,163 | 1304.0362 | 023798722515c64b4157582f25574cd6c78c6006 |
Two Dimensional Incommensurate and Three Dimensional Commensurate Magnetic Order and Fluctuations in La
− X Ba
Department of Physics and Astronomy
Johns Hopkins University
21218BaltimoreMarylandUSA
Cuo
NRC, Chalk River Laboratories
Canadian Neutron Beam Centre
Chalk RiverK0J 1J0OntarioCanada
J J Wagman
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
G Van Gastel
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
K A Ross
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
Department of Physics and Astronomy
Johns Hopkins University
21218BaltimoreMarylandUSA
National Institute of Standards and Technology
20899-6102GaithersburgMarylandUSA
Z Yamani
NRC, Chalk River Laboratories
Canadian Neutron Beam Centre
Chalk RiverK0J 1J0OntarioCanada
Y Zhao
National Institute of Standards and Technology
20899-6102GaithersburgMarylandUSA
Department of Materials Sciences and Engineering
University of Maryland
20742College ParkMarylandUSA
Y Qiu
National Institute of Standards and Technology
20899-6102GaithersburgMarylandUSA
Department of Materials Sciences and Engineering
University of Maryland
20742College ParkMarylandUSA
J R D Copley
National Institute of Standards and Technology
20899-6102GaithersburgMarylandUSA
A B Kallin
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
Department of Physics and Astronomy
University of Waterloo
N2L 3G1WaterlooOntarioCanada
E Mazurek
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
J P Carlo
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
NRC, Chalk River Laboratories
Canadian Neutron Beam Centre
Chalk RiverK0J 1J0OntarioCanada
H A Dabkowska
Brockhouse Institute for Materials Research
McMaster University
L8S 4M1HamiltonOntarioCanada
B D Gaulin
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
Brockhouse Institute for Materials Research
McMaster University
L8S 4M1HamiltonOntarioCanada
Canadian Institute for Advanced Research
180 Dundas St. WM5G 1Z8TorontoOntarioCanada
Two Dimensional Incommensurate and Three Dimensional Commensurate Magnetic Order and Fluctuations in La
We present neutron scattering measurements on single crystals of lightly doped La2−xBaxCuO4, with 0 ≤ x ≤ 0.035. These reveal the evolution of the magnetism in this prototypical doped Mott insulator from a three dimensional (3D) commensurate (C) antiferromagnetic ground state, which orders at a relatively high TN , to a two dimensional (2D) incommensurate (IC) ground state with finite ranged static correlations, which appear below a relatively low effective TN . At low temperatures, the 2D IC magnetism co-exists with the 3D C magnetism for doping concentrations as low as ∼ 0.0125. We find no signal of a 3D C magnetic ground state by x ∼ 0.025, consistent with the upper limit of x ∼ 0.02 observed in the sister family of doped Mott insulators, La2−xSrxCuO4. The 2D IC ground states observed for 0.0125 ≤ x ≤ 0.035 are diagonal, and are rotated by 45 degrees within the orthorhombic basal plane compared with those previously reported for samples with superconducting ground states: La2−xBaxCuO4, with 0.05 ≤ x ≤ 0.095. We construct a phase diagram based solely on magnetic order parameter measurements, which displays much of the complexity of standard high temperature superconductivity phase diagrams discussed in the literature. Analysis of high energy-resolution inelastic neutron scattering at moderately low temperatures shows a progressive depletion of the very low energy dynamic magnetic susceptibility as x increases from 0.0125 to 0.035. This low energy, dynamic susceptibility falls off with increasing temperature on a scale much higher than the effective 2D IC TN appropriate to these materials. Appreciable dynamic 2D IC magnetic fluctuations inhabit much of the "pseudogap" regime of the phase diagram. arXiv:1304.0362v1 [cond-mat.supr-con]
I. INTRODUCTION
The 214 family of cuprates, La 2−x Ba x CuO 4 , and La 2−x Sr x CuO 4 , are among the most studied of the high temperature superconductors (HTS) [1][2][3] . Most of this work has focussed on La 2−x Sr x CuO 4 , which has been available in large, pristine single crystal form for some time 4 . Although La 2−x Ba x CuO 4 was the original HTS family to be discovered 5 , its study has been greatly restricted due to the difficulty of its single crystal growth. These difficulties have now been largely overcome for relatively low doping levels: x ≤ 0.15. While there are many similarities between the magnetic and superconducting properties of the two 214 families of HTS 6,7 , there are also important differences. For example, a low temperature tetragonal phase of La 2−x Ba x CuO 4 exists for 0.05 ≤ x < 0.15 [8][9][10] , and superconductivity is almost completely suppressed at x = 0.125, a phenomenon which is referred to as "the 1/8 anomaly" 11 .
For HTS, the parent, undoped, compounds are Mott insulators which display three dimensional (3D), commensurate (C) antiferromagnetic (AF) ground states 12,13 . This 3D C AF ground state is remarkably sensitive to the presence of mobile, doped holes, and less sensitive to the presence of doped mobile electrons 14 . For hole doping, relevant to Ba in La 2−x Ba x CuO 4 , Sr in La 2−x Sr x CuO 4 and oxygen, in YBa 2 Cu 3 O 6+x , the 3D C AF ground state is very quickly destroyed 15 . This occurs, for example, for x >∼ 0.02 in La 2−x Sr x CuO 4 16,17 . Upon further introduction of holes, a superconducting ground state is obtained for x ∼ 0.05 18,19 . The superconducting T C increases with increased doping and an optimally high superconducting T C is achieved near x ∼ 0.17 [20][21][22] .
Two dimensional (2D) incommensurate (IC) spin structures and dynamics, exhibited by samples with hole-doping concentrations beyond those that destroy the 3D C magnetic order, have been studied in several families of HTS. Inelastic neutron scattering studies are consistent with an "hour glass" dispersion, wherein low energy spin excitations disperse out of IC wavevectors and merge or nearly-merge at the C wavevector to form a resonant spin excitation [23][24][25][26][27][28][29][30] . At higher energies, the excitations disperse out from the C wavevector, before turning over near the Brillouin zone boundaries 23,31,32 . Such a picture has been shown to be relevant even in the relatively low hole-doping regime of La 2−x Sr x CuO 4 , where the insulating ground state is characterized by socalled "diagonal" IC spin order 33 .
The resulting phase diagrams for these families of HTS materials have led some observers to conclude that magnetism and superconductivity are closely linked, as these ground states are either contiguous or almost contiguous to each other. In contrast, others have concluded these ground states compete, as each inhabits a different part of the phase diagram. From either perspective it is important that the microscopic magnetic properties be characterized and well understood across the phase diagram. Even in the underdoped, non-superconducting regime, for x < 0.05 in LBCO and LSCO, the magnetic phase behavior and properties change quickly with doping. This paper seeks to elucidate this evolution of magnetic properties in LBCO, using a variety of neutron scattering techniques. We specifically report on the magnetic structure and dynamics of La 2−x Ba x CuO 4 for doping levels 0 ≤ x ≤ 0.035, and study how the 3D C magnetism evolves into 2D IC magnetism. We construct a phase diagram for La 2−x Ba x CuO 4 based solely on magnetic neutron scattering order parameter measurements and show that it possesses much of the full complexity of conventional HTS phase diagrams based on magnetic and transport measurements. Finally, using time-of-flight neutron scattering techniques, we report on low energy 2D IC spin dynamics in La 2−x Ba x CuO 4 for x ≤ 0.035. We observe the low energy dynamic susceptibility to evolve with temperature on a much higher temperature scale than that given by the effective 2D IC T N for any doping, and show that 2D IC dynamic spin correlations inhabit much of phase diagram associated with the "pseudogap" state 34 .
We will focus our discussion on the 214 cuprate HTS families.
For the low doping levels we are considering, La 2−x Ba x CuO 4 and La 2−x Sr x CuO 4 are isostructural 35,36 . At temperatures that are high relative to room temperature, these crystals display tetragonal crystal structures with space group I4/mmm. As the temperature is lowered, they undergo a structural phase transition to an orthorhombic structure with space group Bmab. This transition occurs near ∼ 308 K for x ∼ 0.08 in LBCO 9 , and increases in temperature with decreasing doping. We will be presenting measurements on LBCO for x ≤ 0.035 and T ≤ 300 K. As such, our samples are orthorhombic at all temperatures measured. The lattice parameters in the orthorhombic basal plane are similar, and this has lead some to treat the orthorhombic cell as tetragonal for convenience 10 . We too shall adopt this simplification. Doing so, we label the C AF wavevector as ( 1 2 , 1 2 , L) and the diagonal IC ordering wavevectors, relevant for LBCO with x < 0.05, as having the form ( 1 2 ± δ, 1 2 ± δ, L) and ( 1 2 ∓ δ, 1 2 ± δ, L).
II. EXPERIMENTAL DETAILS
High quality single crystals of La 2−x Ba x CuO 4 with x = 0, 0.006, 0.0125, 0.025 and 0.035 were grown by floating zone image furnace techniques using a four-mirror optical furnace. The growth method has been reported on previously 37,38 . The resulting samples were cylindrical in shape and weighed ∼ 7 grams each. The crystals were all grown in the same excess oxygen atomosphere resulting in a small oxygen off-stoichiometry. This oxygen off-stoichiometry could be estimated by measuring the 3D C AF phase transition in undoped La 2 CuO 4+δ , as T N is known to be sensitive to the precise value of δ 39 . From a determination that T N ∼ 250 K for our La 2 CuO 4+δ single crystal, we estimate that δ ≈ 0.004. We expect this to be the same for all of our La 2−x Ba x CuO 4+δ samples as they were grown under similar conditions. Hereafter, we will not refer to the oxygen off-stoichiometry in the crystals.
Neutron scattering measurements were performed using both time-of-flight and triple axis spectrometers. These measurements were carried out using several different cryostats, allowing access to the approximate temperature range 1.5 K to 300 K. Three sets of triple axis neutron measurements were performed at two laboratories. All measurements were performed with the horizontal scattering plane coincident with the HK0 plane of the crystal. Two of these triple axis measurements employed a constant final energy E f = 14.7 meV. The first was a set of measurements at the N5 beamline of the NRU reactor at Chalk River Laboratories, which employed a collimation of [open-36'-48'-72'] using the convention of collimation between [source-monochromator, monochromator and sample, sample and analyser, analyser and detector]. The second was a set of measurements using the HB3 instrument at Oak Ridge National Laboratory, which employed [48'-40'-40'-120']. Both sets of measurements employed a pyrolitic graphite filter in the scattered beam to suppress harmonic contamination, and both had an approximate energy resolution of ∼ 1 meV. High resolution, elastic scattering measurements were also made at Chalk River using E f = 5.1 meV and collimation of [open, 12', 12', 72']. Both sets of measurements performed at Chalk River employed a cooled beryllium filter for suppression of higher harmonic incident neutrons. Time-of-flight neutron scattering measurements were performed using the NG4 Disk Chopper Spectrometer (DCS) 40 at the National Institute of Standards and Technology (NIST) Center for Neutron Research. All DCS measurements presented here were performed using an incident neutron wavelenth of 5Å, and a corresponding energy resolution of ∼ 0.09 meV. The measurements at DCS were performed with the HHL plane of the crystals coincident with the horizontal plane. plane for x = 0.0125 taken at 3 K and 125 K. The two temperature data sets have been shifted for ease of viewing. b) Order parameter measurement for x = 0.0125. The solid circle data are measured at the high intensity "hot spot" identified in the (HK0) map in a). The cross hatched data are collected within the ellipse of elastic scattering in a), but away from the "hot spot". c) and d) Low-resolution, elastic scattering measurements of the 2D IC order parameters for x = 0.025 and 0.035 respectively. Dashed lines serve as guides to the eye.
III. RESULTS AND DISCUSSION
A. Magnetic Order Parameter Measurements
Magnetic order parameter measurements of 3D Bragg peaks using triple axis spectrometers are relatively straight forward to perform, compared with the corresponding measurement of 2D Bragg signatures. This is because 3D ordered systems display Bragg spots in reciprocal space, while 2D Bragg signatures appear as rods in reciprocal space. If the strength of the elastic magnetic scattering is otherwise the same, the neutron intensity at a single Q position is much larger in the 3D case as the signal from the 3D ordered state is localised at a resolution-broadened point in reciprocal space, rather than along a rod in the 2D case. Such measurements for a sample of the nominally undoped LBCO x = 0 yield a sharp onset to 3D C Bragg scattering at ( 1 2 , 1 2 , 0) for T N = 250 K. Similar measurements were carried out on x = 0.006 and x = 0.0125 samples. Figure 1 shows the results of such order parameter measurements for x = 0.0125, 0.025, and 0.035 LBCO samples. Figure 1 also shows high resolution elastic triple axis measurements performed with tight collimations and 5.1 meV neutrons for the 3D C Bragg peak near Q = ( 1 2 , 1 2 , 0) in the x = 0.0125 sample. Mesh scans in reciprocal space taken at T = 3 K and T = 125 K are shown in Fig. 1 a), with the two data sets artificially displaced from each other in the figure for clarity. In addition, the intensity scale of each data set has been normalized such that the peak intensity is unity. A broad, elliptical distribution of elastic scattering is observed at all temperatures below ∼ 200 K. However, as can be seen by comparing the T = 3 K map with the intensity-normalized map at T = 125 K in Fig. 1 a), high intensity "hot spot" develop within the ellipse of elastic scattering for temperatures less than ∼ 150 K.
It is possible to follow the temperature dependence of the "hot spot" scattering and that in the weaker periphery of the ellipse. Many such measurements were made. The temperature dependence of the sum of all the Bragg scattering at high intensity hot spots and at low intensity positions within this ellipse are shown in Fig. 1 b). One can see that these two sets of temperature dependencies are the same above ∼ 150 K, where both the hot spots and the periphery of the Bragg positions show upwards curvature as a function of decreasing temperature. Below ∼ 150 K, the two temperature dependencies markedly depart from each other, with the intensity at the hot spot (solid circle data points) positions becoming much stronger than that at the corresponding low intensity positions (cross hatched data points). We therefore identify T N = 150 K for 3D C order in our x = 0.0125 single crystal sample of LBCO.
For the same x = 0.0125 data set at ∼ 25 K, we observe a pronounced drop off in the intensity of the 3D C AF Bragg scattering. This can be seen in the "hot spot" order parameter of Fig. 1 b), which corresponds to the high intensity positions of the reciprocal space map shown in Fig. 1 a). As we will see, this decrease in intensity is associated with the development of coexisting 2D IC elastic scattering, which occurs with an "effective" T N of ∼ 25 K for x = 0.0125. Similar high-resolution elastic magnetic Bragg scattering measurements were performed on La 2−x Ba x CuO 4 with x = 0.025 and 0.035, with no obvious sign of 3D C AF order in either sample. Lower-resolution elastic scattering measurements were performed using 14.7 meV neutrons and relatively coarse collimation, looking explicitly for 2D IC order at appropriate 2D IC diagonal wavevectors Q = ( 1 2 ± δ, 1 2 ± δ, L), with δ ∼ x and L = 0, as required for measurements within the HK0 scattering plane. The scattering at these 2D IC Bragg positions is weak even at low temperatures, as expected for constant-Q elastic scattering measurements of a 2D rod of scattering. Nonetheless effective 2D IC magnetic order parameters were measured and these are shown for x = 0.025 and 0.035 samples in Fig. 1 c) and 1 d) respectively. From these measurements we identify "effective" 2D T N of ∼ 18 K and 23 K for x = 0.025 and x = 0.035, respectively.
B. Time-of-Flight Elastic Neutron Scattering Measurements
The DCS time-of-flight spectrometer was used to measure reciprocal space maps of both the elastic, -0.09 meV ≤hω ≤ 0.09 meV, and inelastic magnetic scattering, 0.09 meV ≤hω ≤ ∼ 0.8 meV, from our lightly-doped LBCO samples, as a function of temperature. Time-offlight data are shown in Figs. 2, 3, and 4. Figure 2 shows maps of the elastic scattering within the HHL scattering plane around ( 1 2 , 1 2 , L) for four different dopings of La 2−x Ba x CuO 4 : x = 0, 0.0125, 0.025, and 0.035. The top panels of Fig. 2 shows these maps taken within the ground state of the samples, at T = 1.5 K. The bottom panels show the same elastic scattering HHL maps for the same four samples, but now taken at T = 35 K. This is still at a low temperature, but above 25K, which is the "effective" 2D T N for x = 0.0125 and is the highest for any of these samples. The ranges of L shown were chosen to avoid complications due to absorbtion by the sample.
Three types of Bragg diffraction features can be seen in these reciprocal space maps. Two of these features are 3D C Bragg peaks of the form ( 1 2 , 1 2 , L = even and L = odd). The ( 1 2 , 1 2 , L = even) 3D C Bragg peaks at L = -2 for x = 0 and x = 0.0125, are nuclear-allowed Bragg peaks. The ( 1 2 , 1 2 , L = odd) 3D C Bragg peak at L = -3 for x = 0 and x = 0.0125 is magnetic in origin. Such 3D C magnetic Bragg peaks are absent at all temperatures for the x = 0.025 and 0.035 samples, and for the x = 0 and x = 0.0125 above their 3D T N s, ∼ 250 and 150 K respectively.
One clearly observes rods of magnetic elastic scattering of the approximate form ( 1 2 , 1 2 , L) for all x except x = 0. These are centred on diagonal IC wavevectors ( 1 2 ± δ, 1 2 ± δ, L). Note that δ ∼ x is small at these low dopings. The rods of scattering are clearly distinct from the Bragg "spots" which signify 3D order. Furthermore, these rods show little or no L dependence, a fingerprint of highly-correlated 2D planes of Cu spin 1 2 magnetic moments, which are largely decoupled from each other. The only L dependence which is observed in our measurements is that associated with self-absorption of the sample in the neutron beam, due to the fact that the cylndrical axis of the crystals is not normal to the scattering plane. Figure 2 shows that the 3D C magnetic order in the x = 0 and 0.0125 samples is largely unaffected by raising the temperature from 1.5 K to 35 K. In the x = 0.0125 sample, 2D IC static correlations co-exist with 3D C AF order at T = 1.5 K, but no signal of the 2D IC static magnetic scattering remains by T = 35 K, leaving only the 3D C AF order. In both the x = 0.025 and 0.035 samples, only 2D IC static magnetic order exists within the ground state, while the 3D C AF order is absent. This is consistent with the low-resolution triple axis measure-ments on the x = 0.025 and 0.035 samples shown in Fig. 1 c) and d), wherein the 2D IC magnetism disappears at relatively low temperatures, but above their effective 2D T N of ∼ 23 K and 18 K, respectively. The appearance of the 2D rods of scattering below ∼ 25 K in the x = 0.0125 sample correlates nicely with the suppression of its 3D C magnetic Bragg scattering shown in Fig. 1 a).
The temperature dependence of the magnetic elastic scattering in the x = 0.0125 samples bears further attention as both 3D C AF Bragg peaks and 2D IC rods of magnetic scattering coexist within the ground state. We have already seen that the temperature dependence of the 3D C AF Bragg peak for x = 0.0125, shown in Fig. 1 b), has a reduction in the scattered intensity below ∼ 25 K. The top set of panels in Figure 3 shows the same ( 1 2 , 1 2 , L) elastic reciprocal space map for x = 0.0125 shown in Fig. 2, now as a function of temperature. For now, we will focus only on the top panels and will return to the bottom panels when we discuss the inelastic scattering in a later section. We clearly see the disappearance of the rod of elastic scattering as the temperature increases to T = 35 K, and that the 3D C AF peak near ( 1 2 , 1 2 , −3) has all but disappeared at T = 160 K, above the 3D C T N ∼ 150 K, indentified in Fig. 1 b) from high-resolution triple axis order parameter measurements. We note that the nuclear Bragg peak at ( 1 2 , 1 2 , −2) is nearly temperature independent over the range of temperature shown. This is as expected for a nuclear Bragg peak, given that all temperatures studied are well removed from the orthorhombic-tetragonal structural phase transition in this material 41 .
The trade-off between 3D C AF and 2D IC static magnetism shown in Fig. 3 and Fig. 1 b) is similar to that reported for La 2−x Sr x CuO 4 at similar doping levels 2 . It implies that the 3D C AF structure forms as the temperature is lowered, but that part of this structure is unstable to the formation of 2D IC order below the 2D effective T N of 25 K in the x = 0.0125 sample. While it is not easy to compare the integrated intensity of the 2D rod scattering to the 3D C AF Bragg scattering, it is straightforward to estimate the reduction of the 3D C AF Bragg peak from saturation shown below ∼ 25 K in Fig. 1 b). This shows that for x = 0.0125 the 2D IC static order accounts for 20% of the elastic magnetic scattering in the ground state. As suggested for La 2−x Sr x CuO 4 , this fraction presumably grows with x until it accounts for 100% of the static elastic magnetic scattering in the ground state for x ≥ 0.02 18 .
The HHL reciprocal space maps around ( 1 2 , 1 2 , L), shown in Fig. 2, all cover the same range in (HH). Note that the L ranges shown differ due to the fact that the self absorbtion for a given position in the (HHL) plane differs for the four crystals. It is clear that at T = 1.5 K (the top panels in Fig. 2) the rod of magnetic scattering broadens in the (HH) direction progressively with increasing doping from x = 0.0125 to 0.035. This is due to the fact that 2D IC static order is expected to change its incommensuration with doping, and the expected dependence is roughly δ ∼ x in the diagonal IC wavevevctor ( 1 2 ± δ, 1 2 ± δ, L) 42,43 . We can explicitly examine the lineshape associated with the 2D IC rods of magnetic scattering and estimate both the δ vs x dependence in the ground state, and the finite-range of the in-plane spin correlations within the 2D IC structure. In Fig. 4, we show cuts in the (HH) direction through the reciprocal space maps displayed in the top row of Fig. 2 appropriate to T = 1.5 K. For the x = 0.0125, 0.025, and 0.035 data sets, the cuts are taken so as to pick out only the 2D IC rod scattering; that is they sample data between the nuclear allowed ( 1 2 , 1 2 , L = even) 3D Bragg peaks for all samples, as well as avoiding the 3D C magnetic Bragg peaks at ( 1 2 , 1 2 , L = odd) for the x = 0.0125 sample. For comparison, we also have a cut through the L = -4 structural Bragg peak in the x = 0.0125 sample as a measure of the instrumental resolution.
These cuts are shown in Fig. 4 a) and b). The x = 0.0125 3D C data set is clearly much narrower in (HH) than that of any of the other three data sets, which exhibit 2D diagonal IC order. The three 2D IC data sets were fit phenomenologically to a functional form of two squared-Lorentzians with identical widths and amplitudes, but centered at different HH positions. Previous studies of such quasi-two dimensional correlations also employed Lorentzian-squared lineshapes to describe the IC elastic scattering 33 . Initially, these data were fit with the sum of two Lorentzians-squared lineshapes wherein their widths were allowed to vary with x. However, the resulting variation of the width with x was small, and the fits were redone using a common width for the Lorentzian-squared lineshapes in all fits.
As the finite (HH) width to the Lorentzian-squared lineshape represents a finite (inverse) correlation length, we conclude that the 2D IC static order is short ranged in La 2−x Ba x CuO 4 , with a correlation length of ∼ 20Å. Over this doping range and to within our resolution, this correlation length is independent of doping. The diag-onal IC wavevector δ can then be extracted from this analysis, and this is shown as a function of x at T = 1.5 K in the inset to Fig. 4 a). We observe a linear relationship δ ∼ x for x = 0.125, 0.025, and 0.035, which extrapolates back through zero at x = 0. This conclusion is somewhat different from that reached in previous studies of La 2−x Sr x CuO 4 , wherein a linear δ ∼ x relationship was also found for sufficiently large x, but δ was ∼ independent of x for very low concentrations < 0.02, which also displayed 3D C AF order 42,44 .
C. Time-of-flight Inelastic Scattering Measurements and Dynamic Susceptibility
The DCS time-of-flight instrument allows the simultaneous measurement of elastic neutron scattering and inelastic neutron scattering. Reciprocal space maps of the inelastic scattering can also be constructed, similar to the elastic scattering data presented in Figs. 2 and 3. The relatively low incident energy, E i , employed in these measurements restricts the accessible inelastic scattering to less than ∼ 1 meV energy transfer, although the magnetic excitations in this system are known to exist to significantly higher energy 31 . We have plotted inelastic scattering for the x = 0.0125 sample as a function of temperature in the bottom panels of Fig. 3. A comparison between this magnetic inelastic scattering from the 0.0125, 0,025, and 0.035 samples, all at T = 35 K, is shown in the top panel of Fig. 5.
The bottom panels of Fig. 3 show reciprocal space maps of the inelastic scattering from the x = 0.0125 sample, integrated in energy from 0.1 meV to 1 meV, and as a function of temperature between T = 1.5 K and T = 300 K. This integrated inelastic magnetic scattering can be compared directly to the same reciprocal space maps of the elastic scattering around ( 1 2 , 1 2 , L) wavevevectors as shown in the top panels for Fig. 3. On this relatively low energy scale, we observe an interesting trend wherein little inelastic scattering is observed at T = 1.5 K, although both the 2D IC elastic rod of magnetic scattering and the 3D C AF Bragg peaks are strong. As the elastic rod of 2D IC scattering for the x = 0.0125 sample fades in intensity above T = 15 K, the inelastic scattering becomes clearly evident. Above the 2D effective T N ∼ 23 K, only the 2D IC inelastic scattering and the 3D C elastic magnetic scattering remain. The intensity of the 2D IC inelastic scattering is prevalent out to 160 K, but has clearly faded at the highest temperature measured, T = 300 K. The lower panels of Fig. 3 show 2D IC dynamic spin fluctuations in the x = 0.0125 sample are present well above the effective 2D T N ∼ 23 K, and only completely disappear above the temperature characteristic of the 3D C T N , ∼ 150 K, in this sample.
The evolution of the low energy inelastic magnetic scattering and the corresponding imaginary part of the dynamic susceptibility as a function of doping is shown in Fig. 5. The top panel of Fig. 5 a) shows the inelastic scattering at T = 35 K for each of the x = 0.0125, 0.025 and 0.035 samples. There data has been integrated in L using −3.5 < L < −1 for x = 0.0125 and 0 < L < 1.9 in x = 0.025 and 0.035. The reason for this choice of L integration is that these regimes avoid complications due to self absorption that arise as the sample is rotated in the beam. This data is plotted in an energy vs. (HH) wavevevector map, over the approximate range in energy from 0.15 meV to 0.8 meV. A temperature of 35 K was chosen for this comparison as it is sufficiently low to approximate the ground state, while high enough such that appreciable magnetic inelastic intensity is evident in all samples. We note that there is little magnetic inelastic scattering evident at T = 1.5K. We take advantage of this and use the T = 1.5K data sets as a measure of the inelastic background for our samples. This will be important in isolating the dynamic magnetic susecptibility from our inelastic scattering data.
The magnetic inelastic scattering, expressed as S(Q, hω, T) is the product of two terms; the Bose population factor which maintains detailed balance, and the imaginary part of the dynamic susceptibility χ . χ is an odd function of energy and characterizes the capacity of the system to absorb energy, thereby creating spin excitations at a particular wavevector and energy. The inelastic magnetic scattering, S(Q,hω, T) is then related to the imaginary part of the dynamic susceptibility through:
S(Q,hω, T ) = [n(hω, T ) + 1] × χ (Q,hω, T )(1)
where [n(hω, T ) + 1] is the Bose population factor:
[n(hω, T )
+ 1] = 1 1 − e −hω k B T(2)
At T = 35 K and for energies ≤ 1 meV, the Bose population factor, [n(hω, T )+1], is sufficiently strong that the overall neutron scattering signal, S(Q,hω) can be easily distinguished from background for all concentrations. One can isolate S(Q,hω) with an appropriate subtraction and divide through by the Bose factor to give the imaginary part of the dynamic susceptibility, χ . This is what is shown in the b) panels of Fig. 5 for x = 0.0125, 0.025, and 0.035, listed from left to right. The a) panels of Fig. 5 show the corresponding S(Q,hω). Focussing on χ (Q,hω) in the b) panels of Fig. 5, we see a suppression of χ (Q,hω) at low energies. This suppression increases with doping between 0.0125 and 0.035. In Fig. 6 we show cuts of χ (Q,hω) made by integrating the data in Fig. 5 b) in (HH) around 0.48 ≤ (HH) ≤ 0.52 and over the relevant L-range so as to capture all the dynamic magnetic susceptibility in this low energy regime. The resulting quantity is then plotted in Fig. 6 as a function of energy for x = 0.0125, 0.025, and 0.035. We see a suppression of the low energy dynamic susceptibility as the doping increases. This can be quantified by fitting the energy dependence of this integrated low energy dynamic susceptibility to the phenomenological form 45 :
χ (E) = A × tan −1 (E/γ)(3)
This allows the extraction of a characteristic energy scale, γ, at which the magnetic dynamic susceptibility, as a function of decreasing energy, turns over and decreases FIG. 6. The energy dependence of χ (Q,hω) integrated over the HH width of the rod of scattering shown in Fig. 5 b), as described in the text. The dashed line shows a fit to a phenomenological model, Eq. 3, describing this energy dependence.
towards zero, as it must in order to be an odd function of energy. The fit is displayed as the dashed lines in Fig. 6, and the appropriate γ value resulting from the fit is displayed in the left corner of each panel.
As expected from Fig. 5 b), γ is lowest for the x = 0.0125 sample, and increases with increased hole doping, x. It is interesting to note that this progression is established in samples that are not superconducting. One might expect this phenomena to be linked to the superconducting gap that is observed in LSCO. There, it is known that for samples of LSCO with superconducting ground states, that is x ≥ 0.05, a spin gap forms for T < T C within the dynamic susceptibility at low energies. For example, for samples with x = 0.16 it is reported that the gap is 7 meV 46 . That said, it has also been reported that no such corresponding spin gap exists in LBCO out to at least x = 1/8 37 . This has been motivated by the absence of a temperature dependence to the low energy dynamic susceptibility in underdoped superconducting samples. However, there has been suggestion that a superconducting gap may exist for higher dopings 10 . To be sure, the present spin gap related phenomena may be different from the related phenomena which occurs in LSCO. But, it is clear that an interesting depletion in dynamic susecptibility as a function of increasing doping seems to be a characteristic of LBCO as well.
We now examine the temperature dependence of χ (Q,hω) for x = 0.0125 and 0.035 samples in Figs. 7, 8, and 9. Figure 7 shows χ (Q,hω) for x = 0.0125 in energy vs. (HH) maps over the range of energy from 0.15 to 0.8 meV, again integrated in L around the ranges appropriate to isolate 2D rods of scattering, as used in Fig. 5. These data sets are at temperatures ranging from 10 K to 300 K, as denoted in the bottom left of each panel, and all data sets used T = 1.5 K data sets as background. Figure 8 shows the same χ (Q,hω) maps for the x = 0.035 sample over the temperature range T = 10 K to T = 35 K, again using the appropriate T = 1.5 K data set as a background.
In both the x = 0.0125 and 0.035 cases, χ (Q,hω) clearly decreases monotonically with increasing temperature over this relatively low energy range. The intriguing behavior seen in the bottom panels of Fig. 3 for the x = 0.0125 sample, wherein the 2D IC inelastic intensity appears to have a temperature dependence complementary to that of the 2D IC elastic scattering, can be understood as a consequence of the temperature dependence of the Bose factor, [n(ω) + 1].
To better understand χ (Q,hω, T ) quantitatively, we integrated the χ (Q,hω) data shown in Figs. 7 and 8 in energy between 0.2 and 0.8 meV. This was then fit to a Gaussian lineshape centred on HH=( 1 2 , 1 2 ) with a linear background. The integrated intensity of the Gaussian gives χ (Q ∼ ( 1 2 , 1 2 , L), 0.2meV ≤hω ≤ 0.8meV ), which is plotted as a functon of temperature on a semi-log scale in Fig. 9 for both x = 0.0125 and 0.035 samples. The signals from the x = 0.0125 and 0.035 samples have been approximately normalized at low temperatures. The values of T N (2DIC) for x = 0.0125 (25K) and 0.035 (15K) as well as T N (3DC) for x = 0.0125 (150K) and x = 0 (250K) are indicated for reference as dashed lines in Fig. 9. We find that the dynamic IC magnetism in both samples is present on a temperature scale that is independent of the static ordering temperatures in either system. χ (Q,hω) is strongest at low temperatures in both materials, and its temperature dependence does not suggest a well defined transition temperature. The phenomenon observed is instead consistent with a cross-over that occurs at some temperature above the 3D C T N for the x = 0.0125 system.
D. Magnetic Phase Diagram
We summarize our elastic and inelastic magnetic neutron scattering measurements on relatively lightly-doped La 2−x Ba x CuO 4 in the phase diagram shown in Fig. 10. It displays three sets of points which represent phase transition temperatures appropriate to 3D C AF order (red circles), 2D diagonal IC static order (yellow triangles), and 2D parallel IC order (blue circles). The latter set of phase transitions occur for concentrations with superconducting ground states for x ≥ 0.05, and comes from our earlier neutron results on magnetic order pa- rameter measurements 6,47 .
We also show extended regions on the phase diagram where 2D dynamic IC magnetism is observed. As seen in Fig. 9, this dynamic 2D IC magnetism gradually fades with increasing temperature and does not display an obvious phase transition. This dynamic 2D IC magnetism occupies the same general region of the HTS phase diagram associated with the "pseudogap phase". The pseudogap phase has been ascribed to several different origins, including phase-incoherent superconducting pairs 48 , and ordering associated with orbital currents 49 .
Whatever other properties it possesses, it is clear that 2D IC spin fluctuations are strong throughout this entire region and that the crossover to a fully paramagnetic state occurs on a high temperature scale.
Coming back to the 3D C and 2D IC phase transitions identified from elastic neutron scattering order parameter measurements, shown as the circles, triangles, and squares respectively in Fig. 10, there are several interesting observations to make. First and foremost, Fig. 10 is compiled exclusively from magnetic order parameter measurements. Nevertheless, it displays much of the full complexity of the HTS phase diagram. In our opinion, such an observation in and of itself leads to the conclusion that the superconducting ground state is intimately related to the magnetic ground state. Second, the fact that the 2D effective T N is so much smaller than the 3D C T N is due to the decrease in dimensionality. This is clear from Fig. 2, which shows the 2D rods of magnetic scattering co-existing with 3D C magnetic Bragg peaks for the x = 0.0125 sample, only at low temperature. In the HTS literature, the region of the phase diagram between 3D C AF order and a superconducting ground state, which is typically 0.02 ≤ x ≤ 0.05, is often refered to as a spin glass regime 2 . This is correct in that the ground state spin correlations within the orthorhombic basal plane of these samples are finite and elastic. However, most importantly, the spin correlation lengths between orthorhombic planes have gone to ≈ zero resulting in distinct rods of magnetic scattering; that is the layers are decoupled. This reduction in magnetic dimensionality from 3D to 2D, on its own, would be expected to strongly suppress any ordering transition in such a layered system, and indeed this is what is observed. It is an interesting observation that the C spin structure within the orthorhombic plane leads to a 3D structure while the IC spin structure within the orthorhomic plane displays a 2D ground state. 6,10 . 3D static C magnetic order gives way to 2D static (on the time scale of high energy resolution neutron measurements) diagonal IC order for x ≥ 0.02, with a co-existence between the two at low temperatures for smaller values of x. At an x ∼ 0.05 quantum critical point, the 2D IC ordering wavevector rotates from diagonal to parallel, relative to the pseudo-tetragonal axes, and this is coincident with the onset of a superconducting ground state 6 . Dynamic 2D IC fluctuations persist to temperatures much higher than those characterizing the onset of static 2D order. These fade continuously with increasing temperature and inhabit much of the phase diagram associated with the "pseudogap" phase.
Finally, although it was first observed some time ago in La 2−x Sr x CuO 4 42,44 and more recently in La 2−x Ba x CuO 4 6 , it bears repeating that the quantum critical point between non-superconducting and superconducting ground states in La 2−x Ba x CuO 4 , near x ∼ 0.05, is coincident with the rotation in the 2D IC spin structure from diagonal to parallel. This also provides strong evidence for an intimate connection between the 2D magnetism and the superconducting properties in these HTS systems.
IV. CONCLUSIONS
We have carried out extensive neutron scattering measurements on the static and low energy, dynamic commensurate (C) and incommensurate (IC) magnetism in lightly-doped La 2−x Ba x CuO 4 (LBCO). We have shown the two dimensional (2D) IC static order to be characterized by the appearance of rods of elastic, diagonal IC scattering with long but finite correlation lengths within the basal plane, and essentially zero correlation length along L. Moreover, below the 2D IC effective ordering temperature, T N (2DIC , these rods are elastic on the energy scale of 0.1 meV, which is ∼ 1 K or less. We can understand the suppression of the 2D IC effective ordering temperature relative to the 3D C ordering temperatures displayed by nearby concentrations as a consequence of the reduction in magnetic dimensionality, rather than being due to proximity to a competing superconducting ground state.
A phase diagram based solely on magnetic order parameter measurements, and constructed using 3D C long range order as well as effective 2D IC static magnetic order transitions for all LBCO samples with x ≤ 0.125 is shown to display much of the same complexity as that corresponding to standard phase diagrams relevant to high temperature superconductivity. This stresses the strong correlation between magnetism and the exotic charge correlation physics, including superconductivity itself, in this family of high temperature superconductors. Our measurements at low temperatures show a systematic suppression of the low energy dynamic susceptibility as a function of increasing doping within the lightly-doped regime x ≤ 0.035, presaging the appearance of superconducting ground states for x ≥ 0.05.
All samples studied in this paper, other than x = 0, display 2D diagonal IC static magnetism at low temperatures within their ground states. Interestingly, we find that the corresponding dynamic IC magnetism exists both at low temperatures as well as on a much higher temperature scale, comparable to nearby 3D C ordering temperatures. The temperature dependence of this dynamic IC magnetism does not change quickly with doping at these low dopings. This 2D dynamic IC magnetism inhabits much of the phase diagram associated with pseudo-gap physics, and there appears to be no characteristic transition temperature associated with these fluctuations; rather their temperature evolution is characteristic of crossover phenomnea.
FIG. 1 .
1Elastic triple axis neutron scattering measurements of La2−xBaxCuO4 for x = 0.0125, 0.025 and 0.035 taken on N5 and HB3. a) Reciprocal space maps in the (HK0)
FIG. 2 .
2Elastic scattering in La2−xBaxCuO4. From left to right are elastic scattering maps of La2−xBaxCuO4 for x = 0, 0.0125, 0.025 and 0.035 respectively. The top row shows data taken at 1.5 K and the bottom row shows data taken at 35 K. All data have an empty cryostat background subtracted from them. FIG. 3. Top row: elastic scattering in La2−xBaxCuO4 for x = 0.0125 shown as a function of temperature. Bottom row: Low energy inelastic scattering for the same x = 0.0125 crystal, S(Q,hω) integrated between 0.15 meV and 0.8 meV. Both sets of data were collected in the same time of flight measurement on DCS and used the same empty cryostat background subtractions. White areas correspond to regions that were not measured.
FIG. 4 .
4Cuts through the elastic magnetic scattering are shown for x = 0.0125, 0.025, and 0.035. Data sets have been normalized to their own maximum intensity for the purposes of qualitative comparison. Solid lines are fits to the data as discussed in the text. a) 3D C structural and 2D IC magnetic peaks are shown in x = 0.0125 at T = 1.5 K. The 2D IC for this doping integrated the data over the [-3.6,-3.2], [-2.7,-2.2],[-1.6,-1.2],[-0.9,-0.65] ranges in L, so as to avoid contributions from 3D C peaks. The 3D C structural peak corresponds to L = -4, and employed a −4.1 ≤ L ≤ −3.9integration. b) 2D IC peaks in x = 0.025 and 0.035 using 0 ≤ L ≤ 1.9 integration for both samples. This range avoids contributions from nuclear Bragg peaks and is minimally affected by self absorbtion. Inset: incommensuration δ as a function of doping as determined from fits of the data. Error bars represent one standard deviation.
FIG. 5 .
5Inelastic scattering for x = 0.0125, 0.025 and 0.035 at T = 35 K. All data sets employed a T=1.5 K data set as background. Panels a) and b) show energy-HH wavevector maps for S(Q,hω) (panel a)) and χ (Q,hω) (panel b)). These data sets employed −3.5 ≤ L ≤ −1 integration for x = 0.0125 and 0 ≤ L ≤ 1.9 integrations for x = 0.025 and 0.035. χ (Q,hω) is related to S(Q,hω) through Eqs. 1 and 2.
FIG. 7 .
7Energy-wavevector maps of χ"(Q,hω) for x = 0.0125 are shown as a function of temperature, from 10 K to 300 K. There is clear monotonic decrease in the spectral weight of the dynamic magnetism with tempertaure.FIG. 8. Energy-wavevector maps of χ"(Q,hω) for x = 0.035 are shown as a function of temperature, from 10 K to 35 K.
FIG. 9 .
9Temperature dependence of the wavevector and low energy (0.2 meV ≤hω ≤ 0.8 meV) integrated χ (Q,hω) for x = 0.0125 and 0.035 is shown as a function of temperature on a semi-log scale. Dashed lines show the 2D IC (for x = 0.035, 0.0125) and 3D C (for x = 0.0125 and 0) magnetic ordering temperatures. The x = 0.0125 and 0.035 data sets have been normalized at low temperatures. Both data sets have employed their 1.5K data set as a background. The temperature scale for the evolution of this low energy dynamic magnetism greatly exceeds the relevant 2D IC magnetic ordering temperatures.
FIG. 10 .
10Magnetic phase diagram for La2−xBaxCuO4 as determined by magnetic order parameter measurements on La2−xBaxCuO4 crystals with x ≤ 0.125.
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| [] |
[
"Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems *",
"Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems *"
] | [
"Iain Smears ",
"Martin Vohralík "
] | [] | [] | We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction-diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction-diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest. | 10.1051/m2an/2020034 | [
"https://arxiv.org/pdf/1812.06678v1.pdf"
] | 119,634,911 | 1812.06678 | 5fea58e42a6419d214ed2714149d8431bcc88244 |
Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems *
December 18, 2018
Iain Smears
Martin Vohralík
Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems *
December 18, 2018Singular perturbationa posteriori error analysislocal efficiencyrobustnessequi- librated flux
We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction-diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction-diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.
Introduction
Let Ω be a polygonal/polyhedral/polytopal domain in R d , d ≥ 1, with a Lipschitz-continuous boundary. Let ε > 0 and κ ≥ 0 be two fixed real parameters, and let f ∈ L 2 (Ω) be a given source term. Consider the problem: find u : Ω → R such that −ε 2 ∆u + κ 2 u = f in Ω, (1.1a)
u = 0 on ∂Ω. (1.1b)
Let a(·, ·) be the symmetric bilinear form defined by a(w, v) := ε 2 (∇w, ∇v) + κ 2 (w, v), w, v ∈ H 1 0 (Ω), (1.2) where (·, ·) denotes the L 2 -inner product of scalar-and vector-valued functions on Ω, with associated norm · . The restriction of the L 2 -inner product to an open subset ω ⊂ Ω is denoted by (·, ·) ω , with associated norm · ω . The weak formulation of problem (1.1) is to find u ∈ H 1 0 (Ω) such that a(u, v) = (f, v) ∀ v ∈ H 1 0 (Ω).
(1.
3)
The energy norm |||·||| associated to problem (1.1) is then the norm induced by the form a(·, ·), namely |||v||| 2 := a(v, v), v ∈ H 1 0 (Ω). (1.4) In this paper, we shall be primarily interested in the case where ε κ, when problem (1.1) is said to be singularly perturbed. Then, the accurate numerical approximation can be challenging due to the typical presence of sharp boundary and/or interior layers in the solution.
In order to present more specifically the focus of this work, let us consider a simplicial mesh T of Ω and let V T := P p (T ) ∩ H 1 0 (Ω) denote the subspace of H 1 0 (Ω) of piecewise polynomial functions of degree at most p, where p ≥ 1 is a fixed integer. The conforming Galerkin finite element approximation of (1.3) consists of finding u T ∈ V T such that
a(u T , v T ) = (f, v T ) ∀ v T ∈ V T . (1.5)
The goal is to find a computable a posteriori error estimator η(u T ) that satisfies |||u − u T ||| ≤ C rel η(u T ), η(u T ) ≤ C eff |||u − u T ||| + data oscillation.
(1.6)
The first inequality in (1.6) is called reliability, while the second inequality is called (global) efficiency. A localized version of the efficiency bound is actually desirable. The quality of the estimator is determined by the product of the two constants C rel and C eff . A key requirement for singularly perturbed problems is to obtain estimators that are robust in the sense that both constants C rel and C eff are independent of the singular perturbation parameters ε and κ. Only such estimates can quantify well the error in the numerical approximation and be reliably used in adaptive algorithms which allow for efficient approximation of the localized features of the solution.
Recently, several methodologies for constructing error estimators that satisfy (1.6) in a robust way have been studied. Verfürth [35] (see also [36] or [38,Section 4.3]) was probably the first to show robust bounds, in the framework of the so-called residual-based estimates. For the problem at hand, these estimators take the form (up to the data oscillation term and possible generic constants) η res (u T ) 2 :=
K∈T α 2 K r T 2 K + F ∈FΩ ε −1 α F j T 2 F ,(1.7)
where the local element and face residuals are defined respectively by
r T | K := (f + ε 2 ∆ T u T − κ 2 u T )| K , (1.8a) j T | F := −ε 2 ∇u T ·n F F ,(1.8b)
and where ∆ T denotes the element-wise Laplacian, ∇u T ·n F F denotes the jump of the normal component of ∇u T over the face F , F Ω stands for the set of internal faces of the mesh T , and the weights (cut-off factors) take the form
α S := min h S ε , 1 κ ,(1.9)
with h S being the diameter of S, where S is either a simplex K or a face F . The resulting estimator η res (u T ) is thus a straightforward extension from the pure diffusion case κ = 0 and is simple to implement in practice. The proof that η res satisfies the second inequality in (1.6) rests on a bubble function technique, where the face bubble functions are defined with respect to a submesh matching the boundary-layer length scales and are possibly very steeply decaying. Their role is to capture the sharp layers caused by the singular perturbation. Note that these bubble functions, and hence the submeshes on which they are defined, are only employed in the analysis; thus they do not need to be constructed in practice. Shortly after, Ainsworth and Babuška [2] extended the method of equilibrated residuals, cf. [3], to satisfy (1.6) in a robust way for lowest-order approximations, i.e. p = 1. In contrast to the residual-based estimators, a boundary-layer adapted submesh in each mesh element needs to be constructed in practice in order to evaluate the estimator. Further progress has been made since, although, to the best of our knowledge, only in the case of lowest-order approximations where the polynomial degree p = 1. Robust estimates that are guaranteed (C rel = 1) and where η(u T ) is fully computable have been obtained in Cheddadi et al. [9]. This remedies that C rel is unknown for residual-based estimates and that exact solutions of some infinite-dimensional boundary value problems on each element (which cannot be performed exactly in practice) are required in the equilibrated residuals approach. The estimator in [9] is based on an equilibrated flux σ T belonging to a discrete subspace of H(div) that satisfies the equilibration identity ∇·σ T + κ 2 u T = f T , where f T is a piecewise polynomial approximation of f . The estimator is then composed of terms of the form
min ε∇u T + ε −1 σ T K , Cε − 1 2 α 1 2 F j T ∂K\∂Ω .
Thus it can be seen as a combination between an equilibrated flux estimator for diffusion problems and the residual-based estimator of [35] for reaction-diffusion problems. No submesh is needed for the construction of the estimator. Subsequently, Ainsworth and Vejchodský [4,5] proceed in two stages. First, equilibrated face fluxes are computed as in [2], and then, equilibrated fluxes are obtained by face liftings, so that the final estimate η(u T ) is also fully computable and the first inequality in (1.6) is guaranteed with C rel = 1. As in [2], though, boundary-layer adapted submeshes appear in the construction of the estimator. The use of a submesh complicates the construction and implementation of the equilibrated flux estimators of [4,5]. Moreover, it is likely to be even more involved when moving beyond lowest-order approximations. In this work, by further developing the idea in [9], we show how to obtain simple, i.e. avoiding any submesh, yet robust equilibrated flux estimators for arbitraryorder approximations. The a posteriori error estimates presented in this paper are based on a locally computable flux σ T and potential approximation φ T , respectively belonging to discrete subspaces of H(div, Ω) and L 2 (Ω) of the current mesh T , that satisfy the key equilibration property
∇·σ T + κ 2 φ T = Π T f,(1.10)
where Π T : L 2 (Ω) → P p (T ) denotes the L 2 -orthogonal projection operator. The upper bound on the error then has the simple form
|||u − u T ||| 2 ≤ K∈T w K ε∇u T + ε −1 σ T K + κ (u T − φ T ) K + w K f − Π T f K 2 , (1.11)
where w K is an elementwise computable weight (cut-off factor) such that
w K = min 1, C * ε κh K , w K = min h K πε , 1 κ ,
with a fixed computable constant C * given by (2.7); see Theorem 3.1 below for further details. The equilibrated flux σ T and approximate potential φ T in (1.10), (1.11) are obtained by an extension of the patchwise equilibration of [12,8], see also [7,19]. Furthermore, we prove robustness and efficiency of the estimator (1.11) by showing that its local contributions are bounded, up to a constant, by the local residual estimators. More precisely, for each K ∈ T , we show that
w 2 K ε∇u T + ε −1 σ T 2 K + κ (u T − φ T ) 2 K K ∈T K α 2 K r T 2 K + F ∈F K ε −1 α F j T 2 F K ∈T K |||u − u T ||| 2 K + α 2 K f − Π T f 2 K , (1.12)
where T K and F K denote the set of elements and faces in a suitable neighbourhood of K and (1.12) are independent of the mesh-sizes h K and problem parameters ε and κ, depending only on the shape-regularity of T , the space dimension d, and the polynomial degree p. Hence, just as for residual-based estimates, equilibrated flux estimates have a straightforward extension from the pure diffusion case κ = 0, based on including appropriate weights (cut-off factors) and not requiring computations of quantities over any submesh or combination with the residual estimators. In light of these results, we believe that the claims in [37,38] of a "structural defect" of the robustness of the equilibrated fluxes estimators are not generally valid.
|||v||| 2 K := ε 2 ∇v 2 K + κ 2 v 2 K K ∈ T ,(1.
As a side result, we also prove in Proposition 5.1 that the weights w K in (1.11) are necessary for robustness of any equilibrated flux estimate involving the terms ε∇u T + ε −1 σ T K whenever σ T is a piecewise polynomial on T (and thus its construction does not involve any submesh), regardless of the precise details of the construction of σ T . This proves that several flux equilibrations proposed in the past cannot be robust with respect to reaction dominance in general (although in many constellations, no loss of robustness may be numerically observed), including those of Repin and Sauter [29], Ainsworth et al. [1], Eigel and Samrowski [16], Eigel and Merdon [15], and Vejchodský [32,34,33].
We only treat isotropic meshes. Results for anisotropic meshes can be found in Kunert [27], Grosman [22], Apel et al. [6], Zhao and Chen [40], or Kopteva [24,25]. Also, we are solely interested in the energy norm. Robust estimates in the maximum norm are obtained in Demlow and Kopteva [11] and, on possibly anisotropic meshes, in Kopteva [23] for p = 1 any in Linss [28] for any order p ≥ 1 in one space dimension. We refer to Stevenson [31] for robust convergence, and we refer to Faustmann and Melenk [21] and the references therein for balanced norms. Finally, extensions to variable coefficients ε and κ can be treated easily as in [5], whereas inhomogeneous Dirichlet and Neumann boundary conditions, mixed parallelepipedal-simplicial meshes, meshes with hanging nodes, and approximations with varying polynomial degree p can be treated as in Dolejší et al. [14].
Construction of the equilibrated flux
We present in this section the construction of our equilibrated flux σ T and of the potential approximation φ T .
Notation
Let T be a matching simplicial partition of the domain Ω, i.e., K∈T K = Ω, any element K ∈ T is a closed simplex (interval when d = 1, triangle when d = 2, tetrahedron when d = 3), and the intersection of two different simplices is either empty, or a vertex, or their common l-dimensional face, 1 ≤ l ≤ d − 1. We denote by σ T > 0 the shape-regularity parameter of the mesh T , i.e.
ϑ T := max K∈T h K ρ K , (2.1)
where ρ K is the diameter of the largest ball contained in K. For each element K ∈ T and for a fixed integer p ≥ 1, let P p (K) denote the space of polynomials of total degree at most p on K. Let P p (T ) := {v ∈ L 2 (Ω), v| K ∈ P p (K) ∀K ∈ T } denote the space of scalar piecewise polynomials of degree at most p over T . Let Π T : L 2 (Ω) → P p (T ) denote the L 2 -orthogonal projection operator from L 2 (Ω) onto P p (T ). We additionally consider L 2 (Ω) := L 2 (Ω; R d ) and RT N p (T ) ⊂ L 2 (Ω) the piecewise Raviart-Thomas-Nédélec space defined by
RT N p (T ) := {v T ∈ L 2 (Ω), v T | K ∈ RT N p (K) ∀K ∈ T }, RT N p (K) := P p (K; R d ) + P p (K)x. (2.2)
For any subset S of Ω, let h S denote the diameter of S. Thus, for instance, h K denotes the diameter of the element K ∈ T . Let V denote the set of vertices of the mesh T . It is partitioned into the set of interior vertices V int := {a ∈ V, a ∈ Ω}, and boundary vertices V ext := V \ V int . For each vertex a ∈ V, the function ψ a is the hat function associated with a, i.e., ψ a ∈ P 1 (T ) ∩ H 1 (Ω) taking value 1 in the vertex a and 0 in the other vertices. The set ω a is the interior of the support of ψ a with associated diameter h ωa . Furthermore, let T a denote the restriction of the mesh T to ω a , and let F a denote the set of interior faces of T a , i.e. the faces of T a that contain the vertex a for a ∈ V int , without those on ∂Ω for a ∈ V ext . For each element K ∈ T , we collect in V K the set of vertices of V belonging to K. We also define T K := a∈V K T a and F K :
= a∈V K F a .
Throughout this work, the notation a b means that a ≤ Cb with a constant C that only depends on the shape-regularity parameter ϑ T of T , on the space dimension d, and on the polynomial degree p, so that it is in particular independent of the mesh-sizes h K and of the problem parameters ε and κ; a b then stands for a b and simultaneously b a.
Trace and inverse inequalities
We first recall two inequalities that we will rely on.
Lemma 2.1 (Trace inequality with explicit constant). For all K ∈ T and for all
v ∈ H 1 (K) that satisfy (v, 1) K = 0, i.e., that have vanishing mean-value on K, there holds v ∂K ≤ C Tr ∇v 1 2 K v 1 2 K , C Tr := ϑ T (d + 1) (2 + d/π). (2.3)
Proof. We refer the reader to [13, Lemma 1.49] for the explicit constants of the trace inequality for general functions in H 1 (K); namely, for each face F ⊂ ∂K,
v 2 F ≤ ϑ T (2 ∇v K + d/h K v K ) v K .
Then, we additionally apply the Poincaré inequality v K ≤ h K /π ∇v K for functions with vanishing mean-value on K, and sum over all the faces F to obtain (2.3).
Lemma 2.2 (Inverse inequalities with explicit constants). For any K ∈ T and any
v ∈ RT N p (K), we have h 1/2 K v · n ∂K ≤ C p+1,d,∂K v K , h K ∇ · v K ≤ C p+1,d,K v K , (2.4)
where the constants C p+1,d,∂K and C p+1,d,K are given by
C p+1,d,∂K := (d + 1)(p + 2)(p + d + 1)ϑ T , (2.5) C p+1,d,K := √ d ϑ T √ 5 4 (2 √ 2) d p(p + 1)(p + 2)(p + 3). (2.6) Proof. See Appendix A.
In practice, possibly sharper constants can be obtained for the inequalities in (2.4) by solving numerically small eigenvalue problems on each mesh element, or on a reference element in combination with bounds for the influence of the affine mapping.
We will need below the following constant composed of the constants of the trace and inverse inequalities (2.3) and (2.4):
C * := 1 √ 2 1 √ π C p+1,d,K + C Tr C p+1,d,∂K . (2.7)
Equilibrated flux σ T and postprocessed potential φ T
The construction of the auxiliary variables σ T and φ T giving the equilibration (1.10) is based on independent local mixed finite element approximations of residual problems over the patches of elements around mesh vertices. For each a ∈ V, let P p (T a ), respectively RT N p (T a ), be the restriction of the space P p (T ), respectively RT N p (T ), to the patch T a around the vertex a. The local mixed finite element spaces V a T and Q a T are defined by
V a T := {v T ∈ H(div, ω a ) ∩ RT N p (T a ), v T ·n = 0 on ∂ω a } if a ∈ V int , {v T ∈ H(div, ω a ) ∩ RT N p (T a ), v T ·n = 0 on ∂ω a \ ∂Ω} if a ∈ V ext , (2.8a) Q a T := P p (T a ) if κ > 0 or a ∈ V ext , {q T ∈ P p (T a ), (q T , 1) ωa = 0} if κ = 0 and a ∈ V int , (2.8b) see Figure 1. Recall that u T ∈ V T with V T = P p (T ) ∩ H 1 0 (Ω)
is the finite element solution given by (1.5). Let C * be the constant composed of the constants of the trace and inverse inequalities and given by (2.7). Our construction is:
Definition 2.3 (Flux σ T and potential φ T ). For each vertex a ∈ V, let (σ a T , φ a T ) ∈ V a T × Q a T
be defined by the local constrained minimization problem Then, extending each σ a T and φ a T by zero outside of the patch ω a , σ T ∈ RT N p (T ) and φ T ∈ P p (T ) are given by
(σ a T , φ a T ) := arg min (v T ,q T )∈V a T ×Q a T ∇·v T +κ 2 q T =Π T (f ψa)−ε 2 ∇u T ·∇ψa w 2 a εψ a ∇u T + ε −1 v T 2 ωa + κ [Π T (ψ a u T ) − q T ] 2 ωa (2.9a) with the weight w a := min 1, C * ε κh ωa . (2.9b) a ∈ V int interior patch ωaσ T := a∈V σ a T , φ T := a∈V φ a T . (2.9c)
We remark that for an interior vertex a ∈ V int , we have
(Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a , 1) ωa = (f, ψ a ) ωa − ε 2 (∇u T , ∇ψ a ) ωa = κ 2 (u T , ψ a ) ωa (2.10)
by Galerkin orthogonality with ψ a ∈ V T as a test function in (1.5). Since (∇·σ a T , 1) ωa = (σ a T ·n ωa , 1) ∂ωa = 0 by Green's theorem and the vanishing normal flux condition imposed in the definition (2.8a) of V a T , it follows that φ a T necessarily satisfies the mean-value property (φ a T , 1) ωa = (ψ a u T , 1) ωa ∀a ∈ V int , whenever κ > 0. If κ = 0 instead, then φ a T is undefined by (2.9a) but one remarks that it is no longer needed anywhere in the paper. In this case, Definition 2.3 coincides with [7, equation (9)], [18, Definition 6.9], or [19,Construction 3.4]; in particular, the Neumann compatibility condition of problem (2.9a) for a ∈ V int follows from (2.10).
In practice, the constrained minimization problem (2.9a) is solved through its Euler-Lagrange equations, which can be reduced to solving a linear system of dimension dim V a T + dim Q a T in the present context. This problem reads:
find (σ a T , φ a T ) ∈ V a T × Q a T with φ a T = γ a T + Π T (ψ a u T ) and (σ a T , γ a T ) ∈ V a T × Q a T such that ε −2 w 2 a (σ a T , v T ) ωa − (γ a T , ∇·v T ) ωa = −w 2 a (ψ a ∇u T , v T ) ωa ∀v T ∈ V a T , (2.11a) (∇·σ a T , q T ) ωa + κ 2 (γ a T , q T ) ωa = (f ψ a − κ 2 ψ a u T − ε 2 ∇u T ·∇ψ a , q T ) ωa ∀q T ∈ Q a T . (2.11b)
Properties of σ T and φ T
We have constructed σ T and φ T such that the following holds: Proposition 2.4 (H(div, Ω)-conformity of σ T , equilibration). Let σ T ∈ RT N p (T ) and φ T ∈ P p (T ) be given by Definition 2.3. Then σ T belongs to H(div, Ω), and σ T and φ T satisfy the equilibration property (1.10).
Proof. First, the H(div, Ω)-conformity of σ T follows from the fact that, for any vertex a ∈ V, the zero extension of σ a T belongs to H(div, Ω) as a result of the vanishing normal flux boundary conditions in the space V a T . Then, to show (1.10), we employ the constraint in (2.9a) together with (2.9c):
∇·σ T + κ 2 φ T = a∈V [∇·σ a T + κ 2 φ a T ] = a∈V Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a = Π T f,
where we have used the fact that the hat functions {ψ a } a∈V form a partition of unity over Ω, i.e. a∈V ψ a = 1.
A computable guaranteed a posteriori error estimate
This section presents our guaranteed and fully computable a posteriori error estimate. The following upper bound on the energy norm of the error builds on [9, Theorems 3.1 and 4.4] and [5, Lemma 2]. It employs additionally the concept of a potential reconstruction φ T that will turn out crucial for a simple and robust flux equilibration. Moreover, it relies on the trace and inverse inequalities of Section 2.2 to make appear the crucial weighs (cut-off factors), with the constant C * given by (2.7). 3) and let u T ∈ V T be its finite element approximation given by (1.5). Let σ T ∈ RT N p (T ) ∩ H(div, Ω) and φ T ∈ P p (T ) be given by Definition 2.3. Then the following upper bound for the energy norm of the error holds:
|||u − u T ||| 2 ≤ K∈T w K ε∇u T + ε −1 σ T K + κ (u T − φ T ) K + w K f − Π T f K 2 , (3.1)
where the weights w K and w K are respectively defined by
w K := min 1, C * ε κh K , w K := min h K πε , 1 κ , K ∈ T . (3.2)
Proof. First, we note that the energy norm of the error |||u − u T ||| is related to the residual
R(u T ) ∈ H −1 (Ω), defined by R(u T ), v := (f, v) − a(u T , v), v ∈ H 1 0 (Ω), through the identity |||u − u T ||| = |||R(u T )||| * , |||R(u T )||| * := sup v∈H 1 0 (Ω), |||v|||=1 R(u T ), v ,(3.3)
cf., e.g., [35, equation (4.1)]. Consider now R(u T ), v for a fixed function v ∈ H 1 0 (Ω). Since σ T ∈ H(div, Ω) and v ∈ H 1 0 (Ω), Green's theorem gives (σ T , ∇v) + (∇·σ T , v) = 0, so
R(u T ), v = (f, v) − a(u T , v) = (f − Π T f, v) + (κ(φ T − u T ), κv) − (ε∇u T + ε −1 σ T , ε∇v), (3.4)
where we have also used the equilibration identity (1.10). We now proceed by estimating each term in (3.4) elementwise.
For each element K ∈ T , we use the identity
(f − Π T f, v) K = (f − Π T f, v − Π T v) K and the Poincaré-Friedrichs inequality on the convex element K, i.e. v − Π T v K ≤ h K
π ∇v K for any v ∈ H 1 (K), together with the energy error definition (1.13), to obtain the following bound
|(f − Π T f, v) K | ≤ f − Π T f K min h K πε ε∇v K , 1 κ κv K ≤ w K f − Π T f K |||v||| K . (3.5)
Here, actually, a little sharper bound is possible by a convex combination of the two possibilities, but we prefer to use the simple form (3.5) with w K in the form of minimum given by (3.2). Next, it is clear that
|(ε∇u T + ε −1 σ T , ε∇v) K | ≤ ε∇u T + ε −1 σ T K |||v||| K (3.6)
for each K ∈ T . However, this is not necessarily the sharpest possible estimate in the singularly perturbed regime κ ε. Therefore, following the idea of [9, Proof of Theorem 4.4], we use Green's theorem elementwise together with the fact that ∇v = ∇(v − v K ), where v K denotes the mean-value of v on K. This gives
(ε∇u T +ε −1 σ T , ε∇v) K = ((ε∇u T +ε −1 σ T )·n, ε(v−v K )) ∂K −(∇· ε∇u T + ε −1 σ T , ε(v−v K )) K . The L 2 (K)-stability of the mean-value, v − v K K ≤ v K , Young's inequality ε∇v 1 2 K κv 1 2 K ≤ 1 √ 2 |||v||| K ,(3.7)
and the multiplicative trace inequality (2.3) altogether lead to
ε v − v K ∂K ≤ C Tr ε 1 2 ε∇v 1 2 K v 1 2 K ≤ C Tr √ 2 h K ε κh K |||v||| K .
Combined with the inverse inequality (2.4), we find that
|((ε∇u T + ε −1 σ T )·n, ε(v − v K )) ∂K | ≤ C p+1,d,∂K C Tr √ 2 ε κh K ε∇u T + ε −1 σ T K |||v||| K . (3.8)
The L 2 (K)-stability of the mean-value, the Poincaré-Friedrichs inequality in the form v − v K K ≤ h K π ∇v K , and (3.7) yield
ε v − v K K ≤ ε v 1 2 K h 1 2 K π − 1 2 ∇v 1 2 K ≤ h K √ 2π ε κh K |||v||| K .
Thus, combined with the inverse inequality (2.4), we find that
|(∇· ε∇u T + ε −1 σ T , ε(v − v K )) K | ≤ 1 √ 2 1 √ π C p+1,d,K ε κh K ε∇u T + ε −1 σ T K |||v||| K . (3.9)
Therefore, combining inequalities (3.6), (3.8), and (3.9), we get
|(ε∇u T + ε −1 σ T , ε∇v) K | ≤ w K ε∇u T + ε −1 σ T K |||v||| K ∀K ∈ T (3.10)
with w K given by (3.2) and C * given in (2.7). As a side remark, it is possible to obtain a slightly sharper, at the expense of making the weight w K more complicated than the simple form given by (3.2).
Finally, we can apply the Cauchy-Schwarz inequality to see that |(κ(φ T − u T ), κv) K | ≤ κ (u T − φ T ) K |||v||| K . Therefore, we deduce from (3.4) and the above inequalities that
| R(u T ), v | ≤ K∈T w K ε∇u T + ε −1 σ T K + κ (u T − φ T ) K + w K f − Π T f K |||v||| K ,
which implies the upper bound on the error (3.1) after another Cauchy-Schwarz inequality, using (3.3) and K∈T |||v||| 2 K = |||v||| 2 .
Efficiency and robustness of the estimate
This section establishes the local (and consequently global) efficiency and robustness of our a posteriori error estimate.
A basic stability result
The main tool in the analysis of efficiency is the following stability result, where, we recall, the broken and the patchwise H(div)-conforming Raviart-Thomas-Nédélec spaces RT N p (T a ) and V a T are respectively given by (2.2) and (2.8). Lemma 4.1 (Stability of patchwise flux equilibration). Let a vertex a ∈ V be fixed, and let g T ∈ P p (T a ) and τ T ∈ RT N p (T a ) be given discontinuous piecewise polynomial functions, with the Neumann compatibility condition (g T , 1) ωa = 0 satisfied if a ∈ V int . Then, there holds
min v T ∈V a T ∇·v T =g T τ T + v T ωa sup v∈H 1 * (ωa) ∇v ωa =1 (g T , v) ωa − (τ T , ∇v) ωa , (4.1)
where H 1 * (ω a ) is the subspace of functions in H 1 (ω a ) that have mean-value zero on the patch subdomain ω a if a ∈ V int is an interior vertex, or that vanish on ∂ω a ∩ ∂Ω if a ∈ V ext is a boundary vertex.
The above result holds for any dimension d ≥ 1, although some additional properties are known for d ≤ 3. Indeed, in the case where d = 2, it is shown in [7, Theorem 7] that the constant in (4.1) is in fact independent of the polynomial degree p, i.e. p-robust. The extension of the p-robustness of the bound to the case of d = 3 was shown in [20, Corollaries 3.3 and 3.6]. It is also possible to extend similar results of this kind to situations with hanging nodes and locally refined submeshes, as shown in [17].
Stability with respect to residual estimators
The next lemma shows that the local contributions of the equilibrated flux a posteriori estimators of Definition 2.3 lie below the local residual estimators as defined in (1.7), with the element residuals r T and face residuals j T are defined by (1.8) and the weights α K and α F defined by (1.9).
Lemma 4.2 (Stability of patchwise flux equilibration with respect to residual estimators).
For each a ∈ V, let σ a T and φ a T be defined by (2.9a). Then
w 2 a εψ a ∇u T + ε −1 σ a T 2 ωa + κ [Π T (ψ a u T ) − φ a T ] 2 ωa K∈Ta α 2 K r T 2 K + F ∈Fa ε −1 α F j T 2 F . (4.2)
Proof. Let a vertex a ∈ V be fixed. Since σ a T and φ a T are defined as minimizers of the functional in the right-hand side of (2.9a), it is enough to prove that there always exists some v * T ∈ V a T and q * T ∈ Q a T that satisfy the constraint ∇·v * T + κ 2 q * T = Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a and that satisfy the bound (4.2) with v * T in place of σ a T and q * T in place of φ a T . The specific construction depends on the mesh size and the problem parameters ε and κ, as we now show.
Case 1, ε/h ωa ≤ κ (reaction dominance). Up to a constant, we have κ −1 h K /ε and κ −1 h F /ε for all elements K ∈ T a and all interior faces F ∈ F a . In this case, we adopt the following construction. Let
ρ a := 1 |ω a | (ψ a r T , 1) ωa = 1 |ω a | (ψ a (f + ε 2 ∆ T u T − κ 2 u T ), 1) ωa , a ∈ V int ,
and ρ a := 0 otherwise. Next, we define
q * T := 1 κ 2 Π T (f ψ a ) + ε 2 ψ a ∆ T u T − ρ a , v * T := arg min v T ∈V a T ∇·v T =g * T ε 2 ψ a ∇u T + v T ωa ,(4.3)
where g * T := −ε 2 (∇u T ·∇ψ a + ψ a ∆ T u T ) + ρ a . It is easy to check that if a ∈ V int , then (g * T , 1) ωa = 0, since the Galerkin orthogonality (take v T = ψ a in (1.5)) implies that
(g * T , 1) ωa = (f, ψ a ) − ε 2 (∇u T , ∇ψ a ) − κ 2 (u T , ψ a ) = 0. (4.4)
Therefore, it follows that q * T ∈ Q a T and v * T ∈ V a T are well-defined and that they satisfy the constraint ∇·v *
T + κ 2 q * T = Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a . We now bound w 2 a ε 2 ψ a ∇u T + v * T 2 ωa and κ [Π T (ψ a u T ) − q * T ] 2 ωa . First, we obtain κ [Π T (ψ a u T ) − q * T ] 2 ωa = 1 κ 2 Π T (ψ a r T ) − ρ a 2 ωa ≤ 1 κ 2 r T 2 ωa K∈Ta α 2 K r T 2 K ,
where we have used the stability of the L 2 -projection (note that ρ a is also the mean value of Π T (ψ a r T ) on ω a for a ∈ V int ) and the fact that ψ a ∞,ωa = 1 to bound Π T (ψ a r T ) − ρ a ωa . Next, we apply Lemma 4.1 to bound w 2 a ε 2 ψ a ∇u T +v * T 2 ωa . Note first that for an interior vertex a ∈ V int , (ρ a , v) ωa = 0 since v ∈ H 1 * (ω a ) implies that v is orthogonal to constant functions on ω a . We find that
ε 2 ψ a ∇u T + v * T ωa sup v∈H 1 * (ωa), ∇v ωa =1 (g * T , v) ωa − (ε 2 ∇u T , ψ a ∇v) ωa = sup v∈H 1 * (ωa), ∇v ωa =1 − (ε 2 ∇u T , ∇(ψ a v)) ωa − (ε 2 ∆ T u T , ψ a v) ωa = sup v∈H 1 * (ωa), ∇v ωa =1 F ∈Fa (j T , ψ a v) F , (4.5)
where the last line follows by elementwise integration by parts. It is then straightforward to deduce from the trace inequalities
v F h − 1 2 K v K + ∇v 1 2 K v 1 2
K and the Poincaré-Friedrichs inequality for functions in H 1 * (ω a ) v ωa h ωa ∇v ωa that
ε 2 ψ a ∇u T + v * T 2 ωa h ωa F ∈Fa j T 2 F . (4.6)
Consequently, using definition (2.9b) of the weight w a
w 2 a εψ a ∇u T + ε −1 v * T 2 ωa ε κh ωa h ωa ε 2 F ∈Fa j T 2 F F ∈Fa ε −1 α F j T 2 F .
Therefore, if ε/h ωa ≤ κ, we have shown that there exists v * T and q * T satisfying the constraint ∇·v * T + κ 2 q * T = Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a and such that
w 2 a εψ a ∇u T + ε −1 v * T 2 ωa + κ [Π T (ψ a u T ) − q * T ] 2 ωa K∈Ta α 2 K r T 2 K + F ∈Fa ε −1 α F j T 2 F .
As explained above, this implies (4.2) in the case ε/h ωa ≤ κ. Case 2, ε/h ωa > κ (diffusion dominance). We select
q * T := Π T (ψ a u T ), v * T := arg min v T ∈V a T ∇·v T =g * T ε 2 ψ a ∇u T + v T ωa , where g * T := Π T (ψ a (f − κ 2 u T )) − ε 2 ∇ψ a ·∇u T .
Notice that Galerkin orthogonality implies that (g * T , 1) ωa = 0 if a ∈ V int as in (4.4), and also ∇·v * T + κ 2 q * T = Π T (f ψ a ) − ε 2 ∇u T ·∇ψ a , so the requested constraint is satisfied. It then follows directly from Lemma 4.1 that
ε 2 ψ a ∇u T + v * T ωa sup v∈H 1 * (ωa), ∇v ωa =1 (Π T (ψ a r T ), v) ωa + F ∈Fa (ψ a j T , v) F ,
where we use the fact that elementwise integration by parts shows that, as in (4.5),
(g * T , v) ωa − (ε 2 ψ a ∇u T , ∇v) ωa = (Π T (ψ a r T ), v) ωa + F ∈Fa (ψ a j T , v) F .
Thus, proceeding as in (4.5)-(4.6) for the face residuals term and using the stability of the L 2 -projection, ψ a ∞,ωa = 1, and the Poincaré-Friedrichs inequality for functions in H 1 * (ω a ), v ωa h ωa ∇v ωa , for the element residuals term, we get
ε 2 ψ a ∇u T + v * T 2 ωa h 2 ωa K∈Ta r T 2 K + h ωa F ∈Fa j T 2 F .
Consequently,
εψ a ∇u T + ε −1 v * T 2 ωa K∈Ta α 2 K r T 2 K + F ∈Fa ε −1 α F j T 2 F .
Hence, on noting that w a ≤ 1 and that κ [Π T (ψ a u T ) − q * T ]) ωa = 0, we see that (4.2) also holds for the case ε/h ωa > κ.
Recall that T K := a∈V K T a and F K := a∈V K F a .
Proposition 4.3 (Bound on flux estimators by the residual estimators).
Let σ T and φ T be given by Definition 2.3. Additionally, let the volume and face residual functions r T and j T be defined by (1.8). Then, for each element K ∈ T , we have the bound
w 2 K ε∇u T + ε −1 σ T 2 K + κ (u T − φ T ) 2 K K ∈T K α 2 K r T 2 K + F ∈F K ε −1 α F j T 2 F . (4.7)
Proof. For each mesh element K ∈ T , we have σ T | K = a∈V K σ a T | K and φ T | K = a∈V K φ a T | K . Furthermore, since {ψ a } a∈V K form a partition of unity over K and since Π T is the elementwise L 2 projection of degree p, it follows that u T | K = Π T u T | K = a∈V K Π T (ψ a u T ) | K . Furthermore, (3.2) and (2.9b) together with the mesh shape regularity imply that w K w a for each a ∈ V K , where the constant depends only on ϑ T . Therefore, we obtain
w 2 K ε∇u T + ε −1 σ T 2 K + κ (u T − φ T ) 2 K a∈V K w 2 a εψ a ∇u T + ε −1 σ a T 2 K + κ[Π T (ψ a u T ) − φ a T ] 2 K .
Therefore, we can use (4.2) for each a ∈ V K to get (4.7).
Local efficiency and robustness of the estimate
We now recall the well-known efficiency and robustness results for residual estimators, see [35,Proposition 4.1] and [38] for details. For each K ∈ T and F ∈ F Ω , there holds
α 2 K r T 2 K |||u − u T ||| 2 K + α 2 K f − Π T f 2 K , (4.8a) ε −1 α F j T 2 F K∈T ,F ⊂∂K |||u − u T ||| 2 K + α 2 K f − Π T f 2 K . (4.8b)
Therefore, the combination of Proposition 4.3 with (4.8) shows that the equilibrated flux estimator of Theorem 3.1 is locally efficient and robust. 3) and let u T ∈ V T be its finite element approximation given by (1.5). Let σ T ∈ RT N p (T ) ∩ H(div, Ω) and φ T ∈ P p (T ) be given by Definition 2.3. Then, for each mesh element K ∈ T , there holds
w 2 K ε∇u T + ε −1 σ T 2 K + κ (u T − φ T ) 2 K K ∈T K |||u − u T ||| 2 K + α 2 K f − Π T f 2 K , (4.9)
where the constant in depends only on the dimension d, the shape-regularity constant ϑ T of T , and on the polynomial degree p, so that it is independent of the parameters ε and κ and the mesh-sizes h K .
5 Necessity of the weights w K in the upper bound Theorems 3.1 and 4.4 show that the estimator w K ε∇u T +ε −1 σ T K + κ (u T − φ T ) K obtained from the flux equilibration of Definition 2.3 is a reliable, locally efficient, and robust energy error estimator for singularly perturbed reaction-diffusion problems. Here we show the necessity of the weight w K for robustness of equilibrated flux estimators that involve only piecewise polynomial vector fields on T . We also recall that an alternative option, related to the approach in [2,4,5,25], is to perform an equilibrations on a submesh.
Necessity of the weights w K
The following proposition applies to any flux equilibration on T : Proposition 5.1 (Best-possible bound by piecewise polynomials of the mesh T ). Let u T ∈ P p (T ) ∩ H 1 0 (Ω) be an arbitrary piecewise p-degree polynomial, p ≥ 1, and let the face residual term j T be defined by (1.8b). Let P p (T ; R d ) denote the space of R d -valued piecewise polynomials of degree at most p over T , where p ≥ 0 is an arbitrary nonnegative integer. Then,
inf v T ∈H(div,Ω)∩P p (T ;R d ) ε∇u T + ε −1 v T κh ε F ∈FΩ ε −1 α F j T 2 F 1 2 , (5.1)
where h := min K∈T h K , and where the constant depends only on the polynomial degrees p and p , the dimension d, and the shape-regularity ϑ T of T .
Proof. Let v T ∈ H(div, Ω) ∩ P p (T ; R d ) be arbitrary. Then, for each interior face F ∈ F Ω , the H(div, Ω)-conformity of v T implies that v T ·n F F = 0, and hence
j T | F = −ε (ε∇u T + ε −1 v T )·n F F . Since (ε∇u T + ε −1 v T )| K ∈ P max(p ,p−1) (K; R d )
for each element K ∈ T , we can apply the triangle inequality and the inverse inequality (analogous to (2.4)) to find that, for any
F ∈ F Ω , ε −1 α F j T 2 F ε κh K∈T ,F ⊂∂K ε∇u T + ε −1 v T 2 K . (5.2)
Therefore, we get (5.1) by summing (5.2) over all faces F ∈ F Ω , and recalling that v T was arbitrary.
The upshot of Proposition 5.1 is that for any problem where the jump estimators are sufficiently dominant, i.e. when
|||u − u T ||| F ∈FΩ ε −1 α F j T 2 F 1 2 , (5.3)
then any error estimator involving a term of the form ε∇u T + ε −1 v T without any weight will necessarily be non-robust when κh/ε takes large values, since (5.1) and (5.3) then imply
inf v T ∈H(div,Ω)∩P p (T ;R d ) ε∇u T + ε −1 v T |||u − u T ||| κh ε . (5.4)
In other words, the effectivity index can be become arbitrarily large in the singularly-perturbed regime when the weight w K is not included. It is then seen that the inclusion of the weight term w K in Theorem 3.1 is necessary when considering flux equilibrations from vector-valued piecewise polynomial subspaces of H(div, Ω) on the mesh T , regardless of the precise details of the construction of the flux. Examples of flux equilibrations proposed in the past that cannot be robust in general include Repin and Sauter [29], Ainsworth et al. [1], Eigel and Samrowski [16], Eigel and Merdon [15], and Vejchodský [32,33,34]. We now present an example of a situation where (5.3) holds and where κh/ε can be arbitrarily large. In fact the example is similar to the one in [2, Section 2.3], albeit with some suitable adjustments. denote the piecewise affine Lagrange interpolant (preserving the point values) of the function x → cos(mπx); it follows from the fact that m is odd that f ∈ H 1 0 (Ω). Note that in the example of [2], the function f was chosen as cos(πx) instead.
Consider now problem (1.1) along with its finite element approximation (1.5) in the space
V T = P 1 (T ) ∩ H 1 0 (Ω).
It is easy to show that
u T = (ε 2 µ h + κ 2 ) −1 f is the discrete solution, where µ h := 6 2 + cos(mπh) 1 − cos(mπh) h 2 ,
as a result of the identity
1/2 −1/2 f v T dx = µ h 1/2 −1/2 f v T dx ∀v T ∈ V T .
Then, noting that interior vertices and faces coincide for problems in one space dimension, it is found that
r T | K = ε 2 µ h ε 2 µ h + κ 2 f | K , j T | xi = −ε 2 u T (x i ) = ε 2 ε 2 µ h + κ 2 2(1 − cos(mπh)) h f (x i ). Now, since lim m→∞ 1−cos(mπh) h = π 2 2 when h = h(m) = 1/(m+1) 2 , we can pick m sufficiently large such that µ h h −1 .
Suppose also henceforth that κh/ε ≥ 1, so that α K given by (1.9) takes the value 1/κ. Then, we find that
K∈T α 2 K r T 2 K = 1 κ 2 ε 2 ε 2 µ h + κ 2 2 µ 2 h f 2 ε 2 ε 2 µ h + κ 2 2 1 κ 2 h 2 .
We also obtain
F ∈FΩ ε −1 α F j T 2 F ε 2 ε 2 µ h + κ 2 2 1 εκ N −1 i=−N +1 |f (x i )| 2 ε 2 ε 2 µ h + κ 2 2 1 εκh ,
where we have used the trigonometric identity
N −1 i=−N +1 |f (x i )| 2 = N −1 i=−N +1 cos mπi 2N 2 = N −1 i=−N +1 cos ( √ 2N − 1)πi 2N 2 = N = 1 2h .
Since εκh ≤ κ 2 h 2 , we see that
K∈T α 2 K r T 2 K F ∈FΩ ε −1 α F j T 2 F ⇐⇒ |||u − u T ||| 2 F ∈FΩ ε −1 α F j T 2 F ,
where we note that there is no data oscillation since f ∈ P 1 (T ). Hence this provides an example where (5.3) holds, and the factor κh/ε can be made arbitrarily large.
Flux equilibration on a submesh
We finish with the following remark:
Remark 5.3 (Flux equilibration on boundary-layer adapted submeshes). The approach in [4,5,25], following [2], can be seen as defining a flux σ T ∈ H(div, Ω) that satisfies an equilibration property similar to (1.10), yet with the key difference that σ T is defined with respect to a submesh T of T with thin elements that are adapted to the parameters ε and κ and local mesh-size (see e.g. [4, Fig. 3]). In this case, the argument in the proof of Proposition 5.1 does not apply, because
the inverse inequality (ε∇u T + ε −1 σ T )·n F F h − 1 2 K ε∇u T + ε −1 σ T K , F ⊂ ∂K, K ∈ T , is not applicable when σ T ∈ P p ( T ; R d ) but σ T / ∈ P p (T ; R d )
. This essentially shows how there are now two different approaches to constructing robust equilibrated flux estimators. Either the flux is computed as a piecewise polynomial vector field with respect to the original mesh, in which case the inclusion of a weight of the form of w K from (3.2) in the upper bound is necessary, or one constructs the flux with respect to some other sufficiently rich subspace of H(div, Ω), such as a piecewise polynomial subspace with respect to an adapted submesh T of T , in which case the weights are not necessary.
A Explicit constants for the inverse inequality
For each polynomial degree p ≥ 0, let C p,1 denote the best constant of the inverse inequality for the unit interval (0, 1), i.e.
v L 2 (0,1) ≤ C p,1 v L 2 (0,1) ∀ v ∈ P p (0, 1), (A.1)
where P p (0, 1) denotes the space of univariate polynomials of degree at most p on (0, 1). It was shown in [26] that, for all p ≥ 0,
C p,1 ≤ 1 √ 2 (p − 1)p(p + 1)(p + 2), (A.2)
where we have taken into account the fact that we consider C p,1 on the unit interval (0, 1) rather than the interval (−1, 1) as in [26]. This improves on earlier bounds, e.g. in [30]. We will show here explicit bounds for the constants of the inverse inequality for hypercubes and simplices in terms of C p,1 .
A.1 Unit hypercube
For an integer d ≥ 1, let {1:d} be a shorthand notation for {1, . . . , d}. Let Q d := {x ∈ R d , |x| ∞ ≤ 1, x i ≥ 0 ∀i ∈ {1:d}} denote the unit hypercube in R d , where |x| ∞ := max i∈{1:d} |x i |. Let P p (Q d ) denote the space of polynomials of total degree at most p on Q d .
Lemma A.1. For all d ≥ 1 and all p ≥ 0, we have v xi L 2 (Q d ) ≤ C p,1 v L 2 (Q d ) ∀ v ∈ P p (Q d ), ∀i ∈ {1:d}. (A.3)
Proof. After a possible re-labelling of the indices, it is enough to show that (A.3) holds for the case i = 1. Then, writing
x = (x 1 , x ) with x ∈ R d−1 , we see that Q d |v x1 | 2 dx = Q d−1 1 0 |v x1 (x 1 , x )| 2 dx 1 dx ≤ C 2 p,1 Q d−1 1 0 |v(x 1 , x )| 2 dx 1 dx = C 2 p,1 Q d |v| 2 dx,
where we use the fact that x 1 → v(x 1 , x ) is in P p (0, 1) for all x .
A.2 Unit simplex
For a parameter t > 0, let K d t := {x ∈ R d , |x| 1 ≤ t, x i ≥ 0 ∀i ∈ {1:d}}, where |x| 1 := d i=1 |x i |, denote the simplex in R d with side-length t. If t = 1, we adopt the simpler notation K d :
= K d 1 . Let C p,d denote the best constant such that v xi L 2 (K d ) ≤ C p,d v L 2 (K d ) ∀ v ∈ P p (K d ), ∀i ∈ {1:d}. (A.4)
We shall obtain here an explicit bound for the constant C p,d in terms of the space dimension d and the constant C p,1 of (A.1).
C p,d ≤ √ 5 4 (2 √ 2) d C p,1 . (A.5)
Proof. The proof is based on an induction on the dimension, where we seek to bound C p,d in terms of C p,d−1 , C p,1 , and d. Without loss of generality, it is enough to consider only the case i = 1 in (A.4), after a possible re-labelling of the indices. Then, writing
x = (x , x d ) with x ∈ R d−1 , we have K d |v x1 | 2 dx = 1 0 K d−1 1−x d |v x1 | 2 dx dx d . Since, for fixed x d ∈ (0, 1), x → v(x , x d ) is a
polynomial of degree at most p on K d−1 1−x d , it would be natural to apply the inverse inequality for simplices of dimension d − 1 after a suitable scaling. However, a difficulty arises for x d close to 1 due to the appearance of a negative power of 1 − x d inside the resulting integral. We can overcome this obstacle using an appropriate subdivision of the unit simplex and a change of variables.
The proof proceeds in two steps. We first treat the case d = 2 and show that (A.5) holds (we actually consider d ≥ 2 below for the sake of generality), and then the induction is carried out on d with a different argument, leading to a sharper bound than that would result from step 1 only.
Step 1. Let d ≥ 2 and consider the partition of K into K * := {x ∈ K, x d < 1 − 1/d} and
K † := K \K * . Then, v x1 2 L 2 (K d ) = v x1 2 L 2 (K * ) + v x1 2 L 2 (K † )
, and the first term can be bounded
as follows: v x1 2 L 2 (K * ) = 1−1/d 0 K d−1 1−x d |v x1 | 2 dx dx d ≤ 1−1/d 0 C 2 p,d−1 (1 − x d ) 2 K d−1 1−x d |v| 2 dx dx d ≤ d 2 C 2 p,d−1 v 2 L 2 (K * ) , (A.6)
where crucially we use the fact that (1 − x d ) −2 ≤ d 2 for x d ≤ 1 − 1/d. In order to bound the second term v x1 2 L 2 (K † ) , we introduce a change of coordinates in terms of the affine map F defined by
F (ξ) := e d + d i=1 (e i−1 − e d )ξ i ,
where e 0 = 0, and e i is the i-th unit vector for
1 ≤ i ≤ d. Letting x = F (ξ), we have x j = ξ j+1 for j ≤ d − 1, and x d = 1 − d i=1 ξ i . The inverse is then given by ξ 1 = 1 − d i=1 x i , and ξ j = x j−1 for 2 ≤ j ≤ d.
It is thus easily seen that F is a bijection from K onto itself, and that F (0) = x d . Thus F corresponds to a change of coordinates on the unit simplex. Additionally, it can be shown that the Jacobian |detDF | = 1.
Let Q d 1/d := {ξ ∈ Q d , |ξ| ∞ ≤ 1/d} be a hypercube with side length 1/d, and let Q † be the parallelepiped obtained as the image of Q d 1/d under the mapping F , i.e. Q † = F (Q d 1/d ). It is then easy, but tedious, to show that K † ⊂ Q † ⊂ K.
(A.7) Figure 2 illustrates the sets K † , Q † , and K for the case d = 3. Now, let v(ξ) = v(F (ξ)) be the pullback of v under F . Since F is affine, v ∈ P p (K d ). It is also easy to check that v x1 = v ξ2 − v ξ1 . Using the change of variables and the fact that |detDF | = 1, it follows from (A.7) that v x1
2 L 2 (K † ) ≤ v x1 2 L 2 (Q † ) = v ξ2 − v ξ1 2 L 2 (Q d 1/d ) ≤ 2( v ξ2 2 L 2 (Q d 1/d ) + v ξ1 2 L 2 (Q d 1/d ) ).
Applying the inverse inequality for hypercubes, namely v ξi 2 L 2 (Q d 1/d ) ≤ d 2 C 2 p,1 v 2 L 2 (Q d 1/d ) , and changing back to the original variables, we then obtain from the second inclusion in (A.7) that v x1 2 L 2 (K † ) ≤ 4d 2 C 2 p,1 v 2 L 2 (Q † ) ≤ 4d 2 C 2 p,1 v 2 L 2 (K d ) .
(A.8) Therefore, combining (A.6) and (A.8), we arrive at v x1 2 L 2 (K d ) ≤ d 2 (C 2 p,d−1 + 4C 2 p,1 ) v 2 L 2 (K d ) , for any v ∈ P p (K d ). This implies C 2 p,d ≤ d 2 (C 2 p,d−1 + 4C 2 p,1 ), and thus (A.1) and an induction argument show that
C p,d ≤ 1 + 4 d−1 j=1 1 (j!) 2 1 2 d! C p,1 (A.9)
for any d ≥ 2. This shows (A.4), but with a worse constant than that of (A.5) for d ≥ 3. For this reason, we proceed in a second step in a different way.
Step 2. Let d ≥ 3. We again subdivide the simplex K, this time as
K = {x ∈ K, x d ≤ 1/2} ∪ {x ∈ K, x d−1 ≤ 1/2}.
Furthermore, for any fixed x d−1 , x d−1 = (x 1 , . . . , x d−2 , x d ) → v(x) is a polynomial of degree at most p on a simplex that is isometric to K d−1 1−x d−1 . Let also x d = (x 1 , . . . , x d−1 ). Crucially, since d ≥ 3 and we subdivide above into two subsets, we can avoid the critical subset K † of Step 1 as v x1
2 L 2 (K d ) ≤ d j=d−1 1/2 0 K d−1 1−x j |v x1 | 2 dx j dx j ≤ d j=d−1 1/2 0 C 2 p,d−1 (1 − x j ) 2 K d−1 1−x j |v| 2 dx j dx j ≤ 8C 2 p,d−1 v 2 K d . (A.10)
It then follows by induction that C p,d ≤ (2 √ 2) d−2 C p,2 for all d ≥ 3. Since C p,2 ≤ 2 √ 5 C p,1 by (A.9), we get (A.5).
Applying Theorem A.2 to the cases d = 2 and d = 3 gives the following explicit bounds C p,2 ≤ 10(p − 1)p(p + 1)(p + 2), C p,3 ≤ 80(p − 1)p(p + 1)(p + 2). (A.11)
A.3 General simplex
Let K be a simplex in R d , d ≥ 2, and letK denote the unit simplex. Let J K denote the differential of the affine transformation mapping T K :K → K. For v ∈ RT N p (K), we define the Piola transformationv ∈ RT N p (K) bŷ
v(x) = |det J K |J −1 K [v • T K (x)]J K 2 ≤ h K ρK , J −1 K 2 ≤ √ 2 ρ K , |det J K | = |K| d |K| d , |∂K| d−1 |K| d ≤ (d + 1)d ϑ T h −1 K . (A.13)
Note that in Lemma A.3, we have used the fact that the diameter of the unit simplex is √ 2 for all d ≥ 2.
Lemma A.4 (Hesthaven & Warburton [39]). Let v ∈ P p (K). Then v ∂K ≤ (p + 1)(p + d)
d |∂K| d−1 |K| d v K .
Therefore, for v ∈ RT N p (K), we have h 1/2 K v · n ∂K ≤ C p+1,d,∂K v K , C p+1,d,∂K := (d + 1)(p + 2)(p + d + 1)ϑ T .
Lemma A.5. Let K be a simplex in R d and v ∈ RT N p (K). Then,
h K ∇ · v K ≤ C p+1,d,K v K , C p+1,d,K := √ 2d ϑ T C p+1,d , (A.14)
where C p,d is characterized in Theorem A.2.
Proof. Using the Piola Transformation, we have ∇ · v = ∇x ·v/|det J K |, therefore, ∇ · v 2 K ≤ |det J K | −1 ∇·v 2K . Then, sincev i ∈ P p+1 (K) for i ∈ {1:d}, we apply Theorem A.2 to obtain ∇·
v 2K ≤ d d i=1 v i,xi 2K ≤ dC 2 p+1,d v 2K .
Then, using the definition of the Piola transformation, it is seen that v 2K ≤ J −1 K 2 2 |det J K | v 2 K . We then use the bound J −1 K 2 ≤ √ 2ϑ T h −1 K from (A.13) to find that h K ∇ · v K ≤ √ 2dϑ T C p+1,d v K , which finishes the proof.
Figure 1 :
1Patches T a , vanishing normal flux conditions in the local Raviart-Thomas-Nédélec spaces V a T , and hat functions ψ a : interior (left) and boundary (right) vertex a ∈ V
Theorem 3. 1 (
1Guaranteed a posteriori error estimate). Let u be the weak solution of problem (1.1) given by (1.
Theorem 4 . 4 (
44Local efficiency and robustness). Let u be the weak solution of problem (1.1) given by (1.
Example 5. 2 (
2Dominant jump estimators). Let Ω := (−1/2, 1/2) and let m be an odd integer that will later on be chosen sufficiently large. Consider a uniform mesh T of Ω with 2N = (m+1) 2 intervals, N := (m + 1) 2 /2, and mesh size h := 1/(2N ) = 1/(m + 1) 2 . Hence, the interior nodes are x i = ih, where i ∈ {−N + 1, . . . , N − 1}. Let f := I T cos(mπx) ∈ P 1 (T ) ∩ H 1 0 (Ω)
Figure 2 :
2Subdivision of the unit simplex used in the proof of Theorem A.2. The unit simplex is shown for d = 3, along with its sub-simplex K † (edges shown in green) and sub-parallelepiped Q † (edges shown in red).Theorem A.2. For all d ≥ 1 and for all p ≥ 0, the best constant C p,d in (A.4) satisfies
. (A.12) Lemma A.3 (Ciarlet [10] Thm 3.1.2 and [13]). There holds
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| [] |
[
"The Earth-Moon system as a typical binary in the Solar System",
"The Earth-Moon system as a typical binary in the Solar System"
] | [
"S I Ipatov [email protected]. \nVernadsky Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences\nKosygina 19119991MoscowRussia; (\n\nSpace Research Institute of Russian Academy of Sciences\nProfsoyuznaya st. 84/32MoscowRussia\n"
] | [
"Vernadsky Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences\nKosygina 19119991MoscowRussia; (",
"Space Research Institute of Russian Academy of Sciences\nProfsoyuznaya st. 84/32MoscowRussia"
] | [] | Solid embryos of the Earth and the Moon, as well as trans-Neptunian binaries, could form as a result of contraction of the rarefied condensation which was parental for a binary. The angular momentum of the condensation needed for formation of a satellite system could be mainly acquired at the collision of two rarefied condensations at which the parental condensation formed. The minimum value of the mass of the parental condensation for the Earth-Moon system could be about 0.02 of the Earth mass. Besides the main collision, which was followed by formation of the condensation that was a parent for the embryos of the Earth and the Moon, there could be another main collision of the parental condensation with another condensation. The second main collision (or a series of similar collisions) could change the tilt of the Earth. Depending on eccentricities of the planetesimals that collided with the embryos, the Moon could acquire 0.04-0.3 of its mass at the stage of accumulation of solid bodies while the mass of the growing Earth increased by a factor of ten.IntroductionIt is supposed by many authors that the Earth-Moon system formed as a result of a collision of the solid Earth with a Mars-sized object. Galimov and Krivtsov [1] presented arguments that such giant impact concept has several weaknesses. Below we discuss that the formation of the Earth-Moon system from a rarefied condensation can be considered similar to formation of trans-Neptunian binaries. Ipatov [2-3] and Nesvorny et al.[4]supposed that trans-Neptunian satellite systems formed by contraction of rarefied condensations. Ipatov[3]concluded that trans-Neptunian satellite systems could get the main fraction of their angular momenta due to collisions of rarefied condensations.Last years, new arguments in favor of the formation of rarefied preplanetesimals (clumps) were presented in several papers. These clumps could include decimeter and meter sized boulders in contrast to dust condensations earlier considered. For example, Johansen et al.[5]found the efficient formation of gravitationally bound clumps, with a range of masses corresponded to radii of contracted objects from 100 to 400 km in the asteroid belt and from 150 to 730 km in the Kuiper belt. Lyra et al.[6]showed that in the vortices launched by the Rossby wave instability in the borders of the dead zone, the solids quickly achieve critical densities and undergo gravitational collapse into protoplanetary embryos in the mass range [0.1M E , 0.6M E ], where M E is the mass of the Earth.Initial Angular Velocities of Rarefied Condensations and Angular Velocities Needed for Formation of Satellite SystemsAccording to Safronov [7], the initial angular velocity of a rarefied condensation (around its center of mass) was 0.2Ω for a spherical condensation, where Ω is the angular velocity of the condensation moving around the Sun. The initial angular velocity is positive and is not enough for formation of satellites. In calculations of contraction of condensations (of mass m and radius r=0.6r H , where r H is the Hill radius) presented in [4], trans-Neptunian objects with satellites formed at initial angular velocities ω о from the range [0.5Ω о , 0.75Ω о ], where Ω о =(Gm/r 3 ) 1/2 , G is the gravitational constant. As Ω o /Ω≈1.73(r H /r) 3/2 , then Ω≈0.27Ω o and 0.2Ω≈0.054Ω o at r=0.6r H .In the 3D calculations of gravitational collapse of the parental condensation for the Earth-Moon system presented in[1], binaries formed at ω о /Ω о from the range [1, 1.46]. The radii of initial condensations used in calculations considered in [1] were much smaller (by about a factor of 40) than their Hill radii. Galimov and Krivtsov [1] considered evaporation of particles that constitute a rarefied condensation in order to explain the formation of the Earth-Moon system from the condensation with the same angular momentum as that of this system. May be such formation could take place in the model without the evaporation if one consider another size of the parental condensation in their calculations? It may be interesting to calculate the contraction of condensations for a wider range of ratios of radii of condensations to their Hill radii than in the calculations presented in[1,4].Formation of Trans-Neptunian BinariesNesvorny et al.[4] calculated contraction of rarefied condensations in the trans-Neptunian | null | [
"https://arxiv.org/pdf/1607.07037v1.pdf"
] | 118,554,798 | 1607.07037 | 514e8167847f4b27ed57fa4d98f80e616b31983e |
The Earth-Moon system as a typical binary in the Solar System
S I Ipatov [email protected].
Vernadsky Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences
Kosygina 19119991MoscowRussia; (
Space Research Institute of Russian Academy of Sciences
Profsoyuznaya st. 84/32MoscowRussia
The Earth-Moon system as a typical binary in the Solar System
Solid embryos of the Earth and the Moon, as well as trans-Neptunian binaries, could form as a result of contraction of the rarefied condensation which was parental for a binary. The angular momentum of the condensation needed for formation of a satellite system could be mainly acquired at the collision of two rarefied condensations at which the parental condensation formed. The minimum value of the mass of the parental condensation for the Earth-Moon system could be about 0.02 of the Earth mass. Besides the main collision, which was followed by formation of the condensation that was a parent for the embryos of the Earth and the Moon, there could be another main collision of the parental condensation with another condensation. The second main collision (or a series of similar collisions) could change the tilt of the Earth. Depending on eccentricities of the planetesimals that collided with the embryos, the Moon could acquire 0.04-0.3 of its mass at the stage of accumulation of solid bodies while the mass of the growing Earth increased by a factor of ten.IntroductionIt is supposed by many authors that the Earth-Moon system formed as a result of a collision of the solid Earth with a Mars-sized object. Galimov and Krivtsov [1] presented arguments that such giant impact concept has several weaknesses. Below we discuss that the formation of the Earth-Moon system from a rarefied condensation can be considered similar to formation of trans-Neptunian binaries. Ipatov [2-3] and Nesvorny et al.[4]supposed that trans-Neptunian satellite systems formed by contraction of rarefied condensations. Ipatov[3]concluded that trans-Neptunian satellite systems could get the main fraction of their angular momenta due to collisions of rarefied condensations.Last years, new arguments in favor of the formation of rarefied preplanetesimals (clumps) were presented in several papers. These clumps could include decimeter and meter sized boulders in contrast to dust condensations earlier considered. For example, Johansen et al.[5]found the efficient formation of gravitationally bound clumps, with a range of masses corresponded to radii of contracted objects from 100 to 400 km in the asteroid belt and from 150 to 730 km in the Kuiper belt. Lyra et al.[6]showed that in the vortices launched by the Rossby wave instability in the borders of the dead zone, the solids quickly achieve critical densities and undergo gravitational collapse into protoplanetary embryos in the mass range [0.1M E , 0.6M E ], where M E is the mass of the Earth.Initial Angular Velocities of Rarefied Condensations and Angular Velocities Needed for Formation of Satellite SystemsAccording to Safronov [7], the initial angular velocity of a rarefied condensation (around its center of mass) was 0.2Ω for a spherical condensation, where Ω is the angular velocity of the condensation moving around the Sun. The initial angular velocity is positive and is not enough for formation of satellites. In calculations of contraction of condensations (of mass m and radius r=0.6r H , where r H is the Hill radius) presented in [4], trans-Neptunian objects with satellites formed at initial angular velocities ω о from the range [0.5Ω о , 0.75Ω о ], where Ω о =(Gm/r 3 ) 1/2 , G is the gravitational constant. As Ω o /Ω≈1.73(r H /r) 3/2 , then Ω≈0.27Ω o and 0.2Ω≈0.054Ω o at r=0.6r H .In the 3D calculations of gravitational collapse of the parental condensation for the Earth-Moon system presented in[1], binaries formed at ω о /Ω о from the range [1, 1.46]. The radii of initial condensations used in calculations considered in [1] were much smaller (by about a factor of 40) than their Hill radii. Galimov and Krivtsov [1] considered evaporation of particles that constitute a rarefied condensation in order to explain the formation of the Earth-Moon system from the condensation with the same angular momentum as that of this system. May be such formation could take place in the model without the evaporation if one consider another size of the parental condensation in their calculations? It may be interesting to calculate the contraction of condensations for a wider range of ratios of radii of condensations to their Hill radii than in the calculations presented in[1,4].Formation of Trans-Neptunian BinariesNesvorny et al.[4] calculated contraction of rarefied condensations in the trans-Neptunian
region and found the cases when the contraction ends in formation of binaries (or triples). They supposed that condensations got their angular momenta when they formed from the protoplanet cloud. Ipatov [3] compared the angular velocities used by Nesvorny et al. [4] as initial data for contracting preplanetesimals with the angular velocity of the condensations formed by the merger between two collided condensations which moved before the collision in circular heliocentric orbits. The angular velocity ω of the condensation formed at the collision of two identical condensations moving in circular heliocentric orbits can be as high as 1.575Ω [3]. At r=r H , ω can be as high as 0.9Ω o , or even a little greater than Ω o , if we take into account the initial 0.2Ω. Ipatov concluded [3] that the initial angular velocity of the parental condensation at which a binary could form at calculations made by Nesvorny et al. [4] could be obtained at the collision of two condensations which moved before the collision in circular heliocentric orbits. For the case when sizes of collided rarefied preplanetesimals (RPPs) are equal to the sizes of RPPs at the moment when they got initial rotation, the typical angular momentum acquired at the collision of two equal condensations is greater by an order of magnitude than the initial angular momentum of the condensation with mass equal to that of the parental condensation.
The role of initial rotation in the angular momentum K s of the parental RPP can be greater if sizes of RPPs changed before the collision. If we consider a collision of two identical spherical RPPs with masses m 1 and radii equal to k col r H , each of which initially formed with radius k in r H and angular velocity equal to 0.2Ω, then the angular momentum of the spherical RPP formed after the collision is
K s ≈(0.96k Θ •k col 2 +0.077•k in 2 )a 1/2 m 1 5/3 G 1/2 M S -1/6
, where a is the semi-major axis of the RPP, and M S is the mass of the Sun. According to [2], the mean value of k Θ is 0.6 and k Θ ≤1. In the above formula at k in /k col >2.7 and k Θ =0.6 (or at k in /k col >3.5 and k Θ =1), the role of initial rotation is greater than the role of the collision. It shows that collisions played the main role in K s only when sizes of preplanetesimals did not differ much (by not more than a factor of 3) from their initial sizes. At k in /k col about 3, masses differ by a factor of about 30. If we consider formation of a condensation at a collision of two RPPs for which the ratio of their radii is equal to k r , then at k Θ =1 and k in =k col , the contribution to K s due to a collision of the RPPs is greater than the contribution to K s due to initial rotation (equal to 0.2Ω) by a factor of 12.5, 3 and 0.8 for k r equal to 1, 2, and 3, respectively. Below we use the term 'similar sizes of RPPs' for the case when the collision of considered RPPs gives the main contribution to the angular momentum of the final RPP, i.e., the ratio of diameters of collided RPPs does not exceed 3 for almost circular heliocentric orbits. At some collisions, the mass of the formed RPP could be smaller than the sum of masses of collided RPPs, and some fraction of the angular momentum of collided RPPs could belong to the material that was not incorporated into the formed RPP.
Prograde and Retrograde Rotation of Trans-Neptunian Binaries
Ipatov [7] studied inclinations i s of orbits of secondaries around 32 objects moving in the trans-Neptunian belt and discussed how such inclinations could form. The below discussion is based on the plots presented in [7], and the plots used the data from http://www.johnstonsarchive.net/astro/astmoons/. Note that i s is considered relative to the ecliptic and differs from the inclination relative to the plane which is perpendicular to the axis of rotation of a primary. For example, i s =96 o for Pluto, though Charon is moving in the plane which is perpendicular to the Pluto's rotational axis. The fraction of objects with i s >90 o equals 13/32≈0.406 at all values of eccentricity e of a heliocentric orbit of a binary, and it is 13/28≈0.464 for e<0. 3. The distribution of i s is in the wide range almost from 0 to 180 o . It shows that a considerable fraction of the angular momentum of the RPPs that contracted to form satellite trans-Neptunian systems was not due to initial rotation of RPPs or to collisions of RPPs with small objects (e.g., boulders and dust), but it was acquired at collisions of the RPPs which masses did not differ much, because else the angular momentum would be positive. Ipatov [2] noted that the angular momentum of collided RPPs could be positive or negative depending on heliocentric orbits of the RPPs. Some excess of the number of discovered binaries with positive angular momentum compared with the number of discovered binaries with negative angular momentum was caused in particular by the contribution of initial positive angular momentum of RPPs and by the contribution of collisions of RPPs with small objects to the angular momentum of the parental RPP that produced the binary. There could be also some excess of positive angular momentum at mutual collisions of RPPs of similar sizes.
We suppose that in the case of two centers of contraction inside the preplanetesimal formed by a merger between two collided rarefied condensations, at the values of the angular momentum of the parental preplanetesimal a little smaller than it is needed for formation of binaries, a solid object consisting of two touching components could form. Some asteroids and comets (e.g., asteroid Itokava and Comet 67P/Churyumov-Gerasimenko) consist of two parts and have a form of dumbbells. Probably, the mass range of initial rarefied condensations in the Solar System was very wide, and their masses could vary from a mass of 1-km solid object to the mass of Mars.
The Angular Momentum of the Condensation Parental for the Earth-Moon System that Formed at a Collision of Two Condensations
The angular velocity of the parental condensation formed at the collision of two identical condensations is a little smaller than Ω o needed for formation of binaries in calculations made by Galimov and Krivtsov [1], but the contraction of the condensation formed at the collision to the size of the condensation considered in [1] can considerably increase the angular velocity. The angular velocity of the condensation of radius r c formed as a result of compression of the condensation, with radius r 1 and the angular velocity ω 1 , equals ω rc =ω 1 (r 1 /r c ) 2 . The angular momentum of the condensation of radius 0.12r H formed at a typical collision of two identical condensations is the same as the angular momentum for the condensation with r=0.025r H considered in [1]. Therefore, any initial angular velocities considered in [1,4] can be reached after contraction of the condensation formed at a collision of condensations not greater than their Hill spheres.
In our opinion, the embryos of the Earth and the Moon could form as a result of contraction of the same parental rarefied condensation. A considerable fraction of the angular momentum of such condensation could be acquired at a collision of two rarefied condensations. Based on formulas presented in [2][3], we have concluded that the present angular momentum of the Earth-Moon system could be acquired at the collision of two identical rarefied condensations with sizes of Hill spheres, which total mass was about 0.1M Е , and which heliocentric orbits were circular. The initial mass of the rarefied condensation that was a parent for the embryos of the Earth and the Moon could be relatively small (0.02M Е or even less) if we take into account the growth of the angular momentum of the embryos at the time when they accumulated planetesimals. There could be also the second main collision of the parental condensation with another condensation, at which the radius of the Earth's embryo condensation was smaller than the semi-major axis of the orbit of the Moon's embryo. The second main collision (or a series of similar collisions) could change the tilt of the Earth to its present value.
The Angular Momentum of the Rarefied Condensation that Formed by Accumulation of Smaller Objects
If the parental condensation with final mass m and radius r=k H r H got all its angular momentum K s as a result of accumulation of smaller objects, then K s ≈0.173k H 2 G 1/2 a 1/2 m 5/3 M S -1/6 ΔK [3], where ΔK is the difference between the fraction of positive increments of angular momentum and the fraction of negative increments. At ΔK=0.9 (a typical value for Hill spheres moving in circular heliocentric orbits), m=M E +M M (the sum of present masses of the Earth and the Moon), k H =1, and a=1 AU, we obtain that K s is greater by a factor of 24.5 than the present angular momentum K sEM of the Earth-Moon system, including the rotational momentum of the Earth. Taking into account that K s is proportional to m 5/3 , we obtain that K s =K sEM at m=(M E +M M )/6.8. The angular momentum of the Earth-Moon system is positive. Therefore, for the mass of the final condensation m≥0.15M E , the angular momentum equal to K sEM could be acquired at any contribution of a collision of two large condensations to the angular momentum of the final condensation. In principle, the angular momentum of the condensation needed for formation of the Earth-Moon system could be acquired by accumulation only of small objects. Nevertheless, we suppose that the collision of two large condensations played a considerable role in the angular momentum of the collapsing parental condensation. Else the parental condensations of Venus and Mars could also get large angular momentum, which was enough for formation of large satellites. The greater was the role of small objects in formation of the condensation that was a parent for the Earth-Moon system, the greater could be the difference in masses of two collided condensations at the main collision. It may be a question whether two condensations which masses differed by not more than an order of magnitude could form at close distances from the Sun. Dust particles and bolders could considerably change distances from the Sun with time and could reach the growing condensation from not close distances if the lifetime of the condensation was not small. In order to get large times of contraction of condensations, it is needed to consider factors preventing fast collapse of condensations.
AcknowledgementsThis study was supported by Program no. 9 of the Presidium of the Russian Academy of Sciences and by the Russian Foundation for Basic Research, project no. 14-02-00319.The Growth of Solid Embryos of the Earth and the MoonSolid embryos of the Earth and the Moon grew to the present masses of the Earth and the Moon (M Е and 0.0123M Е , respectively) by accumulation of smaller planetesimals. For large enough eccentricities of planetesimals, the effective radii r ef of proto-Earth and proto-Moon are proportional to the radius r e of a considered embryo. For such proportionality, we can obtain r Mo =mMo.01170 and f m2 =0.049 at k 2 =0.6 -1/3 . In this case for the growth of the Earth embryo mass by 10 times, the Moon embryo mass increased by the factor of 1.044 and 1.051 at k 2 =1 and k 2 =0.6 -1/3 , respectively. In the above models, depending on eccentricities of planetesimals (i.e., on dependence of r ef on r e ), the Moon could acquire 0.04-0.3 (the lower estimate is for almost circular heliocentric orbits) of its mass at the stage of accumulation of solid bodies during the time when the mass of the growing Earth increased by a factor of ten.In our approach, the influx of the matter to embryos is from the zone around the heliocentric orbit of the Earth-Moon embryos system, but not only from the sphere around the embryos as in[1]. For comparison with[1], in the case of k d =0.6
Origin of the Moon. New concept. / De Gruyter. E M Galimov, A M Krivtsov, 168BerlinGalimov, E. M., Krivtsov, A. M. (2012) Origin of the Moon. New concept. / De Gruyter. Berlin. 168 p.
. S I Ipatov, Mon. Not. R. Astron. Soc. 403Ipatov, S. I. (2010) Mon. Not. R. Astron. Soc., 403, 405-414.
Formation, detection, and characterization of extrasolar habitable planets. S I Ipatov, Proc. IAU Symp. IAU SympCambridge Univ. PressIpatov, S. I. (2014) In Proc. IAU Symp. No. 293 "Formation, detection, and characterization of extrasolar habitable planets", Cambridge Univ. Press, 285-288 (http://arxiv.org/abs/1412.8445).
. D Nesvorny, A N Youdin, D C Richardson, Astron. J. 140Nesvorny, D., Youdin, A. N., Richardson, D. C. (2010) Astron. J., 140, 785-793.
. A Johansen, A N Youdin, Y Lithwick, Astron. Astrophys. 537125Johansen, A., Youdin, A. N., Lithwick, Y. (2012) Astron. Astrophys., 537, A125.
. W Lyra, A Johansen, H Klahr, N Piskunov, Astron. Astrophys. 491Lyra, W., Johansen, A., Klahr, H., Piskunov, N. (2008) Astron. Astrophys., 491, L41-L44.
Evolution of the protoplanetary cloud and formation of the Earth and the planets. V S Safronov, Moscow. Nauka. English translation. 677Safronov, V. S. (1969) Evolution of the protoplanetary cloud and formation of the Earth and the planets, Moscow. Nauka. English translation: NASA TTF-677, 1972.
. S I Ipatov, Lps Xlvi, Ipatov, S.I. (2015) LPS XLVI, # 1512.
| [] |
[
"Evolution of binary seeds in collapsing protostellar gas clouds",
"Evolution of binary seeds in collapsing protostellar gas clouds"
] | [
"Tatsuya Satsuka \nTheoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan\n\nCollege of Science\nIbaraki University\n2-1-1 Bunkyo310-8512MitoIbarakiJapan\n",
"Toru Tsuribe \nCollege of Science\nIbaraki University\n2-1-1 Bunkyo310-8512MitoIbarakiJapan\n",
"Suguru Tanaka \nTheoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan\n",
"Kentaro Nagamine \nTheoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan\n\nDepartment of Physics and Astronomy\nUniversity of Nevada\nLas Vegas, 4505 S. Maryland Pkwy, Las Vegas89154-4002NVUSA\n"
] | [
"Theoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan",
"College of Science\nIbaraki University\n2-1-1 Bunkyo310-8512MitoIbarakiJapan",
"College of Science\nIbaraki University\n2-1-1 Bunkyo310-8512MitoIbarakiJapan",
"Theoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan",
"Theoretical Astrophysics\nDepartment of Earth and Space Science\nGraduate School of Science\nOsaka University\n1-1 Machikaneyama560-0043ToyonakaJapan",
"Department of Physics and Astronomy\nUniversity of Nevada\nLas Vegas, 4505 S. Maryland Pkwy, Las Vegas89154-4002NVUSA"
] | [
"Mon. Not. R. Astron. Soc"
] | We perform three dimensional smoothed particle hydrodynamics (SPH) simulations of gas accretion onto the seeds of binary stars to investigate their short-term evolution. Taking into account of dynamically evolving envelope with non-uniform distribution of gas density and angular momentum of accreting flow, our initial condition includes a seed binary and a surrounding gas envelope, modelling the phase of core collapse of gas cloud when the fragmentation has already occurred. We run multiple simulations with different values of initial mass ratio q 0 (the ratio of secondary over primary mass) and gas temperature. For our simulation setup, we find a critical value of q c = 0.25 which distinguishes the later evolution of mass ratio q as a function of time. If q 0 q c , the secondary seed grows faster and q increases monotonically towards unity. If q 0 q c , on the other hand, the primary seed grows faster and q is lower than q 0 at the end of the simulation. Based on our numerical results, we analytically calculate the long-term evolution of the seed binary including the growth of binary by gas accretion. We find that the seed binary with q 0 q c evolves towards an equal-mass binary star, and that with q 0 q c evolves to a binary with an extreme value of q. Binary separation is a monotonically increasing function of time for any q 0 , suggesting that the binary growth by accretion does not lead to the formation of close binaries. | 10.1093/mnras/stw2709 | [
"https://arxiv.org/pdf/1607.06592v2.pdf"
] | 119,282,040 | 1607.06592 | ffcf43f30d0191755b8fa3ea8e2d3cf786e0efa6 |
Evolution of binary seeds in collapsing protostellar gas clouds
18 November 2016
Tatsuya Satsuka
Theoretical Astrophysics
Department of Earth and Space Science
Graduate School of Science
Osaka University
1-1 Machikaneyama560-0043ToyonakaJapan
College of Science
Ibaraki University
2-1-1 Bunkyo310-8512MitoIbarakiJapan
Toru Tsuribe
College of Science
Ibaraki University
2-1-1 Bunkyo310-8512MitoIbarakiJapan
Suguru Tanaka
Theoretical Astrophysics
Department of Earth and Space Science
Graduate School of Science
Osaka University
1-1 Machikaneyama560-0043ToyonakaJapan
Kentaro Nagamine
Theoretical Astrophysics
Department of Earth and Space Science
Graduate School of Science
Osaka University
1-1 Machikaneyama560-0043ToyonakaJapan
Department of Physics and Astronomy
University of Nevada
Las Vegas, 4505 S. Maryland Pkwy, Las Vegas89154-4002NVUSA
Evolution of binary seeds in collapsing protostellar gas clouds
Mon. Not. R. Astron. Soc
00018 November 2016Accepted xxx. Received xxx; in original form xxx(MN L A T E X style file v2.2)accretion, accretion discs -hydrodynamics -circumstellar matter -binaries: general -stars: formation
We perform three dimensional smoothed particle hydrodynamics (SPH) simulations of gas accretion onto the seeds of binary stars to investigate their short-term evolution. Taking into account of dynamically evolving envelope with non-uniform distribution of gas density and angular momentum of accreting flow, our initial condition includes a seed binary and a surrounding gas envelope, modelling the phase of core collapse of gas cloud when the fragmentation has already occurred. We run multiple simulations with different values of initial mass ratio q 0 (the ratio of secondary over primary mass) and gas temperature. For our simulation setup, we find a critical value of q c = 0.25 which distinguishes the later evolution of mass ratio q as a function of time. If q 0 q c , the secondary seed grows faster and q increases monotonically towards unity. If q 0 q c , on the other hand, the primary seed grows faster and q is lower than q 0 at the end of the simulation. Based on our numerical results, we analytically calculate the long-term evolution of the seed binary including the growth of binary by gas accretion. We find that the seed binary with q 0 q c evolves towards an equal-mass binary star, and that with q 0 q c evolves to a binary with an extreme value of q. Binary separation is a monotonically increasing function of time for any q 0 , suggesting that the binary growth by accretion does not lead to the formation of close binaries.
INTRODUCTION
It is widely recognised that the majority of main sequence stars and pre-main sequence stars are in binary systems and that these binaries have various distributions in mass ratio q, binary separation, and binary frequency (Duquennoy & Mayor 1991;Ghez et al. 1993;Kouwenhoven et al. 2005;Raghavan et al. 2010;Kraus et al. 2011;De Rosa et al. 2014). For main sequence stars, Duquennoy & Mayor (1991) provided statistics of G-dwarf binary systems. They found that binaries have log-normal separation distribution, and the number of binaries with wide separations decreases with increasing q, while the number of binaries with close separations is roughly constant or increase with increasing q. Raghavan et al. (2010) and De Rosa et al. (2014) found a similar distribution of q for A-type binaries as Duquennoy & Mayor (1991).
For young stellar objects (YSOs), the number of binaries with intermediate mass YSOs decreases with increasing q (Kouwenhoven et al. 2005). In contrast, the number of binaries with low-mass YSOs increases with increasing q (Kraus et al. 2011), and have a log-normal separation distribution. Although observers have investigated various distributions of binary properties ⋆ E-mail:[email protected] in detail as mentioned above, formation process of these binaries is not well understood.
The favoured scenario of binary formation is the fragmentation during runaway collapse of cloud core (Boss & Bodenheimer 1979;Miyama et al. 1984;Tsuribe & Inutsuka 1999a,b), and the fragmentation in protostellar disc after the runaway collapse (Williams & Tohline 1988;Adams et al. 1989;Bonnell 1994;Bonnell & Bate 1994;Woodward et al. 1994;Vorobyov 2010). Details of the fragmentation process during cloud core collapse is described in Tohline (2002) and the references therein. In the present paper, we do not discuss the fragmentation process, but focus on the evolution of a seed binary after fragmentation.
After the fragmentation, the binary seeds start to accrete the surrounding envelope and grow towards main sequence stars. Since the initial mass of seed binary is lower than 1 per cent of the stellar mass (Bonnell & Bate 1994), observed physical properties of a binary (i.e., properties at the end of accretion phase) are completely different from those of the seed binaries.
Recently, large-scale hydrodynamical simulations have been performed in order to explain the observed binary properties (Bate et al. 2002a(Bate et al. ,b, 2003Attwood et al. 2009;Bate 2009a,b;Offner et al. 2009). Bate (2009a) simulated a large-scale, homogeneous, isothermal, and turbulent cloud core collapse ignoring radiative and magnetic effects, and succeeded in reproducing the ob-served multiplicity and binary properties such as the frequency of very low-mass binaries, the log-normal separation distribution, and the q-distribution of wide and close binaries. However, they did not mention about how and what physical processes determine the distributions of binaries. In order to understand that, we need to investigate the gas accretion onto a binary in more detail.
A number of simulations of gas accretion onto a binary have been performed using SPH codes in two dimensions (2D) (Dunhill et al. 2015; and 3D (Artymowicz & Lubow 1996;Bate & Bonnell 1997), as well as using grid codes in 2D (Ochi et al. 2005;Hanawa et al. 2010;D'Orazio et al. 2013;Farris et al. 2014). For example, Bate & Bonnell (1997) investigated a steady, isothermal, non-selfgravitating gas accretion onto a binary assuming constant angular momentum and density of gas at the outer boundary. They found that, in the case of low gas angular momentum, the q-value decreases, because the primary is closer to the mass centre of binary than the secondary. On the other hand, in the case of high angular momentum, the q-value increases because the infalling gas encounters the secondary first. Ochi et al. (2005) also investigated in a similar model to Bate & Bonnell (1997), and found contradictory results that the q-value can decrease even when the angular momentum of gas is high. They argued that this discrepancy was caused by the lower numerical resolution in the simulation of Bate & Bonnell (1997), which may have enhanced the accretion onto the secondary. However, there were other differences between the simulations of Bate & Bonnell (1997) and Ochi et al. (2005), such as the gas temperature, computing method, and gravitational potential of the binary, therefore the ultimate cause of the difference was unclear at that time. returned to this problem and investigated the dependence on gas temperature in the same model as Bate & Bonnell (1997) and Ochi et al. (2005), where they fixed the angular momentum of gas and changed only the gas temperature. As a result, they concluded that the disagreement between Bate & Bonnell (1997) and Ochi et al. (2005) was simply caused by the difference in the gas temperature. found that, in the case of cold gas, the gas from circum-binary disc is easily trapped inside the secondary's Roche lobe, whereas in the hot case, the flow from the secondary's Roche lobe to the primary's Roche lobe emerges, suppressing the growth of q-value.
These previous works that we described above were limited to the most simple situations, i.e., isothermal non-self-gravitating gas, constant angular momentum and density of gas at the outer boundary, and an isolated binary (i.e., no growth of binary by accretion). However, if we want to compare the simulations with real binary systems in star-forming regions, we need to consider the unsteady gas accretion caused by the non-uniform distribution of angular momentum and density of the envelope. Motivated by this current situation, we perform SPH simulations to investigate the unsteady gas accretion onto a seed binary considering the non-uniform distribution of angular momentum and density of infalling envelope. Note that we still ignore the self-gravity of gas and the growth of binary by accretion, with the aim of understanding physically how these distributions of gas affect the binary evolution. If the accreted mass exceeds the initial mass of the seed binary, the selfgravity of gas and the growth of binary by accretion might cause non-negligible effects on binary evolution. Therefore, we focus on the short-term evolution until the accreted mass exceeds the initial binary mass. Sections 2 and 3 describe our model and calculation method. We present the results of our simulations in Section 4. In Figure 1. Schematic diagrams of the initial condition of our simulations in the centre-of-mass frame of the envelope and binary. The cross shows the origin, two black circles in the centre are the binary seeds, and the gray regions show the envelope. The top (bottom) panel shows the cross section in a face-on (edge-on) view. The specific angular momentum vector of gas is aligned with z-axis. The angle θ is determined by equation (7).
Section 5, we discuss the results and give estimates of the long-term evolution of seed binaries. We conclude in Section 6.
MODEL OF SEED BINARY AND ENVELOPE
In the present work, we skip the formation process of the seed binary in order to focus on the gas accretion onto the seed binary. The seed binary and the surrounding envelope are set up in the initial condition as we describe below and in Fig. 1.
For the density and specific angular momentum distribution of the envelope, we take the following profiles:
ρ(r) = ρin
Rin r 2 and (1)
j(r) = jin r Rin ,(2)
where r is distance from the centre-of-mass of the envelope, the constants ρin and jin represent the density and specific angular momentum at r = Rin, and Rin is the radius of the inner edge of the envelope. This power-law distribution is motivated by the numerical results often seen for the self-gravitating isothermal cloud core collapse with rotation Matsumoto et al. 1997;Saigo & Hanawa 1998). Within the above gas distribution, we assume that the seed binary is formed via fragmentation of gas, carve out the envelope within r < Rin, and convert its mass into the seed binary while conserving mass and angular momentum. The seed binary is modelled by two point masses, and their masses are Mp and Ms for the primary and secondary, respectively. The total mass of binary is M b ≡ Mp + Ms, the initial mass ratio is q0 ≡ Ms/Mp, and the initial binary separation is a0. We assume that these seeds move circularly around the centre-of-mass with Keplerian velocities. By conservation laws, the mass and specific angular momentum of the seed binary is represented by
M b = R in 0 4πr 2 ρ(r)dr = 4πRin 3 ρin,(3)J b = R in 0 4πr 2 ρ(r)j(r)dr = 1 2 M b jin = q0 (1 + q0) 2 jcircM b ,(4)
where jcirc ≡ √ GM b a0 is the reference specific angular momentum. Once Rin and q0 are determined, jin and ρin are given by
ρin = M b 4πRin 3 ,(5)jin = 2q0 (1 + q0) 2 (GM b a0) 1/2 .(6)
The envelope gas at r > Rin is distributed as equations (1) and (2). We set the velocities of gas at Rin < r < Rout as
j(r) 2 r 3 cyl < GM b r r cyl r ,(7)
such that the gravitational force exceeds the centrifugal force (see Fig. 1). The initial radial velocity of gas is assumed as
vr(r) = 2GM b r − j(r) 2 r cyl 2 1/2 ,(8)
such that the kinetic energy of gas is equal to the gravitational energy, where r cyl is cylindrical radius in the centre-of-mass frame. For simplicity, we assume a non-self-gravitating isothermal gas, and no magnetic fields and radiation. We emphasize that the important difference between our work and the previous ones (Bate & Bonnell 1997;Ochi et al. 2005; is that we aim to investigate unsteady evolution of binary by considering non-uniform distribution of gas density and angular momentum, while we still assume isolated binary and ignore self-gravity of gas.
NUMERICAL METHOD
We use GADGET-3 SPH code (originally described by Springel 2005) in three dimension. In this code, smoothing length of each gas particle is determined by the number of neighbour particles:
(4π/3)h 3 ρ = N ngb m,
where h is the smoothing length, m is the mass of SPH particle, and N ngb is the number of neighbour particles. We adopt N ngb = 50, which corresponds to h ∼ 2.3(m/ρ) 1/3 . For calculation of hydrodynamics, we choose a polytropic index γ = 1 assuming an isothermal gas, and adopt Monaghan-Balsara form of artificial viscosity with the parameters α = 1.0 and β = 2.0 (see Springel 2005 and references therein). All of our simulations are performed with 128 3 SPH particles, which is roughly two orders of magnitude higher than that in Bate & Bonnell (1997). In some of our simulations, a steady circum-secondary disc forms, and the smoothing length is about one-tenth to one-fifth of the scale height at the outer edge of the disc, satisfying the criterion of (see Appendix).
The seed binary is treated as two sink particles with a sink radius of R sink = 0.01a0, following . The SPH particles are removed from the computational domain once they fall into the sink radius of each seed or reach the outer boundary, without any feedback to the seeds.
In the case where the envelope has an angular momentum, circum-stellar discs are formed (Artymowicz & Lubow 1996;Bate 1997;Bate & Bonnell 1997;Ochi et al. 2005). We define our circum-stellar discs as the gas with J < UL1, where J is the Jacobi constant of gas and UL1 is the sum of gravitational and centrifugal potential at L1 point in the corotating frame of the seed binary. The Jacobi constant of gas is written by
J = 1 2 v 2 − GMp rp − GMs rs + c 2 s ln ρ,(9)
where rp, rs, cs are the distance from the primary, the distance from the secondary, and the sound speed of gas, respectively. By our definition of the circum-stellar discs, it is possible that the gas in the circum-stellar discs will be accreted onto each seed via viscous evolution. Therefore we define the time-dependent mass ratio as
q(t) = Ms + ∆Ms(t) Mp + ∆Mp(t) ,(10)
where
∆Mp(t) = Macc,p(t) + M disc,p (t),(11)∆Ms(t) = Macc,s(t) + M disc,s (t).(12)
Here, Macc,p is the mass accreted onto the primary, Macc,s is the mass accreted onto the secondary, M disc,p is the mass of circumprimary disc, and M disc,s is the mass of circum-secondary disc. We terminate our simulations when ∆M b (t) ≡ ∆Mp(t) + ∆Ms(t) exceeds M b , because the self-gravity of gas and the growth of binary cannot be ignored after
∆M b (t) > M b .
With these setup, we compute the gas accretion onto the seed binary for several values of q0 in the range of 0.1 < q0 < 1.0 and the sound speed cs/ GM b /a0 = 0.05 (cold) and 0.25 (hot). Hereafter, we use the units of G = M b = a0 = 1, in which the orbital period of the seed binary corresponds to 2π.
RESULTS
Formation of discs
In Figure 2, we show the time evolution of gas surface density (top and middle panels), and the time evolution of the circumstellar discs and accreted mass onto each seed (bottom panels) for q0 = 0.1 case. For both hot (left column) and cold (right column) cases, the circum-primary disc appears at t ∼ 2π (Fig. 2a,d). At this time, a clear density enhancement with a bridge-like feature exists between the secondary and the primary. Inside this bridge, we find that the Jacobi constant of gas is dissipated and becomes J < UL1, therefore we regard this region as a shock. Neither the circum-secondary disc nor the circum-binary disc appears until the final state (Fig. 2b,e). In Fig. 2c,f, it is seen that the red solid line (∆Mp) is greater than the blue solid line (∆Ms/q). This means Figure 2. Time evolution of gas around the seed binary. The top and middle panels show the face-on logarithmic surface density maps in the centre-of-mass frame for q 0 = 0.1. The hot case is shown in the left column and the cold case on the right column. The top panels are at t = 2π, and the middle panels show the final snapshot at t = 18.1 (panel (b)) and t = 15.1 (panel (e)). The white crosses show the positions of the secondary. The bottom panels show the time evolution of mass accreted onto the primary Macc,p (red dashed), mass accreted onto the secondary Macc,s (blue dashed), mass of the circum-primary disc M disc,p (red dotted), mass of the circum-secondary disc M disc,s (blue dotted), change from the initial mass of primary ∆Mp (red solid), change from the initial mass of secondary ∆Ms (blue solid), change from the initial binary mass ∆M b (black solid). Note that all values for secondary is divided by the initial mass ratio q 0 . All density maps in this paper are produced using SPLASH visualization code (Price 2007) that the mass ratio decreases, as ∆Mp and ∆Ms/q are compared here. In this case of q0 = 0.1, we see that the growth of ∆M b (t) is always dominated by that of the circum-primary disc M disc,p (t). Therefore, the mass ratio monotonically decreases for both hot and cold cases. Figure 3 is the same as Fig. 2, except that it is for q0 = 0.3. The circum-primary disc and the shock between the primary and the secondary appear at t ∼ 2π (Fig. 3a,d), similarly to q0 = 0.1 case. In Fig. 3b,e, the circum-secondary disc appears. From Fig. 3c,f, it is seen that ∆Mp < ∆Ms/q (i.e., mass ratio increases) after t ∼ 3π for both hot and cold cases. Fig. 4 is the same as Fig. 2 and 3, but with q0 = 0.7. In this case, both of the circum-primary disc and the circum-secondary disc are simultaneously formed after t ∼ 2π. At t ∼ 4π (Fig. 4a,d), circum-primary disc and circum-secondary disc are clearly seen. At t ∼ 6π (Fig. 4b,e), the circum-binary disc is seen. After that, gas is accreted onto the circum-binary disc first, before being accreted onto the seeds. Then the gas arrives at circum-stellar discs through L1 or L2 point. (Fig. 5a), gas surface density (Fig. 5b), normalised gravitational force in vertical direction (Fig. 5c), and normalised specific angular momentum (Fig. 5d) of circum-stellar discs at the final state in the hot case with q0 = 0.7. In this figure, each distribution is averaged in each cylindrical shell. In Fig. 5a, the outer edge of the circum-primary disc R disc,p (dashed line) and the outer edge of the circum-secondary disc R disc,s (point-dashed line) are identified by J = UL1. In Fig. 5b, R disc,p (or R disc,s ) is indeed consistent with the location where the surface density of the circum-primary disc (or the circum-secondary disc) start to decline in the outer region. Thus, the circum-stellar discs indeed consists of gas with J < UL1. Another decline in the surface density appears at r cyl ∼ 0.02, which is caused by the sink particle approximation. In Fig. 5c, it is seen that the vertical pressure gradient force roughly balances with gravitational force at the region of 0.02 < r cyl < R disc,p or R disc,s , indicating an equilibrium in the vertical direction. In Fig. 5d, it is seen that angular momentum of gas is approximately Keplerian inside R disc,p or R disc,s . In summary, the circum-stellar discs consist of gas with J < UL1 and are vertically supported by pressure gradi-ent, and rotation velocity of disc is almost Keplerian. This structure is often seen in the standard disc (Shakura & Sunyaev 1973). Figure 6 shows the time evolution of mass ratio q(t). In the cases with q0 = 0.1, q(t) decreases monotonically. With q0 = 0.3, q(t) decreases until t ∼ 3π, but it starts to increase after t ∼ 3π, and eventually exceeds q0. In the cases with q0 > 0.5, q(t) increases monotonically. Taken together, our results show that the time evolu- tion of the mass ratio is qualitatively determined by q0. If q0 < 0.23 (in hot case) or q0 < 0.26 (in cold case), the mass ratio at the final state is smaller than q0. If q0 > 0.23 (in hot case) or q0 > 0.26 (in cold case), on the other hand, the mass ratio at the final state is larger than q0. Dependence on temperature clearly appears at t > 4π in the cases with q0 > 0.7. In these cases, the circumstellar discs settle in a steady state. found that, in steady circum-stellar discs, hot gas easily crosses the L1 point from the secondary's Roche lobe to the primary's Roche lobe compared to the cold gas, and in such a case the growth of q(t) is suppressed. In our simulations, the similar tendency as described above is seen in the cases with q0 > 0.7.
Short-term evolution of mass ratio
Bate (2000) investigated the evolution of seed binaries with various distribution of angular momentum and density of gas, using the protobinary evolution (PBE) code which employs the steady state solutions of Bate & Bonnell (1997). In order to examine the results of PBE code, Bate (2000) also performed three dimensional SPH simulations with NSPH = 1 × 10 5 including angular momentum and density distribution, although limited to one particular case of q0 = 0.6. Although the distributions of ρ ∝ r −1 and j ∝ r 2 adopted in Bate (2000) are different from our equations (1) and (2), our model is similar to one of the models in Bate (2000) because both distributions have the same relation (equation 24). Thus we can compare our results and that of Bate (2000) except for the timescale. Focusing on the short-term evolution until ∆M b < M b , our results mentioned in this subsection is consistent with the results of the SPH simulation and the PBE calculations in Bate (2000). Therefore, our numerical results from SPH simulations confirm the semi-analytical results from PBE calculations in Bate (2000).
DISCUSSION
Categorising the Accreting gas
Focusing on the short-term evolution while ∆M b (t) < = M b , the accreting gas onto the seed binary can be categorised into four different modes as we describe below. To characterise the properties of accreting gas in each mode, it is useful to plot the relation between initial specific angular momentum of gas and q0. In Fig. 7, jin (thick black dotted line) and jout (thick black dot-dashed line) denotes the initial gas specific angular momentum at Rin and Rout, respectively, and jM b (thick black dashed line) denotes the initial gas specific angular momentum at rM b , inside which the gas mass is equal to M b . The specific angular momentum of the secondary and the primary are defined as js (thick red dashed line) and jp (thick red solid line). The specific angular momentum of L1 point is defined as jL1 (blue solid line). The specific angular momentum of the circum-binary disc j cb (green dashed line) is defined as
j cb = 2 1 + q0 jcirc,(13)
such that the centrifugal potential j cb 2 /2r cyl 2 equals to the gravitational potential GM b /r cyl at r cyl = a0/(1+q0) which is the distance of secondary from the mass centre (Ochi et al. 2005). Since the initial specific angular momentum of gas is determined by equation (2), gas with jin is expected to fall first onto the seed binary. At the end of the short-term evolution, gas with jM b is expected to fall onto the circum-stellar discs if we ignore the complex dynamics until the gas falls. In all our simulations, at the end of the short-term evolution, more than 80% of gas in the circum-stellar discs comes from the gas whose initial angular momentum is jin < j < jM b . Thus jM b adequately represents the specific angular momentum of accreted gas at the end of the short-term evolution. Here we define the specific angular momentum of accreted gas as jacc, which is in the region filled by backslash, mesh, and single slash in Fig. 7.
The first mode of accreting gas is the "circum-primary disc mode" (the region filled by backslash in Fig. 7). We can see this mode when jacc < jL1.
For example, when q0 = 0.1, all accreted gas satisfies equation (14), indicating that the gas easily enters inside L1 point where the Jacobi constant of the gas is dissipated by the shock as discussed in Subsection 4.1. Since L1 point and mass centre of the seed binary are in Roche lobe of the primary, the gas forms a circum-primary disc (Fig. 2b,e), the primary seed grows, and the mass ratio decreases monotonically (Fig. 6). The second mode of accreting gas is the "marginal mode" (the region filled by mesh in Fig. 7). This mode is seen when jL1 < jacc < js.
(15) When q0 = 0.2, for example, all accreted gas satisfies equation (15). In this case, most of gas is trapped by the primary similarly to the circum-primary disc mode. In the end, the mass ratio decreases. However, the gas that satisfies equation (15) enters inside secondary's Roche lobe more easily than in the circum-primary disc mode. Inside secondary's Roche lobe, the Jacobi constant of gas is dissipated by the shock. As a result, Macc,s becomes nonnegligible in the end. The third mode of accreting gas is "circum-stellar discs mode" (the region filled by single slash in Fig. 7). We can see this mode when js < jacc < j cb .
In this case, a circum-secondary disc is formed. Once the circumsecondary disc is formed, ∆Ms/q0 dominates, and the mass ratio increases monotonically. We can see this mode when q0 = 0.7, for example (see Fig. 4). The fourth mode is the "circum-binary disc mode" (the region filled by double slash in Fig. 7). We can see this mode when the specific angular momentum of gas is larger than j cb (equation 13),
j > j cb .(17)
In this mode, the majority of gas cannot enter inside each Roche lobe because of the centrifugal barrier, and gas settles down to the circum-binary disc first. Then, the gas enters inside each Roche lobe through L2 or L3 point, and falls onto the circum-stellar discs. This behaviour is seen at t > 6π for q0 = 0.7 (see Fig. 4b,e). Since jM b is lower than j cb for any q0 in our simulations (Fig. 7), the gas in circum-binary disc mode is not accreted by the end of short-term evolution. Therefore, the circum-binary disc mode is irrelevant for the q-evolution in the short term. To investigate q-evolution in this mode, we need to simulate the long-term evolution. In our simulations, the time evolution of mass ratio qualitatively changes at q c,hot = 0.23 (hot case) or q c,cold = 0.26 (cold case). The values of q c,hot and q c,cold roughly correspond to the intersection point of js and jM b in Fig. 7. Therefore, we define a critical initial mass ratio qc at this intersection point, and we find qc = 0.25 from Fig. 7. The value of q c,cold is somewhat closer to qc than q c,hot . This is because the gas flow is closer to a ballistic motion in the cold limit than in the hot case. With a finite gas temperature, pressure gradient force pushes out the gas in radial direction. Therefore, even if jM b < jL1, the rotation radius of gas with jM b can reach jL1 2 /GM b . Since jM b is monotonically increasing function of q0, q c,hot is somewhat lower than qc. The difference between q c,hot and q c,cold is small since this push-out effect is expected to be weak when cs/vK < 1. Here we emphasize that the critical value qc = 0.25 was derived only for a particular distribution of angular momentum and density (equation 24), and that it was evaluated when ∆M b (t) = M b .
In summary, gas accretion onto the primary dominates in the circum-primary disc mode and the marginal mode. While in the circum-stellar discs mode, a circum-secondary disc is formed and accretion onto the secondary becomes significant enough to increase the mass ratio. The gas in circum-binary disc mode forms a circum-binary disc.
Analytic Estimate of Long-term Evolution
In our numerical simulations, we focus on the short-term evolution until ∆M b (t) = M b assuming an isolated binary with no self-gravity. In this subsection, we discuss the long-term evolution of binary separation analytically including binary growth by accretion. There are two effects which change the binary separation by accretion. One is the increase of binary mass. When the binary mass becomes larger and if the angular momentum is conserved, then the binary separation becomes smaller because of stronger gravitational force. The other is the increase of binary angular momentum, which increases the binary separation. The evolution of binary separation is determined by the competition between above two effects. These effects become especially important when ∆M b (t) > M b . First, we formulate the time evolution of binary in our model. Then, we discuss one possibility in which the longterm evolution can be predicted based on our numerical results of short-term evolution.
As for the binary, we define the time-dependent binary mass M b (t), binary separation a(t), mass ratio q(t). The reference specific orbital angular momentum can be written as
jcirc(t) = GM b (t)a(t).(18)
Then the time-dependent orbital angular momentum of binary J b (t) is written by
J b (t) = 2q(t) (1 + q(t)) 2 M b (t)jcirc(t).(19)
We introduce following non-dimensional variables:
M (t) = M b (t) M b ,(20)J(t) = J b (t) J b ,(21)ã (t) = a(t) a0 ,(22)
and jcirc(t) is represented as
jcirc(t) = (1 + q(t)) 2 q(t) q0 (1 + q0) 2J (t) M (t) jcirc.(23)
Note that we stop our simulations when it becomesM = 2.
As for the envelope, in our model (equations 1 and 2), the specific angular momentum of gas j and the gas mass inside the radius r, Mgas, has a relationship j ∝ Mgas ∝ r.
From equations (20) and (24), jin as a function of time is given by
jin(t) =M jin.(25)
From equations (6), (18) and (25), we have
jin = 2q0 (1 + q0) 2 jcirc,(26)jin(t) = 2q(t) (1 + q(t)) 2M 2 (t) J (t) jcirc(t).(27)
Equations (26) and (27) represent the specific angular momentum at the inner edge of the envelope. The power indices ofM andJ in equation (27) reflect the spatial distribution of density and angular momentum in the envelope. If the relatioñ
M 2 (t) J (t) = 1(28)
holds and if q(t) = q0, equations (26) and (27) of M b (t) = a(t) = 1 and M b = a0 = 1. This indicates that the evolution of binary system is self-similar when equation (28) holds and q(t) = q0. Note that, in equations (25) and (27), it is implicitly assumed that all angular momentum and mass of the envelope is converted to the orbital angular momentum and mass of the binary. After the above preparation, we can now discuss the time evolution of binary separation. From equations (4) and (19), we havẽ
a(t) = q(t) q0 −2 1 + q(t) 1 + q0 4J 2 (t) M 3 (t) .(29)
From equation (29), we can see that the separation becomes larger with increasing orbital angular momentum of the binary, and that it becomes smaller with increasing mass. Moreover, the separation also depends on q(t), and this dependence originates from equation (19). For given J b (t) and M b (t), one can see from equation (19) that a(t) inside jcirc depends on q(t). If equation (28) and q(t) = q0 hold, binary separation is proportional to accreted mass in our model:
a(t) =M (t).(30)
The analytic result of equation (30) is consistent with the numerical work by Bate (2000). Here, we discuss one possibility in which the long-term evolution can be predicted by reusing the result of the short-term evolution. From equations (26) and (27), we see that the difference between jin/jcirc and jin(t)/jcirc(t) is caused only by the mass ratio, if equation (28) always holds. According to our simulations, in the hot case with q0 = 0.5, q ≈ 0.7 whenM = 2 from Fig. 6. Under the above assumptions, we can reuse the former result to predict that the mass ratio would be q ≈ 0.9 when it reachesM = 3. Repeating this procedure, we can predict the long-term evolution of a seed binary. We saw in Fig. 6 that, in the short-term evolution, q(t) increases monotonically if q0 > qc, and vice versa. Based on this result and the argument in the previous paragraph, we argue that the long-term evolution of q(t) is qualitatively determined by q0. Fig. 8 plots equation (29) at the end of the short-term evolution (i.e., binary separation atM = 2) using our numerical results of q(t) and equation (28). Fig. 8 shows that the separation reachesã(t end ) = 2 in the cases with q0 → 1.0 and q0 ≃ qc, indicating that the time evolution of a binary is self-similar in these cases (equation 30).
Fig. 8 also shows thatã(t end ) > 1 for any q0, which suggests that the binary separation is a monotonically increasing function of time and therefore close binaries are difficult to form.
Here, we note again that these analytic results are based on the assumption that all angular momentum of the envelope is converted to the orbital angular momentum of the binary. In other words, we are disregarding the division of gas angular momentum into orbital angular momentum of binary and that of circum-stellar discs. In order to investigate the growth of separation more properly, a direct calculation of the binary orbit is needed.
CONCLUSION AND FUTURE WORK
In the present work, we investigate the short-term evolution of a seed binary using the SPH code GADGET-3 in three dimensions. Our simulation setup includes non-uniform distribution of gas and angular momentum with ρ ∝ r −2 and j ∝ r, respectively. In the initial condition, the seed binary is assumed to have formed around the mass centre of the binary by fragmentation, conserving angular momentum and mass. The seed binary is isolated, and selfgravity of gas is ignored. With this setup, we compute the accretion of gaseous envelope onto the seed binary until the binary mass growth exceeds its initial mass, surveying the parameter ranges of 0.1 < q0 < 1.0 and the sound speeds cs/ GM b /a0 = 0.05 (cold) and 0.25 (hot).
As a result, we categorise the gas accretion into four different modes as follows:
(i) "Circum-primary disc mode" is seen when the specific angular momentum of accreting gas is lower than that of L1 point, i.e., jacc < jL1. Most of the gas falls onto the primary and the circumprimary disc, and hence q(t) monotonically decreases. This is because the specific angular momentum is small enough, and the gas with jacc < jL1 enters the primary's Roche lobe.
(ii) "Marginal mode" is seen when jL1 < jacc < js. In this case, although most of the gas is trapped by the primary similarly to the "circum-primary disc mode", the gas is able to enter the secondary's Roche lobe, and the secondary starts to accrete gas. As a result q(t) becomes smaller than q0 after the short-term evolution.
(iii) "Circum-stellar discs mode" is seen when js < jacc < j cb . If the specific angular momentum of gas exceeds that of the secondary, gas starts to rotate around the secondary, and a circumsecondary disc is also formed. Once the circum-secondary disc is formed, q(t) monotonically increases.
(iv) "Circum-binary disc mode" is seen when j cb < j. In this case, gas cannot fall onto the circum-stellar discs directly because of its large angular momentum. Therefore, the gas falls onto the circum-binary disc first, and then later enter the Roche lobes through L2 or L3 point.
We find that the short-term evolution of q-value is qualitatively different according to its initial value q0. If q0 > qc = 0.25, the final mass ratio exceeds q0. This critical value qc is determined by the condition js = jM b in Fig. 7. The critical value qc = 0.25 was derived only for a particular distribution of angular momentum and density (equation 24), and that it was evaluated when ∆M b = M b . In circum-primary disc mode or marginal mode, the dominant accretion onto the primary decreases the q-value. However, once the circum-secondary disc is formed, the accretion onto the secondary becomes significant enough to increase the mass ratio. The value of qc does not differ dramatically depending on gas temperature as long as cs/vK < 1. We also estimate the long-term evolution of a seed binary analytically. Assuming that equation (28) holds, we argue that the evolution of binary system would be self-similar, and the short-term evolution of q(t) from our simulations can be reused just by updating the initial mass ratio. As a result, we find that the binary separation is a monotonically increasing function of time for any q0. This result suggests that close binaries are difficult to form. In the future, we will include direct computations of binary orbit in our simulations in order to investigate the effect of binary growth by accretion.
ACKNOWLEDGMENTS
This work is partially based on the Master's thesis of Suguru Tanaka (Tanaka 2010). We are grateful to Volker Springel for providing us with the original version of GADGET-3 code, and to Kengo Tomida and Fumio Takahara for useful discussions and continuous encouragement. KN acknowledges the partial support by JSPS KAKENHI Grant Number 26247022. Numerical simulations were in part carried out on XC30 at the Centre for Computational Astrophysics, National Astronomical Observatory of Japan.
APPENDIX
Resolution dependence of mass ratio evolution
In order to investigate the resolution dependence of mass ratio evolution, we rerun several simulations in the cases with q0 = 0.7 (hot and cold) and varying number of SPH particles NSPH = 64 3 , 128 3 , 2×128 3 , 4×128 3 , and 256 3 . In a steady circum-stellar disc, pressure gradient and gravitational force are in equilibrium in the vertical direction as mentioned in Subsection 4.1. Therefore, the scale height of this disc H is represented by H(r cyl ) = cs vK(r cyl ) r cyl .
We introduce h/H, the ratio of SPH smoothing length to the scale height at the outer edge of the circum-secondary disc. Here, we define the edge of the circum-secondary disc where J = UL1. In Fig. 9, we plot the mean rate of change of the mass ratio, q = (q(t end ) − q0)/t end , as a function of h(t end )/H. In both hot and cold cases, one can see that the variation ofq is within 25% for different resolution. suggested thatq is independent of resolution if the SPH smoothing length at the outer edge of circum-secondary disc satisfies h/H < 1. Indeed, we also confirm this criterion with our three dimensional simulations.
Figure 3 .
3Same as Fig. 2, but with q 0 = 0.3. Panels (a) and (d) are at t = 2π, panel (b) is at t = 6π, and panel (e) is at t = 4π.
Fig. 5
5shows radial profiles of Jacobi constant
Figure 4 .
4Same asFig. 2, but with q 0 = 0.7. Top panels are at t = 4π, and middle panels at t = 6π.
Figure 5 .
5Properties of circum-stellar discs at the final state of q 0 = 0.7 and cs = 0.25 (hot) case. In each panel, thick red and thin blue solid lines are for the circum-primary and circum-secondary disc, respectively. The vertical black dashed and dot-dashed line denote the edge of circum-primary and circum-secondary disc, respectively. The abscissa is the cylindrical radius from each seed. Panel(a): Jacobi constant, with the horizontal black solid line showing −U L1 . Panel (b): Gas surface density. Panel (c): Magnitude of vertical gravitational force divided by the magnitude of vertical gas pressure gradient force. The black horizontal dotted line represents the equilibrium of vertical forces. Panel (d): Specific angular momentum around each seed, normalised by r cyl v K (1 − c 2 s /v 2 K ).
Figure 6 .
6Time evolution of the mass ratio q(t) in hot (thick red lines) and cold (thin blue lines) cases. The simulations are terminated when the accreted mass reaches ∆M b (t) = M b .
Figure 7 .
7Relation between the initial mass ratio q 0 and the specific angular momentum j of the envelope. Each line shows the specific angular momentum of secondary seed (thick red dashed), primary seed (thick red solid), L1 point (thin blue solid), circum-binary disc (thin green dashed), initial gas specific angular momentum at R in (thick black dotted), at Rout (thick-black dot-dashed), and at r Mb (thick black dashed). The black open circle at the intersection of jp and j M b indicates the critical value qc = 0.25. Each shaded region indicates a different mode of gas accretion: circum-primary disc mode (backslash), marginal mode (mesh), circum-stellar discs mode (single slash), and circum-binary disc mode (double slash).
Figure 8 .
8Binary separation at the end of the short-term evolutionã(t end ) in the cases of hot (thick red line) and cold (thin blue line). The black horizontal line denotesã(t end ) = 2.
Figure 9 .
9Mean rate of mass ratio changeq versus h(t end )/H which is the ratio of SPH smoothing length to the scale height at the outer edge of the circum-secondary disc. The open symbols are for hot case, and the filled symbols are for the cold case with N SPH = 64 3 (triangle), 128 3 (circle), 2 × 128 3 (square), 4 × 128 3 (pentagon), 256 3 (diamond). The value of h(t end )/H is smaller for better resolution. All of our simulations presented in the main text is performed with N SPH = 128 3 .
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| [] |
[
"RANDOM INTERVAL DIFFEOMORPHISMS",
"RANDOM INTERVAL DIFFEOMORPHISMS"
] | [
"Masoumeh Gharaei ",
"Jan Homburg "
] | [] | [] | We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena. | 10.3934/dcdss.2017012 | [
"https://arxiv.org/pdf/1611.07248v1.pdf"
] | 15,371,887 | 1611.07248 | 4f08b48f45da5adb19c7c220501ee17e54c5ac06 |
RANDOM INTERVAL DIFFEOMORPHISMS
Masoumeh Gharaei
Jan Homburg
RANDOM INTERVAL DIFFEOMORPHISMS
We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena.
Introduction
We deal with the dynamics of specific step skew product systems F + : Σ + 2 × I → Σ + 2 × I, where Σ + 2 = {1, 2} N and I = [0, 1], of the form F + (ω, x) = (σω, f ω 0 (x)).
Here σ : Σ + 2 → Σ + 2 is the shift operator; (σω) i = ω i+1 for ω = (ω i ) ∞ 0 and f 1 , f 2 are C 2 diffeomorphisms on I that fulfill the following conditions:
(1) f i (0) = 0, f i (1) = 1 for i = 1, 2;
(2) f 1 (x) < x for x ∈ (0, 1);
(3) f 2 (x) > x for x ∈ (0, 1). We review and present a self-contained study of the dynamics of such skew product systems, characterizing the different possible dynamics. This may seem a restrictive setup, but these systems exhibit a wealth of dynamical behavior that serves as models for dynamics in more general systems.
These step skew product systems provide a setting to study all possible compositions of the two maps f 1 , f 2 in a single framework. Indeed, for initial conditions (ω, x) ∈ Σ + 2 × I, the coordinate in I iterates as
x, f ω 0 (x), f ω 1 • f ω 0 (x), f ω 2 • f ω 1 • f ω 0 (x), ...(1)
The maps f 1 , f 2 simply move points to either smaller or larger values. We will pick the diffeomorphisms f 1 and f 2 randomly, independently at each iterate, with positive probabilities p 1 and p 2 = 1 − p 1 . This corresponds to taking a Bernoulli measure on Σ + 2 from which we pick ω. The obtained random compositions (1) thus form a (nonhomogeneous) random walk on the interval.
The dynamics of the step skew product system depends on the Lyapunov exponents at the boundaries Σ + 2 × {0} and Σ + 2 × {1}. We list the possibilities, which will be worked out in subsequent sections below.
Intermingled basins: With negative Lyapunov exponents at the boundaries, these boundaries are attracting. Their basins are intermingled: any open set in Σ + 2 × I intersects both basins. Master-slave synchronization: With positive Lyapunov exponents at the boundaries, the boundaries are repelling. We find that for almost all fibers {ω} × I, orbits of points in the same fiber converge to each other, i.e. synchronize. On-off intermittency: A zero Lyapunov exponent at a boundary makes that boundary neutral. With the other boundary repelling, a typical time series has long laminary phases where the orbit is close to the "off" state (the neutral boundary) and has bursts where the orbit is in the "on" state, i.e. away from the neutral boundary. Orbits spend a portion of its iterates with full density near the neutral boundary. With two neutral boundaries, orbits spend a portion of its iterates with full density near the union of the neutral boundaries. Random walk with drift: With one attracting and one repelling or neutral boundary, most orbits approach the attracting boundary. Thus we find the most elementary case of the more widespread phenomenon of intermingled basins [1,25], or on-off intermittency [21,38], or master-slave synchronization [36,40].
The setup chosen in this paper is a starting point for research in random dynamics, see e.g. [4], and nonhyperbolic dynamics, see e.g. [10], and has relations to nonautonomous dynamics, see e.g. [30]. The following directions for generalizations give an idea of the many possibilities. We will not give details, but refer to [4,10,30] for more. One may consider other measures than Bernoulli measures to pick random compositions of the interval maps. A natural generalization is also to let the diffeomorphisms on I depend on ω more generally than through ω 0 alone;
(ω, x) → (σω, f ω (x)).
One can further consider parameters ω from other spaces than symbol spaces, with other dynamics than generated by the shift operator. One may then also generalize the skew product structure to maps on fiber bundles, and study perturbations that destroy the skew product structure. A heuristic principle going back to [19] states that phenomena in random dynamics on compact manifolds may also occur for diffeomorphisms of manifolds of higher dimensions. This paper is organized as follows. We start with a section that contains definitions. The next sections form the heart of the paper, describing possible dynamics for the considered class of step skew product systems. An important role in the study of skew product systems is by invariant measures. A basic result gives the connection between invariant measures for skew product systems and their natural extensions. In the appendix this is worked out in the simple context of step skew product systems over one-sided and two-sided shifts.
Acknowledgment. We are indebted to Todd Young for discussions on the topics of this paper, and to Abbas Ali Rashid and Vahatra Rabodonandrianandraina for a careful reading and remarks. Frank den Hollander pointed out relevant work by Lamperti on random walks. Detailed comments from referees have been very helpful in improving the presentation.
Step skew product systems
This section serves to present the setup of this paper and to collect necessary definitions. A skew product system is a dynamical system generated by a map F : Y × X → Y × X of the form F (y, x) = (g(y), f (y, x));
(
if one sees X as the state space of interest, one has dynamics of the x variable that is governed by the map f which depends on the variable y that changes through g. The space Y is the base space, the sets {y} × X are fibers.
We have an interest in skew product systems over full shifts. Write Ω for the finite set of symbols {1, . . . , N }. Let Σ N = Ω Z be the set of bilateral sequences ω = (ω n ) ∞ −∞ composed of symbols in Ω. Let σ : Σ N → Σ N be the shift operator; the map σ shifts every sequence ω ∈ Σ N one step to the left, (σω) i = ω i+1 . We can also consider the shift operator σ acting on the one-sided symbol space Σ + N , i.e. the space of sequences ω = (ω n ) ∞ 0 composed of symbols in Ω. The spaces Σ N and Σ + N are endowed with the product topology. This topology is generated by cylinders like C k 1 ,...,kn ω 1 ,...,ωn for Σ N , C k 1 ,...,kn ω 1 ,...,ωn = {ω ∈ Σ N ; ω k i = ω k i , ∀i = 1, . . . , n}. As it will not lead to confusion, we use the same notation for cylinders in Σ + N . Now let M be a compact manifold, or compact manifold with boundary, and for ω ∈ Σ N , let f ω : M → M be diffeomorphisms depending continuously on ω. Consider skew product systems F :
Σ N × M → Σ N × M ; F (ω, x) = (σω, f ω (x)). Definition 2.1. A skew product system F : Σ N × M → Σ N × M is a step skew product system if it is of the form F (ω, x) = (σω, f ω 0 (x)),
i.e. the fiber maps depend on ω 0 alone.
We denote iterates of a skew product system F (ω, x) = (σω, f ω (x)) as
F n (ω, x) = (σ n ω, f n ω (x)). Here, for n ≥ 1, f n ω (x) = f σ n−1 ω • · · · • f ω (x)
. For a step skew product system this becomes
f n ω (x) = f ω n−1 • · · · • f ω 0 (x). Observe that, if −n < 0, f −n ω (x) = (f n σ −n ω ) −1 .
We also consider (step) skew products over the shift on one-sided symbol sequences. We write F + for the skew product system
F + (ω, x) = (σω, f ω (x)) on Σ + N × M .
Recall that a natural extension of a continuous map is the smallest invertible extension, up to topological semi-conjugacy. The skew product system F on Σ N × M is the natural extension of F + on Σ + N × M . Definition 2.2. Let F be a family of diffeomorphisms on M . The iterated function system IFS (F) is the action of the semigroup generated by F. So a collection of diffeomorphisms f i , 1 ≤ i ≤ N , generates an iterated function system. And an iterated function system IFS {f 1 , . . . , f N } on M corresponds to a step skew product system F + (ω, x) = (σω, f ω 0 (x)) on Σ + N × M . Given an iterated function system IFS (F), a sequence {x n : n ∈ N} is called a branch of an orbit of IFS (F) if for each n ∈ N there is f n ∈ F such that x n+1 = f n (x n ). We say that IFS (F) is minimal if every orbit has a branch which is dense in M .
The appendix collects definitions and basic results on stationary and invariant measures in the context of step skew product systems over shifts. We will make use of the material from the appendix in the following sections.
2.1. Interval fibers. Focus of this paper is the following class of step skew products of diffeomorphisms on I = [0, 1] over the full shift on two symbols {1, 2}, earlier presented in the introduction. Definition 2.3. Let S be the set of step skew product systems F + : Σ + 2 × I → Σ + 2 × I with F + (ω, x) = (σω, f ω 0 (x)), where f 1 , f 2 are C 2 diffeomorphisms that fulfill the following conditions:
(1) f i (0) = 0, f i (1) = 1 for i = 1, 2;
(2) f 1 (x) < x for x ∈ (0, 1);
(3) f 2 (x) > x for x ∈ (0, 1).
So f 1 moves points in (0, 1) to the left, whereas f 2 moves points in (0, 1) to the right. On Σ + 2 we take Bernoulli measure ν + where the symbols 1, 2 have probability p 1 , p 2 , see Appendix A.
The (fiber) Lyapunov exponent of F + at a point (ω,
x) ∈ Σ + 2 × I is lim n→∞ 1 n ln f ω n−1 f n−1 ω (x) · · · f ω 0 (x) = lim n→∞ 1 n n−1 i=0 ln f σ i ω (f i ω (x)) ,
in case the limit exists. Since x = 0, 1 are fixed points of f i , i = 1, 2, by Birkhoff's ergodic theorem, we obtain for x = 0, 1 that
L(x) = lim n→∞ 1 n n−1 i=0 ln f σ i ω (x) = Σ + 2 ln f ω (x) dν + (ω) = 2 i=1 p i ln f i (x)
for ν + -almost all ω ∈ Σ + 2 .
Definition 2.4. The standard measure s on Σ + 2 × I is the product of Bernoulli measure ν + and Lebesgue measure on I. Figure 1. The left frame depicts the graphs of g 1 , g 2 , the diffeomorphisms on I that are conjugate to the maps y → y ± 1 that generate the symmetric random walk. The right frame shows a time series of the iterated function system generated by g 1 , g 2 , both picked with probability 1/2.
A specific example of a step skew product system from S comes from the symmetric random walk. The symmetric random walk is given by translations
k 1 (x) = x − 1, k 2 (x) = x + 1
on the real line, where both maps are chosen randomly with probability 1/2. Now conjugate the symmetric random walk to maps on I as follows. Consider the coordinate change given by the diffeomorphism h : R → (0, 1),
h(x) = e x 1 + e x (note that h −1 (x) = ln(x/(1 − x)))
. Define the step skew product system on Σ 2 × I generated by the fiber diffeomorphisms
g i = h • k i • h −1 with g i (0) = 0, g i (1) = 1, i = 1, 2. We have g 1 (x) = 1 e x 1 + ( 1 e − 1)x ,(3)g 2 (x) = ex 1 + (e − 1)x ,(4)
see Figure 1. We will also refer to the step skew product system generated by g 1 , g 2 as the symmetric random walk. Observe g 1 (0) = 1 e , g 2 (0) = e, g 1 (1) = e, g 2 (1) = 1 e , so that the symmetric random walk has zero Lyapunov exponents at the boundaries Σ + 2 × {0} and Σ + 2 × {1}. Perturbations of g 1 , g 2 that preserve the boundary points 0, 1 lead to diffeomorphisms f 1 , f 2 with various signs of Lyapunov exponents at the boundaries: all cases that are treated in the following sections also occur as small perturbations from the symmetric random walk.
The text book [12] contains a discussion of recurrence properties of random walks on the line with i.i.d. steps. In the same vein one can ask for the iterated function system IFS ({f 1 , f 2 }) to be minimal on (0, 1). The proof of [22,Lemma 3] gives the following result.
Proposition 2.1. Assume that λ = f 1 (0) < 1, µ = f 2 (0) > 1. Assume further that either ln(λ)/ ln(µ) ∈ Q, or f 1 (0) λ 2 − λ = f 2 (0) µ 2 − µ .
Then the iterated function system generated by f 1 , f 2 is minimal on (0, 1). Such minimality is also implied by analogous conditions at the end point 1.
Proof. For the proof we refer to [22]. We add some comments to clarify the conditions. Il'yashenko [22,Lemma 3] considers, for x, y ∈ (0, 1), compositions f l 2 • f k 1 (x) that converge to y for suitable k, l → ∞. Note that this property implies minimality. His analysis uses linearizing coordinates h • f 1 • h −1 (x) = λx with x ∈ [0, s] for an s < 1. Here h is a local diffeomorphism. The two cases where ln(λ), ln(µ) are rationally dependent or not, are distinguished. In case ln(λ), ln(µ) are rationally dependent, the argument works if the second order derivative of h • f 2 • h −1 at 0 is not zero. An explicit calculation shows that this gives the condition in the proposition.
Obviously, the iterated function system generated by g 1 and g 2 , where g 2 = g −1 1 , is not minimal.
Intermingled basins
Kan [25] describes an example of a skew product system on T×I, over an expanding circle map in the base, where the boundary components T × {0} and T × {1} are attractors so that both basins intersect each open set. We will describe his results in the elementary setting of step skew product systems.
The following result describes intermingled basins for step skew product systems F + ∈ S. Let F : Σ 2 × I → Σ 2 × I denote the natural extension of F + . There is an invariant measurable graph ξ : Σ 2 → I that separates the basins: for ν-almost all ω,
lim n→∞ f n ω (x) = 0, if x < ξ(ω), 1, if x > ξ(ω).
We note that Σ + 2 × {0} and Σ + 2 × {1} are attractors in Milnor's sense [34]. The values ξ(ω) depend only on the present and future coefficients (ω i ) ∞ 0 . Before starting the actual proof, we provide a simple argument showing positive standard measure of the basins of Σ + 2 × {0} and Σ + 2 × {1}. Lemma 3.1. Let F + ∈ S and assume L(0) < 0. Let
r(ω) = sup{x ∈ I | lim n→∞ f n ω (x) = 0}.
Then r(ω) > 0 for ν + -almost all ω ∈ Σ + 2 . Proof. The argument follows [25,Lemma 2.2] or [11,Lemma A.1]. For any ε > 0 there exists δ > 0 so that
f i (x) ≤ f i (0) + ε if x < δ, for both i = 1, 2. Write a i = ln(f i (0)+ε).
Recall that L(0) = p 1 ln(f 1 (0))+ p 2 ln(f 2 (0)) is negative by assumption. By Birkhoff's ergodic theorem applied to the function ω → ln(f ω (0) + ε), for ν + -almost all ω,
lim n→∞ 1 n n−1 i=0 a ω i = p 1 a 1 + p 2 a 2 ,
which is negative if ε is small enough. So, for ν + -almost all ω, n−1 i=0 a ω i goes to −∞ as n → ∞ and
A(ω) = max{0, max n≥1 n−1 i=0 a ω i } exists. Take x 0 < δe −A(ω) ≤ δ. Then x n = f n ω (x 0 ) satisfies x n < e n−1 i=0 aω i e −A(ω) δ ≤ δ
for all n ≥ 0 and in fact lim n→∞ x n = 0. This proves the lemma.
Since the function r is positive almost everywhere, the basin of Σ + 2 × {0} has positive standard measure. The same holds for the basin of Σ + 2 × {1}. It is easily seen that any open set in Σ + 2 × I intersects both basins; forward iterations must accumulate onto both Σ + 2 × {0} and Σ + 2 × {1} using that the shift operator is an expansion and 0 and 1 occur as attracting fixed points for f 1 and f 2 respectively. Proof of Theorem 3.1. We prove the theorem by considering the inverse diffeomorphisms, i.e. a step skew product with positive Lyapunov exponents along Σ + 2 × {0} and Σ + 2 × {1}. For the duration of this proof, we consider F + ∈ S with L(0) > 0 and L(1) > 0. The following lemmas deal with this. The theorem will follow by linking the derived statements on the natural extension F of F + and the statements we wish to prove for its inverse.
We write P I for the space of probability measures on I equipped with the weak star topology. As explained in Appendix A, a stationary measure is a fixed point of T : P I → P I given by
T m = p 1 f 1 m + p 2 f 2 m, where f i m is the push forward measure f i m(A) = m(f −1 i (A)).diffeomorphisms f −1 1 (x) = 1−r− √ (1−r) 2 +4rx −2r and f −1 2 (x) = 1+r− √ (1+r) 2 −4rx 2r
. The inverse maps give positive Lyapunov exponents at the end points. The right frame shows a time series for the iterated function system generated by f −1
1 and f −1 2 .
Proof. For small 0 < α < 1, q > 0, and positive c, define
N c = {m ∈ P I ; ∀ 0 ≤ x ≤ q, m [0, x) ≤ cx α and m (1 − x, 1] ≤ cx α }.
The conditions exclude stationary measures supported on the end points 0 or 1. Note that N c depends on α and q; but we do not include this dependence in the notation. We first show that there exist c > 0 and α, q > 0 close to 0 such that
T (N c ) ⊂ N c . Write ρ i = f i (0)
. We claim that there is a small α > 0 such that the assumption
L(0) > 0 implies 2 i=1 p i ρ −α i < 1. Namely, since lim α→0 1−ρ −α i α = ln ρ i for 1 ≤ i ≤ 2, 2 i=1 p i ln ρ i > 0 implies that for sufficiently small α > 0, 2 i=1 p i 1 − ρ −α i α > 0.
Multiplying by α we get
2 i=1 p i − 2 i=1 p i ρ −α i > 0, which implies 2 i=1 p i ρ −α i < 1, because 2 i=1 p i = 1. Thus, there exists a small δ > 0 so that 2 i=1 p i (ρ i − δ) α < 1.(5)
Moreover, for such δ > 0 we are able to choose a sufficiently small q = q(δ) > 0 so that
f −1 i (x) ≤ x ρ i − δ , ∀ 0 ≤ x ≤ q.(6)
Take c with cq α > 1.
Note that this implies that in the definition of N c , m [0, x) ≤ cx α and m (1 − x, 1] ≤ cx α for any 0 ≤ x ≤ 1, and not just for 0 ≤ x ≤ q. Take a measure m ∈ N c .
To prove T m ∈ N c , we must show that for x ≤ q, T m [0, x) ≤ cx α . Knowing that m [0, x) ≤ cx α and applying (5), (6) we obtain the following estimates:
T m [0, x) = 2 i=1 p i f i m [0, x) = 2 i=1 p i m f −1 i [0, x) ≤ 2 i=1 p i m [0, x ρ i − δ ) ≤ 2 i=1 p i c x ρ i − δ α = c 2 i=1 p i (ρ i − δ) α x α ≤ cx α .(7)
Estimates near the right boundary point are treated in the same manner. By the Krylov-Bogolyubov averaging method, for a measure m ∈ N c there is a subsequence of { 1 n n−1 r=0 T r m} n∈N that is convergent to a probability measurê m ∈ N c such that Tm =m.
The following additional reasoning shows that there is an ergodic stationary measure in N c . The set of stationary measures M I is a convex compact subset of P I . The ergodic stationary measures are the extreme points of it. Note that N c ∩ M I is a convex compact subset of M I , which is itself also convex and compact. We claim that the extreme points of N c ∩ M I are also extreme points of M I . Suppose by contradiction that there are n 1 , n 2 ∈ M I \ (N c ∩ M I ) and the convex combination Proof. We follow [4,Theorem 1.8.4]. Consider a µ m and its conditional measures µ m,ω . Let X m (ω) be the smallest median of µ m,ω , i.e. the infimum of all points x for which
m = sn 1 + (1 − s)n 2 ∈ N c ∩ M I . In this case, for 0 ≤ x ≤ q, n 1 ([0, x)) ≤ (c/s)x α and n 1 ((1 − x, 1]) ≤ (c/s)x α and similar estimates for n 2 . That is, x → n i ([0, x))/x α and x → n i ((1 − x, 1])/x α are bounded. As T m = m, we have by (5), (7) that m ∈ Nc for somec < c. It follows that tn 1 + (1 − t)n 2 ∈ N c ∩ M I for t close to s. So s is an interior point of the set of values t for which tn 1 +(1−t)n 2 ∈ N c ∩M I . Since N c ∩M I is closed it follows that n i ∈ N c ∩ M Iµ m,ω ([0, x]) ≥ 1 2 and µ m,ω ([x, 1]) ≥ 1 2 .
The set of medians of µ m is a compact interval and X m :
Σ 2 → I is measurable. Define J − m (ω) = [0, X m (ω)] for which by definition µ m,ω (J − m (ω)) ≥ 1 2 . The set J − m (ω) is invariant: since f 1 and f 2 are increasing, for every x 1 < x 2 and ω we have f ω (x 1 ) < f ω (x 2 ). This implies that x is a median of µ m,ω if and only if f ω (x) is a median of f ω µ m,ω . By invariance of µ m for F we have f ω µ m,ω = µ m,σω . Hence, X m (σω) = f ω (X m (ω)) which implies J − m (σω) = f ω (J − m (ω)). Because µ m is ergodic and J − m (ω) is invariant, µ m,ω (J − m (ω)) = 1, ν-almost surely. Applying the same argument to J + m (ω) = [X m (ω), 1], we obtain µ m,ω ({X m (ω)}) = 1 for {X m (ω)} = J − m (ω) ∩ J + m (ω). Thus µ m,ω = δ Xm(ω) for ν-almost every ω ∈ Σ 2 .
The following lemma shows that the set of stationary measures is the triangle consisting of convex combinations of δ 0 , δ 1 and one other ergodic stationary measure m. Proof. Suppose there are two different such stationary measures m 1 , m 2 . We may take m 1 , m 2 to be ergodic stationary measures. This corresponds to two ergodic invariant measures µ m 1 = µ m 2 for F . By Proposition A.1 and Lemma 3.3 there are measurable functions X m i :
Σ 2 → I and sets D i ⊂ Σ 2 with ν(D i ) = 1, for i = 1, 2, such that lim n→∞ f n σ −n ω m i = δ Xm i (ω) for every ω ∈ D i . From ν(D i ) = 1 we have ν(D 1 ∩ D 2 ) = 1. Since µ m 1 , µ m 2 are mutually singular we have that for a genericω ∈ D 1 ∩ D 2 , X m 1 (ω) = X m 2 (ω). Without loss of generality suppose that X m 1 (ω) < X m 2 (ω).
Observe that the supports of m 1 and m 2 are invariant:
supp (m i ) = f 1 (supp (m i )) ∪ f 2 (supp (m i ))
for i = 1, 2. The convex hulls of the supports of m 1 and m 2 therefore both equal I. We can find generic points (ω, x 1 ) and (ω, x 2 ) for m 1 and m 2 such that
x 1 > x 2 . Because lim n→∞ f n σ −nω (x i ) = X m i (ω)
and f 1 , f 2 are both strictly increasing, we conclude that X m 2 (ω) < X m 1 (ω), contradicting our assumption. Thus, µ m 1 = µ m 2 . for every x ∈ (0, 1). Again since the diffeomorphisms f 1 , f 2 are increasing, for the
By Proposition
inverse diffeomorphisms (f n σ −n ω ) −1 = f −1 ω −n • · · · • f −1 ω −1 this yields lim n→∞ (f n σ −n ω ) −1 (y) = 1 if y > ξ(ω) and lim n→∞ (f n σ −n ω ) −1 (y) = 0 if y < ξ(ω)
. Note that this also shows that for the inverse diffeomorphisms, the union of the basins of attraction of Σ 2 × {0} and Σ 2 × {1} has full standard measure.
Let us give some pointers to further research literature: a discussion of a step skew product system over the full shift on two symbols, with piecewise linear fiber maps is in [2]. Several articles discuss extensions to skew product systems that are not step skew product systems. We refer the reader in particular to [24,28] and [10,Section 11.1]. Further studies that quantify the phenonomenon are [26,35]. See [18,22,23] for related work on so-called thick attractors (attractors of positive standard measure).
Master-slave synchronization
The proof of Theorem 3.1 relies on an analysis of step skew product systems with positive Lyapunov exponents at the boundaries. The following result further discusses such step skew product systems. It describes a synchronization phenomenon that is illustrated in the right frame of Figure 3. We see an example of master-slave synchronization, which refers to synchronization caused by external forcing. It is explained by a single attracting invariant graph for the skew product system [40]. From a slightly different perspective one can also view this as synchronization by noise, where common noise synchronizes orbits with different initial conditions.
In a general context of skew product systems F (y, x) = (g(y), f (y, x)) on a product Y × X of metric spaces Y, X, as in (2), master-slave synchronization is given by the following picture. If {(y, ξ(y)) | y ∈ Y } is a globally attracting graph, then the orbits F n (y, x 1 ) and F n (y, x 2 ) converge to each other. In particular if one observes the dynamics of the x-variable, one has
lim n→∞ d(f n (y, x 1 ), f n (y, x 2 )) = 0,
where d is a metric on X and F n (y, x) = (g n (y), f n (y, x)). An illustrative example, of linear differential equations forced by the Lorenz equations, is given by Pecora and Carroll in [36]. We refer to [37] for an explanation of synchronization in a range of contexts.
The following result describes a similar effect for step skew product systems F + ∈ S; the proof employs a measurable invariant graph for the natural extension F of F + . Theorem 4.1. Let F + ∈ S and assume L(0) > 0 and L(1) > 0. Let x 0 , y 0 ∈ (0, 1). Then for ν + -almost all ω,
lim n→∞ |f n ω (x 0 ) − f n ω (y 0 )| = 0.
Proof. The proof of Theorem 3.1 gives the existence of an invariant measurable graph ξ : Σ 2 → I so that given any x 0 ∈ (0, 1), one has that for ν-almost all ω,
lim n→∞ |f n σ −n ω (x 0 ) − ξ(ω)| = 0.(8)
As ν is invariant for σ, this proves that f n ω (x 0 ) converges to ξ(σ n ω) in probability. This gives the existence of a subsequence n k → ∞ as k → ∞ so that
lim k→∞ |f n k ω (x 0 ) − ξ(σ n k ω)| = 0
(see e.g. [39, Theorem II.10.5]). We have thus obtained the weaker statement that for ν + -almost all ω, lim inf n→∞ |f n ω (x 0 ) − ξ(σ n ω)| = 0. We provide a sketch of the argument to show that this is also true with limit replacing the limit inferior, which would imply the theorem. The measure (id, ξ)ν on Σ 2 ×I, with conditional measures δ ξ(ω) and marginal ν, is invariant for the natural extension F : Σ 2 × I → Σ 2 × I of F + . It corresponds to an invariant measure ν + × m for F + by Proposition A.1 and the observation that ξ(ω) depends on (ω n ) −1 −∞ only. Lemma 4.1. With respect to the measure ν + × m, the system F + has a negative Lyapunov exponent;
λ = 2 i=1 p i I ln(f i (x)) dm(x) < 0.
Proof. One can follow the argument for [29, Theorem 7.1] (an analogue of [8, Theorem 4.2]). We describe the steps. A key idea is the use of the notion of relative entropy; the relative entropy h(m 1 |m 2 ) of a probability measure m 1 on I with respect to a probability measure m 2 on I is given by
h(m 1 |m 2 ) = sup ψ∈C 0 (I) ln I e ψ(x) dm 1 (x) − I ψ(x) dm 2 (x) .
The following properties hold [15].
(i) 0 ≤ h(m 1 |m 2 ) ≤ ∞; (ii) h(m 1 |m 2 ) = 0 if and only if m 1 = m 2 ;
A relation between Lyapunov exponent and relative entropy can be derived for absolutely continuous stationary measures. The argument now involves maps with absolutely continuous noise with shrinking amplitude to approximate the fiber diffeomorphisms. Such a perturbed system admits an absolutely continuous stationary measure. One uses the relation between the Lyapunov exponent and relative entropy for this absolutely continuous stationary measure and considers the limit where the noise amplitude shrinks to zero.
Let ζ be a random variable with values in [0, 1] that is uniformly distributed. For small positive values of ε, let
f i,ζ (x) = (1 − ε)f i (x) + ζε. Note that ζε = f i,ζ (0) ≤ f i,ζ (x) ≤ f i,ζ (1) = 1 − ε + ζε, so that f i,m ε = 2 i=1 p i 1 0 f i,ζ m ε dζ.(9)
Note that m ε is a fixed point of an operator T ε where T ε m ε is defined by the right hand side of (9). One can show that m ε has a smooth density [42]. Moreover, with N c the closed set of probability measures considered in the proof of Lemma 3.3, one has that for for suitable values of α, c, q, m ε ∈ N c for all small positive ε. This is true since T ε maps N c into itself for suitable values of α, c, q. To see this follow the proof of Lemma 3.3 with T ε replacing T . The main calculation analogous to (7) is
straightforward noting that f −1 i,ζ ([0, x)) ⊂ f −1 i,0 ([0, x)) for all ζ ∈ [0, 1]: T ε m ε [0, x) = 2 i=1 p i 1 0 f i,ζ m ε [0, x) dζ = 2 i=1 p i 1 0 m ε f −1 i,ζ [0, x) dζ ≤ 2 i=1 p i 1 0 m ε f −1 i,0 [0, x) dζ ≤ 2 i=1 p i m ε [0, x ρ i − δ ) ≤ 2 i=1 p i c x ρ i − δ α = c 2 i=1 p i (ρ i − δ) α x α ≤ cx α .
A similar argument can be employed near the boundary point 1, for ε small.
The Lyapunov exponent for the stationary measure m ε is given by
λ ε = 2 i=1 p i 1 0 I ln(f i,ζ (x)) dm ε (x)dζ.(10)
Since m ε has a smooth and bounded density, one can prove the relation (see [29,Proposition 7.2])
λ ε = − 2 i=1 p i 1 0 h(f i,ζ m ε |m ε ) dζ.(11)
By (i), λ ε ≤ 0. By (ii), λ ε = 0 if and only if f i,ζ m ε = m ε for all ζ ∈ [0, 1] and i = 1, 2. As the latter is not possible, λ ε < 0. Now take the limit ε → 0. Then m ε → m since T ε is continuous and depends continuously on ε (compare Lemma A.1), convergence is in N c , and m is the unique stationary measure in N c . From (10) one sees that λ ε → λ as ε → 0 and we obtain λ ≤ 0. Since the relative entropy is lower semi-continuous in ε as the supremum of continuous functionals, one finds from (11) that
0 ≤ 2 i=1 p i h(f i m|m) ≤ −λ.
This shows that h(f i m|m) = 0 for i = 1, 2 in case λ = 0. This is clearly not the case by ((ii)), as f 1 m = f 2 m = m. So we have λ < 0.
Because of this lemma, for ν-almost all ω, ξ(ω) from (8) has a stable manifold W s (ω) that is an open neighborhood of ξ(ω) in I. To see this one can refer to general theory for nonuniformly hyperbolic systems as in [7], or apply reasoning as in Lemma 3.1. For each x ∈ W s (ω),
lim n→∞ |f n ω (x) − ξ(σ n ω)| = 0. Write W s (ω) = (r b (ω), r t (ω)).
Then r b and r t are invariant. Hence r b > 0, ν-almost everywhere, or r b = 0, νalmost everywhere, and likewise r t < 1, ν-almost everywhere, or r t = 1, ν-almost everywhere. We will derive a contradiction from the assumption that r t < 1 or r b > 0, ν-almost everywhere. Assume that e.g. r t < 1, ν-almost everywhere. Write
r(ω) = inf{x ∈ I | lim n→∞ f −n ω (x) = 1}.(12)
As L(1) > 0, we have r(ω) < 1 for ν-almost all ω ∈ Σ 2 , compare Lemma 3.1. Since the graphs of r t and r are invariant graphs and also r t < 1, we have r ≥ r t > ξ, ν-almost everywhere.
The measure µ = (id, r)ν on Σ 2 × I with conditional measures δ r(ω) and marginal ν defines an invariant measure for F . It follows from the expression (12) that r(ω) depends on the past ω − = (ω n ) −1 −∞ only. Consequently, µ is a product measure of the form ν + ×ϑ on Σ + 2 ×(Σ − 2 ×I). With Π the natural projection Σ 2 ×I → Σ + 2 ×I, we find that the F + -invariant measure Πµ is a product measure Πµ = ν + ×m on Σ + 2 × I. By Lemma A.2,m is a stationary measure. Since r > ξ, ν-almost everywhere, Proposition A.1 gives that m =m. Lemma 3.4 however prohibits the existence of two different stationary measures with support in (0, 1). The contradiction is derived, establishing that W s (ω) = (0, 1) for ν-almost all ω ∈ Σ 2 .
Under the conditions of Theorem 4.1, the proof of Theorem 3.1 shows that
lim n→∞ |f n σ −n ω (x) − ξ(ω)| = 0
for ν-almost all ω ∈ Σ 2 and any x ∈ (0, 1). This convergence is called pullback convergence. The proof of Theorem 4.1 shows that
lim n→∞ |f n ω (x) − ξ(σ n ω)| = 0
for ν-almost all ω ∈ Σ 2 and any x ∈ (0, 1). This convergence is called forward convergence. So in this case both pullback and forward convergence to ξ holds. In general however forward convergence is not a consequence of pullback convergence. The next section provides an example, involving a zero Lyapunov exponent, with pullback convergence but not forward convergence. See in particular Section 5.1. Section 6 contains a related example, related by going to the inverse skew product system, with forward convergence but not pullback convergence. We refer to [30] for more discussion on conditions for convergence in nonautonomous and skew product systems.
We finish with some pointers to further literature. In [3,14,27,43] synchronization results, similar to Theorem 4.1, for skew product systems with circle diffeomorphisms as fiber maps are treated without employing negativity of Lyapunov exponents. Motivated by Lemma 4.1 for example, one may wonder about other invariant measures than those with Bernoulli measure as marginal. Reference [20] considers, in this direction, the existence of nonhyperbolic measures for step skew product systems with circle fibers.
On-off intermittency
Intermittency in a dynamical system stands for dynamics that exhibits alternating phases of different characteristics. Typically, intermittent dynamics alternates time series close to equilibrium with bursts of global dynamics [9]. In our context, we say that a step skew product system F + ∈ S displays intermittency if the following holds for any sufficiently small neighborhood U of 0:
(1) For all x ∈ (0, 1) and ν + -almost all ω ∈ Σ + 2 :
lim n→∞ 1 n {0 ≤ i < n ; f i ω (x) ∈ U } = 1;
(2) For all x ∈ (0, 1) and ν + -almost all ω ∈ Σ + 2 , f n ω (x) ∈ U for infinitely many n.
Here, for a finite set S, we write |S| for its cardinality.
This kind of intermittency that involves a weakly unstable invariant set, here Σ + 2 × {0} ⊂ Σ + 2 × I, has been called on-off intermittency [21,38]. The occurrence of intermittency in iterated function systems of logistic maps with zero Lyapunov exponent at the fixed point in 0 is treated in [5,6]. See also [11] for a study of specific interval diffeomorphisms over expanding circle maps. (3), (4) and p(x) = 3 10 x(1−x). The corresponding step skew product system has a zero Lyapunov exponent along Σ + 2 × {0} and a positive Lyapunov exponent along Σ + 2 × {1}. The right frame shows a time series for the iterated function system generated by these diffeomorphisms.
x → f i (x) = g i (x)(1 − p(x)), i = 1, 2, with g i (x) as in
In this section we will discuss on-off intermittency for step skew product systems F + ∈ S. Throughout we assume that both diffeomorphisms f 1 , f 2 are picked with probability 1/2. This is for convenience, we expect the more general case with probabilities p 1 , p 2 to go along the same lines. The following two theorems, Theorems 5.1 and 5.2, demonstrate that F + ∈ S with L(0) = 0 and L(1) > 0 displays intermittency. Figure 4 illustrates a typical time series.
Lamperti, in a sequence of papers [31][32][33], developed a general theory of recurrence for nonhomogeneous random walks on the half-line. His results may be used to prove on-off intermittency in our context, see in particular [33, Theorems 3.1 and 4.1]. We will get it by calculating bounds on stopping times, using C 2 differentiability of the generating diffeomorphisms.
Theorem 5.1. Let F + ∈ S and assume L(0) = 0. Let 0 < β be small and x 0 ∈ (0, 1). Then for ν + -almost every ω ∈ Σ + 2 , f n ω (x 0 ) is in [β, 1] for infinitely many values of n.
Proof. We follow the proof of [6,Theorem 1]. Given x 0 ∈ I and ω ∈ Σ + 2 , write x n = x n (ω) = f n ω (x 0 ). It suffices to show that for x 0 < β, ν + ({ω ∈ Σ + 2 | x n ≥ β for some n ≥ 1}) = 1. Let u n = − ln(x n ). We wish to show that for u 0 > K = − ln(β), where n ∧ T = min{n, T }. We claim that z n is a supermartingale;
ν + ({ω ∈ Σ + 2 | u n ≤ K for some n ≥ 1}) = 1. Write f i (x) = c i x/(1 + t i (x)) with t i (x) = O(x) as x → 0. Taking logarithms of x n+1 = f ωn (x n ) we get u n+1 = d ωn + u n + ln(1 + t ωn (e −Lemma 5.1. C z n+1 (ω) dν + (ω) ≤ C z n (ω) dν + (ω)
for cylinders C = C 0,...,n−1 ω 0 ,...,ω n−1 .
Proof. On C, z n (ω) is constant. As further c ω n+1 is independent of c ωn , it suffices to consider n = 0 and to prove
Σ + 2 z 1 (ω) dν + (ω) ≤ ln(u 0 ) for u 0 large enough. Denote h(u 0 ) = Σ + 2 z 1 dν + (ω) − ln(u 0 ).
The zero Lyapunov exponent, L(0) = 0, implies Σ + 2 d ω 0 dν + (ω) = 0. Using this,
h(u) = Σ + 2 ln d ω 0 + u + ln(1 + t ω 0 (e −u )) − ln(u) dν + (ω) = Σ + 2 ln d ω 0 + u + ln(1 + t ω 0 (e −u )) u dν + (ω) = Σ + 2 ln d ω 0 + u + ln(1 + t ω 0 (e −u )) u − d ω 0 u dν + (ω) = Σ + 2 ln d ω 0 u + 1 + ln(1 + t ω 0 (e −u )) u − d ω 0 u dν + (ω).
By developing the integrand of the last expression in a Taylor expansion, this gives
h(u) = − 1 2 Σ + 2 d ω 0 u 2 + o 1 u 2 dν + (ω), u → ∞. So lim sup u→∞ h(u)u 2 < 0,
implying that there existsū so that h(u) < 0 for u ≥ū.
for ν + -almost all ω ∈ Σ + 2 . Let (1) B 1 = {ω ∈ Σ + 2 | z ∞ = lim n→∞ z n < ∞};
(2) B 2 = {ω ∈ Σ + 2 | T = ∞}. We must prove that ν + (B 2 ) = 0. On B 1 ∩B 2 , z n → z ∞ and thus x n → x ∞ ∈ (0, 1) as n → ∞. This is impossible as both f 1 (x ∞ ) = x ∞ and f 2 (x ∞ ) = x ∞ . So B 1 ∩B 2 = ∅. By (13), ν + (B 1 ) = 1. Hence ν + (B 2 ) = 0.
If one assumes L(1) > 0, then a similar, in fact simpler, argument shows that for β small and x 0 ∈ (0, 1), for ν + -almost all ω one finds x n = f n ω (x 0 ) in [0, 1 − β] for infinitely many values of n.
Theorem 5.2. Consider F + ∈ S and assume L(0) = 0 and L(1) > 0. Let 0 < β < 1 and x 0 ∈ (0, 1). Then for ν + -almost every ω ∈ Σ + 2 ,
lim n→∞ 1 n n−1 i=0 1 [0,β) (f i ω (x 0 )) = 1.(14)
Proof. The reasoning is inspired by [6,Theorem 4]. Consider
x n = x n (ω) = f n ω (x 0 ) and y n = ln(x n /(1 − x n )).
We denote
y n+1 = h ωn (y n ).
For β small, K = ln(β/ (1 − β)) is a large negative number. For definiteness assume x 0 ≤ β, i.e. y 0 ≤ K. Define stopping times T 0 = 0, T 2k+1 = inf{n ∈ N | n > T 2k and y n > K}, T 2k = inf{n ∈ N | n > T 2k−1 and y n ≤ K}, see Figure 5. Let
η k = |[T 2k−2 , T 2k−1 )| = T 2k−1 − T 2k−2 , ξ k = |[T 2k−1 , T 2k )| = T 2k − T 2k−1
be the duration of subsequent iterates with y n ≤ K and the duration of subsequent iterates with y n > K, respectively. Lemmas 5.2 and 5.4 determine bounds for the expectation of the stopping times η k (which is shown to be infinite) and ξ k (which is shown to be finite).
Lemma 5.2. Σ + 2 η k (ω) dν + (ω) = ∞;(15f 1 (x) = x/d, f 2 (x) = dx(1 + r(x)),
for some smooth function r(x) = O(|x|), x → 0 (observe that f 1 (0) = 1/f 2 (0) by L(0) = 0). Letz n = ln(x n ) map (0, β] to (−∞,L] withL = ln(β). Then
x n+1 = f ωn (x n ) becomesz n+1 = z n − ln(d),
if ω n = 1, z n + ln(d) + ln(1 + r(ez n )), if ω n = 2.
An additional rescaling z n =z n / ln(d) conjugates this iterated function to
z n+1 = z n − 1,
if ω n = 1, z n + 1 + ln(1 + r(e zn ln(d) ))/ ln(d), if ω n = 2.
Write L =L/ ln(d); we consider z n on (−∞, L]. Let g > 0; g will be chosen large in the sequel. The term ln(1 + r(e zn ln(d) ))/ ln(d) may be bounded from above by Ce −g ln(d) on intervals (−∞, L − g], for some C > 0. On (−∞, L − g] ⊂ (−∞, L] we compare the random walk z n with the random walk
v n+1 = v n − 1, if ω n = 1, v n + 1 + Ce −g ln(d) , if ω n = 2. Given z 0 = v 0 ∈ [L − g − 1, L − g), we define stopping times T z = min{n ∈ N | z n ≥ L − g}, T v = min{n ∈ N | v n ≥ L − g}.
If z 0 , . . . , z n ∈ (−∞, L − g), then z i ≤ v i for all 0 ≤ i ≤ n + 1. Therefore, for each ω ∈ Σ + 2 , T z (ω) ≥ T v (ω). By Wald's identity, see e.g. [39,Section VII.2],
Σ + 2 T v (ω) dν + (ω) = 2 Ce −g ln(d) Σ + 2 v Tv(ω) − v 0 dν + (ω) ≥ ce g ln(d)
for some c > 0, and hence
Σ + 2 T z (ω) dν + (ω) ≥ ce g ln(d) .(17)
Let α > 0 be so that
z n+1 ≤ z n + 1 + α(18)
for L − g ≤ z n ≤ L. Note that we may take α to be small if L is large. Consider the random walk given by (16) with initial point z 0 ∈ (L − 1, L]. Define the stopping time T g = min{n > 0 | z n < L − g or z n > L}.
Lemma 5.3. For α > 0 small enough, there is c, r * = r * (α) < 0, so that
ν + ({ω ∈ Σ + 2 | z Tg < L − g}) ≥ ce gr * . Here r * (α) → 0 as α → 0.
We finish the proof of Lemma 5.2 using this lemma, and then prove Lemma 5.3. Consider the following reasoning. Start with a point z T 2k ∈ (L − 1, L]. Then some iterate of z T 2k will have left [L − g, L], either through the right boundary point L or, with probability determined by Lemma 5.3, through the left boundary point L − g. In the latter case there will be a return time to [L−g, L] after which a further iterate may leave through the right boundary point L. Consequently, combining (17) and Lemma 5.3,
Σ + 2 η k (ω) dν + (ω) ≥ ce gr * e g ln(d)(19)
for some c > 0. For L sufficiently large, α is small enough to ensure e r * e ln(d) > 1, because r * (α) → 0 as α → 0. Then the right hand side of (19) goes to infinity as g → ∞. This concludes the proof of Lemma 5.2.
Proof of Lemma 5.3. Consider the random walk
u n+1 = u n − 1, if ω n = 1, u n + 1 + α, if ω n = 2,
with u 0 ∈ (−1, 0]. Define the stopping time U g = min{n > 0 | u n < −g or u n > 0}.
By (18) we have
ν + ({ω ∈ Σ + 2 | z Tg < L − g}) ≥ ν + ({ω ∈ Σ + 2 | u Ug < −g})
and hence it suffices to prove the estimate ν + ({ω ∈ Σ + 2 | u Ug < −g}) ≥ ce gr * . Write ζ n for the steps u n − u n−1 ; ζ n = −1 or ζ n = 1 + α both with probability 1/2. Write S n = ζ 1 + · · · + ζ n = u n − u 0 and consider the function
G n = e r * Sn ,
where r * < 0 is the solution of 1 2 e −r * + 1 2 e r * (1+α) = 1.
One can check that this equation has a unique solution r * < 0 with r * → 0 as α → 0. Now G n is a martingale as This gives Σ + 2 e r * u Ug dν + (ω) = e r * u 0 .
Observe u Ug ∈ [−g − 1, −g) or u Ug ∈ (0, 1 + α]. Let
A = ν + ({ω ∈ Σ + 2 | u Ug < −g}) be the probability that u Ug < −g. Write Σ + 2 e r * u Ug dν + (ω) = Ae −gr * e −c 1 r * + (1 − A)e c 2 r * , where e −gr * e −c 1 r * = 1 A {ω∈Σ + 2 | u Ug <−g} e r * u Ug dν + (ω), e c 2 r * = 1 1 − A {ω∈Σ + 2 | u Ug >0} e r * u Ug dν + (ω).
In these expressions, 0 ≤ c 1 ≤ 1, 0 ≤ c 2 ≤ 1 + α. A moment of thought gives that c 2 > 0 (c 2 = 0 can only occur if points leave [−g, 0] through the right boundary point 0, but the initial point u 0 ∈ (−1, 0] is mapped with probability 1/2 to a point in (α, 1 + α]). We obtain
A e −gr * e −c 1 r * − e c 2 r * = e u 0 r * − e c 2 r * ,
where this last number is positive.
Similar arguments prove the following lemma.
Lemma 5.4. Σ + 2 ξ k (ω) dν + (ω) < ∞;(20)
ξ k has finite expectation.
Proof. Recall that x n+1 = f ωn (x n ) on I is conjugate to y n+1 = h ωn (y n ) on R through y n = ln(x n /(1 − x n )). We split iterates of y n in [K, ∞) into two sets, namely iterates in [K,K] and iterates in (K, ∞), for some positive and largeK. Near x = 1, write
f i (x) = 1 − a i (1 − x)(1 + r i (1 − x)
) with a i > 0 and r i (u) = O(u), u → 0. The positive Lyapunov condition L(1) > 0 means that ln(a 1 ) + ln(a 2 ) > 0. Calculate
y n+1 = y n − ln(a ωn )+ ln(1 − (1 − x n )) + ln(1 + r ωn (1 − x n )) + ln(1 − a ωn (1 − x n )(1 + r ωn (1 − x n ))),
where 1 − x n = 1/(1 + e yn ). From this expression it is easily seen for any ε > 0 one can pickK large, so that for y n >K, y n+1 ≤ y n − ln(a ωn ) + ε.
Pick ε small enough so that − ln(a 1 ) − ln(a 2 ) + 2ε < 0. For z 0 ∈ (K, h 2 (K)] and z n+1 = h ωn (z n ), let TK = min{n ∈ N | z n ≤K} be the stopping time to leave (K, ∞). As in the proof of Lemma 5.2 one shows that the expectation of TK is finite. To provide the argument, consider the random walk u n+1 = u n − ln(a ωn ) + ε starting at u 0 = z 0 and let T u = min{n ∈ N | u n ≤K}.
Then TK ≤ T u . By Wald's identity, Σ + 2 T u (ω) dν + (ω) < ∞ and hence
Σ + 2 TK(ω) dν + (ω) < ∞.(21)
After these preparations we define the first return map g ω : (−∞,K] → (−∞,K],
g ω (y) = h R(ω,y) ω (y),
where R(ω, y) = min{n ≥ 1 | h n ω (y) ≤K}. By (21), R has finite expectation. In fact, there is C > 0 so that for each y ∈ [K,K],
Σ + 2 R(ω, y) dν + (ω) ≤ C.(22)
The next step is to show that T K = min{n ∈ N | g n ω (y) < K} for y ∈ [K,K] has finite expectation. Consider the skew product G :
Σ + 2 × (−∞,K] → Σ + 2 × (−∞,K], G(ω, y) = (σ R(ω,y) ω, h R(ω,y)
ω (y)) = (σ R(ω,y) ω, g ω (y)).
Let π be the projection π(ω, y) = ω 0 from Σ + 2 × R onto {1, 2}. Given (ω, y) ∈ Σ + 2 × [K,K] we obtain a sequence ρ ∈ Σ + 2 given by ρ i = πG i (ω, y).
It follows from the construction that as the sequence (ω i ) ∞ 0 is independent and identically distributed, also (ρ i ) ∞ 0 is independent and identically distributed with the same distribution: probability 1/2 for both symbols 1, 2. Because f 1 (x) < x, we find h 1 (y) < y and thus that there is a number l < 0 with h 1 (y) < y + l, for y ∈ [K,K]. Hence, for any y ∈ [K,K] and N = (K −K)/l we will have g N 1 (y) = h N 1 (y) < K. The stopping time T K is therefore smaller than the stopping time min{n ∈ N | ρ i = 1 for n − N < i ≤ n}. Note that the expected number of throws of symbols 1, 2 that lead to N consecutive 1's is finite. (In fact it equals 2 N +1 − 2. It is easily bounded by N times the expectation of the first number j so that ω i = 1 for jN ≤ i < j(N + 1); the latter is a geometric distribution with expectation 2 N ). So the expectation of the stopping time T K is finite;
Σ + 2 T K (ω) dν + (ω) < ∞.(23)
Finally we combine (22) and (23): the formula
ξ k (ω) = T K (ω)−1 n=0 R(G n (ω, y T 2k−1 )) implies that Σ + 2 ξ k (ω) dν + (ω) ≤ C Σ + 2 T K (ω) dν + (ω) < ∞.
This proves Lemma 5.4.
We can now finish the proof of Theorem 5.2. Define for n ∈ [T 2k , T 2k+1 ), N η (n) = k, N ξ (n) = k andη (n) = n + 1 − T 2k ,ξ(n) = 0, so thatη counts the number of iterates from T 2k on where y n ≤ K. Likewise define for n ∈ [T 2k+1 , T 2k+2 ),
N η (n) = k + 1, N ξ (n) = k andη (n) = 0,ξ(n) = n + 1 − T 2k+1 .
Soξ counts the number of iterates from T 2k+1 on where y n > K. Finally calculate
1 n n−1 i=0 1 [0,β) (f i ω (x 0 )) = 1 n Nη(n−1) k=1 η k +η(n − 1) = Nη(n−1) k=1 η k +η(n − 1) Nη(n−1) k=1 η k +η(n − 1) + N ξ (n−1) k=1 ξ k +ξ(n − 1) = 1 + N ξ (n−1) k=1 ξ k +ξ(n − 1) Nη(n−1) k=1 η k +η(n − 1) −1 ≥ 1 + N ξ (n−1)+1 k=1 ξ k Nη(n−1) k=1 η k −1 .
By (15) and (20), the last term goes to 1 for ν + -almost all ω, as n → ∞ (note that N η (n − 1) − N ξ (n − 1) ≤ 1).
The next theorem is an immediate consequence of Theorem 5.2. Proof. We will only treat the case L(0) = 0 and L(1) > 0. Suppose there is an ergodic stationary measure m with support in (0, 1). By Lemma A.2, ν + × m is an ergodic invariant measure for F + . By Birkhoff's ergodic theorem, for ν + ×m-almost every (ω, x), we have
lim n→∞ 1 n n−1 i=0 δ f i ω (x) = m.(24)
By Fubini's theorem, there is a subset of I of full m-measure, so that in any Σ + 2 ×{x} with x from this subset, there is a set of full ν + -measure for which (24) holds. This however contradicts (14), since that holds for all β > 0 and applies to all x ∈ I.
The type of reasoning to prove Theorem 5.2 can be used to obtain the following result on iterated functions systems with zero Lyapunov exponents at both end points.
Theorem 5.4. Consider F + ∈ S and assume L(0) = L(1) = 0. Let 0 < β < 1 and x 0 ∈ (0, 1). Then for ν + -almost every ω ∈ Σ + 2 , Figure 1 illustrates a time series of the symmetric random walk, to which this theorem applies. 5.1. Pullback convergence. Theorem 5.1 implies that forward convergence of f n ω (x) to 0 does not hold: it is not true that for ν + -almost all ω ∈ Σ + 2 , f n ω (x) → 0 as n → ∞. The next result stipulates that pullback convergence to 0 does hold. See also [4,Section 9.3.4] for a related example where pullback convergence does not imply forward convergence, in a context of stochastic differential equations.
lim n→∞ 1 n n−1 i=0 1 [β,1−β] (f i ω (x 0 )) = 0.
Theorem 5.5. Let F + ∈ S and suppose L(0) = 0 and L(1) > 0. Take x ∈ (0, 1).
Then for ν-almost all ω ∈ Σ 2 , lim n→∞ f n σ −n ω (x) = 0.(25)
Proof. We reformulate the theorem to the following equivalent statement: for νalmost all ω ∈ Σ 2 , and for all y ∈ (0, 1),
lim n→∞ f −n ω (y) = 1.(26)
Equivalence of the statements (25) and (26) follows from the monotonicity of the interval diffeomorphisms: f −n ω (y) > x precisely if f n σ −n ω (x) < y and thus for ε 1 , ε 2 small positive numbers,
f −n ω (ε 1 ) > 1 − ε 2 precisely if f n σ −n ω (1 − ε 2 ) < ε 1 . To prove (26), consider u(ω) = inf{y ∈ I | lim i→∞ f −i ω (y) = 1}.(27)
As L(1) > 0, by Lemma 3.1 we know that u exists and u < 1, ν-almost everywhere.
Since u is invariant we get that either u > 0, ν-almost everywhere, or u = 0, νalmost everywhere. Assume that u is not identically 0. The measure µ = (id, u)ν on Σ 2 × I with conditional measures δ u(ω) and marginal ν on Σ 2 , defines an invariant measure for F . Denote by Π the natural projection Σ 2 × I → Σ + 2 × I, where Σ 2 = Σ − 2 × Σ + 2 . Expression (27) gives that u(ω) depends on the past ω − = (ω i ) −1 −∞ only. Therefore, the measure µ is a product measure ν + × ϑ on Σ + 2 × Σ − 2 × I . The projection Πµ is therefore a product measure ν + × m on Σ + 2 × I. That is, µ corresponds to an invariant measure ν + × m for F + , see Proposition A.1. Here m is a stationary measure by Lemma A.2. Since 0 < u < 1, ν-almost everywhere, m assigns positive measure to (0, 1). By Theorem 5.3, the only stationary measures are convex combinations of delta measures at 0 and 1. We have obtained a contradiction and proven (26) and hence the theorem. 5.2. Central limit theorem. Under the assumptions of Theorem 5.5, its conclusion that f n σ −n ω (x) → 0 for ν-almost all ω, implies that f n σ −n ω (x) converges to 0 in probability. By σ-invariance of ν, f n ω (x) converges to 0 in probability. Hence, for any a ∈ (0, 1),
lim n→∞ ν + ({ω ∈ Σ + 2 | f n ω (x) ≤ a}) = 1.
We state a central limit theorem that gives convergence of the distribution of the points f n ω (x), after an appropriate scaling. The proof is essentially contained in [32], where a central limit theorem for Markov processes on the half-line is stated.
Theorem 5.6. Let F + ∈ S and assume L(0) = 0 and L(1) > 0. Let x ∈ (0, 1). Then, for a > 0,
lim n→∞ ν + ω ∈ Σ + 2 | f n ω (x) ≥ e −a √ n = a 0 2e −ξ 2 /2 √ 2π dξ.
Proof. Take x n+1 = f ωn (x n ) with x 0 = x. Write y n = − ln(x n ) + ln(x), so that y 0 = 0 and y n ∈ [0, ∞) if x n ∈ (0, x]. Write z n = max{0, y n }.
Lemma 5.5. The moments of z 2 n /n satisfy
lim n→∞ Σ + 2 z 2k n /n k dν + = (2k)! 2 k k! .
Proof. It suffices to follow the proof of [32, Lemma 2.1] and [32,Lemma 2.2]. The proofs in [32] use that the process is null, that is, for any compact interval I ⊂ R, lim n→∞ As in [32, Theorem 2.1], Lemma 5.5 implies that z 2 n /n has a limiting distribution as n → ∞ and
lim n→∞ ν + ω ∈ Σ + 2 | z 2 n n ≤ a 2 = a 0 2e −ξ 2 /2 √ 2π dξ.
We conclude that
lim n→∞ ν + ω ∈ Σ + 2 | y n √ n ≤ a = a 0 2e −ξ 2 /2 √ 2π dξ.
Plugging in y n = − ln(x n ) + ln(x) gives the statement of the theorem.
Random walk with drift
The material in the previous sections treats all possible combinations of signs of L(0) and L(1) except the case where L(0) ≥ 0 and L(1) < 0 (or vice versa). We put the remaining case in the following result. Theorem 6.1. Consider F + ∈ S and assume L(0) ≥ 0 and L(1) < 0. Let x 0 ∈ (0, 1). Then for ν + -almost every ω ∈ Σ + 2 ,
lim n→∞ f n ω (x 0 ) = 1.
Proof. Take the proof of Theorem 5.5 applied to the inverse skew product system F −1 .
Theorem 6.1 establishes forward convergence of f n ω (x 0 ) to 1 under the given assumptions. Consider F + ∈ S with L(0) > 0 and L(1) < 0. Then there is also pullback convergence to 1;
lim n→∞ f n σ −n ω (x 0 ) = 1
for ν-almost all ω ∈ Σ 2 . It follows from the results in Section 5, again by going to the inverse skew product system, that such a pullback convergence does not hold in case L(0) = 0, L(1) < 0. See [13] for considerations on forward versus pullback convergence in an example of random circle dynamics.
Appendix A. Invariant measures for step skew product systems An iterated function system defines a Markov process and as such may admit stationary measures. Their relation with invariant measures for the corresponding one-sided skew product system and its natural extension, the two-sided skew product system, is explored in this section. This is classical material, originating from Furstenberg [17]. A general account of the constructions is found in [4]. We provide a simplified discussion taylored to a setting of step skew product systems over shifts. The reader may also consult the exposition in [41,Chapter 5].
Assume the context from Section 2. So consider Ω = {1, . . . , N } and the family of diffeomorphisms F = {f 1 , . . . , f N } on M . We pick f i with probability p i with 0 < p i < 1 and N i=1 p i = 1. We endow Σ N with the Borel sigma-algebra, denoted by F. Likewise we take the Borel sigma-algebra F + on Σ + N . Given the probabilities p i , we take a Bernoulli measure ν on Σ N which is determined by its values on cylinders;
i (K)) m(f −1 i (A)) − ε. So, lim inf n→∞ f i,n m(A) = lim inf n→∞ m(f −1 i,n (A)) ≥ lim n 0 →∞ m n≥n 0 f −1 i,n (A) ≥ m(f −1 i (K)) ≥ m(f −1 i (A)) − ε.
As this holds for any ε, we get
lim inf n→∞ f i,n m(A) m(f −1 i (A))
and hence that f i,n m converges to f i m. This argument shows that T depends continuously on f 1 , . . . , f N , continuous dependence on parameters p 1 , . . . , p N is clear.
The same type of argument shows that the map T ε , appearing in the proof of Theorem 4.1, is continuous and changes continuously with ε. The set of fixed points of T changes upper semi-continuously in the Hausdorff metric if parameters p 1 , . . . , p N and f 1 , . . . , f N are varied. So if m is a unique fixed point for T , T ε → T as ε → 0 and T ε m ε = m ε , then m ε → m as ε → 0.
Lemma A.2. A probability measure m is a stationary measure if and only if µ + = ν + × m is an invariant measure of F + with marginal ν + on Σ + N .
Proof. Consider the following calculation for product sets C × B ⊂ Σ + N × M of a cylinder C = C 0,...,n−1 i 0 ,...,i n−1 and a Borel set B:
F + (ν + × m)(C × B) = ν + × m((F + ) −1 (C × B)) = N i=1 ν + × m C 0,1,...,n i,i 0 ,...,i n−1 × f −1 i (B) = N i=1 p i ν + (C)m(f −1 i (B)) = N i=1 p i ν + (C)f i m(B).
If m is a stationary measure, then the last expression equals ν + (C)m(B) = ν + × m(C × B), so that F + (ν + × m)(C × B) = ν + × m(C × B). Since the product sets generate the σ-algebra, this proves F + -invariance of ν + × m. Similarly, if ν + × m is F + -invariant, then the last expression equals ν + × m(C × B) = ν + (C)m(B) and
this proves N i=1 p i f i m(B) = m(B).
Let m be a stationary measure for M . We say that m is ergodic if ν + × m is ergodic for F + . A point (ω, x) is said to be a generic point for an ergodic measure ν + × m, if the orbit is distributed according to the measure.
Write π : Σ 2 → Σ + 2 for the natural projection (ω n ) ∞ −∞ → (ω n ) ∞ 0 . The Borel sigma-algebra F + on Σ + N yields a sigma-algebra F 0 = π −1 F + on Σ N . A measure µ on Σ N × M with marginal ν has conditional measures µ ω on the fibers {ω} × M , such that
µ(A) = Σ N µ ω (A ω ) dν(ω)(28)
for measurable sets A, where we have written
A ω = A ∩ ({ω} × M ).
A measure µ + on Σ + N × M with marginal ν + likewise has conditional measures µ + ω . It is convenient to consider ν + as a measure on Σ N with sigma-algebra F 0 and µ + as a measure on Σ N × M with sigma-algebra F 0 ⊗ B. When ω ∈ Σ N we will write µ + ω for the conditional measures µ + πω . The spaces of measures are equipped with the weak star topology.
Invariant measures for F + with marginal ν + correspond to invariant measures for F with marginal ν in a one-to-one relationship, as detailed in Proposition A.1 below. This is a special case of [4,Theorem 1.7.2]. The result implies that stationary measures correspond one-to-one to specific invariant measures for F with marginal ν.
Write Σ N = Σ − N × Σ + N , where Σ − N consists of the past parts (ω i ) −1 −∞ of sequences ω. We have a natural projection
Π : Σ − N × Σ + N × M → Σ + N × M.
Proposition A.1. Let µ + be an F + -invariant probability measure with marginal ν + . Then, there exists an F -invariant probability measure µ with marginal ν and conditional measures µ ω = lim n→∞ f n σ −n ω µ + σ −n ω ,
ν-almost surely. Let µ be an F -invariant probability measure with marginal ν. Then,
µ + = Πµ(30)
is an F + -invariant probability measure with marginal ν + . The correspondence µ ↔ µ + given by (29), (30) is one-to-one. Furthermore, through these relations, F + -invariant product measures µ + = ν + × m correspond one-to-one with F -invariant product measures µ = ν + × ϑ on Σ + N × (Σ − N × M ).
Remark A.1. Consider µ + as a measure on Σ N × M with sigma-algebra F 0 ⊗ B.
Observe that F n (µ + ) has conditional measures f n σ −n ω µ + σ −n ω on {ω} × M . Hence, (29) reads µ = lim n→∞ F n (µ + ).
Remark A.2. From the characterization of ergodic probability measures as extremal points in the set of invariant probability measures, the one-to-one correspondence µ ↔ µ + implies that µ is ergodic if and only if µ + is ergodic.
Proof of Proposition A.1. Note that F s = σ s F 0 are sigma-algebras on Σ N with F s ↑ F. For a Borel set B ⊂ M , define υ t (ω) = f t σ −t ω µ + σ −t ω (B).
Calculate, for A s = σ s A 0 ∈ F s and 0 ≤ s ≤ t,
As υ t (ω) dν(ω) (1) = As f t σ −t ω µ + σ −t ω (B) dν(ω) (2) = A 0 f t σ s−t ω µ + σ s−t ω (B) dν(ω)
= A 0 f s ω µ + ω (B) dν(ω) (4) = As f s σ −s ω µ + σ −s ω (B) dν(ω) (5) = As υ s (ω) dν(ω).
Here (1) and (5) is the definition of υ t , (2) and (4) are by σ-invariance of ν, and (3) is by F + -invariance of µ + (see Lemma A.3 and Corollary A.1 below for a derivation). The above calculation shows that υ t is a martingale with respect to the filtration F t . Hence the limit lim t→∞ υ t (ω) exists. By the Vitali-Hahn-Saks theorem, see [16, Theorem III.10], the limit for varying Borel sets B defines a measure, µ ω . To obtain that the resulting measure µ is F -invariant, we refer to Remark A.1. Since F acts continuous on the space of probability measures, the limit lim n→∞ F n (µ + ) is F -invariant.
It remains to show that µ and µ + are in one-to-one correspondence. We wish to show that, given µ and computing µ + = Πµ, the formula (29) recovers µ. Note, again with A t = σ t A 0 ∈ F t for 0 ≤ t,
At υ t (ω) dν(ω) = At f t σ −t ω µ + σ −t ω (B) dν(ω) = A 0 f t ω µ + ω (B) dν(ω) = A 0 µ + ω ((f t ω ) −1 (B)) dν(ω) = µ + ω∈A 0 {ω} × (f t ω ) −1 (B) = µ ω∈A 0 {ω} × (f t ω ) −1 (B) = µ (F t ) −1 (A t × B) = At µ ω (B) dν(ω).
As F t ↑ F, this shows that υ t converges to µ as t → ∞. If µ is a product measure ν + × ϑ on Σ + N × (Σ − N × M ), then clearly µ + = Πµ is a product measure on Σ + N × M . In the other direction, if µ + = ν + × m, then (29) reads µ ω = lim Since µ ω depends on the past ω − alone, this can be written as µ(A) = ν + × ϑ(C + × B) = ν + (C + )ϑ(B). So µ is a product measure ν + × ϑ on Σ + N × (Σ − N × M ). The following lemma draws conclusions from F + -invariance of µ + . Lemma A.3. For A 0 ∈ F + , B ∈ B, 0 ≤ s ≤ t,
σ s−t A 0 f t ω µ + ω (B) dν + (ω) = σ s−t A 0 f s σ t−s ω µ + σ t−s ω (B) dν + (ω).
Proof. Write A s−t = σ s−t A 0 and compute, using F + -invariance of µ + ,
A s−t f t ω µ + ω (B) dν + (ω) = A s−t f s σ t−s ω f t−s ω µ + ω (B) dν + (ω) = A s−t µ + ω ((f t−s ω ) −1 (f s σ t−s ω ) −1 (B)) dν + (ω) = µ + ω∈A s−t {ω} × (f t−s ω ) −1 (f s σ t−s ω ) −1 (B) = µ + ω∈A 0 {σ s−t ω} × (f t−s σ s−t ω ) −1 (f s ω ) −1 (B) = µ + (F + ) s−t ω∈A 0 {ω} × (f s ω ) −1 (B) = µ + ω∈A 0 {ω} × (f s ω ) −1 (B) = A 0 µ + ω ((f s ω ) −1 (B)) dν + (ω) = A 0 f s ω µ + ω (B) dν + (ω).
As A 0 f s ω µ + ω (B) dν + (ω) = A s−t f s σ t−s ω µ + σ t−s ω (B) dν + (ω) by σ-invariance of ν + , this concludes the argument.
Corollary A.1. The lemma implies that for A 0 ∈ F 0 , B ∈ B, for 0 ≤ s ≤ t,
A 0 f t σ s−t ω µ + σ s−t ω (B) dν(ω) = A 0 f s ω µ + ω (B) dν(ω).
Note that for the natural extension, F -invariance of µ means f t σ s−t ω µ σ s−t ω = f s ω µ ω for 0 ≤ s ≤ t and for ν-almost all ω ∈ Σ N .
Theorem 3 . 1 .
31Let F + ∈ S and assume L(0) < 0 and L(1) < 0. The sets Σ + 2 × {0} and Σ + 2 × {1} attract sets of positive standard measure. Both their basins lie dense in Σ + 2 × I. The union of the basins has full standard measure.
Lemma 3.2. Let F + ∈ S and assume L(0) > 0 and L(1) > 0. Then there exists an ergodic stationary measure m with m({0} ∪ {1}) = 0.
Figure 2 .
2With r = 1/2, the diffeomorphisms f 1 (x) = x − rx(1 − x) and f 2 (x) = x + rx(1 − x) (picked with probabilities 1/2) give negative Lyapunov exponents at the end points 0, 1. Depicted, in the left frame, are the graphs of the inverse
and the claim is proved. Since the extreme points of M I are ergodic stationary measures, we conclude that the extreme points of N c ∩ M I are ergodic stationary measures. Since the Krein-Milman theorem the set of extreme points of N c ∩ M I is nonempty, there are ergodic stationary measures in N c . A stationary measure m gives an invariant measure µ m for the step skew product system F , as explained in Appendix A. Its conditional measures on fibers {ω} × I are denoted by µ m,ω .
Lemma 3 . 3 .
33For every ergodic stationary probability measure m, the conditional measure µ m,ω of µ m is a δ-measure for ν-almost every ω ∈ Σ 2 .
Lemma 3 . 4 .
34There is a unique stationary measure m with m({0} ∪ {1}) = 0.
A. 1 ,
1Lemmas 3.3 and 3.4, there exists a measurable function ξ : Σ 2 → I such that lim n→∞ f n σ −n ω m = δ ξ(ω) , for ν-almost all ω, where m is the stationary measure with m({0} ∪ {1}) = 0. As the convex hull of the support of m equals I and f 1 , f 2 are increasing, this implies lim n→∞ f n σ −n ω (x) = ξ(ω)
Figure 3 .Figure 2 .
32The left frame shows a numerically computed histogram for a time series of the iterated function systems generated by the same diffeomorphisms f The right frame indicates asymptotic convergence of orbits within fibers: it depicts time series for three different initial conditions in I with the same ω.
Figure 4 .
4The left frame depicts the graphs of
un )), with d i = − ln(c i ). Consider the stopping time T = inf{n ≥ 1 | u n ≤ K} (with T = ∞ if u n > K for all n) and write z n = ln(u n∧T ),
Now that Lemma 5.1 gives that z n is a nonnegative supermartingale, by Doob's supermartingale convergence theorem, see e.g.[39, Section VII.4], lim n→∞ z n (ω) < ∞
Figure 5 .
5A sequence of stopping times is defined to label subsequent iterates where y n leaves (−∞, K] or (K, ∞).
C 0 ,
0...,n−1 ω 0 ,...,ω n−1 e r * Sn dν + (ω) = C 0,...,n−1 ω 0 ,...,ω n−1 e r * S n−1 e r * ζn dν + (ω) = e r * S n−1 C 0,...,n−1 ω 0 ,...,ω n−1 e r * ζn dν + (ω) = e r * S ,...,ω n−1 e r * S n−1 dν + (ω). By Doob's optional stopping theorem, see e.g. [39, Theorem VII.2.2], Σ + 2 e r * S Ug dν + (ω) = Σ + 2 e r * S 0 dν + (ω) = 1.
Theorem 5. 3 .
3Let F + ∈ S and assume L(0) ≤ 0 and L(1) > 0. Then the only ergodic stationary measures are the delta measures at 0 and 1.
+ ({ω | z i ∈ I}) = 0.This holds by Theorem 5.2.
ν(C k 1 ,...,kn ω 1 ,...,ωn ) = n i=1p ω i .Write ν + for the Bernoulli measure on Σ + N , defined analogously. For different probability vectors (p 1 , . . . , p N ), the corresponding Bernoulli measures are mutually singular.Denote by B the Borel sigma-algebra on M . For a measure m on M and any B-measurable set A we denote the push-forward measure of m by f i m in whichf i m i (A) = m i (f −1 i (A)). Definition A.1. A stationary measure m on M is a probability measure that equals its average push-forward under the iterated function system IFS (F), i.e. it satisfiesm = N i=1 p i f i m.Write P M for the space of probability measures on M , equipped with the weak star topology. Write T : P M → P M for the mapT m = N i=1 p i f i m.A stationary measure is a fixed point of T .Lemma A.1. The map T is continuous. It also depends continuously on the parameters p 1 , . . . , p N and f 1 , . . . , f N . Proof. Recall that a sequence of measures m n converges in the weak star topology to a measure m precisely if for each open set A, lim inf n→∞ m n (A) m(A), see e.g. [39, Theorem III.1.1]. If m n converges to m it follows that for A open, lim inf n→∞ f i m n (A) = lim inf n→∞ m n (f −1 i (A)) m(f −1 i (A)) since f −1 i (A) is open. That is, lim inf n→∞ f i m n (A) f i m(A) and thus f i m n converges to f i m. This argument also shows that T is continuous. To prove continuous dependence on f 1 , . . . , f N , consider a sequence of maps f i,n converging to f i . By inner regularity, for an open set O ⊂ M one has m(O) = sup C⊂O m(C), where C runs over compact subsets of O. So also, given ε > 0, for A ⊂ M open, there exists compact K ⊂ A with m(f −1 i (K)) m(f −1 i (A)) − ε. Then, for A open, given ε > 0 there are n 0 > 0 and K ⊂ A so that f −1 i (K) ⊂ f −1 i,n (A) for n n 0 and m(f −1
that µ ω does not depend on the future ω + = (ω n ) ∞ 0 of ω. For a product setA = C + × B ⊂ Σ + N × (Σ − N × M ), (28) yields µ(A) = Σ + N Σ − N µ ω (A ω ) dν − (ω − )dν + (ω + ).
Proof.A smooth conjugation brings f 1 near 0 to the linear map x → x/d for some d > 1. More formally, there is a local diffeomorphism h : U → R with U a neighborhood of 0 and h(0) = 0, so that h −1 • f 1 • h(x) = x/d. Replace f 1 by h −1 • f 1 • h and likewise f 2 by h −1 • f 2 • h, near 0.That is, we may assume that on a neighborhood of 0 that contains [0, β],)
η k has infinite expectation.
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| [] |
[
"Heterocedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing",
"Heterocedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing"
] | [
"Luella Fu \nDepartment of Mathematics\nSan Francisco State University\n\n",
"Bowen Gang \nDepartment of Mathematics\nUniversity of Southern California\n\n",
"Gareth M James \nDepartment of Data Sciences and Operations\nUniversity of Southern California\n\n",
"Wenguang Sun \nDepartment of Data Sciences and Operations\nUniversity of Southern California\n\n"
] | [
"Department of Mathematics\nSan Francisco State University\n",
"Department of Mathematics\nUniversity of Southern California\n",
"Department of Data Sciences and Operations\nUniversity of Southern California\n",
"Department of Data Sciences and Operations\nUniversity of Southern California\n"
] | [] | Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We develop a new class of heteroscedasticityadjusted ranking and thresholding (HART) rules that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main idea of HART is to bypass standardization by directly incorporating both the summary statistic and its variance into the testing procedure. A key message is that the variance structure of the alternative distribution, which is subsumed under standardized statistics, is highly informative and can be exploited to achieve higher power. The proposed HART procedure is shown to be asymptotically valid and optimal for false discovery rate (FDR) control. Our simulation results demonstrate that HART achieves substantial power gain over existing methods at the same FDR level. We illustrate the implementation through a microarray analysis of myeloma. | 10.1080/01621459.2020.1840992 | [
"https://arxiv.org/pdf/1910.08107v2.pdf"
] | 212,628,582 | 1910.08107 | c289dadf68181ad5cb8ae78452827e2bcd777e0a |
Heterocedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing
Luella Fu
Department of Mathematics
San Francisco State University
Bowen Gang
Department of Mathematics
University of Southern California
Gareth M James
Department of Data Sciences and Operations
University of Southern California
Wenguang Sun
Department of Data Sciences and Operations
University of Southern California
Heterocedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing
covariate-assisted inferencedata processing and information lossfalse dis- covery rateheteroscedasticitymultiple testing with side informationstructured multiple testing
Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We develop a new class of heteroscedasticityadjusted ranking and thresholding (HART) rules that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main idea of HART is to bypass standardization by directly incorporating both the summary statistic and its variance into the testing procedure. A key message is that the variance structure of the alternative distribution, which is subsumed under standardized statistics, is highly informative and can be exploited to achieve higher power. The proposed HART procedure is shown to be asymptotically valid and optimal for false discovery rate (FDR) control. Our simulation results demonstrate that HART achieves substantial power gain over existing methods at the same FDR level. We illustrate the implementation through a microarray analysis of myeloma.
Introduction
In a wide range of modern scientific studies, multiple testing frameworks have been routinely employed by scientists and researchers to identify interesting cases among thousands or even millions of features. A representative sampling of settings where multiple testing has been used includes: genetics, for the analysis of gene expression levels (Tusher et al., 2001;Dudoit et al., 2003;Sun and Wei, 2011); astronomy, for the detection of galaxies (Miller et al., 2001); neuro-imaging, for the discovery of differential brain activity (Pacifico et al., 2004;Schwartzman et al., 2008); education, to identify student achievement gaps (Efron, 2008a); data visualization, to find potentially interesting patterns (Zhao et al., 2017); and finance, to evaluate trading strategies (Harvey and Liu, 2015).
The standard practice involves three steps: reduce the data in different study units to a vector of summary statistics; standardize the summary statistics to obtain significance indices such as z-values or p-values; and find a threshold of significance that corrects for multiplicity. Given a summary statistic X i with associated standard deviation σ i , traditional multiple testing approaches begin by standardizing the observed data Z i = X i /σ i , which is then used to compute the p-value based on a problem specific null distribution.
Finally, the p-values are ordered, and a threshold is applied to keep the rate of Type I error below a pre-specified level.
Classical approaches concentrated on setting a threshold that controls the family-wise error rate (FWER), using methods such as the Bonferroni correction or Holm's procedure (Holm, 1979). However, the FWER criterion becomes infeasible once the number of hypotheses under consideration grows to thousands. The seminal contribution of Benjamini and Hochberg (1995) proposed replacing the FWER by the false discovery rate (FDR) and provided the BH algorithm for choosing a threshold on the ordered p-values which, under certain assumptions, is guaranteed to control the FDR.
While the BH procedure offers a significant improvement over classical approaches, it only controls the FDR at level (1 − π)α, where π is the proportion of non-nulls, suggesting that its power can be improved by incorporating an adjustement for π into the procedure. Benjamini and Hochberg (2000), Storey (2002) and Genovese and Wasserman (2002) proposed to first estimate the non-null proportion byπ and then run BH at level α/(1 −π). Efron et al. (2001) proposed the local false discovery rate (Lfdr), which incorporates, in addition to the sparsity parameter π, information about the alternative distribution. Sun and Cai (2007) proved that the z-value optimal procedure is an Lfdr thresholding rule and that this rule uniformly dominates the p-value optimal procedure in Genovese and Wasserman (2002). The key idea is that the shape of the alternative could potentially affect the rejection region but the important structural information is lost when converting the z-value to p-value. For example, when the means of non-null effects are more likely to be positive than negative, then taking this asymmetry of the alternative into account increases the power.
However, the sign information is not captured by conventional p-value methods, which only consider information about the null.
Although a wide variety of multiple testing approaches have been proposed, they almost all begin with the standardized data Z i (or its associated p-value, P i ). In fact, in large-scale studies where the data are collected from intrinsically diverse sources, the standardization step has been upheld as conventional wisdom, for it takes into account the variability of the summary statistics and suppresses the heterogeneity -enabling one to compare multiple study units on an equal footing. For example, in microarray studies, Efron et al. (2001) first compute standardized two-sample t-statistics for comparing the gene expression levels across two biological conditions and then convert the t-statistics to z-scores, which are further employed to carry out FDR analyses. Binomial data is also routinely standardized by rescaling the number of successes X i by the number of trials n i to obtain success probabilitiesp i = X i /n i and then converting the probabilities to z-scores (Efron, 2008a,b).
However, while standardization is an intuitive, and widely adopted, approach, we argue in this paper that there can be a significant loss in information from basing hypothesis tests on Z i rather then the full data (X i , σ i ) 1 . This observation, which we formalize later in the paper, is based on the fact that the power of tests can vary significantly as σ changes, but this difference in power is suppressed when the data is standardized and treated as equivalent. In the illustrative example in Section 2.2, we show that by accounting for differences in σ an alternative ordering of rejections can be obtained, allowing one to identify more 1 Unless otherwise stated, the term "full data" specifically refers to the pair (X i , σ i ) in this article. In practice, the process of deriving the pair (X i , σ i ) from the original (full) data could also suffer from information loss, but this point is beyond the scope of this work; see the rejoinder of Cai et al. (2019) for related discussions. true positives at the same FDR level.
This article develops a new class of heteroscedasticity-adjusted ranking and thresholding (HART) rules for large-scale multiple testing that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main strategy of HART is to bypass standardization by directly incorporating (X i , σ i ) into the testing procedure. We adopt a two-step approach. In the first step a new significance index is developed by taking into account the alternative distribution of each X i conditioned on σ i ; hence HART avoids power distortion. This kind of conditioning is not possible for standardized values since the σ i are subsumed under Z i . Then, in the second step the significance indices are ordered and the smallest ones are rejected up to a given cutoff. We develop theories to show that HART is optimal for integrating the information from both X i and σ i . Numerical results are provided to confirm that HART controls the FDR in finite samples and uniformly dominates existing methods in power.
We are not the first to consider adjusting for heterogeneity. Ignatiadis et al. (2016) and Lei and Fithian (2018) mentioned the possibility of using the p-value as a primary significance index while employing σ i as side-information to pre-order hypotheses. Earlier works by Efron (2008a) and Cai and Sun (2009) also suggest grouping methods to adjust for heterogeneous variances in data. However, the variance issue is only briefly mentioned in these works and it is unclear how a proper pre-ordering or grouping can be created based on σ i . It is important to note that the ordering or grouping based on the magnitudes of σ i will not always be informative. Concretely, a large σ i does not generally correspond to a hypothesis that is more or less likely to be a true signal. Our numerical results suggest that ordering by σ i can create a somehow arbitrary ordering of hypotheses, which can even be anti-informative, potentially leading to power loss compared to methods that utilize no side information. In contrast with existing works, we explicitly demonstrate the key role that σ i plays in characterizing the shape of the alternative in simultaneous testing (Section 2.2). Moreover, we develop a principled and optimal strategy, the HART procedure, for incorporating the structural information encoded in σ i into inference. We prove that HART guarantees FDR control and uniformly improves upon all existing methods in asymptotic power.
The findings are impactful for three reasons. First, the observation that standardization can be inefficient has broad implications since, due to inherent variabilities or differing sample sizes between study units, standardized tests are commonly applied to large-scale heterogeneous data to make different study units comparable. Second, our finding enriches the recent line of research on multiple testing with side and structural information (e.g. Cai et al., 2019;Li and Barber, 2019;Xia et al., 2019, among others). In contrast with these works that have focused on the usefulness of sparsity structure, our characterization of the impact of heteroscedasticity, or more concretely the shape of alternative distribution, is new.
Finally, HART convincingly demonstrates the benefits of leveraging structural information in high-dimensional settings when the number of tests is in the thousands or more. Ideas from HART apply to smaller data sets as well, but the algorithm is designed to capitalize on copious data in ways not possible for procedures intended for moderate amounts of data, and thus is most useful in large-scale testing scenarios where the structure can be learned from data with good precision.
The rest of the paper is organized as follows. Section 2 reviews the standard multiple testing model and provides a motivating example that clearly illustrates the potential power loss from standardization. Section 3 describes our HART procedure and its theoretical properties. Section 4 contains simulations, and Section 5 demonstrates the method on a microarray study. We conclude the article with a discussion of connections to existing work and open problems. Technical materials and proofs are provided in the Appendix.
Problem Formulation and the Issue of Standardizing
This section first describes the problem formulation and then discusses an example to illustrate the key issue.
Problem formulation
Let θ i denote a Bernoulli(π) variable, where θ i = 0/1 indicates a null/alternative hypothesis, and π = P (θ i = 1) is the proportion of nonzero signals coming from the alternative distribution. Suppose the summary statistics X 1 , . . . , X m are normal variables obeying distribution
X i |µ i , σ 2 i ind ∼ N (µ i , σ 2 i ),(2.1)
where µ i follows a mixture model with a point mass at zero and σ i is drawn from an unspecified prior:
µ i iid ∼ (1 − π)δ 0 (·) + πg µ (·), σ 2 i iid ∼ g σ (·). (2.2)
In (2.2), δ 0 (·) is a Dirac delta function indicating a point mass at 0 under the null hypothesis, while g µ (·) signifies that µ i under the alternative is drawn from an unspecified distribution which is allowed to vary across i. In this work, we focus on a model where µ i and σ i are not linked by a specific function. The more challenging situation where σ i may be informative for predicting µ i is briefly discussed in Section 6.2.
Following tradition in dealing with heteroscedasticity problems (e.g. Xie et al., 2012;Weinstein et al., 2018), we assume that σ i are known. This simplifies the discussion and enables us to focus on key ideas. For practical applications, we use a consistent estimator of σ i . The goal is to simultaneously test m hypotheses:
H 0,i : µ i = 0 vs. H 1,i : µ i = 0; i = 1, . . . , m. (2.3)
The multiple testing problem (2.3) is concerned with the simultaneous inference of θ θ θ = {θ i = I(µ i = 0) : i = 1, . . . , m}, where I(·) is an indicator function. The decision rule is represented by a binary vector δ = (δ i : 1 ≤ i ≤ m) ∈ {0, 1} m , where δ i = 1 means that we reject H 0,i , and δ i = 0 means we do not reject H 0,i . The false discovery rate (FDR) (Benjamini and Hochberg, 1995), defined as
FDR = E i (1 − θ i )δ i max{ i δ i , 1} , (2.4)
is a widely used error criterion in large-scale testing problems. A closely related criterion is the marginal false discovery rate
mFDR = E { i (1 − θ i )δ i } E ( i δ i ) . (2.5)
The mFDR is asymptotically equivalent to the FDR for a general set of decision rules satisfying certain first-and second-order conditions on the number of rejections (Basu et al., 2018), including p-value based tests for independent hypotheses (Genovese and Wasserman, 2002) and weakly dependent hypotheses (Storey et al., 2004). We shall show that our proposed data-driven procedure controls both the FDR and mFDR asymptotically; the main consideration of using the mFDR criterion is to derive optimality theory and facilitate methodological developments.
We use the expected number of true positives ETP = E ( m i=1 θ i δ i ) to evaluate the power of an FDR procedure. Other power measures include the missed discovery rate (MDR, Taylor et al., 2005), average power (Benjamini and Hochberg, 1995;Efron, 2007) and false negative rate or false non-discovery rate (FNR, Genovese and Wasserman, 2002;Sarkar, 2002). Cao et al. (2013) showed that under the monotone likelihood ratio condition (MLRC), maximizing the ETP is equivalent to minimizing the MDR and FNR. The ETP is used in this article because it is intuitive and simplifies the theory. We call a multiple testing procedure valid if it controls the FDR at the nominal level and optimal if it has the largest ETP among all valid FDR procedures.
The building blocks for conventional multiple testing procedures are standardized statistics such as Z i or P i . Let µ * i = µ i /σ i . The tacit rationale in conventional practice is that the simultaneous inference problem
H 0,i : µ * i = 0 vs. H 1,i : µ * i = 0; i = 1, . . . , m, (2.6)
is equivalent to the formulation (2.3); hence the standardization step has no impact on multiple testing. However, this seemingly plausible argument, which only takes into account the null distribution, fails to consider the change in the structure of the alternative distribution. Next we present an example to illustrate the information loss and power distortion from standardizing.
Data processing and power loss: an illustrative example
The following diagram describes a data processing approach that is often adopted when performing hypothesis tests:
(X i , σ i ) −→ Z i = X i σ i −→ P i = 2Φ(−|Z i |).
(2.7)
We start with the full data consisting of X i and σ 2 i = V ar(X i |µ i ). The data is then standardized, Z i = X i /σ i , and finally converted to a two-sided p-value, P i . Typically these p-values are ordered from smallest to largest, a threshold is chosen to control the FDR, and hypotheses with p-values below the threshold are rejected.
Here we present a simple example to illustrate the information loss that can occur at each of these data compression steps. Consider a hypothesis testing setting with H 0,i : θ i = 0 and the data coming from a normal mixture model, where
µ i iid ∼ (1 − π)δ 0 + πδ µa , σ i iid ∼ U [0.5, 4].
(2.8)
This is a special case of (2.2), where µ i are specifically drawn from a mixture of two point masses, and where we set µ a = 2.
We examine three possible approaches to controlling the FDR at α = 0.1. In the pvalue approach we reject for all p-values below a given threshold. Note that, when the FDR is exhausted, this is the uniformly most powerful p-value based method (Genovese and Wasserman, 2002), so is superior to, for example, the BH procedure. Alternatively, in the z-value approach we reject for all suitably small P(H 0 |Z i ), which is in turn the most powerful z-value based method (Sun and Cai, 2007). Finally, in the full data approach we reject when P(H 0 |X i , σ i ) is below a certain threshold, which we show later is optimal given X i and σ i . In computing the thresholds, we assume that there is an oracle knowing the alternative distribution; the formulas for our theoretical calculations are provided in Section A of the Appendix. For the model given by (2.8) these rules correspond to:
δ δ δ p = {I(P i ≤ 0.0006) : 1 ≤ i ≤ m} = {I(|Z i | ≥ 3.43) : 1 ≤ i ≤ m}, δ δ δ z = {I(P(H 0 |Z i ) ≤ 0.24) : 1 ≤ i ≤ m} = {I(Z i ≥ 3.13) : 1 ≤ i ≤ m}, δ δ δ full = {I(P(H 0 |X i , σ i ) ≤ 0.28) : 1 ≤ i ≤ m},
with the thresholds chosen such that the FDR is exactly 10% for all three approaches.
However, while the FDRs of these three methods are identical, the average powers, AP(δ δ δ) = and full data approach (green line) as a function of Z and σ. Approaches reject for all points above their corresponding line. Right: Heat map of relative proportions (on log scale) of alternative vs null hypotheses for different Z and σ. Blue corresponds to lower ratios and purple to higher ratios. The solid black line represents equal fractions of null and alternative, while the dashed line corresponds to three times as many alternative as null.
1 mπ E ( m i=1 θ i δ i ), differ significantly:
AP(δ p ) = 5.0%, AP(δ z ) = 7.2%, AP(δ full ) = 10.5%.
(2.9)
To better understand these differences consider the left hand plot in Figure 1, which illustrates the rejection regions for each approach as a function of Z and σ 2 . In the blue region all methods fail to reject the null hypothesis, while all methods reject in the black region. The green region corresponds to the space where the full data approach rejects the null while the other two methods do not. Alternatively, in the red region both the z-value and full data methods reject while the p-value approach fails to do so. Finally, in the white region the full data approach fails to reject while the z-value method does reject.
We first compare δ δ δ z and δ δ δ p . Let π + and π − denote the proportions of positive effects and negative effects, respectively. Then π + = 0.1 and π − = 0. This asymmetry of the alternative distribution can be captured by δ δ δ z , which uses a one-sided rejection region. (Note that this asymmetric rejection region is not pre-specified but a consequence of theoretical derivation. In practice δ δ δ z can be emulated by an adaptive z-value approach that is fully data-driven (Sun and Cai, 2007).) By contrast, δ δ δ p enforces a two-sided rejection region that is symmetrical about 0, trading off extra rejections in the region Z i ≤ −3.43 for fewer rejections in the region where 3.13 ≤ Z i ≤ 3.43. As all nonzero effects are positive, negative
z-values are highly unlikely to come from the alternative; this accounts for the 2.2% loss in AP for the p-value method. Next consider δ δ δ full vs δ δ δ z . The full data approach trades off extra rejections in the green space for fewer rejections in the white space. This may seem like a sub-optimal trade-off given that the green space is smaller. However, the green space actually contains many more true alternative hypotheses. Approximately 3.8% of the true alternatives occur in the green region as opposed to only 0.5% in the white region, which accounts for the 3.3% higher AP for the full data approach.
At first Figure 1 may appear counterintuitive. Why should we reject for low z-values in the green region but fail to reject for high z-values in the white region? The key observation here is that not all z-values are created equal. In the green region the observed data is far more consistent with the alternative hypothesis than the null hypothesis. For example, with Z = 4 and σ = 0.5 our observed X is four standard deviations from the null mean but exactly equal to the alternative mean. Alternatively, while it is true that in the white region the high z-values suggest that the data are inconsistent with the null hypothesis, they are also highly inconsistent with the alternative hypothesis. For example, with Z = 4 and σ = 2 our observed X is 8, which is four standard deviations from the null mean, but also three standard deviations from the alternative mean. Given that 90% of observations come from the null hypothesis, we do not have conclusive evidence as to whether this data is from the null or alternative. A z-value of 4 with σ = 0.5 is far more likely to come from the alternative hypothesis than is a z-value of 4 with σ = 2.
The right hand plot of Figure 1 Here we have plotted the density functions of Z under the null hypothesis (black solid) and
alternative hypothesis (red dashed) for different values of σ. The densities have been multiplied by the relative probability of each hypothesis occurring so points where the densities cross correspond to an equal likelihood for either hypothesis. The blue line represents an observation, which is fixed at Z = 2 in each plot. The alternative density is centered at Z = 2/σ so when σ is large the standardized null and alternative are very similar, making it hard to know which distribution Z = 2 belongs to. As σ decreases the standardized alternative distribution moves away from the null and becomes more consistent with Z = 2.
However, eventually the alternative moves past Z = 2 and it again becomes unclear which distribution our data belongs to. Standardizing means that the null hypothesis is consistent for all values of σ but the alternative hypothesis can change dramatically as a function of the standard deviation.
To summarize, the information loss incurred in both steps of data processing (2.7) reveals the essential role of the alternative distribution in simultaneous testing. This structure of the alternative is not captured by the p-value, which is calculated only based on the null.
Our result (2.9) in the toy example shows that by exploiting (i) the overall asymmetry of the alternative via the z-value and (ii) the heterogeneity among individual alternatives via the full data, the average power of conventional p-value based methods can be doubled.
Heteroscadasticity and empirical null distribution
In the context of simultaneous testing with composite null hypotheses, Sun and McLain (2012) argued that the conventional testing framework, which involves rescaling or standardization, can become problematic:
"In multiple testing problems where the null is simple (H 0,i : µ i = 0), the heteroscedasticity in errors can be removed by rescaling all σ i to 1. However, when the null is composite, such a rescaling step would distort the scientific question." Sun and McLain (2012) further proposed the concept of empirical composite null as an extension of Efron's empirical null (Efron, 2004a) for testing composite nulls H 0,i : µ i ∈ [−a 0 , a 0 ] under heteroscedastic models. It is important to note that the main message of this article, which focuses on the impact of heteroscedastiticy on the alternative instead of the null, is fundamentally different from that in Sun and McLain (2012). In fact, we
show that even when the null is simple, the heteroscedasticity still matters. Our finding, which somehow contradicts the above quotes, is more striking and even counter-intuitive.
Moreover, we shall see that our data-driven HART procedure, which is based on Tweedie's formula (or the f -modeling approach, Efron, 2011), is very different from the deconvoluting kernel method (or g-modeling approach) in Sun and McLain (2012) 3 . The new two-step bivariate estimator in Section 3.2 is novel and highly nontrivial; the techniques employed in the proofs of theory are also very different.
HART: Heteroscedasticity Adjusted Ranking and Thresholding
The example in the previous section presents a setting where hypothesis tests based on the full data (X i , σ i ) can produce higher power than that from using the standardized data Z i . In this section we formalize this idea and show that the result holds in general for heteroscedasticity problems. We first assume that the distributional information is known and derive an oracle rule based on full data in Section 3.1. Section 3.2 develops data-driven schemes and computational algorithms to implement the oracle rule. Finally theoretical properties of the proposed method are established in Section 3.3.
The oracle rule under heteroscedasity
Note that the models given by (2.1) and (2.2) imply that
X i |σ i ind ∼ f σ i (x) = (1 − π)f 0,σ i (x) + πf 1,σ i (x), (3.1) where f 0,σ (x) = 1 σ φ(x/σ) is the null density, f 1,σ (x) = 1 σ φ σ x−µ σ g µ (µ)dµ is the alter- native density, φ(x)
is the density of a standard normal variable, and f σ (x) is the mixture density. For standardized data
Z i = X i /σ i , Model 3.1 reduces to Z i iid ∼ f (z) = (1 − π)f 0 (z) + πf 1 (z), (3.2)
where f 0 (z) = φ(z), f 1 (z) is the non-null density, and f (z) is the mixture density of the z-values. As discussed previously, a standard approach involves converting the z-value to
a two-sided p-value P i = 2Φ(−|Z i |), where Φ(·) is the standard normal cdf. The mixture model based on p-values is P i iid ∼ g(p) = (1 − π)I [0,1] (p) + πg 1 (p), for p ∈ [0, 1], (3.3)
where I(·) is an indicator function, and g(·) and g 1 (·) are the mixture density and non-null (2007)].
The oracle FDR procedure for Models 3.2 and 3.3 are both known. We first review the oracle z-value procedure (Sun and Cai, 2007). Define the local FDR (Efron et al., 2001)
Lfdr i = P(H 0 |z i ) = P(θ i = 0|z i ) = (1 − π)f 0 (z i ) f (z i ) . (3.4)
Then Sun and Cai (2007) showed that the optimal z-value FDR procedure is given by
δ z = [I{Lfdr(z i ) < c * } : 1 ≤ i ≤ m], (3.5)
where c * is the largest Lfdr threshold such that mFDR ≤ α. Similarly, Genovese and Wasserman (2002) showed that the optimal p-value based FDR procedure is given by
δ p = [I{P i < c * } : 1 ≤ i ≤ m], (3.6)
where c * is the largest p-value threshold such that mFDR ≤ α.
Next we derive the oracle rule based on m pairs {(x i , σ i ) : i = 1, . . . , m}. This new problem can be recast and solved in the framework of multiple testing with a covariate sequence. Consider Model 3.1 and define the heterogeneity-adjusted significance index 4
T i ≡ T (x i , σ i ) = P(θ i = 0|x i , σ i ) = (1 − π)f 0,σ i (x i ) f σ i (x i ) . (3.7)
Let Q(t) denote the mFDR level of the testing rule [I{T i < t} : 1 ≤ i ≤ m]. Then the oracle full data procedure is denoted
δ full = [I{T i < t * } : 1 ≤ i ≤ m], (3.8) where t * = sup{t : Q(t) ≤ α}.
The next theorem provides the key result showing that δ δ δ full has highest power amongst all α-level FDR rules based on {(x i , σ i ) : i = 1, · · · , m}.
Theorem 1 Let D α be the collection of all testing rules based on {(x i , σ i ) : i = 1, . . . , m} such that mFDR δ ≤ α. Then ETP δ ≤ ETP δ full for any δ ∈ D α . In particular we have
ETP δ p ≤ ETP δ z ≤ ETP δ full .
Based on Theorem 1, our proposed methodology employs a heteroscedasticity-adjusted ranking and thresholding (HART) rule that operates in two steps: first rank all hypotheses according to T i and then reject all hypotheses with T i ≤ t * . We discuss in Section 3.2 our finite sample approach for implementing HART using estimates for T i and t * .
Data-driven procedure and computational algorithms
We first discuss how to estimate T i and then turn to t * . Inspecting T i 's formula (3.7), the null density f 0,σ i (x i ) is known and the non-null proportion π can be estimated byπ using existing methods such as Storey's estimator (Storey, 2002) or Jin-Cai's estimator (Jin and Cai, 2007). Hence we focus on the problem of estimating f σ i (x i ).
There are two possible approaches for implementing this step. The first involves directly estimating f σ i (x i ) while the second is implemented by first estimating f 1,σ i (x i ) and then computing the marginal distribution viâ
f σ i (x i ) = (1 −π)f 0,σ i (x i ) +πf 1,σ i (x i ). (3.9)
Our theoretical and empirical results strongly suggest that this latter approach provides superior results so we adopt this method.
Remark 1
The main concern about the direct estimation of f σ i (x i ) is that the tail areas of the mixture density are of the greatest interest in multiple testing but unfortunately the hardest parts to accurately estimate due to the few observations in the tails. The fact that f σ i (x i ) appears in the denominator exacerbates the situation. The decomposition in (3.9)
increases the stability of the density by incorporating a known part of null density.
Standard bivariate kernel methods (Silverman, 1986;Wand and Jones, 1994) are not
suitable for estimating f 1,σ i (x i ) because, unlike a typical variable, σ i plays a special role in a density function and needs to be modeled carefully. Fu et al. (2018) recently addressed a closely related problem using the following weighted bivariate kernel estimator:
f * σ (x) := m j=1 φ hσ (σ − σ j ) m j=1 φ hσ (σ − σ j ) φ h xj (x − x j ), (3.10) where h = (h x , h σ ) is a pair of bandwidths, φ hσ (σ − σ j )/{ m j=1 φ hσ (σ − σ j )} determines the contribution of (x j , σ j ) based on σ j , h xj = h x σ j is a bandwidth that varies across j, and φ h (z) = 1 √ 2πh exp − z 2 2h 2
is a Gaussian kernel. The variable bandwidth h xj upweights/down-weights observations corresponding to small/large σ j ; this suitably adjusts for the heteroscedasticity in the data.
Let M 1 = {i : θ i = 1}. In the ideal setting where θ j is observed one could extend (3.10) to estimate f 1,σ i (x i ) viã f 1,σ (x) = j∈M 1 φ hσ (σ − σ j ) k∈M 1 φ hσ (σ − σ k ) φ h xj (x − x j ).
(3.11)
Given that θ j is unknown, we cannot directly implement (3.11). Instead we apply a weighted version of (3.11),f
1,σ i (x i ) = m j=1ŵ j φ hσ (σ i − σ j ) m k=1ŵ k φ hσ (σ i − σ k ) φ h xj (x i − x j ) (3.12)
with weightsŵ j equal to an estimate of P (θ j = 1|x j , σ j ). In particular we adopt a two step approach:
1. Computef (0) 1,σ i (x i ) via (3.12) with initial weightsŵ (0) j = (1 −T (0) j ) for all j, wherê T (0) j = min (1−π)f 0,σ j (x j ) f * σ j (x j )
, 1 ,π is the estimated non-null proportion, andf * σ j (x j ) is computed using (3.10).
Computef
(1)
1,σ i (x i ) via (3.12) with updated weightsŵ (1) j = (1 −T (1) j ) wherê T (1) j = (1 −π)f 0,σ j (x j ) (1 −π)f 0,σ j (x j ) +πf (0) 1,σ j (x j )
.
This leads to our final estimate for T i = P(H 0 |x i , σ i ):
T i =T (2) i = (1 −π)f 0,σ i (x i ) (1 −π)f 0,σ i (x i ) +πf (1) 1,σ i (x i )
.
In the next section, we carry out a detailed theoretical analysis to show that bothf σ i (x i )
andT i are consistent estimators with E f σ i − f σ i 2 = E {f σ i (x) − f σ i (x)} 2 dx → 0 and T i P − → T i , uniformly for all i.
To implement the oracle rule (3.8), we need to estimate the optimal threshold t * , which can be found by carrying out the following simple stepwise procedure.
Procedure 1 (data-driven HART procedure) Rank hypotheses by increasing order of
T i . Denote the sorted ranking statisticsT (1) ≤ . . . ≤T (m) and H (1) , . . . , H (m) the correspond- ing hypotheses. Let k = max j : 1 j j i=1T (i) ≤ α .
Then reject the corresponding ordered hypotheses, H (1) , . . . , H (k) .
The idea of the above procedure is that if the first j hypotheses are rejected, then the moving average 1 j j i=1T (i) provides a good estimate of the false discovery proportion, which is required to fulfill the FDR constraint. Comparing with the oracle rule (3.8), Procedure 1 can be viewed as its plug-in version:
δ δ δ dd = {I(T i ≤t * ) : 1 ≤ i ≤ m}, wheret * =T (k) .
(3.13)
The theoretical properties of Procedure 1 are studied in the next section.
Theoretical properties of Data-Driven HART
In Section 3.1, we have shown that the (full data) oracle rule δ δ δ full (3.8) is valid and optimal for FDR analysis. This section discusses the key theoretical result, Theorem 2, which shows that the performance of δ δ δ full can be achieved by its finite sample version δ δ δ dd (3.13) when m → ∞. Inspecting (3.13), the main steps involve showing that bothT i andt * are "close"
to their oracle counterparts. To ensure good performance of the proposed procedure, we require the following conditions.
(C1) supp(g σ ) ∈ (M 1 , M 2 ) and supp(g µ ) ∈ (−M, M ) for some M 1 > 0, M 2 < ∞, M < ∞.
(C2) The kernel function K is a positive, bounded and symmetric function satisfying
K(t) = 1, tK(t)dt = 0 and t 2 K(t)dt < ∞.
The density function f σ (t) has bounded and continuous second derivative and is square integrable.
(C3) The bandwidths satisfy h x = o{(log m) −1 }, lim m→∞ mh x h 2 σ = ∞, lim m→∞ m 1−δ h σ h 2 x = ∞ and lim m→∞ m −δ/2 h 2 σ h −1 x → 0 for some δ > 0. (C4)π p → π.
Remark 2 For Condition (C2), the requirement on f σ is standard in density estimation theory, and the requirements on the kernel K is satisfied by our choice of Gaussian kernel. (1994) and Silverman (1986). Jin-Cai's estimator Jin and Cai (2007) fulfills Condition (C4) in a wide class of mixture models.
Condition (C3) is satisfied by standard choices of bandwidths in Wand and Jones
Our theory is divided into two parts. The next proposition establishes the theoretical properties of the proposed density estimatorf σ and the plug-in statisticT i . The convergence oft * to t * and the asymptotic properties of δ δ δ dd are established in Theorem 2.
Proposition 1 Suppose Conditions (C1) to (C4) hold. Then
E f σ − f σ 2 = E {f σ (x) − f σ (x)} 2 dx → 0,
where the expectation E is taken over (X X X, σ σ σ, µ µ µ). Further, we haveT i P − → T i .
Next we turn to the performance of our data-driven procedure δ δ δ dd when m → ∞. A key step in the theoretical development is to show thatt * P − → t * , wheret * and t * are defined in (3.13) and (3.8), respectively.
Theorem 2 Under the conditions in Proposition 1, we havet * P − → t * . Further, both the mFDR and FDR of δ δ δ dd are controlled at level α + o(1), and ET P δ dd /ET P δ f ull = 1 + o(1).
In combination with Theorem 1, these results demonstrate that the proposed finite sample HART procedure (Procedure 1) is asymptotically valid and optimal.
Simulation
We first describe the implementation of HART in Section 4.
Implementation of HART
The accurate estimation ofT i is crucial for ensuring good performance of the HART procedure. The key quantity is the bivariate kernel density estimatorf 1,σ (x), which depends on the choice of tuning parameters h = (h x , h σ ). Note that the ranking and selection process in Procedure 1 only involves smallT i . To improve accuracy, the bandwidth should be chosen based on the pairs (x i , σ i ) that are less likely to come from the null. We first implement Jin and Cai's method (Jin and Cai, 2007) to estimate the overall proportion of non-nulls in the data, denotedπ. We then compute h x and h σ by applying Silverman's rule of thumb (Silverman, 1986) to the subset of the observations {x i : P i <π}. When implementing HART, we first estimate f σ (x) using the data without (X i , σ i ), and then plug-in the unused data (X i , σ i ) to calculateT i . This method can increase the stability of the density estimator. As shown in the proof of Proposition 1, the asymptotic properties ofT i hold for both the regular and jacknifed approaches.
Comparison in general settings
We consider simulation settings according to Models 2.1 and 2.2, where σ i are uniformly generated from U [0, σ max ]. We then generate X i from a two-component normal mixture model
X i |σ i iid ∼ (1 − π)N (0, σ 2 i ) + πN (2, σ 2 i ).
In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5. Five methods are compared: the ideal full data oracle procedure (OR), the z-value oracle procedure of (Sun and Cai, 2007) (ZOR), the Benjamini-Hochberg procedure (BH), AdaPT (Lei and Fithian, 2018) (AdaPT), and the proposed data-driven HART procedure (DD). Note that we do not include methods that explore the usefulness of sparsity structure (Scott et al., 2015;Boca and Leek, 2018;Li and Barber, 2019;Cai et al., 2019) since the primary objective here is to incorporate structural information encoded in σ i . Also, although Ignatiadis et al. (2016) mention the idea of using σ i as a covariate to construct weighted p-values, no guidance is given on how to do so, and since the way in which σ i are incorporated is particularly important (as illustrated by Section 2.2), we exclude it.
The nominal FDR level is set to α = 0.1. For each setting, the number of tests is m = 20, 000. Each simulation is also run over 100 repetitions. Then, the FDR is estimated as the average of the false discovery proportion FDP Next we discuss some important patterns of the plots and provide interpretations. Panel (a) of Figure 3 shows that all methods appropriately control FDR at the nominal level, with DD being slightly conservative. Panel (b) illustrates the advantage of the proposed HART procedure over existing methods. When π is small, the power of OR can be 60%
(δ) = m i=1 {(1 − θ i )δ i }/( m i=1 δ i ∨ 1)
higher than ZOR. This shows that exploiting the structural information of the variance can be extremely beneficial. DD has lower power compared to OR due to the inaccuracy in estimation. However, DD still dominates ZOR and BH in all settings. We can also see that ZOR dominates BH and the efficiency gain increases as π increases. To explain the power gain of ZOR over BH, let π + and π − denote the proportion of true positive signals and true negative signals, respectively. Then π + = π and π − = 0. This asymmetry can be captured by ZOR, which uses a one-sided rejection region. By contrast, BH adopts a two-sided symmetric rejection region. Under the setting being considered, the power loss due to the conservativeness of BH is essentially negligible, whereas the failure of capturing important structural information in the alternative accounts for most power loss. From the second row of Figure 3, we can again see that all methods control the FDR at the nominal level.
OR dominates the other three methods in all settings. DD is less powerful than OR but Figure 3: Comparison when σ i is generated from a uniform distribution. We vary π in the top row and σ max in the bottom row. All methods control the FDR at the nominal level. DD has roughly the same FDR but higher power compared to ZOR in all settings. has a clear advantage over ZOR with slightly lower FDR and higher power. In most cases,
AdaPT does perform better than BH. However, it is important to note that pre-ordering based on σ i is a suboptimal way for using side information. Moreover, the dominance of AdaPT over BH is not uniform, see section E.1 in the supplement for example. This shows that anti-informative pre-ordering based on σ i can lead to possible power loss for AdaPT.
By contrast, HART utilizes the side information in a principled and systematic way. It uniformly improves competitive methods. Finally, it should be noted that incorporating side information comes with computational costs: conventional methods including BH and ZOR both run considerably faster than DD. However, DD runs faster than AdaPT.
Comparison under a two-group model
To illustrate the heteroscedasticity effect more clearly, we conduct a simulation using a simpler model where σ i takes on one of two distinct values. The example illustrates that the heterogeneity adjustment is more useful when there is greater variation in the standard deviations among the testing units.
Consider the setup in Models 2.1 and 2.2. We first draw σ i randomly from two possible values {σ a , σ b } with equal probability, and then generate X i from a two-point normal mix-
ture model X i |σ i iid ∼ (1 − π)N (0, σ 2 i ) + πN (µ, σ 2 i ).
In this simpler setting, it is easy to show that HART reduces to the CLfdr method in Cai and Sun (2009), where the conditional Lfdr statistics are calculated for separate groups defined by σ a and σ b . As previously, we apply BH, ZOR, OR and DD to data with m = 20, 000 tests and the experiment repeated on 100 data sets. We fix π = 0.1, µ = 2.5, σ a = 1 and vary σ b from 1.5 to 3. The FDRs and powers of different methods are plotted as functions of σ b , with results summarized in the first row of Figure 4. In the second row, we plot the group-wise z-value cutoffs and group-wise powers as functions of σ b for the DD method.
We can see that DD has almost identical performance to OR, and the power gain over ZOR becomes more pronounced as σ b increases. This is intuitive, because more variation in σ tends to lead to more information loss in standardization. The bottom row shows the z-value cutoffs for ZOR and DD for each group. We can see that in comparison to ZOR, which uses a single z-value cutoff, HART uses different cutoffs for each group. The z-value cutoff is bigger for the group with larger variance, and the gap between the two cutoffs increases as the degree of heterogeneity increases. In Panel d), we can see that the power of Group b decreases as σ b increases. These interesting patterns corroborate those we observed in our toy example in Section 2.2.
Data Analysis
This section compares the adaptive z-value procedure (AZ, the data-driven implementation of ZOR, Sun and Cai (2007)), BH, and HART on a microarray data set. The data set measures expression levels of 12, 625 genes for patients with multiple myeloma, 36 for whom magnetic resonance imaging (MRI) detected focal lesions of bone (lesions), and 137 for Figure 4: Two groups with varying σ b from 1.5 to 3. As σ b increases, the cut-off for group a decreases whereas the cut-off for group b increases. The power for tests in group b drops quickly as σ b increases. This corroborates our calculations in the toy example in Section 2.2 and the patterns revealed by Figure 1.
whom MRI scans could not detect focal lesions (without lesions) of bone (Tian et al., 2003). For each gene, we calculate the differential gene expression levels (X i ) and standard errors (S i ). The FDR level is set at α = 0.1.
We first address two important practical issues. The first issue is that the theoretical null N (0, 1) (red curve on the left panel of Figure 5) is much narrower compared to the histogram of z-values. Efron (2004b) argued that a seemingly small deviation from the theoretical z-curve can lead to severely distorted FDR analysis. For this data set, the analysis based on the theoretical null would inappropriately reject too many hypotheses, resulting in a very high FDR. To address the distortion of the null, we adopted the empirical null approach (Efron, 2004b) in our analysis. Specifically, we first used the middle part of the histogram, which contains 99% of the data, to estimate the null distribution as The second issue is the estimation of f σ (x), which usually requires a relatively large sample size to ensure good precision. Figure 6 presents the histogram of S i and scatter plot of S i vs Z i . Based on the histogram, we propose to only focus on data points with S i less than 1 (12172 out of 12625 genes are kept in the analysis) to ensure the estimation accuracy ofT i . Compared to conventional approaches, there is no efficiency loss because no hypothesis with S i > 1 is rejected by BH at α = 0.1 -note that the BH p-value cutoff is 6 × 10 −5 , which corresponds to a z-value cutoff of 5.22; see also Figure 7. (If BH rejects hypotheses with large S i , we recommend to carry out a group-wise FDR analysis, which first tests hypotheses at α in separate groups and then combines the testing results, as suggested by Efron (2008a).) standard deviation Finally we apply BH, AZ and HART to the data points with S i < 1. BH uses the new pvalues P * i based on the estimated empirical null N (0, 1.3 2 ). Similarly AZ uses Lfdr statistics where the null is taken as the density of a N (0, 1.3 2 ) variable. When implementing HART, we estimate the non-null proportion π using Jin-Cai's method with the empirical null taken as N (0, 1.3 2 ). We further employ the jacknifed method to estimate f σ (x) by following the steps in Section 4.1. We summarize the number of rejections by each method in Table 1 and display the testing results in Figure 7, where we have marked rejected hypotheses by each method using different colors.
HART rejects more hypotheses than BH and AZ. The numbers should be interpreted with caution as BH and AZ have employed the empirical null N (0, 1.3 2 ) whereas HART has utilized null density N (0, σ 2 i ) conditioned on individual σ i -it remains an open issue how to extend the empirical null approach to the heteroscedastic case. Since we do not know the ground truth, it is difficult to assess the power gains. However, the key point of this analysis, and the focus of our paper, is to compare the shapes of rejection regions to gain some insights on the differences between the methods. It can be seen that for this data set, the rejection rules of BH and AZ only depend on Z i . By contrast, the rejection region for HART depends on both Z i and S i . HART rejects more z-values when S i is small compared to BH and AZ. Moreover, HART does not reject any hypothesis when S i is large. This pattern is consistent with the intuitions we gleaned from the illustrative example ( Figure 1) and the results we observed in simulation studies (Figure 4, Panel c).
Discussion
Multiple testing with side information
Multiple testing with side or auxiliary information is an important topic that has received much attention recently. The research directions are wide-ranging as there are various types of side information, which may be either extracted from the same data using carefully con- In the context of FDR analysis, the key issue is that the hypotheses become unequal 5 A method is said to have better ranking if it discovers more true positives than its competitor at the same FDR level. Theorem 1 in Section 3.1 shows that the oracle HART procedure has the optimal ranking in the sense that it has the largest power among all FDR procedures at level α in light of side information. Efron (2008a) argued that ignoring the heterogeneity among study units may lead to FDR rules that are inefficient, noninterpretable and even invalid.
We discuss two lines of work to further put our main contributions in context and to guide future research developments.
Grouping, pioneered by Efron (2008a), provides an effective strategy for capturing the heterogeneity in the data. Cai and Sun (2009) showed that the power of FDR procedures can be much improved by utilizing new ranking statistics adjusted for grouping. Recent works along this direction, including Liu et al. (2016), Barber and Ramdas (2017) and Sarkar and Zhao (2017), develop general frameworks for dealing with a class of hierarchical and grouping structures. However, the groups can be characterized in many ways and the optimal grouping strategy still remains unknown. Moreover, discretizing a continuous covariate by grouping leads to loss of information. HART directly incorporates σ i into the ranking statistic and hence eliminates the need to define groups.
Weighting is another widely used strategy for incorporating side information into FDR analyses (Benjamini and Hochberg, 1997;Genovese et al., 2006;Roquain and Van De Wiel, 2009;Basu et al., 2018). For example, when the sparsity structure is encoded by a covariate sequence, weighted p-values can be constructed to up-weight the tests at coordinates where signals appear to be more frequent (Hu et al., 2012;Xia et al., 2019;Li and Barber, 2019).
However, the derivation of weighting functions for directly incorporating heteroscedasticity seems to be rather complicated (Peña et al., 2011;Habiger et al., 2017). Notably, Habiger (2017) developed novel weights for p-values as functions of a class of auxiliary parameters, including σ i as a special case, for a generic two-group mixture model. However, the formulation is complicated and the weights are hard to compute -the methodology requires handling the derivative of the power function, estimating several unknown quantities and tuning a host of parameters.
Open issues and future directions
We conclude the article by discussing several open issues. First, HART works better for large-scale problems where the density with heteroscedastic errors can be well estimated.
For problems with several hundred tests or less, p-value based algorithms such as BH or the WAMDF approach (Habiger, 2017) are more suitable. The other promising direction for dealing with smaller-scale problems, suggested by Castillo and Roquain (2018), is to employ spike and slab priors to produce more stable empirical Bayes estimates (with frequentist guarantees under certain conditions). Second, in practice the model given by (2.2) can be extended to
µ i |σ i ind ∼ (1 − π σ i )δ 0 (·) + π σ i g µ (·|σ i ), σ 2 i iid ∼ g σ (·), (6.1)
where both the sparsity level and distribution of non-null effects depend on σ i ; this setting has been considered in a closely related work by Weinstein et al. (2018). The heterogeneityadjusted statistic is then given by
T i = P(θ i = 0|x i , σ i ) = (1 − π σ i )f 0,σ i (x i ) f σ i (x i ) , (6.2)
where the varying proportion π σ i indicates that σ i also captures the sparsity structure. This
Supplementary Material for "Information Loss and Power
Distortion from Standardizing in Multiple Hypothesis Testing"
A Formulas for the Illustrative Example
Consider Model 2.8 in Section 2.2. We derive the formulas for the oracle p-value, oracle z-value and oracle full data procedures.
• δ δ δ p corresponds to the thresholding rule I(|Z i | > t p ), where
t p = inf t > 0 : 2(1 − π)Φ(t) 2(1 − π)Φ(t) + π Φ (t + µa σ ) +Φ(t − µa σ ) dG(σ) ≤ α ,
withΦ being the survival function of the N (0, 1) variable.
• δ δ δ z is a one-sided thresholding rule of the form I(Z i > t z ), where
t z = inf t > 0 : (1 − π)Φ(t) (1 − π)Φ(t) + π Φ (t − µa σ )dG(σ) ≤ α .
• δ δ δ full is of the form I{P(θ i = 0|x i , σ i ) < λ}. It can be written as
I{Z i > t z,σ (λ)}, where t z,σ (λ) = µ 2 a − 2σ 2 log λπ (1−λ)(1−π) 2µ a σ .
Denote λ * the optimal threshold. Hence δ δ δ full is given by
I{P (θ i = 0|x i , σ i ) < λ * }, where λ * = sup λ ∈ [0, 1] : (1 − π) Φ {t z,σ (λ)}dG(σ) (1 − π) Φ {t z,σ (λ)}dG(σ) + π Φ {t z,σ (λ) − µa σ }dG(σ)
.
The optimal cutoffs can be solved numerically from the above. The powers are given by
AP (δ δ δ p ) = Φ t p + µ a σ +Φ t p − µ a σ dG(σ), AP (δ δ δ z ) = Φ t − µ a σ dG(σ), AP (δ δ δ full ) = Φ t z,σ (λ) − µ a σ dG(σ).
B Proofs of Theorems
B.1 Proof of Theorem 1
We divide the proof into two parts. In Part (a), we establish two properties of the testing rule δ δ δ full (t) = {I(T i < t) : 1 ≤ i ≤ m} for an arbitrary 0 < t < 1. In Part (b) we show that the oracle rule δ δ δ full (t * ) attains the mFDR level exactly and is optimal amongst all FDR procedures at level α.
Part (a). Denote α(t) the mFDR level of δ δ δ full (t). We shall show that (i) α(t) < t for all 0 < t < 1 and that
(ii) α(t) is nondecreasing in t. Note that E { m i=1 (1 − θ i )δ i } = E X,σ ( m i=1 T i δ i ).
According to the definition of α(t), we have
E X,σ m i=1 {T i − α(t)} I(T i ≤ t) = 0. (B.1)
We claim that α(t) < t. Otherwise if α(t) ≥ t, then we must have T i < t ≤ α(t). It follows that the LHS must be negative, contradicting (B.1).
Next we show (ii). Let α(t j ) = α j . We claim that if t 1 < t 2 , then we must have α 1 ≤ α 2 .
We argue by contradiction. Suppose that t 1 < t 2 but α 1 > α 2 . Then
(T i − α 2 )I(T i < t 2 ) = (T i − α 1 )I(T i < t 1 ) + (α 1 − α 2 )I(T i < t 1 ) + (T i − α 2 )I(t 1 ≤ T i < t 2 ) ≥ (T i − α 1 )I(T i < t 1 ) + (α 1 − α 2 )I(T i < t 1 ) + (T i − α 1 )I(t 1 ≤ T i < t 2 ). It follows that E { m i=1 (T i − α 2 )I(T i < t 2 )} > 0 since E { m i=1 (T i − α 1
)I(T i < t 1 )} = 0 according to (B.1), α 1 > α 2 and T i ≥ t 1 > α 1 , contradicting (B.1). Hence we must have
α 1 < α 2 .
Part (b). Letᾱ = α(1). In Part (a), we show that α(t) is non-decreasing in t. It follows that for all α <ᾱ, there exists a t * such that t * = sup{t : α(t * ) = α}. By definition, t * is the oracle threshold. Consider an arbitrary decision rule d = (d 1 , . . . , d m ) ∈ {0, 1} m such
that mFDR(d) ≤ α. We have E m i=1 (T i − α)δ f ull i = 0 and E { m i=1 (T i − α)d i } ≤ 0. Hence E m i=1 (δ f ull i − d i )(T i − α) ≥ 0. (B.2) Consider transformation f (x) = (x − α)/(1 − x). Note that f (x) is monotone, we rewrite δ f ull i = I [{(T i − α)/(1 − T i )} < λ], where λ = (t * − α)/(1 − t * )
. In Part (a) we have shown that α < t OR < 1, which implies that λ > 0. Hence
E m i=1 (δ f ull i − d i ) {(T i − α) − λ(1 − T i )} ≤ 0. (B.3)
To see this, consider the terms where δ f ull i − d i = 0. Then we have two situations: (i)
δ f ull i > d i or (ii) δ f ull i < d i . In situation (i), δ f ull i = 1, implying that {(T i − α)/(1 − T i )} < λ. In situation (ii), δ f ull i = 0, implying that {(T i − α)/(1 − T i )} ≥ λ. Therefore we always have (δ f ull i − d i ) {(T i − α) − λ(1 − T i )} ≤0 ≤ E m i=1 (δ f ull i − d i )(T i − α) ≤ λE m i=1 (δ f ull i − d i )(T i − α) . Finally, since λ > 0, it follows that E m i=1 (δ f ull i − d i )(T i − α) > 0.
Finally, we apply the definition of ETP to conclude that ETP(δ f ull ) ≥ ETP(d) for all d ∈ D α .
B.2 Proof of Theorem 2
We begin with a summary of notation used throughout the proof:
• Q(t) = m −1 m i=1 (T i − α)I{T i < t}. • Q(t) = m −1 m i=1 (T i − α)I{T i < t}.
• Q ∞ (t) = E{(T OR − α)I{T OR < t}}.
• t ∞ = sup{t ∈ (0, 1) : Q ∞ (t) ≤ 0}: the "ideal" threshold.
For T (k)
OR < t < T (k+1)
OR , define a continuous version of Q(t) as
Q C (t) = t − T (k) OR T (k+1) OR − T (k) OR Q k + T (k+1) OR − t T (k+1) OR − T (k) OR Q k+1 , where Q k = Q T (k)
OR . Since Q C (t) is continuous and monotone, its inverse Q −1 C is welldefined, continuous and monotone. Next we show the following two results in turn: (i)
Q(t) p → Q ∞ (t) and (ii) Q −1 C (0) p → t ∞ .
To show (i), note that Q(t) p → Q ∞ (t) by the WLLN, so that we only need to establish that Q(t) − Q(t) p → 0. We need the following lemma, which is proven in Section D.
Lemma 1 Let U i = (T i − α)I(T i < t) and U i = (T i − α)I{T i < t}. Then E U i − U i 2 = o(1).
By Lemma 1 and Cauchy-Schwartz inequality,
E U i − U i U j − U j = o(1). Let S m = m i=1 U i − U i . It follows that V ar m −1 S m ≤ m −2 m i=1 E U i − U i 2 +O 1 m 2 i,j:i =j E U i − U i U j − U j = o(1).
By Proposition 1, E(m −1 S m ) → 0, applying Chebyshev's inequality, we obtain m −1 S m =
Q(t) − Q(t) p → 0. Hence (i) is proved. Notice that Q ∞ (t) is continuous by construction, we also have Q(t) p → Q C (t).
Next we show (ii). Since Q C (t) is continuous, for any ε > 0, we can find η > 0 such
that Q −1 C (0) − Q −1 C Q C (t ∞ ) < ε if Q C (t ∞ ) < η. It follows that P Q C (t ∞ ) > η ≥ P Q −1 C (0) − Q −1 C Q C (t ∞ ) > ε .
Proposition 1 and the WLLN imply that Q C (t) Define the continuous version of Q(t) as Q C (t) and the corresponding threshold as Q −1 C (0). Then by construction, we have
p → Q ∞ (t). Note that Q ∞ (t ∞ ) = 0. Then P Q C (t ∞ ) > η → 0. Hence we have Q −1 C (0) p → Q −1 C Q C (t ∞ ) = t ∞ ,δ dd = I T i ≤ Q −1 C (0) : 1 ≤ i ≤ m and δ f ull = I T i ≤ Q −1 C (0) : 1 ≤ i ≤ m .
Following the previous arguments, we can show that Q −1
C (0) p → t ∞ . It follows that Q −1 C (0) = Q −1 C (0) + o p (1). By construction mFDR(δ f ull ) = α. The mFDR level of δ dd is mFDR(δ dd ) = P H 0 T i ≤ Q −1 C (0) P T i ≤ Q −1 C (0)
. From Proposition 2,T i p → T i . Using the continuous mapping theorem, mFDR δ δ δ dd = mFDR δ δ δ f ull + o(1) = α + o(1). The desired result follows.
Finally, using the fact thatT i p → T i and Q −1
C (0) p → Q −1 C (0), we can similarly show that ETP(δ δ δ dd )/ETP(δ δ δ f ull ) = 1 + o(1).
C Proof of Proposition 1 Summary of notation
The following notation will be used throughout the proofs:
•f * σ (x) = m j=1 φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i ) φ hxσ j (x − x j ). •f * 1,σ (x) = m j=1 φ hσ (σ − σ j )I(θ j = 1) m i=1 φ hσ (σ − σ i )I(θ i = 1) φ hxσ j (x − x j ). •f 1,σ (x) = m j=1 φ hσ (σ − σ j )P (θ j = 1|x j , σ j ) m i=1 φ hσ (σ − σ i )P (θ i = 1|x i , σ i ) φ hxσ j (x − x j ). •f 1,σ (x) = m j=1 φ hσ (σ − σ j )P (θ j = 1|x j , σ j ) m i=1 φ hσ (σ − σ i )P (θ i = 1|x i , σ i ) φ hxσ j (x − x j ). •f σ (x) = (1 −π)f 0,σ (x) +πf 1,σ (x).
The basic idea is that a consistent one-step estimator constructed viaf * σ (x) leads to a consistent two-step estimator viaf σ (x). By Condition (C4) and the triangle inequality, it is sufficient to show that
E f 1,σ (x) − f 1,σ (x) 2 dx → 0. (C.4) Let u j = φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i ) . A direct consequence of condition (C1 ) is 0 < C 1 m ≤ Eu j ≤ C 2 m < ∞ for some positive constants C 1 and C 2 . Let C = min(1, C 1 ).Consider event A = m i=1 θ j − mπ < C 2 mπ . (C.5)
By Hoeffding's inequality and Condition (C2 ),
P (A C )O(h −2 x ) ≤ exp(−C 2 m/2)O(h −2 x ) → 0.
Therefore it suffices to prove (C.4) under A. We establish the result in three steps:
1. E {f * 1,σ (x) − f 1,σ (x)} 2 dx → 0. 2. E {f 1,σ (x) −f * 1,σ (x)} 2 dx → 0. 3. E {f 1,σ (x) −f 1,σ (x)} 2 dx → 0.
The proposition then follows from the triangle inequality.
C.1 Proof of Step (a)
Let b * j = φ hσ (σ − σ j )I(θ j = 1) m i=1 φ hσ (σ − σ i )I(θ i = 1)
. It is easy to show that
f * 1,σ (x) − f 1,σ (x) 2 = m j=1 m k=1 b * j b * k {φ hxσ k (x − x k ) − f 1,σ (x)} φ hxσ j (x − x j ) − f 1,σ (x) .
Under condition (C1 ) and event A,
we have E(b * j b * k ) = O(m −2 )
. Using standard arguments in density estimation theory (e.g. Wand and Jones (1994) page 21), and the fact that
E m j=1 (b * j ) 2 = O(m −1 h −1 σ ), we have E {f * 1,σ (x) − f 1,σ (x)} 2 dx = O (mh σ h x ) −1 + h 4
x . Under condition (C2) and (C3) the RHS → 0, establishing Step (a).
C.2 Proof of Step (b)
Let b j = φ hσ (σ − σ j )P (θ j = 1|x j , σ j ) m i=1 φ hσ (σ − σ i )P (θ i = 1|x i , σ i )
. Then
f 1,σ (x) −f * 1,σ (x) 2 = m j=1 (b * j − b j ) 2 φ 2 hxσ j (x − x j ) + (j,k):j =k (b * j − b j )(b * k − b k )φ hxσ j (x − x j )φ hxσ k (x − x k ).(C.6) We first bound E(b * j − b j ) 2 . Write E(b * j − b j ) 2 = {E(b * j − b j )} 2 + V ar(b * j − b j ). It is clear that E(b * j ) and E(b j ) are both O(m −1 ). Hence {E(b * j − b j )} 2 = O(m −2 ). Next consider V ar(b * j − b j ) = V ar(b * j ) + V ar(b j ) − 2Cov(b * j , b j ). We have by condition (C3) V ar(b * j ) = V ar I(θ j = 1)φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) ≤ E(b * j ) 2 = O(m −2 h −1 σ ). Similarly V ar(b j ) = O(m −2 h −1 σ ). It follows that Cov(b * j , b j ) = O(m −2 h −1 σ ). Therefore V ar(b * j − b j ) = O(m −2 h −1 σ ) and E(b * j − b j ) 2 = O(m −2 h −1 σ ). Using the fact that φ 2 hxσ j (x − x j )dx = O(h −1 x ), we have E m j=1 (b * j − b j ) 2 φ 2 hxσ j (x − x j )dx = O{(mh x h σ ) −1 } → 0. (C.7) Next we bound E{(b * j − b j )(b * k − b k )} for j = k. Consider the decomposition E{(b * j − b j )(b * k − b k )} = E(b * j − b j )E(b * k − b k ) + Cov(b * j − b j , b * k − b k ). (C.8) Our goal is to show that E{(b * j − b j )(b * k − b k )} = O(m −3 h −2 σ ) + O(m −4 h −4 σ ). It suffices to show E θ|σ,x θ|σ,x θ|σ,x {(b * j − b j )(b * k − b k )} = O(m −3 h −2 σ ) + O(m −4 h −4 σ ). (C.9)
Observe that V ar
1 mh −1 σ m i=1 φ hσ (σ − σ j )I(θ i = 1)|σ, x σ, x σ, x = O(m −1 ) and E θ|σ,x θ|σ,x θ|σ,x 1 mh −1 σ m i=1 φ hσ (σ − σ j )I(θ i = 1) = 1 mh −1 σ m i=1 φ hσ (σ − σ j )P (θ i = 1|σ i , x i ).
Applying Chebyshev's inequality,
1 mh −1 σ m i=1 φ hσ (σ − σ j )I(θ i = 1) − 1 mh −1 σ m i=1 φ hσ (σ − σ j )P (θ i = 1|σ i , x i ) p − → 0.
It follows that for any > 0,
P m i=1 φ hσ (σ − σ j )I(θ i = 1) − m i=1 φ hσ (σ − σ j )P (θ i = 1|σ i , x i ) < mh −1 σ → 1. Under A defined in (C.5), we have m i=1 φ hσ (σ − σ j )I(θ i = 1) > h −1 σ C 3 m for some C 3 , and P m i=1 φ hσ (σ − σ j )P (θ i = 1|σ i , x i ) < h −1 σ C 3 m → 0. (C.10)
The boundedness of b * j and b j and (C.10) imply that we only need to prove (C.9) on the event
A * = (x x x, σ σ σ) : m i=1 φ hσ (σ − σ j )P (θ i = 1|σ i , x i ) ≥ h −1 σ C 3 m . We shall consider E θ|σ,x θ|σ,x θ|σ,x (b * j − b j ) and Cov(b * j − b j , b * k − b k |σ σ σ, x x x) in turn. Write E θ|σ,x θ|σ,x θ|σ,x (b * j ) = E θ|σ,x θ|σ,x θ|σ,x I(θ j = 1)φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) = P (θ j = 1|x j , σ j )E θ|σ,x θ|σ,x θ|σ,x φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) + Cov θ j , φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) σ, x σ, x σ, x . Let Y = φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1)
. We state three lemmas that are proven in Section D.
Lemma 2 Under event
A * , we have E θ|σ,x θ|σ,x θ|σ,x (Y )−E θ|σ,x θ|σ,x θ|σ,x (Y |θ j = 1) = O(m −2 ) and E θ|σ,x θ|σ,x θ|σ,x (Y )− E θ|σ,x θ|σ,x θ|σ,x (Y |θ j = 0) = O(m −2 h −1 σ ).
Lemma 3 Under event A * , we have
E θ|σ,x θ|σ,x θ|σ,x φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) = φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )P (θ i = 1|x i , σ i ) + O(m −2 h −2 σ ).
Lemma 4 Under event
A * , we have Cov(b * j − b j , b * k − b k |σ σ σ, x x x) = O(m −3 h −2 σ ).
According to Lemma 2, we have
Cov θ j , φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) σ, x σ, x σ, x (C.11) = {P (θ j = 0|x j , σ j )P (θ j = 1|x j , σ j )}{y − E θ|σ,x θ|σ,x θ|σ,x (Y )}f Y |θ j =1,σ,x σ,x σ,x (y)dy − {1 − P (θ j = 1|x j , σ j )} 2 {y − E θ|σ,x θ|σ,x θ|σ,x (Y )}f Y |θ j =0,σ,x σ,x σ,x (y)dy = O(m −2 h −1 σ ).
Together with Lemma 3, we have
E θ|σ,x θ|σ,x θ|σ,x I(θ j = 1)φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )I(θ i = 1) = P (θ j = 1|x j , σ j )φ hσ (σ − σ j ) m i=1 φ hσ (σ − σ i )P (θ i = 1|x i , σ i ) + O(m −2 h −2 σ ). (C.12) It follows that E(b * j − b j ) = O(m −2 h −2 σ ) and E(b * j − b j )E(b * k − b k ) = O(m −4 h −4 σ ).
The decomposition (C.8) and Lemma 4 together imply E{
(b * j − b j )(b * k − b k )} = O(m −3 h −2 σ ) + O(m −4 h −4 σ ). It follows that E (j,k):j =k (b * j −b j )(b * k −b k )φ hxσ j (x−x j )φ hxσ j (x−x k )dx = O{(mh x h 2 σ ) −1 +O (mh σ ) −2 → 0. (C.13)
Combing (C.6), (C.7) and (C.13), we conclude that E {f 1,σ (x) −f * 1,σ (x)} 2 dx → 0.
C.3 Proof of Step (c)
Let q j = P (θ j = 1|σ j , x j ),q j =P (θ j = 1|σ j , x j ) = min (1 −π)f 0,σ j (x j ) f * σ j (x j ) , 1 andf 1,σ (x) = m j=1 φ hσ (σ − σ j )q j m i=1 φ hσ (σ − σ i )q i φ hxσ j (x − x j ).
Writeq j = q j + a j , then |a j | ≤ 1 and a j = o P (1). We have
E f 1,σ (x) −f 1,σ (x) 2 dx = O h −1 x m 2 E φ hσ (σ − σ j )q j m i=1 φ hσ (σ − σ i )q i − φ hσ (σ − σ j )q j m i=1 φ hσ (σ − σ i )q i 2 = O h −1 x h 2 σ Ea 2 j .
Next we explain why the last equality holds. Let
c i = φ hσ (σ − σ i )h σ . Then E φ hσ (σ − σ j )q j m i=1 φ hσ (σ − σ i )q i − φ hσ (σ − σ j )q j m i=1 φ hσ (σ − σ i )q i 2 = E a j m i=1 φ hσ (σ − σ i )q i − q j m i=1 φ hσ (σ − σ i )a i { m i=1 φ hσ (σ − σ i )q i }{ m i=1 φ hσ (σ − σ i )q i } 2 = h 2 σ 1 m 4 O E a j m i=1 c i q i − q j m i=1 c i a i 2 = h 2 σ m 4 O E m 2 a 2 j − 2ma j m i=1 a i + m i=1 a i 2 = h 2 σ m 2 O E a 2 j .
The last line holds by noting that
E (a j a i ) ≤ E(a 2 j )E(a 2 i ) = O{E(a 2 j )}. The next step is to bound E(a 2 j ). Note that a j = O f 0,σ j (x j ){f * σ j (x j ) − f σ j (x j )} f σ j (x j )f * σ j (x j )
.
By the construction ofq j , we havef σ j (x j ) ≥ (1 −π)f 0,σ j (x j ). Hence
a j = O 1 −f * σ j (x j ) f σ j (x j ) and E(a 2 j ) = O E 1 −f * σ j (x j ) f σ j (x j ) 2 . Let K j = −σ j √ δ √ log m − M, σ j √ δ √ log m + M .
By the Gaussian tail bound, P {x j ∈ K j } = O(m −δ/2 ). By the boundedness of a 2 j and the fact that h −1 x h 2 σ m −δ/2 → 0 (Condition (C3 )),
we only need to consider E
1 − 2f * σ j (x j ) f σ j (x j ) + f * σ j (x j ) f σ j (x j ) 2 x j for x j ∈ K j . Let f σ (x j ) = φ σ (x) {(1 − π)δ 0 (x j − x) + πg µ (x j − x)} dx.
Define a jacknifed version off * ,(j) σ j that is formed without the pair (σ j , x j ). It follows that
E{f * ,(j) σ j (x j )|x j } = φ σ 2 +h 2 x σ 2 j (x) {(1 − π)δ 0 (x j − x) + πg µ (x j − x)} g σ (σ j )dσ j dx.
By the intermediate value theorem and Condition (C1),
E{f * ,(j) σ j (x j )|x j } = φ √ σ 2 +h 2 x c (x) {(1 − π)δ 0 (x j − x) + πg µ (x j − x)} dx
for some constant c. Next consider the ratio
E{f * ,(j) σ j (x j )|x j } f σ j (x j ) = φ √ σ 2 +h 2 x c (x) {(1 − π)δ 0 (x j − x) + πg µ (x j − x)} dx φ σ (x) {(1 − π)δ 0 (x j − x) + πg µ (x j − x)} dx . By Condition (C1), supp(g µ ) ∈ (−M, M ) with M < ∞, we have inf x j −M <x<x j +M φ √ σ 2 +h 2 x c (x) φ σ (x) ≤ E{f * ,(j) σ j (x j )|x j } f σ j (x j ) ≤ sup x j −M <x<x j +M φ √ σ 2 +h 2 x c (x) φ σ (x)
.
Note that x j ∈ (−σ j √ δ √ log m − M, σ j √ δ √ log m + M ).
The above infimum and supremum are taken over
x ∈ K = (−σ j √ δ √ log m − 2M, σ j √ δ √ log m + 2M ). Using Taylor expansion φ √ σ 2 +h 2 x c (x) φ σ (x) = σ σ 2 + h 2 x c 1 + ∞ k=1 1 k! h 2 x cx 2 2(σ 2 + h 2 x c) k , we have inf x∈K φ √ σ 2 +h 2 x c (x) φ σ (x) = σ σ 2 + h 2 x c = 1 + O(h 2 x ). Similarly, sup x∈K σ σ 2 + h 2 x c 1 + ∞ k=1 1 k! h 2 x cx 2 2(σ 2 + h 2 x c) k = 1 + O(h 2 x ) + O sup ∞ k=1 1 k! h 2 x cx 2 2(σ 2 + h 2 x c) k .
It follows that
E{f * ,(j)
σ j (x j )|x j } f σ j (x j ) = 1 + O(h 2 x ) + O sup ∞ k=1 1 k! h 2 x cx 2 2(σ 2 + h 2 x c) k , (C.14) 1 − 2 E{f * ,(j) σ j (x j )|x j } f σ j (x j ) = −1 + O(h 2 x ) + O sup ∞ k=1 1 k! h 2 x cx 2 2(σ 2 + h 2 x c) k . (C.15) Next consider E f * ,(j) σ j (x j ) f σ j (x j ) x j 2 = E{f * ,(j) σ j (x j )|x j } f σ j (x j ) 2 + V ar f * ,(j) σ j (x j ) f σ j (x j ) x j .
By the same computation on page 21 of Wand and Jones (1994),
V ar f * ,(j) σ j (x j ) f σ j (x j ) x j = O (mh x ) −1 f σ j (x j ) −1 + o (mh x ) −1 f σ j (x j ) −2 .
Since x j ∈ K j , f σ j (x j ) ≥ C 3 m −δ/2 for some constant C 3 , together with Condition (C3 ), we (1). It follows from (C.14) and (C.15) that
have h −1 x V ar f * ,(j) σ j (x j ) f σ j (x j ) x j = oh −1 x − 2h −1 x E f * ,(j) σ j (x j ) f σ j (x j ) x j + h −1 x E f * ,(j) σ j (x j ) f σ j (x j ) x j 2 = O h x + sup ∞ k=1 1 k! h 2k−1 x c k x 2k 2 k (σ 2 + h 2 x c) k + o(1). (C.16)
By condition (C2) and the range of x, the RHS goes to 0.
Let S j = m i=1 φ hσ (σ j − σ i ) and S − j = m i =j φ hσ (σ j − σ i ). Some algebra showsf * σ j (x j ) = S − j S jf * ,(j) σ j (x j ) + 1 2S j πh σ h x σ j
. We use the fact that f σ j (x j ) ≥ C 3 m −δ/2 for some constant C 3
and Condition (C3) to claim that on A * ,
h −1 x E{f * σ j (x j )|x j } f σ j (x j ) = h −1 x E{f * ,(j) σ j (x j )|x j } f σ j (x j ) + o(1). (C.17)
Similar computation shows that
h −1 x E f * σ j (x j ) f σ j (x j ) x j 2 = h −1 x E f * ,(j) σ j (x j ) f σ j (x j ) x j 2 + o(1U i − U i 2 = T i − T i 2 I T i ≤ t, T i ≤ t + T i − α 2 I T i ≤ t, T i > t + (T i − α) 2 I T i > t, T i ≤ t .
Denote the three sums on the D.4 Proof of lemma 4
Consider b * j = φ hσ (σ − σ j )I(θ j = 1) m i=1 φ hσ (σ − σ i )I(θ i = 1) defined in Section C.1. Letb j = θ j m i=1 θ i . By Condition (C1 ), Cov(b * j , b * k |σ σ σ, x x x) = O{h −2 σ Cov(b j ,b k |σ σ σ, x x x)}. Note that Cov(b j ,b k |σ σ σ, x x x) = E θ|σ,x θ|σ,x θ|σ,x (b jbk ) − E θ|σ,x θ|σ,x θ|σ,x (b j )E θ|σ,x θ|σ,x θ|σ,x (b k ).
Using similar argument as in the proof for (C.12), we
have E θ|σ,x θ|σ,x θ|σ,x (b j ) = P (θ j = 1|σ σ σ, x x x) m i=1 P (θ i = 1|σ σ σ, x x x) +O(m −2 ) and E θ|σ,x θ|σ,x θ|σ,x (b k ) = P (θ k = 1|σ σ σ, x x x) m i=1 P (θ i = 1|σ σ σ, x x x) +O(m −2 ). It follows that E θ|σ,x θ|σ,x θ|σ,x (b j )E θ|σ,x θ|σ,x θ|σ,x (b k ) = P (θ j = 1|σ σ σ, x x x)
E Supplementary Numerical Results
E.1 Non-Gaussian alternative
We generate σ i uniformly from [0.5, σ max ], and generate X i according to the following model:
X i |σ i iid ∼ (1 − π)N (0, σ 2 i ) + πN (µ i , σ 2 i ), µ i iid ∼ 0.5N (−1.5, 0.1 2 ) + 0.5N (2, 0.1 2 ).
In the first setting, we fix σ max = 2 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 1.5 to 2.5. Five methods are compared: the ideal full data oracle procedure (OR), the z-value oracle procedure of (Sun and Cai, 2007) (ZOR), the Benjamini-Hochberg procedure (BH), AdaPT (Lei and Fithian, 2018) (AdaPT), and the proposed data-driven HART procedure (DD). The nominal FDR level is set to α = 0.1.
For each setting, the number of tests is m = 20, 000. Each simulation is also run over 100
repetitions. The results are summarized in Figure 8. All methods can control the FDR at the nominal level with BH slightly conservative. DD performs almost as well as OR. The ordering information from σ i seems to help AdaPT in some cases, but in other cases, it causes AdaPT to underperform BH. There is a clear power gap between DD and ZOR.
E.2 Unknown σ i
This section investigates the robustness of our method when σ i is unknown. In some applications, the exact value of σ i is unknown but can be estimated. For this simulation, we independently generate 200 copies of X i using the following model:
X i |σ i iid ∼ (1 − π)N (0, σ 2 i ) + πN (2/ √ 200, σ 2 i ), σ i iid ∼ U [0.5, σ max ].
For fair comparison we replace ZOR by AZ, the data driven version of ZOR described in (Sun and Cai, 2007 Figure 8: Comparison when σ i is generated from a uniform distribution and µ i is generated from a Gaussian mixture. We vary π in the top row and σ max in the bottom row. All methods control the FDR at the nominal level. DD performs almost as well as OR and has a significant power advantage over ZOR,BH and AdaPT.
the sample size is relatively large, normal distribution serves as a good approximation. The number of tests is m = 20, 000. Each simulation is run over 100 repetitions. In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5. The results are summarized in Figure 9.
We can see that all data-driven methods have been adversely affected. However, the inflation of the FDR for DD is less severe compared to the other three data-driven methods.
The gap in power between OR and DD becomes larger but the power advantage of DD over other methods is maintained. Figure 9: Comparison when σ i is unknown. We vary π in the top row and σ max in the bottom row. All data-driven methods are adversely affected. But the effect on DD is the smallest.
E.3 Weak dependence
We investigate the robustness of our method under weak dependence. We consider two weak dependence models:
Case 1 : We use the following model: Here M 11 is a 4000 × 4000 matrix, the (i, i) entry is σ 2 i , the (i, i + 1) and (i + 1, i) entries are 0.5σ i σ i+1 , the (i, i + 2) and (i + 2, i) entries are 0.4σ i σ i+2 . The rest of the entries are 0. M 22 is a 16000 × 16000 diagonal matrix, with the (i, i) entry being σ 2 i+4000 . In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5. The results are summarized in Figure 10. The number of tests is m = 20, 000. We can see that the weak dependence has little effect on the pattern. This demonstrates the robustness of DD.
Case 2 : We use the following model: Here M 11 is a 4000 × 4000 matrix, the (i, j) entry of M 11 is 0.5 |i−j| σ i σ j . M 22 is a 16000 × 16000 diagonal matrix with diagonal being (σ 2 4001 , ..., σ 2 20000 ). In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5. The results are summarized in Figure 11.
We can see again, there is no noticeable difference in pattern from the independent setting. This shows the robustness of DD under positive dependence. This seems to be consistent with existing results in the literature.
E.4 Non-Gaussian noise
We study the performance of our method when the noise follows a heavy-tailed distribution.
For this simulation, we independently generate 200 copies of X i using the following model:
X i = µ i + σ i i , µ i iid ∼ (1 − π)δ 0 (·) + πδ 2/ √ 200 (·), σ i iid ∼ U [0, σ max ], i iid ∼ t 5 .
We use the sample standard deviation of x i (denotedσ i ) as an estimate of σ i for all five methods. We then apply the testing procedures to the pairs ( √ 200x i ,σ i ). The number of tests is m = 20, 000. Each simulation is run over 100 repetitions. Since the precise distribution of x i /σ i is hard to compute, we replace ZOR by AZ. Note that in this case, the model is mis-specified even for OR. But OR has access to the distribution of µ i and π while other data-driven methods do not. In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5.
The results are summarized in Figure 12. We can see that the pattern is similar to the Gaussian-noise case.
E.5 Unknown null distribution
We study the performance of our method when the z-values do not follow a standard normal distribution under the null hypothesis. For this simulation, we use the following model:
Z i iid ∼ N (0, 0.8 2 ), σ i iid ∼ U [0, σ max ], X i = Z i σ i + µ i , µ i iid ∼
(1 − π)δ 0 (·) + πδ 2 (·).
For the data-driven methods, we estimate the null distribution of z i 's using the method described in Jin and Cai (2007). Let σ 0 be the estimated variance of Z i . For DD, f 0,σ j is now the density function of N (0, σ 2 0 σ 2 j ). In the first setting, we fix σ max = 4 and vary π from 0.05 to 0.15. In the second setting, we fix π = 0.1 and vary σ max from 3.5 to 4.5. The results are summarized in Figure 13.
We can see that the variance of the estimator of null variance has a noticeable effect on all data-driven methods. But DD still performs the best. Figure 13: Comparison when the null distribution is estimated. We can see that all the data-driven methods show some instability: this is due to the variance of the estimator of null variance. But DD still out-performs other data-driven methods
Figure 1 :
1Left: Rejection regions for the p-value approach (black line), z-value approach (red line)
Figure 2 :
2makes this clear. Here we have plotted (on a log scale) the relative proportions of alternative vs null hypotheses for different Z and σ. Blue corresponds to lower ratios and purple to higher ratios. The solid black line represents equal fractions of null and alternative, while the dashed line corresponds to three times as many alternative as null. Clearly, for the same z-value, alternative hypotheses are relatively more common for low σ values. Notice how closely the shape of the dashed line maps the Plots of the density functions of Z under the null hypothesis (black solid) and alternative hypothesis (red dashed) for different values of σ. The blue line represents an observation at Z = 2.green rejection boundary in the left hand plot, which indicates that the full data method is correctly capturing the regions with most alternative hypotheses. By contrast, the p-value and z-value methods fail to correctly adjust for different values of σ.
Figure 2
2provides one further way to understand the effect of standardizing the data.
density of the p-values, respectively. Models 3.2 and 3.3 provide a powerful and flexible framework for large-scale inference and have been used in a range of related problems such as signal detection, sparsity estimation and multiple testing [e.g.Efron et al.
and the average power is estimated as the average proportion of true positives that are correctly identified, m i=1 (θ i δ i )/(mp), both over the number of repetitions. The results for differing values of π and σ max are respectively displayed in the first and second rows of
Figure 3 .
3Figure 3.
Figure 5 :
5Left: histogram of z-values: the estimated empirical null N (0, 1.3 2 ) (green line) seems to provide a better fit to the data compared to the theoretical null N (0, 1) (red line). Middle: histogram of original p-values. Right: histogram of estimated p-values based on the empirical null. The z-value histogram suggests that the theoretical null is inappropriate (too narrow, leading to too many rejections). The use of an empirical null corrects the non-uniformity of the histogram of the p-values.
N
(0, 1.30 2 ) [seeEfron (2004b) for more details]. The new p-values are then converted from the z-values based on the estimated empirical null:P * i = 2Φ * (−2|Z i |),where Φ * is the CDF of a N (0, 1.30 2 ) variable. We can see fromFigure 5that the empirical null (green curve) provides a better fit to the histogram of z-values. Another evidence for the suitability of the empirical null approach is that the histogram of the estimated p-values looks closer to uniform compared to that of original p-values. The uniformity assumption is crucial for ensuring the validity of p-value based procedures.
Figure 6 :
6Histogram of S i (left), scatter plot of (Z i , S i ) (right)
Figure 7 :
7Scatter plot of rejected hypotheses by each method. Green: BH, blue: AZ, red: HART. AZ and BH reject every hypothesis to the right of the dashed line. The rejection region for HART depends on both z and σ.
structed auxiliary sequences or gleaned from secondary data sources such as prior studies, domain-specific knowledge, structural constraints and external covariates. The recent works by Xia et al. (2019), Li and Barber (2019) and Cai et al. (2019) have focused on utilizing side information that encodes the sparsity structure. By contrast, our work investigates the impact of the alternative distribution, showing that incorporating σ i can be extremely useful for improving the ranking and hence the power in multiple testing 5 .
is possible, for example, in applications where observations from the alternative have larger variances compared to those from the null. An interesting, but challenging, direction for future research is to develop methodologies that can simultaneously incorporate both the sparsity and heterocedasticity structures into inference. Third, the HART-type methodology can only handle one covariate sequence {σ i : 1 ≤ i ≤ m}. It would be of great interest to develop new methodologies and principles for information pooling for multiple testing with several covariate sequences. Finally, our work has assumed that σ i are known in order to illustrate the key message (i.e. the impact of alternative distribution on the power of FDR analyses). Although this is a common practice, it is desirable to carefully investigate the impact of estimating σ i on the accuracy and stability of large-scale inference.Tusher, V. G., Tibshirani, R., and Chu, G. (2001). Significance analysis of microarrays applied to the ionizing radiation response. Proc. Natl. Acad. Sci. U. S. A., M. P. and Jones, M. C. (1994). Kernel Smoothing, volume 60 of Chapman and Hall CRC Monographs on Statistics and Applied Probability. Chapman and Hall CRC. Weinstein, A., Ma, Z., Brown, L. D., and Zhang, C.-H. (2018). Group-linear empirical Bayes estimates for a heteroscedastic normal mean. Journal of the American Statistical Association, 113:698-710. Xia, Y., Cai, T. T., and Sun, W. (2019). Gap: A general framework for information pooling in two-sample sparse inference. Journal of the American Statistical Association, 0(just-accepted):to appear. Xie, X., Kou, S., and Brown, L. D. (2012). Sure estimates for a heteroscedastic hierarchical model. Journal of the American Statistical Association, 107(500):1465-1479. Zhao, Z., De Stefani, L., Zgraggen, E., Binnig, C., Upfal, E., and Kraska, T. (2017). Controlling false discoveries during interactive data exploration. In Proceedings of the 2017 ACM International Conference on Management of Data, SIGMOD '17, pages 527-540, New York, NY, USA. ACM.
0 .
0Summing over the m terms and taking the expectation yield (B.3). Combining (B.2) and (B.3), we obtain
FDR(δ δ δ dd ) = FDR(δ δ δ f ull ) + o(1) = α + o(1), we only need to show mFDR(δ δ δ dd ) = mFDR(δ δ δ f ull ) + o(1). The result then follows from the asymptotic equivalence of FDR and mFDR, which was proven inCai et al. (2019).
∼Figure 10 :
10(1 − π)δ 0 (·) + πδ 2 (·), σ i iid ∼ U [0, σ max ], x x x ∼ N (µ µ µ, Comparison under weak dependence case 1. The dependence structure has little effect on all the methods. The pattern is almost the same as in the independent case.
∼Figure 11 :
11(1 − π)δ 0 (·) + πδ 2 (·), σ i iid ∼ U [0, σ max ], x x x ∼ N (µ µ µ, Comparison under weak dependence case 2. Again, the dependence structure has little effect on all the methods. The pattern is almost the same as in the independent case.
Figure 12 :
12Comparison when the noise is non-Gaussian. The non-Gaussian noise has little effect on the overall pattern.
1. Section 4.2 presents results for the general setting where σ i comes from a continuous density function. In Section 4.3, we further investigate the effect of heterogeneity under a mixture model where σ i takes on one of two distinct values. Simulation results for additional settings, including a non-Guassian alternative, unknown σ i , weak dependence structure, non-Gaussian noise, and estimated empirical null, are provided in Section E of the Supplementary Material.
Table 1 :
1Numbers of genes (% of total) that are selected by each method.α-level BH
AZ
HART
0.1
8 (0.07%) 25 (0.2%) 122 (1%)
0 andStep (c) is established.T i p → T i then follows from Lemma A.1 and Lemma A.2 inSun and Cai (2007).Using the definitions of U i and U i , we can show that).
(C.18)
(C.16), (C.17) and (C.18) together implies h −1
x h 2
σ E{a 2
j |x j } → 0. Hence E
f 1,σ (x) −f 1,σ (x)
2
dx →
D Proof of Lemmas
D.1 Proof of lemma 1
). We use the sample standard deviation of x i (denoted s i ) as an estimate of σ i for DD, AZ, BH and AdaPT. OR has access to the exact value of σ i . We then apply the testing procedures to the pairs ( √ 200x i , s i ). The z-value is computed as √ 200x i s i and the p-value is computed using 1 2 {1 − Φ(|z i |)} where Φ is the CDF for standard normal distribution. Strictly speaking the z-values should follow a t distribution, but since0.06
0.08
0.10
0.12
0.14
0.00
0.05
0.10
0.15
0.20
a)FDR Comparison varing π
π
FDR
BH
ZOR
OR
DD
AdaPT
0.06
0.08
0.10
0.12
0.14
0.06 0.08 0.10 0.12 0.14 0.16
b)Power Comparison varing π
π
Power
1.6
1.8
2.0
2.2
2.4
0.00
0.05
0.10
0.15
0.20
c)FDR Comparison varing σ max
σ max
FDR
1.6
1.8
2.0
2.2
2.4
0.06
0.10
0.14
0.18
d)Power Comparison varing σ max
σ max
Power
The p-value method will also reject for large negative values of Z but, to keep the figure readable, we have not plotted that region.
The deconvoluting kernel method has an extremely slow convergence rate. Our numerical studies show that the method inSun and McLain (2012) only works for composite nulls where the uncertainties in estimation can be smoothed out over an interval[−a0, a0]. However, the deconvoluting method is highly unstable and does not work well when testing simple nulls H0,i : µi = 0. Our numerical results show that the two-step method in Section 3.2 works much better.
Note that the oracle statistic P (θi = 0|Xi, σi) is equivalent to P (θi = 0|Zi, σi) since the pairs (Xi, σi) and (Zi, σi) contain the same amount of information. We use the pairs (Xi, σi) in the next formula just to facilitate the estimation methodology.
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By Proposition 2, E(I) = o(1). I Rhs As, I I , ; Consider P T I ≤ T, T I > T ≤ P T I ≤ T, T I ∈ (t, T + Ε) + P T I ≤ T, T I ≥ T + Ε, Let ε > 0RHS as I, II, and III respectively. By Proposition 2, E(I) = o(1). Let ε > 0. Consider P T i ≤ t, T i > t ≤ P T i ≤ t, T i ∈ (t, t + ε) + P T i ≤ t, T i ≥ t + ε
| [] |
[
"Spintronic Emitters for Super-Resolution in THz-Spectral Imaging",
"Spintronic Emitters for Super-Resolution in THz-Spectral Imaging"
] | [
"Finn-Frederik Stiewe \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Tristan Winkel \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Yuta Sasaki \nDepartment of Applied Physics\nGraduate school of Engineering\nTohoku University\nSendaiJapan\n\nWPI Advanced Institute for Materials Research (AIMR)\nTohoku University\nSendaiJapan\n",
"Tobias Tubandt \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Tobias Kleinke \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Christian Denker \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Ulrike Martens \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Nina Meyer \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Tahereh Sadat Parvini \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Shigemi Mizukami \nWPI Advanced Institute for Materials Research (AIMR)\nTohoku University\nSendaiJapan\n\nCenter for Spintronic Research Network (CSRN)\nTohoku University\nSendaiJapan\n\nCenter for Science and Innovation in Spintronics (CSIS)\nTohoku University\nSendaiJapan\n",
"Jakob Walowski \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n",
"Markus Münzenberg \nInstitut für Physik\nUniversität Greifswald\nGreifswaldGermany\n"
] | [
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Department of Applied Physics\nGraduate school of Engineering\nTohoku University\nSendaiJapan",
"WPI Advanced Institute for Materials Research (AIMR)\nTohoku University\nSendaiJapan",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"WPI Advanced Institute for Materials Research (AIMR)\nTohoku University\nSendaiJapan",
"Center for Spintronic Research Network (CSRN)\nTohoku University\nSendaiJapan",
"Center for Science and Innovation in Spintronics (CSIS)\nTohoku University\nSendaiJapan",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany",
"Institut für Physik\nUniversität Greifswald\nGreifswaldGermany"
] | [] | THz-spectroscopy is an attractive imaging tool for scientific research, especially in life science, offering non-destructive interaction with matter due to its low photon energies. However, wavelengths above 100 µm principally limit its spatial resolution in the far-field by diffraction to this regime, making it not sufficient to image biological cells in the micrometer scale. Therefore, superresolution imaging techniques are required to overcome this restriction. Near-field-imaging using spintronic emitters offers the most feasible approach because of its simplicity and potential for wideranging applications. In our study, we investigate THz-radiation generated by fs-laser-pulses in CoFeB/Pt heterostructures, based on spin currents, detected by commercial LT-GaAs Auston switches. The spatial resolution is evaluated applying a 2D scanning technique with motorized stages allowing scanning steps in the sub-micrometer range. By applying near-field imaging we can increase the spatial resolution to the dimensions of the laser spot size in the micrometer scale. For this purpose, the spintronic emitter is directly evaporated on a gold test pattern separated by a 300 nm spacer layer. Moving these structures with respect to the femtosecond laser spot which generates the THz radiation allows for resolution determination using the knife-edge method. We observe a fullwidth half-maximum THz beam diameter of 4.9 ± 0.4 µm at 1 THz. The possibility to deposit spintronic emitter heterostructures on simple glass substrates makes them an interesting candidate for nearfield imaging for a large number of applications. | 10.1063/5.0076880 | [
"https://arxiv.org/pdf/2111.05023v1.pdf"
] | 243,860,883 | 2111.05023 | 7d6378ea5530d52053be77df098af112ca8a3c2c |
Spintronic Emitters for Super-Resolution in THz-Spectral Imaging
Finn-Frederik Stiewe
Institut für Physik
Universität Greifswald
GreifswaldGermany
Tristan Winkel
Institut für Physik
Universität Greifswald
GreifswaldGermany
Yuta Sasaki
Department of Applied Physics
Graduate school of Engineering
Tohoku University
SendaiJapan
WPI Advanced Institute for Materials Research (AIMR)
Tohoku University
SendaiJapan
Tobias Tubandt
Institut für Physik
Universität Greifswald
GreifswaldGermany
Tobias Kleinke
Institut für Physik
Universität Greifswald
GreifswaldGermany
Christian Denker
Institut für Physik
Universität Greifswald
GreifswaldGermany
Ulrike Martens
Institut für Physik
Universität Greifswald
GreifswaldGermany
Nina Meyer
Institut für Physik
Universität Greifswald
GreifswaldGermany
Tahereh Sadat Parvini
Institut für Physik
Universität Greifswald
GreifswaldGermany
Shigemi Mizukami
WPI Advanced Institute for Materials Research (AIMR)
Tohoku University
SendaiJapan
Center for Spintronic Research Network (CSRN)
Tohoku University
SendaiJapan
Center for Science and Innovation in Spintronics (CSIS)
Tohoku University
SendaiJapan
Jakob Walowski
Institut für Physik
Universität Greifswald
GreifswaldGermany
Markus Münzenberg
Institut für Physik
Universität Greifswald
GreifswaldGermany
Spintronic Emitters for Super-Resolution in THz-Spectral Imaging
(Dated: October 28, 2021)1
THz-spectroscopy is an attractive imaging tool for scientific research, especially in life science, offering non-destructive interaction with matter due to its low photon energies. However, wavelengths above 100 µm principally limit its spatial resolution in the far-field by diffraction to this regime, making it not sufficient to image biological cells in the micrometer scale. Therefore, superresolution imaging techniques are required to overcome this restriction. Near-field-imaging using spintronic emitters offers the most feasible approach because of its simplicity and potential for wideranging applications. In our study, we investigate THz-radiation generated by fs-laser-pulses in CoFeB/Pt heterostructures, based on spin currents, detected by commercial LT-GaAs Auston switches. The spatial resolution is evaluated applying a 2D scanning technique with motorized stages allowing scanning steps in the sub-micrometer range. By applying near-field imaging we can increase the spatial resolution to the dimensions of the laser spot size in the micrometer scale. For this purpose, the spintronic emitter is directly evaporated on a gold test pattern separated by a 300 nm spacer layer. Moving these structures with respect to the femtosecond laser spot which generates the THz radiation allows for resolution determination using the knife-edge method. We observe a fullwidth half-maximum THz beam diameter of 4.9 ± 0.4 µm at 1 THz. The possibility to deposit spintronic emitter heterostructures on simple glass substrates makes them an interesting candidate for nearfield imaging for a large number of applications.
Terahertz (0.6 − 10 THz) radiation provides a great potential for biomedical applications such as biosensing for cancer imaging due to its low non-ionizing photon energies and thus a nondestructive interaction of THz radiation with tissue [1] [2] [3] [4]. In addition, THz spectroscopy determines specific spectral fingerprints for numerous materials which are not interfering with other spectral ranges and especially water-based materials show a strong absorption [5] [6]. This imaging technique is already used in airports, to distinguish macroscopically sized materials from human cell tissue. Increasing the imaging resolution to the micrometer scale will provide new insights into biological cells with diameters from 1 μm to 100 μm [7] or even the determination of impurities and intoxicants enriched inside them. Further improvement will enable the observation of genetic information (DNA) [8] [9] [10].
In accordance with Abbe´s diffraction limit, the spatial resolution of microscopy techniques is defined by half of the wavelength which for THz radiation yields to a few hundred micrometers. The simplest approach to overcome this limitation is near-field imaging [11] [12] [13]. For this purpose, the light source spot needs to be limited by an aperture and simultaneously, the THz radiation needs to be generated in direct vicinity to the investigated structure. Where both the aperture and the distance to the structure need to be much smaller than the generated wavelength. In principle, the dimensions of these two factors limit the possible resolution. Several studies have already demonstrated the huge potential of THz near-field imaging for the investigation of optical near-fields close to metallic structures, such as small apertures [14] [15] [16] [17] [18], or microresonators and metamaterials [19] [20]. In those investigations, THz pulses with large beam dimensions are generated in the far-field and propagate through apertures with subwavelength dimensions to achieve reduced beam diameters. The diffracted waves need to be detected behind those apertures at distances smaller than the wavelengths, before diffraction effects occur [21] [22] [23] [24]. Further, there is development in near field imaging by illuminating whole arrays and reconstructing the images by calculating the signal correlation [25]. In this regard, spintronic THz emitters exhibit several possibilities for implementation with biological samples in combination with near field imaging. They are low cost in production and can be mass produced by depositing the bilayer heterostructures on a huge variety of glass substrates. Besides this, they offer polarization control and therefore the choice of materials to be examined can be extended to magnetic particles e.g., in functionalized cell tissue. In this study, we investigate spintronic THz emitters which can be placed in direct proximity to the imaged structure. THz radiation is generated by femtosecond laser pulses focused down to the micrometer scale, and thus confining the spatial expansion of the THz source spot to those dimensions. The specially designed and lithographically prepared test structure consists of an adjacent patterned gold film, separated from the emitter by an insulating spacer layer. The THz pulse generation in direct vicinity of the imaged object in general permits a near-field independent detection because the imaging itself is not affected by diffraction effects. By systematically moving the sample, a THz pulse is generated, and the gold pattern is imaged. The results confirm a high near-field spatial resolution of 4.9 ± 0.4 µm for the frequency of 1 THz, which is in the range of the laser spot size. With this approach, the capability for near-field imaging at high spatial resolutions overcomes the Abbe limit by 30 times [26] [27] [28]. a) b) Figure 1: a) Scheme of the optical setup. A Ti:sapphire laser system (central wavelength 810 nm, repetition rate 80 MHz, pulse duration 40 fs) is used as a femtosecond optical source. The laser is split into two parts, a pump beam and a probe beam. The pump beam is guided to the spintronic emitter (sample) which emits the THz-radiation. It is modulated by a chopper with a frequency of 1.5 kHz to improve the signal-to-noise ratio with a lock-in amplifier. The lock-in amplifier is connected to the detector, an Auston switch (photoconductive antenna (PCA)). The lens in front of the spintronic emitter confines the pump beam to the micrometer range. The probe beam is guided through a variable path for probe beam delay adjustment and focused onto the backside of the PCA. For zero delay between the probe laser pulse exciting free charge carriers and the THz-pulse generating an electric field, the resulting electric current in the PCA yields the measurement signal. b) Schematic of sample layout together with principle of spintronic terahertz emitter. A spin current is excited by the femtosecond laser beam in the non-magnetic/ ferromagnetic (NM/FM) heterostructure, and then transient charge currents are generated by the inverse spin Hall effect (ISHE), leading to THz emission out of the structure.
The schematic of the optical setup used for near-field THz imaging is shown in Figure 1a). A Ti:Sa laser (Coherent Vitara) with 40 fs pulse duration at 810 nm central wavelength and a repetition rate of 80 MHz is used as an optical light source for THz pulse generation and stroboscopic detection. The laser beam is split into a pump and a probe beam. The intense pump beam is guided to the emitter to generate THz radiation. The probe beam is directed through an adjustable temporal delay, which allows for systematic THz pulse reconstruction in the detection scheme. The THz emitter is a spintronic heterostructure consisting of a ferromagnetic FM and a heavy metal non-magnetic layer NM [29] [30] [31]. Upon laser excitation in the FM hot electrons are generated and propagate towards the NM layer (zdirection). The majority electrons experience less scattering than the minority electrons and thus have longer lifetimes in the excited state [32]. This ensures, a spin-polarized current s arriving in the NM layer. The high spin polarization leads to a motion deflection perpendicular to the propagation direction caused by the inverse Spin-Hall-Effect (ISHE) of the electrons in the NM layer [29] [33] [34] [35]. After cooling down, by exchanging energy with the lattice, the electrons relax back to their original positions. The process induces a charge current pulse An Auston switch (photo-conductive antenna (PCA)) is used as a detector for the THzradiation. The femtosecond laser pulse excites electrons in the PCA gap shorting it. The THz pulse electric field arriving at the gap sets a potential for the excited electrons generating a current and thus, the detection signal. The current direction is set by the electric field oscillation.
C = Φ SH s × | | , where Φ SH
For spin polarization enhancement, the sample is placed in a saturating magnetic field of 10 mT. Its direction determines the phase and polarization of the THz field, which is perpendicular to the direction of the magnetic field. Thus, the THz field polarization is incorporated into the experiment and completely controllable by the magnetic field [29]. The 2D scanning capability is implemented by two motorized μm-stages, moving the sample horizontally and vertically (x-and y-direction) with a minimum stepsize of 200 nm perpendicular to the laser and the THz beam propagation in z-direction. The frequencies contained in the measured THz pulses are extracted using fast Fourier transformation (FFT) [41].
As schematically depicted in figure 1b), the sample used to study THz near-field imaging consists of two different units fabricated in two steps. The first unit is a spintronic emitter (Co 40 Fe 40 B 20 (2 nm)/ Pt (2 nm)). The second unit is a Cr (5 nm)/Au (100 nm) test pattern layer for imaging and resolution determination. Both are separated by a SiO 2 (300 nm) spacer layer, which is much thinner than the THz wavelengths. This layout ensures near-field imaging and at the same time electric insulation between the emitter and Au layer. In this construction a constant distance between both units throughout the whole sample provides the necessary condition for identical THz pulses and propagation path at each position. The first unit is fabricated in the first step, where the CoFeB layer is magnetron sputtered to establish a homogeneously smooth interface with the e-beam evaporated heavy metal Pt layer. For an equidistant separation of both units, the SiO 2 layer is prepared by e-beam evaporation within the first preparation step. The imaging pattern is created lithographically in the second preparation step and subsequent deposition of the Cr and Au layers, where Cr serves as an adhesive layer for the Au. Using FFT, the included frequencies are extracted from each spectrum. The extracted frequencies are normed to the electric field amplitude of the 1 THz contribution in the signal (red), whereby, the 0.5 THz and the 1.5 THz signals (white) are generated with about half the amplitude of the 1 THz signal. For all three THz frequencies, the slit structure is clearly imaged, with the Au layer completely blocking all THz radiation (blue). In the next step, the resolution is determined using the Rayleigh criterion. According to this, considering Airy disk shape light distribution, it is possible to distinguish between two equal point sources if they are separated by a distance equal to the length between the center and the first minimum. That means, the maximum of the second point source must be located in the minimum of the first Airy disk. In this case, the amplitude in the overlap region drops down to 73.5% of both sources maximum amplitude providing a contrast of 15% and thus discriminability [42]. In another view, the fullwidth half-maximum (FWHM) diameter of the light source determines the lower limit for the resolution. In our approach the emitted THz pulses have a Gaussian amplitude distribution around the propagation axis, originating from the shape of the exciting laser beam. Therefore, the FWHM diameter of the emitted THz waves in the near-field limit determines the possible resolution. In figure 2c) the amplitudes extracted at the cross-section along the white lines in figure 2b) show multiple transitions between the THz radiation blocking and transmitting areas. This corresponds to knife-edge measurements, which are a valid method to determine the spot size [43] [44] [45]. To support the experimental approach, the THz beam broadening during propagation in the intermediate distance between THz generator and imaged structure, and its propagation towards the detector are investigated by simulation. In the first step, we study the beam expansion on the 1 = 300 nm path through the SiO 2 spacer layer, corresponding to the zdirection in the experiment. For this purpose, the wave equation derived from Maxwell equations is solved for a THz pulse propagating through the SiO 2 spacer layer, by iterating the time from = 0 ps in steps of = 10 as up to = 1 ps after generation
2 ⃑ 2 = 2 ⃑ 2 ⃑ .
Here, ⃑ is the THz beam electric field amplitude, = 0 is the phase velocity of light considering the medium refractive index and ⃑ denotes the spacial wave propagation. The used parameters are, a THz pulse with a duration of 250 fs, a Gaussian beam spot with a FWHM = 4.7 μm and a refractive index for THz wavelengths in SiO 2 of = 2 [46] [47]. at the point of generation and broadens insignificantly after the propagation distance = , as depicted in the extracted profiles in b).
The simulation results for the THz wave amplitude distribution within the SiO 2 spacer layer up to the imaged structure are illustrated in figure 3a), the snapshot is taken 500 fs after emission from the Pt layer. The propagation through the SiO 2 clearly shows, that the beam width increases only insignificantly from the emission point to the imaging pattern. The trivial amplitude broadening at propagation distances, 1 = 150 nm and 1 = 300 nm, relative to the point of generation can be seen in figure 3b) indicating a very slight broadening below. Gauss fits to the amplitude distributions obtained throughout the whole time-interval reveal that the initial FWHM increases by around 1 % at this distance. This confirms, that defocusing happens on larger length scales and the THz beam stays focused within the spacer layer. Further simulations for larger distances even show, that propagation through spacer thicknesses in the micrometer range lead to a beam width expansion just below 10 %, if the refractive index of the spacer material stays below = 2. . At peak distances from edge < , the beam is fully (blue) or more than half (grey) transmitted, while at distances ≥ , the beam is half (grey dashed) to entirely (yellow dashed) blocked. b-d) Calculated E-field amplitudes hitting the sensor after passing the test pattern plane depicted for three different detector distances . At small distances around = between test pattern and sensor, the arriving amplitude is high because the broadening of the beam is small. With increasing distance , the THz wave broadens by three orders of magnitude, and the e-field amplitude decreases. For each sensor distance, the first 3 waves (x-position < ) represented by blue colors of each distance are the largest, because they aren't cut off from the edge of the structure. The fifth wave (at x-position = ) is half blocked by the structure, so that the amplitude is decreased to the half. The following waves (x-position > ) are mostly or completely blocked, so that almost no or no more amplitude reaches the sensor.
The second simulation describes the behavior of the THz-beam after diffraction at the test pattern and propagating to the detector through air. This part of calculation is performed using energy conservation, Huygens-Fresnel principle and the superposition principle with complexphase-corrected addition. Starting with a Gaussian electric field distribution in the test pattern plane, the electric field distribution arriving in the detector plane for different detector distances Beam spots with a Gaussian electric field amplitude distribution whose center is in the transmitting region are fully or partially transmitted. Whereas beam spots whose center is located in the blocking region, are partially or fully blocked. Therefore, in this scenario, the transmitted beams are represented by the blue Gaussian distribution curve with the center position further away from the edge in the transmitting region than the beams FWHM. The beams partially blocked, with the center position closer to the knife-edge than the beams FWHM, are represented by the brown curve. The beams fully blocked, entering the pattern plane further away from the edge than their FWHM, are represented by the yellow curve. At this stage, the Gaussian beam has a FWHM = 4.7 μm. For the simulation, the Gaussian beam is moved from x-position −10 μm to +10 μm against the knife-edge and the diffraction pattern is calculated for an incrementally increasing sensor distance 2 . The resulting THz beam efield amplitudes arriving at the sensor for each of the three sensor distances ( 2 = 500 nm, 300 μm, 2 mm) are depicted in figure 4b-d) respectively. For each sensor distance, the electric field amplitudes arriving at the sensor grouped in three groups, those fully transmitted are plotted in blue colors, those partially blocked are plotted in brown colors and those fully blocked are plotted in yellow colors. At the first sensor distance, 500 nm behind the test pattern, shown in figure 4b), only a small broadening occurs, and the peak electric field amplitude is in the range of the incoming wave. At larger detector distances in the order of the wavelength of 300 μm, the THz e-field distribution broadens into the millimeter scale (see figure 4c), and even further for the 2 mm sensor distance shown in figure 4d). At the same time, the THz peak electric field amplitude decreases with the sensor distance ∼ 1/ 2 , as known for spherical waves. Moreover, the amplitudes on the sensor decrease, when the x-position is changed from the transmitting to the blocking region across the knife-edge. In figures 4(b-d) this decrease is visible as a function of the x-position entering the test pattern plane. The width of this decrease corresponds to the width of the incoming beam set to FWHM = 4.7μm, as confirmed by errorfunction fits.
Both simulation results, support the experimental data, showing, that the resolution of THz imaging with wavelengths > 100 μm, in the near-field approach can be enhanced to the micrometer range, limited only by the optical properties of the excitation light waves. Therefore, by using objectives for the pump laser beam, a resolution even below the micrometer scale will be possible. Besides this, the simulations show a very homogeneous and almost spherical wave propagation, which means, there are no interference minima that need to be avoided while adjusting the detector. Further, two parameters are revealed whose control will lead to a significant gain of measurement signal. First, the intensity of the THz wave propagating to the detector decreases by ∼ 1/ 2 with the detector distance. This requires detector positioning within a vicinity of a few millimeters behind the emitter. Second, the intensity distribution broadens significantly due to the strong focusing of the exciting light beam for THz wave generation. Therefore, implementing collimating micro lenses directly into the plane of the imaged pattern would improve the signal. Lenses with such dimensions can be produced using three-dimensional lithography techniques directly on top of the emitters.
The results demonstrate a resolution enhancement for our approach applying near-field technique compared to the common far-field methods. For practical use, the spintronic THz emitters must be deposited directly on the planes or attached to the objects for imaging. This might be achieved by depositing THz emitters on glass surfaces of microscope slides or the corresponding coverslips. In addition, further improvement will be realized with the implementation of stronger focusing lenses and the application of other detector techniques to optimize the signal-to-noise ratio. However, the obtained results present the possibility to investigate significantly smaller structures by spectroscopy techniques in the future.
Our results clearly demonstrate the potential for near-field THz imaging using spintronic heterostructure based emitters. Their resolution can be enhanced below 1 μm and the emitter heterostructures can be deposited directly on glass surfaces e.g., microscope slides or coverslips. Adding nanometer thick spacer layers to those emitters, the distance to the imaged objects can be kept constant, ensuring identical conditions for THz wave generation on each spot and thus offering ideal conditions for imaging using the two-dimensional scanning technique. This provides the possibility for investigation of biological cells or e.g., blood platelets. In the future, more data revealing the spectral absorption of polymers in this range, will make this technique available for investigations of plastic nanoparticle residues in human cells. Also, insights into the specifics of spectral absorption will allow distinguishing between cell types and thus make the identification of e.g., cancer cells or blood platelet malformations possible. Besides this, this technique still offers signal-to-noise ratio improvement, by implementing collimating nanolenses to avoid beam divergence after the transmission through the imaged objects. This will not only increase signal intensity on the detector but also permit for more flexible detector positioning at larger distances to the samples. In addition, the implementation of other detection techniques, like electro-optic detection with ZnTe crystals will expand the investigated spectral range.
is the spin-Hall angle, and the magnetization. As depicted in figure 1b), flows in y-direction and occurs on the picosecond time scale corresponding to a Hertzian dipole emitting in the THz frequency range in z-direction. This method to generate THz radiation is meanwhile well established in various applications[36] [37][38] [39][40].
Figure 2 :
2a) Optical microscope image of the designed aperture pattern. b) Two-dimensional images extracted from near-field THz scanning measurements for 0.5, 1.0 and 1.5 THz. The white dashed lines indicate the position of the intensity profiles shown in c), respectively. The error function fits (lines) to the extracted data (dots) determine the beam dimensions in the order of 5 μm. d) Additional small step size knife-edge-method data for precise resolution determination. The error function fits (erf-Fit, lines) to the amplitude profiles determine the FWHM in the plane of the gold structure on the sample for selected THz frequencies, as indicated by the arrows.
Figure 2a )
2adepicts a light microscope image of the test pattern deposited in direct vicinity to the spintronic emitter layer. The golden area depicts the Au layer blocking the THz radiation, whereas the three slits ensuring full THz transmission are shown in brown color. This triangular inter digital pattern allows to adjust the slot size continuously up to ~50 µm width.
Figure 2b )
2bdisplays two-dimensional THz images of the test pattern in scale with figure 2a) for three extracted THz frequencies obtained by scanning the sample in x-and y-direction using a step size of 5 µm. At each xy-position, a temporal spectrum of the THz pulse is recorded.
Their transition widths are determined by fitting error functions (lines) to the experimental data (dots) providing the FWHM diameter of the THz beams. ), where 0 is the amplitude difference between transmitted and blocked THz signal, 0 is the slit position offset and off takes the background noise into account. The extracted FWHM values are for the 1.5 THz beam 5.8 ± 1.1 μm, for the 1.0 THz beam 6.1 ± 1.0 μm and for the 0.5 THz beam 5.9 ± 1.3 μm. Because of the large step size used for the imaging process, the extracted beam parameters are in the same magnitude, but are expected to be smaller, due to stronger laser beam focusing. For a final verification, additional one-dimensional knife-edge measurements at various positions, using the minimum step size of 200 nm confirm the spot size with a FWHM diameter close to and below 5 μm.Figure 2d)shows the extracted amplitude data for the corresponding positions as dots together with the fitted function by solid lines for 1 THz (red), 0.5 THz (black) and 1.5 THz (blue), respectively. The smallest FWHM in the plane of the Au layer is measured for 1.0 THz with 4.9 ± 0.4 µm. At 0.5 THz the FWHM increases to 5.1 ± 0.5 µm and at 1.5 THz to 5.0 ± 0.5 µm. The expected resolution limit for imaging in the far-field is in the range from 100 μm to 300 μm for radiation from 1 THz to 3 THz. Here the application of the near-field imaging technique improves the resolution by a factor of 20 to 60.
Figure 3 :
3Simulation of the THz beam propagation through the thick SiO2 layer. a) Snapshot of the THz beam e-field amplitude distribution after generation at . The Gaussian beam profile has a = .
Figure 4 :
4Intensity calculation for THz wave propagating the distance from slit to detector through air. a) Schematic depiction for three Gaussian beams passing the test pattern plane (orange) at three distances from the edge, showing different transmission scenarios (see text)
Figure 4a )
4aillustrates the beam passing the test pattern plane and leaving the sample. In this knife-edge scenario, the region with negative x-positions is fully transmitting (white) and for positive x-positions fully blocking the beam (orange). The knife-edge is located at x-position =
is calculated. The electric fields are normed to the amplitude in the test pattern plane at a distance of 1 = 300 nm from the emitter after the propagation distance through the SiO 2 . The computed results are imaged infigure 4.
AcknowledgementsThe authors gratefully acknowledge the financial support from the BMBF, MetaZIK PlasMark-T (FKZ:03Z22C511).Y.S. acknowledges the Graduate Program in Spintronics (GP-Spin) at Tohoku University.Data AvailabilityThe data that support the findings of this study are available from the corresponding author upon reasonable request.
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| [] |
[
"CONVERGENCE OF MEAN CURVATURE FLOW IN COMPACT HYPERKÄHLER MANIFOLDS",
"CONVERGENCE OF MEAN CURVATURE FLOW IN COMPACT HYPERKÄHLER MANIFOLDS"
] | [
"Keita Kunikawa ",
"Ryosuke Takahashi "
] | [] | [] | Inspired by the work of Leung-Wan [LW07], we study the mean curvature flow in compact hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the twistor energy by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperkähler complex structure. In particular, our result implies some kind of energy gap theorem for hyperkähler manifolds which have no complex Lagrangian submanifolds. | 10.2140/pjm.2020.305.667 | [
"https://arxiv.org/pdf/1808.06997v4.pdf"
] | 56,279,311 | 1808.06997 | 8fcd0141b76e95027b9635c67693abb1011da40a |
CONVERGENCE OF MEAN CURVATURE FLOW IN COMPACT HYPERKÄHLER MANIFOLDS
Keita Kunikawa
Ryosuke Takahashi
CONVERGENCE OF MEAN CURVATURE FLOW IN COMPACT HYPERKÄHLER MANIFOLDS
Inspired by the work of Leung-Wan [LW07], we study the mean curvature flow in compact hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the twistor energy by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperkähler complex structure. In particular, our result implies some kind of energy gap theorem for hyperkähler manifolds which have no complex Lagrangian submanifolds.
Introduction
Let (M, g) be a compact hyperkähler 4n-manifold, i.e. , the holonomy group is contained in Sp(n). Or equivalently, there exist distinct, g-compatible complex structures {J d } d=1,2,3 which satisfy the quaternion relations: Typical examples of compact hyperkähler manifolds are a K3 surface and a compact torus T 4 (In fact, any Calabi-Yau 4-manifold is hyperkähler since SU(2) Sp(1) and these are only compact 4-dimensional examples). Beauville [Bea83] constructed two distinct deformation classes of hyperkähler's in 4n-dimension for every n > 1. Moreover, Grady (cf. [Gra99], [Gra03]) constructed two additional deformation classes in dimensions 12 and 20. Each deformation class has representatives which are moduli spaces of semistable sheaves on projective K3 surfaces or abelian surfaces or modifications of such moduli spaces.
In this paper, we show the existence and convergence result for the mean curvature flow (MCF) in compact hyperkähler manifolds when the initial data is very small. It is no doubt that for studying the MCF, Lagrangian is one of the good class of submanifolds in a Kähler-Einstein manifold. Indeed, from Smoczyk's result [Smo96], the Lagrangian property is preserved under the MCF, and it gives a lot of benefits for computations of evolution equations, by identifying the extrinsic normal bundle with the intrinsic tangent bundle via the complex structure. Nevertheless, we would like to consider another class of submanifolds, called "hyper-Lagrangian submanifolds" as displayed below. This class includes Lagrangian submanifolds in compact hyperkähler 4-manifolds.
1.1. Main result. A natural counterpart of the Lagrangian condition in compact hyperkähler manifolds is the "complex Lagrangian": for J ∈ S 2 , let Ω J be a holomorphic symplectic form (i.e. nowhere non-degenerate J-holomorphic 2-form) with respect to J. For a 2n-dimensional real submanifold L ⊂ M , we say that L is complex Lagrangian if Ω J | L = 0 for some J ∈ S 2 . From a basic fact of hyperkähler geometry, we find that there exists a J-orthogonal element K ∈ S 2 such that Ω J can be expressed as
Ω J = ω JK − √ −1ω K ,
where ω JK = g(JK·, ·), ω K = g(K·, ·) are real symplectic forms for JK and K respectively. So the condition Ω J | L = 0 means that two symplectic forms ω JK and ω K vanish at the same time for any J-orthogonal K ∈ S 2 . However, this "bi-Lagrangian" condition is so strong that any complex Lagrangian submanifold L in M automatically becomes a (minimal) complex submanifold (cf. [Hit99]). So, following the idea of Leung-Wan [LW07], we relax the assumption by using rich geometry on M . We say that L is hyper-Lagrangian if Ω Ψ(x) | L = 0 at every point x ∈ L for some varying complex structure Ψ : L → S 2 . Then this map Ψ is called the complex phase. In particular, complex Lagrangian is a special case when we can take Ψ as a constant map. In [LW07], they showed that if the initial submanifold L 0 is hyper-Lagrangian, then L t := F t (L) is still hyper-Lagrangian under the MCF F t : L → M , and then the complex phase Ψ t evolves according to the coupled flow equation:
d dt F t = H t d dt Ψ t = ∆ t Ψ t , (1.1)
where ∆ t Ψ t denotes the tension field of Ψ t with respect to the evolving metric g t := F * t g. We would like to call (1.1) the hyper-Lagrangian mean curvature flow (HLMCF). Like other success stories of coupled flows (cf. [Mul10], [Smo01]), the two geometric flows (1.1) can interact with each other to reveal better properties than it had by itself. For any hyper-Lagrangian submanifold F : L → M , we introduce the twistor energy of L as the Dirichlet energy of the complex phase Ψ w.r.t. the induced metric g := F * g:
T (L) := L |∇Ψ| 2 dµ,
where dµ denotes the Riemannian volume of g. Intuitively, the twister energy measures the deviation from L being complex Lagrangian. We can show that any hyper-Lagrangian submanifold which is "almost" complex Lagrangian can be deformed to a genuine one in the following sense: Theorem 1.1 (Convergence of the HLMCF). Let (M, g) be a compact hyperkähler 4n-manifold. Suppose L is a hyper-Lagrangian submanifold with the complex phase Ψ 0 which is smoothly immersed into M . Then for any V 0 , Λ 0 and δ 0 > 0, there exists ε 0 = ε 0 (n, V 0 , Λ 0 , δ 0 , Rm, inj(M )) > 0 such that if L satisfies
Vol(L 0 ) V 0 , |A|(0) Λ 0 , λ 1 (∆ L )(0) δ 0 , T (L 0 ) ε 0 ,
then the hyper-Lagrangian mean curvature flow (1.1) starting from L converges smoothly, exponentially fast to a complex Lagrangian submanifold in M for one of the hyperkähler complex structure on M .
In the above theorem, we need not assume that M has a complex Lagrangian submanifold, so it also gives an existence result for such a submanifold as well as the stability along the MCF. Although generic K3 surfaces do not have holomorphic curves at all, it is also interesting to understand this situation from geometric analytic point of view. Applying our theorem, one can immediately see that the twistor energy causes some gap: for any V 0 , Λ 0 and δ 0 > 0 we define
L(V 0 , Λ 0 , δ 0 ) := L ⊂ M L is a hyper-Lagrangian submanifold Vol(L) V 0 , |A| Λ 0 , λ 1 (∆ L ) δ 0 .
Then we have the following:
Corollary 1.2 (Energy gap theorem). Assume that a 4n-dimensional compact hyperkähler manifold M has no complex Lagrangian submanifolds. Then for any V 0 , Λ 0 and δ 0 > 0, there exists a constant c = c(n, V 0 , Λ 0 , δ 0 , Rm, inj(M )) > 0 such that
inf L∈L(V 0 ,Λ 0 ,δ 0 ) T (L) c.
The proof of Theorem 1.1 is based on [Li12] for the Lagrangian mean curvature flow (LMCF). In [Li12], the crucial step is to establish the exponential estimate for the L 2 -norm of mean curvature vector H by using the fact that each submanifold L t is Lagrangian, which is not valid for our case. Instead, we take an alternative approach from the view point of the theory of harmonic map flow. A key observation is that the L 2 -norm of H is bounded by the twistor energy at each time (cf. Proposition 2.4):
Lt |H t | 2 dµ t 2T (L t ).
So the problem comes down to establishing the exponential estimate for the twistor energy, which is indeed, possible along the same line as the usual harmonic map flow (cf. Lemma 3.4). Then we will face another problem: for the harmonic map flow into positively curved targets, the flow possibly forms singularities in a finite time even if it has small initial Dirichlet energy (cf. [CD90]). One can overcome this by using another key observation (cf. Proposition 2.5), and see that the bootstrapping arguments along the generalized harmonic map flow go well assuming the uniform C 0 bound for the second fundamental form A. With all of these observations, we can obtain the desired result.
1.2.
Examples and relation to other results. Unfortunately, the authors do not know the examples of hyper-Lagrangian submanifolds with a non-constant complex phase for n > 1. However, there are abundance histories when n = 1 and the concept of hyper-Lagrangian is universal, enables us to make a systematic study of several conditions for submanifolds preserved under the MCF. We can see that every surface L in an oriented compact hyperkähler 4-manifold M admits a canonical complex phase map Ψ : L → S 2 defined by
J Ψ e 1 = e 2 , J Ψ e 3 = −e 4 ,
where {e 1 , e 2 , e 3 , e 4 } is any oriented orthonormal frame on T M such that {e 1 , e 2 } is an oriented frame on T L and {e 3 , e 4 } is an orthonormal frame for the normal bundle. Indeed, the map Ψ is independent of the choice for such a frame. In the following, we will explain the each class of submanifolds separately while considering what shape the each complex phase is (see also [LW07]).
1.2.1. Simplectic mean curvature flow. First, we consider symplectic surfaces. It was asked by Yau (for instance, see [Wan01]) that how can a symplectic submanifold be deformed to a holomorphic one? Since a symplectic surface remains to be symplectic along the MCF in a Kähler-Einstein surface (cf. [CL01], [Wan01]), one expects that the symplectic mean curvature flow (SMCF) is applicable to Yau's question. However, there are many difficulties to show the long-time existence and convergence of the SMCF for arbitrary initial data, so it is natural to start with the case when the initial submanifold is sufficiently close to a holomorphic one. It seems that the convergence of the SMCF with small initial data has not been accomplished yet in the general case, whereas we know several partial results for this issue. For instance, our theorem generalizes Han-Sun's result [HS12, Corollary 4.6]: we express Ψ as a map a : L → R 3 , i.e. , a is a coefficient of Ψ with respect to {J d }
J Ψ = d a d J d , a := (a 1 , a 2 , a 3 ).
By using the quaternion relations, we see that cos α := ω J 3 (e 1 , e 2 ) = g(J 3 e 1 , e 2 ) = a 3 .
(1.2)
Hence the condition that L is symplectic w.r.t. ω J 3 is equivalent to say that the image Ψ(L) is contained in the hemisphere S 2 + := {(c 1 , c 2 , c 3 ) ∈ S 2 ⊂ R 3 |c 3 > 0}. Then the (local) angle α defined by (1.2) is called the kähler angle. Applying the maximum principle to the evolution equation of a, we find that the hemisphere condition is preserved under the HLMCF (cf. Corollary 3.2), which is essentially a restatement of the fact as explained above that if the initial surface is symplectic, then the surface is still symplectic along the mean curvature flow. In [HS12], they showed the convergence of the SMCF under the stronger assumption that the ambient Kähler surface M has zero sectional curvature and the initial L 2 -norm of A is very small. Also there is a convergence result for the SMCF in Kähler-Einstein surfaces with positive Ricci curvature by Han-Li [HL05], where the positivity of the extrinsic curvature was essentially used. Anyways, Theorem 1.1 indicates that the MCF method is still valid for Yau's question, and makes the first step in this direction.
1.2.2. Lagrangian mean curvature flow. Next, we explain the Lagrangian case. If L is Lagrangian with respect to ω Jo for a fixed J o ∈ S 2 , then without loss of generality, we may assume J 3 = J o . By the Lagrangian condition, we find that L has the J 3 -orthogonal complex phase J Ψ which can be expressed as
J Ψ (x) = cos θ(x)J 1 + sin θ(x)J 2 (1.3)
for some multi-valued function θ : L → R. Moreover, the function θ and ω Jo are related by the formula
i H ω Jo = dθ. (1.4)
So θ is nothing but the Lagrangian angle. In particular, we often consider the following special cases:
(1) The form i H ω Jo is exact, or equivalently, θ is a single-valued function.
(2) The submanifold L is almost calibrated, i.e. , L satisfies (1) and cos θ > 0. As it is for Lagrangian, these two conditions are preserved under the MCF (cf. [Smo99], [CL01], [Wan01]). The convergence result for the LMCF with small initial data was obtained by Li [Li12, Theorem 1.2]. He showed the similar convergence result to Theorem 1.1 under the assumption (1) (but, we need not assume (2)) and that the initial L 2 -norm of H is very small. So Theorem 1.1 is still meaningful even if L 0 is Lagrangian since we need not assume (1) in our theorem. Finally, we again emphasize the benefit of the hyper-Lagrangian submanifolds. In fact, the hyper-Lagrangian structure gives one a comprehensive view point to understand the concepts of symplectic surfaces or (almost calibrated) Lagrangian submanifolds in compact hyperkähler 4-manifolds. Figure 1 shows the correspondence between each of these concepts and the image of the complex phase map Ψ : L → S 2 . is non-empty. Moreover, Chen [Che99] proved the existence of infinitely many holomorphic curves on general K3 surfaces. Then we can take any small perturbation of the holomorphic curves as an initial data in Theorem 1.1.
1.3. Organization of the paper. Our article will be organized as follows. We will first recall some results discovered by Leung-Wan [LW07] and prove formulas relating the mean curvature vector (or second fundamental form) with the complex phase which are needed in the rest of the article. In Section 3, we study the behavior of the twistor energy and first eigenvalue along the HLMCF, and then establish some parabolic estimates. Finally, we give the proof of Theorem 1.1 in the last part of Section 3.
Acknowledgment. The authors express their gratitude to Shigetoshi Bando for helpful conversations. We also would thank Ryoichi Kobayashi for pointing out Corollary 1.2. R.T was supported by Grant-in-Aid for JSPS Fellows Number 16J01211.
Hyper-Lagrangian submanifolds
In this section, we recall some results about hyper-Lagrangian submanifolds studied in [LW07]. Let M be a compact hyperkähler 4n-manifold and L ⊂ M a real submanifold of dimension 2n. In this section, we promise that the indices (i, j, α, β, etc. ) run in the following manner i, j = 1, . . . , 2n, α, β = 2n + 1, . . . , 4n, A, B = 1, . . . , 4n, ν, λ = 1, . . . n, µ, ρ = n + 1, . . . , 2n.
Definition 2.1. A submanifold L is called hyper-Lagrangian if Ω Ψ(x) | L = 0 at every point x ∈ L for some Ψ : L → S 2 .
Then Ψ is called the complex phase. In particular, a hyper-Lagrangian submanifold is called complex Lagrangian if we can take Ψ as a constant map.
Let Φ : L → S 2 be a smooth map such that Φ(x) is orthogonal to Ψ(x) for each x ∈ L. We can take a special orthonormal frame {e i } for T L satisfying
J Ψ e 2ν−1 = e 2ν .
Then {e i+2n := J Φ e i } is an orthonormal frame for the normal bundle satisfying
J Ψ e 2µ−1 = −e 2µ .
Then {e A } defines a frame of T M . For a hyper-Lagrangian submanifold L with the complex phase Ψ, we denote the associated almost-complex structure by J Ψ . Then the complex phase J Ψ acts on T L, and determines an almost-complex structure on L. However, hyper-Lagrangian is a strong condition which imposes a lot of restrictions on the structural equations. For instance, let {ϕ AB } be the connection forms with respect to {e A }, i.e. ∇e A = ϕ AB e B . Then the structure theorem of hyper-Lagrangian submanifolds (cf. [LW07, Theorem 4.1]) implies
ϕ 2ν−1,2λ−1 = ϕ 2ν,2λ , ϕ 2ν,2λ−1 = −ϕ 2ν−1,2λ , ϕ 2µ−1,2ρ−1 = −ϕ 2µ,2ρ , ϕ 2µ,2ρ−1 = ϕ 2µ−1,2ρ .
(2.1)
As a consequence, we obtain the following:
Theorem 2.2 ([LW07]
, Corollary 4.2). The complex phase Ψ induces an integrable Kähler structure (J Ψ , g| L ) on L with holomorphic normal bundle.
We set
e ν = 1 2 (e 2ν−1 − √ −1e 2ν ), e ν = 1 2 (e 2ν−1 + √ −1e 2ν ), e µ = 1 2 (e 2µ−1 + √ −1e 2µ ), e µ = 1 2 (e 2µ−1 − √ −1e 2µ ).
Then {e ν , e µ } defines a complex basis referred as the canonical frame adapted to (Ψ, Φ). Correspondingly, we take the basis {ζ A } dual to {e A } and set
ζ ν = ζ 2ν−1 + √ −1ζ 2ν , ζ ν = ζ 2ν−1 − √ −1ζ 2ν , ζ µ = ζ 2µ−1 − √ −1ζ 2µ , ζ µ = ζ 2µ−1 + √ −1ζ 2µ .
With this basis, Ω Ψ can be written by
Ω Ψ = − √ −1 ν,µ ζ ν ∧ ζ µ .
Leung-Wan (cf. [LW07, Theorem 4.5]) found the formula relating the mean curvature vector H and the complex phase Ψ as follows:
Proposition 2.3. We have i H Ω Ψ + 2 √ −1∂Ψ = 0. (2.2)
In particular, the above proposition shows that a hyper-Lagrangian submanifold L is minimal if and only if the complex phase Ψ is anti-holomorphic. Meanwhile, by using the formula (2.2), one can obtain a bound for |H| by means of the energy density of the complex phase Ψ:
Proposition 2.4. We have |H| 2 2|∇Ψ| 2 .
Proof. For a fixed x ∈ L, we set
J 1 = J Ψ (x), J 2 = J Φ (x), J 3 = J 1 J 2 .
We would like to call it the canonical basis adapted to (Ψ, Φ) at x. Then we set the coefficient a = (a 1 , a 2 , a 3 ) as
J Ψ = d a d J d .
We take a local representation of Ψ:
Θ(p) = a 1 (p) + √ −1a 2 (p) 1 − a 3 (p)
via stereographic projection. Then the formula (2.2) yields that
i H Ω Ψ + 2 √ −1∂Θ = 0 at x.
From the construction, we know that
a 1 (x) = 1, a 2 (x) = a 3 (x) = 0, Θ(x) = 1.
Also since L is hyper-Lagrangian with the complex phase Ψ, the derivative ∇J Ψ is spanned by J 2 and J 3 at x, so
da 1 | x = 0. Thus we have ∂Θ| x = √ −1∂a 2 | x + ∂a 3 | x ,
|∂Θ| 2 2(|∂a 2 | 2 + |∂a 3 | 2 ) = |da 2 | 2 + |da 3 | 2 = |∇a | 2 at x. On the other hand, if we set H = − α H α e α , one can easily observe that
i H Ω Ψ = − √ −1 µ (H 2µ−1 − √ −1H 2µ )ζ µ , |i H Ω Ψ | 2 = 2|H| 2 .
So we have
|H| 2 = 2|∂Θ| 2 2|∇a | 2 .
We note that a and Θ heavily depend on the choice of the basis (J 1 , J 2 , J 3 ) whereas a only depends on the background basis (J 1 , J 2 , J 3 ). However, the point is that the norm |∇a | 2 is independent of the choice of an orthogonal basis (J 1 , J 2 , J 3 ) since the Euclidean metric on R 3 is invariant under the standard O(3)-action. So we have |∇a | = |∇a| = |∇Ψ| and |H| 2 2|∇Ψ| 2 .
We also remark that the quantity |∇Ψ| has the following three equivalent definitions:
• We regard the complex phase Ψ as a map a : L → S 2 ⊂ R 3 , and define |∇Ψ| as the energy density of a:
|∇a| 2 = d |∇a d | 2 g .
• We define |∇Ψ| as the energy density of Ψ : L → S 2 , i.e. , a map into S 2 (also see (3.2)). • We define |∇Ψ| as the norm of the covariant derivative of J Ψ along L:
|∇J Ψ | 2 = i,A,B g((∇ i J)(e A ), e B ) 2 ,
where ∇ denotes the Levi-Civita connection on the ambient space (M, g). Then, taking account into the fact that {J d } is parallel and J d , J e g = 4nδ de , we have
∇J Ψ = d da d ⊗ J d and |∇J Ψ | = 2 √ n|∇a|.
As for the relation to the second fundamental form A, we have the following:
Proposition 2.5. In the canonical frame adapted to (Ψ, Φ), the quantity |∇J Ψ | 2 is expressed as
|∇J Ψ | 2 = 4 i,ν,µ (h 2µ−1 2ν,i − h 2µ 2ν−1,i ) 2 + (h 2µ−1 2ν−1,i + h 2µ 2ν,i ) 2 ,
where h α ij := g(e i , ∇ j e α ). In particular, we have |∇Ψ| c(n)|A|.
Proof. Set J i,A,B := g(∇ i J Ψ (e A ), e B ) for simplicity. We compute
(∇J Ψ )(e 2ν−1 ) = ∇(e 2ν ) − J Ψ (∇e 2ν−1 ) = j ϕ 2ν,j e j + α ϕ 2ν,α e α − J Ψ j ϕ 2ν−1,j e j + α ϕ 2ν−1,α e α .
By using (2.1), we know that the first and third terms cancel each other out. So we have
(∇J Ψ )(e 2ν−1 ) = µ (ϕ 2ν,2ν−1 − ϕ 2ν−1,2µ )e 2µ−1 + (ϕ 2ν,2µ + ϕ 2ν−1,2µ−1 )e 2µ ,
and hence
J i,2ν−1,j = 0, J i,2ν−1,2µ−1 = −h 2µ−1 2ν,i + h 2µ 2ν−1,i , J i,2ν−1,2µ = −h 2µ 2ν,i − h 2µ−1 2ν−1,i .
In the same way, we can compute other terms by using (2.1) as follows:
J i,2ν,j = 0, J i,2ν,2µ−1 = h 2µ 2ν,i + h 2µ−1 2ν−1,i , J i,2ν,2µ = −h 2µ−1 2ν,i + h 2µ 2ν−1,i , J i,2µ−1,α = 0, J i,2µ−1,2ν−1 = h 2µ−1 2ν,i − h 2µ 2ν−1,i , J i,2µ−1,2ν = −h 2µ 2ν,i − h 2µ−1 2ν−1,i , J i,2µ,α = 0, J i,2µ,2ν−1 = h 2µ−1 2ν−1,i + h 2µ 2ν,i , J i,2µ,2ν = h 2µ−1 2ν,i − h 2µ 2ν−1,i .
So we obtain the desired formula.
3. hyper-Lagrangian mean curvature flow 3.1. Evolution of the coefficient vector. We regard the complex phase Ψ as a map into S 2 ⊂ R 3 and write a = (a 1 , a 2 , a 3 ). We compute the evolution equation of a when Ψ evolves along the generalized harmonic map flow d dt Ψ = ∆ t Ψ. Proof. We take a polar coordinate (θ, ϕ) of S 2 and express a as
a = cos Ψ θ sin Ψ ϕ sin Ψ θ sin Ψ ϕ cos Ψ ϕ ,
where we write Ψ θ = θ • Ψ, Ψ ϕ = ϕ • Ψ for simplicity. Then
d dt Ψ = d dt Ψ θ · ∂ ∂θ • Ψ + d dt Ψ ϕ · ∂ ∂ϕ • Ψ.
Let (x 1 , . . . , x 2n ) be a local coordinate in L. Recall the definition of the tension field of Ψ:
∆Ψ = n i,j=1 g ij ∇ i ∇ j Ψ θ · ∂ ∂θ • Ψ + n i,j=1 g ij ∇ i ∇ j Ψ ϕ · ∂ ∂ϕ • Ψ ∈ C ∞ (Ψ −1 T S 2 ),
where ∇ denotes the canonical connection on Ψ −1 T S 2 associated to g and the standard metric g on S 2 . Then
∇ i ∇ j Ψ α = ∇ i ∇ j Ψ α + β,γ=θ,ϕ Γ α βγ (Ψ) ∂Ψ β ∂x i · ∂Ψ γ ∂x j , α = θ, ϕ,
where Γ α βγ denotes the Christoffel symbol w.r.t. g. We can easily compute g θθ = sin 2 ϕ, g θϕ = 0, g ϕϕ = 1,
Γ θ θθ = Γ ϕ ϕϕ = 0, Γ θ θϕ = cos ϕ sin ϕ , Γ ϕ θθ = − sin ϕ cos ϕ. This implies that d dt Ψ θ = n i,j=1 g ij ∇ i ∇ j Ψ θ = ∆Ψ θ + cos Ψ ϕ sin Ψ ϕ · ∇Ψ θ , ∇Ψ ϕ g , d dt Ψ ϕ = n i,j=1 g ij ∇ i ∇ j Ψ ϕ = ∆Ψ ϕ − sin Ψ ϕ cos Ψ ϕ · |∇Ψ θ | 2 g . Since ∆a 3 = − sin Ψ ϕ · ∆Ψ ϕ − cos Ψ ϕ · |∇Ψ ϕ | 2 g , |∇a| 2 = sin 2 Ψ ϕ · |∇Ψ θ | 2 g + |∇Ψ ϕ | 2 g , (3.2) we have d dt a 3 = − sin Ψ ϕ · d dt Ψ ϕ = ∆a 3 + |∇a| 2 a 3 .
We can compute the evolution equation of a 1 and a 2 in the similar way.
Applying the maximum principle to (3.1), we obtain Corollary 3.2 (also see [LW07], Theorem 5.1). If L 0 satisfies a 3 > c for some constant c ∈ (0, 1) then a 3 > c holds along the HLMCF L t for all t ∈ [0, T ]. In particular, the hemisphere condition Ψ(L) ⊂ S 2 + is preserved under the HLMCF. 3.2. L 2 -estimates. Let L ⊂ M be a hyper-Lagrangian submanifold with the complex phase Ψ. By using (3.1), we can obtain the exponential estimate for the twistor energy:
Lemma 3.4 (Exponential estimate for the twistor energy). For the HLMCF L t , we have
d dt T (L t ) (−2λ 1 (t) + C(n) max Lt |H||A| + 2 max Lt |∇Ψ| 2 ) · T (L t ),
where λ 1 (t) > 0 denotes the first eigenvalue of the Laplacian ∆ t .
Proof. First, we recall the evolution of the Riemannian metric on L (for instance, see [CL01]): d dt g ij = −2H α h α ij . By using this and the expression of the energy density as the norm of the coefficient vector
|∇Ψ| 2 = |∇a| 2 , we compute d dt L |∇a| 2 dµ t = 2 L ∇ d dt a, ∇a dµ t + L d d dt g ij ∇ i a d ∇ j a d dµ t − L |∇a| 2 |H| 2 dµ t .
We estimate each term separately. The first term is
2 L ∇ d dt a, ∇a dµ t = 2 L ∇((∆ + |∇a| 2 )a), ∇a dµ t = −2 L |∆a| 2 dµ t − 2 L |∇a| 2 a, ∆a dµ t −2λ 1 L |∇a| 2 dµ t + 2 L |∇a| 4 dµ t −2λ 1 L |∇a| 2 dµ t + 2 max Lt |∇a| 2 L |∇a| 2 dµ t ,
where we used the formula 0 = d dt a, a = (∆ + |∇a| 2 )a, a = ∆a, a + |∇a| 2 , which can be proved easily by differentiating |a| 2 = 1 in t. For the second term, we have
L d d dt g ij ∇ i a d ∇ j a d dµ t = − 2 L d H α h α ij ∇ i a d ∇ j a d dµ t C(n) max Lt |H||A| · L |∇a| 2 dµ t .
This completes the proof of the Lemma.
The above lemma says that we need to control λ 1 in order to obtain a bound for the twistor energy. So we establish the exponential estimate for λ 1 as follows:
Lemma 3.5 (Exponential estimate for the first eigenvalue). Along the HLMCF, the first eigenvalue λ 1 (t) satisfies d dt
λ 1 −(max Lt |H| 2 + C(n) max Lt |H||A|) · λ 1 .
Proof. Let f be an eigenfunction w.r.t. λ 1 , i.e. f satisfies
−∆ t f = λ 1 f, L f 2 dµ t = 1.
Then the first eigenvalue λ 1 is
λ 1 = L |∇f | 2 dµ t . Differentiating L f 2 dµ t = 1 in t, we have L 2 d dt f · f − f 2 |H| 2 dµ t = 0.
Thus we can compute
d dt λ 1 = 2 L ∇ d dt f, ∇f dµ t + L d dt g ij ∇ i f ∇ j f dµ t − L |∇f | 2 |H| 2 dµ t = −2 L d dt f · ∆f dµ t − 2 L H α h α ij ∇ i f ∇ j f dµ t + L f ∆f · |H| 2 dµ t + L f ∇f, ∇|H| 2 dµ t .
Using the relation −∆f = λ 1 f , we find that the first term and the third term cancel each other out. The second term can be estimates as
− 2 L H α h α ij ∇ i f ∇ j f dµ t C(n) max Lt |H||A| · λ 1 .
The fourth term is
L f ∇f, ∇|H| 2 dµ t = − L (f ∆f + |∇f | 2 )|H| 2 dµ t = λ 1 L f 2 |H| 2 dµ t − L |∇f | 2 |H| 2 dµ t − max Lt |H| 2 · λ 1 .
Thus we obtain the desired result.
3.3. C 0 -estimates. In order to get the C 0 -estimates from the L 2 , the notion of noncollapsing geodesic ball is convenient. Roughly speaking, it says that the volume of each geodesic ball in L is bounded from below by that of the Euclidean geodesic ball of the same radius. Let N be a compact Riemannian m-manifold.
Definition 3.6. We say that Proof. Assume that |σ| attains its maximum at a point x 0 ∈ N and the statement does not hold, i.e. , |σ(x 0 )| > (Λ + κ −1/2 )ε 1 m+2 . Then by setting δ := ε 1 m+2 , we get
(1) A geodesic ball B(x, ρ) in N is called κ-noncollapsed ifΛδ = Λε 1 m+2 < |σ(x 0 )|. Thus for any x ∈ B(x 0 , δ), we have |σ(x)| |σ(x 0 )| − Λδ > 0. Integrating on B(x 0 , δ) yields that ε B(x 0 ,δ) |σ| 2 dµ (|σ(x 0 )| − Λδ) 2 Vol(B(x 0 , δ)) (|σ(x 0 )| − Λδ) 2 κδ m ,
where we used δ = ε 1 m+2 r and the assumption that N is κ-noncollapsed on the scale r in the last inequality. So putting δ = ε 1 m+2 into the above yields that |σ(x 0 )| (Λ + κ −1/2 )ε 1 m+2 , contradicting the assumption. This completes the proof. Now we go back to our situation, so let L t be the HLMCF in a compact hyperkähler 4n-manifold M . The above lemma indicates that it is important to study the evolution of the volume ratio along the flow.
Lemma 3.8 (Volume ratio estimate). If L 0 is κ 0 -noncollapsed on the scale r 0 , then for any small geodesic ball B t (x, ρ) in L t with radius ρ ∈ (0, r 0 ), we have
Vol(B t (x, ρ)) κ 0 e −(2n+1)E(t) ρ 2n ,
where E(t) is given by
E(t) := t 0 (max Ls |H| 2 + max Ls |A||H|)ds.
Proof. Let γ t be a length minimizing unit-speed geodesic w.r.t. g(t) joining p to q ∈ B t (p, ρ). Then for every t 0 we have
d t (p, q) = Length g(t) (γ t ) Length g(t) (γ t 0 ),
and equality holds when t = t 0 , which implies that
d dt d t (p, q)| t=t 0 = d dt Length g(t) (γ t )| t=t 0 = d dt Length g(t) (γ t 0 )| t=t 0 .
Thus we can compute
d dt d t (p, q) = 1 2 dt(p,q) 0 L dg t dt d ds γ t , d ds γ t dµ t ds − dt(p,q) 0 |H| 2 dµ t ds, d dt d t (p, q) E(t) · d t (p, q).
This implies that
e −E(t) d 0 (p, q) d t (p, q) d 0 (p, q)e E(t) , dµ t e −E(t) µ 0 .
Since L 0 is κ 0 -noncollapsed on the scale r 0 , for ρ r 0 , we have
Vol(B t (p, ρ)) = Bt(p,ρ) dµ t B 0 (p,e −E(t) ρ) e −E(t) dµ 0 κ 0 e −(2n+1)E(t) ρ 2n .
The lemma is proved.
3.4. Some parabolic estimates for the HLMCF. In this subsection, we prove some parabolic estimates for the HLMCF. The first lemma is Short-time stability, which says that the HLMCF does not change a lot in short time intervals.
Lemma 3.9 (Short-time stability). If L 0 satisfies
|A|(0) Λ, |∇Ψ|(0) P, λ 1 (0) δ,
then there exists T = T (n, Λ, Rm) such that the HLMCF L t satisfies
|A|(0) 2Λ, |∇Ψ|(t) 2P, λ 1 (t) 2 3 δ, t ∈ [0, T ].
Proof. The estimate of |A| follows from [HS12, Lemma 2.2]. Then the estimate of λ 1 follows from the exponential estimate for λ 1 . Finally, we establish the estimate for |∇Ψ|. By the Bochner identity, Gauss equation and Proposition 2.5, we can compute
d dt − ∆ t |∇Ψ| 2 = −2|∇ 2 Ψ| 2 + Rm S 2 * (∇Ψ) 4 + Rm * (∇Ψ) 2 + A 2 * (∇Ψ) 2 C(n, Λ, Rm)|∇Ψ| 2 .
Applying the maximum principle, we obtain |∇Ψ|(t) e 1 2 C(n,Λ,Rm)t |∇Ψ|(0) e 1 2 C(n,Λ,Rm)t P, so we may take T 2 log 2 C(n,Λ,Rm) . We can obtain not only the usual smoothing estimates for A, but also for Ψ with the help of Proposition 2.5. for some T > 0. Then for each l 1, there exist constants Λ l = Λ l (n, Λ, Rm, T ) such that
sup Lt |∇ l A| Λ l t l/2 , t ∈ (0, T ].
Moreover, for any t 0 ∈ (0, T ], there exist constants P l = P l (n, Λ, Rm, t 0 , T ) such that
sup Lt |∇ l Ψ * | P l , t ∈ [t 0 , T ],
where Ψ * = ∇Ψ is the differential map of the complex phase Ψ : L → S 2 .
Proof. The estimate of A follows from [HS12, Theorem 3.1]. Then for any t 0 ∈ (0, T ] we have
sup Lt |∇ l A| Λ l (t 0 /2) l/2 , t ∈ [t 0 /2, T ].
We use this estimate to show the estimate of Ψ * . Note also that |Ψ * | has a uniform bound |Ψ * | c(n)|A| c(n)Λ by Proposition 2.5.
In order to derive the estimate of Ψ * , we first compute the time derivative of |∇ l Ψ| 2 along the generalized harmonic map flow. A straight calculation shows that for each l 0 we get the formula:
d dt ∇ l Ψ * = ∆(∇ l Ψ * ) + r+i+j+k=l ∇ r Rm S 2 * (Ψ * ) r * ∇ i Ψ * * ∇ j Ψ * * ∇ k Ψ * + r+i i +···+i l +j=l ∇ r Rm * ∇ i 1 −1 A * · · · * ∇ i l −1 A * ∇ j Ψ * + i+j+k=l ∇ i A * ∇ j A * ∇ k Ψ * ,
where ∇ denotes the Levi-Civita connection on T S 2 . It follows that for
t ∈ [t 0 /2, T ] we have d dt |∇ l Ψ * | 2 = A 2 * (∇ l Ψ * ) 2 + 2 d dt ∇ l Ψ * , ∇ l Ψ * ∆|∇ l Ψ * | 2 − 2|∇ l+1 Ψ * | 2 + C 0 i+j+k l |∇ i Ψ * ||∇ j Ψ * ||∇ k Ψ * ||∇ l Ψ * |, (3.3)
where C = C(n, Λ, Rm, t 0 , T ) is a constant. From (3.3) we have
d dt |Ψ * | 2 ∆|Ψ * | 2 − 2|∇Ψ * | 2 + c 1 and d dt |∇Ψ * | 2 ∆|∇Ψ * | 2 − 2|∇ 2 Ψ * | 2 + c 2 |∇Ψ * | 2 + c 3 ,
where c k = c k (n, Λ, Rm, t 0 , T ) (k = 1, 2, 3) are constants. Set
F := (t − t 0 /2)|∇Ψ * | 2 + α|Ψ * | 2 ,
where α is a constant which will be determined later. It is not difficult to see
d dt − ∆ F (−2α + 1 + T c 2 )|∇Ψ * | 2 + αc 1 + T c 3 .
Then we choose α = (1 + T C 2 )/2 to get
d dt − ∆ F 1 + T c 2 2 c 1 + T c 3 .
Applying the maximum principle, we have
F (t) F (0) 1 + T c 2 2 Λ 2 = C 1 (n, Λ, Rm, t 0 , T ), t ∈ [t 0 /2, T ].
Hence we get
|∇Ψ * | 2 C 1 t − t 0 /2 , t ∈ (t 0 /2, T ].
It follows
sup Lt |∇Ψ * | 6C 1 t 0 = P 1 (n, Λ, Rm, t 0 , T ), t ∈ [2t 0 /3, T ].
This proves the case l = 1. For l 2, we prove it by induction. Assume that the following estimate holds for each 0 m l − 1:
sup Lt |∇ m Ψ * | (m + 1)(m + 2)C m (n, Λ, Rm, t 0 , T ) t 0 , t ∈ [((m + 1)/(m + 2))t 0 , T ].
Then by (3.3) we have
d dt |∇ l−1 Ψ * | 2 ∆|∇ l−1 Ψ * | 2 − 2|∇ l Ψ * | 2 + c 4 and d dt |∇ l Ψ * | 2 ∆|∇ l Ψ * | 2 − 2|∇ l+1 Ψ * | 2 + c 5 |∇ l Ψ * | 2 + c 6 ,
for t ∈ [(l/(l + 1))t 0 , T ], where c k = c k (n, Λ, Rm, t 0 , T ) (k = 4, 5, 6) are constants which are controlled by the lower order estimates. By the same way as l = 1, using maximum principle we see |∇ l Ψ * | 2 C l (n, Λ, Rm, t 0 , T ) t − (l/(l + 1))t 0 , t ∈ ((l/(l + 1))t 0 , T ].
Therefore we obtain the desired bound |∇ l Ψ * | 2 (l + 1)(l + 2)C l (n, Λ, Rm, t 0 , T ) t 0 =: P l (n, Λ, Rm, t 0 , T ) for t ∈ [((l + 1)/(l + 2))t 0 , T ].
Remark 3.11. From the smoothing estimates, for any t 0 ∈ (0, T ) we have
sup Lt |∇ l A| Λ l (n, Λ, Rm, t 0 ), sup Lt |∇ l Ψ| P l (n, Λ, Rm, t 0 ), t ∈ [t 0 /2, t 0 ].
In particular, we have bounds for the derivatives |∇ l A| and |∇ l Ψ| for l 1 at t = t 0 .
On the other hand, as in the proof of the above lemma, it is not difficult to see that we have bounds which only depend on n, A(t 0 ) and Ψ * (t 0 ) (including their higher order derivatives)
sup Lt |∇ l A| Λ l (n, A(t 0 ), Rm), sup Lt |∇ l Ψ| P l (n, A(t 0 ), Ψ * (t 0 ), Rm), t ∈ [t 0 , T ].
Combining the both estimates on [t 0 , T ], we obtain T -independent estimates sup Lt |∇ l A| Λ l (n, Λ, Rm, t 0 ), sup Lt |∇ l Ψ| P l (n, Λ, Rm, t 0 ), t ∈ [t 0 , T ].
We often use this property without mentioning in later arguments.
3.5. Convergence of the flow. Now we are ready to prove the main theorem.
Theorem 3.12 (Theorem 1.1). Let (M, g) be a compact hyperkähler 4n-manifold. Suppose L is a hyper-Lagrangian submanifold with the complex phase Ψ 0 which is smoothly immersed into M . Then for any V 0 , Λ 0 and δ 0 > 0, there exists ε 0 = ε 0 (n, V 0 , Λ 0 , δ 0 , Rm, inj(M )) > 0 such that if L satisfies
Vol(L 0 ) V 0 , |A|(0) Λ 0 , λ 1 (∆ L )(0) δ 0 , T (L 0 ) ε 0 ,
then the hyper-Lagrangian mean curvature flow starting from L converges smoothly, exponentially fast to a complex Lagrangian submanifold in M for one of the hyperkähler complex structure on M .
Proof.
Step 1. (Reduction from L 2 to C 0 ): In the first step, we see that after a short period of time, the parabolicity of the flow improves the initial L 2 -condition for ∇Ψ to the C 0 -condition. From Proposition 2.5 and Lemma 3.9, we know that L t satisfies
|A|(t) 2Λ 0 , |∇Ψ|(t) c(n)Λ 0 , λ 1 (t) 2 3 δ 0 , t ∈ [0, T 0 ]
for T 0 = T 0 (n, Λ 0 , Rm). So Lemma 3.4 implies the following exponential estimate for the twistor energy:
T (L t ) e ct T (L 0 ) ε 0 e ct , t ∈ [0, T 0 ]
for some c = c(n, Λ 0 ) > 0. Therefore we can choose t 0 = t 0 (n, Λ 0 ) ∈ (0, T 0 ] so that
T (L t ) 2ε 0 , t ∈ [0, t 0 ].
On the other hand, by the smoothing estimates, we know that for any l 1,
|∇ l A|(t) C l (n, Λ 0 , Rm), t ∈ [t 0 /2, t 0 ],(3.4)
and also |∇ 2 Ψ|(t) c(n, Λ 0 , Rm), t ∈ [t 0 /2, t 0 ].
In order to get the estimate for the energy density |∇Ψ|, we need to establish the noncollapsing estimate for L t at first. By [CH10, Proposition 2.2] and (3.4), we know that the injectivity radius of L is bounded from below along the HLMCF
inj(L t ) ι(n, Λ 0 , Rm, inj(M )) > 0, t ∈ [t 0 /2, t 0 ].
Meanwhile, the Gauss equation implies that
| Rm | C(Λ 0 , Rm), t ∈ [t 0 /2, t 0 ].
So in the same way as the proof of [Li12, Theorem 1.1], the volume comparison theorem yields that there exists κ = κ(n, Λ 0 , Rm, inj(M )) and r = r(n, Λ 0 , Rm, inj(M )) such that L t is κ-noncollapsed on the scale r for all t ∈ [t 0 /2, t 0 ]. So Lemma 3.7 implies that
|∇Ψ|(t) (c + κ −1/2 )(2ε 0 ) 1 2n+2 =: η, t ∈ [t 0 /2, t 0 ],
where we take ε 0 sufficiently small so that 2ε 0 r 2n+2 .
Step 2. (Self-improving estimates): We set
A(κ, r, Λ, P, δ) := L is a hyper-Lagrangian submanifold L ⊂ M L is κ-noncollapsed on the scale r |A| Λ, |∇Ψ| P, λ 1 (∆ L ) δ .
Without loss of generality, we regard L t 0 /2 as the initial data of the HLMCF, so we have
L t ∈ A(κ, r, Λ, η, δ), t ∈ [0, t 0 /2],
where Λ := 2Λ 0 , η := (c + κ −1/2 )(2ε 0 ) 1 2n+2 , δ := 2 3 δ 0 . So the short-time stability (cf. Lemma 3.9) combining with the volume ratio estimate (cf. Lemma 3.8) implies that we can choose a small T * > 0 such that
L t ∈ A 1 3 κ, r, 6Λ, 2η 1 2n+2 , 1 3 δ , t ∈ [0, T * ].
Let T * be the maximal time such that the above estimate holds. Then in order to prove the long-time existence of the flow, it suffices to prove the following:
Claim 3.13. There exists a small η > 0 (and hence small ε 0 > 0) such that
L t ∈ A 2 3 κ, r, 3Λ, η 1 2n+2 , 1 2 δ , t ∈ [0, T * ].
Indeed, if T * < ∞ then from the claim we have L t ∈ A( 2 3 κ, r, 3Λ, η 1 2n+2 , 1 2 δ) for t ∈ [0, T * ]. By using the short-time stability and volume ratio estimate again, we find that there exists T > T * such that L t ∈ A( 1 3 κ, r, 6Λ, 2η 1 2n+2 , 1 3 δ) for t ∈ [0, T ], contradicting the maximality of T * .
First, we establish an estimate for |∇Ψ|. We know that
λ 1 (t) 1 3 δ, t ∈ [0, T * ].
So if we choose η > 0 small so that
λ 1 (t) 1 4 δ + C(n) · 3Λ · 2η 1 2n+2 + (2η 1 2n+2 ) 2 , t ∈ [0, T * ],
then the exponential estimate for the twistor energy (cf. Lemma 3.4) implies
T (L t ) e − δ 2 t T (L 0 ) η 2 V 0 e − δ 2 t , t ∈ [0, T * ].
By Lemma 3.9, there exists some t * = t * (n, Λ, Rm) ∈ (0, T * ) such that
|∇Ψ| 2η η 1 2n+2 , t ∈ [0, t * ],
for η 1 2 . On the other hand, since |A|(t) 6Λ for t ∈ [0, T * ], the smoothing estimates imply that |∇ 2 Ψ| C(n, Λ, Rm), t ∈ [t * , T * ].
Thus we obtain |∇Ψ|(t) C(n, Λ, κ, r, V 0 , Rm) · η 1 n+1 e − δt 4n+4 , t ∈ [t * , T * ].
(3.5) So we can choose η > 0 small so that C(n, Λ, κ, r, V 0 , Rm) · η 1 2n+2 1 and obtain |∇Ψ|(t) η 1 2n+2 , t ∈ [0, T * ]. Next, we compute |A|. By the smoothing estimates, for any l 1, we have
|∇ l A| C l (n, Λ, Rm), t ∈ [t * , T * ].
Thus we also have |∇ l H| C l (n, Λ, Rm), t ∈ [t * , T * ]. From Proposition 2.4 and (3.5), we know that |H| also decreases exponentially fast. So integrating by parts, we have We recall the evolution equation of A along the MCF (cf. [CL01])
d dt h α ij = ∇ i ∇ j H α + H β h β jk h α ik + H β R αjβi + h β ij b β α ,
where b β α = g( d dt e α , e β ) = g(∇ H e α , e β ). Note that b β α is anti-symmetric since 0 = d dt (g(e α , e β )) = b β α + b α β . So integrating (3.6) in t and using the exponential decay of |H|, we have |A|(t) |A|(t * ) + c(n) t t * (|∇ 2 H| + |H|| Rm | + |H||A| 2 )ds 2Λ + c(n) cη 1 2(n+1) 2 16(n + 1) 2 δ + (C(Rm) + 64Λ 2 ) · cη 1 n+1 8(n + 1) δ .
Thus we can take η > 0 sufficiently small so that |A|(t) 3Λ, t ∈ [0, T * ].
Then we establish the estimate for λ 1 (t). Since λ 1 (0) δ, Lemma 3.9 shows that λ 1 (t) 2 3 δ, t ∈ [0, t * ].
Thus the exponential estimate for λ 1 combining with the exponential decay of |H| imply that
λ 1 (t) exp − t t * (max Ls |H| 2 + C(n) max Ls |H||A|)ds λ 1 (t * )
exp − c 2 η 2 n+1 4(n + 1) δ − C(n) · 3Λ · cη 1 n+1 8(n + 1) δ λ 1 (t * ).
If we take η > 0 sufficiently small, then
λ 1 (t) 1 2 δ, t ∈ [0, T * ].
We can prove a non-collapsing estimate of L t in the same way as λ 1 , by using the volume ratio estimate.
Step 3. (Exponential convergence of the flow): From
Step 2, we have a uniform bound for A. So the standard bootstrapping arguments combining with Simon's theorem [Sim83] imply the smooth convergence of the MCF L t → L ∞ . Moreover, we have already seen that for a fixed sufficiently small η > 0, we have |∇Ψ(t)| C(n, Λ, κ, r, V 0 , Rm) · η 1 n+1 e − δt 4n+4 0.
In particular, Proposition 2.4 implies that H t converges exponentially fast to H ∞ = 0, and hence L ∞ is minimal.
As for the generalized harmonic map flow, we have also the uniform bounds |∇ l Ψ| C l for all l 1. Thus there exists a subsequence {Ψ t i } which converges to a smooth map Ψ ∞ : L → S 2 and L ∞ inherits a hyper-Lagrangian structure with the complex phase Ψ ∞ . Since |∇Ψ ∞ | = 0, the map Ψ ∞ should be a constant. Finally, we show that the complex phase Ψ ∞ which arises from the generalized harmonic map flow does not depend on the choice of the subsequence {Ψ t i } by contradiction. So we assume that there exists a two constant phase maps Ψ ∞ and Ψ ∞ which arise in this way. We take a small geodesic ball in B ⊂ S 2 centered at Ψ ∞ so that Ψ ∞ ∈ B. Since {Ψ t i } converges to Ψ ∞ we know that Ψ t i (L) ⊂ B for i large enough. We fix such an i and consider the generalized harmonic map flow Ψ t starting from the data (L t i , Ψ t i ). Then a simple maximum principle argument (cf. Corollary 3.2) shows that Ψ t (L) ⊂ B for all t ∈ [0, ∞) whereas {Ψ t } should have a convergent subsequence to Ψ ∞ ∈ B, so contradiction. This completes the proof.
J 1 J 2 J 3 = −Id. Then each compact hyperkähler manifold M admits a 2-sphere of complex structures called the twistor family d c d J d for (c 1 , c 2 , c 3 ) ∈ S 2 ⊂ R 3 .
Figure 1 .
1Image of the complex phase Ψ in S 2
1. 2 . 3 .
23Holomorphic curves in K3. On any polarized K3 surface (M, H) (with H O M ), it is known that there exists at least 1 holomorphic curve which belongs to the linear system |mH| for all m 1 (Bogomolov, Mumford, Mori-Mukai [MM83]). Due to the Lefschetz theorem, the existence of such an H is equivalent to say that the Néron-Severi lattice NS(M ) := H 1,1 (M ) ∩ H 2 (M, Z)
Lemma 3 . 1 .
31Along the HLMCF, a satisfies d dt − ∆ t a = |∇a| 2 a. (3.1)
Definition 3. 3 .
3we define the twistor energy of L as the Dirichlet energy of the complex phase: T (L) := L |∇Ψ| 2 dµ.
whenever B(y, s) ⊂ B(x, ρ).(2) A compact Riemannian manifold N is called κ-noncollapsed on the scale r if every geodesic ball B(x, s) is κ-noncollapsed for s r.
Lemma 3. 7 .
7Let (E, h, D) be a vector bundle with a fiber metric h and a compatible connection D over a compact Riemmanian manifold N . Assume that N is κ-noncollapsed on the scale r. For any smooth section σ ∈ C ∞ (E), if |Dσ| Λ, N |σ| 2 dµ ε r m+2 ,
Lemma 3. 10 (
10Smoothing estimates). Suppose along the HLMCF, we have sup Lt |A| Λ, t ∈ [0, T ]
0 ,
0H|dµ t C(n, Λ, κ, r, V 0 , Rm)η 1 n+1 e − δt 4n+4for t ∈ [t * , T * ]. So we have |∇ 2 H| c(n, Λ, κ, r, V Rmn+1) 2 , t ∈ [t * , T * ].
Then it follows H||A| + |H||A|| Rm | + |H||A| 3 ).Dividing both sides by |A|, we haved dt |A| c(n)(|∇ 2 H| + |H|| Rm | + |H||A| 2 ).|A|(t) 2Λ, t ∈ [0, t * ].h α
ij h β
ij b α
β = 0.
So we compute
2|A|
d
dt
|A| =
d
dt
|A| 2 c(n)(|∇ 2 (3.6)
Meanwhile, Lemma 3.9 shows that
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| [] |
[
"Generation of measures by statistics of rotations along sets of integers",
"Generation of measures by statistics of rotations along sets of integers"
] | [
"E Lesigne ",
"A Quas ",
"J Rosenblatt ",
"M Wierdl "
] | [] | [] | Let S := (s 1 < s 2 < . . . ) be a strictly increasing sequence of positive integers and denote e(β) := e 2πiβ . We say S is good if for every real α the sequence 1 N ∑ n≤N e(s n α) N∈N of complex numbers is convergent.Equivalently, the sequence S is good if for every real α the sequence (s n α) possesses an asymptotic distribution modulo 1. We are interested in finding out what the limit measure µ S,α := lim N 1 N ∑ n≤N δ s n α can be. In this first paper on the subject, we investigate the case of a single irrational α. We show that if S is a good set then for every irrational α the limit measure µ S,α must be a continuous Borel probability measure. Using random methods, we show that the limit measure µ S,α can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on T. On the other hand, if ν is the uniform probability measure supported on the Cantor set, there are some irrational α so that for no good sequence S can we have the limit measure µ S,α equal ν. We leave open the question whether for any given singular, continuous Borel probability measure ν there is an irrational α and a good sequence S so that µ S,α = ν. | null | [
"https://export.arxiv.org/pdf/2210.02233v1.pdf"
] | 252,715,633 | 2210.02233 | ef5346014d425fc209014d48e6edb181d4f97a53 |
Generation of measures by statistics of rotations along sets of integers
October 6, 2022
E Lesigne
A Quas
J Rosenblatt
M Wierdl
Generation of measures by statistics of rotations along sets of integers
October 6, 2022
Let S := (s 1 < s 2 < . . . ) be a strictly increasing sequence of positive integers and denote e(β) := e 2πiβ . We say S is good if for every real α the sequence 1 N ∑ n≤N e(s n α) N∈N of complex numbers is convergent.Equivalently, the sequence S is good if for every real α the sequence (s n α) possesses an asymptotic distribution modulo 1. We are interested in finding out what the limit measure µ S,α := lim N 1 N ∑ n≤N δ s n α can be. In this first paper on the subject, we investigate the case of a single irrational α. We show that if S is a good set then for every irrational α the limit measure µ S,α must be a continuous Borel probability measure. Using random methods, we show that the limit measure µ S,α can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on T. On the other hand, if ν is the uniform probability measure supported on the Cantor set, there are some irrational α so that for no good sequence S can we have the limit measure µ S,α equal ν. We leave open the question whether for any given singular, continuous Borel probability measure ν there is an irrational α and a good sequence S so that µ S,α = ν.
Haar-Lebesgue measure λ We denote by λ the Haar-Lebesgue probability measure on the torus. e(θ), e p (θ) We use Weyl's notation 1 1 Weyl 1916. e(θ) := exp(2πiθ), for θ ∈ T (3.2) and e p (θ) := exp(2πipθ), for p ∈ Z, θ ∈ T (3.3) so e = e 1 .
[N] We borrow the following convenient notation from combina- Counting measure #S We denote by # the counting measure on N, so #S = #(S) = ∑ n∈N 1 S (n) is the number of elements in S ⊂ N.
Arithmetic average A S f For a finite set S and for a function f defined on S, we denote by A S f the arithmetic average of f on S,
A S f := 1 #S ∑ s∈S f (s) (3.6)
σ-average A σ S f If σ is a non-identically 0, finite measure on a set S, we then denote by A σ S f the σ-average of f on S,
A σ S f := 1 σ(S) S f dσ.
(3.7) generation of measures by statistics of rotations along sets of integers 4
So A σ is simply the normalized integral with respect to the measure σ, and hence the integral of f with respect to σ has to make sense. We usually encounter this notation for S ⊂ N, and in this case it takes the form
A σ S f = 1 σ(S) ∑ s∈S σ(s) f (s).
(4.1)
Good set
In this paper we are interested in those sets S ⊂ N for which the averages A s∈S(N) e(sα) converge for every α ∈ T as N → ∞.
Definition Good set
Let S ⊂ N be an infinite set. We say that S is a good set if for every α ∈ T the limit lim N A s∈S(N) e(sα) (4.2)
exists.
We see immediately that the set N is good. Since the work of Weyl,2 we know that if S is the set of kth powers, S = n k n ∈ N 2 Weyl 1916, Satz 9; Kuipers and Niederreiter 1974, Theorem 3.2.
for a k ∈ N, then S is good. That the set of primes is good follows from the work of de la Vallée Poussin and Vinogradov. 3 In both 3 The case of rational α is equivalent with the prime number theorem for arithmetic progressions which is proved in Vallée Poussin 1896; the case of irrational α follows from the corresponding uniform distribution result of Vinogradov, see Vinogradow 1937. cases, for irrational α the limit in eq. (4.2) is 0. We find that it's more intuitive to describe the meaning of the limit in eq. (4.2) in terms of Borel probability measures on T. To do this, we need some more notations.
Nth average measure A s∈S(N) δ s We call A s∈S(N) δ s , where δ s is Dirac's delta function at s the Nth average measure along S.
Nth transform measure A s∈S(N) δ sα We call A s∈S(N) δ sα the Nth transform measure along S. ν(φ) It is often convenient for us to use the functional notation for the integral with respect to a Borel measure ν on T, so for a νintegrable T → C function φ we denote ν(φ) := T φ dν (4.3)
With this notation, for a Borel measurable set B ⊂ T we have
ν(B) = ν(1 B ).
For a fixed α, the transform measure A s∈S(N) δ sα is a Borel probability measure on the torus. Notice that if the set S is good, that is the limit in eq. (4.2) exists for every α ∈ T, then, by Weierstrass's theorem on the density of trigonometric polynomials in the space of continuous functions on T, for every continuous function φ defined on T the limit lim N A s∈S(N) φ(sα) exists for every α ∈ T. This means that for a fixed α ∈ T, the sequence A s∈S(N) δ sα N of transform measures converges weakly to a Borel probability measure on T.
Limit transform measure µ S,α For a good set S and α ∈ T, we define the limit measure µ S,α along S at α by , µ S,α := lim N A s∈S(N) δ sα (5.1)
where the limit is in the weak sense, that is, for every continuous T → C function φ we have
µ S,α (φ) = lim N A s∈S(N) φ(sα) (5.2)
Let us record what we have established as a proposition.
Proposition Good set in terms of limit measures
Let S ⊂ N be an infinite set. Then S is a good set if and only if the weak-limit measure µ S,α := lim N A s∈S(N) δ sα exists for every α ∈ T.
Note that if α is irrational and lim N A s∈S(N) e(psα) = 0 for every nonzero p ∈ Z, then the limit measure µ S,α is λ.
Our main object in this paper will be to try to figure out what the limit transform measure µ S,α can be, and for this it's convenient to introduce the following definition.
Definition Representable measure at α
Let S be a good set, and let ν be a finite Borel measure on T.
We say that S represents ν at α ∈ T if µ S,α = 1 ν(T) ν. We say ν is representable at α if there is a good set which represents ν at α.
It's possible to reformulate the concept of a good set in terms of dynamical systems 4 . By the spectral theorem, we have the following 4 By a dynamical system, we mean a probability space (X, p) equipped with a measurable, measure preserving transformation T of X.
proposition.
Proposition Good set in terms of a dynamical systems
Let S ⊂ N be infinite. Then the set S is good if and only if in every dynamical system (X, p, T) for every f ∈ L 2 (X) the sequence A s∈S(N) f • T s N∈N is convergent in L 2 -norm.
generation of measures by statistics of rotations along sets of integers 6
Mean M( f ) The mean M( f ) of f ∈ C N is the limit of the sequence Relative mean M R ( f ) Let R ⊂ N be an infinite set. The relative mean M R ( f ) of f ∈ C N in R is the limit of the sequence A R(N) f N∈N as N → ∞ if the limit exists,
A [N] f N∈N as N → ∞ ifM R ( f ) := lim N A R(N) f (6.2)
If f is the indicator of a set S ⊂ R, we then may write M R (S) in place of M R (1 S ). Of course, M R (S) is the relative density of S in R.
Sequences with mean M We denote by M the collection of f ∈ C N for which the mean exists and is finite
M := { f | f ∈ C N , M( f ) exists and is finite } (6.3)
Weights, W, good weights, µ w,α The sequence w ∈ R N is called a weight if w is unsigned and ∑ n w(n) = ∞. A weight can be considered a measure on N and in that case for S ⊂ N we may briefly write w(S) in place of ∑ s∈S w(s).
The set of all weights is denoted by W,
W := { w | w ∈ R N , w ≥ 0, ∑ n w(n) = ∞ } (6.4)
Let R ⊂ N be an infinite set. We often consider R as a strictly increasing sequence (r n ) of integers, in which case we sometimes write w(n) instead of w(r n ), so in this way we view w as supported on N.
For a weight w supported on R we say w is good if the weak limit of the sequence A w n∈[N] δ r n β N of measures exists for every β ∈ T. We denote this limit by µ w,β ,
µ w,β := lim N A w n∈[N] δ r n β = lim N 1 w([N]) ∑ n∈[N]
w(n)δ r n β (6.5)
In the special case of a good set S, we have µ S,α = µ 1 S ,α since the weighted averages with weight w := 1 S correspond to the averages along S.
Let ν be a Borel probability measure on T and let α ∈ T.
We say the weight w represents ν at α if w is good and µ w,α = ν.
generation of measures by statistics of rotations along sets of integers 7
Seminorms 1 , 1,R We define the Besicovitch type seminorm 1 for all complex valued sequences by
f 1 := lim sup N A [N] | f |, f ∈ C N (7.1)
The number 1 in the subscript of 1 expresses the similarity of this norm to the L 1 norm.
For a set S ⊂ N, we may use the notation S 1 instead of 1 S 1 . It is the upper density of S.
For an infinite set R ⊂ N we define the relative 1-norm 1,R by
f 1,R := lim sup N A R(N) | f |, f ∈ C R (7.2)
If the set R is given as a strictly increasing sequence (r n ) and for an f defined on R we define F by F(n) := f (r n ), then f 1,R = F 1 .
Seminorms M We define the M-seminorm M for all complex valued sequences by
f M := lim sup N A [N] f , f ∈ C N (7.3) Variation distance ν 1 − ν 2 V For finite Borel measures ν 1 , ν 2 on T we denote by ν 1 − ν 2 V their variation distance. ν 1 − ν 2 V := sup B∈B |ν 1 (B) − ν 2 (B)| (7.4)
where B is the family of Borel subsets of T. Note that we have
ν 1 − ν 2 V = sup φ∈C + |ν 1 (φ) − ν 2 (φ)| (7.5)
where C + denotes the set of [0, 1]-valued continuous functions on T,
C + := { φ | φ : T → [0, 1], continuous } (7.6)
We summarize our notations in table 1.
Main results
To appreciate the concept of a good set, note the following. Suppose we are given an irrational number α ∈ T and let ν be any Borel probability measure on T. Then we can always find a set S ⊂ N so that lim N A s∈S(N) δ sα = ν. To do this, take an iid sequence (X n ) of T-valued random variables, each with law ν. Then the sequence (e p •X n ) is also an iid sequence of random variables generation of measures by statistics of rotations along sets of integers 8 Table 1: Notations for every p ∈ Z, and by the strong law or large numbers we have lim N A n∈[N] e p •X n = ν(e p ) with full probability for every p ∈ Z. This implies that there is an ω so that lim N A n∈[N] δ X n (ω) = ν. Then, using the density of the sequence (nα) n (mod 1), we can select a strictly increasing sequence (s n ) of integers so that lim n (s n α − X n (ω)) = 0 (mod 1). We finally take S := { s n | n ∈ N }. It is particularly simple to get a point-mass as a limit measure. For example, to get the Dirac measure at 1/2, so µ = δ 1/2 , take a strictly increasing sequence (s n ) of natural numbers so that s n α converges to 1/2 mod 1, and let S := { s n | n ∈ N }. In contrast to this example, for good sets we have, generation of measures by statistics of rotations along sets of integers 9
Symbol Definition Parameters Name N {1, 2, 3, . . . } Natural numbers T [0, 1) torus λ λ(T) Haar-Lebesgue measure on T e(θ) exp(2πiθ) θ ∈ T e p (θ) e(pθ) p ∈ Z [N] {1, 2, . . . , N} N ∈ N S(N) S ∩ [N] S ⊂ N initial segment of S #S(N) ∑ s∈S(N) 1 S ⊂ N counting function of S A S f 1 #S ∑ s∈S f (s) set S is finite average of f on S A σ S f 1 σ(S) S f dσ σ is a finite measure on set S σ-average of f on set S µ S,α lim N A s∈S(N) δ sα S ⊂ N, α ∈ T limit transfer measure of S at α µ w,α lim N A w n∈[N] δ r n β weight w on (r n ), α ∈ T limit transfer measure of w at α ν(φ) T φ dν M( f ) lim N A [N] f f ∈ C N mean of f M R ( f ) lim N A R(N) f f ∈ C N , R ⊂ N relative mean of f M { f | f ∈ C N , M( f ) exists and is finite } sequences with mean W { w | w ∈ R N , w ≥ 0, ∑ n w(n) = ∞ } set of weights f 1 lim sup N A [N] | f | f ∈ C N 1-seminorm f 1,R lim sup N A R(N) | f | R ⊂ N, f ∈ C R relative 1-seminorm f M lim sup N A [N] f f ∈ C N M-seminorm C + { φ | φ : T → [0.1], continuous } ν 1 − ν 2 V sup φ∈C + (ν 1 (φ) − ν 2 (φ)) ν i finite Borel measures on T variation distance
Theorem Only continuous measures can be represented at irrational points
Let S ⊂ N be a good set and α be an irrational number. Then the limit Borel probability measure
µ S,α = lim N A s∈S(N) δ sα (9.1)
is a continuous measure. In other words, only continuous measures can be represented at an irrational number.
The obvious question in turn is if every continuous Borel probability measure can be represented at a given irrational number. The answer is no, as the next result shows.
Theorem Some continuous measures cannot be represented at every irrational point
Let ν be a Borel probability measure on T so that its Fourier coefficients do not converge to 0, so lim sup p→∞ µ e p > 0 (9.2)
Then there is a set A ⊂ T of full Lebesgue measure so that ν cannot be represented at any α ∈ A.
Since a measure ν is called a Rajchman measure 5 if lim p µ e p = 5 Lyons 1995. 0, we can rephrase theorem 9.2 that if ν is representable at every irrational α then it must be a Rajchman measure. 6 A well known non-6 After discussions on the content of the present paper, Christophe Cuny and François Parreau constructed a non-Rajchman measure which is representable at uncountably many α's. This appears in the preprint Cuny and Parreau 2022.
Rajchman continuous measure is the uniform measure on the triadic Cantor set.
The following questions remain open.
Question Is every continuous measure representable somewhere?
Let ν be a continuous Borel probability measure on T.
Is there an irrational α so that ν is representable at α?
Question Is a Rajchman measure representable at every α?
Let ν be a Rajchman probability measure on T and let α be irrational.
Is ν representable at α?
These questions can also be addressed for particular examples of singular measures such as the uniform probability measure on the triadic Cantor set (a non-Rajchman measure) or its modification by Menshov who gave the first example of a singular Rajchman mea-generation of measures by statistics of rotations along sets of integers 10 sure. 7 7 Menchoff 1916; see also Lyons 1995. The next result says that the answer to question 9.4 is yes if ν is absolutely continuous with respect to the Lebesgue measure on the torus T.
Theorem Absolutely continuous measures are representable at every irrational point
Let ν be a Borel probability measure on T which is absolutely continuous with respect to the Lebesgue probability measure on T.
Let α be an irrational number. Then ν is representable at α.
Our proof of theorem 10.1 is flexible and enables us to show a more general result.
Theorem Absolutely continuous measures can be represented by subsets of a good set
Let R be a good set which is "sublacunary", that is, it satisfies the growth condition
lim N log N #R(N) = 0. (10.1)
Let α an irrational number, and let the Borel probability measure ν be absolutely continuous with respect to µ R,α . Then there is a good set S ⊂ R which represents ν at α.
As a consequence of theorem 10.2, every measure which is absolutely continuous with respect to the Lebesgue measure can be represented at any given irrational α by a subset of the primes, squares, or n 2 log n n ∈ N .
In fact, the proof of theorem 10.2 reveals a close connection between the Radon-Nikodym derivative ρ of ν with respect to µ R,α and the mean of the set S representing ν.
Theorem Connection between dν
dµ R,α , M R (S) and S 1,R Let R be a sublacunary good set.
1.
For an irrational α let the unsigned function ρ ∈ L 1 (µ R,α ) with µ R,α (ρ) = 1 be bounded so ρ L ∞ (µ R,α ) < ∞.
Then there is a good set S ⊂ R representing the measure ρ · µ R,α at α and satisfying M R (S) = 1 ρ L ∞ ( µ R,α ) .
2.
Let S be a good subset of R with positive upper density in R, so S 1,R > 0.
Then for every irrational β the limit measure µ S,β is absolutely continuous with respect to µ R,β . Furthermore, the
Radon-Nikodym derivative ρ β := dµ S,β dµ R,β is a bounded function satisfying ρ β L ∞ (µR,β) ≤ 1 S 1,R .
Item 2 of theorem 10.3 has the following consequence.
generation of measures by statistics of rotations along sets of integers 11 11.1 Corollary If the RN derivative ρ is unbounded, then M R (S) = 0 Let R be a good set and α an irrational number. Suppose the unsigned function ρ ∈ L 1 (µ R,α ) with µ R,α (ρ) = 1 is unbounded, and that the good set S ⊂ R represents the measure ρ · µ R,α at α. Then S must have 0 mean in R, M R (S) = 0.
In contrast to good sets, the representation of absolutely continuous measures by weights can always be accomplished by weights with positive, finite mean. In fact, the representing weight has an additional property.
Definition Integrable weight
We call the weight w integrable if it can be approximated arbitrary closely in the seminorm 1 by bounded, good weights: for every
> 0 there is a good weight v with v ∞ < ∞ so that v − w 1 < .
Theorem Representation by weights
Let R be a good set.
1.
For an irrational α let the unsigned function ρ ∈ L 1 (µ R,α ) satisfy µ R,α (ρ) = 1.
Then there is an integrable weight w on R with M R (w) = 1 which represents the measure ρ · µ R,α at α. If ρ ∈ L ∞ (µ R,α ) then the representing weight w can also satisfy ρ L ∞ (µ R,α ) = w ∞ .
2.
Let w be a good, integrable weight supported on R which satisfies w 1,R > 0.
Then for every β the limit measure µ w,β is absolutely continuous with respect to µ R,β .
In contrast to the case of irrational points, representation of measures at rational points is completely resolved. Consider the rational number α = a q where q ∈ N and a ∈ [q] and let S be a good set. Then, as can be seen readily, the limit probability measure µ S, a q is supported on the set T q of qth roots of unity
T q := j q j ∈ [0, q − 1] . (11.1)
The next result says that this limit measure µ S, a q can be any probability measure supported on T q . generation of measures by statistics of rotations along sets of integers 12
Theorem Every probability measure on T q can be represented
Let q and a be positive integers with gcd(a, q) = 1, and let ν be a probability measure supported on T q . Then ν can be represented at a q , that is, there is a good set S so that µ S, a q = ν.
The techniques developed in this paper allow one to address the simultaneous representability of probability measures at several different points of the torus, and we plan to explore this in a future work. But which family { µ α | α ∈ T } of measures can be represented by a single good set remains open even if we restrict the family to absolutely continuous measures with respect to the Lebesgue probability measure λ. What we can say at this point is that for a given good set S, the set of α ∈ T where the limit measure µ S,α is not the Lebesgue measure is small: it is both of first Baire category and of 0 measure under every Rajchman measure 8 on T.
Basic example for representation
In this section we want to work out a rather simple but instructive example, which will then motivate and form the basis of many of our constructions later on. When we are done with presenting this example, we in fact proved theorem 10.2 in case the Radon-Nikodym derivative is the indicator of a Jordan measurable set.
Let α be irrational and let I ⊂ T be an interval. We want to show that if a probability measure ν is absolutely continuous with respect to λ with the Radon-Nikodym derivative equal 1 I , the indicator of I, then there is a set S which represents ν at α. Probably the simplest way 9 to define such a set S is by taking 9 We could also define such a set by taking { n | n ∈ N, n 2 α ∈ I (mod 1) } or { p | p ∈ Π, pα ∈ I (mod 1) } where Π is the set of primes. Here is a dynamically generated good set: in a dynamical system let A be a set with positive measure and consider { n | n ∈ N, T n x ∈ A }. By the Wiener-Wintner theorem, this set is good for a.e. x.
S = { n | n ∈ N, nα ∈ I } (12.1)
There are two things to verify. First, that S is indeed a good set, and to do that, we need to show that the weak limit µ S,β = lim N A s∈S(N) δ sβ exists for every β. Second, we then have to verify that µ S,α = 1 λ(I) 1 I · λ. The second one, in fact, is almost instantaneous to do since it follows from the uniform distribution of (nα) n∈N (mod 1). To see how it follows, it's enough to show that for every interval J ⊂ T we have
µ S,α (J) = λ 1 J · 1 λ(I) 1 I , that is lim N A s∈S(N) 1 J (sα) = λ 1 J · 1 λ(I) 1 I (12.2)
The right hand side is 1 λ(I) λ(J ∩ I). The left hand side can be written generation of measures by statistics of rotations along sets of integers 13
as lim N A s∈S(N) 1 J (sα) = lim N N #S(N) A n∈[N] 1 I (nα)1 J (nα) since lim N #S(N) N = λ(I) by the uniform distribution of (nα) n∈N (mod 1), = 1 λ(I) lim N A n∈[N] 1 I∩J (nα)
again by the unifom distribution of (nα) n∈N (mod 1)
= 1 λ(I) λ(I ∩ J).
To show that the weak limit µ S,β = lim N A s∈S(N) δ sβ exists for every β, it's enough to show that lim N A s∈S(N) e(sβ) exists for every β. Since
A s∈S(N) e(sβ) = N #S(N) A n∈[N] 1 I (nα) e(nβ) (13.1) and since lim N #S(N) N = λ(I),
it's enough to show that the limit lim N A n∈[N] 1 I (nα) e(nβ) exists for every β ∈ T. To see this, first note that if we replace 1 I by the character e k the limit of A n∈[N] e k (nα) e(nβ) = A n∈[N] e(n(kα + β)) as N → ∞ exists and is as follows
lim N A n∈[N] e k (nα) e(nβ) = 1 if β = −kα (mod 1) 0 otherwise. (13.2)
From this we get that if we replace 1 I by a trigonometric polynomial φ, the limit of A n∈[N] φ(nα) e(nβ) exists and can be given explicitly as 10 10 Notice that in eq. (13
.3) λ(φ e k ) is the kth Fourier coefficient of φ. lim N A n∈[N] φ(nα) e(nβ) = λ(φ e k ) if β = −kα (mod 1) 0 otherwise. (13.3)
Using Weierstrass' theorem on being able to uniformly approximate a continuous function by trigonometric polynomials, we can verify that in eq. (13.3) we can take φ to be any continuous function. Figure 1: Approximating the indicator 1 I of the interval I by continuous functions φ a (from above) and φ b (from below).
I φ b φ a 2 2
Now, for a given > 0 let us choose unsigned continuous func-
tions φ a , φ b so that φ b ≤ 1 I ≤ φ a and λ(φ a − φ b ) < (see fig. 1). We then have A n∈[N] 1 I (nα) e(nβ) − A n∈[N] φ b (nα) e(nβ) = A n∈[N] 1 I (nα) − φ b (nα) e(nβ) ≤ A n∈[N] 1 I (nα) − φ b (nα) ≤ A n∈[N] φ a (nα) − φ b (nα) .
generation of measures by statistics of rotations along sets of integers 14
It follows, since the sequence (nα) n∈N is uniformly distributed
mod 1 and λ(φ a − φ b ) < , that lim sup N A n∈[N] 1 I (nα) e(nβ) − A n∈[N] φ b (nα) e(nβ) < . (14.1) Denoting L( , β) := lim N A n∈[N] φ b (nα) e(nβ), we can rewrite eq. (14.1) as lim sup N A n∈[N] 1 I (nα) e(nβ) − L( , β) < . (14.2) Since lim →0 L( , β) = λ(1 I e k ) if β = −kα (mod 1) 0 otherwise, (14.3) the limit of A n∈[N] 1 I (nα) e(nβ)
as N → ∞ exists and is given by
lim N A n∈[N] 1 I (nα) e(nβ) = λ(1 I e k ) if β = −kα (mod 1) 0 otherwise. (14.4) We finally get, since µ S,β (e) = lim N A s∈S(N) e(nβ) = 1 λ(I) lim N A n∈[N] 1 I (nα) e(nβ), µ S,β (e) = 1 λ(I) λ(1 I e k ) if β = −kα (mod 1) 0 otherwise. (14.5)
The above shows that µ S,β (e) can be nonzero only if β is an integer multiple of α, and we recognize λ(1 I e k ) as the kth Fourier coefficient of the function 1 I , that is, 1 λ(I) λ(1 I e k ) is the kth Fourier coefficient of the measure 1 λ(I) 1 I λ. One can rather easily extend this example in two ways. First, the proof can be repeated almost verbatim for the case when we take any Jordan measurable set B in place of the interval I. Indeed, all we need to remark is that a set B is Jordan measurable iff, for every given > 0, its indicator function 1 B can be approximated by a pair of unsigned, continuous functions φ a and φ b so that φ b ≤ 1 B ≤ φ a and λ(φ a − φ b ) < . This approximability by continuous functions both below and above is also equivalent with saying that the boundary of the set has zero Lebesgue measure.
Definition ν-Riemann integrability
Let ν be a finite Borel measure on T and let φ be a Borel measurable T → C function. We call the function φ ν-Riemann integrable if it's continuous at ν-almost every point. We call the Borel measurable set B ν-Jordan measurable if its indicator function 1 B is ν-Riemann integrable.
generation of measures by statistics of rotations along sets of integers 15
As it is well known, the equivalence of approximability by continuous functions and the boundary having zero measure carries over to the setting of any finite Borel measure on the torus. We can thus extend the example to the setting when the Lebesgue measure is replaced by an arbitrary finite Borel measure.
We record our findings in the following result.
Proposition The Radon-Nikodym derivative can be the indicator of a Jordan measurable set
Let R be a good set, α be an irrational number and let B ⊂ T be
µ R,α -Jordan measurable with µ R,α (B) > 0.
Then the measure 1 B µ R,α , which is absolutely continuous with respect to µ R,α , can be represented at α by the good set S defined by
S := { r | r ∈ R, rα ∈ B } (15.1) so we have µ S,α = 1 µ R,α (B) 1 B µ R,α .
We also have µ R,α (B) = M R (S) and that if β is independent from α over the rationals then
µ S,β = µ R,β .
Let us go back to trying to represent measures which are absolutely continuous with respect to the Lebesgue measure λ. New ideas are needed to cover the case when we want to represent the measure 1 B λ when B is a Borel set which is not Jordan measurable. What is the new difficulty? We'd like to think that we could just again take the "visit set" S = { n | n ∈ N, nα ∈ B }, but this is not the case anymore. Indeed, take B to be an open set with λ(B) < 1 and containing all integer multiples of our irrational α. This open set is not Jordan measurable anymore. The set S cannot represent the measure 1 B λ anymore since S = N. In fact, we show in section 12.3 that for any given irrational α, one can construct an open set B so that the visit set of B doesn't even have mean. So we definitely need new ideas.
We also need new ideas even for the case when we try to represent a measure which is absolutely continuous with respect to the Lebesgue measure with a Radon-Nikodym derivative which is not an indicator function. We need these new ideas even if this Radon-Nikodym derivative is a continuous function.
Proof of theorem 10.2 for indicators
Strictly speaking, we have already begun the proof of theorem 10.2 in the previous section, when we proved that at an irrational number every measure with Jordan measurable Radon-Nikodym derivative can be represented. Our fixed set up in this section is that we are given a good "base" set R ⊂ N and an irrational number α. Since the generation of measures by statistics of rotations along sets of integers 16 set R is fixed throughout the section, we suppress the set R from our notations, so we write
µ β := µ R,β , for every β (16.1) M := M R (16.2)
When we consider the elements of R arranged in the increasing sequence (r n ), we identify a subset S of R with the index set { n | r n ∈ S }, and we denote this index set also by S. The sets we consider in this section have positive mean in R. For such a set S, the non-normalized averages A n∈[N] 1 S (n)δ r n β are easier to handle than the normalized ones A n∈S(N) δ r n β . The convergence or divergence properties of the two averages are equivalent since they are connected as
lim N A n∈[N] 1 S (n)δ r n β = M(S) lim N A n∈S(N) δ r n β (16.3)
Since our focus is to widen the class of the Radon-Nikodym derivatives with respect to the base limit measure µ α , the following definition will simplify our language.
Definition Representing a function, a Borel set
Let ρ ∈ L 1 (T, µ α ) be unsigned and µ α (ρ) > 0. We say that the good set S ⊂ N represents ρ at α if the weak limit of the sequence A n∈[N] 1 S (n)δ r n α N is the measure ρµ α .
If ρ is the indicator of a Borel measurable set B ⊂ T, we then say S ⊂ N represents B at α.
In section 4 we proved that if B is µ α -Jordan measurable, then it can be represented by the set S B defined by
S B = { n | r n α ∈ B } (16.4)
and we have the relation
M(S B ) = µ α (B) (16.5)
We also indicated that this definition of S B may not give a good set if B is not Jordan measurable. The idea of extending the representation to any Borel measurable set is via a limit procedure. To explain what we mean by "a limit procedure", consider the case when B is an open set, and write it as a disjoint union of open intervals, B = ∪ j I j . Defining B k := ∪ j∈[k] I j for every k ∈ N, each B k is Jordan measurable and the sequence (B k ) increases monotonically to B. We have lim k µ α (B k ) = µ α (B). Denoting S k := S B k , the sequence (S k ) also increases to a set S ⊂ N, but M(S) not only may not be equal lim k M(S k ) but M(S) may not even exist. The limit procedure which is suitable for our purposes is determined by the seminorm f 1 .
Our main tools will be two lemmas. The first one is modeled after a result of Marcinkiewicz 11 on the completeness of Besicovitch 11 Marcinkiewicz 1939. spaces.
Lemma Cauchy sequence of sets is convergent in the seminorm 1
Let (S k ) be a Cauchy sequence of subsets of N with respect to the seminorm 1 , so they satisfy
lim k sup l≥k 1 S l − 1 S k 1 = 0 (17.1) Then there is a set S ⊂ N satisfying lim k 1 S k − 1 S 1 = 0 (17.2)
Proof. We construct the set S by pasting together finite pieces of the S k . More precisely, we recursively define a fast enough increasing sequence N 1 < N 2 < . . . of indices, and then we define S to be equal S k on the interval (N k , N k+1 ]
1 S := ∑ k 1 S k 1 (N k ,N k+1 ] (17.3)
For the recursive definition of the (N k ), define first the sequence ( k ) by
k := 2 sup l≥k 1 S l − 1 S k 1 (17.4)
We can assume, without loss of generality, that k > 0 for every k, since k = 0 for some k would imply 1 S l − 1 S k 1 = 0 for l ≥ k hence we could take S = S k .
In the first step of the recursion, let N 1 = 1.
In the second step, let N 2 > N 1 to be large enough to satisfy
N 1 N 2 < 1 (17.5) A [N] 1 S 1 − 1 S 2 < 1 for every N ≥ N 2 (17.6)
and
A [N] 1 S 1 − 1 S 3 < 1 for every N ≥ N 2 (17.7)
Complete the second step of the recursion by defining S to be equal S 1 on the interval (N 1 , N 2 ]. Let k > 2 and assume that we have defined N 1 < N 2 < · · · < N k−1 and S to be equal S j on the interval (N j , N j+1 ] for j ∈ [k − 2]. For step k of the recursion let N k > N k−1 be generation of measures by statistics of rotations along sets of integers 18
large enough to satisfy 1 N k ∑ n∈[N k−1 ] 1 S j (n) − 1 S (n) < j , for every j ∈ [k − 2] (18.1) A [N] 1 S j − 1 S k−1 < j for every N ≥ N k , j ∈ [k − 2] (18.2)
and
A [N] 1 S j − 1 S k < j for every N ≥ N k , j ∈ [k − 2] (18.3)
Complete the kth step of the recursion by defining S to be equal S k−1 on the interval (N k−1 , N k ]. Let us fix j and let N be large enough so that for some k ≥ j + 2 we have
N k ≤ N < N k+1 (18.4)
We want to show that
A [N] 1 S j − 1 S < 3 j (18.5) Let us estimate A [N] 1 S j − 1 S as, A [N] 1 S j − 1 S = 1 N ∑ n∈[N k−1 ] 1 S j (n) − 1 S (n) (18.6) + 1 N ∑ n∈(N k−1 ,N k ] 1 S j (n) − 1 S (n) (18.7) + 1 N ∑ n∈(N k ,N] 1 S j (n) − 1 S (n) (18.8)
We can estimate the term in eq. (18.6), using eq. (18.1) and that N ≥ N k , as 1
N ∑ n∈[N k−1 ] 1 S j (n) − 1 S (n) < j (18.9)
For the term in eq. (18.7) we have
1 N ∑ n∈(N k−1 ,N k ] 1 S j (n) − 1 S (n) < j (18.10) This follows from eq. (18.2) since S = S k−1 on the interval (N k−1 , N k ]. For the term in eq. (18.8) we have 1 N ∑ n∈(N k ,N] 1 S j (n) − 1 S (n) < j (18.11)
This follows from eq. (18.3) since S = S k on the interval (N k , N].
Putting the estimates in eqs. (18.9) to (18.11) together we obtain eq. (18.5).
generation of measures by statistics of rotations along sets of integers 19
The second lemma shows that the family M of sequences with mean is closed with respect to the seminorm M defined in eq. (7.3).
Lemma M is closed with respect to the seminorm M
Let ( f j ) be a sequence from M. Suppose that ( f j ) converges to f ∈ C N in the seminorm M , so lim j f j − f M = 0 (19.1) Then f ∈ M and M( f ) = lim j M( f j ) (19.2)
Proof. First note that, as a consequence of eq. (19.2), the sequence ( f j ) is a Cauchy sequence, meaning that for a given > 0 there is J so that
f j − f J M < for every j ≥ J (19.3) Since M( f j ) − M( f J ) = M( f j − f J ) = f j − f J M we see, M( f j ) − M( f J ) < for every j ≥ J (19.4)
so the sequence M( f j ) of means is a Cauchy sequence of numbers. Denote L := lim j M( f j ). We want to show that M( f ) = L. For a given > 0, choose a j so that M( f j ) − L < and f − f j M < . We then have, for an arbitrary N,
A [N] f − L ≤ A [N] ( f − f j ) + A [N] f j − L (19.5)
Taking lim sup N of both sides, we get Since µ α (B) > 0, we have µ α (B k ) > 0 for large enough k. For simplicity, we assume that µ α (B k ) > 0 for every k. The sets B k increase to B monotonically, hence, in particular, we have lim k µ α (B k B) = 0. According to proposition 15.1, the set B k can be represented by the set S k defined by S k := { n | r n α ∈ B k } (19.7) generation of measures by statistics of rotations along sets of integers 20
lim sup N A [N] f − L ≤ f − f j M + M( f j ) − L (19.6) Since f − f j M < and M( f j ) − L < , we get lim sup N A [N] f − L < 2 . Since > 0 was arbitrary, we have lim N A [N] f − L = 0 which means M( f ) = L = lim j M( f j ).
and we have M(S k ) = µ α (B k ). Since for every k, l the set B k B l is Jordan measurable, we also have
M 1 S k − 1 S l = M(S k S l ) = µ α (B k B l ) (20.1) Since (B k )
is a Cauchy sequence, so lim k sup l≥k µ α (B k B l ) = 0, the isometry in eq. (20.1) implies that (S k ) is also a Cauchy sequence, so we have lim k sup l≥k M(S k S l ) = 0. According to lemma 17.1, there is a set S to which the (S k ) converges, so lim k 1 S k − 1 S 1 = 0, and by lemma 19.1, M(S) = lim k M(S k ) > 0. We want to show that the set S is good and it represents B at α. To this end, let β ∈ T be arbitrary and define the sequences f β k and f β by This is equivalent with saying that S represents B at α. We record the general idea we used as item 2 in proposition 21.1 below.
f β k (n) := 1 S k (n) e(r n β) for n ∈ N (20.2) f β (n) := 1 S (n) e(r n β) for n ∈ N (20.3) Since each set S k is good with M(S k ) > 0, we have f β k ∈ M for every k, β. The fact that for every β, the sequence ( f β k ) converges to f β in the norm M follows from the uniform estimate f β k − f β M ≤ 1 S k − 1 S 1 for every β(
generation of measures by statistics of rotations along sets of integers 21
Proposition Limit of good sets with positive mean is good
Let (S k ) be a sequence of good subsets of R with mean which converge to S ⊂ R in 1 -seminorm, that is, lim k S k S 1 = 0. Assume that lim sup k M(S k ) > 0. Then we have the following.
1. lim k M(S k ) exists and M(S) = lim k M(S k ) > 0.
2. S is a good set.
3. The sequence µ S k ,β k of limit measures converge to µ S,β in variation distance and uniformly in β,
lim k sup β µ S k ,β − µ S,β V = 0 (21.1)
4. Let ν be a Borel measure on T.
If for some α, µ S k ,α is absolutely continuous with respect to ν with Radon-Nikodym derivative ρ k for every k, then µ S,α is also absolutely continuous with respect to ν with Radon-Nikodym derivative ρ which satisfies
lim k ρ k − ρ L 1 (ν) = 0 (21.2)
Proof. The proof of item 1 follows from the triangle inequality for the 1 -seminorm, since we then have
|M(S k ) − M(S)| = | S k 1 − S 1 | ≤ S k S 1
and just use the assumption that lim k S k S 1 = 0. The argument we gave just before the enunciation of our proposition proves that S is a good set.
For the proof of item 3 note that in the argument preceding our proposition we proved that the sequence µ S k ,β k of measures converges weakly to µ S,β for every β but an estimate similar to eq. (20.4) enables us to draw the stronger conclusion of eq. (21.1).
The following lemma gives us the estimates we need.
generation of measures by statistics of rotations along sets of integers 22
22.1 Lemma 1 dominates V and L 1 Let v 1 , v 2 be good weights on R = (r n ). Assume that
max{ v 1 1 , v 2 1 } > 0 (22.1)
Then we have the following.
(a)
sup β µ v 1 ,β − µ v 2 ,β V ≤ 2 max{ v 1 1 , v 2 1 } v 1 − v 2 1 (22.2) (b)
If, for some α, the limit measures µ v 1 ,α and µ v 2 ,α are absolutely continuous with respect to a Borel measure ν on T with Radon-Nikodym derivatives ρ 1 and ρ 2 , respectively, then
ρ 1 − ρ 2 L 1 (ν) ≤ 4 max{ v 1 1 , v 2 1 } v 1 − v 2 1 (22.3)
Proof. To prove item a, that is, the inequality in eq. (22.2), fix β and φ ∈ C + , so φ is a continuous T → C function with 0 ≤ φ ≤ 1. We can assume without loss of generality that max{ v 1 1 , v 2 1 } = v 1 1 . Let (N l ) l be a strictly increasing sequence of indices so that
lim l A [N l ] v 1 = v 1 1 (22.4)
Let us estimate as
A v 1 n∈[N l ] φ(r n β) − A v 2 n∈[N l ] φ(r n β) = 1 A [N l ] v 1 A n∈[N l ] v 1 (n)φ(r n β) − 1 A [N l ] v 2 A n∈[N l ] v 2 (n)φ(r n β) adding 0 = − 1 A [N l ] v 1 A n∈[N l ] v 2 (n)φ(r n β) + 1 A [N l ] v 1 A n∈[N l ] v 2 (n)φ(r n β)
inside the absolute value and using the triangle inequality,
≤ 1 A [N l ] v 1 A n∈[N l ] v 1 (n)φ(r n β) − A n∈[N l ] v 2 (n)φ(r n β) + 1 A [N l ] v 1 − 1 A [N l ] v 2 A n∈[N l ] v 2 (n)φ(r n β) ≤ 1 A [N l ] v 1 A [N l ] |v 1 − v 2 | + A [N l ] |v 1 − v 2 | A [N l ] v 1 A [N l ] v 2 A [N l ] v 2 = 2 A [N l ] v 1 A [N l ] |v 1 − v 2 | so we have A v 1 n∈[N l ] φ(r n β) − A v 2 n∈[N l ] φ(r n β) ≤ 2 A [N l ] v 1 A [N l ] |v 1 − v 2 | (22.5)
generation of measures by statistics of rotations along sets of integers 23
Since lim l A v i n∈[N l ] φ(r n β) = µ v i ,β (φ), lim l A [N l ] v 1 = v 1 1 and lim sup l 2 A [N l ] v 1 A [N l ] |v 1 − v 2 | ≤ 2 v 1 1 v 1 − v 2 1 , we get µ v 1 ,β (φ) − µ v 2 ,β (φ) ≤ 2 v 1 1 v 1 − v 2 1 (23.1)
which is independent of β and φ ∈ C + , proving eq. (22.2). To prove item 3 observe that, since µ v i ,α = ρ i ν and ρ 1 ν − ρ 2 ν V = 1 2 ρ 1 − ρ 2 L 1 (ν) , we have
µ v 1 ,α − µ v 2 ,α V = 1 2 ρ 1 − ρ 2 L 1 (ν) (23.2)
and then use eq. (22.2). Now, using eq. (22.2) with v 1 = 1 S k and v 2 = 1 S , we get
sup β µ S k ,β − µ S,β V ≤ 2 max{ S k 1 , S 1 } S k S 1 (23.3)
Using the assumption that lim k S k S 1 = 0 and that, by item 1, we have lim k S k 1 = lim k M(S k ) = M(S) = S 1 > 0, we get eq. (21.1). For the proof of item 4, by item 1, we can assume, without loss of generality that M(S k ) > 0 for every k. Using eq. (22.3) with v 1 = S k and v 2 = S l we get
ρ l − ρ k L 1 (ν) ≤ 4 max{ S k 1 , S l 1 } S k S l 1 (23.4)
This implies, since the sequence (S k ) is convergent in 1 -seminorm and hence is Cauchy, that the sequence (ρ k ) is Cauchy in L 1 (ν)-norm.
Since L 1 (ν) is complete and ν(ρ k ) = 1 for every k, there is a ρ ∈ L 1 (ν) with ν(ρ) = 1 so that
lim k ρ k − ρ L 1 (ν) = 0 (23.5) Since ρ k − ρ L 1 (ν) = 2 ρ k ν − ρν V and ρ k ν = µ S k ,α , we get lim k µ S k ,α − ρν V = 0 (23.6)
But by item 3 we also have lim k µ S k ,α − µ S,α V = 0 hence we must have µ S,α = ρν.
We can use proposition 21.1 in an argument similar to the one we used to show that any open set can be represented at α to prove that if a G δ set B has positive µ α -measure then it can be represented at α. Only the initial setup of the proof is different. This time let (B k ) be a decreasing sequence of open sets which converges to B. Let S k ⊂ R generation of measures by statistics of rotations along sets of integers 24 represent B k at α. We again have the isometry eq. (20.1) from which everything follows: the existence of a good set S which represents B at α and M(S) = µ α (B).
Since every Borel measurable set differs from a G δ set on a set of µ α -measure zero, we in fact showed that every Borel set of positive µ α -measure can be represented. So we proved the following more precise version of theorem 10.2 for the case when the Radon-Nikodym derivative of a measure with respect to µ α is an indicator.
Proposition Theorem 10.2 for indicator
Let R ⊂ N be a good set, α be an irrational number, and let B be a Borel set with µ α (B) > 0. Then B can be represented at α by a set S ⊂ R which satisfies
M R (S) = µ α (B) > 0 (24.1) 6
Measures that cannot be represented at every irrational α
For this section, we suspend the proof of theorem 10.2 just to see how proposition 24.1 can be used to prove theorem 9.1. We will also prove theorem 9.2.
Proof of theorem 9.1
In this section we want to prove that if the Borel probability measure ν has a point-mass at a point γ ∈ T and α is irrational then ν cannot be represented at α. The proof is by contradiction: let us assume that for some γ ∈ T, ν({γ}) > 0 and that ν can be represented by the set R at α, so µ R,α = ν. Then the Dirac mass δ γ is absolutely continuous with respect to µ R,α with Radon-Nikodym derivative equal 1 ν(γ) 1 {γ} . By proposition 24.1 there is a good set S ⊂ R which represents δ γ at α, so µ S,α = δ γ . Let us define the function φ : T → C as
φ(β) := µ S,β (e) (24.2)
Then, by the definition of µ S,β (e), φ is the limit of the sequence (φ N ) of continuous functions defined by φ N (β) := A n∈[N] e(s n β) where (s n ) is the elements of S arranged in increasing order. Since for every p ∈ Z we have µ S,pα (e) = µ S,α (e p ) and µ S,α (e p ) = e p (γ), we have hold simultaneously for the limit of continuous functions.
Proof of theorem 9.2
So in this section we want to prove that if ν is a Borel probability measure on T with lim sup p→∞ ν(e p ) > 0 then there is an irrational α where ν cannot be represented. In fact the set of such α's is of full Lebesgue measure. From the assumption that lim sup p→∞ ν(e p ) > 0 it follows that there is an > 0 and a infinite sequence p 1 < p 2 < . . . of indices so that |ν e p k | > for k ∈ N (25.
A := { α | (p k α) k∈N is uniformly distributed (mod 1) } (25.2)
has full λ measure. We want to show that A is a subset of those α's at which the measure ν cannot be represented. Let α ∈ A, and suppose the measure ν can be represented at α, say, by the set S = (s n ), that is, µ S,α = ν. Let us define the function
φ : T → C as φ(β) := µ S,β (e) (25.3)
Then, by the definition of µ S,β (e), φ is the limit of the sequence (φ N ) of continuous functions defined by φ N (β) := A n∈[N] e(s n β). Since for every p ∈ Z we have µ S,pα (e) = µ S,α (e p ) and µ S,α (e p ) = ν e p , by eq. (25.1) we have µ S,p k α (e) > for every k ∈ N (25.4)
Since α ∈ A, the sequence (p k α)is uniformly distributed mod 1, hence dense in T. So we have that |φ| > on the dense set { p k α | k ∈ N }. In this section, we fix the good set R and the irrational number α, and we continue in the tradition of section 5 suppressing the set R in our notation, so µ α = µ R,α and M = M R . In trying to extend the class of representable functions ρ from indicators, we first consider an easier problem. Instead of representing by sets, we represent by weights, that is, unsigned sequences w.
Definition Function represented by a weight
Let ρ be an unsigned L 1 (T, µ α ) function with µ α (ρ) > 0. We say the weight w represents ρ at α if w is good and we have lim N A n∈[N] w(n)δ r n α = ρµ α (26.1)
Note that in this case we have M(w) = µ α (ρ) and w represents the measure 1 µ α (ρ) ρµ α at α.
In section 4 we have already seen that if ρ is an unsigned continuous function with µ α (ρ) > 0 then the weight w defined by
w(n) := ρ(r n α) (26.2)
is good, unsigned and it represents ρ at α. Since every unsigned µ α -integrable function can be approximated arbitrary closely by unsigned continuous functions in L 1 (T, µ α )-norm, the proof of item 1 of theorem 11.3 requires only an approximation argument similar to what we had in section 5. We restate item 1 of theorem 11.3 in the following form for the readers convenience.
Proposition Any integrable function is representable with weights
Let ρ be an unsigned function from L 1 (T, µ α ) with µ α (ρ) > 0. Then there is a weight w which represents ρ at α. In particular, we have
M(w) = µ α (ρ) (26.3)
Furthermore, if ρ is a bounded function then the weight w can be chosen to be bounded.
The following analog of lemma 17.1 for weights 17 is the main 17 Marcinkiewicz 1939. ingredient for proving proposition 26.2.
generation of measures by statistics of rotations along sets of integers 27
Lemma Cauchy sequence of weights is convergent in the seminorm 1
Let (w k ) be a Cauchy sequence of weights with respect to the seminorm 1 , so they satisfy
lim k sup l≥k w l − w k 1 = 0 (27.1)
Then there is a weight w satisfying
lim k w k − w 1 = 0 (27.2)
Furthermore, if the sequence (w k ) is uniformly bounded, so sup k w k ∞ < ∞, then we can also have w ∞ ≤ sup k w k ∞ .
The proof of lemma 27.1 is the same as the proof of lemma 17.1: we recursively define a fast enough increasing sequence N 1 < N 2 < . . . of indices, and then we define w to be equal w k on the interval
(N k , N k+1 ] w := ∑ k w k 1 (N k ,N k+1 ] (27.3)
The details of the proof are safely left for the reader. The form of w in eq. (27.3) guarantees that w is unsigned, since each w k is an unsigned sequence, and if the (w k ) is uniformly bounded, then w is a bounded weight. Lemma 27.1, combined with lemmas 19.1 and 22.1, gives the following analog of proposition 21.1 generation of measures by statistics of rotations along sets of integers 28
Proposition Limit of good weights with positive mean is good
Let (w k ) be a sequence of good weights with mean which converge to the weight w in 1 -seminorm, so lim N w k − w 1 = 0. Assume that lim sup k M(w k ) > 0. Then we have the following.
1. lim k M(w k ) exists and lim k M(w k ) = M(w) > 0.
2. w is a good weight.
3. The sequence µ w k ,β k of limit measures converge to µ w,β in variation distance and uniformly in β,
lim k sup β µ w k ,β − µ w,β V = 0 (28.1)
Let ν be a Borel measure on T.
If for some α, µ w k ,α is absolutely continuous with respect to ν with Radon-Nikodym derivative ρ k for every k then µ w,α is also absolutely continuous with respect to ν with Radon-Nikodym derivative ρ which satisfies
lim k ρ k − ρ L 1 (ν) = 0 (28.2)
With this proposition, we can complete the proof of proposition 26.2 exactly as we proved proposition 24.1, using a sequence (ρ k ) of unsigned continuous functions that converge to ρ in L 1 (µ α )-norm. We need to remark only that if ρ is a bounded function, then the sequence (ρ k ) of continuous functions can be chosen to be uniformly bounded.
8 Proof of theorem 10.2 for bounded ρ In this section, we still are working with a fixed good set R of positive integers, an irrational number α, but now we also fix a bounded Borel measurable, unsigned function ρ with µ α (ρ) > 0. We proved in section 7 that ρ can be represented at α by a good, bounded weight w. In this section we will show that there is a good set S ⊂ R which also represents ρ at α, hence proving theorem 10.2 for bounded ρ. It follows from the definition of representation that if the good weight w represents ρ then so does the weight cw for every positive constant c. In particular, we can assume that the weight w representing ρ is bounded by 1. We will show that then there is a set S ⊂ R so that The "construction" of S satisfying eq. (28.3) is done randomly. Our random method requires that we limit the growth of the set R; we need to assume that R is sublacunary. 18
Definition Sublacunary set and weight
The set R is sublacunary if its counting function #R(N) satisfies
lim N log N #R(N) = 0 (29.1)
The weight w supported on the set R is sublacunary if it satisfies
lim N log N w(R(N)) = 0 (29.2)
Our main tool in this section is the following.
Proposition There is a set representing the same measures as a bounded weight
Let w = (w(r)) r∈R be a bounded, sublacunary weight supported on R.
Then there is a set S ⊂ R so that
lim N max β∈T A s∈S(N) e(sβ) − A w r∈R(N) e(rβ) = 0 (29.3)
As a consequence, if the weight w is good then so is the S and we have µ S,β = µ w,β for every β (29.4)
Proof. Since we can always assume that the bound of the weight w is 1, proposition 29.2 follows from the following lemma.
Lemma Random selection of a good set
let σ be a weight on R bounded by 1. We assume that for a constant b > 0 we have
lim inf N σ R(N) log N > b (29.5)
Let (Ω, P) be a probability space and and let (X r ) r∈R be a sequence of totally independent Ω → {0, 1} random variables indexed by R and with distribution P(X r = 1) = σ(r).
Then we have P ω sup N max β∈T ∑ r∈R(N) X r (ω) − σ(r) e(rβ) (log N)σ R(N) < ∞ = 1 (29.6)
generation of measures by statistics of rotations along sets of integers 30
To see that proposition 29.2 indeed follows from lemma 29.3, let
σ = w w ∞
, so σ is bounded by 1. Here we make a bit more complicated argument than needed to show that there is a rate of convergence in eq. (29.3).
The sublacunarity assumption on w implies that σ is sublacunary. We then have, as a consequence of eq. (29.6), that there is an Ω 1 with P(Ω 1 ) = 1 so that for every ω ∈ Ω 1 there is a finite positive constant
C ω with max β∈T 1 σ(R(N)) ∑ r∈R(N) X r (ω) e(rβ) − 1 σ(R(N)) ∑ r∈R(N) σ(r) e(rβ) ≤ C ω log N σ R(N) (30.1) For β = 0, we then have 1 σ(R(N)) ∑ r∈R(N) X r (ω) − 1 ≤ C ω log N σ R(N) (30.2) This implies that if we replace σ(R(N)) by ∑ r∈R(N) X r (ω) in 1 σ(R(N)) ∑ r∈R(N) X r (ω) e(rβ) we make a O log N σ R(N)
error, hence eq. (30.1) implies
max β∈T 1 ∑ r∈R(N) X r (ω) ∑ r∈R(N) X r (ω) e(rβ) − 1 σ(R(N)) ∑ r∈R(N) σ(r) e(rβ) ≤ C ω log N σ R(N) (30.3) Defining S ω ⊂ R by S ω := { r | r ∈ R, X r (ω) = 1 } (30.4)
we can write eq. (30.3) as
max β∈T A s∈S ω (N) e(sβ) − A σ r∈R(N) e(rβ) ≤ C ω log N σ R(N)
for every ω ∈ Ω 1 (30.5) Since σ is a constant multiple of w, we can replace σ by w in eq. (30.5),
max β∈T A s∈S ω (N) e(sβ) − A w r∈R(N) e(rβ) ≤ C ω w ∞ log N w R(N) for every ω ∈ Ω 1 (30.6) Since lim N w ∞ log N w R(N)
= 0, due to the sublacunarity assumption on the weight w, we get eq. (29.3) if we take S = S ω for any ω ∈ Ω 1 . where we'll choose the constant c appropriately later. By the Borel-Cantelli lemma, it's enough to prove
∑ N P max β∈T |Z N (β)| ≥ t N < ∞ (31.1)
The first idea in proving eq. (31.1) is that we do not have to take the maximum over all β ∈ T, but over a finite subset B of T which contains N 3 elements 19 . Since the degree of the trigonometric poly-19 In fact, we can take a set B with as few elements as 10N, but in our applications, 10N won't improve anything over N 3 .
nomial Z N (β) is at most N, we can readily see that sup β∈T |Z N (β)| ≤ N 2 sup β∈T |Z N (β)|. It follows that if we take B N ⊂ T to be an arithmetic progression with |B N | = N 3 then
max β∈T |Z N (β)| ≤ 2 max β∈B N |Z N (β)| (31.2) Hence we have P max β∈T |Z N (β)| ≥ t N ≤ P max β∈B N |Z N (β)| ≥ t N /2 (31.3)
Using the union estimate, we get This estimate says that if Y k , k ∈ [K], are totally independent, mean zero, complex valued random variables with |Y k | ≤ 1, then
P max β∈B N |Z N (β)| ≥ t N /2 ≤ N 3 max β∈B N P(|Z N (β)| ≥ t N /2)(P ∑ k∈[K] Y k ≥ t ≤ 4 max exp − t 2 /8 ∑ k∈[K] E|Y k | 2 , exp(−t/3) for every t > 0.
(31.7) Take K = #R(N) and Y r (β) := (X r − σ(r))e(rβ) for r ∈ R(N). Then |Y r (β)| ≤ 1 so the Y r satisfy the assumption in Bernstein's inequality, hence, with t = t N /2, we get the estimate
P |Z N (β)| ≥ t N /2 ≤ 4 max exp − t 2 N /32 ∑ r∈R(N) E|Y r | 2 , exp(−t N /6) (31.8)
generation of measures by statistics of rotations along sets of integers 32
Since E|Y r (β)| 2 = σ(r)(1 − σ(r)) we have ∑ r∈R(N) E|Y r (β)| 2 ≤ σ R(N) (32.1)
Using that t N = c · (log N)σ R(N) , we get
t 2 N /32 ∑ r∈R(N) E|Y r (β)| 2 = (c 2 /32)(log N)σ R(N) ∑ r∈R(N) E|Y r (β)| 2
using the estimate in eq. (32.1)
≥ (c 2 /32)(log N)σ R(N) σ R(N) = (c 2 /32)(log N) hence exp − t 2 N /32 ∑ r∈R(N) E|Y r (β)| 2 ≤ e −(c 2 /32)(log N) (32.2)
In order to get e −(c 2 /32)(log N) ≤ N −5 = e −5 log N , we need to have c 2 /32 ≥ 5, so it enough to have, since √ 160 < 13, c ≥ 13.
√ b ≥ 5, that is, c ≥ 30 √ b (32.5)
Thus choosing the constant c large enough to satisfy both eqs. (32.3) and (32.5), the estimate in eq. (31.8) implies the one in eq. (31.6).
Notes to lemma 29.3
The type of method we used in lemma 29.3 to estimate trigonometric polynomials goes back to generation of measures by statistics of rotations along sets of integers 33 9 Absolute continuity and positive mean
The general theme of this section is that if a good set or weight has positive mean then it can represent only an absolutely continuous measure. To be specific, we want to prove item 2 of theorems 10.3 and 11.3. Our standing assumption is that R is a sublacunary good set, and hence we suppress it in our notation: We write µ α instead of µ R,α , M instead of M R and S 1 instead of S 1,R . For a given f defined on R, if we consider R as the strictly increasing sequence of integers (r n ) and define F by F(n) := f (r n ) then we actually have M R ( f ) = M(F) and f 1,R = F 1 .
Proof of Item 2 of theorem 10.3
Item 1 of theorem 10.3 says that if ρ is an unsigned L ∞ (µ α ) function with µ α (ρ) > 0 and α is an irrational number then ρ can be represented at α with a good set S ⊂ R satisfying M(S) = µ α (ρ)
ρ L ∞ (µα )
. We have proved this in section 8.
Item 2 of theorem 10.3 says that the converse is also true: if the good set S ⊂ R satisfies S 1 > 0 then the limit measure µ S,β is absolutely continuous with respect to µ β with a bounded Radon-Nikodym derivative ρ β which must satisfy
ρ β L ∞ (µβ) ≤ 1 S 1 for every β (33.1)
This is what we intend to prove now. Since β ∈ T is fixed, we suppress it in our notation, so for example we write µ for µ β and µ S for µ S,β . Let S ⊂ R be such that S 1 > 0. Let us first show that for every β, the limit measure µ S is absolutely continuous with respect to µ. This will follow if we show that for every Borel set B we have
µ S (B) ≤ 1 S 1 µ(B) (33.2)
To see this, it's enough to show that for every unsigned, continuous function φ on T we have
µ S (φ) ≤ 1 S 1 µ(φ) (33.3)
Let φ be such a function and let N 1 < N 2 < . . . be a sequence of generation of measures by statistics of rotations along sets of integers 34
indices for which lim k A n∈[N k ] 1 S (n) = S 1 . We can then estimate as
µ S (φ) = lim N A n∈S(N) φ(r n β) = lim k A n∈S(N k ) φ(r n β) = lim k 1 A n∈[N k ] 1 S (n) A n∈[N k ] 1 S (n)φ(r n β) ≤ lim sup k 1 A n∈[N k ] 1 S (n) A n∈[N k ] φ(r n β) since lim k 1 A n∈[N k ] 1 S (n) = 1 S 1 and lim N A n∈[N] φ(r n β) exists, = 1 S 1 lim N A n∈[N] φ(r n β) = 1 S 1 · µ(φ) proving eq. (33.3). Now, inequality µ ρ β 1 B ≤ 1 S 1 µ(B) applied to the Borel set B = ρ β > 1 S 1
readily gives eq. (33.1).
Proof of item 2 of theorem 11.3
So we need to prove that if the good weight w has positive 1-norm and it is integrable, that is, it can be approximated arbitrary closely by bounded, good weights in 1 -seminorm, then for every irrational β the limit measure µ w,β is absolutely continuous with respect to µ β . Let (w k ) be a sequence of good, bounded weights which converges to w in 1 -seminorm, lim k w k − w 1 = 0. Since | w k 1 − w 1 | ≤ w k − w 1 , we have lim k w k 1 = w 1 > 0, and hence we can assume without loss of generality that w k 1 > 0 for every k. That for every k the measure µ w k ,β is absolutely continuous with respect to µ β for every β follows from
µ w k ,β (B) ≤ w k ∞ w k 1 µ β (B) for every Borel set B (34.1)
The proof of this inequality is almost identical to the proof of the inequality in eq. (33.2), hence we omit it. Now the rest of the proof of theorem 11.3 follows from lemma 22.1.
10 Proof of theorem 10.2 for unbounded ρ
In this section we work with a fixed, sublacunary good set R ⊂ N which we view as a sequence (r n ) arranged in increasing order. As a consequence, we omit R from our notation, so we write µ β instead generation of measures by statistics of rotations along sets of integers 35
of µ R,β and M(w)instead of M R (w). We also fix an irrational number α. Let ρ ∈ L 1 (µ α ). We want to find a sublacunary good set S ⊂ R which represents ρ at α. According to proposition 26.2 there is a good weight w on R which represents ρ at α. Since this weight w has positive relative mean with respect to R, it's a sublacunary weight. The problem is that, as per construction, w is not a bounded weight if ρ is unbounded, hence we cannot use our proposition 29.2 to construct the desired set S. Our main job in this section hence will be to construct a good weight v satisfying the following properties
• v is bounded by 1;
• v is sublacunary;
• v represents the same measure at every β as w, so µ v,β = µ w,β for every β.
Once we have such a good weight v, we can use proposition 29.2 to "construct" the desired good set S.
The weight v will be of the form σw where the weight σ is a decreasing weight, that is, σ(n) ≥ σ(n + 1) for every n ∈ N. That a weight v of this form represents the same measures everywhere is a consequence of the following general but probably familiar resultour main new tool in this section.
First recall the definition of a dissipative sequence of measures on N.
Definition Dissipative sequence of measures
Let (v N ) N∈N be a sequence of finite measures on N. We say, the sequence (
v N ) N∈N is dissipative if lim N v N (j) v N (N) = 0, for every j ∈ N (35.1)
generation of measures by statistics of rotations along sets of integers 36
Proposition Decreasing weights preserve limits
Let w be a weight, (σ N ) N∈N be a sequence of finite measures on N and let x = (x n ) be a sequence from a normed space (X, ).
Denoting v N := σ N · w, we assume the following 1. Each σ N has finite support.
2. The sequence (v N ) is dissipative. 3. For each N the measure σ N is decreasing, σ N (1) ≥ σ N (2) ≥ . . . .
4.
The sequence A w n∈ [N] x n N converges to some y ∈ X,
lim N A w n∈[N] x n = y (36.1)
Then, the sequence A v N j∈N x j N of averages converge to the same limit as the w-weighted averages,
lim N A v N j∈N x j = y (36.2)
At the heart of this result is the following quantitative estimate:
For a given > 0, if K is such that A w n∈[j] x n − y < for j ≥ K then we have A v N j∈N x j − y ≤ + max j∈[K] A w n∈[j] x n − y · v N ([K]) v N (N) (36.3)
for every N ≥ K.
Note that the estimate in eq. (36.3) indeed implies the conclusion of the proposition in eq. (36.2). To see this, let N → ∞ in eq. (36.3).
Then, since (v N ) is a dissipative sequence so lim N v N ([K]) v N (N) = 0, we get that lim sup N A v N j∈N x j − y ≤ . Since > 0 is arbitrary, we get lim N A v N j∈N x j − y = 0.
Proof of proposition 36.1. The main idea of the proof is to write A v N j∈N x j as an average of the w-averages with respect to another measure
q N on N A v N j∈N x j = A q N j∈N A w n∈[j] x n for all N (36.4)
These measures q N will also satisfy q N (N) = v N (N) for every N ∈ N (36.5)
The measure q N appears during performing summation by parts:
generation of measures by statistics of rotations along sets of integers 37 setting σ N (0) := 0, w(0) := 0 and x 0 := 0, we have
A v N j∈N x j = 1 v N (N) ∑ j∈N σ N (j)w(j)x j = 1 v N (N) ∑ j∈N σ N (j) ∑ n∈[j] w(n)x n − ∑ n∈[j−1] w(n)x n = 1 v N (N) ∑ j∈N σ N (j) − σ N (j + 1) ∑ n∈[j] w(n)x n = 1 v N (N) ∑ j∈N σ N (j) − σ N (j + 1) · w([j]) · A w n∈[j] x n
Thus, defining the measure q N by
q N (j) := σ N (j) − σ N (j + 1) · w([j]), for j ∈ N (37.1)
we get the identity in eq. (36.4) once we show that q N really is a measure satisfying eq. (36.5). That q N (j) is unsigned follows from the assumption that the sequence (σ N (j)) j∈N is decreasing for fixed N. That q N (N) = v N (N) follows by setting x j = 1 for every j in the summation by parts argument above since then we get exactly q N (N) = v N (N):
1 = A v N j∈N 1 = 1 v N (N) ∑ j∈N σ N (j) − σ N (j + 1) · w([j]) · A w n∈[j] 1 = 1 v N (N) ∑ j∈N q N (j) · 1 = 1 v N (N) · q N (N)
Using the now obvious identity y = A q N j∈N y together with eq. (36.4), we can now write A
v N j∈N x j − y as A v N j∈N x j − y = A q N j∈N A w n∈[j] x n − y (37.2)
Let > 0. Since we assumed lim N A w n∈ [N] x n = y, there is an
K = K( ) so that A w n∈[j] x n − y < , for j ≥ K (37.3)
Splitting the summation on j in A q N j∈N A w n∈ [j] x n − y into two parts at generation of measures by statistics of rotations along sets of integers 38
K and using the triangle inequality, we get the estimate
A q N j∈N A w n∈[j] x n − y ≤ 1 q N (N) ∑ j∈[K] q N (j) A w n∈[j] x n − y + 1 q N (N) ∑ j>K q N (j) A w n∈[j] x n − y (38.1)
We can estimate the first term as
1 q N (N) ∑ j∈[K] q N (j) A w n∈[j] x n − y ≤ max j∈[K] A w n∈[j] x n − y · q N ([K]) q N (N) (38.
2) Using the definition of q N (j) as given in eq. (37.1), we can estimate
q N ([K]) as q N ([K]) = ∑ j∈[K] σ N (j) − σ N (j + 1) · w([j]) = ∑ j∈[K] σ N (j) w([j]) − w([j − 1]) − σ N (K + 1)w([K]) = ∑ j∈[K] σ N (j)w(j) − σ N (K + 1)w([K]) = ∑ j∈[K] v N (j) − σ N (K + 1)w([K]) ≤ v N ([K])
Using this estimate and that q N (N) = v N (N) in eq. (38.2) we get
1 q N (N) ∑ j∈[K] q N (j) A w n∈[j] x n − y ≤ max j∈[K] A w n∈[j] x n − y · v N ([K]) v N (N) (38.
3) The second term in eq. (38.1) can be estimated, using eq. (37.3), as
1 q N (N) ∑ j>K q N (j) A w n∈[j] x n − y ≤ (38.4)
Putting the estimates in eqs. (38.3) and (38.4) into eq. (38.1) and using the identity in eq. (37.2) we get eq. (36.2).
generation of measures by statistics of rotations along sets of integers 39
Corollary Decreasing weights preserve limit measures of weights
Let w and σ be weights. Denoting v := σ · w, we assume the following
1. v(N) = ∞.
2. The weight σ is decreasing σ(1) ≥ σ(2) ≥ . . . .
3.
The weight w is good.
Then the weight v is good and represents the same measures everywhere as w, and (x n ) defined by x n := e(r n β) (39.4)
µ v,β = µ w,
Let us now go back to our good weight w which represents ρ at α. Since we assumed that R is sublacunary and M(w) > 0, the weight w is also sublacunary. Recall that we obtained w as the limit of a sequence (w k ) of bounded good weights. In fact, we pasted w together from the w k piece by piece in a sense that after choosing indices N 1 < N 2 < . . . , we define w to be equal w k on the interval (N k , N k+1 ]
w := ∑ k w k 1 (N k ,N k+1 ] .
(39.5)
In order to obtain a good weight v which is bounded by 1 and would represent the same measures as w, we could do the following. Define σ by
σ := 1 max j∈[k] w j ∞ · 1 (N k ,N k+1 ] (39.6)
Then σ is a decreasing and the weight v := σw is bounded by 1. The remaining issue is that v may not be sublacunary. But in the recursive process of choosing the indices (N k ) if we choose N k large enough compared to N k−1 we can ensure that v is sublacunary. Here is what we need to do for this.
generation of measures by statistics of rotations along sets of integers 40
We set F(N) := log(r N ). Since the weight w is sublacunary and have positive mean, we have
lim N w([N]) F(N) = ∞ (40.1)
So now, if we make sure that our weight v, bounded by 1, satisfies
lim N v([N]) F(N) = ∞ (40.2)
then it will be a sublacunary weight. We want to show that we can choose the indices N k so that we will have eq. (40.2).
In the proof of lemma 27.1 (and in lemma 17.1) we can see that, during the recursive construction of the sequence (N k ) we can choose N k arbitrary large compared to N k−1 . For our purposes, we just need to choose N k large enough to satisfy the following additional crite-
rion N max j∈[k] w j ∞ > kF(N) for every N ≥ N k (40.3)
which is possible since the set R is sublacunary, since eq. (40.3) ensures that the weight v = σw is itself sublacunary.
That v represents the same measures as w at every β follows from corollary 39.1. As in the last step of our proof of theorem 10.2, we use proposition 29.2 to show the existence of a good set S ⊂ R which represents the same measures as v at every β, hence at β = α we have µ S,α = ρµ α .
11
The limit measure at rational points In this section we want to prove theorem 12.1. The base set is N which we suppress in our notation, so we write µ β instead of µ N,β .
Given the probability measure ν on T q and the rational number a q , gcd(a, q) = 1, let us see what properties a good set S would need to have so that µ S,a/q = ν.
Introducing the sets S j by
S j := { s | s ∈ S, sa ≡ j (mod q) }, for every 0 ≤ j ≤ q − 1 (40.4)
let us write, using that the S j are pairwise disjoint, Since µ S,a/q is supposed to be equal ν, we get
A s∈S(N) = 1 #S(N) ∑ s∈S(N) δ sa/q = 1 #S(N) ∑ 0≤j≤q−1 ∑ s∈S j (N) δ j/q = ∑ 0≤j≤q−1 #S j (N) #S(N) δ j/qlim N #S j (N) #S(N) = ν(j/q) (41.2)
This gives us the idea how to construct S: we start out from the set R j defined by
R j := { n | na ≡ j (mod q) }, for every 0 ≤ j ≤ q − 1 (41.3)
Note that R j is a full residue class mod q, namely, if j denotes the unique solution to the congruence j a ≡ j (mod q), then R j is the arithmetic progression { kq + j | k ∈ N }. Note that R j is a good set, as are all arithmetic progressions. We clearly have
M(R j ) = 1 q for every 0 ≤ j ≤ q − 1 (41.4)
Now what remains is to find a set S j ⊂ R j with relative mean ν j q and make sure that S j is a good set. Let γ be an irrational number and consider S j := r r ∈ R j , rγ ∈ 0, ν j q for every 0 ≤ j ≤ q − 1 (41.5)
Using proposition 15.1 with α = γ and R = R j , we deduce that S j is a good set with M R j (S j ) = ν j q , as desired. We finally define S as
S := 0≤j≤q−1 S j (41.6)
The set S is good since it's the finite union of pairwise disjoint good sets with mean. Indeed, we have M(S j ) = 1 q · ν j q and hence M(S) = 1 q .
Examples
12.1 Two good sets, but their intersection has no mean.
Here we construct randomly two good sets, R, S with M(R) = M(S) = 1/2 but M(R ∩ S) doesn't exist.
Let (X n ) be a iid sequence of random variables on the probability space (Ω, P), modeling fair coin flipping, so with distribution P(X n = generation of measures by statistics of rotations along sets of integers 42 1) = P(X n = 0) = 1/2. Let us also consider another sequence of random variables (Y n ) defined by Y n = X n if n ∈ [2 k , 2 k+1 ) for even k 1 − X n if n ∈ [2 k , 2 k+1 ) for odd k (42.1)
The (Y n ) is also an iid sequence with the same distribution as the (X n ). Define the sets R ω , S ω by R ω := { n | X n (ω) = 1 } and S ω := { n | Y n (ω) = 1 }. By lemma 29.3 both R ω and S ω are good sets almost surely with M(R ω ) = M(S ω ) = 1/2. We claim that M(R ω ∩ S ω ) almost surely doesn't exists. To see this, denote T ω := R ω ∩ S ω and observe that if M(T ω ) existed then lim k T ω ∩[2 k ,2 k+1 ) 2 k would exist. But, denoting by O the odd numbers and by E the even numbers, we almost surely have
lim k∈O T ω ∩ [2 k , 2 k+1 ) 2 k = 0 lim k∈E T ω ∩ [2 k , 2 k+1 ) 2 k = 1 2
12.2 R 1 ∪ R 2 and R 1 ∩ R 2 have means but are not good
Here is an example of two good sets R 1 and R 2 each with mean 2/3, M(R 1 ∩ R 2 ) = 1/2 but R 1 ∩ R 2 is not good and M(R 1 ∪ R 2 ) = 5/6 but R 1 ∪ R 2 is not good. Both sets will be defined in blocks of intervals . Partition N into a sequence of disjoint intervals I n so that their lengths go to infinity but slower than the left endpoints go to infinity. For example, I n = [n 2 , (n + 1) 2 ) will do.
The first good set R 1 will contain all iNtegers from I 1 , then only Odd numbers from I 2 then Even numbers from I 3 then repeat this pattern for I 4 , I 5 , I 6 etc:
NOENOE . . . (42.2)
The set R 2 is defined similarly, except it will have one pattern in intervals J k := [3 k , 3 k+1 ) for even k and another for odd k. Both of these sets are good and they represent the same (uniform) measure at every β.
The intersection R 1 ∩ R 2 has the patterns EOEEOE . . . for even k, Clearly M(R 1 ∩ R 2 ) = 1/2 but the average of e(n/2) is different on J k for even k from those on odd k: for even k the average will go to 1/3 while for odd k it goes to −1/3.
As for the union R 1 ∪ R 2 , it has the patterns NONNON . . . for even k, Clearly M(R 1 ∪ R 2 ) = 5/6 but the average of e(n/2) is different on J k for even k from those on odd k: for even k the average will go to −1/3 while for odd k it goes to 1/3.
12.3
Open set U with visit set { n | nα ∈ U } not good
Let α be an irrational number in the torus R/Z. We show that there exists an open subset U of the torus such that the sequence A n∈[N] 1 U (nα) N does not converge when N goes to infinity.
The construction does not use at all the group structure or the dimensional properties of the torus. This can be extended in a general context of a sequence in a compact metric space with a non purely atomic asymptotic distribution.
We want to construct an open subset U of the torus and an increasing sequence of positive integers (N k ) k≥0 such that the averages A n∈[N 2k ] 1 U (nα), k = 0, 1, 2, . . . , with even indices are large whereas the averages A n∈[N 2k+1 ] 1 U (nα), k = 0, 1, 2, . . . with odd indices are small.
The sequence (N k ) will be constructed by induction and each N k will be associated to k := 1/(2 k+4 N k ). In this induction process, we construct also a sequence of open subsets (U k ) k≥0 .
We start with N 0 > 1 fixed and we define
U 0 := n∈[N 0 ]
(nα − 0 , nα + 0 )
We have of course A n∈[N 0 ] 1 U 0 (nα) = 1 and 0 < λ(U 0 ) ≤ 2N 0 0 This is the initial step of our construction. In order to be understandable, let us describe the two next steps.
By the uniform distribution of the sequence (nα) n in the torus, there exists a number N 1 > N 0 such that A n∈[N 0 ] 1 U 0 (nα) ≤ 2λ(U 0 )N 1 ≤ 4(N 0 0 )N 1 We fix such a N 1 . To any n ∈ [N 1 ] ∩ U 0 c we associate a real δ n that 0 < δ n ≤ 1 and (nα − δ n , nα + δ n ) ∩ U 0 = ∅.
generation of measures by statistics of rotations along sets of integers 44
We define U 1 := n∈[N 1 ]∩U 0 c (nα − δ n , nα + δ n )
We have A n∈[N 1 ] 1 U 1 (nα) ≥ 1 − 4N 0 0 and 0 < λ(U 1 ) ≤ 2N 1 1 Note also that by construction U 0 ∩ U 1 = ∅. By the uniform distribution of the sequence (nα) n in the torus, there exists a number N 2 > N 1 such that A n∈[N 2 ] 1 U 1 (nα) ≤ 2λ(U 1 )N 2 ≤ 4(N 1 1 )N 2 We fix such a N 2 . To any n ∈ [N 2 ] ∩ U 1 c we associate a real δ n that 0 < δ n ≤ 2 and (nα − δ n , nα + δ n ) ∩ U 1 = ∅ Note that the values of the δ n 's are reinitialized.
We define
U 2 := U 0 ∪ n∈[N 2 ]∩U 1 c (nα − δ n , nα + δ n )
We have A n∈[N 2 ] 1 U 2 (nα) ≥ 1 − 4N 1 1 and λ(U 2 ) ≤ 2N 0 0 + 2N 2 2 Note also that by construction U 2 ∩ U 1 = ∅ and U 0 ⊂ U 2 . Let us state now our induction hypothesis. Suppose that, for a fixed integer k > 0 we have already constructed two sequences (U ) 0≤ ≤k and N 0 < N 1 < N 2 < . . . < N k such that • U 0 ⊂ U 2 ⊂ U 4 ⊂ . . . and U 1 ⊂ U 3 ⊂ U 5 ⊂ . . .,
• If is even and is odd, then U and U are disjoint,
• Each U is a finite union of open intervals,
• If 0 ≤ 2 ≤ k, then λ(U 2 ) ≤ 2(N 0 0 + N 2 2 + . . . + N 2 2 ) and A n∈[N 2 ] 1 U 2 (nα) ≥ 1 − 4(N 1 1 + N 3 3 + . . . + N 2 −1 2 −1 ) generation of measures by statistics of rotations along sets of integers 45
• If 1 ≤ 2 + 1 ≤ k, then λ(U 2 +1 ) ≤ 2(N 1 1 + N 3 3 + . . . + N 2 +1 2 +1 ) and A n∈[N 2 +1 ] 1 U 2 +1 (nα) ≥ 1 − 4(N 0 0 + N 2 2 + . . . + N 2 2 )
Here begins the induction process. By the uniform distribution of the sequence (nα) n in the torus, there exists a number N k+1 > N k such that
A n∈[N k+1 ] 1 U k (nα) ≤ 2λ(U k )N k+1
We fix such a N k+1 . To any n ∈ [N k+1 ] ∩ U k c we associate a real δ n that 0 < δ n ≤ k+1 and (nα − δ n , nα + δ n ) ∩ U k = ∅
We define
Note that the values of δ n 's are reinitialized at each induction step.
U k+1 := U k−1 ∪ n∈[N k+1 ]∩U k c (nα − δ n , nα + δ n )
The items of the induction hypothesis are now satisfied by the sequences (U ) 0≤ ≤k+1 and (N ) 0≤ ≤k+1 .
We can consider these sequences as infinite, and we define U := k≥0 U 2k . Recalling our choice N k k = 2 −k−4 , we obtain
A n∈[N 2k ] 1 U (nα) ≥ A n∈[N 2k ] 1 U 2k (nα) ≥ 1 − 4 ∑ N 2 +1 2 +1 = 5/6 and A n∈[N 2k+1 ] 1 U (nα) ≤ A n∈[N 2k+1 ] 1 U 2k+1 c (nα)
≤ 4 ∑ N 2 e 2 = 1/3
2
Notations, definitions Natural numbers N The set {1, 2, 3, . . . } of natural numbers is denoted by N. Torus T We identify the torus T := R/Z with the unit right-open interval, T := [0,
torists [N] := {1, 2, . . . , N} (3.4) Nth initial segment S(N) We denote by S(N) the Nth initial segment of S ⊂ N S(N) := S ∩ [N] (3.5)
is the indicator of a set S ⊂ N, we then often write M(S) in place of M(1 S ). Of course, M(S) is the density of S.
do we now show that every open set can be represented? Let B ⊂ T be open with positive µ α measure, let B = ∪ j I j be its decomposition into pairwise disjoint open intervals I j and set B k := ∪ j∈[k] I j .
in particular, S is good. Let now β = α. Since the sequence (B k ) converges to B in L 1 (µ α )-norm we have lim k µ α e p 1 B k = µ α e p 1 B for every p ∈ Z (20.6)Since M( f pα k ) = µ α e p 1 B k and, by eq. (20.5), lim k M( f pα k ) = M( f pα ), eq. (20.6) implies that M( f pα ) = µ α e p 1 B for every p ∈ Z (20.7)
's theorem, 15 φ = 0 on a set of full Lebesgue measure, so, as 15 Weyl 1916, Satz 21; Kuipers and Niederreiter 1974, Theorem 4.1. 's theorem, 16 eqs. (25.5) and (25.6) cannot be true together for 16 Baire 1995, Page 83. the limit of continuous functions.
A
n∈[N] 1 S (n) e(r n β) − A n∈[N] w(n) e(r n β) = 0 (28.3) generation of measures by statistics of rotations along sets of integers 29
Proof of lemma 29.3. To see clearly what we need to do, denoteZ N (β) := ∑ r∈R(N) X r (ω) − σ(r) e(rβ)
P
|Z N (β)| ≥ t N /2 < 2 N 5 for every β ∈ T (31.6)To prove eq. (31.6), we use Bernstein-Chernoff exponential estimate. 20 20 Tao and Vu 2010, Exercise 1.3.4 with t = λσ.
N ≤ N −5 = e −5 log N which poses the requirement (c/6)
EONEON . . . for even k, (42.3) ONEONE . . . for odd k. (42.4)
1 )
1By Weyl's result, 14 the set A ⊂ T defined by14 Weyl 1916, Satz 21; Kuipers and
Niederreiter 1974, Theorem 4.1.
Recent developments21 Salem and Zygmund 1954, Chapter
IV.
have been given for example by Weber 22 and by Cohen-Cuny. 23
22 Weber 2000.
23 Cohen and Cuny 2006.
Proof. We need to show that for a given β we have to do this, use proposition 36.1 with σ N as σ restricted to the set [N],β for every β.
(39.1)
lim
N
A v
n∈[N] e(r n β) = µ w,β (e)
(39.2)
σ N (n) := σ(n)1 [N] (n)
(39.3)
generation of measures by statistics of rotations along sets of integers 41If we make the assumption 24 that lim N #S j (N) #S(N) exists for every j then,24 In fact, the existence of lim N follows from S being a good set.letting N → ∞, we get µ S,a/q = ∑#S j (N)
#S(N)
0≤j≤q−1
δ j/q lim
N
#S j (N)
#S(N)
(41.1)
5 )
5generation of measures by statistics of rotations along sets of integers 43 OOEOOE . . . for odd k, (43.1)
2 )
2NNENNE . . . for odd k, (43.3)
Lyons 1985, Theorem 3; see alsoLyons 1995.
Jones, Lacey, and Wierdl 1999, Theorem B.
Leçons sur les fonctions discontinues. Les Grands Classiques Gauthier-Villars. René Baire, Gauthier-Villars Great ClassicsBaire, René (1995). Leçons sur les fonctions discontinues. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics].
isbn: 2-87647-124-8. generation of measures by statistics of rotations along sets of integers 46. Reprint of the 1905 original. Éditions Jacques Gabay. Sceaux65Reprint of the 1905 original. Éditions Jacques Gabay, Sceaux, pp. viii+65. isbn: 2-87647-124-8. generation of measures by statistics of rotations along sets of integers 46
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"Karen Livescu [email protected] \nToyota Technological Institute at Chicago\n60637ChicagoILUSA\n"
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"Toyota Technological Institute at Chicago\n60637ChicagoILUSA",
"Toyota Technological Institute at Chicago\n60637ChicagoILUSA",
"Toyota Technological Institute at Chicago\n60637ChicagoILUSA"
] | [
"Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing"
] | We present CHARAGRAM embeddings, a simple approach for learning character-based compositional models to embed textual sequences. A word or sentence is represented using a character n-gram count vector, followed by a single nonlinear transformation to yield a low-dimensional embedding. We use three tasks for evaluation: word similarity, sentence similarity, and part-of-speech tagging. We demonstrate that CHARAGRAM embeddings outperform more complex architectures based on character-level recurrent and convolutional neural networks, achieving new state-of-the-art performance on several similarity tasks. 1 | 10.18653/v1/d16-1157 | [
"https://www.aclweb.org/anthology/D16-1157.pdf"
] | 3,202,289 | 1607.02789 | 87f1bf0d961668346ab028b04c7b5d99a1522b5c |
CHARAGRAM: Embedding Words and Sentences via Character n-grams
November 1-5, 2016
John Wieting [email protected]
Toyota Technological Institute at Chicago
60637ChicagoILUSA
Mohit Bansal [email protected]
Toyota Technological Institute at Chicago
60637ChicagoILUSA
Kevin Gimpel [email protected]
Toyota Technological Institute at Chicago
60637ChicagoILUSA
Karen Livescu [email protected]
Toyota Technological Institute at Chicago
60637ChicagoILUSA
CHARAGRAM: Embedding Words and Sentences via Character n-grams
Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing
the 2016 Conference on Empirical Methods in Natural Language ProcessingAustin, TexasNovember 1-5, 2016
We present CHARAGRAM embeddings, a simple approach for learning character-based compositional models to embed textual sequences. A word or sentence is represented using a character n-gram count vector, followed by a single nonlinear transformation to yield a low-dimensional embedding. We use three tasks for evaluation: word similarity, sentence similarity, and part-of-speech tagging. We demonstrate that CHARAGRAM embeddings outperform more complex architectures based on character-level recurrent and convolutional neural networks, achieving new state-of-the-art performance on several similarity tasks. 1
Introduction
Representing textual sequences such as words and sentences is a fundamental component of natural language understanding systems. Many functional architectures have been proposed to model compositionality in word sequences, ranging from simple averaging (Mitchell and Lapata, 2010;Iyyer et al., 2015) to functions with rich recursive structure (Socher et al., 2011;Zhu et al., 2015;Tai et al., 2015;Bowman et al., 2016). Most work uses words as the smallest units in the compositional architecture, often using pretrained word embeddings or learning them specifically for the task of interest (Tai et al., 2015;He et al., 2015).
Some prior work has found benefit from using character-based compositional models that encode arbitrary character sequences into vectors. Examples include recurrent neural networks (RNNs) and convolutional neural networks (CNNs) on character sequences, showing improvements for several NLP tasks (Ling et al., 2015a;Kim et al., 2015;Ballesteros et al., 2015;dos Santos and Guimarães, 2015). By sharing subword information across words, character models have the potential to better represent rare words and morphological variants.
Our approach, CHARAGRAM, uses a much simpler functional architecture. We represent a character sequence by a vector containing counts of character n-grams, inspired by Huang et al. (2013). This vector is embedded into a low-dimensional space using a single nonlinear transformation. This can be interpreted as learning embeddings of character n-grams, which are learned so as to produce effective sequence embeddings when a summation is performed over the character n-grams in the sequence.
We consider three evaluations: word similarity, sentence similarity, and part-of-speech tagging. On multiple word similarity datasets, CHARAGRAM outperforms RNNs and CNNs, achieving state-ofthe-art performance on SimLex-999 (Hill et al., 2015). When evaluated on a large suite of sentencelevel semantic textual similarity tasks, CHARA-GRAM embeddings again outperform the RNN and CNN architectures as well as the PARAGRAM-PHRASE embeddings of Wieting et al. (2016). We also consider English part-of-speech (POS) tagging using the bidirectional long short-term memory tagger of Ling et al. (2015a). The three architectures reach similar performance, though CHARAGRAM converges fastest to high accuracy.
We perform extensive analysis of our CHARA-GRAM embeddings. We find large gains in performance on rare words, showing the empirical benefit of subword modeling. We also compare performance across different character n-gram vocabulary sizes, finding that the semantic tasks benefit far more from large vocabularies than the syntactic task. However, even for challenging semantic similarity tasks, we still see strong performance with only a few thousand character n-grams.
Nearest neighbors show that CHARAGRAM embeddings simultaneously address differences due to spelling variation, morphology, and word choice. Inspection of embeddings of particular character ngrams reveals etymological links; e.g., die is close to mort. We release our resources to the community in the hope that CHARAGRAM can provide a strong baseline for subword-aware text representation.
Related Work
We first review work on using subword information in word embedding models. The simplest approaches append subword features to word embeddings, letting the model learn how to use the subword information for particular tasks. Some added knowledge-based morphological features to word representations (Alexandrescu and Kirchhoff, 2006;El-Desoky Mousa et al., 2013). Others learned embeddings jointly for subword units and words, defining simple compositional architectures (often based on addition) to create word embeddings from subword embeddings (Lazaridou et al., 2013;Botha and Blunsom, 2014;Qiu et al., 2014;Chen et al., 2015).
A recent trend is to use richer functional architectures to convert character sequences into word embeddings. Luong et al. (2013) used recursive models to compose morphs into word embeddings, using unsupervised morphological analysis. Ling et al. (2015a) used a bidirectional long shortterm memory (LSTM) RNN on characters to embed arbitrary word types, showing strong performance for language modeling and POS tagging. Ballesteros et al. (2015) used this model to represent words for dependency parsing. Several have used character-level RNN architectures for machine translation, whether for representing source or target words (Ling et al., 2015b;Luong and Man-ning, 2016), or for generating entire translations character-by-character (Chung et al., 2016). Sutskever et al. (2011) and Graves (2013) used character-level RNNs for language modeling. Others trained character-level RNN language models to provide features for NLP tasks, including tokenization and segmentation (Chrupała, 2013;Evang et al., 2013), and text normalization (Chrupała, 2014).
CNNs with character n-gram filters have been used to embed arbitrary word types for several tasks, including language modeling (Kim et al., 2015), part-of-speech tagging (dos Santos and Zadrozny, 2014), named entity recognition (dos Santos and Guimarães, 2015), text classification (Zhang et al., 2015), and machine translation (Costa-Jussà and Fonollosa, 2016). Combinations of CNNs and RNNs on characters have also been explored (Józefowicz et al., 2016).
Most closely-related to our approach is the DSSM (instantiated variously as "deep semantic similarity model" or "deep structured semantic model") developed by Huang et al. (2013). For an information retrieval task, they represented words using feature vectors containing counts of character n-grams. Sperr et al. (2013) used a very similar technique to represent words in neural language models for machine translation. Our CHARAGRAM embeddings are based on this same idea. We show this strategy to be extremely effective when applied to both words and sentences, outperforming character LSTMs like those used by Ling et al. (2015a) and character CNNs like those from Kim et al. (2015).
Models
We now describe models that embed textual sequences using their characters, including our CHARAGRAM model and the baselines that we compare to. We denote a character-based textual sequence by x = x 1 , x 2 , ..., x m , which includes space characters between words as well as special start-of-sequence and end-of-sequence characters. We use x j i to denote the subsequence of characters from position i to position j inclusive, i.e., x j i = x i , x i+1 , ..., x j , and we define x i i = x i . Our CHARAGRAM model embeds a character sequence x by adding the vectors of its character n-grams followed by an elementwise nonlinearity:
g CHAR (x) = h b + m+1 i=1 i j=1+i−k I x i j ∈ V W x i j
(1) where h is a nonlinear function, b ∈ R d is a bias vector, k is the maximum length of any character ngram, I[p] is an indicator function that returns 1 if p is true and 0 otherwise, V is the set of character ngrams included in the model, and W x i j ∈ R d is the vector for character n-gram x i j . The set V is used to restrict the model to a predetermined set (vocabulary) of character n-grams. Below, we compare several choices for V . The number of parameters in the model is d + d|V |. This model is based on the letter n-gram hashing technique developed by Huang et al. (2013). One can also view Eq. (1) (as they did) as first populating a vector of length |V | with counts of character ngrams followed by a nonlinear transformation.
We compare the CHARAGRAM model to two other models. First we consider LSTM architectures (Hochreiter and Schmidhuber, 1997) over the character sequence x, using the version from Gers et al. (2003). We use a forward LSTM over the characters in x, then take the final LSTM hidden vector as the representation of x. Below we refer to this model as "charLSTM."
We also compare to convolutional neural network (CNN) architectures, which we refer to below as "charCNN." We use the architecture from Kim (2014) with a single convolutional layer followed by an optional fully-connected layer. We use filters of varying lengths of character n-grams, using two primary configurations of filter sets, one of which is identical to that used by Kim et al. (2015). Each filter operates over the entire sequence of character n-grams in x and we use max pooling for each filter. We tune over the choice of nonlinearity for both the convolutional filters and for the optional fullyconnected layer. We give more details below about filter sets, n-gram lengths, and nonlinearities.
We note that using character n-gram convolutional filters is similar to our use of character ngrams in the CHARAGRAM model. The difference is that, in the CHARAGRAM model, the n-gram must match exactly for its vector to affect the representa-tion, while in the CNN each filter will affect the representation of all sequences (depending on the nonlinearity being used). So the CHARAGRAM model is able to learn precise vectors for particular character n-grams with specific meanings, while there is pressure for the CNN filters to capture multiple similar patterns that recur in the data. Our qualitative analysis shows the specificity of the learned character ngram vectors learned by the CHARAGRAM model.
Experiments
We perform three sets of experiments. The goal of the first two (Section 4.1) is to produce embeddings for textual sequences such that the embeddings for paraphrases have high cosine similarity. Our third evaluation (Section 4.2) is a classification task, and follows the setup of the English part-of-speech tagging experiment from Ling et al. (2015a).
Word and Sentence Similarity
We compare the ability of our models to capture semantic similarity for both words and sentences. We train on noisy paraphrase pairs from the Paraphrase Database (PPDB; Ganitkevitch et al., 2013) with an L 2 regularized contrastive loss objective function, following the training procedure of Wieting et al. (2015) and Wieting et al. (2016). More details are provided in the supplementary material.
Datasets
For word similarity, we focus on two of the most commonly used datasets for evaluating semantic similarity of word embeddings: WordSim-353 (WS353) (Finkelstein et al., 2001) and SimLex-999 (SL999) (Hill et al., 2015). We also evaluate our best model on the Stanford Rare Word Similarity Dataset (Luong et al., 2013).
For sentence similarity, we evaluate on a diverse set of 22 textual similarity datasets, including all datasets from every SemEval semantic textual similarity (STS) task from 2012 to 2015. We also evaluate on the SemEval 2015 Twitter task and the SemEval 2014 SICK Semantic Relatedness task (Marelli et al., 2014). Given two sentences, the aim of the STS tasks is to predict their similarity on a 0-5 scale, where 0 indicates the sentences are on different topics and 5 indicates that they are completely equivalent.
Each STS task consists of 4-6 datasets covering a wide variety of domains, including newswire, tweets, glosses, machine translation outputs, web forums, news headlines, image and video captions, among others. Most submissions for these tasks use supervised models that are trained and tuned on provided training data or similar datasets from older tasks. Further details are provided in the official task descriptions (Agirre et al., 2012;Agirre et al., 2013;Agirre et al., 2014;Agirre et al., 2015).
Preliminaries
For training data, we use pairs from PPDB. For word similarity experiments, we train on word pairs and for sentence similarity, we train on phrase pairs. PPDB comes in different sizes (S, M, L, XL, XXL, and XXXL), where each larger size subsumes all smaller ones. The pairs in PPDB are sorted by a confidence measure and so the smaller sets contain higher precision paraphrases. PPDB is derived automatically from naturally-occurring bilingual text, and versions of PPDB have been released for many languages without the need for any manual annotation (Ganitkevitch and Callison-Burch, 2014).
Before training the CHARAGRAM model, we need to populate V , the vocabulary of character n-grams included in the model. We obtain these from the training data used for the final models in each setting, which is either the lexical or phrasal section of PPDB XXL. We tune over whether to include the full sets of character n-grams in these datasets or only those that appear more than once.
When extracting n-grams, we include spaces and add an extra space before and after each word or phrase in the training and evaluation data to ensure that the beginning and end of each word is represented. We note that strong performance can be obtained using far fewer character n-grams; we explore the effects of varying the number of n-grams and the n-gram orders in Section 4.4.
We used Adam (Kingma and Ba, 2014) with a learning rate of 0.001 to learn the parameters in the following experiments.
Word Embedding Experiments
Training and Tuning For hyperparameter tuning, we used one epoch on the lexical section of PPDB XXL, which consists of 770,007 word pairs. We used either WS353 or SL999 for model selection (reported below). We then took the selected hyperparameters and trained for 50 epochs to ensure that all models had a chance to converge. Full details of our tuning procedure are provided in the supplementary material. In short, we tuned all models thoroughly, tuning the activation functions for CHARAGRAM and charCNN, as well as the regularization strength, mini-batch size, and sampling type for all models. For charCNN, we experimented with two filter sets: one uses 175 filters for each ngram size ∈ {2, 3, 4}, and the other uses the set of filters from Kim et al. (2015), consisting of 25 filters of size 1, 50 of size 2, 75 of size 3, 100 of size 4, 125 of size 5, and 150 of size 6. We also experimented with using dropout (Srivastava et al., 2014) on the inputs to the final layer of charCNN in place of L 2 regularization, as well as removing the last feedforward layer. Neither variation significantly improved performance on our suite of tasks for word or sentence similarity. However, using more filters does improve performance, apparently linearly with the square of the number of filters.
Architecture Comparison
The results are shown in Table 1. The CHARAGRAM model outperforms both the charLSTM and charCNN models, and also outperforms recent strong results on SL999.
We also found that the charCNN and charLSTM models take far more epochs to converge than the CHARAGRAM model. We noted this trend across experiments and explore it further in Section 4.3.
Comparison to Prior Work We found that performance of CHARAGRAM on word similarity tasks can be improved by using more character n-grams. This is explored in Section 4.4. Our best result from these experiments was obtained with the largest Model SL999 Hill et al. (2014) 52 Schwartz et al. (2015) 56 Faruqui and Dyer (2015) 58 Wieting et al. (2015) 66.7 CHARAGRAM (large) 70.6 Mrkšić et al. (2016), but the setting is not comparable to ours as they started with embeddings tuned on SL999.
Lastly, we evaluated our model on the Stanford Rare Word Similarity Dataset (Luong et al., 2013), using SL999 for model selection. We obtained a Spearman's ρ of 47.1, which outperforms the 41.8 result from Soricut and Och (2015) and is competitive with the 47.8 reported by Pennington et al. (2014), which used a 42B-token corpus for training.
Sentence Embedding Experiments
Training and Tuning We did initial training of our models using one pass through PPDB XL, which consists of 3,033,753 unique phrase pairs. Following Wieting et al. (2016), we use the annotated phrase pairs developed by Pavlick et al. (2015) as our validation set, using Spearman's ρ to rank the models. We then take the highest performing models and train on the 9,123,575 unique phrase pairs in the phrasal section of PPDB XXL for 10 epochs.
For all experiments, we fix the mini-batch size to 100, the margin δ to 0.4, and use MAX sampling (see supplementary material). For CHARA-GRAM, V contains all 122,610 character n-grams (n ∈ {2, 3, 4}) in the PPDB XXL phrasal section. Other tuning settings are the same as Section 4.1.3.
For another baseline, we train the PARAGRAM-PHRASE model of Wieting et al. (2016), tuning its regularization strength over {10 −5 , 10 −6 , 10 −7 , 10 −8 }.
The PARAGRAM-PHRASE model simply uses word averaging as its composition function, but outperforms many more complex models.
In this section, we refer to our model as CHARAGRAM-PHRASE because the input is a character sequence containing multiple words rather than only a single word as in Section 4.1.3. Since the vocabulary V is defined by the training data sequences, the CHARAGRAM-PHRASE model includes character n-grams that span multiple words, permitting it to capture some aspects of word order and word co-occurrence, which the PARAGRAM-PHRASE model is unable to do.
We encountered difficulties training the char-LSTM and charCNN models for this task. We tried several strategies to improve their chance at convergence, including clipping gradients, increasing training data, and experimenting with different optimizers and learning rates. We found success by using the original (confidence-based) ordering of the PPDB phrase pairs for the initial epoch of learning, then shuffling them for subsequent epochs. This is similar to curriculum learning (Bengio et al., 2009). The higher-confidence phrase pairs tend to be shorter and have many overlapping words, possibly making them easier to learn from.
Results An abbreviated version of the sentence similarity results is shown in Table 3; the supplementary material contains the full results. For comparison, we report performance for the median (50%), third quartile (75%), and top-performing (Max) systems from the shared tasks. We observe strong performance for the CHARAGRAM-PHRASE model. It always does better than the char-CNN and charLSTM models, and outperforms the PARAGRAM-PHRASE model on 15 of the 22 tasks. Furthermore, CHARAGRAM-PHRASE matches or exceeds the top-performing task-tuned systems on 5 tasks, and is within 0.003 on 2 more. The charLSTM and charCNN models are significantly worse, with the charCNN being the better of the two and beating PARAGRAM-PHRASE on 4 of the tasks.
We emphasize that there are many other models that could be compared to, such as an LSTM over word embeddings. This and many other models were explored by Wieting et al. (2016). Their PARAGRAM-PHRASE model, which simply learns word embeddings within an averaging composition function, was among their best-performing models. We used this model in our experiments as a stronglyperforming representative of their results. Lastly, we note other recent work that considers a similar transfer learning setting. The Fast-Sent model (Hill et al., 2016) uses the 2014 STS task in its evaluation and reports an average Pearson's r of 61.3. On the same data, the C-PHRASE model (Pham et al., 2015) has an average Pearson's r of 65.7. 2 Both results are lower than the 74.7 achieved by CHARAGRAM-PHRASE on this dataset.
POS Tagging Experiments
We now consider part-of-speech (POS) tagging, since it has been used as a testbed for evaluating architectures for character-level word representations. It also differs from semantic similarity, allowing us to evaluate our architectures on a syntactic task. We replicate the POS tagging experimental setup of Ling et al. (2015a). Their model uses a bidirectional LSTM over character embeddings to represent words. They then use the resulting word representations in another bidirectional LSTM that predicts the tag for each word. We replace their character bidirectional LSTM with our three architectures: char-CNN, charLSTM, and CHARAGRAM.
We use the Wall Street Journal portion of the Penn Treebank, using Sections 1-18 for training, 19-21 for tuning, and 22-24 for testing. We set the dimensionality of the character embeddings to 50 and that of the (induced) word representations to 150. For optimization, we use stochastic gradient descent with a mini-batch size of 100 sentences. The learning rate and momentum are set to 0.2 and 0.95 respectively. We train the models for 50 epochs, again to ensure that all models have an opportunity to converge.
The other settings for our models are mostly the same as for the word and sentence experiments (Section 4.1). We again use character n-grams with n ∈ {2, 3, 4}, tuning over whether to include all 54,893 in the training data or only those that occur more than once. However, there are two minor differences from the previous sections. First, we add a single binary feature to indicate if the token contains a capital letter. Second, our tuning considers rectified linear units as the activation function for the CHARAGRAM and charCNN architectures. 3 The results are shown in Table 4. Performance is similar across models. We found that adding a second fully-connected 150 dimensional layer to the CHARAGRAM model improved results slightly. 4
Convergence
One observation we made during our experiments was that different models converged at significantly different rates. Figure 1 plots ging. We only show the first 10 epochs of training in the tagging plot. The plots show that the CHARAGRAM model converges quickly to high performance. The charCNN and charLSTM models take many more epochs to converge. Even with tagging, which uses a very high learning rate, CHARAGRAM converges significantly faster than the others. For word similarity, it appears that charCNN and charLSTM are still slowly improving at the end of 50 epochs. This suggests the possibility that these models could eventually surpass CHARAGRAM with more epochs. However, due to the large training sets available from PPDB and the computational requirements of these architectures, we were unable to explore the regime of training for many epochs. We conjecture that slow convergence could also be the reason for the inferior performance of LSTMs for similarity tasks as reported by Wieting et al. (2016).
Model Size Experiments
The default setting for our CHARAGRAM and CHARAGRAM-PHRASE models is to use all character bigram, trigrams, and 4-grams that occur in the training data at least C times, tuning C over the set {1, 2}. This results in a large number of parameters, which could be seen as an unfair advantage over the comparatively smaller charCNN and char-LSTM similarity models, which have up to 881,025 and 763,200 parameters respectively (including 134 character embeddings for each). However, for a given sequence, very few parameters in the CHARAGRAM model are actually used. For charCNN and charLSTM, by contrast, all parameters are used except character embeddings for characters not present in the sequence. For a 100character sequence, the 300-dimensional CHARA-GRAM model uses approximately 90,000 parameters, about one-tenth of those used by charCNN and charLSTM for the same sequence.
We performed a series of experiments to investigate how the CHARAGRAM and CHARAGRAM-PHRASE models perform with different numbers and lengths of character n-grams. For a given k, we took the k most frequent character n-grams for each value of n in use. We experimented with k values in {100, 1000, 50000}. If there were fewer than k unique character n-grams for a given n, we used all of them. For these experiments, we did very little tuning, setting the regularization strength to 0 and only tuning the activation function. For word similarity, we report performance on SL999 after 5 training epochs on the lexical section of PPDB XXL. For sentence similarity, we report the average Pearson's r over all 22 datasets after 5 training epochs on the phrasal section of PPDB XL. For tagging, we report accuracy on the tuning set after 50 training epochs.
The results are shown in Table 5. When using extremely small models with only 100 n-grams of each order, we still see relatively strong performance on tagging. However, the similarity tasks require far more n-grams to yield strong performance. Using 1000 n-grams clearly outperforms 100, and 50,000 n-grams performs best. We also found that models converged more quickly on tagging than on the similarity tasks. We suspect this is due to differences in task complexity. In tagging, the model does not need to learn all facets of each word's semantics; it only needs to map a word to its syntactic categories. Therefore, simple surface-level features like affixes can help tremendously. However, learning representations that reflect detailed differences in word meaning is a more fine-grained endeavor and this is presumably why larger models are needed and convergence is slower.
Analysis
Quantitative Analysis
One of our primary motivations for character-based models is to address the issue of out-of-vocabulary (OOV) words, which were found to be one of the main sources of error for the PARAGRAM-PHRASE model from Wieting et al. (2016). They reported a negative correlation (Pearson's r of -0.45) between OOV rate and performance. We took the 12,108 sentence pairs in all 20 SemEval STS tasks and binned them by the total number of unknown words in the pairs. 5 We computed Pearson's r over each bin. The results are shown in Table 6 The CHARAGRAM-PHRASE model has better performance for each number of unknown words. The PARAGRAM-PHRASE model degrades when more unknown words are present, presumably because it is forced to use the same unknown word embedding for all unknown words. The CHARAGRAM-PHRASE model has no notion of unknown words, as it can embed any character sequence.
We next investigated the sensitivity of the two models to length, as measured by the maximum 5 Unknown words were defined as those not present in the 1.7 million unique (case-insensitive) tokens that comprise the vocabulary for the GloVe embeddings available at http://nlp.stanford.edu/projects/glove/. The PARAGRAM-SL999 embeddings, used to initialize the PARAGRAM-PHRASE model, use this same vocabulary. of the lengths of the two sentences in a pair. We binned all of the 12,108 sentence pairs in the 20 SemEval STS tasks by length and then again found the Pearson's r for both the PARAGRAM-PHRASE and CHARAGRAM-PHRASE models. The results are shown in Table 7 Both models are robust to sentence length, achieving the highest correlations on the longest sentences. We also find that CHARAGRAM-PHRASE outperforms PARAGRAM-PHRASE at all sentence lengths. Aside from OOVs, the PARAGRAM-PHRASE model lacks the ability to model word order or cooccurrence, since it simply averages the words in the sequence. We were interested to see whether CHARAGRAM-PHRASE could handle negation, since it does model limited information about word order (via character n-grams that span multiple words). We made a list of "not" bigrams that could be represented by a single word, then embedded each bigram using both models and did a nearest-neighbor search over a working vocabulary. 6 The results, in Ta est neighbor is a paraphrase for the bigram and the next neighbors are mostly paraphrases as well. The PARAGRAM-PHRASE model, unsurprisingly, is incapable of modeling negation. In all cases, the nearest neighbor is not, as it carries much more weight than the word it modifies. The remaining nearest neighbors are either the modified word or stalled. We did two additional nearest neighbor explorations with our CHARAGRAM-PHRASE model. First, we collected nearest neighbors for words that were not in the training data (i.e., PPDB XXL), but were in our working vocabulary. These are shown in the upper part of Table 9. In the second, we collected nearest neighbors of words that were in our training data, shown in the lower part of Table 9.
Qualitative Analysis
Several kinds of similarity are being captured simultaneously. One kind is similarity in terms of spelling variation, including misspellings (vehicals, vehicels) and repetition for emphasis (babyyyyyyy). Another kind is similarity in terms of morphological variants of a shared root (e.g., journeying and journey). We also find many synonym relationships without significant amounts of overlapping characters (e.g., vehicles, cars, automobiles). Words in the training data, which tend to be more commonly used, do tend to have higher precision in their nearest neighbors (e.g., neighbors of huge). We see occasional mistakes for words that share many characters but are not paraphrases (e.g., litered, a likely misspelling of littered).
Lastly, since our model learns embeddings for character n-grams, we show an analysis of character n-gram nearest neighbors in Table 10. They appear to be grouped into themes, such as death (row 1), food (row 2), and speed (row 3), but have different granularities. The n-grams in the last row appear in paraphrases of 2, whereas the second-to-last row shows n-grams in words related to language.
Conclusion
We performed a careful empirical comparison of character-based compositional architectures on three NLP tasks. We found a consistent trend: the simplest architecture converges fastest to high performance. These results, coupled with those from Wieting et al. (2016), suggest that practitioners should begin with simple architectures rather than moving immediately to RNNs and CNNs. We release our code and trained models so they can be used by the NLP community for general-purpose, character-based text representation.
Figure 1 :
1the performance of the word similarity and tagging tasks as a function of training epoch. For word similarity, we plot the oracle Spearman's ρ on SL999, while for tagging we plot accuracy on the tuning set. We evaluate every quarter epoch (approximately every 194,252 word pairs) for word similarity and every epoch for tag-Plots of performance versus training epoch for word similarity and POS tagging.
Table 1: Word similarity results (Spearman's ρ × 100). The inter-annotator agreement is the average Spearman's ρ between a single annotator and the average of all others.Model
Tuned on
WS353
SL999
charCNN
SL999
26.31
30.64
WS353
33.19
16.73
charLSTM
SL999
48.27
54.54
WS353
51.43
48.83
CHARAGRAM
SL999
53.87
63.33
WS353
58.35
60.00
inter-annotator agreement
-
75.6
78
Table 2 :
2Spearman's ρ × 100 on SL999.CHARAGRAM (large) refers to the CHARAGRAM model described in Section 4.4.This model contains 173,881 character n-grams, more than the
100,283 in the CHARAGRAM model used in Table 1.
model we considered, which contains 173,881 n-
gram embeddings. When using WS353 for model
selection and training for 25 epochs, this model
achieves 70.6 on SL999. To our knowledge, this is
the best result reported on SL999 in this setting; Ta-
ble 2 shows comparable recent results. Note that a
higher SL999 number is reported by
Table 3 :
3Results on SemEval textual similarity datasets (Pearson's r × 100). The highest score in each row is in boldface (omitting the official task score columns). The last row shows the average performance over all 22 textual similarity datasetsModel
Accuracy (%)
charCNN
97.02
charLSTM
96.90
CHARAGRAM
96.99
CHARAGRAM (2-layer)
97.10
Table 4 :
4Results on part-of-speech tagging.
Table 5 :
5Results of using different numbers and different com-binations of character n-grams.
.Number of
Unknown Words
N
PARAGRAM-
PHRASE
CHARAGRAM-
PHRASE
0
11,292
71.4
73.8
1
534
68.8
78.8
2
194
66.4
72.8
≥ 1
816
68.6
77.9
≥ 0
12,108
71.0
74.0
Table 6 :
6Performance (Pearson's r × 100) as a function of the number of unknown words in the sentence pairs over all 20 SemEval STS datasets. N is the number of sentence pairs.
.Max Length
N
PARAGRAM-
PHRASE
CHARAGRAM-
PHRASE
≤ 4
71
67.9
72.9
5
216
71.1
71.9
6
572
67.0
69.7
7
1,097
71.5
74.0
8
1,356
74.2
74.5
9
1,266
71.7
72.7
10
1,010
70.7
74.2
11-15
3,143
71.8
73.7
16-20
1,559
73.0
75.1
≥ 21
1,818
74.5
75.4
Table 7 :
7Performance (Pearson's r × 100) as a function of the maximum number of tokens in the sentence pairs over all 20 SemEval STS datasets. N is the number of sentence pairs.
Table 8 :
8Nearest neighboring words of selected bigrams under CHARAGRAM-PHRASE and PARAGRAM-PHRASE embeddings.
ble 8, show how the CHARAGRAM-PHRASE embeddings model negation. In all cases but one, the near-Word Nearest Neighbors vehicals vehical, vehicles, vehicels, vehicular, cars, vehicle, automobiles, car serious-looking serious, grave, acute, serious-minded, seriousness, gravity, serious-faced near-impossible impossible, hard/impossible, audacious-impossible, impractical, unable growths growth, grow, growing, increases, grows, increase, rise, growls, rising litered liter, litering, lited, liters, literate, literature, literary, literal, lite, obliterated journeying journey, journeys, voyage, trip, roadtrip, travel, tourney, voyages, road-trip babyyyyyy babyyyyyyy, baby, babys, babe, baby.i, babydoll, babycake, darling adirty dirty, dirtyyyyyy, filthy, down-and-dirty, dirtying, dirties, ugly, dirty-blonde refunding refunds, refunded, refund, repayment, reimbursement, rebate, repay reimbursements, reimburse, repaying, repayments, rebates, rebating, reimburses professors professor, professorships, professorship, teachers, professorial, teacher prof., teaches, lecturers, teachings, instructors, headteachers, teacher-student huge enormous, tremendous, large, big, vast, overwhelming, immense, giant formidable, considerable, massive, huger, large-scale, great, daunting
Table 9 :
9Nearest neighbors of CHARAGRAM-PHRASE embeddings. Above the double horizontal line are nearest neighbors of words that were not in our training data, and below it are nearest neighbors of words that were in our training data.
n - gram
-Nearest Neighbors die dy, die, dead, dyi, rlif, mort, ecea, rpse, d aw foo foo, eat, meal, alim, trit, feed, grai, din, nutr, toe pee peed, hast, spee, fast, mpo , pace, vel, loci, ccel aiv waiv, aive, boli, epea, ncel, abol, lift, bort, bol ngu ngue, uist, ongu, tong, abic, gual, fren, ocab, ingu 2 2 , 02, 02 , tw, dua, xx, ii , xx, o 14, d .2
Table 10 :
10Nearest neighbors of character n-gram embeddings from trained CHARAGRAM-PHRASE model. The underscore indicates a space, which signals the beginning or end of a word.
Trained models and code are available at http://ttic. uchicago.edu/˜wieting.
Both the results for FastSent and C-PHRASE were computed fromTable 4in(Hill et al., 2016).
We did not consider ReLU for the similarity experiments because the final embeddings are used directly to compute cosine similarities, which led to poor performance when restricting the embeddings to be non-negative.4 We also tried adding a second (300 dimensional) layer for the word and sentence similarity models and found it to hurt performance.
This has all words in PPDB-XXL, our evaluations, and two other datasets: SST andSNLI (Bowman et al., 2015), resulting in 93,217 unique (up-to-casing) tokens.
AcknowledgmentsWe thank the anonymous reviewers for their valuable comments. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. We thank the developers of Theano (Theano Development Team, 2016) and NVIDIA Corporation for donating GPUs used in this research.
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[] | [
"Jon A Sjogren \nHOMOGENEOUS SYSTEMS AND EUCLIDEAN TOPOLOGY\nTowson University\n\n"
] | [
"HOMOGENEOUS SYSTEMS AND EUCLIDEAN TOPOLOGY\nTowson University\n"
] | [] | Introductory Remarks on the Topology of Euclidean SpaceFundamental facts that are characteristic of finite dimensional Euclidean space R n , a real vector space endowed with the Pythagorean metric include Invariance of Domain (that a locally injective mapping is open) and the Jordan-Brouwer theorem (that a topological S n−1 as a subspace of standard R n , when removed leaves one bounded and one unbounded component).These theorems have been approached from several points of view. Certainly, Brouwer's Fixed-Point theorem, with generalizations, is a powerful implement toward these important results. In addition, there exist now several proofs that use only elementary calculus and are easy to comprehend. The theorem known by names of H. Poincaré and C. Miranda (but understood earlier by Hadamard, Kronecker and others), is "equivalent" to BFPT and sometimes makes a more direct application to the problem at hand.Also renowned is the Borsuk-Ulam Theorem in n dimensions. This theorem directly implies BFPT, so it may well the correct tool to use. This utility has been observed more often in texts on non-linear analysis, see [Deimling] than those on topology. So we aim at a suitable proof of the Borsuk-Ulam (B-U) Theorem. The various versions of B-U will not formally be listed; they can be found, together with the Lusternik-Schnirel'mann covering theorem in the book of [Matoušek] and the notes of [Suciu]. We wish to avoid most of the proofs commonly cited, that require high-powered theory, complicated constructors, or subtle concepts that are extraneous to the problem at hand.The pathway we choose starts with transforming the problem from one of "continuous mapping" to one of "solve a collection of homogeneous multi-nomials" by means of the Weierstraß Approximation Theorem. The latter result is quite effective as seen from an analytic solution to the Heat equation, or one of the formulas that yield the multinomial coefficients, such as the expressions due to Bernstein or Landau, see [Sjogren, Iterated].It turns out that we have arrived at a purely algebraic problem exposited by A. Pfister. The result actually has meaning for any real-closed field R that is ground field to a vector space, not only for the standard reals R. For those topological analysts whose facility in the homological theory of commutative rings may not rise to the level achieved by Prof. Pfister, there is a way to simplify the proof in the Typeset by A M S-T E X | null | [
"https://arxiv.org/pdf/1708.00985v1.pdf"
] | 119,147,639 | 1708.00985 | c5fca39030d053cef8b720359ed6e40be19a852a |
3 Aug 2017 24 July 2017
Jon A Sjogren
HOMOGENEOUS SYSTEMS AND EUCLIDEAN TOPOLOGY
Towson University
3 Aug 2017 24 July 2017
Introductory Remarks on the Topology of Euclidean SpaceFundamental facts that are characteristic of finite dimensional Euclidean space R n , a real vector space endowed with the Pythagorean metric include Invariance of Domain (that a locally injective mapping is open) and the Jordan-Brouwer theorem (that a topological S n−1 as a subspace of standard R n , when removed leaves one bounded and one unbounded component).These theorems have been approached from several points of view. Certainly, Brouwer's Fixed-Point theorem, with generalizations, is a powerful implement toward these important results. In addition, there exist now several proofs that use only elementary calculus and are easy to comprehend. The theorem known by names of H. Poincaré and C. Miranda (but understood earlier by Hadamard, Kronecker and others), is "equivalent" to BFPT and sometimes makes a more direct application to the problem at hand.Also renowned is the Borsuk-Ulam Theorem in n dimensions. This theorem directly implies BFPT, so it may well the correct tool to use. This utility has been observed more often in texts on non-linear analysis, see [Deimling] than those on topology. So we aim at a suitable proof of the Borsuk-Ulam (B-U) Theorem. The various versions of B-U will not formally be listed; they can be found, together with the Lusternik-Schnirel'mann covering theorem in the book of [Matoušek] and the notes of [Suciu]. We wish to avoid most of the proofs commonly cited, that require high-powered theory, complicated constructors, or subtle concepts that are extraneous to the problem at hand.The pathway we choose starts with transforming the problem from one of "continuous mapping" to one of "solve a collection of homogeneous multi-nomials" by means of the Weierstraß Approximation Theorem. The latter result is quite effective as seen from an analytic solution to the Heat equation, or one of the formulas that yield the multinomial coefficients, such as the expressions due to Bernstein or Landau, see [Sjogren, Iterated].It turns out that we have arrived at a purely algebraic problem exposited by A. Pfister. The result actually has meaning for any real-closed field R that is ground field to a vector space, not only for the standard reals R. For those topological analysts whose facility in the homological theory of commutative rings may not rise to the level achieved by Prof. Pfister, there is a way to simplify the proof in the Typeset by A M S-T E X
case of the standard reals, as noted in [Lang, Places] and by others. The Annals of Mathematics paper of S. Lang does not directly refer to the B-U Theorem however.
We express the Borsuk-Ulam Theorem in a minimalist form: any odd (antipodepreserving) mapping ϕ : S n → S n is essential (meaning not nil-homotopic, not contractible within the image space S n ). Of course the version stating that the Brouwer degree deg ϕ is an odd integer, is sharper. A statement, only apparently more general, is that a "Z 2 -equivariant" self-map of the sphere is essential.
Our version of B-U allows one immediately to proceed to the Invariance of Domain Theorem without using any numerical invariants. In particular we avoid formulas for the computation of mapping degree. Also we will not have used the Leray Product Formula, Borsuk's separation thesis, various simplicial approximations, or characterization of connected components, amongst other subtle concepts of general topology. We say "subtle", noting in particular that classical Domain Invariance from [Hurewicz & Wallman] was covered in [Dugundji], but has not further been explained in recent texts, except by rote. Prof. T. Tao gives a concise proof in his on-line journal [Tao, blog], related to work by W. Kulpa, using only metric topology.
Borsuk-Ulam via Projective Varieties
Our main supporting result is a polynomial verion of B-U in the spirit of [Knebusch], [Arason] and [Pfister]. This result applies to any real-closed ground field R, not only the standard reals. From there to reach the usual B-U theorem, then to Invariance of Domain, we need basic observations about R of an analytic nature. This is analogous to the saying, "the Fundamental Theorem of Algebra (that a real polynomial of degree ≥ 3 is reducible) cannot be proven using algebra only". In fact, given Brouwer's Invariance of Domain, a topological proof of "FTA" is readily derived, [Sjogren, Domain], in the real form as stated, not mentioning complex numbers.
In other words, a completeness property of R such as the Bolzano theorem on the least upper bound is required. Furthermore, one must employ "compactness" in the sense that "the space of lines (real projective space) is compact" in finite dimension. This tells the Analyst that an accumulation point of a subspace lies in the projective space. Hence we may find a point y ∈ S n that maps to 0 ∈ R n by ϕ : S n → R n , a given continuous mapping.
In the standard case of ground field R, Prof. Lang could simplify his quasireal Bézout theorem for polynomials ("multi-nomials") of odd total degree. The connection to the Borsuk-Ulam question was not understood until later. So let us state a "multi-nomial version" of the B-U Theorem, and indicate a proof by algebraic methods. Then it is not surprising that the Weierstraß approximation leads to the full "continuous" B-U result.
Theorem 1 Given a quantity n of polynomials over the field R in n + 1 variables q 1 (x 1 , . . . , x n+1 ), . . . , q n (x 1 , . . . , x n+1 ), which are all "odd", namely q j (−x 1 , −x 2 , . . . , −x n+1 ) = −q j (x 1 , . . . , x n+1 ) for j = 1, . . . , n. Then there exists a ray consisting of all vectors λ(b 1 , . . . , b n+1 ) where λ > 0 and b is not the zero vector, such that q j (λ b) = 0 for each j = 1, . . . , n.
We will see that in Theorem 1, the standard reals R can be replaced by any other real-closed field R. We defer the proof until Theorem 3 has been stated, which is actually the principal result of the Section. It seems that the general B-U theorem for α continuous mapping is not true for any real-closed field R other than R.
Theorem 2 (Borsuk-Ulam) Given f : S n → R n , an odd continuous mapping, so that for X ∈ S n there holds f (−x) = −f (x), then there exists y ∈ S n such that f (y) = 0 ∈ R n .
Remark Another formulation is that any continuous g : S n → R n yields up some y ∈ S n satisfying g(y) = g(−y) ∈ R n .
Sketch of proof Let f i : S n → R, i = 1, . . . , n be the coordinates of f . Then taking {p i (x)} to be ǫ-approximations to {f i (x)}, where p i (x) is the restriction to S n of a real multi-nomial p i (x 1 , . . . , x n+1 ), we may actually replace p i (x) by
q i (x) = 1 2 [p i (x) − p i (−x)] and obtain for all 1 ≤ i ≤ n |f i (x) − q i (x)| < ǫ for x ∈ S n .
This inequality holds since f i is odd, and we also know that q i (x) is odd from its definition. Next, if f i on S n is bounded away from φ by δ > 0, then all {q j } are bounded away from φ in modulus by δ − ǫ > 0, where we chose ǫ > 0 small enough. By continuity of {q j } and compactness of S n , one may infer that the {f j } have no common zero (as a ray), which contradicts Theorem 1.
This proof uses the well-known (to analysts) "compactness argument" whereby a sequence of values in a compact space gives rise to a "convergent sub-sequence" or equivalently an "accumulation point". We need to use the compactness argument again in this section.
We now state the form of Bézout's theorem "over a real-closed field" that is required. We use the standard real numbers R as our prototype or main exemplar of a real-closed field.
Theorem 3 For n ≥ 1, let f 1 , . . . , f n ∈ R [x 1 , . . . , x n+1 ] be homogeneous multinomials (forms) of respective degrees d 1 , . . . , d n , with each d i an odd natural number. Then there exists a non-zero real solution vector a = (a 1 , . . . , a n+1 ) ∈ R n+1 , that is, satisfying f j (a 1 , . . . , a n+1 ) = 0 for j = 1, . . . , n. In fact, a generates a solution ray {λ a}, λ = 0, λ real.
Remark Several components of a proof are indicated, which may be selected and assembled according to the taste of the reader. The proof should be "algebraic enough" still to hold for other real-closed fiels.
The "simplest" proof is perhaps constituted by the observation that Theorem 3 is exactly the Theorem 1 given on page 239 of [Shafarevich].
Thus the reader who accepts certain results "modulo the algebra" now has the Borsuk-Ulam theorem fully in hand (once the derivation of our Theorem 1 is completed as a Corollary). The treatment in [Shafarevich] is straightforward based on the theory of algebraic divisors. Nevertheless, we proceed to redo parts of this work based on the concept of Resultant Systems ( [Macaulay], [Kapferer], [vd Waerden 1927], [Behrend]), which embodies Algebraic Geometry of a generation or two prior to Basic Algebraic Geometry, Vol. I. In volume II the same learned author Prof. Shararevich treats the contemporaneous theory of schemes developed by Serre-Grothendieck. In the continuation, which is largely based on B.L. van der Waerden's foundational articles and chapters, we intend for definitions to be reasonably concrete. For example, "multiplicity of a solution" should be calculable from Polynomial Ideal Theory. Now recall that we wished to establish B-U theorem at least for multi-nomial functions.
Proof of Theorem 1 See [Pfister]. We have a quantity n of polynomials {q j } which are odd as functions in their n + 1 arguments x 1 , . . . , x n+1 , but we may homogenize the q j by throwing in an additional variable to achieve the required total degree.
For example, q(x 1 , x 2 , x 3 ) = 2x 1 − x 2 x 2 3 + x 3 1 x 2 x 3 − 3x 1 x 2 3 + x 2 2 x 3 satisfies q(−x 1 , −x 2 , −x 3 ) = −q(x 1 , x 2 , x 3 ).
Note that the degree of each term is odd, so the needed power of x 0 is always even. Takẽ
q = (x 0 , x 1 , x 2 , x 3 ) = 2x 4 0 x 1 − x 2 0 x 2 x 2 3 + x 3 1 x 2 x 3 − 3x 2 0 x 1 x 2 3 − x 2 0 x 2 2 x 3 .
It is not difficult to show that the above observation on degrees holds in general. Now for each j = 1, . . . , n, replace any factor x 2 0 by x 2 1 + · · · + x 2 n+1 in q j . Doing so yields a quantity n of odd-degree homogeneous polynomials (or multi-nomials) q j (x 1 , . . . , x n+1 ) which by Theorem 3 above possess a common solution valid on a ray in R n+1 that is generated by a non-zero real vector (a 1 , . . . , a n+1 ). By homogenity of theq j , we may choose the solution vector b = a a ∈ S n . Also − b is an acceptable solution. Either can be taken as the point on the nsphere (or on RP n ) sought by the Borsuk-Ulam theorem (expressed also in Theorem 2).
Concerning the "homotopy" interpretation of B-U Theorem
Strong versions of the theorem exist, in the form of "an antipode-preserving mapping g : S n → S n has odd Brouwer degree".
Homotopy Borsuk-Ulam Theorem Such an odd mapping (commuting with the canonical involution of S n ) is essential. That is, g is not contractible to a point in the image sphere S n w . The conclusion once again is that g is not homotopic within S n w to any constant mapping.
Proof We deduce this from Theorem 2. Also the result implies Theorem 2 directly, [Matoušek]. For g to be inessential or nil-homotopic means that there is an extensiong : B n+1 → S n of g whose domain is the Euclidean ball B n+1 with boundary S n . That is,g restricted to ∂B n+1 is just g, see [Dugundji]. Next we may define the projection π : S n+1 + → B n+1 from the "upper hemisphere" of ∂B n+2 by means of π(x 1 , . . . , x n+2 ) = (x 1 , . . . , x n+1 ) where x n+2 > 0 and n+2 i=1 x 2 i = 1. Thus we have a continuous mapping f :g•π : S n+1 → S n and similarly f : S n+1 → S n on the lower hemisphere, defined by f (x) = −g • π(−x). Sinceg |S n is antipode-preserving (odd), the mapping f : S n → S n is well-defined, continuous and antipode-preserving, hence it is also such a mapping S n+1 → R n+1 not meeting the origin, which violates Theorem 2.
For future use, we note a simple Homotopy Fact: suppose that for g, h : S n−1 → S n−1 , g ∼ h (considered as mappings to R n ) by a homotopy H : S n−1 × I → R n . Then if H(s, t) never attains 0 ∈ R n , where s ∈ S n , t ∈ [0, 1], then h is homotopic to g considered as mappings to S n−1 w .
Proof If H exists, it may be modified by pushing away from 0 and ∞ so that all of its values lie on S n−1 . Thus we have a homotopyH : g ∼ h within S n−1 . In particular g is essential if and only if h is essential. One consequence of this Fact is that a mapping g : B z → B w of one ball to another ball, which restricts to and essential map ∂g : ∂B z → ∂B w must itself be surjective onto B w .
Classical Domain Invariance
Background for Brouwer's Invariance of Domain can be found in [Dugundji], [Deimling] and [Tao, blog]. This famous theorem on the topology of Euclidean space, from around 1910, can be stated:
Theorem IVD1 Let Ω ⊂ R n be an open set. Then any (continuous) mapping h : Ω → R n that is locally one-to-one, is an open mapping.
By way of explanation of the terminology, we quote an equivalent but more concrete statement.
Proposition IVD2 Suppose g : B n z → B n w is a one-to-one mapping with g( 0 z ) = 0 w . Then there exists an open subset U ⊂ g(B z ) with 0 ∈ U . Here B z , B w are the open unit balls at the Origin, distinguishing "domain" from "range". Taking a ball of smaller radius, we could regard g as defined and continuous on B z (1), the closed unit ball.
Proof Now consider the homotopy
H : B z × I → R n w defined by H(x, t) = g x 1 + t − g −tx 1 + t .
For all 0 ≤ t ≤ 1, H maps the Origin 0 z to the Origin 0 w . Also Im(H) ⊂ B w (1), though to avoid a calculation, H could be scaled radially so that its image fits into B w (1). An important fact is that for x ∈ ∂B(ρ), ρ > 0, H(x, t) is never 0 w : the homotopy restricted to any sphere of radius ≤ 1 cannot cross the origin. This follows from the assumption of injectivity for g. Thus on each "central sphere" S n−1 p = ∂B n (ρ), the mapping g is homotopic to
φ(x) = H(x, 1) = g x 2 − g −x 2
by the restriction of y = H(x, t), where as t varies, y never crosses the Origin. We note that φ on every central sphere S n−1 ρ is odd (Z 2 -equivariant or antipodepreserving). By compactness of S n−1 (1), g and φ attain their infinum in norm g(x) and φ(x) , x ∈ S n−1 z (1). Choose a radius σ > 0 smaller than both of these positive infina. Next we consider a deformation retraction G : B n w × I → B n w (σ) given by
G(y, t) = σt + y (1 − t) y y , for y ≥ σ,
y, for y < σ.
One notes that G is "piece-wise linear" and not generally smooth on S a (σ). In the following Figures
S z S z B z B z O z O z O w O w g g t = 0 φ(x) t = 1 φ(S) 0 ≤ t ≤ 1 x ≤ 1 H(x, t) = g x 1 + t − g −tx 1 + t never meets O w except for x = 0 B w B w g(S z ) Inf inf = σ φ is Z 2 equivariant Radius σ, least radial distance attained by H on S z (1) × I
Figure 1
For each t ∈ I, the radial ray containing y is kept invariant (in terms of its z-and w-coordinates). The mapping G is a homotopy between the "identity": B n z (1) → B n w (1) and the "radial retraction": B n z (1) → B n w (σ) that keeps the smaller ball point-wise fixed.
Now define L g (x) = G(y, 1) • g(x) and L φ (x) = G(y, 1) • φ(x), both of which map B n z (1) to B n w (σ). Furthermore both L g and L φ , restricted to ∂B n z (1), have Deformation Refract B w (1) O w ×I G(y, t) B w (σ) radius σ G(y, 0) is id on B w G(y, 1) is the canonical retraction "σ smaller than both infina" B w (1) → B w (σ) Figure 2 "H(x, t) never attains O w except at x = O z , which it does for all t ∈ I" O z O z L g (x) = G( y, 1) • g(x ) L φ(x) = G( y, 1) • φ(x )
is a radially shrunken version of
g x 2 − g −x 2 O w O w B w (σ) B w (σ) The homotopy H(s, t) on S z (1) avoids this ball B w (σ)
Both mappings L g and L φ take image contained in ∂B n w (σ), the "small image sphere". We observe that L φ restricted to the "big z-sphere" S n−1 z (1) is actually antipode-preserving (also if restricted to other central spheres). Hence by the homotopy form of the B-U theorem above, we conclude that L φ : S n−1 z (1) → S n−1 w (σ) is an essential mapping. Also L g : S n−1 z (1) → S n w (σ), though not an injective mapping, is homotopic to L φ within this space of mappings, since the homotopy H(x, t) followed by G(y, s) avoids the w-Origin, so these homotopies may be projected radially to the w-sphere of radius σ. It follows that L g restricted to S n−1 z (1) is also essential and by the Homotopy Fact above, L g :
S z to S w (σ) = ∂B w (σ) still Z 2 -equivalentB n z (1) → B n w (σ) is a surjection. A given b ∈ B w (σ) is therefore in the image of L g = G • g, but it is not moved under G(· , t). Hence b = g(a)
for some a ∈ B z (1). Since b was chosen arbitrarily, we have found an open neighborhood B w (σ) of 0 w in the image, confirming that g must be an open mapping.
Bézout's Theorem and Solution Multiplicity
Theorem 3 above follows from Theorem 4, which allows a more general "ground field", see [vd Waerden, Algebra II], section 83.
Theorem 4 (Bézout) If a system F 1 , . . . , F n of homogeneous equations (F j = 0), in n + 1 variables x 1 , . . . , x n , with x j ∈ R( √ −1), with coefficients in the real-closed field, has only finitely many distinct solutions (x i ) = 0, then there holds a formula for their multiplicity. Consider as before the solutions generating lines (or rays)
over C = R( √ −1). Defining ∆ = n i=1 deg F j , we obtain (B) ∆ = P mult(P ),
where {P } runs though the distinct solution rays and mult(P ) is the multiplicity of the solution to the given and algebraic definition.
Finally, we are looking for Theorem 3 as a corollary. We may write Theorem 3 again as: Theorem 5 With the hypotheses of Theorem 4, given the homogeneous system F 1 , . . . , F n with coefficients in the real-closed R, suppose that each degree (F j ) = d j is odd, j = 1, . . . , n, then we conclude that there exists a solution (ξ 0 : ξ 1 : · · · : ξ n ) defining a ray, with all ξ j ∈ R.
To finish a proof of Theorem 3, we specialize R in Theorem 5 to the "standard" real numbers R. For our purposes, we also need only consider the case (as in [vd Waerden, Algebra II] p. 16, where there exist only finitely many solution (rays). Furthermore, for the application to the B-U theorem, we may assume that the coefficients of the equation system are transcendental, and algebraically independent over the rationals Q.
A result similar to Theorem 5 has been considered from several points of view, as is seen in the section below.
Algebra and Topology in Theorem Five
We mentioned that on Chapter III of [Shafarevich], Book 1, the theory of the divisor class group of a variety is applied to prove Bézout's theorem in the form we need, our Theorem 3 or 4. As a matter of fact, this algebraic method uses a general position argument concerning the equations F 1 , . . . , F n of our system (which is fulfilled if then coefficients are algebraically independent or generic). For the standard real numbers R as coefficients, the usual limiting inference (by compactness RP n ) gives Theorem 5 more generally. This discussion shows in a rough manner the trade-off between the power of using the metric on R, and achieving the Theorem for an arbitrary system (not necessarily generic).
Using divisors on a variety was originally beyond our scope, so we examine proofs that use algebraic geometry of a nature even more elementary. Now a rather pure form of Theorem 5, wholly algebraic in statement and proof, is given in the book of [ Pfister], p. 57. The author's remarks point toward an interpretation into geometry of his module-theoretic argument (valid for any real-closed R). It is argued that, the greater degree to which the proof is intuited geometrically, the less it is convincing in its rigor.
Our point of view is that by throwing in a bit of the order or the topology of R, we obtain a proof of Theorem 5 that is predominantly algebraic but uses commonly known facts. On the other hand, for R, work of Borsuk and Hopf from the 1930s on the B-U theorem itself, leads to a purely topological proof (with almost nothing about polynomials). Readers are invited to revisit this part of the history, [Hopf], where the demonstrations may not be obvious to the contemporary scholar. By means of the modern machinery of algebraic topology, such proofs can be downsized; we indicate the section ahead covering the earlier work of L. Lusternik and L. Schnirel'mann.
An early algebraic proof of Theorem 5 is reputed to be that of [Behrend]. Here the result is stated for coefficients in R, which is our case of interest. The author is looking at our system F 1 , . . . , F n where the latter are dependent on several rows of indeterminates, not only x 0 , . . . , x n but some other sequence y 0 , . . . , y s as well. So the existence of a real solution is proved in more general circumstances. A sequence of homogeneous systems is constructed, each of which can be decomposed into linear factors that are in general position. The latter given condition is an algebraic one.
Each of these systems will have finitely many (hence an odd number) of solutions, with any non-real solution paired with its conjugate. But the equations F j and F ′ j (the new one) can be connected by a homotopy to yield a system valid over some algebraic closure Λ of R(t). By considering the simplicity of solutions (coming from [vd Waerden, Einführung]) in this field, a real solution can be pulled back from the finitely many solutions now seen to exist over the (real-closed) field of real Puiseux series. One should consult the article [Behrend] for details.
We perceive formula (B) as arising, in Bézout's Theorem (Thm 4) for a sum of multiplicities of solution rays. Such a situation, for a quantity of equations equal to one less than the number of homogeneous variables, would be easier to deal with in case each solution had unit multiplicity. This is indeed the case when the coefficients are generic (algebraically independent over Q). At least when we are allowed to operate over C or R as coefficients, it would seem that we could nudge them one by one into genericity while homing in on the "specialized" solution that we seek over RP n . This strategy best fits the approach from [Lang, Places] which exhibits both a "more algebraic" and "more topological" version to finish off the proof of Theorem 5. The above-mentioned work of F. A. Behrend reduces the problem to one of simple solutions, coming from a classical criterion for simplicity which we will refer to again. Relevant background is described in [vd Waerden, Einführung], section 39.
The following gives us a result that would be a sufficient alternative. It comes from the same textbook of van der Waerden, Dover edition (1945) or Springer-Verlag edition (1973). In section 41 we read "The intersection of an irreducible ddimensional variety of reduced degree γ, with a quantity k ≤ d generic hypersurfaces of degrees e 1 , e 2 , . . . , e k respectively, has degree γ k j=1 e j . Hence in case k = d, this variety consists of (this many) points".
A similar statement from the earlier book of [Macaulay], p. 16 indicates that "the number of solutions is either L = l 1 · l 2 · · · l n , or infinite, the latter being the case when F 0 (a resultant of the system with respect to x 1 , . . . , x n ) vanishes identically".
Finally, in [Cox AG], it is proved using an explicit construction that "the equations F 1 = · · · = F n = 0 when generic, have d 1 · · · d n distinct solutions". The discussion is in Chapter 3, Section 5, including Exercise 6. The proof involves projective elimination theory and the use of Macaulay's resultant (which is effective if inefficient). In the sequel it will be seen that we do not need the hard "generic" precondition on coefficients to first finish Theorem 5 and hence the B-U Theorem over R. We will wish however to avoid the "Ausnahmefall" (infinitely many solutionways). With this in mind, we do use Resultant applications from both [Cox AG] and [vd Waerden, Algebra II].
We now point out that these transcendental constructions, say in Lang's method can be gotten around in a sense. With the concern that the solutions not be infinite in number, and actually all possess unit multiplicity, it comes down to whether certain resultants (integer multi-nomials in the coefficients) can possibly vanish. But for given degrees d 1 , . . . , d n , the "size" of these resultants is definitely bounded. Thus we don't need transcendental numbers, we merely construct sets that are "sufficiently" independent. For example, to approximate α ∈ R, we could use α+ ǫ where ǫ is a small transcendent, on we could use ǫ ′ = (p) 1 q for large enough primes p, q ∈ N. The proof of any of these assertions goes far beyond our intentions.
The alternative offered by Lang at the end of the 1953 Annals paper is to use the more familiar mathematics of the standard R.
In finding a real solution to F 1 (x 0 , . . . , x n ) = 0, F 2 (x 0 , . . . , x n ) = 0, F n (x 0 , . . . , x n ) = 0, we have noted several "algebraic" proofs of the past, including those of Macaulay, Behrens, the theory of "faithful specializations" (with which van der Waerden replaced a heavy reliance on the explicit use of classical resultants), Pfister's module-theoretic approach, and finally (in our narrative), the method of real places introduced by S. Lang. We saw how to gain an advantage (through the full complement of simple solutions) by approximating the given coefficients of {F j } by a set of algebraically independent coefficients. As [Lang, Places] points out, thus can be done by embedding the real-closed coefficient field R into a real-closed domain Ω having many transcendental elements that are infinitesimal with respect to R. Such constructions are algebraic and do not use the order-topology of R.
After mentioning the work of these authors, we assure the loyal Reader that we quickly finish up this approach to Theorem 5. Multiplicity of solutions is allowed (the coefficients can be specialized), so the remaining component is an algebraic description of "multiplicity" from [vd Waerden, Algebra II]. This is based on the theory of [Kapferer]
The Real Solution-Ray
We work with the system of homogeneous equations in x 0 , . . . , x n over the standard reals R F 1 = 0, . . . , F n = 0, although we emphasize the algebraic aspects of the problem. We avoid the full power of Lang's real-closed domain Ω by allowing for the metric closeness of R. We avoid the need to work with systems where each solution has to be simple, and thereby also avoid explicit resultant constructions coming from Elimination Theory. We do wish to use the methods leading to the statement of Bézout's theorem on page 16 of [vd Waerden, Algebra II], Section 83. Thus we must ensure that the system (S) possesses only finitely many solutions. We saw how this would come about in case the collection of all coefficients were generic, as it is taken in [Lang, Places], see also [Cox AG], Chapter 3. The number of distinct monomials is something like (see [Ryser]
), n j=1 (n + d j )! n!d j ! .
Instead we propose to take all of these coefficients to lie in Q (the rationals), except for one coefficient, which is chosen to be transcendental. Even better, this final real number can be chosen as algebraic but of such an unreachable algebraic order (such as we noted, some (p) 1 q ) that it could never be canceled in the resultant evaluation that arises.
More specially, we may examine (S) for "points at infinity" by specializing x 0 = 0. Now we obtain a system (still homogeneous) Hence (S) is amenable to the theory of Inertial Forms of H. Kapferer (1927). In our case of interest this boils down to saying that there exists a multi-nomial R(u 11 , . . . , u 1n , . . . , u nn ) in the coefficients of S that vanishes precisely when a solution-ray to S exists (projective solution). Amongst other properties, R(u) is homogeneous in the vector of coefficients for F 1 , of total degree d 2 · · · d n−1 · d n , and for F j , of total degree d 1 · · ·d j · · · d n .
For R to equal 0 for a particular specialization cannot happen for the case we have chosen of "all coefficients rational" (except for the one of them which is chosen transcendental). Hence by this theorem of [Macaulay], there are no common solutions for S, hence no solutions "at infinity" for (S). A modern and algorithmic account of the Macaulay resultant is available in [Kalorkoti]; see also [Canny], [CLO] and [Jou].
An ideal-theoretic definition and description of solution-multiplicity in given in Chapter XI of [vd Waerden, Algebra II] and in [vd Waerden 1927]. We add to S the linear equation with independent coefficients F 0 (u) = u 0 x 0 + · · · + u n x n in order to form the "u-resultant" of system S. The Kapferer (or Inertial) resultant ideal is generated by multi-nomials b 1 (u), . . . , b n (u), so that this b-system vanishes at (u 0 , . . . , u n ) exactly when a solution x = (ξ 0 , . . . , ξ n ) of S exists such that also L = u 0 ξ 0 + · · · + u n ξ n = 0.
For each such solution ξ p = (ξ p 0 , . . . , ξ p n ), we have a linear form L p . Each b i (u) has roots in the variety defined by Λ = p L p (u). Since we operate over an algebraically closed field (an extension of R), we may apply the strong form of Hilbert's Nullstellensatz to obtain b i (u) τi ∈ Λ. Actually the roots of the b-system and of Λ(u) are the same so we also have
Λ(u) τ ∈ (b 1 (u), . . . , b r (u)) .
By the theory of Inertial ideals, the greatest common divisor R(u) = gcd (b 1 (u), . . . , b r (u)) decomposes into the linear factors as indicated:
R(u) = p L sp p (u).
Thus, the linear forms L p which determine the solution rays of (S) constitute the irreducible factors of the u-resultant R(u). The exponents {s p } in the factorization are the solution multiplicities. Since it is known that the generator R(u) of the (principal) Inertial ideal has total homogeneous degree D = n j=1 d j , we again have Bézout's theorem, valid for when the solution-rays for (S) are finite in number:
s p = D.
Now we finish our intended proof that a real solution of (S) exists. The point is that for any non-real solution p, its multiplicity and that of its complex conjugate solution are the same: mult(p) = mult(p), or s p = s p . This observation is made by sheer logic, as the algebraic operations used in calculating s p do not depend on how an imaginary coordinate was named, ı or − ı. In other words, one may re-label a value ξ = (−i, π + 7i, 4) as ξ ′ = (i, π − 7i, 4) without affecting the solution algorithm. All ideals, resultants and multiplicities come up again with a superficial change of symbolism. This same fact can be expressed more geometrically of course, as in Chapter IV, 2.2 of [Shafarevich].
What remains as far as the use of Bézout's theorem is concerned in to see how we have avoided the "Ausnahmefall" of infinitely many zeros. In that case, Bézout's theorem holds true and since the degree D is a product of odds, and the non-real solutions are paired up, we must obtain a real solution. This again is what is needed in our approach to the Borsuk-Ulam Theorem and Invariance of Domain.
The issue of solutions at ∞ (where x 0 = 0) comes down to the Macaulay resultant taking on a (scalar) value of zero. For the genericity that we have built into the coefficients of S), this is not possible. We picked one coefficient to be transcendental in R and the rest algebraic over Q (or even rational). Since the resultant construction treats coefficients without prejudice, an equation R(c ij ) = 0 would lead to an algebraic relation not leaving out the chosen "generic" one. Therefore, given that (S) has no solution-rays at infinity, we infer that the quantity of solution rays is finite. This is a well-known proposition in projective geometry, to which there are several approaches, the more analytical and the more algebraic.
Closed Variety Away from ∞ By making the system (S), {F 1 , . . . , F n } generic enough, we avoided solutions (rays) at infinity, so in fact (S) and its associated variety V can be expressed by:
G 1 (x 1 , . . . , x n ) = F 1 (1, x 1 , . . . , x n ) = 0 . . . (S 1 ) G n (x 1 , . . . , x n ) = F n (1, x 1 , . . . , x n ) = 0.
Hence V is an affine variety, in particular the {G j } are generally non-homogeneous multi-nomials. We are working in a situation where we need not be concerned with "real" fields. The field K of coefficients of {F 1 , . . . , F n } should be algebraically closed.
We will prove what is required to complete the argument for Theorem 5. The case of interest is where K = C, so we begin with an argument that uses the ordertopology of C. Subsequently we review an argument from elementary algebraic geometry showing that for any K, it is also true that the system (S 1 ) : {G 1 = 0, . . . , G n = 0} also has only finitely many solutions.
Considering first the complex case K = C, we note that the variety V is a "projective algebraic set" and hence compact in the C-topology. Now we change the affine coordinates of {G j } if necessary. For parameters λ j ∈ C, j = 1, . . . , n − 1 set
x ′ i = x i + λ i x n and x ′ n = x n .
Proposition A A compact, complex affine variety must be a finite set.
Proof The C(x 1 , . . . , x n )-ideal generated by {G 1 , . . . , G n } is called I(G) and its zero-set ⊂ C n is called Z(G). It is known how to define the "first elimination ideal" J = I ∩ C[x 2 , . . . , x n ]. An induction hypothesis is that "Z(J ) is bounded in the C-norm, implies that Z(J ) is finite". The base of induction, with one variable say x n , provides of course finitely many solutions. Now let g ∈ I be any multinomial of the ideal. By breaking g up into its homogeneous pieces, it is possible to find parameters λ 1 , . . . , λ n−1 so that
( †) g(x ′ 1 , . . . , x ′ n ) = γx ′ m 1 + lower degree terms in x ′ 1 with coefficients h α (x ′ 2 , . . . , x ′ n ),
where m is the highest total degree of a monomial in g.
Such a coefficient γ is actually equal to g m (1, λ 2 , . . . , λ n ) where g m is the homogeneous part of highest degree. In an infinite field, this expression cannot always equal 0 unless g m is identically 0, which gives a contradiction.
Next we convert all expressions of the problem into the coordinates {x ′ 1 , . . . , x ′ n }, then remove the "prime" for legibility. At this point we have performed a version of the Noether normalization lemma, see [Arrondo]. Now a "long solution" (c 1 , · · · , c n ) ∈ Z projects into a "short solution" (c 2 , . . . , c n ) ∈ Z(J ). We could assume that J gives rise to an unbounded set of short solutions, or else only a finite quantity of them. If they tend out to infinity, so do the long solutions arising from ( †). If they are finite in number, ( †) shows also that the long solutions are finite in number. Hence if Z(J ) is compact it is finite.
One may phrase this result to say that a "variety" over K can only be both affine and projective, when it consists of finitely many solution points. An algebraic set coming from a finitely generated ideal is the union of irreducible algebraic sets, also called "varieties" by some authors. So we may consider a variety X that is also an affine set in K n .
Consider now the field of regular functions on X consisting of quotients h/g of homogeneous terms h, g ∈ K[x 0 , x 1 , . . . , x n ] having the same total degree. But g should be non-zero everywhere, so must be a constant, hence also h has to be a constant.
Proposition B The field of regular functions on an (irreducible) projective variety X is a field of constants ≃ K. See [Shafarevich] p. 59.
Proof Elaborating on our previous argument, we know that X is "affine" and hence its "coordinate ring" is O(X) = K[x 1 , . . . , x n ]/I(G).
But this quotient gives the field K only if I is a maximal ideal, which by Hilbert's Nullstellensatz only holds true (over algebraically closed K) when I is the ideal (x 1 − b 1 , x 2 − b 2 , . . . , x n − b n ) whose solution zero is the single point b, as was to be proved. See [Atiyah].
Finally we re-work this last result that X must be a finite solution-set, in somewhat greater detail where we employ a "compactness" argument modified from the case of ground field = C. The new argument applies also to general (closed) fields. Similar material may be found in a classical exposition, [Shafarevich].
Consider a regular mapping f : X → Y of one closed projective set to another. Thus locally, f is defined by a polynomial map. The graph of f is the set of pairs
Γ f = {(x, f (x))} ⊂ X × Y . Proposition 1 For a regular mapping f , the graph Γ f is (Zariski-) closed in X × Y . Proof if ı is the identity ı : Y → Y , it is seen that Γ f equals the inverse image of Γ ı under (f, ı) : X × Y → X × Y , hence is closed if we know that Γ ı is closed. But the "diagonal" Γ ı ⊂ Y × Y is defined by polynomial equations, hence is closed.
Proposition 2 If X is a projective variety, and Y is a projective or affine variety, then the projection π : X ×Y → Y onto the second factor maps closed sets to closed sets.
Remarks This "Main Theorem of Elimination Theory" is covered in textbooks as well as the computational manual [CLO], Chapter 8, Section 5.
We have referred previously to polynomial conditions (the resultant systems) whose zero-sets define the parameter values (in Y ) for which a set of equations have solutions in X. We saw the following result earlier on.
Corollary 1 If ϕ is a regular function on an irreducible projective variety then ϕx = c for all x ∈ X, so ϕ may be considered as a field element (scalar constant).
Proof Similar to before, ϕ can be viewed as a map to P 1 that misses the infinity point. We have from the Proposition that ϕ(x) is closed in P 1 , since ϕ(x) equals the projection to P 1 of the graph Γ ϕ ⊂ X × P 1 . But a closed set in A 1 ⊂ P 1 is a finite set, which must be a singleton since X is irreducible.
Corollary 2 If a projective set variety X is embedded in an affine Y , X consists of finitely many points.
Proof If Y ⊂ A m , the coordinates of image-points of each irreducible component must be constant by Corollary 1. Since there are finitely many components, X ⊂ Y is a finite set.
This settles again the issue needed for Bézout's theorem, that a projective variety avoiding points at infinity must be finite (and 0-dimensional).
We address this question one final time, letting the Reader pursue the matter further. A finite mapping ϕ : X → Y is a regular mapping whose image is (Zariski-) open, and which satisfies an integrality condition on the induced inclusion of coordinate ring K[Y ] ⊂ K[X]. For our purposes, it is enough to know that when K = C, ϕ must be a finite-to-one continuous mapping of spaces in the C-topology (a finite covering with some branch points). This is itself a formulation of the Noether Normalization Theorem on C. First the result:
Proposition 3 Over general K, algebraically closed, an irreducible affine variety X can be mapped to some affine A m by a finite mapping.
Proof See [Shafarevich] p. 65.
We complete our remarks concerning the complex case. For general affine X, we use the finite mapping given by Proposition 3 to construct a particular finite mapping ϕ : X → A m . As we saw, for K = C such a mapping is continuous and finite-to-one. In particular ϕ is proper and onto, so if X were compact in the trascendental topology, C m would be too, which gives a contradiction.
The article [Kalorkoti] gives an effective algorithm precisely in the case of "no zeros at infinity" and produces the u-resultant and in principle its factors. Thus are derived the finitely many solutions, with multiplicity, to the original system F 1 , F 2 , . . . , F n .
The B-U Theorem according to Lusternik and Schnirel'mann The authors of [L-S] introduced a natural number cat(M ) which for our purposes applies to a compact manifold of finite dimension. It turns out that cat is actually an invariant of homotopy type [James]. The paper "Méthodes Topologiques..." seeks to introduce a sharpening of Morse's inequalities [Milnor], and to study geodesics on a Riemannian manifold.
The usual definition of cat(M ) = k is to say that M can be covered by a quantity k open subsets {U i }, each of which is contractible to a point ambiently within M (the inclusion ı i : U i → M is nil-homotopic).
An inequality cat(RP n ) ≤ n+ 1 follows from general considerations of dimension (see below). The more challenging assertion is that cat(RP n ) ≥ n+1, which follows from the fact that RP 1 ⊂ RP 2 ⊂ · · · ⊂ RP n is a chain of similar subspaces, where each inclusion is homologically non-trivial.
The importance of cat(RP n ) = n + 1 is seen by Proposition 4 If this calculation holds true, then in every covering of S n by quantity n + 1 open sets, one of the sets contains an antipodal pain of points {x, −x}, x ∈ S n . Thus the Lusternik-Schnirel'mann theorem, see [Matoušek], would be demonstrated.
Proof Let q : S n → RP n be the canonical double covering (quotient) mapping. If {U 0 , . . . , U n } covers S n with no U i containing any antipodal pair, then q(U 1 ), . . . , q(U n ) must cover RP n . Indeed, if ξ ∈ RP n is not in their union, then for some y ∈ U 0 , we have q(y) = ξ. Since −y / ∈ U 0 , we get −y belonging to another U j , j = 0. However, q(−y) = ξ so ξ ∈ q(U j ) ⊂ i =0 q(U i ) after all. A nil-homotopy in S n of U i ⊂ S n induces a nil-homotopy of q(U i ) ⊂ RP n , so we see that RP n has a nil-homotopic cover of size n which gives a contradiction to the hypothesis.
The upper bound we need on cat(M ), that is, one plus the dimension of M , is obtained by means of finding a categorical sequence [Fox] for M . When M is a finite simplicial complex, it is not difficult to produce such a sequence by means of a "Balls, Beams, Plates" construction similar to that used with Haken manifolds. We illustrate this in the specific case where M is a 3-dimensional pseudo-manifold (each two-simplex is the boundary of exactly two three-simplexes). Assume that M is topologically connected. The vertices of the 3-complex ∆ with |∆| ≃ M are thickened into 3-balls, called "Shot". Since M is connected, the Shot is contractible into a point of M . Next the 1-simplices are thickened into "Beams" which are separated near the vertices, so the collection of Beams is also ambiently contractible. The same holds for the thickened faces, or "Plates". Finally, all the open interiors of the 3-simplices of ∆ are united to form the "Stuffing" whose inclusion into M is nil-homotopic.
We have covered M with four contractible open sets, confirming the formula cat(M ) ≤ dim M + 1. The same method applies to any connected pseudo-manifold of a higher dimension. See Figure 4.
A "homology" version Hcat(M ) was introduced by [Schnirel'mann]. This number is not greater than cat(M ). Suppose that for j ≤ 0, · · · , n, L j is a manifold of dimension j, satisfying L 0 ⊂ L 1 ⊂ · · · L n = M , with the following homological condition: any 1-cycle mod Z 2 of L j that bounds a Z 2 -chain in M already bounds in L j . Then we may deduce the following.
Proposition 5 Under the above conditions, it follows that Hcat(M ) ≥ n + 1 and hence cat(M ) ≥ n + 1.
Proof See [Fox] and [Schnirel'mann].
In the case of M = RP n , we may define L j = RP j , canonically embedded in M , and verify the hypotheses of Proposition 5. Thus we give witness to an earlier proof of the Lusternik-Schnirel'mann theorem, [L-S], and hence the Borsuk-Ulam theorem, this time based to an extent on "chain-level intersection" in homology.
Work in the cohomology ring has largely replaced a historical fashion for chainlevel intersection. One shows that the nilpotency index of the ring gives a lower bound for cat(M ). This integer k is the least such that all k-fold cup-products vanish, see [James]. The computation of the ring H * (RP n , Z 2 ) seems more involved than the proofs of [Fox] or [Schnirel'mann] sketched above.
In [Goresky-MacPherson], the authors encourage a return to geometric intersection products as an alternative to the cohomology ring. Another article, by Mc-Clure, asserts that these theories are "probably" the same as given in the manual [Lefschetz]. Prof. Lefschetz' intersection calculus utilizing "looping coefficients" has not often been applied, though there is a monograph [Keller] from Leipzig (1969) that thoroughly addresses such issues. One hopes that some of the contemporary authorities have read this work. In any case, the old geometric intersection theory (of chains) seems not yet to be fully integrated with a modern homological version. Further discussion can be found in [McClure].
We suggest that a reworking of the "intersection-level" Borsuk-Ulam proof of [L-S], based on a specific triangulation and dual triangulation of the real projective space would be of interest, especially to combinatorial mathematicians.
Application to Banach Geometry
The concept of defect or gap between two operators on a (real) Banach space, as developed by M. A. Krasnoselskii and co-workers in [KKM], proved to have fundamental implications concerning the geometry of a Banach space. If M and N are subspaces of finite dimension in a Hilbert space H, and dim M < dim N , then there is a vector u ∈ N that is orthogonal to all of M . This fact is not hard to see, since in a Hilbert space one can project M into N by a projection π, where the image is then a linear space of lesser dimension. Some vector u ∈ N that is orthogonal to Im(π(M )) will then also be orthogonal to M itself.
If alternatively M and N are subspaces of a normed linear (or Banach) space, the analogous result is less obvious. For one thing, it is necessary to define the "orthogonality" of a given vector u with some subspace M . We may adopt the definition d(u, M ) ≡ inf { u − y : y ∈ M } .
Thus the distance from u to the subspace should be minimized as the distance to the 0 subspace, giving the norm of u, that is, u . Call this result (the existence of u ∈ N orthogonal to M ⊂ N ) the Theorem on the Deviation of Subspaces [Brown]. In fact the statement is logically equivalent (by a short derivation) to the Borsuk-Ulam theorem.
We indicate some features of the proof of Deviation of Subspaces from the B-U theorem. Without loss of generality, one may assume that dim N = dim M + 1. For a first case take it that the Banach space E is just the (finite-dimensional) sum of M and N , and that E is strictly convex. This means that for two linearly independent vectors u, v ∈ E, we have u + v < u + v . Now a derivation from the elementary theory of normed vector spaces shows that every u ∈ E has a nearest vector ψ(u) ∈ M , and that ψ : E → M is continuous in the norm topology. In case E is not a Hilbert space, ψ might not be a linear mapping, but it does satisfy ψ(−u) = −ψ(u), so is antipode-preserving on the sphere of "norm one" vectors of N . Hence, by the Borsuk-Ulam theorem, see [Matoušek], there exists u ∈ N with u = 1 and ψ(u) = 0. As alluded to above, this vector in N is the one we seek, it is orthogonal in the Banach sense, to all of M . For the general case where E is not strictly convex, given ǫ > 0, the experts (see [Gohberg-Krein]) construct a new metric 0 on E which satisfies v ≤ v 0 ≤ (1 + ǫ) v for all v ∈ E. It turns out that the sphere {v : v 0 = 1} is strictly convex. With the new norm, one can find u of norm = 1 that is orthogonal to M . Actually, this u depends on the choice of norm and should be written u ǫ . As ǫ → 0, one picks out a convergent subsequence of the u, where ǫ = 2 −k (the "original" norms of these vectors go to 1), and this vector is then shown to be orthogonal to M .
Figure 3
3Figure 3
and the "u-resultant". F. S. Macaulay attributes the u-construction to Liouville. Solution of systems by means of variations on the u-resultant figure importantly in Computational Algebra [Cox AG], [CanMan], [D'Andrea].
F 1 F
1(x 1 , . . . , x n ) = F 1 (0, x 1 , . . . , x n ) n (x 1 , . . . , x n ) = F n (0, x 1 , . . . , x n ) = 0 in which the number of variables equals the number of equations.
Figure 4
we suppress the dimensions of the Spheres and other spaces that are depicted.. . . .
.
.
.
.
1-1 continuous
Homotope to φ:
no central circle crosses
the origin
A. L. Brown proved the converse in [Brown]. We already have a proof of the B-U theorem, but he applies Deviation of Subspaces to the space E = C(S n ) of continuous real-valued functions on the n-sphere, equipped with the "supremum" (or "uniform") norm. Let N ⊂ E be generated by the coordinate functions of R n+1 , S n ⊂ R n+1 . Let M be generated by the n coordinate functions on R n , after applying ϕ : S n → R n , a continuous, antipode-preserving mapping.One need only show that there is a vector w ∈ S n with ϕ(w) = 0 ∈ R n . But if z ∈ N can be found, orthogonal to M as asserted by the Deviation theorem, z is actually a linear functional on R n+1 that attains its norm in C(S n ) at a (unique) antipodal pair {w, −w}. This choice of w ∈ S n turns out to provide the "Borsuk-Ulam" vector that is required.
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| [] |
[
"Topological bound states of a quantum walk with cold atoms",
"Topological bound states of a quantum walk with cold atoms"
] | [
"Samuel Mugel \nMathematical Sciences\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUnited Kingdom\n\nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain\n",
"Alessio Celi \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain\n",
"Pietro Massignan \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain\n",
"János K Asbóth \nInstitute for Solid State Physics and Optics\nWigner Research Centre\nHungarian Academy of Sciences\nP.O. Box 49H-1525BudapestHungary\n",
"Maciej Lewenstein \nICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain\n\nICREA-Institució Catalana de Recerca i Estudis Avançats\nE-08010BarcelonaSpain\n",
"Carlos Lobo \nMathematical Sciences\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUnited Kingdom\n"
] | [
"Mathematical Sciences\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUnited Kingdom",
"ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain",
"ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain",
"ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain",
"Institute for Solid State Physics and Optics\nWigner Research Centre\nHungarian Academy of Sciences\nP.O. Box 49H-1525BudapestHungary",
"ICFO-Institut de Ciencies Fotoniques\nThe Barcelona Institute of Science and Technology\n08860Castelldefels (Barcelona)Spain",
"ICREA-Institució Catalana de Recerca i Estudis Avançats\nE-08010BarcelonaSpain",
"Mathematical Sciences\nUniversity of Southampton\nHighfieldSO17 1BJSouthamptonUnited Kingdom"
] | [] | We suggest a method for engineering a quantum walk, with cold atoms as walkers, which presents topologically non-trivial properties. We derive the phase diagram, and show that we are able to produce a boundary between topologically distinct phases using the finite beam width of the applied lasers. A topologically protected bound state can then be observed, which is pinned to the interface and is robust to perturbations. We show that it is possible to identify this bound state by averaging over spin sensitive measures of the atom's position, based on the spin distribution that these states display. Interestingly, there exists a parameter regime in which our system maps on to the Creutz ladder. | 10.1103/physreva.94.023631 | [
"https://arxiv.org/pdf/1604.06082v1.pdf"
] | 33,644,217 | 1604.06082 | 96a2dd0fa7d63e89e5320067322d71032be33ed7 |
Topological bound states of a quantum walk with cold atoms
Samuel Mugel
Mathematical Sciences
University of Southampton
HighfieldSO17 1BJSouthamptonUnited Kingdom
ICFO-Institut de Ciencies Fotoniques
The Barcelona Institute of Science and Technology
08860Castelldefels (Barcelona)Spain
Alessio Celi
ICFO-Institut de Ciencies Fotoniques
The Barcelona Institute of Science and Technology
08860Castelldefels (Barcelona)Spain
Pietro Massignan
ICFO-Institut de Ciencies Fotoniques
The Barcelona Institute of Science and Technology
08860Castelldefels (Barcelona)Spain
János K Asbóth
Institute for Solid State Physics and Optics
Wigner Research Centre
Hungarian Academy of Sciences
P.O. Box 49H-1525BudapestHungary
Maciej Lewenstein
ICFO-Institut de Ciencies Fotoniques
The Barcelona Institute of Science and Technology
08860Castelldefels (Barcelona)Spain
ICREA-Institució Catalana de Recerca i Estudis Avançats
E-08010BarcelonaSpain
Carlos Lobo
Mathematical Sciences
University of Southampton
HighfieldSO17 1BJSouthamptonUnited Kingdom
Topological bound states of a quantum walk with cold atoms
(Dated: April 21, 2016)numbers: 0365Vf3710Jk6785-d
We suggest a method for engineering a quantum walk, with cold atoms as walkers, which presents topologically non-trivial properties. We derive the phase diagram, and show that we are able to produce a boundary between topologically distinct phases using the finite beam width of the applied lasers. A topologically protected bound state can then be observed, which is pinned to the interface and is robust to perturbations. We show that it is possible to identify this bound state by averaging over spin sensitive measures of the atom's position, based on the spin distribution that these states display. Interestingly, there exists a parameter regime in which our system maps on to the Creutz ladder.
I. INTRODUCTION
Random walks have found extensive use in modelling intrinsically random systems as well as for designing computer algorithms. The quantum analogue of the random walk, the quantum walk, is obtained by replacing the walker by a quantum particle, where path interference effects result in a myriad of new properties [1] and make quantum walks relevant to quantum algorithms [2][3][4][5].
Beyond quantum algorithms, the study of quantum walks has been motivated by the study of fundamental phenomena. They were found to be intimately related to the path integral formalism [6] and the Dirac equation [7]. It was even found that some many body systems can be well modelled using a quantum walk on a one dimensional (1D) semi-infinite lattice, the sites of which represent the system's energy levels [8]. Experimental implementations of quantum walks have been realised with photons [9][10][11][12][13][14][15][16], and single and multiple cold atoms [17][18][19][20][21][22] and ions [23,24], which were relevant to the study of Anderson localisation, decoherence and reversibility in strongly interacting systems.
Despite their simplicity, quantum walks present rich topological phenomena [25][26][27], as they can realise all known topological classes in one and two dimensions [28]. For a detailed explanation of the emergence of topological phenomena, we refer the reader to Ref. [29]. Discrete time quantum walks, being periodically driven systems, can present topological invariants which are not found in the topological classification of Hamiltonians, as was shown in [30,31]. Thanks to these properties, they form an ideal platform for realising Floquet bound states, which are bound as a result of the system's dynamics. These states further differ from bound states in static systems by presenting constrained dynamics when considered at half time-steps [32]. The topological prop-erties of quantum walks and of Floquet bound states have been explored by means of photonic experiments in one and two dimensions [12,[14][15][16].
In this paper, we suggest an experimental scheme for realising a quantum walk with cold atoms in a 1D optical lattice, and show that this system is topologically non-trivial. Here and in the following, we define a timestep in a discrete time quantum walk as the sequence of a translation operation, which transports a right-and leftwalker in opposite directions, and of a coin operation, which brings the system into a superposition of rightand left-walkers. Cold atomic gases appear as a natural candidate for this type of application, as it is possible to exert extremely fine control over them. Additionally, cold atoms suffer from few losses relative to photons, and optical lattices have scalable size, allowing for a very long evolution with many time-steps. It was shown in Refs. [33][34][35] that 1D atomic gases are a powerful tool for generating topological bound states in static systems, or for building topologically protected edge states by using a synthetic dimension [36][37][38]. Inspired by these results, we suggest a method to generate Floquet bound states by spatially controlling the system's parameters. We show that these topologically protected states have a heavily constrained spin distribution. Thanks to this property, a spin sensitive measure of the system's probability distribution is sufficient to identify the system's topological bound states. This information can be retrieved by averaging over measurements of individual atom positions at a time t. This can be done in a single measurement, by performing the experiment multiple times in an array of 1D tubes, then observing the atoms' position using e.g: a quantum gas microscope. It would also become possible to study the Floquet bound states' robustness to interactions and to faults in the periodic driving, thereby providing useful information on systems which are still arXiv:1604.06082v1 [cond-mat.quant-gas] 20 Apr 2016 poorly understood.
In this paper's second section, we suggest an experimental protocol to realise a quantum walk with a single, two-state atom in a 1D optical lattice. The idea is to use the particle's internal degree of freedom to bring it into a superposition of going right and left simultaneously, which we do by driving the system periodically with a spin mixing operation. In Sec. III, the equations governing the time evolution are presented. We use these operators to perform numerical simulations of the system, and show that the atoms have the dynamics of a quantum walk. In Sec. IV, we show that this system is topologically non-trivial and derive its phase diagram. We find that the topological phase can be changed by changing the spin mixing angle, allowing us to generate a topological boundary. In Sec. V, we populate the bound state that appears at this interface, and suggest a method for measuring its presence. In Sec. VI, we consider an interesting limit of this Hamiltonian, which also presents bound states, despite being topologically trivial. We explain this by showing that these are in fact a pair of Jackiw-Rebbi states, and study the mechanism according to which they can hybridise. In Appendix A, we show that, in a certain parameter regime, the system maps onto the Creutz ladder. In Appendix B, we show that the protocol we suggest can be understood as a superposition of two independent quantum walks. In Appendix C, we explain the method used to find the system's symmetries.
II. EXPERIMENTAL PROPOSAL
Our idea to implement a quantum walk is as follows: particles are allowed to evolve in a medium which accommodates right movers and left movers. By periodically applying a pulsed operation which has amplitude to interconvert right movers and left movers, we obtain path interference phenomena which are consistent with a quantum walk.
The specific background which is needed to obtain the topological properties we desire is the 1D lattice represented in Fig. 1. We will consider the dynamics of an atom with two internal degrees of freedom, which we can refer to as the particle's spin. Spin up (↑) particles see a superlattice with two sites per unit cell, with intra-cell hopping amplitude J−δ and inter-cell hopping J+δ. This type of lattice is obtained by superimposing two standing waves l 1 and l 2 , with wavelengths λ 1 = 2λ 2 . The easiest way to obtain l 2 is to frequency double l 1 . The distance between neighbouring sites is d = λ 2 /2, and the size of a full unit cell is 2d.
Spin down (↓) particles see the same lattice as ↑ particles, but shifted by d. This can be done by making l 1 attractive for ↑ particles but repulsive for ↓ particles, effectively shifting it by a phase of π. To do this, set λ 1 to be the so called anti-magic wavelength of the atom, such that the lattice has an equal and opposite detuning for ↑ Figure 1. 1D superlattice used to generate the spatial translation operation of the atomic quantum walk. The grey shaded boxes represent unit cells, in which there exists A and B sublattice sites. This lattice geometry is obtained by superimposing two lasers with wavelengths λ1 and λ2 such that λ1 = 2λ2; the resulting intersite distance is d = λ2/2. Spin up (↑) particles see the orange lattice (top), which has intra-cell hopping J −δ (dashed arrow) and inter-cell hopping J +δ (solid arrow). Spin down (↓) particles see the blue lattice (bottom), which is identical to the orange lattice, but shifted by d. In the figure, J and δ are depicted as positive parameters. Particles in this lattice are subject to the HamiltonianĤS Eq. (3). and ↓ states.
Interesting candidates to play the role of our ↑ and ↓ states are the clock states of either Ytterbium or Strontium atoms. These electronic states have narrow transitions, making them long lived, and are well separated in energy, such that their anti-magic wavelength λ 1 is readily accessible, while l 2 remains at approximately the same amplitude for both species. Additionally, these states can be coupled using Raman beams without significantly heating the system. We drive this system by periodically applying two laser pulses, which induce Raman transitions between ↑ and ↓ states. This is the coin operationĈ θ of the atomic quantum walk. The amplitude of the coupling, controlled by the angle θ, is proportional to the intensity of the lasers. Additionally, when the angle between the two lasers is non-zero,Ĉ θ applies a momentum kick.
We will show in Sec. IV that the topological properties of this system can be modified by changing the value of θ or δ. We suggest to vary θ spatially to create a boundary between two regions with different topological properties.
One way to do this is to create a gradient in the intensity of the lasers which generate the Raman operation. Because these beams can have beam waists of order the length of the system itself, this can be done simply by focussing the laser away from the centre of the lattice. The amplitude of the spin mixingĈ θ is proportional to the intensity of the Raman pulses; thus if the Raman lasers' intensity varies over the length of the system, the spin mixing angle θ will vary accordingly.
III. MODEL
In this section, we present the model realizing the atomic quantum walk, and the operator controlling its time evolution. Using this knowledge, we perform numerical simulations to determine the system's properties. Finally, we introduce a convenient unitary transformation which simplifies the description of the system.
As shown in Fig. 1, we divide the system into unit cells (indexed by n), and assume that the atoms only ever populate four quantum states per unit cell: within the nth unit cell, the atom can have spin ↑ or ↓, and can reside in the motional ground state of the left/right potential well -we will refer to this as sublattice A/B. We gather these internal degrees of freedom into a formal vector, and define vector creation operators aŝ
c † n = (c † n↑A , c † n↓A , c † n↑B , c † n↓B ).(1)
We use σ j and τ j to denote the Pauli matrices acting in spin space (↑ and ↓) and sublattice space (A and B) respectively, with j = {1, 2, 3}, and τ 0 and σ 0 are 2 × 2 identity matrices. We will also use τ ± to represent the sublattice index raising and lowering operators,
τ ± = 1 2 (τ 1 ± iτ 2 ).(2)
Shift operation on wavepackets
Consider first the Hamiltonian of the 1D bichromatic optical lattice without the Raman pulses. This readŝ
H S = nĉ † n τ 1 ⊗ (Jσ 0 + δσ 3 )ĉ n + n ĉ † n+1 τ + ⊗ (Jσ 0 − δσ 3 )ĉ n + H.c .(3)
The parameters J and δ control the particles' hopping amplitudes, as shown in Fig. 1. Since the Hamiltonian of Eq. (3) is translation invariant, its eigenstates are plane waves with well defined quasimomentum k ∈ [−π/(2d), π/(2d)]. These states can be chosen to be fully polarised, either ↑ or ↓. Note that each energy eigenvalue is doubly degenerate, since the system presents two identical lattices (one for each spin) with no tunnelling between them (see Fig. 1). The eigenstates ofĤS can be chosen to be fully polarised (either ↑ or ↓), and the ↑ and ↓ bands overlap exactly; the value of the band gap is 4δ. Note that the dispersion is symmetric about E = 0. This means that the slope of the two circled regions of the dispersion are equal and opposite. Thus the wavepackets existing in these regions, which are the states centred around k = π/(4d), with energies E and −E respectively, move on average at equal and opposite velocities. The spin mixingĈ θ couples states in these regions (illustrated by an arrow).
The object that undergoes a quantum walk is a wavepacket that is broad in position space but restricted in momentum space to the vicinity of the wavenumber k = π/(4d). For this quasimomentum the Hamiltonian is almost dispersionless, as shown in Fig. 2. Thus a wavepacket constructed with states from the lower branches of the dispersion relation, with k ≈ π/(4d), is translated with a uniform velocity to the right, and its real-space width grows only very slowly. It is therefore a right-mover. A wavepacket similarly constructed, but belonging to the upper branches of the dispersion relation, is a left-mover.
Rotation operation using the Raman pulses
Consider next the effect of the two Raman lasers on the system, which are switched on for a brief but intense pulse of duration . Assuming the laser pulses are intense enough,Ĥ S can be neglected during the time they are switched on, and we havê
H(t) = n Ω(n, t)ĉ † n τ 3 ⊗ σ 2ĉn ,(4)
where Ω(n, t) is the Rabi frequency of the lasers at the position of the nth unit cell. By setting the two Raman lasers at an appropriate angle, the pulse applies a π/d quasimomentum kick which results in the τ 3 operator appearing in Eq. (4). We define the area of the pulse at unit cell n by θ(n):
θ(n) = /2 − /2 Ω(n, t)dt,(5)
such that the effect of the whole pulse is given by:
C θ = nĉ † n exp(−iθ(n)τ 3 ⊗ σ 2 )ĉ n ≡ exp(−iĤ θ ). (6)
The effect ofĈ θ is to couple right-moving states from the bottom branch of the dispersion relation to left-moving states of the top branch, as indicated in Fig. 2. Interestingly,Ĉ θ also couples states on the same branch of the dispersion relation. We will discuss the consequences of this later in the paper.
Complete sequence
We obtain a quantum walk by switching on the Raman lasers for brief intense pulses of duration which follow each other periodically, with period T . In the following, we will use dimensionless units where T / = 1. The unitary time evolution operator for one complete period, U, readsÛ
= e −iĤ θ /2 e −iĤ S e −iĤ θ /2 .(7)
In writing down Eq. (7), we have chosen the origin of time such that the sequence of operations defining the walk has an inversion point around which it is symmetric in time, as discussed in detail in Ref. [31]. While this choice has little effect on the system's properties, the form of Eq. (7) makes it easier to find the symmetries ofÛ. It is however important to always pick the same origin of time when averaging over multiple runs of the experiment, otherwise important details could be averaged out. The tunnelling amplitudes induced byĤ S andĤ θ are sketched in the figure 3(a). Finally, we define the Floquet HamiltonianĤ F as:
U = exp(−iĤ F ).(8)
This static Hamiltonian describes the motion, integrated over a time step. This allows us to compute the system's spectrum. Because the eigenvalues ofĤ F are defined from Eq. (8), the band structure ofĤ F is 2π periodic. We will refer to the eigenvalues ofĤ F as the system's quasienergies.
At the end of this section we will find a change of basis which simplifiesĤ F . Despite this, we will prefer to work in the basis defined above when performing numerical simulations, so that our results can be easily compared to experimental results from the protocol described in Sec. II. (6), respectively. The system has two sites per unit cell. (b) We flip the spins on every second site. the time evolutionÛ is controlled byĤ S andĤ θ , given by Eqs. (11) and (12). The lattice has one site per unit cell. The system maps onto the Creutz ladder when J = δ.
Quantum Walk
As discussed above, repeated application of the timestep operatorÛ of Eq. (7) on a wavepacket can be described as a quantum walk. We verified this using numerical simulations. As an example, we present the atom's final density distribution after 60 time steps in Fig. 4. For this simulation, we initiated the system with a ↑ polarised Gaussian wavepacket |ψ t=0 with width 4d centred around k = π/(4d); we set J = π/3, δ = 0.42 and θ = 0.15. The final density distribution shows sharp peaks where density is furthest from the origin. The probability to find the particle in any other region is inhibited due to back travelling waves, which interfere destructively with forward travelling ones. The inset shows the standard deviation of the position of the particle, which can be seen to increase linearly with time. Both the destructive interference and the ballistic expansion are well known feature of quantum walks (in contrast with classical random walks, which show diffusive expansion). The interference pattern that the density distribution forms is typical of quantum walks. The walker's initial positions were sampled from the Gaussian wave function centred at the site n = 0, with width 4d and average quasimomentum k = π/(4d). We used the Hamiltonian's parameters: J = π/3, δ = 0.42 and θ = 0.15. Inset: the variance of the wavepacket scales linearly with time, which is a characteristic feature of quantum walks.
Gauge transformation for a smaller unit cell
By inspection of Fig. 3(a), we notice that it is possible to simplify the system's description by introducing the new vector creation operators:
c † 2n = (c † n↑A , c † n↓A ); (9) c † 2n+1 = (c † n↓B , c † n↑B ).(10)
Notice the inversion of the order of ↑ and ↓ on the odd sites. In this basis, the tunnellings induced by the atomic quantum walk are represented in Fig. 3(b). Under this transformation, we see that the atomic quantum walk is reminiscent of the Creutz ladder [39][40][41], a 1D model which is known to support a non-zero winding number. We will make this correspondence more obvious in Appendix A.
In the basis introduced above, the HamiltoniansĤ S andĤ θ become:
H S →Ĥ S = nĉ † n+1 (Jσ 1 − iδσ 2 )ĉ n + H.c; (11) H θ →Ĥ θ = nĉ † n θ(n)σ 2ĉn .(12)
WhileĤ θ Eq. (12) is already in diagonal form, we can diagonalizeĤ S by Fourier transforming Eq. (11):
H S (k) = 2J cos(kd)σ 1 + 2δ sin(kd)σ 2 ,(13)
with k ∈ [−π/d, π/d]. Because of the smaller unit cell, the dispersion has now two non-degenerate branches. It is interesting to redefine our right-and left-walkers in this basis, such that the mapping of the system to a quantum walk can be made more obvious. This is done in Appendix B, where we show that the system has dynamics which is more complex than the standard quantum walk considered by Refs. [17,20,26]. By analogy with Eqs. (7) and (8), we can introduce the time evolution operator and the Floquet Hamiltonian in the new basis:
U = e −iĤ θ /2 · e −iĤ S · e −iĤ θ /2 = exp(−iĤ F ).(14)
In the sections that follow, all analytical results will be obtained in the simplified basis ofĤ F . In the next section, we shall see that the system is topologically nontrivial, and that these topological properties are reflected inĤ F .
IV. TOPOLOGICAL PROPERTIES OF THE ATOMIC QUANTUM WALK
As we have seen in Eq. (6),Ĉ θ simultaneously brings the system into a superposition of right movers and left movers, and in a superposition of the spin degree of freedom. This results in spin-orbit coupling terms when the dynamics are averaged over a period of the motion, a fact which is essential for the non-trivial topological properties of the Floquet Hamiltonian to appear. We will see in this section thatĤ F can have non-zero winding numbers, which can be modified by changing the value of θ and δ.
To understand the topological properties of this system, it is important to analyse the symmetries ofĤ F , the Floquet Hamiltonian in the new basis. The method for finding the symmetries ofĤ F is explained in detail in Appendix C. By inspection of Eqs. (12) and (13), we find thatĤ F has chiral symmetry (CS) implemented by the operatorΓ = σ 3 .
The presence of CS implies thatĤ F anti-commutes with a unitary, Hermitian matrixΓ. Importantly, the CS operator acts only within single unit cells. This will later allow us to break translational invariance without breaking CS. The existence of CS implies that every eigenstate ofĤ F has a chiral partner with equal and opposite quasienergy:
H F |ψ = E|ψ ⇒Ĥ F Γ |ψ = −E Γ |ψ .(15)
States with quasienergies such that E = −E are special because they can transform into themselves under CS. This means that these states can exist without a chiral partner. When this is the case, they cannot be moved away from the quasienergy satisfying E = −E without breaking CS. These states can therefore not be coupled to other states in the system by chiral symmetric perturbations. In a Floquet system the quasienergies are 2π-periodic. There are therefore two quasienergies, E = 0 and E = π, which satisfy the condition E = −E. As we will show shortly, this implies that bound states which are topologically protected as a result of CS can appear at E = 0 or E = π.
A system which presents CS but neither time reversal symmetry nor particle hole symmetry belongs to the AIII class of the topological classification of Hamiltonians [28]. Floquet Hamiltonians in 1D which belong to this class can have two non-zero topological invariants, the winding numbers ν 0 and ν π [31]. For a detailed physical interpretation of the physical significance of these quantities, we refer the reader to Ref. [32].
Consider connecting two regions R 1 , R 2 which have different winding numbers and present spectral gaps. The topological invariants ν 0 , ν π cannot change without closing the spectral gap in E = 0, E = π respectively. We assume that the boundary between the two bulks is smooth and slowly varying, such that the spectral gaps of R 1 , R 2 are densely populated in this region. If we now make the interface between R 1 and R 2 sharper, fewer states can live in the boundary region, meaning that fewer states can populate the spectral gap of R 1 , R 2 . To illustrate this, the energy of the three states closest to E = 0 at a topological boundary are plotted versus the boundary sharpness in Fig. 5. In the infinitely sharp boundary limit, all states have been lifted out of the spectral gap except for those which do not have a chiral partner. Thus, if a system presents an interface between regions R 1 , R 2 with different winding numbers ν 0 , there must exist an E = 0 state at this interface. Because R 1 and R 2 must be gapped to have well defined winding number, the E = 0 state can only exist at the topological interface. Correspondingly, if the system presents an interface between two values of ν π , an E = π bound state must exist where ν π changes.
In fact, the number of times that the spectral gap closes at the interface between two regions is (at least) the difference in their winding numbers. The difference in ν 0 (ν π ) tells us how many topologically protected E = 0 (E = π) bound states exist at this interface. In the following, we will follow Refs. [31,32] to calculate the winding numbers ν 0 and ν π from the time evolutionÛ , and find the parameter regime which makes the atomic quantum walk topologically non-trivial.
The time evolution operator Eq. (14) has the form:
U =Γ ·Ĝ † ·Γ ·Ĝ,(16)withĜ = e −iĤ S /2 · e −iĤ θ /2 = a(k) b(k) c(k) d(k) ,(17)
and where a(k), ..., d(k) are the entries ofĜ, which are, in general, complex functions of k. In general, the winding number of a function z(k) can be evaluated using the formula: Figure 5. Energy of the three states closest to E = 0 at a topological boundary between two regions R1, R2 versus the boundary sharpness, controlled by the boundary width ξ. The system is chiral symmetric and regions R1, R2 have a spectral gap [−∆E/2, ∆E/2]; these energies are indicated by dashed yellow lines. As ξ is reduced, the interface between R1 and R2 can accommodate fewer states, such that, when ξ < ∼ 1, only topological bound states can exist in the spectral gap of R1, R2.
ν[z] = 1 2πi π/d −π/d dk d dk log z(k).(18)
In the basis whereΓ = σ 3 , we have
ν 0 = ν[b] and ν π = ν[d], with ν[b]
and ν[d] the winding numbers of the b(k) and d(k) functions respectively [31].
Due to the anti-commutation property of Pauli matrices, we have that for any vector v:
e iv·σ = σ 0 cos(|v|) + i v · σ |v| sin(|v|).(19)
This allows us to compute the exponential forms of Eqs. (12) and (13). Substituting into Eq. (17), we find:
b(k) = cos(ε(k)) sin(θ/2) − sin(ε(k)) cos(θ/2) δ sin(kd) + iJ cos(kd) ε(k) ,(20)
and:
d(k) = cos(ε(k)) cos(θ/2) + sin(ε(k)) sin(θ/2) δ sin(kd) − iJ cos(kd) ε(k) ,(21)
where ε(k) are the energy eigenvalues ofĤ T /2:
ε(k) = ± J 2 cos 2 (kd) + δ 2 sin 2 (kd).(22)
The simplest way to visualize the winding numbers ν 0 and ν π is to plot b(k) and d(k) in the complex plane as k goes from −π/d to π/d (both bands have the same winding number). The system is topologically non-trivial if at least one of the curves winds around the origin. The curves that b(k) and d(k) form in the complex plane are presented for J = π/3, δ = 0.42, and θ = 0.15 in Fig. 6. It is clear from this figure that the system can present non-zero winding numbers. The winding number ν 0 cannot change unless both the real and imaginary parts of b(k) vanish simultaneously (see Fig. 6). This happens when δ = ±θ/2 + nπ, n ∈ Z, for which values the band gap closes either at quasimomentum k = π/(2d) or k = −π/(2d). Similarly, ν π cannot change unless d(k) vanishes, which happens when δ = ±θ/2 + (n + 1/2)π, n ∈ Z, at k = π/(2d) or k = −π/(2d). This allows us to construct the topological phase diagram of the atomic quantum walk, which is presented in Fig. 7. We evaluate the winding numbers in each region using the general formula Eq. (18). It is also possible for band gap closing events to occur in θ = nπ, n ∈ Z without changing either of the winding numbers. When J = π/3 however (value used in the simulations presented in this paper), no such events occur in the interval considered in the Fig. 7.
As is visible from Fig. 7, the winding number of the atomic quantum walk is a function of δ and θ. In the following section, we will use this to create two regions with distinct topological properties.
V. DETECTION OF THE TOPOLOGICAL BOUND STATE
As we saw in Sec. IV, the topological properties of the atomic quantum walk can be modified by changing the spin mixing angle θ. By using spatially inhomogeneous Raman lasers, it is therefore possible to create two regions in the system with distinct topological properties. At the boundary between two regions which have different winding numbers, there lives a robust bound state which is protected by the system's symmetries and pinned either at E = 0 or E = π. In the following, we suggest a method for experimentally generating this topological bound state, and identifying it using its characteristic spin distribution. The density distribution can either be retrieved in a single measurement, if the quantum walk is performed with a gas of non-interacting atoms, or by repeating the experiment many times and averaging if a single atom is used. We will be interested in the spin populations of sublattices A and B, meaning that the position measurement must have single site resolution and be sensitive to the atom's spin. The location of A and B sites is fixed by the lasers generating the optical lattice (see Fig. 1), so that their position will remain the same from one experiment to the next.
When performing this experiment, we expect a portion of the density to remain pinned to the topological boundary. This corresponds to the part of the initial state which overlaps with the bound state wave function. The rest of the density is translated ballistically away from the topological interface. We verify this numerically by performing simulations, where θ is varied according to:
θ(n) = θ max + θ min 2 + θ max − θ min 2 tanh nd ξ ,(23)
where ξ determines the width of the region where θ(n) changes value. This function is represented in Fig. 8.
Note that the existence of the bound state only requires that θ(n) crosses 2δ, and does not depend otherwise on the precise form of Eq. (23). Our simulations are performed in the basis whereĤ S andĤ θ have the form given by Eqs. (3) and (6) respectively. Fig. 9(a) shows an example of the evolution of the atomic density during 50 time-steps. As expected, we find that a portion of the density remains pinned to the region of n = 0, which is the location of the topological boundary. Atoms which do not populate the bound state are transported ballistically away from n = 0. The spreading is anisotropic due to the initial state we chose, which is fully ↑ polarised. The spreading density shows interference patterns between forward and backward travelling atoms, as is characteristic for quantum walks (indeed, we already observed this behaviour in Fig. 4).
The density at specific moments in time is represented in Fig. 9(b), (c) and (d). We observe that at late times, the probability distribution is exponentially peaked at the location of the topological boundary. Importantly, the relative population of this state does not decrease at later times. We have verified this numerically by computing the overlap between the atom's wave function and the E = 0 eigenstate ofĤ F , and found that it is time independent.
Experimentally, this can be verified by plotting the total density in the neighbourhood of the topological boundary, which is displayed in Fig. 10. We see that at early times, the total density near the origin decreases rapidly as atoms which do not populate the bound state are transported away from the topological boundary. During this period, we notice that the total density presents oscillations; these are due to the interference of ballistically transported atoms. At late times, the total density near the topological boundary converges to a non-zero value, which is a sign that atoms populating the bound state do not leak into other states of the system. Another experimental signature is the mapping of the phase boundaries. Indeed, Fig. 7 show the parameter regimes in which we expect to find topological bound states, and knowledge of the function with which θ varies determines the position around which these states are centred. By varying δ and θ, we can explore the system's parameter space and verify that these exponentially bound states occur at all topological boundaries. This provides a straightforward way to verify that we are indeed observing topological bound states, and relies only on imaging the atoms' probability density function.
An alternative method for verifying that a system is a topological bound state is to verify that it is an eigenstate of the chiral symmetry (CS) operatorΓ. Indeed, we saw in Sec. IV that if a state verifies the following conditions, it is a topological bound state:
1. The state is an eigenstate simultaneously ofΓ and the Floquet HamiltonianĤ F .
2. The state has vanishing overlap with any other state of equal energy which satisfies condition 1.
In the following, we describe a method to identify an eigenstate ofΓ, thereby providing a strong way of identifying a topological bound state. We can verify that the bound state we are observing is a single eigenstate ofĤ F directly from the time evolution of probability density distribution. Indeed, if this state was a superposition of states, we would observe Rabi oscillations, while it is clear from figure Fig. 9(a) that the density distribution near the origin remains the same after a period of driving. Additionally, it can be seen from the inset of Fig. 9(d) that the state which is found near the origin at late times has an extremely interesting structure. Indeed, we find that ↑ states only occur on even sites, while ↓ states only occur on odd sites. This is a direct result of being an eigenstate of CS, which has the formΓ = τ 3 ⊗ σ 3 in this basis, leading to a constrained spin distribution. Importantly, all other eigenstates ofĤ F must transform into their chiral partner under the action of CS. The only states that can do this have equal ↑ and ↓ density on each site. Consequently, the state we are measuring near n = 0 can only be a topological bound state.
A spin sensitive density measurement therefore provides a direct method to identify an E = 0 or an E = π bound state protected by CS in a 1D system. Using the exact same measurement at half time-steps, it is possible to discriminate between E = 0 and E = π energy states. A half time-step is performed by applying a spin rotation θ/2, followed by time evolving withĤ S for a time T /2. It was shown in [32] that the dynamics of the bound state between two time-steps is sensitive to the state's energy. As shown in Fig. 11(a), at integer time-steps, the ↑ (↓) density of E = π states only has support on even (odd) sites. At half time-steps, the spin structure of E = π states is reversed, with spin ↑ (↓) states only on odd (even) sites. This is shown in Fig. 11(b). This contrasts with the behaviour of E = 0 states, which keep the same spin structure at integer time-steps and half integer timesteps. Thus, performing a position measurement with single site resolution which is sensitive to the spin state does not only identify a topological bound state of the system, but it also provides a method to differentiate an E = 0 from an E = π state.
To recapitulate, we suggested a method to generate topologically protected bound states experimentally, and simulated this protocol numerically. We saw that it is possible to identify topological bound states by correlating their occurrence with Fig. 7 and mapping out the phase boundaries by changing θ, δ. Alternatively, we can recognise topological bound states thanks to their unique density distribution, which can also be used to differentiate E = 0 from E = π states. In the next section, we will see that exponentially localised states with E ≈ 0, π can also appear at trivial band gap closings (ones where the winding number does not change). It is therefore important to know for which values of δ, θ these occur when exploring the parameter space of the atomic quantum walk.
VI. PAIR OF MOMENTUM SEPARATED JACKIW-REBBI STATES
In this section, we will see that not all E = 0, π states which are exponentially localised are topologically protected. We will consider the atomic quantum walk in the limit δ = 0. As can be seen from Fig. 7, when following the path in parameter space indicated by a dashed green arrow, a topologically trivial band gap closing occurs when θ change sign. Indeed, the winding numbers {ν 0 , ν π } are the same in both regions. As a result of this, the Hamiltonian in this limit also presents E ≈ 0 and E ≈ π bound states. By finding an approximate expression for the Floquet HamiltonianĤ F , we will understand that the E ≈ 0 bound states correspond to solutions of the Jackiw-Rebbi model in the continuum case. Because these states occur in pairs, they can hybridise and move symmetrically away from E = 0, thus destroy- ing the states' topological protection. Importantly, these trivial bound states are not eigenstates of chiral symmetry, and therefore do not have the same spin distribution as the states presented in Fig. 11.
In the limit of δ = 0,Ĥ S from Eq. (11) andĤ θ from Eq. (12) have the form:
H S = n Jĉ † n+1 σ 1ĉn + H.c;(24)H θ = − nĉ † n θ(n)σ 2ĉn .(25)
Using the method presented in Appendix C, we find the symmetries of the system in this limit. These are presented in the Table I. From the expression of the chiral symmetry, time reversal symmetry and particle hole symmetry operators, we find that we are in the BdG symmetry class CI of the classification of single-particle Hamiltonians. Thus, the atomic quantum walk is topologically trivial in 1D in the limit δ = 0 [28]. Despite this, the system displays E ≈ 0 and E ≈ π bound states when the spin mixing angle θ is varied spatially. We can verify this numerically by considering a chain of length L and θ = θ(n), a continuous function of position, going from θ min to θ max over a length scale ξ according to Eq. (23). We assume d ξ L, where d is the lattice spacing. The spectrum ofĤ F from Eq. (14) is shown in Fig. 12. We observe that a band gap is open around E = 0, except for θ max = 0 (mod 2π). A pair of chiral partner zero-energy states appears in the spectral gap whenever θ min < 0 < θ max , exponentially localised around the site where θ(n) = 0. Similarly, π energy states, centred at θ(n) = π, appear if θ min < π < θ max .
We will now derive an approximate expression ofĤ F in the limit d ξ L, and use it to explain the origin of the E ≈ 0 bound states in this model. We will find it useful to go to the continuous limit, where we are able to find an exact eigenstate such thatĤ F |ψ = 0. We will then obtain a discrete ansatz from this solution and evaluate its energy, which we can compare to results from numerical simulations.
For constant θ(n) = θ, the band gap closes when θ = 0 (mod 2π) at ±k bs = ±π/(2d), where d is the lattice spacing. For slowly varying θ(n) near θ(n) = 0, we will assume the eigenstates ofĤ F such that E ≈ 0 have the form: where |n is the state which is well localised at site n, and ϕ ± (n) is the envelope function spinor, which we assume to be slowly varying. WhenĤ S acts on |ψ ± , we obtain:
|ψ ± = n e ±ik bs nd ϕ ± (n)|n ,(26)H S |ψ ± = ± iJ n e ±ik bs nd σ 1 · ( ϕ ± (n + 1) − ϕ ± (n − 1)) |n .(27)
Assuming that ϕ ± (n) varies slowly compared to the lattice spacing d, we can take the continuous limit of Eq. (27) by sending d → 0, n → ∞ such that nd = x is constant:
H S |ψ ± → ±2iṽσ 1 dx e ±ik bs x ∂ x ϕ ± (x)|x ,(28)
where we have introduced the velocity parameterṽ = Jd. The length scale associated with ∂ x ϕ ± (x) is much larger than the length scale over which exp(±ik bs x) varies, and ϕ ± (x) decays exponentially away from where θ(x) vanishes, allowing us to expandĤ F to first order inĤ S and θ(x):Ĥ
F ≈Ĥ approx =Ĥ S +Ĥ θ ,(29)
when acting on states near E = 0 which are localised in the neighbourhood of θ(x) = 0. Thus, we have derived a static, approximate Hamiltonian,Ĥ approx . Using Eq. (28), we can apply this Hamiltonian to |ψ ± :
H approx |ψ ± = dx e ±ik bs x ± 2iṽσ 1 · ∂ x ϕ(x) − θ(x)σ 2 · ϕ(x) |x .(30)
It was shown by Jackiw and Rebbi that there always exists a zero solution to the right hand side of Eq. (30) when the sign of θ(x) is different for x → +∞ and x → −∞ [42]. When this is the case,Ĥ approx has a E = 0 solution. If lim x→±∞ sign(θ(x)) = ±1, then ϕ ± (x) takes the form:
ϕ ± (x) = ψ 0 exp − 1 2ṽ x 0 dx θ(x ) |∓ ,(31)
where we have defined the spin states: |+ = (1, 0) T and |− = (0, 1) T , and ψ 0 is the normalization of the wave function. In the following, we will restrict our study to θ(x) varying as:
θ(x) = α + β tanh x ξ ,(32)
with β > α and β > 0. This is the continuous version of Eq. (23), with 2α = θ max + θ min and 2β = θ max − θ min . From Eqs. (31), (32), we see that ϕ ± (x) has the form:
ϕ ± (x) = ϕ(x)|∓ ,(33)ϕ(x) = ψ 0 exp − αx 2ṽ cosh x ξ −βξ/(2ṽ) .(34)
By discretising the above result, sending x → nd, we obtain an ansatz wave function for the two zero energy states that we are observing:
|ψ ± = n e ±ik bs nd ϕ n |n, ∓ ,(35)ϕ n = ψ 0 exp − αn 2J cosh nd ξ −βξ/(2Jd) .(36)
Let's take a moment to recapitulate what we have done so far. We have defined two states, |ψ + and |ψ − , which are centred around the momenta k = ±k bs . We have shown that when θ(x) changes sign, these states are eigenstates of the Hamiltonian with eigenvalue E = 0. In the Jackiw-Rebbi model, there is only one such state, which is pinned at E = 0 by chiral symmetry. In our system, spatially varying θ(x) leads to terms which mix the states |ψ + and |ψ − , causing them to hybridise and move symmetrically away from E = 0.
We can find the energy of these hybrid states by studying the eigenvalues ofĤ red , the projected Hamiltonian on the basis of states |ψ + and |ψ − .
H red = ψ + |Ĥ approx |ψ + ψ + |Ĥ approx |ψ − ψ − |Ĥ approx |ψ + ψ − |Ĥ approx |ψ − .(37)
This matrix has eigenvalues:
E ± = ± n (−1) n ϕ * n (J(ϕ n+1 − ϕ n−1 ) + θ(n)ϕ n ) .(38)
The energy |E ± | from Eq. (38) is plotted versus ξ in Fig. 13. Alongside this estimate of the lowest eigenstate's energy, we have diagonalisedĤ approx and plotted its eigenvalues. Visibly there is a good agreement between the estimate Eq. (38) and the exact energies, suggesting that Figure 13. Energy of the Jackiw-Rebbi states |E±| versus ξ, the inverse rate of change of θ(n). Blue dots: value of |E±| from Eq. (38). Yellow triangles: Eigenvalues ofĤ approx closest to E = 0 (obtained numerically). The lines are exponential fits to the data. Both these curves approach each other exponentially (which is clear because the scale of the y axis becomes exponentially small). In both cases, α = −1/4, β = 1.
our hypothesis was indeed correct, and that the states we are observing are indeed two Jackiw-Rebbi states separated in momentum, the energies of which behave as:
|E ± | ≈ J exp − cξ d ,(39)
where c is a positive constant.
From the results presented here, we can therefore conclude that the system presents a pair of zero energy bound states when θ(n) changes sign, despite being in a trivial topological configuration. We find that when θ(n) changes slowly (ξ d), we have two non-overlapping (in k space) bound states, one associated to each Dirac cone. The overlap increases for faster changing θ(n) (smaller ξ), resulting in both states hybridising and moving away from E = 0 without breaking chiral symmetry. For this reason, these bound states do not benefit from the robustness displayed by topologically bound states.
This type of behaviour is generally observed in systems which present two degenerate energy levels. When the two states are coupled, their energies split symmetrically, proportionally to the matrix element between them (the off diagonal term of Eq. (37)). In our system, we are able to directly control the tunnelling between these states by modifying the length scale over which the potential varies. Thus our model gives us access to a single parameter which controls the states' degree of hybridisation.
VII. CONCLUSIONS
In Sec. II, we suggested an experiment in which spin 1/2 cold atoms realise a topologically non-trivial quantum walk. This protocol relies on trapping the atoms in a 1D optical lattice which is spin dependent and has two sites per unit cell. We drive the system by periodically coupling the atoms' spin states. When the dynamics are averaged over one full period of the motion, we see from the operators in Sec. III that this driving results in spinorbit coupling terms. These terms are the key to realising topologically non trivial properties. Specifically, we verified analytically in Sec. IV that the atomic quantum walk can realise either E = 0 or E = π bound states, which are associated to two separate topological invariants. These states appear when the amplitude of the spin coupling varies spatially; they are bound to the location where one of the topological invariants changes its value.
We showed numerically in Sec. V that topologically protected bound states can be isolated by performing the atomic quantum walk for a large number of steps (t/T ≈ 50). Atoms that do not populate the bound state then leave the region of the boundary, such that only an exponentially localised density peak remains at the topological interface. To verify that exponential density peaks such as this one correspond to topological bound states, we suggest exploring the full parameter space, and verifying that exponentially bound states exist whenever the system presents a topological boundary. As an alternative method for identifying topological bound states, we suggested searching for eigenstates of the chiral symmetry operator. These states have a spin distribution which is heavily constrained, allowing us to identify E = 0, π eigenstates of the Hamiltonian by averaging over spin sensitive measures of the atom's position. By performing the same measure at half time-steps, we saw that we can discriminate between E = 0 and E = π bound states, and thereby measure both of the system's topological invariants.
Finally, we mentioned that it is possible for exponentially bound states to appear at band gap closings where none of the topological invariants change value. In Sec. VI, we studied a limit of the atomic quantum walk where this happens; in this case, our model admits two Jackiw-Rebbi states which are separated in quasimomentum. These states can move away from E = 0 by hybridising, and the degree of their hybridisation is controlled by the rate of change of the spin coupling amplitude. Importantly, these states are not eigenstates of the chiral symmetry operator, meaning that they cannot be confused with topological bound states when performing a spin sensitive measure of the atoms' density distribution.
We have not considered the possibility of spatially varying the tunnelling amplitude δ. This could be done by varying the detuning of spin ↑ and ↓ states spatially, as was experimentally realised in Ref. [35]. By varying δ spatially, two bound states could be generated at the same location and at the same energy. As can be seen from Fig. 7, when the driving is set to θ = 0 and δ changes sign, the winding ν 0 changes by ∆ν 0 = 2, implying that there must exist two zero energy bound states at the topological interface. This situation is interesting because, despite the fact that these states overlap spatially, they cannot hybridise.
Aside from this particular parameter regime, it would be interesting to explore the robustness of the topological bound states, when for instance interactions or faults in the periodic driving are introduced. Interactions, as long as they are small and preserve chiral symmetry, should not in principle destroy the system's topological properties. Similarly, the derivation in Appendix C suggests that the bound states should be robust to (perturbative) faults in the periodic driving. It would be interesting to verify theoretically and experimentally to what extent this is the case, as this study would provide a test of how applicable the Floquet description is to real world systems.
H C (k) = −v sin(kd)σ 1 + (w − g cos(kd))σ 2 ,(A2)
where we have set the ladder's unit cell size to d. The Creutz ladder belongs to the BDI class of the topological classification of Hamiltonians [41]. As a result, this system admits a non-zero winding number ν C , the value of which depends on the system's parameters as:
ν C = 1 2 (sign(w + g) − sign(w − g)).(A3)
We will now show that we can map the atomic quantum walk to the Creutz ladder. We consider the translational invariant atomic quantum walk in the two-state basis presented in Sec. III, and change the origin of momentum k → k + π/(2d). In this basis, Eq. (11) becomes:
H S (k) →H S (k) = −2J sin(kd)σ 1 + 2δ cos(kd)σ 2 . (A4)
The HamiltonianĤ θ given by Eq. (12) is not modified by this transformation. Note that this is a trivial Gauge transformation that cannot change the topological properties of the system. Interestingly, in this basis, the system has CS, TRS and PHS. The operators implementing these symmetries are detailed in the Table II. The method used to determine the system's symmetries is detailed in the Appendix C.
From Table II, we find that the system presents CS, TRS and PHS, all of which square to the identity. This tells us immediately that we are in the BDI class of the topological classification of Hamiltonians, i.e the same symmetry class as the Creutz ladder.
We will now proceed to show thatH F , the Floquet Hamiltonian, maps onto the Creutz ladder when J = δ. We can findH F by substituting Eqs. (12) and (A4) into Eq. (14). In the limit J = δ,H F is:
H F = ±Ẽ (k) sin(Ẽ(k))
− sign(δ) sin(2δ) sin(kd)σ 1 + (− cos(2δ) sin(θ) + sign(δ) cos(θ) sin(2δ) cos(kd))σ 2 .
(A5)
Apart from the upfrontẼ(k)/ sin(Ẽ(k)), this is exactly the Creutz ladder Hamiltonian Eq. (A2), with: v = sign(δ) sin(2δ), (A6) w = −sign(δ) cos(2δ) sin(θ), (A7) g = − cos(θ) sin(2δ).
(A8) By substituting these values in Eq. (A3), we can calculate the winding number ν C and deduce the phase diagram in this time frame, which is represented in Fig. 15. While we expect the upfront termẼ(k)/ sin(Ẽ(k)) to deform the band structure, it does not change the symmetry properties ofH F , and therefore cannot change its topological properties as long as it does not close the band gap. In expressingÛ in the symmetric form for Eq. (14), we made a choice of time frame. There exists another time frame which has an inversion point in time and has the form:Û T = e −iĤ S /2 e −iĤ θ e −iĤ S /2 .
(A9)
While the system in this time frame does not map onto the Creutz ladder, it does have a winding number ν T , which is the second topological invariant represented on Fig. 15. By comparing Figs. 7 and 15 when δ > 0, we notice that:
ν C = ν 0 + ν π , (A10) ν T = ν 0 − ν π ,(A11)
in agreement with Ref. [32]. When J < 0 all winding numbers from Fig. 7 change sign. Taking this into account, we see that the winding numbers from Fig. 15 obey Eqs. (A10) and (A11) also when J = δ < 0. Remember that the phase of the atomic quantum walk is independent of the absolute value of J. Thus the Creutz ladder accurately gives the winding number of the atomic quantum walk even when J = δ.
metries. We are interested in whether or not the atomic quantum walk presents time reversal symmetry (TRS), particle hole symmetry (PHS) and chiral symmetry (CS), which determine the system's topological class. While we study the case specific to our system, a more general and complete study of symmetries and their relevance to topological phases can be found in Ref. [43]. For simplicity, we will work in the basis whereÛ is given by Eq.
The system has CS if there is a unitary operatorΓ acting within a single unit cell which anti-commutes with the Hamiltonian:
Γ·Ĥ F (k)·Γ † = −Ĥ F (k) ⇒Γ·Û (k)·Γ † =Û (k) † . (C1)
Thanks to the symmetric form of Eq. (14), if an opera-torΓ simultaneously anti-commutes withĤ θ andĤ S , it automatically satisfies Eq. (C1): U † = e iĤ θ /2 · e iĤ S · e iĤ θ /2 ⇒Γ ·Û † ·Γ † = e iΓ·Ĥ θ ·Γ † /2 · e iΓ·Ĥ S ·Γ † · e iΓ·Ĥ θ ·Γ † /2 =Û .
(C2)
Thus, ifΓ is a valid CS forĤ θ andĤ S , it is also a CS operator forÛ . By inspection of Eqs. (12) and (13), we find thatΓ = σ 3 simultaneously anti-commutes withĤ θ andĤ S , and is therefore the CS operator in this basis. But is it the only operator which satisfies Eq. (C1)? Given a matrixÛ which has the form Eq. (14), it can happen thatΓ satisfies Eq. (C1) without simultaneously anti-commuting withĤ S andĤ θ . Assuming this is true, however, leads to strong constraints on the form ofĤ S andĤ θ . In particular, whenĤ S is a function of quasimomentum (as in the present case), there will in general exist no additional CS operator. This result must remain valid when θ varies spatially. Indeed, breaking translational invariance cannot introduce new symmetries in the system.
We can now turn to the other symmetries of the system, starting with TRS. The system has TRS symmetry if there is an anti-unitary operatorT which commutes with the Hamiltonian, and acts only within a single unit cell. Without loss of generality, we can expressT as the product of a unitary operatorτ andK, the complex conjugation operator:T =τK. The complex conjugation operator is an anti-unitary operator which acts as:
K n|Kψ = ψ(n) * .(C3)
Using |k = n exp(−iknd)|n , we find that:
K k|Kψ = n e −iknd ψ(n) * = ψ(−k) * .
ThusK sends k → −k. Searching for a TRS operator therefore amounts to findingτ such that:
τ ·Ĥ F (−k) T ·τ † =Ĥ F (k) ⇒τ ·Û (−k) T ·τ † =Û (k). (C5) As previously, if an anti-unitaryT simultaneously commutes withĤ S (k) andĤ θ , it automatically satisfies Eq. (C5). In this case however, no such operator exists. This is not sufficient to say that the system does not have TRS. As was the case with CS however, the existence ofT which satisfies Eq. (C5) without simultaneously commuting withĤ S (k) andĤ θ would imply strong constraints on these matrices. These are in general not satisfied when H S (k) andĤ θ are functions of independent variables. We can confirm numerically that the system does not have TRS by plotting the dispersion ofĤ F . Indeed, TRS implies that any eigenstate ofĤ F has a partner eigenstate with equal energy and opposite quasimomentum. Because the system's spectrum is not symmetric about k = 0, we can conclude that the system does not present TRS. Finally, the system has PHS symmetry if there is a anti-unitary operatorP which anti-commutes with the Hamiltonian. We defineˆ , the unitary part ofP, such that:P =ˆ K . This operator satisfies:
·Ĥ F (−k) T ·ˆ † = −Ĥ F (k) ⇒ˆ ·Û(−k) T ·ˆ † =Û(k) † .
(C6) We already know that there is no such operator due to the absence of TRS. Indeed, ifÛ admitted both CS and PHS, their product would yield an anti-unitary matrix which commutes with the Hamiltonian, and this operator would satisfy Eq. (C5). As no such operator exists, we can conclude that PHS is also absent from this system.
The presence of only chiral symmetry implies that the atomic quantum walk belongs to the AIII class of the classification of topological phases. Hamiltonians in this symmetry class can in general have non-zero winding numbers [28].
As a closing remark, we remind the reader that in Sec. III, we approximated the time evolution by Eq. (7). We were able to do this by saying that the spin mixing pulse is so short and intense thatĤ S is negligible during this period. We now point out that, because the CS operator anti-commutes simultaneously withĤ S andĤ θ , even when our approximation breaks down, CS is not broken. Finally, note thatΓ = σ 3 is an operator which acts within a single unit cell. This implies that we can break translational invariance without breaking chiral symmetry.
Figure 2 .
2Dispersion ofĤS Eq. (3) for J = π/3 and δ = 0.42.
Figure 3 .
3Sketch of the tunnelling amplitudes in the atomic quantum walk. Tunnelling amplitudes are different along single and double lines. The tunnelling amplitudes are indicated by curved arrows, the colour of which is unimportant. Unit cells are represented by grey shaded boxes. (a) The time evolutionÛ is controlled byĤS andĤ θ , Eqs. (3) and
Figure 4 .
4Spatial probability distribution after 60 time-steps.
Figure 6 .
6Plots of functions b(k) Eq. (20) (full blue) and d(k) Eq. (21) (dashed yellow) in the complex plane as k goes from −π/d to π/d for J = π/3, δ = 0.42, and θ = 0.15. The number of times that b(k) (d(k)) winds around the origin corresponds to the topological invariant ν0 (νπ). In this case we can read off ν0 = −1 and νπ = 0.
Figure 7 .
7Phase diagram of the atomic quantum walk. The function b(k) (d(k)) vanishes along the full blue (dashed yellow) lines, indicating a boundary along which the topological invariant ν0 (νπ) can change its value. The winding numbers for each region of the plot have been calculated using Eq.(18), and are specified on the figure as {ν0, νπ}; similarly coloured regions have the same winding numbers. In Sec. V, we produce an E = 0 bound state by varying θ spatially along the path in parameter space indicated by a red arrow. The topological boundary θ = 2δ is crossed in this process. In Sec. VI, we cross a trivial band gap closing which occurs for δ = 0. This path in parameter space is indicated by a dashed green arrow.
Figure 8 .
8Topological interface created by varying θ spatially from θmin to θmax. In yellow: θ varies spatially according to Eq. (23). The topological boundary occurs at n = 0, which is the point where θ = 2δ. In blue: topological bound state occurring at this boundary, obtained by exact diagonalisation ofĤF for J = π/3, δ = 0.42, θmin = 0.15, θmax = 1.5 and ξ = 10d. This path in parameter space is indicated onFig. 7by a red arrow.
Figure 9 .
9(a) Atom density in position space versus time. After t/T = 50 time-steps, some density has escaped ballistically to infinity but the probability density function remains sharply peaked around the origin. (b), (c) and (d) Probability density function at times t1 = 10, t2 = 30 and t3 = 50 respectively, with ↑ represented in blue (grey) and ↓ in black. The inset of (d) shows the density in the interval n ∈ [0, 10]. We observe that ↑ (↓) states have non-zero density only on even (odd) sites. This simulation was realised with J = π/3, δ = 0.42, and θ varying spatially from θmin = 0.15 to θmax = 1.5. The initial state was a Gaussian wavepacket centred around site n = 0 with mean quasimomentum k = π/(2d), in ↑ state with equal support on A and B sublattices.
Figure 10 .Figure 11 .
1011Full atomic density, obtained from the simulationFig. 9, for 50 time-steps in the region n ∈ [−15,15]. This interval corresponds to the neighbourhood of the topological boundary, andP is the projector onto this region. At early times we observe oscillations as atoms which are not trapped in the bound state leave the region of the boundary. At late times, we see that the total density converges to a non-zero value. (a) Density distribution of an E = π topological bound state (plotted on a log scale), obtained by diagonalisinĝ HF (notĤ F ). ↑ (↓) density is represented in blue (black). The state is exponentially localised in the neighbourhood of n = 0. ↑ (↓) states are only found on even (odd) sites. Only eigenstates of the chiral symmetry operator have this complex spin distribution. (b) The same state, after evolution through half a time-step. ↑ (↓) states are only found on odd (even) sites. For clarity, a grey vertical line has been drawn at n = 0. States with E = π show an inversion in their spin density distribution at half time-steps.
Figure 12 .
12Eigenvalues ofĤ F for fixed θmin = −π/2, as a function of θmax, with open boundary conditions on an N = 300 site lattice. Pairs of E ≈ 0 and E ≈ π states appear in the band gaps and are exponentially localised around the lattice site where θ = 0 and θ = π, respectively.
Figure 15 .
15Winding numbers {νC , νT } of the atomic quantum walk when J = δ. In the time-frame Eq. (14), the system maps onto the Creutz ladder and νC is accurately predicted by Eq. (A3). νT is the winding number in the other symmetric time frame Eq. (A9). These are related to the winding numbers ν0, νπ through Eqs. (A10) and (A11).
Table I .
IIn the first column, the operators implementing various symmetries ofĤ F are listed in the limit δ = 0. We list the squares of these operators in the second column.symmetry
operator square
Chiral symmetry
σ3
σ0
Time reversal symmetry σ1K
σ0
Particle hole symmetry −iσ2K −σ0
Table II .
IIIn the first column, the operators implementing various symmetries ofĤ F are listed. We list the squares of these operators in the second column.symmetry
operator square
Chiral symmetry
σ3
σ0
Time reversal symmetry σ3K
σ0
Particle hole symmetryK
σ0
all real parameters, and block-diagonalise Eq. (A1) by
Fourier transformation:
ACKNOWLEDGMENTSThe authors would like to thank A. Dauphin and G. Richardson for fruitful advice and conversations, and L. Tarruell for insightful comments. This project was supported by the University of Southampton as host of the Vice-Chancellor Fellowship scheme. We also acknowledge support of the Spanish MINECO (SEVERO OCHOA Grant SEV-2015-0522 and FOQUS FIS2013-46768), the Generalitat de Catalunya (SGR 874), Fundació Privada Cellex, and ERC AdG OSYRIS. P. M. acknowledges support from the Ramón y Cajal programme.Appendix A: mapping to theCreutz ladderIn this section, we will show that, under a simple change of basis, the system maps exactly onto a well known topologically non-trivial 1D system, the Creutz ladder[39][40][41]. We will then use this knowledge to rederive the phase diagram shown inFig. 7.The Creutz ladder describes a spinless particle hopping in a 1D ladder, as sketched inFig. 14.The particularity of this model is that the particle has amplitude to hop in the diagonal directions. We have chosen the basis of spin space such thatFig. 14is reminiscent ofFig. 3(b).The Hamiltonian of the Creutz ladder has the form:As previously,ĉ † n (ĉ n ) creates (annihilates) a particle with two internal states on site n. The σ i matrices denotes the Pauli matrices acting in the space of sites perpendicular to the axis of the ladder (represented in the vertical direction inFig. 14), with i ∈ {1, 2, 3}. We will assume that the hopping amplitudes v, w and g are Appendix B: double atomic quantum walk As we explained in section III, the system has the dynamics of a quantum walk. As we mentioned in this same section however, our protocol is slightly more complex than the standard quantum walk considered in Refs.[17,21,26]. Indeed, we saw that each eigenstate ofĤ S is doubly degenerate, implying that two wavepackets exist in each circled region ofFig. 2. As a result of this, our quantum walk has four distinct walkers, and the coin operationĈ θ can be shown to couple all four. In the following, we will start by mapping the system in the simplified (two state) basis to a standard quantum walk. We then show that the system which we simulated in Sec. III has in reality the dynamics of two independent quantum walks.The HamiltonianĤ S Eq. (13) can readily be diagonalised, yielding the eigenstates:where |+ , k , |− , k are the eigenstates belonging to the top and bottom bands respectively, and "arg" denotes the phase of the complex number. From this definition, we define a pair of walkers in this basis as wavepackets centred narrowly around k = π/(4d), such that the leftand right-walker belong to the top and bottom bands respectively, as represented onFig. 16. As previously, thanks to the symmetry of the spectrum about E = 0, these are translated in real space byĤ S at equal and opposite average velocities.To understand in what sense this system describes a quantum walk, we expressĈ θ in the basis |+ , k , |− , k :This corresponds to a rotation by an angle θ about the axis (0, − cos(φ(k)), sin(φ(k))).To recapitulate, we have defined left-and rightwalkers, which are translated in real space at average equal and opposite velocities. Our walkers are periodically coupled by theĈ θ operation, as indicated inFig. 16. In this sense the protocol fits exactly our definition of a quantum walk. Note however that no choice of θ can in general fully interchange right-and left-walkers; in this respect it is different from the quantum walks considered in Refs.[17,21,26].An additional subtlety comes from the choice of initial state. Let |+, k, ↑ , |+, k, ↓ be the eigenstates ofĤ S Eq. (3) in the superlattice basis, corresponding to the state with quasimomentum k belonging to the top band with spin ↑ or ↓ respectively, and |−, k, ↑ , |−, k, ↓ their bottom band counterparts. In general, these states are related to the eigenstates ofĤ S through:where only the σ i matrices act on the states |± , k .When simulating the atomic quantum walk, we chose as an initial state the Gaussian wavepacket narrowly centred around k = π/(4d) in the superlattice basis, as circled onFig. 2. This state can be expressed in the simpler basis ofĤ S andĈ θ as a superposition of wavepackets centred about k = π/(4d) and k = −3π/(4d), as represented onFig. 16by solid and dotted circles respectively. With this choice of initial state, we are performing a quantum walk with four walkers (two right-and two left-walkers, seeFig. 16). As states which have different quasimomentum are not coupled byĈ θ , it is clear that two independent quantum walks are simultaneously being performed. This, however, should have no effect on the outcome of the simulation, as the slope at k = π/(4d) and k = −3π/(4d) is exactly the same for any choice of parameters.Appendix C: symmetries of the atomic quantum walkIn general in 1D, a system can display non-trivial topological behaviour only if it is constrained by certain sym-
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The wavepackets centred around k = π/(4d), where the slope is locally linear, are circled by a solid line. Due to the symmetry of the dispersion about E = 0, these states move at equal and opposite average velocities. The regions circled in Fig. 2 correspond to a superposition of the wavepackets centred at k = π/(4d) and the ones centred at k = −3π/(4d) (circled by a dotted line). The couplings induced byĈ θ are illustrated by arrows. When both the states in the region of k = π/(4d) and k = −3π. 0.42Figure 16. Dispersion ofĤ S Eq. 4d) are populated, the system performs two simultaneous. independent quantum walksFigure 16. Dispersion ofĤ S Eq. (13) for J = π/3 and δ = 0.42. The wavepackets centred around k = π/(4d), where the slope is locally linear, are circled by a solid line. Due to the symmetry of the dispersion about E = 0, these states move at equal and opposite average velocities. The regions circled in Fig. 2 correspond to a superposition of the wavepackets centred at k = π/(4d) and the ones centred at k = −3π/(4d) (circled by a dotted line). The couplings induced byĈ θ are illustrated by arrows. When both the states in the region of k = π/(4d) and k = −3π/(4d) are populated, the system performs two simultaneous, independent quantum walks.
| [] |
[
"Coexisting Z-type charge and bond order in metallic NaRu 2 O 4",
"Coexisting Z-type charge and bond order in metallic NaRu 2 O 4"
] | [
"Arvind Kumar \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea\n\nDepartment of Physics and Astronomy\nSeoul National University\n08826SeoulKorea\n\nUGC-DAE Consortium for Scientific Research\nIndore-452001India\n",
"Yogi ",
"Alexander Yaresko \nMax-Planck-Institut für Festkörperforschung\n70569StuttgartGermany\n",
"C I Sathish \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea\n\nDepartment of Physics and Astronomy\nSeoul National University\n08826SeoulKorea\n",
"Hasung Sim \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea\n\nDepartment of Physics and Astronomy\nSeoul National University\n08826SeoulKorea\n",
"Daisuke Morikawa \nInstitute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan\n",
"J Nuss \nMax-Planck-Institut für Festkörperforschung\n70569StuttgartGermany\n",
"Kenji Tsuda \nFrontier Research Institute for Interdisciplinary Sciences\nTohoku University\n980-8577SendaiJapan\n",
"Y Noda \nInstitute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan\n\nJ-PARC Center\nJapan Atomic Energy Agency\n2-4 Shirakata319-1195TokaiIbarakiJapan\n",
"Daniel I Khomskii \nInstitute of Physics II\nUniversity of Cologne\n50937CologneGermany\n",
"Je-Geun Park [email protected]&[email protected] \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea\n\nDepartment of Physics and Astronomy\nSeoul National University\n08826SeoulKorea\n\nCenter for Quantum Materials\nSeoul National University\n08826SeoulKorea\n\nInstitute of Applied Physics\nSeoul National University\n08826SeoulKorea\n"
] | [
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea",
"Department of Physics and Astronomy\nSeoul National University\n08826SeoulKorea",
"UGC-DAE Consortium for Scientific Research\nIndore-452001India",
"Max-Planck-Institut für Festkörperforschung\n70569StuttgartGermany",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea",
"Department of Physics and Astronomy\nSeoul National University\n08826SeoulKorea",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea",
"Department of Physics and Astronomy\nSeoul National University\n08826SeoulKorea",
"Institute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan",
"Max-Planck-Institut für Festkörperforschung\n70569StuttgartGermany",
"Frontier Research Institute for Interdisciplinary Sciences\nTohoku University\n980-8577SendaiJapan",
"Institute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan",
"J-PARC Center\nJapan Atomic Energy Agency\n2-4 Shirakata319-1195TokaiIbarakiJapan",
"Institute of Physics II\nUniversity of Cologne\n50937CologneGermany",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulKorea",
"Department of Physics and Astronomy\nSeoul National University\n08826SeoulKorea",
"Center for Quantum Materials\nSeoul National University\n08826SeoulKorea",
"Institute of Applied Physics\nSeoul National University\n08826SeoulKorea"
] | [] | How particular bonds form in quantum materials has been a long-standing puzzle. Two key concepts dealing with charge degrees of freedom are dimerization (forming metal-metal bonds) and charge ordering (CO) [1,2]. Since the 1930s, these two concepts have been frequently invoked to explain numerous exciting quantum materials [3-6], typically insulators. Here we report dimerization and CO within the dimers coexisting in metallic NaRu2O4. By combining high-resolution x-ray diffraction studies and theoretical calculations, we demonstrate that this unique phenomenon occurs through a new type of bonding, which we call Z-type ordering. The low-temperature superstructure has strong dimerization in legs of zigzag ladders, with short dimers in legs connected by short zigzag bonds, forming Z-shape clusters: simultaneously, site-centered charge ordering also appears. Our results demonstrate the yet unknown flexibility of quantum materials with the intricate interplay among orbital, charge, and lattice degrees of freedom. | 10.1038/s43246-022-00224-8 | [
"https://arxiv.org/pdf/2202.06437v1.pdf"
] | 246,048,684 | 2202.06437 | 1c689bb2f949c22883238077d4fef02acfd2bb9d |
Coexisting Z-type charge and bond order in metallic NaRu 2 O 4
Arvind Kumar
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulKorea
Department of Physics and Astronomy
Seoul National University
08826SeoulKorea
UGC-DAE Consortium for Scientific Research
Indore-452001India
Yogi
Alexander Yaresko
Max-Planck-Institut für Festkörperforschung
70569StuttgartGermany
C I Sathish
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulKorea
Department of Physics and Astronomy
Seoul National University
08826SeoulKorea
Hasung Sim
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulKorea
Department of Physics and Astronomy
Seoul National University
08826SeoulKorea
Daisuke Morikawa
Institute of Multidisciplinary Research for Advanced Materials
Tohoku University
980-8577SendaiJapan
J Nuss
Max-Planck-Institut für Festkörperforschung
70569StuttgartGermany
Kenji Tsuda
Frontier Research Institute for Interdisciplinary Sciences
Tohoku University
980-8577SendaiJapan
Y Noda
Institute of Multidisciplinary Research for Advanced Materials
Tohoku University
980-8577SendaiJapan
J-PARC Center
Japan Atomic Energy Agency
2-4 Shirakata319-1195TokaiIbarakiJapan
Daniel I Khomskii
Institute of Physics II
University of Cologne
50937CologneGermany
Je-Geun Park [email protected]&[email protected]
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulKorea
Department of Physics and Astronomy
Seoul National University
08826SeoulKorea
Center for Quantum Materials
Seoul National University
08826SeoulKorea
Institute of Applied Physics
Seoul National University
08826SeoulKorea
Coexisting Z-type charge and bond order in metallic NaRu 2 O 4
1 * Authors with equal contributions # Corresponding authors:
How particular bonds form in quantum materials has been a long-standing puzzle. Two key concepts dealing with charge degrees of freedom are dimerization (forming metal-metal bonds) and charge ordering (CO) [1,2]. Since the 1930s, these two concepts have been frequently invoked to explain numerous exciting quantum materials [3-6], typically insulators. Here we report dimerization and CO within the dimers coexisting in metallic NaRu2O4. By combining high-resolution x-ray diffraction studies and theoretical calculations, we demonstrate that this unique phenomenon occurs through a new type of bonding, which we call Z-type ordering. The low-temperature superstructure has strong dimerization in legs of zigzag ladders, with short dimers in legs connected by short zigzag bonds, forming Z-shape clusters: simultaneously, site-centered charge ordering also appears. Our results demonstrate the yet unknown flexibility of quantum materials with the intricate interplay among orbital, charge, and lattice degrees of freedom.
Chemical bonding lies at the heart of all natural sciences like condensed matter physics, chemistry, and even biology. As such, the understanding of chemical bonding was a great puzzle for a long time. In the long and rich history of condensed matter physics, one cannot emphasize strongly enough the importance of the two basic concepts: dimerization [1] and charge ordering (CO) [2], both born in the 1930s. However, these two phenomena have been rarely found to coexist in one compound, much less in metallic systems.
Many studies in the late 1990s and early 2000s demonstrated, one example after another, how the new patterns emerge out of simple structures: hexamers in ZnCr2O4 [3], octamers in CuIr2S4 [4], spontaneous dimensionality reduction in Tl2Ru2O7 [5], and the formation of trimeron in Fe3O4 [6], to name only a few. Despite the various properties, an ever occurring theme is that these phenomena typically happen near the metal-insulator boundary, with the ordered states usually insulating. This characteristic feature is naturally captured by Mott-Hubbard physics [7], and it is particularly prominent among frustrated magnets [8][9][10][11]. It is relatively less well-known how this rich physics plays out in metallic systems; it remains unchartered territory to date. One can imagine that this question may become more important in the newly emerging field of quantum materials that increasingly include metallic systems [12].
Ru has a special place in this extensive search because of its two characteristic features: the modest spin-orbit coupling, not too small nor too large compared to the Coulomb energy and the bandwidth; another is a relatively stronger importance of orbital physics [7]. This combination of the two has been behind the active researches on Ru oxides; for instance, superconducting Sr2RuO4 and ferromagnetic SrRuO3 are two prominent examples [13,14]. In comparison, there have been few studies on Na-Ru-O systems, including NaRu2O4. Using neutron diffraction, the authors of Ref. [15] determined that its room-temperature structure has the space group of Pnma, the typical orthorhombic structure among oxides: to our best knowledge, it was initially reported in 1975 [16] before being re-examined in 2006. It was reported to have a paramagnetic susceptibility down to the lowest temperature. While studying another Na compound, Na2.7Ru4O9 [17] which was also investigated in the same Ref. [15], we realized that the high-temperature phase of these Na-Ru-O systems could exhibit exciting new unreported properties and hence deserve careful studies.
In this report, we made extensive studies of both structure and physical properties of NaRu2O4 over a wide temperature range up to 600 K above which it becomes chemically unstable. We made the high-resolution XRD study above room temperature to find a surprising first-order phase transition during this study. We found that NaRu2O4 undergoes a metal-metal phase transition below 535 K with a distinct structure change, from a CaFe2O4-type structure with the orthorhombic Pnma symmetry at a higher temperature to a monoclinic structure P1121/a below Tc. By combining experimental and theoretical studies, we found that at this transition we have simultaneously a site-centered charge ordering coexisting with dimerization.
Results
Bulk properties. Electrical resistivity (ρ) measurements were carried out using a homemade system equipped with a furnace (300 to 685 K) and a pulsed-tube cryostat with the base temperature of 3 K. The electrical resistance was measured in a four-point geometry, where contacts were made using silver paint and 25 µm gold wire as shown in the inset of Fig. 1a. The current was applied perpendicular to the single crystal length, which is the crystallographic b-axis. As the single crystals are all in a think needle form, we could not measure the resistivity along the other crystallographic directions. We tested several long rod-shaped as-grown crystals and discovered the same transition (see also Fig. S1). The resistivity data shows that a phase transition occurs in NaRu2O4 at Tc = 535 K with small hysteresis in the electrical resistivity, a sign of a weak first-order phase transition at Tc. We will extensively discuss this transition below; it is the main feature of NaRu2O4.
The resistivity (T) data measured down to 3.3 K indicates metallic behavior with a room temperature resistivity of ~ 40 -cm. To have a quantitative understanding, we analyzed the results theoretically using the Bloch-Grüneisen (BG) model. The following formula gives the expression for the temperature-dependent part of the metallic resistivity:
≈ ℏ Mv ∫| ( ))| (ℏ ⁄ ) dq (exp(ℏ ⁄ ) ) exp( ℏ ⁄ ) (1),
where v(q) is the Fourier transform of the potential associated with one lattice site and vs being the sound velocity. When Eq. (1) was considered for the acoustic phonon contribution, it leads to the BG function as shown below:
ph ( , ) = 4 ac ( ⁄ ) × ∫ ( − 1) (1 − ) dx ⁄ (2),
where x = /kBT, D is Debye temperature, and Aac is a proportionality constant. The total resistivity of a material can be written as, where is the temperature-independent residual resistivity and the second term represents the electron-phonon scattering. The B-G model is found to work well for our study, and the theoretically estimated is quite consistent with the experimental data for NaRu2O4. The characteristic Debye temperature () and residual resistivity (0) is found to be 452.2 K and 35.9 -cm, respectively. We also measured magnetic susceptibility and specific heat of NaRu2O4 in the accessible temperature range of 2 to 300 K. The susceptibility () for NaRu2O4 was measured in the temperature range between 1.9 T 300 K with an 0.5 T of the external field applied perpendicular to the length of the asgrown crystal as in Fig. 1a. No magnetic ordering was observed in the measured temperature range. It is noticeable that the susceptibility is almost temperature-independent over the wide temperature range, showing a sizeable paramagnetic behavior consistent with the reported susceptibility data on the polycrystalline sample [15]. The low-temperature upturn is apparently due to some magnetic impurities, and it can be explained by using a modified Curie-Weiss formula (see the red line in Fig. 1b ( ) = + ph ( , ), and SI Note II). The fit yields the following values: Curie-Weiss temperature CW = -1.8 K, observed moment eff = 0.04 B per Ru atoms and 0 = 1.15×10 -7 (cm 3 / Ru mol) -1 , where the latter is the temperature-independent van Vleck contribution to the susceptibility. The CW-fit gives a significantly less observed moment, indicating negligible localized magnetic moments in NaRu2O4, similar to the reported polycrystalline sample [15]. The heat capacity was measured between 1.9 T 300 K and is displayed in Fig. 1c. The heat capacity did not show any -type anomaly over the entire temperature range, confirming no longrange magnetic order from 300 to 1.9 K. Low-temperature heat capacity data were analyzed using the usual formula of Cp = T + T 3 , where is the electronic specific-heat coefficient and is the phonon contribution at low temperature (see the inset of Fig. 1c). The electronic contribution to the specific heat () for NaRu2O4 was 3.93 mJ/mol K 2 , consistent with the metallic behavior.
Single-crystal XRD data & crystal structures at high and low temperatures. To understand the hightemperature phase transition seen in the resistivity data, we conducted high-resolution single-crystal diffraction experiments to obtain the most informative data. We performed the temperaturedependent SC-XRD experiment from 300 to 575 K using a single crystal diffractometer (XtaLAB P200, Rigaku) (see Fig. 2). In our room-temperature data, we observed q = (0, ½, 0) superlattice peak (satellite reflections). It is important to note that this presence of the satellite reflection cannot possibly be explained by the Pnma space group proposed in Ref. [15]. While searching battling for a possible true structure at room temperature, we decided to employ the CBED (Convergence Beam Electron Diffraction) as it is the most powerful technique in determining the space group and gives crucial information about the lattice's symmetry [18] (see Fig. 3). The weak superlattice reflections in the x-ray diffraction experiments already form strong evidence for broken symmetry (both translation and rotational) and lattice distortion in a crystal. In addition, the superlattice peak at q = (0, ½, 0) disappears upon heating just above Tc. We collected CBED diffraction patterns along the various [100], [130], [001], and [201] axes as shown in Fig. 3(a-d) at room temperature. The CBED experiment revealed P1121/a symmetry (no. 14 with setting unique axis c [19]) at room temperature, a subgroup of the reported Pnma symmetry. Therefore, a modified structure model under group-subgroup relation is used to solve the room temperature SC-XRD data. We used the Shelx software [20] to address the twin problem in our refinement. In the twin refinement for NaRu2O4 SC-XRD data, we used a monoclinic merohedral twin model under twin law as (-100) (010) (00-1) to get better refinement parameters and statistics as well (see SI Note III). All the reflection patterns collected at 300 K could be indexed as monoclinic P1121/a symmetry with a primitive lattice of a = 9.273 (6) Å, b = 5.643 (3) Å, and c = 11.17 (7) Å.
Table-1 summarizes the SC-XRD refinement results for the LT-phase (300 K). All the crystallographic sites were considered to be fully occupied and were kept fixed during the refinement. We carried out the structure analysis by including the twin structure of monoclinic cells. The structural parameters were finally determined from the least-squares refinement of the single-crystal XRD data. For the single-crystal refinement, we used a total of 5021 reflections, and all the reflections were well indexed by a monoclinic a0 × 2b0 × c0 cell. We used P1121/a (no. 14) with a table setting choice: c1. No constraints were placed on the atomic positions. The comparison of calculated and observed intensity (F 2 ) after the structural refinement at room temperature is shown in Fig. 2a.
As regards the HT phase, our single-crystal data could not uniquely solve the structure above Tc (HT-phase) due to the sizeable diffuse scattering. Most probably, at such high temperatures, lighter Na ions located in 1D tunnels of NaRu2O4 start to move within the tunnels. In this situation, we decided to begin with positional parameters of distorted phase (Table-2, LT-phase with a02b0c0 P1121/a from Shelxl refinement with twin), and generate undistorted positional parameters (undistorted P1121/a a0b0c0). The undistorted positional parameters of P1121/a symmetry are used to get positional parameters for orthorhombic HT-phase Pnma symmetry under cell-symmetry and group-subgroup relation [19]. Expected coordinates at HT Pnma phase based on LT P1121/a structure (a0b0c0) are shown in Table-2. Excellent relation is seen between expected positions and reported one [15] (by missing superlattice reflections: a0b0c0), as shown in Table-3.
The crystal structure of NaRu2O4 is close to that of the prototype compound CaFe2O4 with similar coordination environments. In our revised crystal structure for NaRu2O4, the most characteristic feature we found is the edge shared octahedral double RuO6 chain, which runs along the crystallographic b-axis. These double chains form a two-leg zigzag ladders along the crystallographic b-axis. The RuO6 octahedra are edge-shared within each chain, tied to the neighboring chain through the corner oxygen. This leads to an interesting lattice geometry with pseudo triangular tunnels, in which a single Na atom can be accommodated, as shown in Fig. 4(a), which is also found in various CaFe2O4-type lattices [21][22][23][24][25][26]. Our detailed single-crystal x-ray diffraction analysis shows a transition from the HT (high-temperature) Pnma structure to the LT (low-temperature) P1121/a structure at Tc = 535 K, as shown in Fig. 4(b, c): which is consistent with the results of resistivity measurements shown in Fig.1a.
According to our diffraction experiments done between 300 T 575 K, the q = (0, ½, 0) superlattice peak (satellite reflection) persist right up to 550 K as shown in Fig. 5a. The line cut from the temperature-dependent reciprocal image analysis for peak q = (0, ½, 0) shows clearly the suppression of intensity as a function of temperature as in the main article Fig. 5(b,c). The q = (0, ½, 0) superlattice peak is substantially suppressed just above the first-order transition (Tc = 535 K).
Analysis and discussion
The structure of NaRu2O4 can be visualized as the double chain of edge-sharing RuO6 octahedra running in the b-direction and forming 2-leg zigzag ladders (see Fig. 6a). The corner-sharing oxygen atoms connect these ladders, also having a zigzag pattern. In effect, there appear in this system, not one, but three different types of zigzag ladders: those in edge-sharing double chains (ladders-2 marked by red in Fig. 6b. For shortness, we will call them "red" ladders), and two types of corner-sharing zigzag ladders, marked in Fig. 6(b,c) by blue and green. When describing this class of materials, one often pays primary attention to the phenomena occurring in the "red" ladders (see [27]). But the cornersharing ladders are sometimes found to play a more critical role in systems like hollandite K2Cr8O16 [28]. As we will see below, these edge-sharing ladders also play a crucial role in NaRu2O4.
Here it is worth while to think about possibility of certain effect due to off-stoichiometry. In particular, one can ask whether the observed metallic behaviour is related to a sample quality issue.
For instance, it is known that such stoichiometry plays an important role in determining the lowtemperature properties of RuP [29], whose high-quality powder samples exhibit a metal-insulator transition. On the other hand, single crystal of RuP, presumably off-stoichiometric, has a metallic ground state. For NaRu2O4, we can rule out this issue as our powder sample also shows a metallic behaviour (see also Fig. S2). Another point is that our heat capacity measurements taken on two different samples consistently show a finite gamma value at low temperature, which is consistent with the data reported in [15].
An exciting picture now emerges from the detailed structural study. The HT phase has two independent Ru sites (Ru1 and Ru2), equally spaced (2.82 Å) with the lattice constant b0 along two Ru chains. The most prominent feature of the structural changes is the appearance of strong dimerization in Ru chains (legs of zigzag ladders), leading to cell-doubling in the b-direction (see Fig. 7a). Ru-Ru distance in short Ru dimers is 2.60 Å-even shorter than the Ru-Ru distance in Ru metal (2.65 Å). It is a clear sign of direct metal-metal bonding in such dimers -which quite unexpectedly coexist with metallic conductivity. Simultaneously with the dimerization in 1D Ru chains -legs of zigzag ladders, Ru sites themselves become inequivalent, Ru(A) and Ru(B), a clear sign of a site-centered CO in NaRu2O4. The coexistence of bond-and site-centered orderings is extremely rare in metallic systems, making NaRu2O4 especially interesting.
Another essential feature of this superstructure is the difference of distances in the inter-chain bond length (interdimer), both in edge-sharing "red" and in corner-sharing "green" and "blue" ladders ( Fig. 6b). The Ru1(A)-Ru1(A) and Ru2(A)-Ru2(A) distance in the edge-sharing ladder is 3.10 and 3.09 Å, while the Ru1(A)-Ru1(B) and Ru2(A)-Ru2(B) distance are 3.12 and 3.11 Å, respectively, with the remaining distance between dimers (Ru1(B)-Ru1(B) and Ru2(B)-Ru2(B)) being 3.16 Å (see Figs. 7(a,b)). From the viewpoint of the bond order, short bonds have a shape like a "letter-Z" (see Fig. 8). The same is true for two other types of ladders, edge-sharing "blue" and "green" ladders. We thus see that the dimers also form Z-type orbital clusters in corner-sharing ladders. The relative difference of interdimer bonds in these is even more prominent than in the "red" edge-sharing ladders: it is ~ 1.7% compared to ~ 0.7% in the "red" ladders. This already hints that what happens in corner-sharing ladders is probably more critical than in edge-sharing ones, as in hollandite K2Cr8O16 [28]. According to our abinitio calculations based on the charge valence of our choice [Ru1(A) 3+ , Ru1(B) 4+ , Ru2(B) 4+ , Ru2(A) 3+ ), the nn hoppings in corner-sharing blue and green zigzags are: for blue corner sharing bonds t=0.40 eV, for green corner sharing ones t=0.29 eV, and for red edge sharing bonds t=0.35 eV. For other CO the values are somewhat different but the blue corner sharing bonds are always the strongest. It shows the dominance of the blue bond among all relevant bonds.
For the determination of the oxidation states of cations (Na and Ru), we carried out calculations by using the bond valence sum (BVS) method with an inbuilt program in Fullprof software [30]. The principle of BVS method in determining the valence V from different cation sites [31,32] can be given by an expression V = i [exp(d0-di)/0.37]. Here, di is the experimental bond lengths to the surrounding ions, and d0 is a tabulated empirical value characteristic for the cation-anion pair [31,32]. The BVS result for NaRu2O4 was determined for the LT phase. The charge at different Ru sites is illustrated by different colors (brown-Ru 4+ and green-Ru 3+ ) in the main article Figs. 6 and 7. The BVS's for the eight oxygen sites were between 1.86 and 2.20. The BVS's for the four Ru cations sites for different charge configurations (we call them four different models) are shown below in Table-4. The respective reliability parameter as Global Instability Index (GII) is also listed for each BVS model for NaRu2O4.
To be more technical, an important question is how different Ru valence states are distributed in the low-temperature phase. Specifically, one has to know which ions, in which charge state, are connected by these shorter zigzag rungs in orbital Z clusters. For that, we used the results from the bond valence sum (BVS) analysis at each Ru, an empirical measure of its charge or its valence state [33,34]. In the HT phase, all Ru are equivalent, and BVS gives the valence Ru 3.5+ . In contrast, Ru ions in a short dimer are indeed inequivalent in the LT phase, with the valence estimated from BVS being 3.536 and 3.106; that is, one can think of Ru 4+ and Ru 3+ in each dimer. However, from our structural data alone, although collected with high-resolution single-crystal x-ray diffraction and the total number of 5021 Bragg peaks, we cannot uniquely discriminate the four possible CO patterns, mainly because an x-ray experiment fundamentally suffers from the weak scattering signals of oxygen. The reliability parameters (Global Instability Index (GII)) for four different CO patterns are comparable to one another, about 10-11% (see Table 4). Still, a slightly better index is obtained for charge distribution with Ru 4+ ions at the ends of short diagonal bonds in blue and green ladders (reliability factor 10.43 % for model 3 in Table 4). In contrast, these are Ru 3+ in edge-sharing red ladders. Note that different ladders are connected in the structure of NaRu2O4 so that the charges at short zigzags in neighboring ladders are opposite (see Fig. 8).
Theoretical Studies
As we commented above, there is unavoidable ambiguity about our determination of the charge valence using the BVS method although we used the total number of 5021 Bragg peaks from the highresolution single-crystal x-ray diffraction experiment. Therefore we employed ab-initio LSDA and LSDA+U calculations to clarify the charge pattern favored by the LT crystal structure. We used different Coulomb repulsion U values in the range 2.7 -4.2 eV and Hund's coupling JH = 0.7 eV as estimated from LSDA. All calculations were performed for the experimental HT and LT crystal structures. No attempts to optimize lattice geometry have been made. The effect of spin-orbit coupling (SOC) on Ru t2g bands was relatively weak. In the following, for simplicity, we discuss the results of scalar-relativistic calculations. Our theoretical studies show that different self-consistent spin and charge-ordered solutions can be stabilized for sufficiently large U (3.7 eV). These solutions have the following common features: Ru dxy states (in local coordinates, with the axes directed from Ru to O) are strongly split into occupied bonding and unoccupied antibonding molecular orbital states (see Fig. 7c-f). Second, each dimer's Ru ions become formally Ru 3+ , with its dxz, dyz orbitals being doubly occupied. Another Ru ion of the dimer, Ru 4+ , has one half-occupied orbital, approximated by a linear combination of dxz and dyz lying in the plane perpendicular to the b-axis. The Ru 4+ ion acquires a spin moment of about 0.8 B.
Orbital-resolved density of Ru d states illustrates the charge disproportionation. With decreasing U, the hybridization becomes more robust between the unoccupied dxz, dyz orbital of Ru 4+ and occupied t2g orbitals of Ru 3+ , with a gap separating the unoccupied dxz, dyz band from the top of the valence band closes, and for U ~ 2.7 eV the charge disproportionation between Ru 3+ and Ru 4+ ions disappears (see Fig. 7c-f). It should be noted that this is only a rough estimate for the critical U value which may change if optimized crystal structure were used in the calculations and is in general implementation specific.
Out of the four CO patterns allowed for the LT structure, three are nearly degenerate, with the energy difference being less than 3 meV/f.u. This finding agrees with the BVS analysis results, which give comparable reliability factors for different CO. Still, the lowest total energy is found for one CO pattern: for which the lowest value of GII was obtained experimentally from BVS. The final charge pattern is that the short zigzag bonds in Z-cluster in corner-sharing "blue" ladders connect Ru 4+ -Ru 4+ ions (see Fig. 8).
Finally, we would like to note that the structure of NaRu2O4 is composed of frustrated 2-leg zigzag ladders with a spin singlet ground state, which makes this material quite interesting (see SI Note IV). As shown in Fig. 9(a,b), the orthorhombic Pnma crystal structure of NaRu2O4 has two ruthenium atoms Ru1 and Ru2. The edge-shared bond between the two RuO6 octahedra for each dimer 1 (2.60 Å), dimer 2 (2.61 Å) and Z diagonal-bond (3.10 Å) give rise to two Ru-O-Ru bonds which are deviated from 90˚ for each and also all of the edge-shared octahedral shows significant trigonal-distortion. The schematic representation shows the double chain-forming two-leg zigzag ladders (solid thick lightgreen, reddish-brown, and sky-blue lines) running along the crystallographic b-axis in crystal. The 2leg zigzag ladders are connected in two different forms: the corner (blue or green ladders) or the edge (red ladders) shared RuO6 octahedra. The t2g energy-level sketch with respective electron fillings is shown for a charge-ordered state, so-called Z-order (left panel) and Ru-dimer (right panel) in Fig. 9d.
Conclusions
Thus the following final picture emerges from both the experimental and theoretical studies. NaRu2O4 has very strong dimerization below Tc = 535 K (in other words, strong metal-metal bonding with the use of xy-orbitals) in one-dimensional chains and legs of zigzag ladders (this is a bond-centered CDW). With the average valence Ru 3.5+ , one active electron (or one hole) remains per a dimer. And simultaneously with this dimerization (actually the orbitally-driven Peierls dimerization [35][36][37][38]), these extra electrons order in each dimer-there appears also a site-centered charge ordering or sitecentered CDW. The remaining electrons form one stronger diagonal bond between dimers in neighboring legs, creating Z-clusters (Fig. 8). The detailed form of orbitals in forming this short zigzag bond in the "blue" corner-sharing ladders is shown in Fig. 9c. But this extra bonding is not strong enough to make the system insulating, and it remains metallic below Tc due to the multi-orbital effect. These Ru-Ru dimers probably survive above Tc, forming dimer liquid, e.g., in Li2RuO3 [39]; this dimer liquid phase in Li2RuO3 was found stable upon doping too [40]. This could be checked by probes like EXAFS or PDF (Pair Distribution Function) analysis.
To put our results in a broader perspective, we would like to comment on why the dimerization and the site-centered CO coexistence are favorable for metallic NaRu2O4, while it has been rarely seen in other transition metal compounds: another example is IrTe2 [41]. Most importantly, the Ru 4d bands of NaRu2O4 exhibit the delicate balance between the correlations, the Coulomb U, and the bandwidth W. Two other factors play a crucial role: orbital freedom, leading to orbitally-driven dimerization, and the mixed-valence of Ru, Ru 3.5+ , promoting site-centered charge ordering. These features seem to be the key to realizing CO and dimer coexistence in a metallic NaRu2O4. Going further, we anticipate that our observations will prompt renewed efforts towards the hitherto scarcely investigated Mott-Hubbard physics in a metallic regime.
Acknowledgments
We would like to thank Juan Rodríguez-Carvajal, Sang-Wook Cheong, and Masahiko Isobe for fruitful discussions. Work at the Center for Quantum Materials was supported by the Leading Researcher
Competing financial interests
The authors declare no competing financial interests.
Methods
Sample preparations: Polycrystalline NaRu2O4 samples were synthesized by solid-state reaction of preheated RuO2 (99.999%, Aldrich) and Na2CO3 (99.999%, Aldrich) under an Ar-gas environment at 950 ˚C for 90 h with several intermediate grindings and pelletization. Subsequently, high-quality single crystals of NaRu2O4 were grown from this polycrystalline powder via a modified self-flux vapor transport reaction under flowing Ar-gas (ultra-pure 99.999%). Long needle-shaped high-quality single crystals (1 × 0.1 × 0.1 mm 3 ) were obtained from the final products. The synthesized polycrystalline sample's phase purity was checked using a Bruker D8 Discover diffractometer with a Cu-Kα source with no impurity peaks observed. Elemental analysis was subsequently done confirming the samples' stoichiometry: we used a COXI EM-30 scanning electron microscope equipped with a Bruker QUANTAX 70 energy dispersive x-ray system.
Bulk properties: Electrical resistivity (ρ) measurements were carried out using a homemade system equipped with a furnace (300 to 685 K) and a pulsed-tube cryostat (down to 3 K, Oxford). The electrical resistance was measured in the four-point geometry on a static sample holder, where the contacts to the sample were made using silver paint and 25 µm gold wire. The current was applied perpendicular to the single crystal length, the crystallographic b-axis. Magnetic susceptibility χ(T) measurements were taken using an MPMS-SQUID magnetometer (Quantum Design). Heat capacity Cp(T) measurements were made using the commercial Physical Property Measurement System (PPMS, Quantum Design).
Structure analysis: The temperature-dependent single-crystal x-ray diffraction (SC-XRD) was performed from 300 to 575 K using a single crystal diffractometer (XtaLAB P200, Rigaku). The roomtemperature SC-XRD data exhibit superstructures q = (0, ½, 0). The crystal structure was refined by using Fullprof suite software and the ShelX program with a modified monoclinic merohedral twin model under twin law as (-100) (010) (00-1) (see SI Note III). We also carried out temperaturedependent SC-XRD measurements on NaRu2O4 crystal to confirm the structural phase transition. Besides, we made a convergent beam electron diffraction (CBED) experiment, which is the most accurate in determining the correct symmetry of materials, especially regarding the loss of the inversion center. This CBED measurement allows us to determine the exact space-group of NaRu2O4. The CBED result for NaRu2O4 gives P1121/a symmetry and works well for the observed commensurate superstructures with q = (0, ½, 0). LDA band calculations: Band structure calculations were performed for the experimentally obtained high-and low-temperature crystal structures using the LMTO method as implemented in the PY LMTO computer code [42] and the Perdew-Wang [43] parameterization of the exchange-correlation potential in the local spin density approximation (LSDA). To account for Coulomb repulsion within Ru d shell, we used a rotationally invariant formulation of the LSDA+U method [44] with the doublecounting term in the fully localized limit. An LDA calculation for the HT crystal structure gave a nonmagnetic metallic solution with partially occupied Ru1 and Ru2 t2g orbitals. The band structure's characteristic feature is Ru t2g bands with stable quasi-one-dimensional dispersion along the b direction. These bands originate from Ru dxy orbitals and are evidence of strong direct Ru dxy-dxy hopping along the b chains. Here and in the following, Ru orbital is defined in local frames with the axes directed approximately to three nearest O ions in such a way that the local [110] direction always points along the crystallographic b-axis. The LDA band structure calculated for the LT structure is qualitatively different. Although it remains nonmagnetic and metallic, the shortening of the Ru-Ru distance within dimers leads to the splitting of Ru dxy drives bands into completely filled bonding states 2 eV below EF and unoccupied antibonding ones 1 eV above EF. The bonding dxy bands accommodate one electron per Ru while the remaining 3.5 d electrons fill bands are formed by Ru dxz, dyz states. The single-crystal x-ray diffraction measurement of NaRu2O4 at 300 K is used for the refinement. The experimental structural result at room temperature has been determined by twin analysis (with twin (0.43748)) using the ShelX program [20]. The P1121/a symmetry is used for the refinement as obtained from CBED Experiments. The reliable parameters of refinement are found to be: wR2 = 0.0821, R1 = 0.0344, GoF = S = 1.162: a = 99.256(1) Å, b = 5.634(1) Å, and c = 11.154(1) Å; = 90.10 (1)˚; volume = 581.7(1) Å 3 . (Fig. 1 & 3a). The GII is the reliability parameter as Global Instability Index. The average valence on Ru sites is 3.5+. poly-crystalline Figure S2: Electrical transport measurement of NaRu2O4 on poly-crystals bar sample. The temperaturedependent resistivity (r) curve is given as a function of temperature for poly-crystal rectunglar bar sample indicateing metallic nature in whole tempreture range.
Figure Captions
Atoms
Model
4
II. Modified Curie-Weiss fit
We fitted the measured ( ) by the following expression:
( ) = + ( + ) , where is the temperature-independent contribution.
The temperature dependence of the inverse susceptibility ( -1 ) is strongly nonlinear, i.e., it does not follow the Curie-Weiss behavior as shown in the inset of Fig. 1b. Regarding the Curie behavior at low temperature, we analyzed the data using a modified Curie-Weiss law in between 50 to 100 K. The fit yields the following values: Curie-Weiss temperature CW = -1.8 K, observed moment meff = 0.04 mB per Ru atoms and 0 = 1.15×10 -7 (cm 3 / Ru mol) -1 , where the latter is the temperature-independent van Vleck contribution to the susceptibility. The CW-fit gives a significantly less observed moment, indicating negligible localized magnetic moments in NaRu2O4, similar to the reported polycrystalline sample [1].
III. Merohedral twin refinement
Our SC-XRD refinement using Fullprof software [2] reveals a bit large value of reliable parameters. Therefore, we turn to adapt the twin-law in our SC-XRD refinement using the Shelx software [3]. The twin-law defect is a symmetry operator of the crystal system. It is never the point group or Laue group of the crystal system. In the case of such existing twinning in the crystal system, one may get perfect overlap or close to the reflections from both domains, as shown by a schematics representation of reflections from twin domains in Fig. S3.
Moreover, the monoclinic lattice is crystallographically unique as it sometimes has angle very close to 90°. If twinning occurs, the unit cells in one domain may be rotated by 180° about the crystallographic aor c-axes. Sometimes twinning does exist in a single-crystal due to energetically competitive inter-molecular interactions across twin domains. Due to this reason, twinning commonly occurs if a high-symmetry phase of material undergoes a phase transition to a lower-symmetry phase upon lowering or increasing of temperature [4]: a broken-symmetry element that is equivalent in the high-symmetry phase can show the twin-law in the low-symmetry phase. This is what we observed from our structural analysis. By applying the twin law [(-100), (010), and (00-1)], we improved the refinement results as well as SC-XRD data fitting. This means that the crystal measured was a merohedral twin, showing inversion symmetry in an experimentally determined space group (P1121/a) in the lower Laue class. The observed twin ratio was found to be 0.43748 for NaRu2O4 crystal.
Program of the National Research Foundation of Korea (Grant No. 2020R1A3B2079375) with partial funding by the Institute for Basic Science, Republic of Korea. The work of D. Kh. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project number 277146847 -CRC 1238). A.K.Y. acknowledges financial support from the Institute for Basic Science of the Republic of Korea and was partially supported by the Research Fellowships of the Max-Planck Institute Foundation, Germany.Authors ContributionsJGP initiated and supervised the project. AKY, HS, and CIS prepared the samples and carried out the bulk measurements. AKY did the structural analysis under the supervision of YN. DM and KT did the CBED measurements. JN did the merohedral-twin refinement by using Shelx software. AY made the theoretical studies of LDA calculations. DK and JGP interpreted the data with the help of AKY, AY, and YN. AKY, AY, YN, DK, and JGP wrote the manuscript with inputs from all authors.
Figure 1 :
1Bulk properties. (a) The electrical resistivity data as a function of temperature is taken over single-crystal with the solid red line for the BG model. Inset shows the single crystal used for the fourprobe resistivity method. (b) Magnetic susceptibility () is measured at an applied magnetic field B = 0.5 T. The inset shows inverse susceptibility with the modified Curie-Weiss (CW) (the solid red line). (c) Temperature dependence of the specific heat Cp measured at zero fields (B = 0 T). Solid black circles are the raw data with Cp/T shown in the inset of the figure and the solid red line for the corresponding Cp/T = + T 2 fitting.
Figure 2 :
2Room-temperature SC-XRD refinement for LT-phase: Single-crystal XRD reciprocal-lattice images and lattice parameters across the structural transition. (a) The SC-XRD refinement includes a modified structural model with a new space group (P1121/a) determined from CBED experiments. The refinement is performed by including a monoclinic merohedral twin model under twin law as (-100) (010) (00-1). The solid red and blue circles are fundamental and superlattice reflections, respectively. The inset of the figure shows the optical image of the measured crystal. (b) Single-crystal x-ray diffraction data were measured at various temperatures below and above the phase transition (Tc = 535 K) for NaRu2O4. (c) The cell parameters with error bars of NaRu2O4 at different temperatures are extracted from single-crystal XRD data. The vertical line corresponds to Tc.
Figure 3 :
3The Converge Beam Electron Diffraction (CBED) patterns at room temperature in NaRu2O4 crystal. (a) The room temperature CBED patterns for NaRu2O4 were taken along the reciprocal axis directions [100] (top panel) and (b) [130] (bottom panel). The black vertical solid lines are the position of the mirror plane (m), and solid red arrows are respective reciprocal axis directions. An arrowhead in (a, b) indicates the dynamical extinction of the glide plane and screw axis. (c) From the selected area electron diffraction pattern and CBED pattern along [001], we found whole pattern symmetry as 2z, which lacks both mx and my symmetries and a c//2-fold symmetry. (d) Similarly, the diffraction pattern along [201] found the whole pattern symmetry without my.
Figure 4 :
4The HT and LT crystal structures. (a) The HT-phase in NaRu2O4 has an orthorhombic structure with a Pnma space group, similar to the CaFe2O4-type structure[1]. The edge shared octahedral and Ru atoms are represented in light brown and O atoms in red color. (b) The HT-phase crystal structure (Pnma) of NaRu2O4 in bc-plane. (c) The LT-phase crystal structure of NaRu2O4 has monoclinic symmetry P1121/a.
Figure 5 :
5Single-crystal x-ray diffraction (SC-XRD). (a) Reciprocal lattice maps of the SC-XRD data for NaRu2O4 obtained at 470 K and 575 K with the first order transition at Tc = 535 K. (b) The line cut for various temperatures below and above the first-order phase transition (> Tc) as extracted from the reciprocal space analysis. (c) Temperature dependence of the (4 3 0) super-lattice peak.
Figure 6 :
6Crystal structure of the tunnel compound NaRu2O4. (a) Top view of the low-temperature structure. (b) Illustration of different types of zigzag ladders: "red" ladders with edge-sharing RuO6 octahedra; "green" and "blue" ladders with corner-sharing octahedra. (c) The detailed structure of corner-sharing and edge-sharing ladders.
Figure 7 :
7Superstructure in the low-temperature phase and the results of the LSDA+U band structure calculations. (a) The unit-cell of the low-temperature with two dimerized Ru bonds and the valence of Ru as determined from bond valence sum (BVS) calculations. (b) Structure of the blue corner-sharing ladder with different valence states of Ru and different bond lengths marked. (c-f) Orbital resolved DOS for four nonequivalent Ru atoms calculated within LSDA+U with U = 3.7 eV and J = 0.7 eV, assuming the charge order shown in (a). Ru orbitals are defined in local frames with the axes directed to the nearest O ions.
Figure 8 :
8Z-order in the two-leg corner-and edge-sharing RuO6 zigzag ladders. In the low-temperature phase, the sketch of self-organized Z-clusters formed within the edge-sharing red and cornersharing blue ladders of NaRu2O4 lattice.
Figure 9 :
9CaFe2O4 type tunnel structure and edge shared RuO6 2 leg ladders in NaRu2O4 lattice. (a) The orthorhombic Pnma crystal structure. (b) The edge-shared bond between the two RuO6 octahedra for each dimer 1 (2.60 Å), dimer 2 (2.61 Å), and Z diagonal-bond (3.10 Å). (c) Schematic representation of the double chain with two-leg zigzag ladders (solid thick light-green, reddish-brown, and sky-blue lines). (d) The 2-leg zigzag ladders in two different forms, either corner (blue or green ladders) or edge (red ladders), shared RuO6 octahedra.
Figure 2
2Figure 2
Figure 3
3Figure 3
Figure 4
4Figure 4
Figure 5
5Figure 5
Figure 6
6Figure 6
Figure 7
7Figure 7
Figure 8
8Figure 8
FigureFigure S1 :
S1Electrical transport measurements of NaRu2O4 on different single-crystals. The temperaturedependent resistivity (r) curves are given as a function of temperature for two different single-crystals with a first-order phase transition at TC = 535 K, marked by a downward arrow.
Figure S3 :
S3The twin-law applied in NaRu2O4 crystal. (a) The schematic representation of Bragg reflections from twin domains is shown by red and blue color arrows. Some possible twin reflections are demonstrated for H (H') and K (K') planes. (b) The phase transition (Tc = 535 K) from HT-phase to LT-phase and cell doubling relation under group-subgroup is shown along the b-axis.
Figure S4 : 7 Figure S5 :
S47S5Electronic structure, orbitals, and dimerization(Peierls-type) in 1D tunnel-lattice of NaRu2O4. (a) NaRu2O4 band-structure is calculated by using LSDA+U, and the calculations were done for various sets of U = 2.7, 3.2, and 3.7 and J = 0.7 eV values (Ueff = U -J = 2.0, 2.5, and 3.0 eV). (b) The schematic representation of Ru orbitals in red and blue ladders. (c) The schematic of lattice distortion in NaRu2O4 below 535 K is shown in a metallic one-dimensional (1D) lattice. Density of states (DOS) at high-temperature (HT) and low-temperature (LT) phase and consequence of Peierls transition in 1D tunnel-lattice of NaRu2O4. (a-f) DOS for NaRu2O4 at HT-and LTphase obtained from the LDA band-structure calculation. (g) The schematics of energy levels at HTphase (top panel) and LT-phase (bottom panel) for NaRu2O4. The formation of energy-gap (Eg) at the Fermi level (EF) due to unit-cell doubling along crystallographic b-axis breaks HT symmetry (Pnma) which results in Ruthenium (R) CO at LT-phase (P1121/a).
Table 1 :
1
Table 2 :
2Expected coordinates at HT Pnma phase based on LT P1121/a structure ( a0b0c0).Atoms
Table 3 :
3Reported LT Pnma structure (a0b0c0). Atoms are at 4c site[19] Atoms
x
1/4
z
-x+1/2
3/4
z+1/2
-x
3/4
-z
x+1/2
1/4
-z+1/2
Ru1
0.0603
0.25
0.1152
0.4397
0.75
0.6152
-0.0603
0.75
-0.1152
0.5603
0.25
0.3848
Ru2
0.0848
0.25
0.6036
0.4152
0.75
0.1036
-0.0848
0.75
-0.6036
0.5848
0.25
-0.1036
Na1
0.2399
0.25
0.3397
0.2601
0.75
-0.1603
-0.2399
0.75
-0.3397
0.7399
0.25
0.1603
O1
0.2946
0.25
0.6594
0.2054
0.75
0.1594
-0.2946
0.75
-0.6594
0.7946
0.25
-0.1594
O2
0.3847
0.25
0.9751
0.1153
0.75
0.4751
-0.3847
0.75
-0.9751
0.8847
0.25
-0.4751
O3
0.4730
0.25
0.2181
0.0270
0.75
-0.2819
-0.4730
0.75
-0.2181
0.9730
0.25
0.2819
O4
0.0870
0.25
0.9347
0.4130
0.75
0.4347
-0.0870
0.75
-0.9347
0.5870
0.25
-0.4347
Table 4 :
4The four different models corresponding to four options for two dimers: counting from the left, (3+, 4+) in the upper dimer, (3+, 4+) in the lower dimer. The summary of Ru valences from the Bond-Valence Sum (BVS) calculations for different Ru sites in NaRu2O4 at 300 K shows different models below. The model-3 looks consistent as per the observed dimerization on the two-leg zigzag ladder and our detailed LSDA+U calculations. Accordingly, we have used charges on the respective Ru sites in the NaRu2O4 crystal structure
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. Max-Planck, Sendai; JapanInstitut für Festkörperforschung, 70569 Stuttgart, Germany 5 Institute of Multidisciplinary Research for Advanced Materials, Tohoku UniversityMax-Planck-Institut für Festkörperforschung, 70569 Stuttgart, Germany 5 Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan
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| [] |
[
"Correct traction boundary conditions in the indeterminate couple stress model",
"Correct traction boundary conditions in the indeterminate couple stress model"
] | [
"Patrizio Neff ",
"Ionel-Dumitrel Ghiba ",
"Angela Madeo ",
"Ingo Münch "
] | [] | [] | In this paper we consider the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model. The main aim is to show that the traction boundary conditions were not yet completely deduced. As it turns out, and to our own surprise, restricting the boundary condition framework from the strain gradient models to the couple stress model does not reduce to Mindlin's set of accepted boundary conditions. We present therefore, for the first time the complete, consistent set of traction boundary conditions. | null | [
"https://arxiv.org/pdf/1504.00448v1.pdf"
] | 117,106,097 | 1504.00448 | eb826e9d79ba0ac0cb908b372a9831da5f6596a7 |
Correct traction boundary conditions in the indeterminate couple stress model
2 Apr 2015 April 3, 2015
Patrizio Neff
Ionel-Dumitrel Ghiba
Angela Madeo
Ingo Münch
Correct traction boundary conditions in the indeterminate couple stress model
2 Apr 2015 April 3, 2015generalized continuastrain gradient elasticitymodified couple stress modelconsistent traction boundary conditions
In this paper we consider the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model. The main aim is to show that the traction boundary conditions were not yet completely deduced. As it turns out, and to our own surprise, restricting the boundary condition framework from the strain gradient models to the couple stress model does not reduce to Mindlin's set of accepted boundary conditions. We present therefore, for the first time the complete, consistent set of traction boundary conditions.
Introduction
Higher gradient elasticity models are nowadays increasingly used to describe mechanical structures at the microand nano-scale or to regularize certain ill-posed problems by means of higher gradient contributions [1,2]. One of the very first among such models is the so called indeterminate couple stress model [3,4,5,6] in which the higher gradient contributions only enter through gradients on the continuum rotation, i.e. the total elastic energy can be written as W (∇u, ∇(∇u)) = W e (sym∇u) + W curv (∇(curlu)).
The question of boundary conditions in higher gradient elasticity models has been a subject of continuous attention. The crux of the matter in higher gradient models is the impossibility to vary the test function and its gradient independently. A suitable split into tangential and normal parts must always be considered. This is well known in general higher gradient models, see e.g. [7,8]. The indeterminate couple stress model has been investigated in this respect as well. A first answer has been given by Mindlin and Tiersten [4] as well as Koiter [6] who established (correctly) that only 5 geometric and 5 traction boundary conditions can be prescribed due to the dependence of the curvature energy only on gradients of rotations. We agree that there are 5 traction boundary conditions in the indeterminate couple stress model which may be independently prescribed. However, we show in [9,10] that the correct traction boundary conditions are not those proposed by Mindlin and Tiersten [4] and which are currently used in the literature. Since all papers dealing with the indeterminate couple stress model use this incomplete set of boundary conditions we will not refer further to any specific one.
The indeterminate couple stress model
We consider a body which occupies a bounded open set Ω of the three-dimensional Euclidian space R 3 and assume that its boundary ∂Ω is a piecewise smooth surface. An elastic material fills the domain Ω ⊂ R 3 and we refer the motion of the body to rectangular axes Ox i , i = 1, 2, 3. For vector fields v with components
v i ∈ H 1 (Ω), i = 1, 2, 3, we define ∇ v = (∇ v 1 ) T , (∇ v 2 ) T , (∇ v 3 ) T T ,
while for tensor fields P with the rows P i ∈ H(div ; Ω), i = 1, 2, 3, we define Div P = (div P 1 , div P 2 , div P 3 )
T . Equivalently, in index notation: (∇v) ik = v i,k and (Div P ) i = P ij,j . In the remainder of the paper, sym X and skew X denote the symmetric and the skew symmetric part of the matrix X, respectively, tr(X) denotes the trace of the matrix X, X is the Frobenius norm of the matrix X. The identity tensor on R 3×3 will be denoted by 1 1. We also use the operator anti :
R 3 → so(3), so(3) := {X ∈ R 3×3 |X T = −X}, defined by (anti(v)) ij = −ε ijk v k , ∀ v ∈ R 3 ,
where ε ijk is the totally antisymmetric third order permutation Levi-Civita tensor. We use the curl operator, curl v = ε ijk v k,j , ∀ v ∈ R 3 and denote respectively by · , : and ·, · a simple and double contraction and the scalar product between two tensors of any suitable order 1 . Everywhere we adopt the Einstein convention of sum over repeated indices if not differently specified.
The Grioli-Koiter-Mindlin-Toupin isotropic indeterminate couple stress model [3,4,5,6] considers the curvature energy W curv (∇(curl u)) = α1 4 sym ∇curl u 2 + α2 4 skew ∇curl u 2 and the classical elastic energy W e (sym ∇u) = µ sym∇u 2 + λ 2 [tr(sym∇u)] 2 , where µ, λ, α 1 and α 2 are constitutive coefficients. The correct and accepted strong form of the Euler-Lagrange equations are
Div (σ − 1 2 anti(Div m)) + f = 0, equilibrium of forces σ = D sym ∇u W e (sym ∇u) = 2 µ sym ∇u + λ tr(∇u)1 1, symmetric Cauchy-stress m = D ∇curl u W curv (∇curl u) = α 1 sym(∇curl u) + α 2 skew(∇curl u), couple stress tensor. (2.1)
Note that the couple stress tensor m is a second order and trace free tensor. Having the Euler-Lagrange equation, the question of which boundary conditions may be prescribed arises.
The incomplete boundary conditions considered in literature
We want to stress the fact that in the framework of a complete second gradient theory we can arbitrarily prescribe u and the normal derivative of the displacement ∇u · n on the Dirichlet boundary Γ. This means that one has 6 independent geometric (or kinematical) boundary conditions that can be assigned on the boundary of the considered second gradient medium. Analogously, one can assign 6 traction (or natural) conditions on the force (in duality of u) and double force (in duality of ∇u · n), respectively, at ∂Ω \ Γ. The situation is slightly different in the indeterminate couple stress model since only a certain linear combination of second derivatives, i.e. ∇curl u, is controlled. Mindlin and Tiersten [4] concluded that the geometric boundary conditions on Γ ⊂ ∂Ω are the five independent conditions
u Γ = u 0 , (1 1 − n ⊗ n) · curl u Γ = (1 1 − n ⊗ n) · curl u 0 ,(3.2)
for a given vector function u 0 at the boundary, where n is the unit normal vector on ∂Ω and ⊗ denotes the dyadic product of two vectors. The latter condition, in fact, prescribes only the tangential component of curl u. Therefore, one may prescribe only 5 independent boundary conditions. The possible traction boundary conditions on the remaining boundary ∂Ω \ Γ given first by Mindlin and Tiersten [4] are
σ − 1 2 anti(Div m) · n − 1 2 n × ∇[ n, (sym m) · n ] ∂Ω\Γ = t, (1 1 − n ⊗ n) · m · n ∂Ω\Γ = (1 1 − n ⊗ n) · g,(3.3)
for prescribed vector functions t and g at the boundary, where ·, · denotes the scalar product of two vectors. Mindlin and Tiersten [4] have correctly concluded that the maximal number of independent traction boundary conditions is also 5. The same conclusion has been arrived at by Koiter [6]. These traction boundary conditions (3.3) have been rederived again and again. However, they are erroneous.
The correct boundary conditions in the indeterminate couple stress model
The prescribed traction boundary conditions (3.3) proposed by Mindlin and Tiersten [4] do not remain independent, in the sense that g leads to a further energetic conjugate, besides t, of u. From this reason and looking back to the clear and correct boundary conditions considered in the more general second gradient elasticity model, in order to prescribe independent geometric boundary conditions and their corresponding completely independent energetic conjugate (traction boundary conditions), we have to prescribe u and (1 1 − n ⊗ n) · (∇u · n). Let us remark that prescribing u Γ = u 0 and (1 1 − n ⊗ n) · (∇u · n) Γ = (1 1 − n ⊗ n) · ∇ u 0 · n is fully equivalent with prescribing u Γ = u 0 and (1 1 − n ⊗ n) · curl u Γ = (1 1 − n ⊗ n) · curl u 0 , which is (3.2). However, in the formulation of the principle of virtual power, the energetic conjugate of (1 1 − n ⊗ n) · curl u is not equal to the energetic conjugate of (1 1 − n ⊗ n) · (∇u · n). Using the principle of virtual power proposed by Mindlin and Tiersten [4,Eq. (5.13)], but now suitably applying the surface divergence theorem [9,11,12], we arrive at the following traction boundary conditions on
∂Ω \ Γ σ − 1 2 anti(Div m) · n − 1 2 n × ∇[ n, (sym m) · n ] − 1 2 ∇ [ anti ( (1 1 − n ⊗ n) · m · n ) · (1 1 − n ⊗ n) ] : (1 1 − n ⊗ n) ∂Ω\Γ = t, (1 1 − n ⊗ n) · anti[(1 1 − n ⊗ n) · m · n] · n ∂Ω\Γ = (1 1 − n ⊗ n) · g,(4.4)
together with the traction boundary conditions on ∂Γ
{([anti[(1 1 − n ⊗ n) · m · n]] + − [anti[(1 1 − n ⊗ n) · m · n]] − ) · ν} ∂Γ = π,(4.5)
where t and g are prescribed vector functions on ∂Ω \ Γ, while π is a prescribed vector function on ∂Γ. Here, ν is a vector tangential to the surface Γ and which is orthogonal to its boundary ∂Γ. (1 1 − n ⊗ n) which also performs work against u. On the other hand, we show in [9] that when the higher gradient contributions only enter through gradients on the continuum rotation, i.e., ∇curl u, the independent traction boundary conditions which are coming from the representation in terms of a particular case of second gradient elasticity model written with third order moment tensors coincide with our novel traction boundary conditions (4.4) and (4.5), and not with the traction boundary conditions (3.3) proposed by Mindlin and Tiersten. Our renewed interest in traction boundary conditions in the indeterminate couple stress model was triggered by the controversial papers [13,14]. There, the authors have made far reaching claims on the possible antisymmetric nature of the second order couple stress tensor m. Their reasoning is based on physically plausible assumptions similar to a Cosserat or micromorphic theory [15] which led them to require a total split of the effect of force and moment tensors in (3.3). In (3.3), this can be achieved if and only if m is skew-symmetric and this constitutes the essence of their claim. However, since (3.3) is incomplete, their conclusion is misleading, see also [16]. The couple stress tensor in the indeterminate couple stress theory is not necessarily skew-symmetric! Quite to the contrary, the couple stress tensor may be chosen to be symmetric [17,18].
The term [anti[(1 1 − n ⊗ n) · m · n]] + − [anti[(1 1 − n ⊗ n)· m · n]] − measures the discontinuity of anti[(1 1 − n ⊗ n) · m · n] across ∂Γ. Comparing (3.3) and (4.4), we remark that in the Mindlin and Tiersten formulation (3.3) 2 it remains a missing boundary term − 1 2 ∇ [ anti ( (1 1 − n ⊗ n) · m · n ) · (1 1 − n ⊗ n) ] :
For example, (A · v) i = A ij v j , (A · B) ik = A ij B jk , A : B = A ij B ji , (C · B) ijk = C ijp B pk , (C : B) i = C ijp B pj , v, w = v · w = v i w i , A, B = A ij B ij etc.
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| [] |
[
"All-order results for soft and collinear gluons",
"All-order results for soft and collinear gluons"
] | [
"Lorenzo Magnea [email protected] \nDipartimento di Fisica Teorica\nUniversità di Torino\nINFN\nSezione di Torino Via P\nGiuria 1I-10125TorinoItaly\n\nIntroduction\n\n"
] | [
"Dipartimento di Fisica Teorica\nUniversità di Torino\nINFN\nSezione di Torino Via P\nGiuria 1I-10125TorinoItaly",
"Introduction\n"
] | [] | I briefly review some general features and some recent developments concerning the resummation of long-distance singularities in QCD and in more general non-abelian gauge theories. I emphasize the field-theoretical tools of the trade, and focus mostly on the exponentiation of infrared and collinear divergences in amplitudes, which underlies the resummation of large logarithms in the corresponding cross sections. I then describe some recent results concerning the conformal limit, notably the case of N = 4 superymmetric Yang-Mills theory.1 | 10.1007/s12043-009-0006-y | [
"https://arxiv.org/pdf/0806.3353v1.pdf"
] | 16,185,033 | 0806.3353 | 99cf3faed8a06f8be5f60d4888fa2f2a54006cf3 |
All-order results for soft and collinear gluons
20 Jun 2008 June 2008
Lorenzo Magnea [email protected]
Dipartimento di Fisica Teorica
Università di Torino
INFN
Sezione di Torino Via P
Giuria 1I-10125TorinoItaly
Introduction
All-order results for soft and collinear gluons
20 Jun 2008 June 2008
I briefly review some general features and some recent developments concerning the resummation of long-distance singularities in QCD and in more general non-abelian gauge theories. I emphasize the field-theoretical tools of the trade, and focus mostly on the exponentiation of infrared and collinear divergences in amplitudes, which underlies the resummation of large logarithms in the corresponding cross sections. I then describe some recent results concerning the conformal limit, notably the case of N = 4 superymmetric Yang-Mills theory.1
Introduction
Massless gauge field theories, which are classically conformal invariant, are characterized by the fact that all length scales enter in the calculation of any given physical process. Consider for example the perturbative calculation of a scattering amplitude A(p 1 , . . . , p n ) with momentum invariants characterized by a common scale Q. Just like in any quantum theory, when we compute A beyond the leading perturbative order, we must allow for exchanges of virtual quanta of arbitrarily high energy, E ≫ Q, corresponding to processes happening at arbitrarily small length scales. These exchanges are responsible for the ultraviolet (UV) problems we encounter in perturbative calculations, and must be dealt with, when possible, with renormalization. If our theory is massless (or if the masses are negligible with respect to the scale Q), a similar problem arises at the other end of the spectrum: we must allow for exchanges of very low-energy quanta, E ≪ Q, which happen at very large distances. Such processes may or may not, in a general field theory, endanger our calculational framework, but they certainly do so in the case of gauge theories, even abelian ones. This is the origin of the infrared (IR) problems of perturbative calculations, which are usually dealt with using factorization.
In either case, it is worth recalling that the appearance of infinities in our perturbative calculations is due to the fact that we have stretched our approximations beyond their limits of applicability. In the UV regime, we have tacitly assumed that our theory should be applicable at extreme short distances, where in fact we do not even know what the relevant degrees of freedom might be. In the case of renormalizable theories, we are forgiven our arrogance, since we can show that physics at very short distances effectively decouples from the calculation of our amplitude at the scale Q: the contributions of high-energy quanta factorize, and can be absorbed into rescalings of the local couplings of our original theory. In the IR regime, for gauge theories, our mistake is more subtle: massless gauge bosons mediate long-range interactions, which cannot be switched off even at asymptotycally large distances; hence, it is not correct to formulate our perturbative expansion in a Hilbert space of Fock states built with creation and annihilation operators of free fields. The true asymptotic states of the theory are coherent states containing an infinite number of massless quanta, and the price of stretching our approximation is that the S matrix actually does not exist in our Hilbert space: all scattering amplitudes diverge because of virtual IR exchanges.
In QED, and to a more limited extent in QCD, IR problems are alleviated by the KLN theorem [1,2]. When we compensate for our inadequate choice of Hilbert space by constructing physically measurable probabilities, which are obtained by summing over all Fock states that are degenerate in energy, all IR divergences must cancel. This is sufficient to solve the IR problem in QED, where actually a sum over final-state degeneracies is enough to achieve the required cancellation. In an unbroken non-abelian gauge theory like QCD, things are considerably more complicated: because of confinement, the relationship between partonic Fock states and the true non-perturbative asymptotic states of the theory (which are color-singlet hadrons) is highly non-trivial, and beyond the range of our techniques; this is reflected at the perturbative level in the fact that a sum over initial state degeneracies is both necessary to cancel divergences, and impossible to perform in practice, since initial state partons are far from free at large distances.
In order to rescue the applicability of perturbative methods, one must resort once again to factorization, coupled with asymptotic freedom. One exploits the quantum-mechanical incoherence of processes happening at different distance scales to show that high-energy inclusive cross sections can be written as convolutions of short-distance finite partonic cross sections, which can be computed in perturbation theory thanks to asymptotic freedom, with long-distance factors (parton distributions or fragmentation functions), which are non-perturbative but universally associated with hadronic wave functions.
Proofs of factorization are highly non-trivial in perturbation theory [3], but they pay big dividends. First of all, they underpin essentially all perturbative QCD predictions for high-energy cross sections, from Deep Inelastic Scattering, to Drell-Yan production of electroweak vector bosons and Higgs bosons, to general jet cross sections. There are, furthermore, several other applications, important both for theory and for phenomenology, some of which will be reviewed below.
• Factorization leads to evolution equations. All factorization theorems introduce intermediate arbitrary scales separating the momentum space regions one wishes to disentangle. Demanding that physical observables be independent of these arbitrary scales leads to evolution equations for individual contributions to the factorized observable. A prime example is of course Altarelli-Parisi [4] evolution of parton distributions.
• Solving the evolution equations dictated by factorization leads to the resummation of classes of logarithmic contributions to all orders in perturbation theory. Renormalization group evolution and collinear evolution of parton distributions are examples, but also Sudakov resummation, both for threshold and transverse momentum logarithms, can be derived in this way (see [5] and, for a review, [6]). Such resummations are important, in some cases essential, for the phenomenological success of perturbative QCD. Alternatively, having an established a factorization theorem, one can apply effective field theory methods to derive the same results in a different systematical way (see, for example, [7]).
• Resummation, in turn, probes the all-order structure of perturbation theory and helps identify leading non-perturbative corrections to many cross sections.
The study of power-suppressed corrections to QCD factorization theorems has in fact developed into an active subfield of research, with many phenomenological applications and new interesting theoretical ideas. The literature is vast, involving renormalon techniques as well as resummations; for reviews and further references on some of these ideas see for example [8] and [9].
• At the amplitude level, resummation of IR singularities displays universal features of gauge theories which find application both in theory and in phenomenology. Understanding the structure of infrared and collinear poles at high orders is instrumental to construct subtraction algorithms to compute efficiently multiparticle cross sections at colliders. On the other hand, remarkably, the all-order structure of infrared poles, uncovered in QCD, has recently found application in the context of supersymmetric gauge theories, and most notably for N = 4 Super-Yang-Mills (SYM) theory, which is conjectured to be equivalent to string theory on the background of an AdS 5 × S 5 space-time.
Gluon amplitudes can now be computed in weak-coupling N = 4 SYM to very high order, and can also, in some cases, be computed at strong ('t Hooft) coupling using directly the AdS-CFT correspondence [10]. The universal structure of infared and collinear singularities of these amplitudes places powerful constraints on their structure, and helps identify the relevant anomalous dimensions. This is a very active and fast-developing field, recently reviewed in [11] and [12].
In the following sections, I will briefly describe some of the field theory tools that are used to study the universal structure of gauge theories at long distances. I will begin in Section 2 by discussing qualitatively the connection between factorization, evolution ad resummation; then, in Section 3, I will introduce some of the technical tools that are needed to make precise all-order statements. In Section 4, I will focus on the simplest example of nonabelian exponentiation of infrared singularities, the form factor of a massless particle in a gauge theory. Finally, in Section 5, I will describe some applications of these results to conformal gauge theories. I will include in the discussion several new results recently obtained in [13].
Factorization leads to resummation
A factorization theorem is a sufficient (though not necessary) condition to perform a resummation of perturbation theory, typically through the solution of an appropriate evolution equation. Clearly, the difficult work is proving factorization to all orders in perturbation theory, which requires a detailed diagrammatic analysis and often a delicate implementation of the symmetry properties of the theory. Once factorization is established, evolution equations automatically follow, and their solutions entail an all-order organization of certain perturbative contributions. Let's look at three classical examples, following [5].
• The prototypical perturbative factorization is the renormalization of UV divergences. In this case the difficult step is to prove, to all orders, that the dependence of a generic Green function of the theory on the cutoff scale can be associated with a finite number of universal multiplicative factors. If that's the case, then one may write, for an n-point bare correlator G
(n) 0 , G (n) 0 (p i , Λ, g 0 ) = n i=1 Z 1/2 i (Λ/µ, g(µ)) G (n) R (p i , µ, g(µ)) ,(1)
where g 0 is the bare coupling, and Λ the UV cutoff, which can be interpreted as the scale of 'new physics', or the scale at which the effective low-energy theory under consideration breaks down. In order to perform the factorization in Eq. (1) it has been necessary to introduce and intermediate, arbitrary scale µ. Bare Green functions, however, do not depend on µ, so that we can write
dG (n) 0 dµ = 0 → d log G (n) R d log µ = − n i=1 γ i (g(µ)) ,(2)where γ i (g) ≡ d log Z i /d log µ 2 .
Solving the renormalization group equation, Eq. (2), resums the logarithmic dependence on the renormalization scale µ. Furthermore, since renormalized Green functions, up to their overall engineering dimension, depend on µ only through ratios such as p i · p j /µ 2 , the same equation can be used to extract information about the dependence of G (n) R on external momenta. Notice that, in order to enforce the cancellation in Eq.
(2), the anomalous dimensions γ i (g) can only depend on arguments that are common to Z i and to G (n) R , the only one in this case being the renormalized coupling g.
• Another familiar example is collinear factorization in high-energy QCD cross sections, most topically DIS. In this case the difficult task is to show that collinear singularities in the cross section can be absorbed into universal factors associated with the wave functions of initial state hadrons. For DIS structure functions, say F 2 (x, Q 2 ), this is true in the form of a convolution, which becomes an ordinary product upon taking a Mellin transform. One writes then
F 2 N, Q 2 m 2 , α s = C N, Q 2 µ 2 F , α s f N, µ 2 F m 2 , α s .(3)
Here m is a label for a collinear regulator, say a light quark mass; C is a perturbatively computable coefficient function, free of collinear sensitivity, while f is a universal (but non-computable) parton distribution. Again, to perform factorization it has ben necessary to introduce an arbitrary scale µ F , and one can exploit the fact that the structure function F 2 does not depend in the choice of µ F . One derives
d F 2 dµ F = 0 → d log f d log µ F = γ N (α s ) ,(4)
where γ N (α s ) ≡ −d log C/d log µ F are the Mellin moments of the appropriate Altarelli-Parisi splitting function. The anomalous dimension γ N can only depend on arguments common to C and f , in this case N and α s . Solving Eq. (4) resums logarithms of the factorization scale, and allows to evolve parton distributions to the scales appropriate for applications to other high-energy cross sections.
• Let us finally turn to the most significant (and difficult) case of Sudakov resummation. In the previous two cases one was dealing with the resummation of single logarithms, arising from single (UV or collinear) poles of the corresponding amplitudes. Sudakov resummation involves double (IR and collinear) poles, which requires in principle a more elaborate factorization (see, however, the arguments in [14]). Since our focus below will be on amplitudes rather than cross sections, let us begin by considering the simplest scattering amplitude which is affected by such double poles, the form factor of any massless particle minimally coupled to a massless gauge boson. Using a massless quark as an example one can define
Γ µ (p 1 , p 2 ; µ 2 , ǫ) ≡ p 1 , p 2 |J µ (0)|0 = u(p 1 )γ µ v(p 2 ) Γ Q 2 µ 2 , α s (µ 2 ), ǫ ,(5)
corresponding to pair creation of a qq pair out of the QCD vacuum by means of a source (an off-shell photon) of mass Q. It can be shown (as reviewed in [15]) that this amplitude factorizes into the product of different functions, each one responsible for a specific set of singularities. In dimensional regularization, the precise form of this factorization can be written as [13] Γ
Q 2 µ 2 , α s (µ 2 ), ǫ = C Q 2 µ 2 , (p i · n i ) 2 n 2 i µ 2 , α s (µ 2 ), ǫ × S β 1 · β 2 , α s (µ 2 ), ǫ × 2 i=1 J (p i ·n i ) 2 n 2 i µ 2 , α s (µ 2 ), ǫ J (β i ·n i ) 2 n 2 i , α s (µ 2 ), ǫ .(6)
Here β i are four velocities associated with the quark and the antiquark (so that p ν i ∝ Qβ ν i ), while n i are auxiliary vectors associated with Wilson lines (to be described below), which are introduced in order to factorize wide-angle soft radiation from collinear one. The function S is an eikonal function responsible for the radiation of soft gluons, while the jet function J is associated with radiation collinear to either the quark or the antiquark, and C is finite as ǫ → 0. Including both S and J double counts the soft-collinear region, which is compensated for by introducing eikonal versions of the jets, J . In order to derive evolution equations, one can now exploit the renormalization group invariance of Γ, which must not depend on the scale µ, as well as the manifest independence on the auxiliary vectors n i . Depending on how the calculation is performed, the independence of Γ on n i can be understood either as gauge invariance (if working in an axial gauge), or as Lorentz invariance (if working in Feynman gauge, as we will do below).
In order to move forward and be more precise we need at this point to step back and introduce some technical tools. Specifically, in order to give precise operator expressions for the functions entering Eq. (6), we will need to introduce Wilson line operators, which are also instrumental in the mapping to strong coupling which becomes possible for N = 4 SYM. Furthermore, in order to perform a consistent resummation in dimensional regularization, we need to define the running coupling in d = 4 − 2ǫ dimensions. We briefly turn to these issues in the next section.
3 Tools of the trade 3.1 Dimensional regularization for the strong coupling Dimensional regularization (DR), in its various flavors, is a unique tool for the study of non-abelian gauge theories. It can be used both for UV and IR divergences, it preserves gauge invariance, it is by far the simplest scheme to use from the computational point of view. In the context of all-order calculations, it has further virtues. In this case, one starts with the renormalized theory, and regulates long-distance singularities by taking d = 4 − 2ǫ, with ǫ < 0. One must then recall that RG equations acquire ǫ dependence in d = 4. The coupling, for example, runs according to
µ ∂α ∂µ ≡ β(ǫ, α) = − 2ǫα +β(α) ,β(α) = − α 2 2π ∞ n=0 b n α π n .(7)
The ǫ dependence of the β function is a consequence of the engineering dimension of the bare coupling, and it implies that the running coupling behaves like a power of its scale, α s (µ 2 )/α s (µ 2 0 ) ∼ (µ 2 /µ 2 0 ) −ǫ : in fact, in d > 4, the β function has an IR-free fixed point at α s = 0, where it vanishes with a positive derivative. As a consequence, α s (µ 2 = 0) = 0. The RG equation, Eq. (7), is easily solved at one loop, yielding
α µ 2 , ǫ = α s (µ 2 0 ) µ 2 µ 2 0 ǫ − 1 ǫ 1 − µ 2 µ 2 0 ǫ b 0 4π α s (µ 2 0 ) −1 .(8)
Note that α depends only on the scale µ 2 and on ǫ, but not on the chosen initial condition µ 2 0 . At higher orders an explicit analytic solution such as Eq. (8) is not available, but one may still expand α in powers of the coupling at a fixed reference scale, as
α ξ 2 , ǫ = α s ξ −2ǫ + α 2 s ξ −4ǫ b 0 4πǫ 1 − ξ 2ǫ + α 3 s ξ −6ǫ 1 8π 2 ǫ b 2 0 2ǫ 1 − ξ 2ǫ 2 + b 1 1 − ξ 4ǫ + O α 4 s .(9)
The key advantage of Eq. (7) is that it provides a simple initial condition for the solution of evolution equations for amplitudes, basically stating that all radiative corrections vanish when the scale vanishes. This fact was first exploited to give an explicit exponentiated expression for the Sudakov form factor in [16]. A further advantage of Eq. (7) is that the Landau pole for the running coupling acquires a non-vanishing imaginary part when ǫ < −b 0 α s (µ 2 0 )/(4π), a fact that can be exploited to evaluate resummed amplitudes explicitly as analytic functions of the coupling and of ǫ [17].
Wilson lines and the eikonal approximation
An important feature of Eq. (6) is the fact that all singular factors comprising the form factor have well-defined operator expressions. This is especially significant when one is trying to make a connection to non-perturbative features of the theory, as is the case for N = 4 SYM. It is well-known, and easily verified, that in the soft approximation, relevant for the calculation of infrared poles, gluon interactions with other hard partons can be completely expressed in terms of correlators of Wilson lines: energetic partons do not recoil against soft radiation, so that the only effect of interactions with soft gluons is the buildup of an eikonal phase on the parton field; soft gluons, in turn, are only sensitive to the direction and color representation of the hard parton, but not to its spin and energy. The situation with collinear gluons is not as simple: it can be shown that they couple eikonally to hard partons moving in different light-cone directions, but they retain to some extent the spin and energy dependence of the coupling to partons belonging to their own jet. In either case, eikonal lines play a major role in factorization formulas such as Eq. (6). Defining the Wilson line operator as Φ n (λ 2 , λ 1 ) = P exp ig
λ 2 λ 1 dλ n · A(λn) ,(10)
we can give explicit operator expressions for all the functions appearing in Eq. (6). The soft function S is just the eikonal approximation of the full form factor
S β 1 · β 2 , α s (µ 2 ), ǫ = 0|Φ β 2 (∞, 0) Φ β 1 (0, −∞) |0 ;(11)
the jet functions J, on the other hand, couple a hard parton to an eikonal line off the light cone, along an arbitrary space-like direction n µ ,
J (p · n) 2 n 2 µ 2 , α s (µ 2 ), ǫ u(p) = 0 |Φ n (∞, 0) ψ(0) |p .(12)
Eikonal jets J , finally, represent the soft approximation of the partonic jets J, so that the parton field ψ is replaced by its own Wilson line,
J (β · n) 2 n 2 , α s (µ 2 ), ǫ = 0|Φ n (∞, 0) Φ β (0, −∞) |0 .(13)
As we will briefly summarize below, while the evolution equations (2) and (4) were stemming from the invariance of the observable before factorization with respect to variations of a mass scale, in the case of Eq. (6) one derives evolution by demanding invariance with respect to the choice of the 'factorization vectors' n µ i .
Resummation for the form factor
The derivation of the evolution equation for the form factor can be understood by considering n µ i dependence in Eq. (6). Clearly, all such dependence is through the dimensionless ratio x i ≡ (β i · n i ) 2 /n 2 i . Demanding that ∂ log Γ/∂ log x i = 0, and noting that Γ depends on x i only through the jet functions and the finite coefficient function C, one derives
x i ∂ ∂x i log J i = − x i ∂ ∂x i log C + x i ∂ ∂x i log J i ≡ 1 2 G i x i , α s (µ 2 ), ǫ + K α s (µ 2 ), ǫ ,(14)
where the second line defines the functions G i and K i . The key feature of Eq. (14) is that the n µ i dependence of the partonic jets has been organized in a function G i , which carries all the kinematic dependence, but is finite as ǫ → 0 (because C is finite), plus a function K i , which on the contrary is a pure counterterm (because J is), but carries no kinematic dependence. Note also that, while J i has a double pole, its derivative with respect to x i must have only a single pole, since the double pole is independent of kinematics. Integrating Eq. (14) thus leads to one of the key features of resummation: double pole observables exponentiate, and their logarithms contain only single poles.
It is not difficult to generalize the argument leading to Eq. (14) to the full form factor [15]. One finds an equation of the same form
Q 2 ∂ ∂Q 2 log Γ Q 2 µ 2 , α s (µ 2 ), ǫ = 1 2 K ǫ, α s (µ 2 ) + G Q 2 µ 2 , α s (µ 2 ), ǫ ,(15)
where G is finite as ǫ → 0 and carries the full Q 2 dependence, while K is a Q 2independent pure counterterm. In order to solve Eq. (15), we still need three ingredients.
• Renormalization group invariance of the form factor requires
µ ∂ ∂µ + β(ǫ, α s ) ∂ ∂α s G Q 2 µ 2 , α s (µ 2 ), ǫ = − β(ǫ, α s ) ∂ ∂α s K ǫ, α s (µ 2 ) ≡ γ K α s (µ 2 ) ,(16)
where in the second equation we have used the fact that K is a pure counterterm, and thus has no explicit scale dependence. Eq. (16) defines the anomalous dimension γ K , and allows one to solve for the µ dependence of G in terms of an initial condition, say at µ = Q.
• The infrared freedom of the theory for ǫ < 0 provides us with a simple initial condition for Eq. (15),
α(µ 2 = 0, ǫ < 0) = 0 → Γ 0, α s (µ 2 ), ǫ = Γ (1, α (0, ǫ) , ǫ) = 1 .(17)
• By the same token, the counterterm K can be expressed directly as an integral of the anomalous dimension γ K , using Eq. (16) and the vanishing of the coupling at µ 2 = 0. One verifies that
K ǫ, α s (µ 2 ) = − 1 2 µ 2 0 dλ 2 λ 2 γ K ᾱ(λ 2 , ǫ) .(18)
Putting these ingredients together, one can express the solution to Eq. (15) in a simplified form, displaying the fact that infrared and collinear poles to all orders are generated by just two functions of the coupling, G and γ K . One finds [13] Γ Q 2 , ǫ = exp 1 2
−Q 2 0 dξ 2 ξ 2 G − 1, α ξ 2 , ǫ , ǫ − 1 2 γ K α ξ 2 , ǫ log −Q 2 ξ 2 .(19)
In light of recent developments, both in QCD and in N = 4 SYM, it is worth emphasizing that the form factors play an important role also in the much more general case of fixed-angle scattering amplitudes with any number of external legs, for massless gauge theories. Such amplitudes also factorize in a manner similar to Eq.
M [m] L β j , Q 2 µ 2 , α s (µ 2 ), ǫ = m i=1 J i Q ′2 µ 2 , α s (µ 2 ), ǫ S [m] LK β j , Q ′2 µ 2 , Q ′2 Q 2 , α s (µ 2 ), ǫ × H [m] K β j , Q 2 µ 2 , Q ′2 Q 2 , α s (µ 2 ) .(20)
Eq. (20) is expressed in terms of velocity four-vectors β i for each external leg, and the restriction to fixed-angle scattering has been exploited to extract from particle momenta a common hard scale Q; the scale Q ′ , on the other hand, plays the role of a factorization scale separating infrared and collinear momenta; collinear singularities are organized into the m 'jet' functions J i , each characterized only by the properties of the originating parton; soft gluons, on the other hand, can mix the color components of the hard scattering and thus are organized into a matrix S K . One may now exploit the fact that the jets J i collect the same collinear and infrared-collinear singular regions as the form factors Γ i for the same parton species: soft wide-angle radiation would be different, but one can make use of the fact that the soft matrix S is defined up to multiplication times a multiple of the identity matrix in order to reconstruct the appropriate soft emission structure. In other words, there exists a factorization scheme such that one can define
J i Q ′2 µ 2 , α s (µ 2 ), ǫ = Γ i Q ′2 µ 2 , α s (µ 2 ), ǫ 1 2 .(21)
This factorization is especially useful because it teaches us that the structure of collinear singularities of fixed-angle multi-leg scattering amplitudes is completely captured by partonic form factors. Furthermore, in this factorization scheme, the matrix S becomes proportional to the identity matrix in the planar, N c → ∞ limit. This feature simplifies considerably the analysis in the interesting case of planar N = 4 SYM, where a continuation of the amplitude to strong coupling has, in some cases, become possible.
Beyond QCD: conformal gauge theories
One striking feature of Eq. (19) is the fact that the logarithm of the form factor is expressed in terms of two finite functions of the coupling, G and γ K . All infrared and collinear poles are generated by the explicit integration over the scale of the running coupling. In QCD, and for ǫ < 0, the scale dependence of the coupling (see for example Eq. (9)) is such that poles up to 1/ǫ p+1 are generated at order α p s . By contrast, in a conformal gauge theory such as N = 4 SYM, regularized by dimensional continuation, the coupling runs simply according to its engineering dimension in d = 4 − 2ǫ; as a consequence, the integration in Eq. (19) yields at most double poles. Expanding γ K and G in powers of α s /π, and denoting their perturbative coefficients by γ (n) K and G (n) (ǫ) respectively, one easily finds [19] log Γ
Q 2 µ 2 , α s (µ 2 ), ǫ = − 1 2 ∞ n=1 α s (µ 2 ) π n µ 2 −Q 2 nǫ γ (n) K 2n 2 ǫ 2 + G (n) (ǫ) nǫ = − 1 2 ∞ n=1 α s (Q 2 ) π n e −iπnǫ γ (n) K 2n 2 ǫ 2 + G (n) (ǫ) nǫ ,(22)
where in the second line I note that the logarithm of the form factor displays exact renormalization group invariance, as expected. Using Eq. (22), it is possible to study the analytic continuation of the form factor from time-like to space-like kinematics, in the conformal case. This continuation is of practical interest in QCD: in fact, as shown to all orders in [16], the modulus of the ratio of the time-like to the space-like form factor is finite in d = 4; furthermore, this ratio is closely related to physically observable cross sections: for example, it resums a class of large constant contributions to the Drell-Yan cross section in the DIS factorization scheme [20,21,22]. In the present case, having constructed a finite quantity, one may take ǫ → 0 and compute the ratio in the four-dimensional theory with exact conformal invariance. One finds [13] Γ
(Q 2 ) Γ(−Q 2 ) 2 = exp π 2 4 γ K α s (Q 2 ) .(23)
Eq. (23) resums perturbation theory for finite quantities which admit a nonperturbative definition in terms of operator matrix elements. Thus, it can be conjectured to be an exact result. It would be of great interest if a strong-coupling analogue could be derived. Combining Eq. (20), Eq. (21) and Eq. (22) places strong constraints on the allorder structure of scattering amplitudes in dimensionaly-regularized N = 4 SYM. In fact, after a reanalysis of one-and two-loop results for the four-point planar MHV amplitude [23], Bern, Dixon and Smirnov (BDS) performed the highly nontrivial calculation of the same amplitude at three loops [19], and found an intriguing pattern of exponentiation, consistent with Eq. (22), but extending to non-singular, ǫ-independent terms. In order to illustrate this pattern, note that in the planar limit the all-order factorized matrix element in Eq. (20) has color structure proportional to the tree-level amplitude; in this case the soft matrix S can be taken to be diagonal. Defining then, in shorthand notation, a reduced matrix element M [m] (ǫ), by dividing out the tree-level result, one can show that the non-abelian exponentiation following from Eq. (20) and Eq. (21) leads to the expression [19] M [m] (ǫ) = exp in a general gauge theory depends both on the number of particles m and on their momenta k i . BDS [19] observed that the finite remainder h (4) p (k i ) is a constant, independent of kinematics, for p ≤ 3; using also results on collinear limits derived in [23], they conjectured that this property might remain true to all orders in the 't Hooft coupling and for any number m of particles. On the other hand, Alday and Maldacena [10], computing the four-point amplitude at strong 't Hooft coupling by means of the AdS-CFT correspondence, found a structure closely matching Eq. (24). These results have lead to a sustained effort by several groups, employing rather different theoretical tools, to study the structure of amplitudes in N = 4 SYM and related theories, and to constrain and compute the anomalous dimensions γ K and G that govern their singularities (for references, see the reviews in [11,12]). To summarize very briefly the status of these efforts to date, the BDS conjecture is now expected to hold for the four-and -five point amplitudes, while it is known to break down for the six-point amplitude, starting at two loops [24,25]; an ansatz exists [26] for the function γ K (λ), which reproduces all available perturbative results, both at weak and at strong coupling [27,28]; the function G has also been analyzed in detail [13], in the general case of an arbitrary massless gauge theory, expressing it in terms of anomalous dimensions of operators involving Wilson lines and fundamental fields, plus running coupling contributions. For a conformal gauge theory, one finds the very simple result G(1, α s , ǫ = 0) = 2B δ (α s ) + G eik (α s ) ,
where G eik is a subleading anomalous dimension associated with Wilson lines, and thus in principle amenable, like γ K , to studies with non-perturbative techniques, while B δ is the virtual contribution to the Altarelli-Parisi splitting kernel. B δ involves matrix elements of local fields as well as Wilson lines, so it would be quite interesting to to see how an equation of the form of Eq. (26) might arise at strong coupling.
Conclusion
The study of long-distance singularities of gauge theories began more than seventy years ago [29], yet it remains an active and fertile field of research. In QCD, all-order results for soft and collinear gluons are instrumental for phenomenology, providing nontrivial tests of finite order calculations, and forming the basis for the resummation of several classes of large logarithms that would otherwise hinder the applicability of perturbation theory. From a theoretical standpoint, studying long-distance effects to all orders in perturbation theory opens a window on non-perturbative effects, which are suppressed by powers of the hard scale but may still be very relevant for high-energy cross sections in certain kinematical regimes. When tools are available for a quantitative study of a gauge theory at strong coupling, as is the case for maximally supersymmetric Yang-Mills theory, soft and collinear singularities of amplitudes still provide a bridge between weak coupling and non-perturbative regimes.
Recent progress in the study of N = 4 SYM has been especially remarkable; bringing together tools from perturbation theory, string theory and integrable models, it has been possible to reach results that point towards a very ambitious goal: a full and detailed understanding of a non-trivial four-dimensional gauge theory. We may indeed look forward to new developments and applications, both on the practical side of collider phenomenology, and on our way to a deeper understanding of quantum field theory.
( 6 )
6, albeit with a more complicated color structure. Indeed, an amplitude with m external colored legs, M {a i } , i = 1, . . . , m, can be written as a vector in the space of available color configuration, with components M (m) L in a suitable basis of color tensors c L {a i } . One may then write[18]
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| [] |
[
"RecipeSnap -a lightweight image to recipe model",
"RecipeSnap -a lightweight image to recipe model"
] | [
"Jianfa Chen \nGeorgia Institute of Technology\n225 North AvenueAtlantaGA\n",
"Yue Yin \nGeorgia Institute of Technology\n225 North AvenueAtlantaGA\n",
"Yifan Xu \nGeorgia Institute of Technology\n225 North AvenueAtlantaGA\n"
] | [
"Georgia Institute of Technology\n225 North AvenueAtlantaGA",
"Georgia Institute of Technology\n225 North AvenueAtlantaGA",
"Georgia Institute of Technology\n225 North AvenueAtlantaGA"
] | [] | In this paper we want to address the problem of automation for recognition of photographed cooking dishes and generating the corresponding food recipes. Current image-to-recipe models are computation expensive and require powerful GPUs for model training and implementation. High computational cost prevents those existing models from being deployed on portable devices, like smart phones. To solve this issue we introduce a lightweight image-to-recipe prediction model, RecipeSnap, that reduces memory cost and computational cost by more than 90% while still achieving 2.0 MedR, which is in line with the state-of-the-art model. A pre-trained recipe encoder from [8] was used to compute recipe embeddings. Recipes from Recipe1M [6] dataset and corresponding recipe embeddings are collected as a recipe library(Figure 1), which are used for image encoder training (Figure 2) and image query(Figure 3) later. We use MobileNet-V2 as image encoder backbone, which makes our model suitable to portable devices. This model can be further developed into an application for smart phones with a few effort. A comparison of the performance between this lightweight model to other heavy models are presented in this paper. Code, data and models are publicly accessible 1 . | 10.48550/arxiv.2205.02141 | [
"https://arxiv.org/pdf/2205.02141v1.pdf"
] | 248,512,568 | 2205.02141 | 4dcb963cae799a10316252a418d810f25c016d8c |
RecipeSnap -a lightweight image to recipe model
Jianfa Chen
Georgia Institute of Technology
225 North AvenueAtlantaGA
Yue Yin
Georgia Institute of Technology
225 North AvenueAtlantaGA
Yifan Xu
Georgia Institute of Technology
225 North AvenueAtlantaGA
RecipeSnap -a lightweight image to recipe model
In this paper we want to address the problem of automation for recognition of photographed cooking dishes and generating the corresponding food recipes. Current image-to-recipe models are computation expensive and require powerful GPUs for model training and implementation. High computational cost prevents those existing models from being deployed on portable devices, like smart phones. To solve this issue we introduce a lightweight image-to-recipe prediction model, RecipeSnap, that reduces memory cost and computational cost by more than 90% while still achieving 2.0 MedR, which is in line with the state-of-the-art model. A pre-trained recipe encoder from [8] was used to compute recipe embeddings. Recipes from Recipe1M [6] dataset and corresponding recipe embeddings are collected as a recipe library(Figure 1), which are used for image encoder training (Figure 2) and image query(Figure 3) later. We use MobileNet-V2 as image encoder backbone, which makes our model suitable to portable devices. This model can be further developed into an application for smart phones with a few effort. A comparison of the performance between this lightweight model to other heavy models are presented in this paper. Code, data and models are publicly accessible 1 .
Introduction
Introduction and Background
With the rapid spread and easy access of social networks nowadays, countless pictures for cooking dishes and recipes are created and shared with everyone. As almost nothing in Figure 1. Build recipe library. A pre-trained recipe encoder from [8] is used to compute recipe embeddings. Recipes from Recipe1M [6] dataset and corresponding recipe embeddings are collected as a recipe library. Figure 2. Image encoder training. We use MobileNet V2 as image encoder backbone. Images from Recipe1M training partition combined with corresponding pre-computed recipe embeddings in recipe library are serving as input for model training. The loss function is same as the pair loss function in [8]. Figure 3. Predict recipe from image. A query image is passed to pre-trained image encoder to get image embedding. Then we compute the cosine similarity between image embedding and recipe embedding array (numpy array). Only top k (here we use k=5) recipe embedding ids are collected. Corresponding top k recipes can be easily retrieved from recipe dictionary with embedding ids. the world other than food has such a profound and tremendous impact in people's life this has created urging needs for machine learning algorithms to automatically retrieve associated recipes given input images of food or dishes. The mass quantity of user submitted cooking recipes and food images online brings the possibility of training networks to find digital cooking recipes (ingredient lists, cooking instructions etc.) relevant to the specific food image. Although it is also possible to reverse the image-to-recipe procedure that is to find relevant food images given recipes such tasks are unlikely to occur in real world. Usually people would search food images with keyword or recipe title in a searching engine.
After searching for and studying on some of the recent published works, we found that there are a couple of existing frameworks established in this particular fields. More details are illustrated in the following Section 1.2. However, all of them require large memory and powerful computation resource, like GPUs. Heavy model size prevent these models from deploying on mobile devices. After conducting a comprehensive analysis quantitatively and qualitatively, we decided to build a lightweight cross-modal framework that is suitable for mobile devices.
Related Work
To start our work in the development of the lightweight prediction model to handle image-to-recipe retrieve tasks, we reviewed a couple of relevant papers. We mainly built our work upon the study of a couple of recent published papers that utilize multi-model framework to solve similar problems and here provided a brief summary of their work.
In [6] the authors use bi-directional LSTM model in the design of the two major components, ingredients and instructions, of recipe encoder. The two-stage structure has taken consideration of both forward and backward orderings and better suits the recipe representation than a simple LSTM model. For the image encoding authors adopt two major deep convolutional networks that has achieved proven record of success, VGG-16 and ResNet-50 models.
In [2] the authors got inspired by the success of Transformers in NLP field and conducted experiments on using standard Transformer model directly on image inputs. To realize their goal with minimum modification the authors processed the original images into patches and a sequence of linear embedding resulted is input into the Transformer. According to the experiment results provided from the paper, this model only achieved better performance when trained on larger size of dataset which range from 14M to 300M images. Among all that their Vision Transformer (ViT) produced excellent results when pre-trained on sufficient amount of data-points and transferred to smaller-scale tasks.
Amaia et al. introduced a hierarchical Transformer model which encodes each recipe components eg. titles, ingredients and instructions individually [6]. In addition, the author proposed a self-supervised loss function computed based on pairs of the individual recipe components to leverage the semantic relationships within recipes. Whereas in [8] the author developed a neural with joint embedding learned on the recipes and images in common space. In the model a high-level classification task was added to further improve the classification performance. [7] Hao and others proposed a end-to-end trainable Adversial Cross-Model Embedding (ACME) validated using the Recipe1M dataset [11] to resolve the food retrieval task where their specific goal is to develop a constructive mapping space from recipes to matching food images. As shown in the paper, the proposed network achieved outperformance over all baseline models. Specifically, the authors proposed an refined triplet loss methodology which increased the convergence and accuracy of the neural network. In addition, an adversarial loss scheme was utilized to align the distribution of the encoded features from recipes and food images.
We noticed that these papers use more and more complex neural networks to achieve better performance. In the meanwhile, prerequisite on hardware, like memory and computation power, is being more and more difficult to meet. Given the prevalence of portable devices, like smart phones and tablets, we believe it will be greatly advanced if these models can be deployed on portable devices. To build a lightweight image-to-recipe model that can be easily trained and deployed on smart phones, we also reviewed some existing models that are suitable for smart phones.
Mark et al. introduced the MobileNet V2 model in [4] which is a mobile framework based on a inverted residual structure. This model is specifically designed to provide a convenient and efficient network for actual usage. The approaches that's been taken by the author to meet their goal including: within the structure a shortcut path are built to connect the bottleneck layers; they use depth-wise convolution filters to extract features; they remove non-linearity from narrow layers to obtain better representing power. Additionally the paper described valid ways of application of this mobile network framework to object detection. Model size and complexity comparison is disclosed in Table 2.
Dataset
Our model was training on the Recipe1M dataset, which is a large dataset contains paired data points -cooking recipes and dish images. This dataset was first introduced and collected by Javier et al. [11] by scraping from a variety of popular cooking websites. The overall dataset contains over one million cooking recipes and about 900K food images. As shown in Table 1 the data has been broken down in a 70%,15%,15% manner for the purposes of training/validation/testing of the model.
The cooking recipes and linked images are initially scraped from over two dozen widely used cooking websites and handled via a pipeline process. Only related content is generated from the raw HTML that is to say unnecessary blank space, non-ASCII characters etc. have been left out during the extraction process. After removing the duplicates or close-matches, the final dataset is consolidated into a compressed JSON schema and ready for use.
Due to the limited time and resource we have for this project, we believe the use of the Recipe1M dataset should fulfill our goal given the following considerations:
• This dataset has over one million recipe and image data which is among the largest publicly available datasets ever been used in related work.
• Recipe1M contains paired recipe and image data points which well suits the purpose of training a multimodel framework.
• The dataset is clean and consolidated with any duplicates and near-matches taken out which save us time for more model experiments.
Model
Param. FLOPs AlexNet [5] 58.3M 725M VGG16 [5] 134.2M 15.5G ResNet50 [5] 23.5M 3.80G GoogLeNet [5] 6.0M 1.57G ViT(base) [1] 86M 15.85G MobileNetV2 [9] 3.34M 319M
Approach
Our ultimate goal is to build an application that can carry out the image-to-recipe tasks on standard mobile devices. Therefore, we naturally face the trade-off between a deeper and more complicated network model with higher accuracy vs. a less-powered model with lighter weight and low latency. To understand the trade-off better, we have explored and compared a few different approaches. We first start with the models that deliver the state-of-theart (SOTA) performance on accuracy. Specifically, we focus on two models that frame the problem as a cross-modal recipe retrieval task [6] [8]. The main difference between these two models lies in the design of recipe encoder: 1) one uses a two-stage LSTM [6] while the other uses hierarchical transformers [6]. 2) [6] introduces an auxiliary self-supervised task and loss to learn the semantic relationship between recipe components, strengthening the recipe encoder. For image encoder, [6] uses the visual transformer ViT [2] while [8] uses the ResNet-50 [3]. We train these two models from scratch to compare the performance and cost. For evaluation, we follow the procedure described in [8], which uses cosine similarity in the common space for ranking the relevant recipes and perform im2recipe retrieval on validation dataset in Recipe 1M [11]. The performance metrics we report include median rank (MedR) and recall rate at top K (R@K, K = 1, 5, 10) for all the retrievals. We measure the cost by the avg training time per epoch and convergence speed. We want the model to be easy and fast to train because we want our solution to be friendly for other developers to iterate and build upon. Table 3 demonstrates that the hierarchical transformer model in [6] is converging faster: it achieves MedR = 16.2 at second Epoch while the LSTM model needs to complete 20 epochs to achieve a better performance (MedR = 14.15, at Epoch 19 the MedR is 17.9). But it takes the transformer model 9.5 hours to complete just one epoch, which is very costly for most mobile application development. We need a more efficient network architecture. That's why we decide to move to MobileNets [4], a family of small, low-latency, low-power neural network models parameterized to meet the resource constraints on mobile device or embedded applications. Specifically, we use MobileNet-v2
Experiments and Results
Computation Resource
Most of our experiments were conducted on Google Cloud Platform. We created a Google Cloud Storage bucket to restore training data and created a Deep Learning VM instance with one NVIDIA Tesla K80 GPU and 2 CPUs to train the model. We followed the getting started guide to set up VM instance.
Reproduce Previous Works
We started with reproducing results from LSTM model [6] and hierarchical transformer model [8]. It took us some efforts to set up the environment correctly. However, we quickly realized that training from the scratch is too slow and too expensive for us. As Table 3 shows, LSTM model consumes 1 epoch data in 2.5 hours but converges slowly and hierarchical transformer converges fast but is slow in data consumption. Training a model on par with the best performance on the paper would charge us more than one thousand dollars. Hence we stopped reproducing previous works from scratch and decided to utilize the best checkpoint provided by authors.
Build Recipe Library
As we explained above, we adopted the recipe encoder and best checkpoint from [8]. As Figure 1 shows, recipes from Recipe1M were passed to recipe encoder to extract embeddings. A embedding vector has a dimension of 1 × 2014. All recipe embeddings were stacked into a N × 1024 array, where N is the number of recipes. Using embedding id as key, corresponding recipes were stored in a recipe dictionary . A recipe library consists of a recipe embedding array and a recipe dictionary. We built two in-dividual recipe libraries based on the training and validation partition in Recipe1M dataset. The training library was used to train image encoder and the validation library was used for recipe prediction. As Table 1 shows, the number of recipes in training partition is about five times of those in validation partition. To reduce memory cost, validation library was adopted for image query. We also provided a training library option for users.
Train Image Encoder
As we explained in Section 2, we used MobileNet-V2 as image encoder backbone. Pre-computed recipe embeddings served as ground true labels. In this way, image encoder was forced to learn projecting a image into an embedding close to recipe embedding ( Figure 2). MobileNet-V2 was initialized with the parameters pre-trained on ImageNet. In this way, we succeeded to train a converged image encoder in 10 hours with MedR=2.0 (3).
We used bi-directional triplet loss function for model training. We only briefly introduce the loss function here. Readers can check [8] for more details. A triplet loss function can be written as follows:
L cos (a, p, n) = max(0, c(a, n) − c(a, p) + m (1) where a, p, and n refer to the anchor, positive, and negative samples, c(.) is the cosine similarity metric, and m is the margin (m=0.3 in our work). A bi-directional triplet loss function on feature sets a and b can be defined as:
L bi (i, j) = L cos (a n=i , b n=i , b n=j ) + L cos (b n=i , a n=i , a n=j )(2)
Predict Recipe From Image
As Figure 3 shows, the process is quite straightforward: 1) convert query images into image embedding with image encoder, 2) compute the cosine similarity with recipe embedding array, 3) retrieve top k (here we use k=5) recipe ids, and 4)query top k recipes from recipe dictionary with recipe ids. We list some image query results in Table 4. The retrieved recipe is very close to what we queried.
Expand Recipe Library
We created a RecipeSnap class integrating all important modules, such as image encoder loading, recipe encoder loading, image preprocessing, recipe preprocessing, recipe prediction, and recipe library update. With this class object, users can easily query recipes from certain image or add new recipe to the recipe library.
Collecting all recipes was an impossible mission. Our model makes this task feasible. Smart phone is accessible to almost everyone. If our model can be deployed on smart phones, it would be very easy for everyone to contribute new recipes to our recipe library.
Discussion
People are looking for easier ways to keep cooking fit into their busy schedules in today's fast-paced life. As a result, many meal-planning services are booming. Those services try to simplify the cooking by planning everything for their customers ahead. But people who use these services may get stuck with things they don't like. We believe it's essential to keep the joy for people to discover attractive dishes by themselves and let them feel the fulfillment from cooking beautiful and tasty food. So we want to build this application where people can customize their recipe library and easily query the recipe and ingredients from their phone with any dish images they like.
In the future, we plan add more features to advance this purpose. For example, people usually face the dilemma of seeing a few ingredients left in the fridge and wondering how they can utilize them. An ingredients-to-recipe func-tion can be quite applicable for this use case. Beyond that, people may want the feature to rank or filter the queried recipe, given their nutrition and budget requirements. We can also add a feedback module to let the application learn the user's preference so that the query results can be more intelligent. To sum up, many interesting features can be added to what we have built now, and we believe they will benefit foodies who are busy but still pursue the fun of cooking.
Conclusion
In this paper, we introduce RecipeSnap, a lightweight image-to-recipe retrieve model developed to handle the the problem of automation for recognition of images of cooking dishes and finding the relevant recipes containing information including ingredient lists, instructions etc.. The experiments conducted using multiple existing solutions and RecipeSnap imply that our model design help reduce memory cost and computational expense for model training and implementing while obtain a compatible performance comparing to current state-of-art neural networks. The use of the pre-trained recipe encoder as part of the recipe encoder and the cooked recipe library collected using recipes from a large scaled dataset Recipe1M make RecipeSnap easier and faster to train and deploy. We adopt MobileNet-V2 instead of other deeper complex networks as the backbone for image encoder, which makes RecipeSnap suitable to portable devices and platforms like smart phones and tablets.
This model can easily be developed into applications for smart phone and tablets where potential users can customize their recipe library and easily query the recipes from their phones with any dish images they are interested at. In the future, we plan to keep contribute to this tool by exploring more meaningful features to advance this purpose.
Work Division
Individual contributions are summarized in Table 5. Yifan Xu Implementation, Analysis and Reporting Explore and study related works in the specific field. Experiment with alternative ways of model implementation and deployment. Implement recipe preprocessing module. Conduct unit testing with model code. Contribute to the introduction and conclusion sections of the report.
Table 1 .
1Dataset SizesPartition
Num of Recipes Num of Images
Training
720,639
619,508
Validation
155,036
133,860
Testing
154,045
134,338
Total
1,029,720
887,706
Table 2 .
2Comparison of model size and complexity
Table 3 .
3Comparison between two SOTA modelsModel
Performance
Cost (one NVIDIA K80 GPU)
MedR Recall@1 Recall @5 Recall @10 Avg. Training Time per Epoch Epoch
LSTM [6]
5.1
0.24
0.52
0.64
-
-
Hierarchical Transformer [8]
1.0
0.60
0.876
0.929
-
-
LSTM (ours)
14.15
0.1165
0.3149
0.4409
2.5 hours
20
Hierarchical Transformer (ours)
16.20
0.0969
0.2870
0.4063
9 hours
2
MobileNet-v2 + Transformer (ours)
2
0.4123
0.7187
0.8210
3.5 hours
3
[10] in our final model architecture. MobileNet-v2 pushes
the state of the art in accuracy vs. latency trade-off for mo-
bile visual recognition tasks, which is a perfect fit for this
project. Table 2 shows that MobileNe-v2 significantly re-
duces the number of parameters and Floating Point Oper-
ations Per Second (FLOPS) compared to the main-stream
networks with regular convolutions.
Table 4 .
4Image2Recipe Retrieval ExampleQuery Image
True Ingredients
Retrieved Title
Retrieved Ingredients@1
Title: Osso Buco
4 ounces pancetta, diced into 1/4 inch cubes (see recipe note)
2 1/2 to 3 pounds veal shanks (4 to 6 pieces 2 to 3 inches thick)
1/2 cup diced carrot (1/4-inch dice)
1/2 cup diced celery (1/4 inch dice)
1 medium onion (1/4 inch dice)
2 tablespoons chopped garlic (about 4 cloves)
3 to 4 sprigs fresh thyme (or 1 teaspoon dried)
1 cup dry white wine
1 to 2 cups chicken or veal stock
Flour for dusting the meat before browning
Salt and pepper
1. braised veal shanks
2. chipotle lamb chops
3. beef short ribs in chipotle and green
chili sauce
4. haitian pork griot
2 cups veal demiglace (16 fl oz)*
4 (1-lb) meaty cross-cut veal shanks (osso buco), each tied with
kitchen string 1 teaspoon salt
1/4 teaspoon black pepper
1/4 cup all-purpose flour
1 1/2 tablespoons olive oil
3 tablespoons unsalted butter
2 medium onions, cut into 1/4-inch dice (2 cups)
2 medium carrots, cut into 1/4-inch dice (1 cup)
2 celery ribs, cut into 1/4-inch dice (1 cup)
1 garlic clove, finely chopped
1 1/2 cups dry red wine
1 (28-oz) can whole tomatoes in juice, drained and coarsely chopped
1 turkish or 1/2 california bay leaf (preferably fresh)
2 teaspoons chopped fresh flat-leaf parsley
1 teaspoon finely grated fresh orange zest
3/4 teaspoon finely chopped fresh rosemary
3/4 teaspoon finely chopped fresh thyme
accompaniment: wild mushroom risotto
Title: Iced Matcha Bubble Tea
Ingredients
1 teaspoon matcha green tea powder
2 Tablespoons hot water
2 Tablespoons honey, or other sweetener, to taste
1 cup (240 ml) Almond Milk
½ cup (120 g) ice cubes, plus more for serving
¼ cup (50 g) cooked tapioca pearls
1. avocado aperitif
2. super green juice
3. pineapple mojito gelatin shot
4. kiwi, apple and mint juice recipe
5. green smoothie with spinach and
greek yogurt
Ingredient:
1 medium avocados
1 large cucumbers
1 each lemon juice of
1/2 cup parsley leaves fresh, chopped
2 cups ice cubes crushed
1 slices lemon or cucumber peel
Table 5 .
5Contributions of team members. Propose the idea of lightweight model. Build recipe library. Train image encoder with MobileNet V2 backbone. Major contributor to RecipeSnap code. Contribute to the writing of abstract, experiments and results, and conclusion section.Yue YinModel implementation and experimentation Propose extension ideas and writing Run experimentation to train and compare SOTA models. Implement the image pre-processing module. Propose ideas for future extension. Contribute to the writing of Approach and Discussion.Individual
Contributed Aspects
Details
Jianfa Chen
Propose idea, model training,
implementation, and writing
https://github.com/jianfa/RecipeSnap-a-lightweight-image-to-recipemodel.git
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Harmonious attention network for person re-identification. Wei Li, Xiatian Zhu, Shaogang Gong, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionWei Li, Xiatian Zhu, and Shaogang Gong. Harmonious attention network for person re-identification. In Proceed- ings of the IEEE conference on computer vision and pattern recognition, pages 2285-2294, 2018. 3
Recipe1m+: A dataset for learning cross-modal embeddings for cooking recipes and food images. Javier Marín, Aritro Biswas, Ferda Ofli, Nicholas Hynes, Amaia Salvador, Yusuf Aytar, Ingmar Weber, Antonio Torralba, IEEE Trans. Pattern Anal. Mach. Intell. 431Javier Marín, Aritro Biswas, Ferda Ofli, Nicholas Hynes, Amaia Salvador, Yusuf Aytar, Ingmar Weber, and Antonio Torralba. Recipe1m+: A dataset for learning cross-modal embeddings for cooking recipes and food images. IEEE Trans. Pattern Anal. Mach. Intell., 43(1):187-203, 2021. 1, 2, 3, 4
Inverse cooking: Recipe generation from food images. Amaia Salvador, Michal Drozdzal, Xavier Giró-I-Nieto, Adriana Romero, abs/1812.06164CoRRAmaia Salvador, Michal Drozdzal, Xavier Giró-i-Nieto, and Adriana Romero. Inverse cooking: Recipe generation from food images. CoRR, abs/1812.06164, 2018. 2
Revamping cross-modal recipe retrieval with hierarchical transformers and self-supervised learning. Amaia Salvador, Erhan Gundogdu, Loris Bazzani, Michael Donoser, IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual. Computer Vision Foundation / IEEE, 2021. 1, 2, 3, 4Amaia Salvador, Erhan Gundogdu, Loris Bazzani, and Michael Donoser. Revamping cross-modal recipe retrieval with hierarchical transformers and self-supervised learning. In IEEE Conference on Computer Vision and Pattern Recog- nition, CVPR 2021, virtual, June 19-25, 2021, pages 15475- 15484. Computer Vision Foundation / IEEE, 2021. 1, 2, 3, 4
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Mobilenetv2: Inverted residuals and linear bottlenecks. Mark Sandler, Andrew G Howard, Menglong Zhu, Andrey Zhmoginov, Liang-Chieh Chen, 2018 IEEE Conference on Computer Vision and Pattern Recognition. Salt Lake City, UT, USAMark Sandler, Andrew G. Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018, pages 4510- 4520. Computer Vision Foundation / IEEE Computer Soci- ety, 2018. 4
Learning cross-modal embeddings with adversarial networks for cooking recipes and food images. Hao Wang, Doyen Sahoo, Chenghao Liu, Ee-Peng Lim, Steven C H Hoi, IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019. Long Beach, CA, USA23Computer Vision Foundation / IEEEHao Wang, Doyen Sahoo, Chenghao Liu, Ee-Peng Lim, and Steven C. H. Hoi. Learning cross-modal embeddings with adversarial networks for cooking recipes and food images. In IEEE Conference on Computer Vision and Pattern Recogni- tion, CVPR 2019, Long Beach, CA, USA, June 16-20, 2019, pages 11572-11581. Computer Vision Foundation / IEEE, 2019. 2, 3
| [
"https://github.com/jianfa/RecipeSnap-a-lightweight-image-to-recipemodel.git"
] |
[
"Constraining new physics in b → c ν transitions",
"Constraining new physics in b → c ν transitions"
] | [
"Martin Jung [email protected] \nExcellence Cluster Universe\nBoltzmannstr. 285748GarchingGermany\n",
"David M Straub [email protected] \nExcellence Cluster Universe\nBoltzmannstr. 285748GarchingGermany\n"
] | [
"Excellence Cluster Universe\nBoltzmannstr. 285748GarchingGermany",
"Excellence Cluster Universe\nBoltzmannstr. 285748GarchingGermany"
] | [] | B decays proceeding via b → c ν transitions with = e or µ are tree-level processes in the Standard Model. They are used to measure the CKM element V cb , as such forming an important ingredient in the determination of e.g. the unitarity triangle; hence the question to which extent they can be affected by new physics contributions is important, specifically given the long-standing tension between V cb determinations from inclusive and exclusive decays and the significant hints for lepton flavour universality violation in b → cτ ν and b → s decays. We perform a comprehensive model-independent analysis of new physics in b → c ν, considering all combinations of scalar, vector and tensor interactions occuring in single-mediator scenarios. We include for the first time differential distributions of B → D * ν angular observables for this purpose. We show that these are valuable in constraining non-standard interactions. Specifically, the zero-recoil endpoint of the B → D ν spectrum is extremely sensitive to scalar currents, while the maximum-recoil endpoint of the B → D * ν spectrum with transversely polarized D * is extremely sensitive to tensor currents. We also quantify the room for e-µ universality violation in b → c ν transitions, predicted by some models suggested to solve the b → cτ ν anomalies, from a global fit to B → D ν and B → D * ν for the first time. Specific new physics models, corresponding to all possible tree-level mediators, are also discussed. As a side effect, we present V cb determinations from exclusive B decays, both with frequentist and Bayesian statistics, leading to compatible results. The entire numerical analysis is based on open source code, allowing it to be easily adapted once new data or new form factors become available.arXiv:1801.01112v3 [hep-ph] 19 Dec 2018In the context of the Standard Model (SM), the element V cb of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be determined in various ways:• from a global fit including e.g. meson-antimeson mixing observables [1, 2],, where X c is any charmed hadronic final state,• from exclusive b → c ν transitions, specifically B → D ν and B → D * ν ( = e, µ) decays (see e.g.[4] for a recent review).When considering the SM as the low-energy limit of a more fundamental theory of "new physics" (NP), these determinations are not applicable model-independently in general. Specifically, flavour-changing neutral current processes like meson-antimeson mixing could easily be affected by NP, invalidating the global fit. In this case, a potential disagreement between the global fit and the tree-level determinations from semi-leptonic B decays can signal the presence of NP. However, even the tree-level processes could in principle be significantly affected by NP. This possibility has been considered in the past in particular in view of the long-standing tensions between V cb from different decay channels. Clearly, these tensions could be due to statistical fluctuations or underestimated theoretical uncertainties. While the inclusive decay can be computed to high precision in an expansion in α s and 1/m c,b , see e.g.[3] for a recent overview, the exclusive decays require the knowledge of hadronic form factors. Lattice QCD (LQCD) calculations of B → D form factors are now available also at non-zero recoil[5,6]for the relevant SM operators, but for B → D * a full lattice calculation at non-zero recoil is still lacking[7,8]. In both cases, the dependence on the chosen form factor parametrization has received considerable interest recently[6,[9][10][11][12][13][14][15][16], indicating that at least part of the tension between inclusive and exclusive decays might stem from an underestimation of the systematic or theoretical uncertainties. Nevertheless, entertaining the possibility of NP as the origin of the tensions between SM and data is certainly worthwhile, since these analyses are suggestive, but do not provide proof that the form factor parametrization is indeed the reason for the observed tension. Additional interest in NP modifying b → c ν with light leptons was generated by the signficant deviations from SM expectations in decays based on the b → cτ ν transition, including in B → Dτ ν, B → D * τ ν, and, most recently, B c → J/ψτ ν[17][18][19][20][21][22][23]. Many NP models have been proposed to explain these tensions. Depending on the flavour structure of the model, b → c ν with light leptons can be affected as well. This is true in particular for models explaining simultaneously the apparent deviations from lepton flavour universality (LFU) in B → K and B → K * , with = e or µ (see e.g.[24][25][26][27]). Hence an important question is to what extent LFU is tested in b → c ν, independent of any tension between b → c ν data and SM predictions.Recent analyses of NP in b → c ν include[28][29][30][31][32][33][34]. Most of them have focused on individual operators or specific subsets and only used experimental information from the measurements of total branching ratios in exclusive decays. Recently however, differential distributions of exclusive measurements, including angular observables in B → D * ν, have been released by the BaBar and Belle collaborations[35][36][37][38]. As will be shown in section 5, these data contain valuable information that allows to independently constrain different types of non-standard | 10.1007/jhep01(2019)009 | [
"https://arxiv.org/pdf/1801.01112v3.pdf"
] | 84,839,140 | 1801.01112 | 4658645223236a3eb64b6488d759106a13972c7f |
Constraining new physics in b → c ν transitions
19 Dec 2018
Martin Jung [email protected]
Excellence Cluster Universe
Boltzmannstr. 285748GarchingGermany
David M Straub [email protected]
Excellence Cluster Universe
Boltzmannstr. 285748GarchingGermany
Constraining new physics in b → c ν transitions
19 Dec 2018
B decays proceeding via b → c ν transitions with = e or µ are tree-level processes in the Standard Model. They are used to measure the CKM element V cb , as such forming an important ingredient in the determination of e.g. the unitarity triangle; hence the question to which extent they can be affected by new physics contributions is important, specifically given the long-standing tension between V cb determinations from inclusive and exclusive decays and the significant hints for lepton flavour universality violation in b → cτ ν and b → s decays. We perform a comprehensive model-independent analysis of new physics in b → c ν, considering all combinations of scalar, vector and tensor interactions occuring in single-mediator scenarios. We include for the first time differential distributions of B → D * ν angular observables for this purpose. We show that these are valuable in constraining non-standard interactions. Specifically, the zero-recoil endpoint of the B → D ν spectrum is extremely sensitive to scalar currents, while the maximum-recoil endpoint of the B → D * ν spectrum with transversely polarized D * is extremely sensitive to tensor currents. We also quantify the room for e-µ universality violation in b → c ν transitions, predicted by some models suggested to solve the b → cτ ν anomalies, from a global fit to B → D ν and B → D * ν for the first time. Specific new physics models, corresponding to all possible tree-level mediators, are also discussed. As a side effect, we present V cb determinations from exclusive B decays, both with frequentist and Bayesian statistics, leading to compatible results. The entire numerical analysis is based on open source code, allowing it to be easily adapted once new data or new form factors become available.arXiv:1801.01112v3 [hep-ph] 19 Dec 2018In the context of the Standard Model (SM), the element V cb of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be determined in various ways:• from a global fit including e.g. meson-antimeson mixing observables [1, 2],, where X c is any charmed hadronic final state,• from exclusive b → c ν transitions, specifically B → D ν and B → D * ν ( = e, µ) decays (see e.g.[4] for a recent review).When considering the SM as the low-energy limit of a more fundamental theory of "new physics" (NP), these determinations are not applicable model-independently in general. Specifically, flavour-changing neutral current processes like meson-antimeson mixing could easily be affected by NP, invalidating the global fit. In this case, a potential disagreement between the global fit and the tree-level determinations from semi-leptonic B decays can signal the presence of NP. However, even the tree-level processes could in principle be significantly affected by NP. This possibility has been considered in the past in particular in view of the long-standing tensions between V cb from different decay channels. Clearly, these tensions could be due to statistical fluctuations or underestimated theoretical uncertainties. While the inclusive decay can be computed to high precision in an expansion in α s and 1/m c,b , see e.g.[3] for a recent overview, the exclusive decays require the knowledge of hadronic form factors. Lattice QCD (LQCD) calculations of B → D form factors are now available also at non-zero recoil[5,6]for the relevant SM operators, but for B → D * a full lattice calculation at non-zero recoil is still lacking[7,8]. In both cases, the dependence on the chosen form factor parametrization has received considerable interest recently[6,[9][10][11][12][13][14][15][16], indicating that at least part of the tension between inclusive and exclusive decays might stem from an underestimation of the systematic or theoretical uncertainties. Nevertheless, entertaining the possibility of NP as the origin of the tensions between SM and data is certainly worthwhile, since these analyses are suggestive, but do not provide proof that the form factor parametrization is indeed the reason for the observed tension. Additional interest in NP modifying b → c ν with light leptons was generated by the signficant deviations from SM expectations in decays based on the b → cτ ν transition, including in B → Dτ ν, B → D * τ ν, and, most recently, B c → J/ψτ ν[17][18][19][20][21][22][23]. Many NP models have been proposed to explain these tensions. Depending on the flavour structure of the model, b → c ν with light leptons can be affected as well. This is true in particular for models explaining simultaneously the apparent deviations from lepton flavour universality (LFU) in B → K and B → K * , with = e or µ (see e.g.[24][25][26][27]). Hence an important question is to what extent LFU is tested in b → c ν, independent of any tension between b → c ν data and SM predictions.Recent analyses of NP in b → c ν include[28][29][30][31][32][33][34]. Most of them have focused on individual operators or specific subsets and only used experimental information from the measurements of total branching ratios in exclusive decays. Recently however, differential distributions of exclusive measurements, including angular observables in B → D * ν, have been released by the BaBar and Belle collaborations[35][36][37][38]. As will be shown in section 5, these data contain valuable information that allows to independently constrain different types of non-standard
Introduction
interactions in b → c ν. The main aim of this paper is to perform a comprehensive modelindependent analysis of all possible types of NP effects in b → c ν, making use of the wealth of experimental data. This paper is organized as follows: In section 2, the effective Hamiltonian for b → c ν is defined and relations among the operators implied by SM gauge invariance are discussed. In section 3, we discuss our treatment of B → D and B → D * form factors that are crucial ingredients in the analysis of exclusive b → c ν decays. In section 4, we perform fits to V cb from B → D ν and B → D * ν. Reproducing the values in the literature, this step is useful as a crosscheck of our numerics. We also perform the analysis with frequentist and Bayesian statistics, explicitly demonstrating their agreement. In section 5, we perform the NP analyses, starting with a discussion of NP models with tree-level mediators and their characteristic patterns of Wilson coefficients, and subsequently discussing each of the relevant operator combinations. Section 6 contains our conclusions.
An important feature of our analysis is that it is entirely based on open-source code. We have implemented all observables of interest as well as our predictions for B → D and B → D * form factors in the flavio flavour physics package [39]. This has three benefits: First, it makes our analysis transparent and reproducible. Second, it allows anyone to update the best-fit values of V cb or the allowed ranges for the Wilson coefficients when new experimental data or new lattice form factor computations become available. Third, it easily allows to study the viability of more involved NP models with multiple Wilson coefficients, that cannot be easily visualized in two-dimensional plots. Additionally, to corroborate the reliability of our results, we have obtained all numerical results with a completely independent Mathematica code.
Effective Hamiltonian
The effective Hamiltonian for b → c ν transitions can be written as 1
H b→c ν eff = 4G F √ 2 V cb O V L δ + i C i O i + h.c. ,(1)
where the sum runs over the following operators:
O V L = (c L γ µ b L )(¯ L γ µ ν L ) , O S R = (c L b R )(¯ R ν L ) , O T = (c R σ µν b L )(¯ R σ µν ν L ) , O V R = (c R γ µ b R )(¯ L γ µ ν L ) , O S L = (c R b L )(¯ R ν L ) ,(2)
with in general charged-leptonand neutrino-flavour-dependent coefficients. Since we are focusing on decays with light leptons in the final state, we only consider = e or µ, but allow for = e, µ, τ , which cannot be distinguished experimentally. We have defined the coefficients C i in (2) such that they vanish in the SM. A lepton-flavour universal and diagonal NP effect in C V L can always be absorbed by a shift in V cb , since V cb is a free parameter in the SM and presently not meaningfully constrained by CKM unitarity. In the following, we will use the shorthands
C i ≡ C i ,Ṽ cb = V cb (1 + C V L ) ,C i = C i /(1 + C V L ) ,(3)
where appropriate.
The operators in the effective Hamiltonian arise from more funadamental interactions at or above the electroweak scale. The available high-energy data from LHC indicate the existence of another energy gap between the electroweak scale and that of NP. In such a scenario interactions beyond the SM can be cast into another effective theory, with operators symmetric under the full SM gauge group. For linearly realized electroweak symmetry breaking this effective theory is called Standard Model effectivie field theory (SMEFT), the operators of which can be ordered in terms of their mass dimension, with those at dimension six giving the dominant contributions here [41,42]. The tree-level matching of SMEFT operators onto the effective Hamiltonian (2) reads [43][44][45][46]
C V L = −v 2 V ci V cb C (3) i3 lq + v 2 V ci V cb C (3)i3 φq δ , C V R = v 2 2 C 23 φud δ ,(4)C S R = − v 2 2 V ci V cb C 3i ledq , C S L = − v 2 2 V ci V cb C (1) 3i lequ ,(5)C T = − v 2 2 V ci V cb C (3) 3i lequ ,(6)
where the definition of the SMEFT operators can be found in [42] 2 and we are employing a weak basis for SMEFT where down-type and charged-lepton mass matrices are diagonal. An important prediction of this framework is that the Wilson coefficient C V R is lepton flavour universal and diagonal [45,47]. A deviation from this prediction would hence indicate a nonlinear realization of electroweak symmetry breaking [45]. Presently such a deviation is not observed, however, and we will use C V R ≡ C V R δ in the remainder of this paper. Another implication of SMEFT is that that the Wilson coefficients C 3i ledq , that give rise to C S R , also generate neutral-current operators of the form (q L b R )(¯ R L ), where q = d, s, b. For q = s or d, these operators are constrained very strongly by the leptonic decays B d,s → , that are forbidden for = and strongly helicity suppressed for = in the SM. From existing bounds and measurements, we find that the SMEFT Wilson coefficients C 31 ledq and C 32 ledq can induce effects in C S R at most at the per mil level, which would not lead to any visible effects in b → c ν at the current level of precision. However, sizable effects in C S R cannot be excluded, since the coefficients C 33 ledq contributing to the sum in (5) only generate the flavour-conserving operators (b L b R )(¯ R L ), that are allowed to be sizable. 3 Finally, the availability of the full anomalous dimension matrix for SMEFT dimension-six operators [49][50][51] allows for the prediction of operator mixing due to electroweak renormalization effects; this will be discussed briefly at the end of section 5.1.
B → D ( * ) form factors
The hadronic form factors of the B → D ( * ) transitions are crucial both for the determination of the CKM element V cb in the SM and for constraining NP contributions to b → c ν. An important difference between the two scenarios is that in the SM V cb only changes the overall normalization of the rates, but does not modify the shapes of differential distributions. NP contributions on the other hand can modify these shapes and can also involve additional form 2 We use a normalization LSMEFT = i CiOi, i.e. dimension-6 operators have dimensions of inverse mass squared. 3 Note that these operators do not contribute to (and thus are not constrained by) leptonic decays of Υ(nS) [48].
factors, in particular tensor form factors. Since our main interest is constraining NP in b → c ν, we want to use as much information on the form factors from theory as possible, while at the same time remaining conservative enough not to introduce ficticious tensions with the precise experimental data due to too rigid parametrizations. We therefore use information from four complementary methods to determine the B → D ( * ) form factors:
• LQCD. We use all available unquenched LQCD calculations of B → D ( * ) form factors.
The B → D vector and scalar form factors have been computed by the HPQCD [5] and the Fermilab/MILC collaborations [6] at several values of q 2 , constraining the shape of these form factors in addition to their normalizations. The FLAG collaboration has performed an average of these two computations, fitted to the BCL parametrization [12]. For the B → D * form factors, so far only calculations at zero hadronic recoil have been reported [7,8]; we use their average calculated in [8].
• QCD light-cone sum rules allow to compute the form factors in the region of large hadronic recoil, depending on B meson light-cone distribution amplitudes as non-perturbative input [52]. This method is complementary to LQCD, being valid in the opposite kinematic limit. We use the form factor values and ratios obtained in Ref. [52] at 4 q 2 = 0, and extract from their values given for ρ 2 D,D * two more, correlated pseudo-datapoints at w(q 2 ) = 1.3.
• Heavy Quark Symmetry and Heavy Quark Effective Theory (HQET). Treating both the bottom and charm quark as heavy compared to a typical scale of QCD interactions, QCD exhibits a symmetry among the heavy quarks [53,54]. As a consequence, all B → D ( * ) form factors either vanish or reduce to a single function -the leading Isgur-Wise function -in the infinite mass limit [55]. Perturbative QCD and power corrections to this limit are partly calculable [13,[56][57][58][59][60][61][62][63][64], to be discussed below.
• By crossing symmetry, the form factors describing the semi-leptonic transition also describe the pair production of mesons. Unitarity can then be used to impose constraints on the form factors. Taking into account contributions from other two-body channels with the right quantum numbers leads via a conservative application of HQET to the strong unitarity constraints [65,66]. We employ the updated bounds given in [9] for B → D and the simplified bounds derived in [11] for B → D * .
We proceed by using the HQET parametrization of all B → D ( * ) form factors, including corrections of O(α s , Λ QCD /m c,b ), in the notation of [13], with two differences:
1. Instead of using the CLN relation between slope and curvature of the leading Isgur-Wise function, we include both as independent parameters in the fit.
2. The treatment of higher-order corrections of O(Λ 2 QCD /m 2 c ) has recently been shown to have a significant effect on the extraction of V cb in the SM, see the discussions in [9][10][11][12][13][14][15][16]. We make these corrections explicit by including in addition to the subleading Isgur-Wise functions at order Λ QCD /m c,b corrections of order Λ 2 QCD /m 2 c in those HQET form factors that are protected from O(Λ QCD /m c ) corrections. 5 From comparison of the HQET predictions at O(α s , Λ QCD /m b,c ), using the three-point sum rule results for the subleading 4 The kinematic variable w is related to q 2 as w = (m 2 B + m 2 D ( * ) − q 2 )/(2mBm D ( * ) ). 5 Note that two of these corrections are implicitly included in [13] when the normalizations of the B → D ( * ) form factors hA 1 ,+ are decoupled from their HQET values.
Isgur-Wise functions [60][61][62] with LQCD results, we observe a shift of about −10% from these corrections in h A 1 (1), while the corrections in h + (1) are only a few per cent. For h T 1 (1) we allow for an independent correction of 10% which is not constrained by lattice data. Note that these corrections are in principle obtainable from LQCD in all form factors, however, so far only results for those appearing in the SM are available.
We then perform Bayesian and frequentist fits of this parametrization to pseudo data points corresponding to the described inputs. The result is a posterior probability distribution or profile likelihood for all the form factor parameters, respectively. These can be interpreted as theory predictions for all B → D ( * ) form factors in the entire (semi-leptonic) kinematic range. This theory prediction is then used in our numerical analysis as a prior on (in Bayesian fits) or a pseudo-measurement of (in frequentist fits) the form factor parameters.
V cb from exclusive decays in the Standard Model
Fits for the CKM element V cb from the exclusive decays B → D ν and B → D * ν measured at the B factories BaBar and Belle have already been performed in the literature (see e.g. [6,[9][10][11][12][13][14][15][16] for recent fits). Here, we repeat this exercise to define our assumptions on form factors and our experimental input. Furthermore, all of our fits are reproducible using open source code, allowing them to be adapted or modified with different theoretical or experimental inputs. We perform fits to B → D ν and B → D * ν decays, where the theoretical uncertainties are dominated by the hadronic form factors. To study the impact of different statistical treatments of these "nuisance" parameters, we consider three different fits:
• A frequentist fit where the theoretical knowledge of form factors is treated as a pseudomeasurement and the individual parameters are profiled over.
• A Bayesian fit where the theoretical knowledge of form factors is treated as a prior probability distribution and the individual parameters are marginalized over.
• A "fast fit" where the theoretical uncertainty on each bin is determined by varying the form factor parameters according to the theoretical constraints and is added in quadrature with the experimental uncertainties.
4.1. V cb from B → D ν
The BaBar collaboration has measured the differential branching ratio of B → D ν, reconstructed with hadronic tagging [35], in ten bins, averaged over electrons and muons as well as charged and neutral B decays. Since only statistical uncertainties are given, we follow [9] and add a fully correlated systematic uncertainty of 6.7% on the rate. To avoid a bias towards lower values of V cb caused by underfluctuations in individual bins ("d'Agostini bias" [71]), we always treat relative systematic errors as relative to the SM predictions rather than the experimental central values. In addition to this differential measurement, we also include a BaBar measurement of the total branching ratio from a global fit, split by electrons and muons [67], that is statistically independent of the former; following Ref. [11], we assume the measurement of the total branching ratio to be unaffected by the form factor parametrization. We take into Table 1: Experimental analyses considered. The analyses labeled by B → D ( * ) ν do not differentiate between the lepton species and are hence not used when analyzing scenarios with non-universal coefficients.
B → D * (e, µ)ν BR BaBar global fit [67] B → D * ν BR BaBar hadronic tag [68] B → D * ν BR BaBar untagged B 0 [69] B → D * ν BR BaBar untagged B ± [70] B → D * (e, µ)ν dΓ L,T /dw Belle untagged [36] B → D * ν dΓ/d(w, cos θ V , cos θ l , φ) Belle hadronic tag [38]
account the significant correlation with the D * modes extracted in the same analysis. Following HFLAV [72], we apply a rescaling of −3.7 % to the published branching ratio to account for updated D branching ratios.
We note that we cannot use the global HFLAV average of the B → D ν branching ratio because
• it contains older measurements that assumed a particular form factor parametrization and would be inconsistent to use in a NP analysis,
• it contains measurements from the anaylses that we include in binned form, such that including it would amount to double-counting,
• it only considers the average of the electronic and muonic branching ratios, so we cannot use it for the lepton-flavour dependent NP analysis.
The Belle collaboration has measured the differential branching ratios separately for electrons and muons as well as charged and neutral B decays in ten bins each [37], specifying the full correlation matrix. This measurement does not rely on a specific form factor parametrization. The combined fit to Belle and BaBar data with the three different statistical approaches, using the form factors described in section 3, yields
V B→D ν cb = (3.96 ± 0.09) × 10 −2 (frequentist fit),(7)V B→D ν cb = (3.96 ± 0.09) × 10 −2 (Bayesian fit),(8)V B→D ν cb = (3.96 ± 0.09) × 10 −2 (fast fit),(9)
where the errors in the frequentist fit refer to a change by 0.5 in the profile likelihood and in the Bayesian case correspond to the highest posterior density interval with 68.3% credibility. The agreement of the central value between the frequentist and the Bayesian fit is not surprising as the minimum of the two likelihoods coincides (the prior probability for theory parameters has the same mathematical form as the "pseudo measurements" used in the frequentist fit). That We observe good compatibility with other recent extractions from this mode [9,13,16]. A direct comparison is not possible, since neither the form factor parametrizations nor the data sets used are identical. The inclusion of additional data in our case is responsible for the slightly smaller uncertainties. Note that the inclusion of the measurement [67] shifts V cb to smaller values compared to [9].
In our numerical analysis of new physics effects in section 5, when allowing for lepton flavour non-universal effects, we only use measurements where electron and muon samples are separated, since the generally different but unknown electron and muon efficiencies prohibit a consistent interpretation of the combined measurements in such scenarios. It is instructive to extract the value of V cb only from these subsets of measurements. Using the frequentist approach, we find
V B→Deν cb = (4.00 +0.07 −0.17 ) × 10 −2 ,(10)V B→Dµν cb = (3.96 +0.11 −0.10 ) × 10 −2 .(11)
We observe consistency among these values and with the global fit, albeit with larger uncertainties.
4.2. V cb from B → D * ν
Since the D * is a vector meson and further decays to e.g. Dπ, a four-fold differential decay distribution in three angles and w can be measured. Belle has recently published an analysis with one-dimensional distributions in all four kinematic quantities with full error correlations, based on hadronic tagging [38]. An earlier (and statistically independent) untagged analysis by Belle [36] considers the w-differential branching ratios into longitudinally or transversely polarized D * , that can be extracted from the angular distribution. Since the error correlations are not publicly available, we simply assume the statistical uncertainties to be fully uncorrelated and the systematic ones to be fully correlated. In addition, we rescale the systematic uncertainties, that are given as relative uncertainties with respect to the measured central values, into relative uncertainties with respect to the SM prediction instead, again to avoid the D'Agostini bias mentioned before. In addition to the differential measurements, we include four measurements of the total branching ratio by BaBar, listed in table 1, which are statistically independent to a good approximation [73]. As mentioned in section 4.1, we do however take into account the significant correlation between the B → D * (e, µ)ν and B → D(e, µ)ν measurements of the BaBar "global fit". As in the case of B → D ν, we cannot use the HFLAV average for the B → D * ν branching ratio, but we apply the same rescalings as HFLAV to account e.g. for changes in D * branching ratios. The central value of the BaBar global fit is hardly modified, but the other three branching ratio measurements are reduced by 5-6% compared to the published values.
The combined fit to these data, using our form factor parametrization with the constraints discussed in Sec. 3, yields
V B→D * ν cb = (3.90 ± 0.07) × 10 −2 (frequentist fit),(12)V B→D * ν cb = (3.90 ± 0.07) × 10 −2 (Bayesian fit),(13)V B→D * ν cb = (3.90 ± 0.07) × 10 −2 (fast fit).(14)
Again, we observe excellent agreement of the three different approaches and the same comments as in section 4.1 apply. We conclude that the extraction of V cb from exclusive decays does not depend in a relevant way on the statistical approach taken. The extracted values are also comparable to those in the recent literature. Again we observe good compatibility with other recent extractions from this mode [10,13,14,16], and the same comments regarding comparability apply as in B → D. The inclusion of additional data in our case is responsible for the slightly smaller uncertainty and lower central value. The latter is also related to a shift from using the experimental values from HEPData, as discussed below.
As for B → D ν at the end of section 4.1, we also repeat the extraction of V cb using only the measurements that separate the electron and muon samples, since these measurements are used in lepton flavour non-universal NP scenarios in section 5. Using the frequentist approach, we find
V B→D * eν cb = (3.89 ± 0.10) × 10 −2 ,(15)V B→D * µν cb = (3.76 ± 0.11) × 10 −2 .(16)
While consistent with each other and with the global fit, we observe that the muonic data prefer a value of V cb that is lower by about one standard deviation. Finally we would like to comment on the robustness of the extracted V cb value from binned data. The value obtained from the Belle 2017 data alone depends on the precise inputs used: the difference between the binned data given on HEPData [74] and that in the publication, where the values are rounded to two significant digits, yields a shift in V cb of about one standard deviation. 6 This is problematic, since the uncertainties and correlations themselves have an uncertainty that is likely to be larger than the difference between these inputs. This should be kept in mind when analyzing the unfolded spectrum. We proceed using the inputs given on HEPData where available.
New physics
Having extracted the CKM element V cb from data assuming the SM, we next proceed to constrain NP effects. As discussed in section 2, there are five Wilson coefficients per leptonflavour combination, with relations for the right-handed vector current C V R ≡ C V R δ , i.e. up to 25 independent Wilson coefficients for b → c(e, µ)ν transitions. Since the ones for = are indistinguishable, these are effectively reduced to 17, and in the lepton-flavour diagonal case only 9 operators remain. Before analyzing them in more detail, we discuss in section 5.1 all possible simplified models that can be generated by the exchange of a single new mediator particle, implying specific combinations of these Wilson coefficients.
While we focus on exclusive modes in the SM V cb fits, for the NP fits we also compare to the inclusive decays B → X c ν. Since the full inclusive analysis involves fitting several moments of the spectrum simultaneously with V cb , quark masses, and HQET parameters, reproducing it is beyond the scope of our present analysis. Rather, we use a simplified approach where we approximate the total rate in the presence of new physics as
Γ(B → X c eν) ≈ Γ(B → X c eν) SM Γ(B → X c eν) LO NP Γ(B → X c eν) LO SM ,(17)
where Γ(B → X c eν) SM is the full state-of-the-art SM prediction [3] and the rates with superscript "LO" refer to the partonic leading-order calculations. We confirm the known expressions for the LO contributions in the m → 0 limit [28,34,75,76]:
Γ LO (B → X c ν) = Γ LO SM (B → X c ν) |1 + C V L | 2 + |C V R | 2 + 1 4 |C S L | 2 + |C S R | 2 + 12|C T | 2 + Γ LO Int (B → X c ν) Re (1 + C V L )C * V R − 1 2 Re(C S L C * S R ) , with(18)Γ LO SM (B → X c ν) = Γ 0 1 − 8x c − 12x 2 c log x c + 8x 3 c − x 4 c and(19)Γ LO Int (B → X c ν) = − 4Γ 0 √ x c 1 + 9x c + 6x c (1 + x c ) log x c − 9x 2 c − x 3 c ,(20)
and Γ 0 = G 2 F m 5 b |V cb | 2 /(192π 3 ). 1/m 2 c,b and α s corrections to the NP contributions are partly known and can be sizable [28,34,[76][77][78][79][80]. Specifically for scalar operators the α s corrections can qualitatively change the result [76]. However, since we do not include effects on the spectra in any case, we stick to the simple expressions given above; we therefore do not consider the inclusive constraints used here as on the same footing as the exclusive ones and do not combine them in a global fit.
Finally, we include the constraint from the total width of the B c meson, which can be modified significantly in scenarios with scalar couplings [81,82].
Tree-level models
Since the b → c ν transition is a tree-level process in the SM, NP models with sizable effects in these modes typically involve tree-level contributions as well. The known quantum numbers of the involved fermions allow to determine all possible mediator quantum numbers. These correspond to the following simplified models:
• New vector-like quarks modifiying the W -couplings of the SM quarks,
• tree-level exchange of a heavy charged vector boson (W ),
• tree-level exchange of a heavy charged scalar (H ± ),
• tree-level exchange of a coloured vector or scalar boson (leptoquark).
Vector-like quarks are among the simplest renormalizable extensions of the SM. They are also present, e.g., in models with partial compositeness in the form of fermionic resonances of the strongly interacting sector. An SU (2) L singlet vector-like quark with mass m Ψ , coupling to the SM quark doublet and the Higgs with strength Y , can generate a modified left-handed W coupling of order Y 2 v 2 /m 2 Ψ , while an SU (2) L doublet, coupling to the SM quark singlet and the Higgs, can generate a right-handed W coupling. The resulting contributions to the Wilson coefficients C V L or C V R , respectively, are lepton-flavour universal.
The tree-level exchange of W , H ± , or leptoquarks has been studied extensively in the literature in the context of explanations of the b → cτ ν anomalies (see e.g. [24,[83][84][85][86][87][88][89][90][91][92].) The case of leptoquarks is particularly diverse as there are six different representations -three scalars and three vectors -that can in principle contribute to b → c ν at tree level. The b → c ν Wilson coefficients generated in each of the tree-level models considered are shown in table 2. Crosses correspond to a possible tree-level effect at the matching scale, corresponding to the scale of new physics. If two crosses appear in a line, the contributions are governed by independent parameters in the model.
An interesting observation concerning table 2 is that none of the models generates the tensor operator O T on its own. At the matching scale, it is only present in the two scalar leptoquark models S 1 and R 2 in a characteristic correlation with the operator O S L .
Since table 2 is valid at the new physics scale, it is also important to consider renormalization group (RG) effects between this scale and the b quark scale that could possibly change this picture. Normalized as in Eq. (2), it can be easily seen that the vector operators are not renormalized under QCD as they correspond to conserved currents; the same is true for the combinations m b O S L,R , such that the Wilson coefficients renormalize under QCD like the quark mass. Thus, the operator O T , which is mildly QCD-renormalized, cannot mix with the other operators under QCD. Therefore the only qualitatively relevant RG mixing effects among the five operators of interest could come from electroweak renormalization in the SMEFT above the electroweak scale. Indeed, the operators O lequ , that match onto O S L and O T , respectively, mix with each other through the weak coupling constants [93]. The RG-induced values of the respective coefficients at the electroweak scale v can be written in leading logarithmic approximation as
C S L (v) ⊃ 3 8π 2 (3g 2 + 5g 2 ) ln v Λ NP C T (Λ NP ) ,(21)C T (v) ⊃ 1 128π 2 (3g 2 + 5g 2 ) ln v Λ NP C S L (Λ NP ) ,(22)
Model Table 2: Pattern of Wilson coefficients generated by tree-level models. Crosses correspond to a possible effect at the matching scale. If two crosses appear in a line, the contributions are governed by independent parameters in the model.
C V L C V R C S R C S L C T C S L = 4C T C S L = −4C T Vector-like singlet × Vector-like doublet × W × H ± × × S 1 × × R 2 × S 3 × U 1 × × V 2 × U 3 ×
where the self-mixing contributions have been omitted for brevity, but are included in the numerical analysis. For Λ NP = 1 TeV, the numerical prefactors are roughly −0.1 and −0.002, respectively, so the effect is small unless the scale separation is very large, in particular given that the tree-level models already predict C S L C T at the matching scale. We thus conclude that table 2 is useful as classification of tree-level models in terms of low-energy effects and is qualitatively stable under quantum corrections. Of course, a realistic model may combine several of the discussed particles and thus lead to a more diverse pattern of effects.
Right-handed currents
C V R has been considered as a possible source for a tension between inclusive and exclusive determinations in V cb for a long time [94]. However, updated analyses based on total rates alone have already shown that the scenario is disfavoured as a solution to the present tension [33].
The main novelty of the present analysis is the inclusion of the B → D * ν angular observables in the fit. For the total rate, an effect in C V R can always be absorbed by an appropriate shift in V cb (orṼ cb ), such that constraining right-handed currents requires considering several modes. Including angular observables changes this picture, as the shape of these observables is directly sensitive to right-handed currents. As a consequence, a fit to B → D * ν data alone is able to constrain right-handed currents directly, as shown 7 in figure 2, together with the constraints from B → D ν and the inclusive decay, both constraining only combinations of V cb and C V R . In this plot we assume lepton-flavour universality as discussed above.
We observe that the tensions between the V cb determinations from B → D ν, B → D * ν, and B → X c ν, that are anyway milder than in the past, cannot be completely removed by postulating new physics in right-handed currents. What is new is that even B → D * ν alone cannot be brought into perfect agreement with B → X c ν for any value of C V R .
Lepton flavour universality violation
In view of the observed tensions with SM expectations in b → cτ ν and b → s transitions, investigating e-µ universality in b → c ν with light leptons is important. Specific new physics models suggested as solutions to the b → cτ ν anomalies actually predict such violation. Some of the experimental analyses assume LFU to hold. These analyses cannot be used in a modelindependent fit allowing for LFU violation. This is because the measurements are not simply averages of the respective electron and muon observables, but linear combinations with weights depending on the experimental efficiencies that can differ between electrons and muons even as a function of kinematical variables. Thus it is of paramount importance that experimental collaborations present their results separately for electrons and muons. In the meantime, the existing analyses that already include separate results for electrons and muons (see table 1) can be used to perform a fit with a non-universal modification of the SM operator, i.e. C e V L = C µ V L . The fit result in terms of the lepton-flavour-dependent effective CKM elementsṼ cb is shown in figure 3. Both for B → D ν and B → D * ν the fit not only shows perfect agreement with LFU, but also implies a stringent constraint on departures from the LFU limit. Given the good agrement of the constraints from B → D ν and B → D * ν, we have also performed a combined Bayesian fit of the scenario to both decay modes, marginalizing over all nuisance parameters. We find with a small correlation of −10%. Equivalently, we find
1 2 Ṽ e cb +Ṽ µ cb = (3.87 ± 0.09)% ,(23)1 2 Ṽ e cb −Ṽ µ cb = (0.022 ± 0.023)% ,(24)V e cb V µ cb = 1.011 ± 0.012 ,(25)
which can be used as a generic constraint on e-µ universality violation in b → c ν processes. This is the first global combination of LFU constraints in b → c ν transitions and provides a significantly stricter bound on LFU than the recent measurement in Ref. [38] alone. As already discussed in section 2, a violation of LFU can also manifest itself as a contribution from a "wrong-neutrino" decay generated by C V L with = . However, this case does not have to be discussed separately as it merely leads to a rescaling of all observables that can be absorbed by definingṼ
cb = V cb |1 + C V L | 2 + = |C V L | 2 1/2 .(26)
Scalar operators
The interference of scalar contributions with the SM amplitude in exclusive and inclusive decays arises only from lepton-mass-suppressed terms. Consequently, scalar contributions always lead to an increase in the rates for a fixed value of V cb . The suppression of interference terms also implies that there is no qualitative difference between operators with = or = , so we focus on the former for simplicity. Again the differential distributions contain valuable information about possible scalar contributions. The most striking example is the endpoint of the B → D ν differential decay rate, see also, e.g., [95]. In the SM, close to while in presence of NP contributions to scalar operators, it behaves as
q 2 = q 2 max = (m B − m D ) 2 , it behaves as dΓ(B → D ν) dq 2 ∝ f 2 + q 2 − q 2 max 3/2 ,(27)dΓ(B → D ν) dq 2 ∝ f 2 0 |C S R + C S L | 2 q 2 − q 2 max 1/2 .(28)
This implies an exceptional sensitivity of the highest q 2 (or lowest w) bin in B → D ν to the sum of scalar Wilson coefficients, C S R + C S L . To illustrate this effect, in figure 4 we show on the left the predictions for the SM (using V cb from eq. (7)) and a scenario with a large NP contribution in C S L for fixed V cb , compared to the experimental data. The scenarios are chosen such that they give the same prediction for the total branching ratio. The qualitatively different endpoint behaviour can clearly distinguish between them, excluding such large scalar contributions. As a consequence, a fit to the differential B → D ν rates leads to a more stringent constraint on scalar operators than B → D * ν or the inclusive decay. While the total rates of these modes constrain only combinations of |V cb | and |C S L ± C S R | or |C S L | 2 + |C S R | 2 , these are accessible individually with differential distributions. This is demonstrated in figure 4 on the right, where the constraint resulting from the total rate alone is compared to the one resulting from all available information on that channel. Importantly, including the information from the differential rate,Ṽ cb and Re(C S R ) can be determined individually.
The constraining power of B → D ν and the complementary dependence of B → D ν and B → D * ν on the scalar Wilson coefficients C S L and C S R is also demonstrated in figures 5 and 6. In these figures, the constraints are shown separately for the modes involving electron and muons, as there is in general no reason to expect LFU to hold for scalar contributions. Furthermore, here and in the following only the measurements of electronic modes are used to constrain the electronic coefficients and the measurements of muonic modes for the muonic coefficients. We stress that more stringent constraints would be obtained by assuming the coefficient with the other lepton flavour to be free from NP and using both sets of data. In this respect, our constraints are conservative.
In figure 5, corresponding to scenarios with a U 1 or V 2 leptoquark, the electronic modes do not show any preference for non-zero C e S R , while B → D * µν prefers a sizable value of C µ S R slightly over the SM, but such values are excluded by B → Dµν in this scenario, and also disfavoured by the B c lifetime. A second interesting scenario is having NP in both C S R and C S L , as can arise in models with a charged Higgs boson (cf. table 2). The resulting constraints are shown in figure 6, again separately for electronic and muonic modes. In addition to the form factor parameters, also V cb is varied as a nuisance parameter. Consequently, the inclusive decay alone, only entering our analysis via its total rate, cannot constrain the coefficients individually and is not shown. In principle, the same qualification applies to the leptonic decays B c → ν. To give an indication, we nevertheless show the constraints that would apply for a conservatively low value of V cb = 0.035. The feature that B → D ν only constrains the sum and B → D * ν only the difference of these Wilson coefficients is clearly seen. In the muonic case, we observe a mild preference for non-zero values of both Wilson coefficients from B → D ( * ) ν. In constrast to the scenario in figure 5, the constraint from B → D ν does not exclude sizable NP effects if C S R ∼ C S L . However, the constraint from the B c lifetime excludes such a scenario, even when allowing for very large modifications of ∼ 40% and a small value of V cb = 0.035.
Tensor operator
In contrast to the scalar operators, for tensor operators B → D * ν leads to a much more stringent constraint. This is due to the existing lepton-specific data on the transverse and longitudinal decay rates (cf. table 1). In the SM the D * is fully longitudinally polarized at maximum recoil (q 2 min = m 2 ). This is no longer true in the presence of tensor operators. Indeed, in the limit of q 2 → q 2 min ≈ 0, the differential rate to transversely polarized D * in the presence of NP contributions to C V L and C T behaves as
dΓ T (B → D * ν) dq 2 ∝ q 2 |1 + C V L | 2 A 1 (0) 2 + V (0) 2 + 16m 2 B |C T | 2 T 1 (0) 2 + O m 2 D * m 2 B ,(29)
where the kinematic relation T 2 (0) = T 1 (0) was used and the D * mass was neglected for simplicity. As a consequence, this observable is exceptionally sensitive to tensor operators near the maximum recoil point. To illustrate this fact, in figure 7 on the left we show the predictions for the SM and a scenario with a sizable NP contribution in C T compared to the experimental data. The scenarios are chosen such that they give the same prediction for the total branching ratio. The different behaviour at q 2 = 0 allows to clearly distinguish them and disfavours tensor contributions of this size. On the right of figure 7 we demonstrate again this qualitative difference by comparing the constraints from the total B → D * µν rate alone and including the differential distribution. An additional observable that would be very sensitive to the tensor operator, but has not been measured yet, is the "flat term" in B → D ν. The normalized differential decay rate as a function of the angle θ between the charged lepton and the B in the lepton-neutrino mass frame can be written as
1 Γ dΓ d cos θ dq 2 = 3 4 1 − F H (q 2 ) sin 2 θ + 1 2 F H (q 2 ) + A FB (q 2 ) cos θ .(30)
In complete analogy to the B → K + − decay, the observable F H vanishes in the SM up to tiny lepton mass effects, but can be sizable in the presence of new physics in the tensor operator. Neglecting the lepton masses and allowing for NP in C T and C V L , one finds
F H (q 2 ) ≈ 18q 2 f 2 T (q 2 ) m 2 B f 2 + (q 2 ) |C T | 2 |1 + C V L | 2 .
(31) Figure 8 shows the constraints on the tensor and left-handed scalar operators, which always appear together in models with a tree-level mediator, see Table 2, specifically in leptoquark models. The displayed constraints from B → D ν and B → D * ν, shown separately for electrons and muons, demonstrate clearly the strong sensitivity of B → D * ν to tensor contributions. While the individual modes B → D * eν, B → Dµν, and B → D * µν show a slight preference for non-zero NP contributions in either C S L or C T , the combination of B → D ν and B → D * ν constraints allows neither of these solutions and leads to a strong constraint on both operators.
Conclusions
Semi-leptonic charged-current transitions b → c ν with = e or µ are traditionally used to measure the CKM element V cb . In principle, this transition could be affected by new physics with vector, scalar, or tensor interactions, possibly violating lepton flavour universality. This is motivated by the long-standing tensions between inclusive and exclusive determinations of V cb , but also by hints of a violation of lepton-flavour universality in b → cτ ν and b → s transitions. We have conducted a comprehensive analysis of general new-physics effects in b → c ν transitions, considering for the first time the full operator basis and employing for the first time in a new physics analysis measurements of B → D * ν angular observables. Our main findings can be summarized as follows.
Re C µ T flavio C µ SL = ±4C µ T @ 160 GeV C µ SL = ±4C µ T @ 1 TeV B → Dµν B → D * µν
• Extracting the absolute value of the CKM element V cb from fits to the full sets of B → D ν and B → D * ν data in Table 1 yields values consistent with the recent literature and no significant tension between determinations from B → D ν vs. B → D * ν.
• We find no dependence of the V cb extraction on the statistical approach, but find a significant dependence on the treatment of systematic uncertainties in binned observables due to the "d'Agostini bias".
• We find that NP in right-handed currents cannot improve the agreement between inclusive and exclusive determinations of V cb . Thanks to our use of differntial and angular dsitributions, this conclusion can even be drawn considering B → D * ν vs. B → X c ν alone.
• We find strong constraints on violations of e-µ universality, specifically for C V L .
• We demonstrate that the zero-recoil endpoint of the B → D ν spectrum is exceptionally sensitive to NP in scalar operators.
• We demonstrate that the maximum-recoil endpoint of the transverse B → D * ν spectrum is exceptionally sensitive to NP in tensor operators.
Our analysis could be improved in several respects. The treatment of experimental data had partly to rely on crude estimates of the systematic uncertainties or correlations, where these were not public. We urge the experimental collaborations to publish this information for future and also existing analyses. Our treatment of the inclusive decay is also approximate, as discussed at the beginning of section 5. Clearly, a full fit to the moments of the inclusive mode would be interesting, but is beyond the scope of our present analysis. Finally, our treatment of B → D * form factors had to some extent to rely on the heavy quark expansion, with only partial inclusion of 1/m 2 c,b contributions. A full calculation of the q 2 dependence from lattice QCD, ideally including tensor form factors, would make these constraints much more reliable. We emphasize again that our analysis can be easily improved once this information becomes available as all of our code is open source.
A. Numerical results for B → D ( * ) form factors
In this appendix, we give details on the parametrization of form factors discussed in section 3 and present the numerical results of our fit to lattice and LCSR calculations employing the HQET parametrization and unitarity bounds. The B → D ( * ) form factors h i in the HQET basis 8 can be found in ref. [13]. In the heavy quark limit, they vanish or reduce to a common form factor, the leading Isgur-Wise function ξ(w). It is thus convenient to write the form factors as h i (w) = ξ(w)ĥ(w). The expressions for allĥ(w) at next-to-leading order in α s and next-to-leading power in b,c =Λ/2m b,c can be found in ref. [13]. As discussed in section 3, we modify these expressions by allowing for an additional O( 2 c ) correction to the form factors that are protected from O( c ) corrections:
h A 1 (w) →ĥ A 1 (w) + 2 c δ h A 1 ,ĥ T 1 (w) →ĥ T 1 (w) + 2 c δ h T 1 ,ĥ + (w) →ĥ + (w) + 2 c δ h + .(32)
We neglect a possible w dependence of the δ i terms. The leading order Isgur-Wise function ξ can be written to second order in the z expansion as ξ(z) = 1 − 8ρ 2 z + (64c − 16ρ 2 )z 2 .
We then perform a Bayesian fit (a Markov Chain Monte Carlo employing flavio [39] and emcee [97]) to the theory constraints described in section 3 of the ten parameters parametrizing the functionsĥ i and ξ. 9 The mean, standard deviation and correlation matrix of nine of those parameters is
χ 2 (1) χ 2 (1) χ 3 (1) η(1) η (1) ρ 2 c δ h A 1 δ h + =
−0.058 ± 0.019 −0.001 ± 0.020 0.035 ± 0.019 0.358 ± 0.043 0.044 ± 0.125 1.306 ± 0.059 1.220 ± 0.109 −2.299 ± 0.394 0.485 ± 0.269
(34) ρ = .(35)
The tenth parameter, δ T 1 , is not constrained by the fit, thus its posterior is equal to the prior, which we conservatively take to be a Gaussian with mean 3 centered around 0. These form factors have been implemented and set as defaults in flavio version 0.26.
Figure 1 :
1Fit results for V cb from exclusive decays in the Standard Model. Left: frequentist profile likelihood vs. "fast fit" method; the horizontal dotted line delineates the 1σ region. Right: Bayesian posterior probability vs. "fast fit" method.the errors from the profile likelihood and the posterior marginalization agree is less trivial, but is a consequence of all theory and experimental uncertainties being close to Gaussian. The excellent agreement of the two more sophisticated approaches with the "fast fit" approach indicates that the fitted values of the nuisance parameters are close to the theoretical central values (otherwise a mismatch would be observed in the fast fit). The full one-dimensional profile likelihood and posterior probability distribution are shown in figure 1.
Figure 2 :
2Constraints on right-handed currents from inclusive and exclusive decays, assuming LFU.
Figure 3 :
3Constraints on LFU violation in the left-handed vector current from exclusive decays. The vertical line corresponds to the LFU limit.
Figure 4 :
4Left: Prediction for the differential B → Dµν branching ratio in the SM (blue band) and a scenario with new physics in C S L (orange band) vs. the Belle measurement, demonstrating the different endpoint behaviour at zero recoil (q 2 ≈ 11.6 GeV 2 ). Both scenarios predict the same total B → Dµν branching ratio. Right: Comparison of the constraint on the scalar coefficientC µ S R vs.Ṽ µ cb from the total B → Dµν branching ratio measurements only (dashed) and using all B → Dµν measurements (solid).
Figure 5 :
5Constraints on the scalar coefficientsC e,µ S R vs.Ṽ e,µ cb , absorbing a potential vector coefficient C V L , from inclusive and exclusive decays as well as the total width of the B c meson, seperately for electrons (left) and muons (right).
Figure 6 :
6Constraints on the scalar Wilson coefficients from exclusive decays and the total width of the B c meson, seperately for electrons (left) and muons (right).
Figure 7 :
7Left: Prediction for the transverse differential B → D * µν branching ratio in the SM (blue band) and a scenario with new physics in C µ T (orange band) vs. the Belle measurement, demonstrating the different endpoint behaviour at maximum recoil (q 2 = 0). Both scenarios predict the same total B → D * µν branching ratio. Right: Comparison of the constraint on the tensor coefficientC µ T vs.Ṽ µ cb from the total B → D * µν branching ratio measurements only (dashed) and using all B → D * µν measurements (solid).
Figure 8 :
8Constraints on C e,µ S R vs. C e,µ T , profiling over V cb . The black dotted and dash-dotted lines show the expected ratio of the Wilson coefficients in the two leptoquark scenarios S 1 and R 2 (cf. table 2), assuming the relation C S L = ±4C T to hold at the scale 160 GeV or 1 TeV.
We do not consider scenarios with light right-handed neutrinos. This general form for NP in b → c ν transitions has been first considered in Ref.[40].
This might also explain the difference between the values of V cb obtained from unfolded and folded data in this article.
This plot and all the following Wilson coefficient plots correspond to the two-dimensional profile likelihood in the space of the Wilson coefficients shown. For observables only constraining a single combination of Wilson coefficients, the bands correspond to −2∆ ln L = 1 and 4, respectively, otherwise to −2∆ ln L = 2.30 and 6.18, thereby accounting for the different degrees of freedom.
The relation of the HQET form factors to the traditional form factor basis Vi, Ai, Ti and fi can be found in appendix A of ref.[96].9 Again these values have also been obtained by an independent frequentist implementation.
AcknowledgmentsWe thank Florian Bernlochner, Christoph Bobeth, Jorge Martin Camalich, Danny van Dyk, Paolo Gambino, Thomas Kuhr, Marcello Rotondo, Stefan Schacht, and Christoph Schwanda for useful discussions. The work of M. J. and D. S. is supported by the DFG cluster of excellence "Origin and Structure of the Universe". The numerical analysis has been carried out on the computing facilities of the Computational Center for Particle and Astrophysics (C2PAP).
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| [] |
[
"INTERSECTIONS OF THICK COMPACT SETS IN R d",
"INTERSECTIONS OF THICK COMPACT SETS IN R d"
] | [
"Kenneth Falconer ",
"Alexia Yavicoli "
] | [] | [] | We introduce a definition of thickness in R d and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in R d with thickness τ , there is a number N (τ ) such that the set contains a translate of all sufficiently small similar copies of every set in R d with at most N (τ ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.2020 Mathematics Subject Classification. MSC 11B25, MSC 28A12, MSC 28A78, and MSC 28A80. | 10.1007/s00209-022-02992-y | [
"https://arxiv.org/pdf/2102.01186v2.pdf"
] | 231,749,865 | 2102.01186 | 2d629427007f8b41499169e378994f3be080c7cd |
INTERSECTIONS OF THICK COMPACT SETS IN R d
Kenneth Falconer
Alexia Yavicoli
INTERSECTIONS OF THICK COMPACT SETS IN R d
We introduce a definition of thickness in R d and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in R d with thickness τ , there is a number N (τ ) such that the set contains a translate of all sufficiently small similar copies of every set in R d with at most N (τ ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.2020 Mathematics Subject Classification. MSC 11B25, MSC 28A12, MSC 28A78, and MSC 28A80.
compact sets C 1 , C 2 ⊂ R d (2) dim H (C 1 ∩ (C 2 + x)) ≤ max{0, dim H (C 1 × C 2 ) − d} for Lebesgue almost-all x ∈ R d ; the right-hand side can be replaced by max{0, dim H (C 1 )+ dim H (C 2 ) − d} if, for example, either C 1 or C 2 has equal Hausdorff and upper boxcounting dimension, see [17]. On the other hand, for all > 0,
(3) dim H (C 1 ∩ σ(C 2 )) ≥ max{0, dim H (C 1 ) + dim H (C 2 ) − d − }
for a set of similarities σ of positive measure with respect to the natural measure on the group of similarities σ on R d . The similarity group may be replaced by the group of isometries if dim H C 1 > (d + 1)/2 (it is not known if this condition is necessary if d ≥ 2), see [12,17]. The disadvantage of these results is that they are measure theoretic, and tell us nothing about which particular similarities or isometries these inequalities are valid for. At the other extreme, there are classes C of 'limsup sets' of Hausdorff dimension 0 < s < d which are dense in R d with the property that the intersection of any countable collection of similar copies of sets in C still has Hausdorff dimension s, see for example [8].
It is natural to ask for specific conditions on compact sets that are 'close enough' to each other that guarantee non-empty intersection, or even give a lower bound for the dimension of their intersection. For subsets of the real line Newhouse [19] introduced a notion of thickness, see Definition 1, which depends on the relative sizes of the complementary open intervals of the set and showed that two Cantor-like sets, with neither contained in a gap of the other, must intersect if the product of their thickness is greater than 1, see Theorem 2. In this paper we propose a definition of thickness for compact subsets of R d for all d ≥ 1. We obtain a higher dimensional gap lemma, and show that given several compact sets in R d (d ≥ 1) that are not too far apart in a sense that will be made precise, if their thicknesses are large enough then they have non-empty intersection, and we obtain a lower bound for the Hausdorff dimension of this intersection.
We first review the definition of thickness for subsets of the real line. Recall that every compact set C on the real line can be constructed by starting with a closed interval I ≡ I 1 (the convex hull of C) and successively removing disjoint open complementary intervals (they are the path-connected components of the complement of C). Clearly there are finitely or countably many disjoint open complementary intervals (G n ) n , which we may assume are ordered so that their lengths |G n | are non-increasing; if several intervals have the same length, we order them arbitrarily. The two unbounded pathconnected components of R \ C are not included. For each n ∈ N the interval G n is a subset of some closed path-connected component I n of I \ (G 1 ∪ · · · ∪ G n−1 ). We say that such a G n is removed from I n .
Definition 1 (Thickness in R). Let C ⊂ R be compact with convex hull I, and let (G n ) n be the ordered sequence of open intervals comprising I \ C. Each G n is removed from a closed interval I n , leaving behind two closed intervals L n and R n ; the left and right intervals of I n \ G n . The thickness of C ⊂ R is defined as τ (C) := inf n∈N min{|L n |, |R n |} |G n | .
The sequence of complementary intervals (G n ) n may be finite, in which case the infimum is taken over the finite set of indices. The thickness of a single point is taken to be 0, and that of a non-degenerate interval to be +∞.
If there are several complementary intervals of equal length, then the ordering of them does not affect the value of τ (C). See [1,11,20,22] for more information on Newhouse thickness and alternative definitions.
Theorem 2 (Newhouse's Gap Lemma). Given two compact sets C 1 , C 2 ⊂ R, such that neither set lies in a gap of the other, if τ (C 1 )τ (C 2 ) > 1 then
C 1 ∩ C 2 = ∅.
Theorem 2 was proved only for subsets of R and it does not guarantee positive Hausdorff dimension of the intersection, nor does it generalise in any simple way to intersections of 3 or more sets.
Here we give a definition of thickness for compact subsets of R d that enables us to generalize Theorem 2 to higher dimensions, and also obtain lower bounds for the Hausdorff dimension of the intersection of several sets. For a different definition of thickness for certain dynamically defined subsets of the complex plane see [3].
Our setting throughout the paper is as follows. Given a compact subset C of R d , we define (G n ) ∞ n=1 to be the (at most) countably many open bounded path-connected components of C C and E to be the unbounded open path-connected component of C C (except when d = 1 when E consists of two unbounded intervals). We call E together with G n (n ∈ N) the gaps of C. We may assume that the sequence of gaps (G n ) ∞ n=1 is ordered by non-increasing diameter. Note that we make no assumption about the connectedness or simply connectedness of C.
We write dist for the usual distance between points or non-empty subsets of R d and diam for the diameter of a non-empty subset of R d .
Definition 3 (Thickness in R d ). We define the thickness of C to be
τ (C) := inf n∈N dist(G n , 1≤i≤n−1 G i ∪ E) diam(G n ) ,
provided that E is not the only path-connected component of C.
When the only complementary path-connected component is E, we define
(4) τ (C) := +∞ if C • = ∅ 0 if C • = ∅ We say C is thick if τ (C) > 0.
If the sequence of complementary intervals (G n ) n is finite then the infimum is taken over the finite set of indices. Moreover, thickness is well-defined in the sense that if two gaps have the same diameter, interchanging their positions in the ordering does not change the definition of thickness.
Note that τ ∈ [0, +∞]. Also, τ is invariant under homothetic maps, and agrees with the usual definition of thickness in the real line (recall Definition 1).
Observation 4. If C ⊂ R d is a thick compact set, then either there are finitely many gaps (G n ) n or lim n→∞ diam G n = 0. To see this we can assume that E is not the only complementary path-connected component. If diam G n ≥ c > 0 for infinitely many n, taking points x n ∈ G n with dist(x n , x i ) ≥ cτ (C) for 1 ≤ i < n contradicts the sequential compactness of E C .
In Section 2, we obtain a higher dimensional gap lemma, Theorem 10. The gap lemma does not generalize in any simple way to intersections of three or more sets, so we need to use other methods to study such intersections. To achieve this we obtain lower bounds for the Hausdorff dimension of the intersection of several thick compact sets in terms of their thicknessess, which is easy to estimate in many cases.
Our main theorem, Theorem 6 which will follow from Theorem 18 which relates thickness to 'winning sets'.
The following constants appear in many of our results:
Definition 5. In R d (d ≥ 1), let (5) K 1 := 2d(24 √ d) d log(16 √ d) 1 − 1 2 d and K 2 := (24 √ d) d (1 + 4 d 2) 1 − 1 2 d 2 .
We now state our main theorem which will follow from applying Theorem 18 on 'winning sets' to thickness. We write E i for the unbounded open path-connected component of C C i (the union of two unbounded intervals when d = 1). Theorem 6 (Intersection of compact sets in R d ). Let (C i ) i be a family of countably many compact sets in R d , where C i has thickness τ i > 0, such that:
(i) sup i diam(C i ) < +∞, (ii) there is a ball B such that B ∩ E i = ∅ for every i, where E i is the unbounded component of C C i , (iii) there exists c ∈ (0, d) such that i τ −c i ≤ 1 K 2 β c (1 − β d−c ) where β := min 1 4 , diam(B) sup i diam(C i ) .
Then
dim H B ∩ i C i ≥ d − K 1 i τ −c i d/c β d | log(β)| > 0.
Note that condition (iii) comes from Theorem 18 and is needed both to obtain the lower bound for the dimension of the intersection and to ensure that this bound is positive.
The significance of Theorem 6 is that a condition on thicknesses can give a lower bound for the dimension of intersection of a finite or countable collection of sets in R d so ensure that the intersection is non-empty. In practice, the thicknesses needed are rather large as a consequence of the large constants K 1 and K 2 .
A very active research area involves finding conditions on a set that guarantees the set contains homothetic copies of a given finite set of points, called a pattern in this context. It will follow from Theorem 6 that a set contains homothetic copies of any given pattern in R d provided it is sufficiently thick. Patterns and intersections are related: the set C contains a homothetic copy of A := {a 1 , . . . , a n } if and only if there exists λ = 0 such that 1≤i≤n (C − λa i ) = ∅.
A consequence of the Lebesgue density theorem is that any set E ⊂ R d of positive Lebesgue measure contains a homothetic copy of every finite set at all sufficiently small scales, so it is natural to seek conditions on sets of zero Lebesgue measure form which this remains true. Perhaps the most natural notion of size to consider is Hausdorff dimension but there are constructions (see for example [6,13,14,16,18,21]) which indicate that Hausdorff dimension cannot, in itself, detect the presence or absence of patterns in sets of Lebesgue measure zero, even in the most basic case of points in arithmetic progressions.
Laba and Pramanik [15] showed that if, in addition to having large Hausdorff dimension, a subset of R supports a probability measure with appropriate Fourier decay, then it contains arithmetic progressions of length 3. The hypotheses were relaxed and the family of patterns covered greatly enlarged in subsequent papers [5,9,10]. This work uses harmonic analysis, and such methods do not work easily for longer arithmetic progressions. Moreover, the hypotheses may be difficult to check, and are not even known to hold for natural classes of fractals such as central self-similar Cantor sets.
Yavicoli [22] showed that Newhouse thickness, Definition 1, allows the detection of homothetic and more general copies of patterns inside fractal sets in the real line. Newhouse thickness is easy to compute or estimate for many classical fractal sets such as self-similar sets or sets defined in terms of continued fraction coefficients. Our notion of thickness in higher dimensions, Definition 3, enables such results to be extended to R d .
Theorem 7. Let C ⊂ R d be a compact set with thickness τ := τ (C), such that E C contains a ball B. Then C contains a homothetic copy of every set A with at most and K 2 is as in (5).
Moreover, for all λ ∈ 0, diam(B) 16 diam(A) , there exists a set X of positive Hausdorff dimension (depending on A, B, C and λ) such that
x + λA ⊆ C for all x ∈ X.
We also discuss the relationship between Hausdorff dimension and thickness of a set. It is shown in [20, p.77] that for C ⊂ R, (7) dim H (C) ≥ log 2 log(2 + 1/τ (C)) , and in Section 6 we obtain analogous lower bounds for C ⊂ R d .
A Gap Lemma in R d
In this section we extend Theorem 2, Newhouse's gap lemma on R, to R d for d ≥ 2. We first study a particular case when the gaps are either linked or do not intersect; in this setting we can use an analogous argument to Newhouse's proof.
We denote the boundary of U ⊂ R d by ∂U .
Definition 8. We say that U ⊆ R d and V ⊆ R d are linked gaps if U ∩ V = ∅, (∂U ) \ V = ∅ and (∂V ) \ U = ∅.
We say that C 1 and C 2 are linked compact sets in R d if for every pair of gaps G 1 and G 2 of C 1 and C 2 respectively we have that either their intersection is empty or they are linked gaps.
We first obtain the conclusion when C 1 and C 2 are linked compact sets and then in Theorem 10 we reduce to this case the weaker condition that neither C 1 or C 2 is contained in any gap of the other. Figure 1 illustrates how gaps may satisfy the hypotheses of Theorem 10 but not of Proposition 9. Proposition 9. Let C 1 and C 2 be linked compact sets in R d , with τ (C 1 )τ (C 2 ) > 1, then
C 1 ∩ C 2 = ∅.
Proof. By definition of τ ,
τ 1 := τ (C 1 ) := inf m dist G 1 m , 1≤i≤m−1 G 1 i ∪ E 1 diam(G 1 m ) and τ 2 := τ (C 2 ) := inf n dist G 2 n , 1≤i≤n−1 G 2 i ∪ E 2 diam(G 2
n ) where C 1 and C 2 have gaps G 1 n and G 2 n and external path-connected components E 1 and E 1 respectively.
We assume that C 1 ∩ C 2 = ∅ and will obtain a contradiction. Then, We will construct inductively a sequence (U i , V i ) i∈N of pairs of linked bounded gaps that occur in the construction of C 1 and C 2 respectively, such that either diam
C 1 ⊆ C C 2 = i G 2 i ∪ E 2 and C 2 ⊆ C C 1 = i G 1 i ∪ E 1 .U i → 0 or diam V i → 0 (or both).
To start the induction: We will define (U 1 , V 1 ) linked bounded gaps.
Since E 1 and E 2 are linked gaps, there is
x 1 ∈ ∂E 1 \ E 2 ⊆ C 1 \ E 2 ⊆ C C 2 \ E 2 , so there is a bounded gap V 1 := G 2 n 1 of C 2 such that x 1 ∈ G 2 n 1 .
Since E 1 and V 1 intersect and C 1 and C 2 are linked compact sets, E 1 and V 1 are linked gaps. Hence, there is
x 2 ∈ ∂V 1 \ E 1 ⊆ C 2 \ E 1 ⊆ C C 1 \ E 1 , so there is a bounded gap U 1 := G 1 m 1 of C 1 such that x 2 ∈ G 1 m 1 .
Since U 1 and V 1 are gaps that intersect, and C 1 and C 2 are linked compact sets, U 1 and V 1 are linked.
Inductive step: Given that we have defined a pair of linked gaps (U k , V k ) of C 1 and C 2 defined, we now define (U k+1 , V k+1 ).
Since
(U k , V k ) is a pair of linked gaps, there is a k ∈ ∂U k \ V k . Since a k ∈ ∂U k , we have a k ∈ C 1 , hence by assumption a k / ∈ C 2 , so there is a gap G 2 n k of C 2 such that a k ∈ G 2 n k . Note that (U k , G 2
n k ) are linked because they intersect and C 1 and C 2 are linked. In the same way, since
(U k , V k ) is a pair of linked gaps there is b k ∈ ∂V k \ U k . Since b k ∈ ∂V k , then b k ∈ C 2 , hence b k / ∈ C 1 , so there is G 1 m k a gap of C 1 such that b k ∈ G 1 m k Again (G 1 m k , V k ) are linked. We will show that we can choose (U k+1 , V k+1 ) to be either (U k , G 2 n k ) or (G 1 m k , V k )
in such a way the diameters of either U k or V k tends to 0.
We observe that for a fixed pair n, m ∈ N the following two inequalities cannot hold simultaneously:
• dist(G 1 m , 1≤i≤m−1 G 1 i ∪ E 1 ) ≤ diam(G 2 n ) • dist(G 2 n , 1≤i≤n−1 G 2 i ∪ E 2 ) ≤ diam(G 1 m ).
For if both hold, then by definition of thickness,
diam(G 2 n ) ≥ τ 1 diam(G 1 m ) and diam(G 1 m ) ≥ τ 2 diam(G 2 n ). Using the hypothesis that τ 1 τ 2 > 1, diam(G 1 m ) ≥ τ 2 diam(G 2 n ) ≥ τ 1 τ 2 diam(G 1 m ) > diam(G 1 m ), which is a contradiction.
The gaps U k and V k can be identified as U k := G 1 m and V k := G 2 n for some n, m ∈ N.
In the case dist(G 1 m , 1≤i≤m−1 G 1 i ∪ E 1 ) > diam(G 2 n ), we also know that (U k , V k ) are linked, so V k does not intersect E 1 or G 1 i for every 1 ≤ i ≤ m − 1. Also b k ∈ ∂V k \ U k ⊆ ( 1≤i≤m−1 G 1 i ∪ E 1 ) C ∩ C C 1 . Then b k belong to a bounded gap G 1 m k with m k > m, and we take (U k+1 , V k+1 ) := (G 1 m k , V k ). In the case dist(G 1 m , 1≤i≤m−1 G 1 i ∪ E 1 ) ≤ diam(G 2 n ), by the previous observation we have dist(G 2 n , 1≤i≤n−1 G 2 i ∪ E 2 ) > diam(G 1 m ).
Analogously to the previous case a k belong to a bounded gap G 2 n k with n k > n, and we take (U k+1 , V k+1 ) := (U k , G 2 n k ). Since one or other of these cases occurs infinitely many times, we get a sequence (U k , V k ) of linked gaps of C 1 and C 2 , where at least one of the diameter sequences tends to 0. Assume, by symmetry,
that diam(U k ) → 0. Take x k ∈ ∂U k ⊆ C 1 , and y k ∈ U k ∩ ∂V k ⊆ C 2 . Then, dist(x k , y k ) ≤ diam(U k ) → 0. Since (x k ) k ⊆ C 1 there exists (x k j ) j a subsequence (x k j ) j convergent to x ∈ C 1 . Since (y k j ) j ⊆ C 2 , we also get (y k j ) j → x ∈ C 2 . So x ∈ C 1 ∩ C 2 contradicting the assumption that C 1 ∩ C 2 = ∅.
We can now relax the hypotheses of Proposition 9.
Theorem 10 (Gap Lemma in R d ). Let C 1 and C 2 be compact sets in R d such that neither of them is contained in a gap of the other and τ (C 1 )τ (C 2 ) > 1. Then C 1 ∩C 2 = ∅.
Proof. We write τ 1 := τ (C 1 ) and τ 2 := τ (C 2 ). By hypothesis, C 1 and C 2 are thick compact sets.
We will show that if the theorem is not trivially true then there are sets C 1 and C 2 with thicknessesτ 1 ≥ τ 1 andτ 2 ≥ τ 2 that satisfy the conditions of Proposition 9 such that C 1 ∩ C 2 = C 1 ∩ C 2 , from which the theorem follows immediately. We do this using a sequence of steps to modify the sets so that we can assume that the sets satisfy such stronger conditions. Note that in these steps G 1 will always be a gap of C 1 and G 2 will be a gap of C 2 ; such gaps may be unbounded unless stated otherwise.
(0) We may assume that there is at least one bounded gap in the construction of C 1 , and similarly for C 2 . Otherwise C 1 = E C 1 . But by hypothesis C 2 is not contained in E 1 , so C 1 ∩ C 2 = ∅ and the theorem is trivially true.
(1) We may assume that ∂G 1 ∩ ∂G 2 = ∅ for all gaps G 1 and G 2 of C 1 and C 2 respectively. Otherwise there exist gaps G 1 and G 2 of C 1 and C 2 such that ∂G 1 ∩ ∂G 2 = ∅, so C 1 ∩ C 2 = ∅ and the theorem is trivially true.
(2) We may assume that ∂E 1 E 2 and ∂E 2 E 1 . Otherwise ∂E 1 ⊆ E 2 (or vice-versa). Since C 1 and C 2 are compact, there exists a closed ball B R (x) such that C 1 ∪ C 2 ⊆ B R (x). We define r := R/(2τ 2 + 1) ∈ (0, R),x ∈ R d such that dist(x,x) > 2R and C 2 := C 2 ∪ B R (x) \ B r (x). Thus the external path-connected component of C 2 is E 2 = E 2 \ B R (x)
, and there is a new gap G 2 := B r (x) that was not in the construction of C 2 . Then τ 2 =τ 2 by definition of r.
We taker ∈ (0, r) and defineC 1 :
= C 1 ∪ Br(x). Then the external path- connected component ofC 1 isẼ 1 := E 1 \ Br(x) andτ 1 = τ 1 .
By construction C 1 and C 2 are compact sets, with the same thicknesses as C 1 and C 2 , such that
C 1 ∩ C 2 = C 1 ∩ C 2 , and ∂ E 1 E 2 and ∂ E 2 E 1 . (3)
We may assume that no bounded gap of C 1 is contained in a gap of C 2 , and vice-versa. If there are bounded gaps G 1 i of C 1 contained in bounded gaps G 2 j of C 2 , we set
C 1 := C 1 ∪ j G 1 i ⊆G 2 j G 1 i ;
thus C 1 is obtained from C 1 by 'filling in' the gaps that are contained in a gap of C 2 . Then C 1 is compact withτ 1 ≥ τ 1 and C 1 ∩ C 2 = C 1 ∩ C 2 and no gaps of C 1 are contained in gaps of C 2 . Now we can apply the same argument to C 2 and C 1 (filling in certain gaps of C 2 ) to obtain a set C 2 . Hence, C 1 and C 2 are compact sets such that τ 1 τ 2 > 1 and C 1 ∩ C 2 = C 1 ∩ C 2 . (4) We may assume that there are no bounded gaps G 1 of C 1 such that ∂G 1 ⊆ G 2 and G 1 G 2 for any gap G 2 of C 2 , and vice-versa. If this is not the case, we can inductively replace each gap
G 2 of C 2 by G 2 := G 2 ∪ ∂G 1 j ⊆G 2 G 1 j (since ∂G 1 ⊆ G 2 this is intuitively G 2 with some holes filled in). Then diam(G 2 ) = diam( G 2 )
, possibly infinity. Note that a priori in this new sequence of gaps we could have gaps contained in another gap, but that can be easily fixed by removing (in order of the sequence) the gaps that are contained in a previous gap. In this way we obtained a compact set C 2 that satisfiesτ 2 ≥ τ 2 and C 1 ∩
C 2 = C 1 ∩ C 2 .
In a symmetric manner, we may repeat this procedure with C 1 and C 2 to obtain C 1 and C 2 withτ 1 ≥ τ 1 andτ 2 ≥ τ 2 and C 1 ∩ C 2 = C 1 ∩ C 2 , and such that all gaps satisfy condition (4). (There may remain gaps of C 1 fully contained in gaps of C 2 or vive-versa, and these may be removed by Step (3).) (5) We may assume that ∂G 1 G 2 for every bounded gap G 1 of C 1 and every gap G 2 of C 2 , and vice-versa. This combines Steps 3 and 4.
(6) We may assume that ∂G 1 G 2 and ∂G 2 G 1 for all gaps G 1 of C 1 and all gaps G 2 of C 2 . This means that we can assume that C 1 and C 2 satisfy the hypothesis of Theorem 9. We consider in turn the cases when G 1 and G 2 are unbounded and bounded gaps.
• Case G 1 = E 1 and G 2 = E 2 : was proved in Step 2.
• Case G 1 bounded and G 2 = E 2 : By Step 5 we have that ∂G 1 E 2 . To check that ∂E 2 G 1 , note that if ∂E 2 ⊆ G 1 with G 1 a bounded gap of C 1 , then C 2 ⊆ G 1 , contradicting the gap containment hypothesis of this Theorem.
• Case G 2 bounded and G 1 = E 1 : as in the previous case.
• Case G 1 and G 2 bounded: was proved in Step 5.
Thus we can replace C 1 and C 2 by a pair of sets with the same intersection and at least the same thicknesses which satisfy the hypotheses of Proposition 9, and applying it completes this proof.
Thickness and winning sets
Schmidt's game and its variants are a powerful tool for investigating properties of intersections of sequences of sets, see [2] for a survey. We will define a game and prove that every set with positive thickness can be seen as a winning set with certain parameters for the game. We will show that game has good properties, for example monotonicity in its parameters, invariance under similarities, and that the intersection of winning sets is again a winning set with different parameters. Theorem 18, proved in the Appendix, gives a lower bound for the Hausdorff dimension of winning sets for this game and this leads to Theorem 6 on the dimension of intersections.
Definition of the Game. We define a game in R d similar to the potential game from [4] but adapted to our purposes: Definition 11. Given α, β, ρ > 0 and c ≥ 0, Alice and Bob play the (α, β, c, ρ)-game in R d under the following rules:
• For each m ∈ N 0 Bob plays first, and then Alice plays. • lim m→∞ ρ m = 0 (Note that this is a non-local rule for Bob. One can define the game without this rule, adding that Alice wins if lim m→∞ ρ m = 0. But to make the definitions simpler we added this condition as a rule for Bob.)
Alice is allowed not to erase any set, or equivalently to pass her turn.
There exists a single point x ∞ = m∈N 0 B m called the outcome of the game. We say a set S ⊂ R d is an (α, β, c, ρ)-winning set, or just a winning set when the game is clear, if Alice has a strategy guaranteeing that if x ∞ / ∈ m∈N 0 i A i,m , then x ∞ ∈ S. Note that the conditions B 0 ⊇ B 1 ⊇ · · · and lim m→∞ ρ m = 0 imply β < 1.
Good properties of the game.
Proposition 12 (Countable intersection property). Let J be a countable index set, and for each j ∈ J let S j be an (α j , β, c, ρ)-winning set, where c > 0. Then, the set S := j∈J S j is (α, β, c, ρ)-winning where α c = j∈J α c j (assuming that the series converges).
To see this, it is enough to consider the following strategy for Alice: in the turn k she plays the union over j of all the strategies of turn k.
Proposition 13 (Monotonicity). If S is (α, β, c, ρ)-winning andα ≥ α,β ≥ β,c ≥ c andρ ≥ ρ, then S is (α,β,c,ρ)-winning.
This holds because so Alice can answer in the (α,β,c,ρ)-game using her strategy to answer from the (α, β, c, ρ)-game. This holds by "translating" the strategies being played through f .
Remark 15 (Relationship with the potential game in [4]). Let P be the set of singletons in R d . Since every set A is contained in a ball of radius diam(A), if S ⊆ R d is an (α, β, c, ρ)-winning set, then it is an (α, β, c, ρ, P)-potential winning set in the game defined in [4].
Relationship between thickness and winning sets. We now establish the key property that relates winning sets to thickness. Proof. We first describe a strategy for Alice. Given a move B by Bob, how does Alice respond? If there exists n ∈ N such that B intersects G n and diam(B) < dist(G n , 1≤i≤n−1 G i ∪ E), then B ∩ G n = ∅ and B ∩ G k = ∅ for all 1 ≤ k < n and B ∩ E = ∅. Alice erases G n if it is a legal move, otherwise Alice does not erase anything.
To show that this strategy is winning, suppose that x ∞ / ∈ m A m . We want to show that x ∞ ∈ S. Otherwise x ∞ / ∈ S so there exists n such that x ∞ ∈ G n . We will show that Alice erases G n at some stage of the game. By definition x ∞ ∈ B m for all m ∈ N 0 , and we assumed x ∞ ∈ G n , so x ∞ ∈ B m ∩ G n for all m ∈ N 0 . Since τ > 0, then dist(G n , 1≤i≤n−1 G i ∪ E) > 0. Also lim m→∞ diam(B m ) = 0, so taking m n ∈ N 0 to be the smallest integer such that dist(G n , 1≤i≤n−1 G i ∪ E) > diam(B mn ), we know that B mn ∩ G n = ∅ and B mn ∩ G k = ∅ for all 1 ≤ k < n. If m n = 0, then
diam(B 0 ) = 2ρ 0 ≥ 2ρ = β diam(C) ≥ βdist G n , 1≤i≤n−1 G i ∪ E . If m n > 0, then diam(B mn ) ≥ β diam(B mn−1 ) ≥ βdist G n , 1≤i≤n−1 G i ∪ E . So diam(B mn ) ≥ βdist G n , 1≤i≤n−1 G i E . Hence, diam(G n ) ≤ 1 τ dist G n , 1≤i≤n−1 G i ∪ E ≤ 1 τ β diam(B mn ).
This means that it is legal for Alice to erase G n in the m n -th turn, and her strategy specifies that she does so. Finally, if m i = m j then the first gap intersecting B m i = B m j is G j and also G i , so i = j; thus the elements of {m n : n ∈ N} are all different.
Observation 17. Let C be a compact set in R d and τ := τ (C) > 0. Then, by Proposition 16 and monotonicity, S := C ∪ E is a 1 τ β , β, c, β 2 diam(C) -winning set for all β ∈ (0, 1) and all c ≥ 0.
A lower bound for the dimension of intersections of thick compact sets in R d
Whilst the gap lemma, Theorem 10, concerns the intersection of just two sets, it is of interest to obtain conditions that ensure that finitely many, or even countably many compact subsets of R d have non-empty intersection. Using the game introduced in Definition 11 we not only obtain conditions involving thickness that ensure that such collection of sets in has non-empty intersection, but also get a lower bound for the Hausdorff dimension of this intersection, as stated in Theorem 6.
To achieve this we use the following technical theorem that gives a lower bound for the dimension of winning sets, based on [4, Theorem 5.5] and [22,Theorem 4] and proved in the Appendix A. The parameters of a winning set provide a measure of its size and we translate this in terms of thickness which is a single number that is easy to compute and work with.
Theorem 18. Let S ⊆ R d be an (α, β, c, ρ)-winning set with c < d and β ≤ 1 4 . Then for every ball B 0 of radius larger than ρ,
dim H (S ∩ B 0 ) ≥ d − K 1 α d | log(β)| > 0 if α c ≤ 1 K 2 (1 − β d−c ),
where K 1 and K 2 are as in (5).
We now prove Theorem 6 by combining Theorem 18 with the fact that sets of positive thickness can be regarded as winning sets.
Proof of Theorem 6. By Observation 17, for each i
S i := E i ∪ C i is a 1 τ i β , β, c, β 2 diam(C i ) -winning set
for all β ∈ (0, 1) and all c ≥ 0. We fix c ∈ (0, d) and β ∈ (0, 1 4 ] from the hypothesis (iii). We define ρ := β 2 sup i diam(C i ) which is a finite number by hypothesis (i). By monotonicity, Proposition 13, S i a 1 τ i β , β, c, ρ -winning set. Hence, by Proposition 12,
S := i S i is a (α, β, c, ρ)-winning set, where (8) α := i (τ i β) −c 1/c = 1 β i τ −c i 1/c .
By hypothesis (ii) there exists a ball B such that B ∩ E i = ∅ for all i and we take r to be the radius of B. By definition of ρ and β, we have r ≥ ρ. By hypothesis (iii) and equation (8)
we have α c ≤ 1 K 2 (1 − β d−c )
, hence we can apply Theorem 18 to get
dim H (S ∩ B) ≥ d − K 1 α d | log(β)| > 0,
and we know by definition of α that d − K 1
α d | log(β)| = d − K 1 ( i τ −c i ) d/c β d | log(β)| . Since B does not intersect any E i , S i ∩ B ⊆ S i ∩ E C i ∩ B = C i ∩ B for every i, so S ∩ B ⊆ B ∩ i C i .
The conclusion follows.
Application: Patterns in thick compact sets of R d
In this section we deduce Theorem 7 on the existence of small copies of pattens in sufficiently thick sets from Theorem 6 and illustrate this in the case of Sierpiński carpets.
Proof of Theorem 7. We write B 0 := 1 8 B for the ball with the same centre as B but with radius 1 8 rad(B). Given a finite set A and λ ∈ 0, diam(B) 16 diam(A) we seek translates of λA := {b 1 , · · · , b n } with b i ∈ R d where we can assume b 1 = 0. As diam(λA) < diam(B) 16 then λA ⊆ B(0, diam(B) 16 ). We define C i := C −b i which is a compact set with thickness τ for every 1 ≤ i ≤ n. By hypothesis there is a ball
B ⊆ E C , so there is a ball B ⊆ 1≤i≤n (B − b i ) ⊆ 1≤i≤n E C i of diameter diam(B)(1 − 1 16 ) = 15 16 diam(B)
. We take β := min{ 1 4 , diam(B) diam(C) }, α := 1/τ β and c := d − 1/ log(τ β). Then α c = eα d and d − c = 1/ log(τ β).
By Theorem 6, if
(9) nα c ≤ 1 K 2 (1 − β d−c ) or equivalently n ≤ 1 K 2 α −c (1 − β d−c ), then dim H ( B ∩ 1≤i≤n C i ) > 0.
By definition of α, β and c, and using that f (τ ) := log(τ )(1−β 1/ log(τ β) ) is a decreasing function with lim τ →∞ f (τ ) = | log(β)|,
1 K 2 α −c (1 − β d−c ) = 1 eK 2 (τ β) d (1 − β 1/ log(τ β) ) = 1 eK 2 τ d log(τ ) β d log(τ )(1 − β 1/ log(τ β) ) ≥ 1 eK 2 β d | log(β)| τ d log(τ ) Setting N (τ ) := β d | log(β)| eK 2 τ d log(τ ) it follows from if (9) that if n ≤ N (τ ) then dim H ( B ∩ 1≤i≤n C i ) > 0. If x ∈ X := B ∩ 1≤i≤n C i , then x+b i ∈ C i +b i = C for every 1 ≤ i ≤ n, so C ⊇ x+{b 1 , · · · , b n } = x+λA as required.
Sierpiński carpets and sponges. Sierpiński carpets and sponges provide examples of sets for which thickness is easily found and which satisfy Theorem 7 .
Let n 1 , · · · , n d ∈ N ≥3 be odd natural numbers. Let D = i := (i 1 , . . . , i d ) : 1 ≤ i k ≤ n k , with (i 1 , . . . , i d ) = 1 2 (n 1 + 1), . . . , 1 2 (n d + 1) .
The family of affine maps
f i : R d → R d : i ∈ D , where f i (x 1 , . . . , x d ) = x 1 + i 1 − 1 n 1 , . . . , x d + i d − 1 n d ,
forms an iterated function system, which defines a unique non-empty compact set C ⊂ R d such that C = i∈D f i (C), see [7,Chapter 9]. Then C is a self-affine Sierpiński sponge (carpet if d = 2) which can also be realised iteratively by repeatedly substituting the coordinate parallelepipeds obtained by dividing the unit cube [0, 1] d into n 1 ×· · ·×n d smaller parallelepipeds, with the central one removed, into themselves. In other words
C = ∞ k=0 i 1 ,...,i k ∈D f i 1 • · · · • f i k ([0, 1] d ).
We will find the thickness of C. Each parallelepiped at the kth step of the iterative construction has side-lengths 1/n k i (1 ≤ i ≤ d). Thus the central parallelepipeds that are removed and which form gaps at the kth step have diameter
diam k := 1≤k≤d 1 n 2k i .
The minimal distance of a gap removed at the kth step from the previous gaps and the external complementary component E is
dist k := min 1≤i≤d 1 n k i n i − 1 2 .
Hence, the thickness of C is
τ := τ (C) = inf k∈N dist k diam k = inf k∈N min 1≤i≤d 1 n k i n i −1 2 1≤k≤d 1 n 2k i .
Thus, with β := min{ 1 4 , 15
16 √ d }, Theorem 7 gives that C contains homothetic copies of every pattern with at most N (τ ) points where N (τ ) is given by (6).
For example, the self-similar Sierpiński carpet C n in R 2 , taking n 1 = n 2 = n above, has thickness τ = (n−1) 2 √ 2, so there is a homothetic copy in C n of every configuration of up to N (τ ) points. Because K 2 is large, n needs to be large to guarantee even that similar copies of all triangles can be found in C n . On the other hand, for C n to contain copies of all k-point configurations, n = O((k log k) 1/2 ) which does not increase too rapidly for large k.
Thickness and Hausdorff dimension
In this section we obtain two different lower bounds for the Hausdorff dimension of sets in R d in terms of their thickness.
Firstly, Theorem 6 yields a lower bound by taking a single set C.
Corollary 19. Let C be a compact set in R d with positive thickness τ (so diam(C) < +∞ and there is a ball B such that B ∩ E = ∅). If there exists c ∈ (0, d) such that
τ −c ≤ 1 K 2 β c (1 − β d−c ) for β := min{ 1 4 , diam(B) diam(C) } then dim H (C) ≥ dim H (B ∩ C) ≥ d − K 1 τ −d β d | log(β)| > 0.
Secondly, we can get a lower bound in the case of convex sets with convex gaps by considering 1-dimensional sections.
Proposition 20. Let C 0 be a proper compact convex set in R d where d ≥ 2, and let
C = C 0 \ ∞ k=1 G k , where {G k } k
are open convex gaps ordered by decreasing diameters. Then τ (C ∩ L) ≥ τ (C) for every straight line L that properly intersects C 0 .
Proof. Let L be a straight line that properly intersects
C 0 . Let {I i } ∞ i=1 be the (countable or finite) set of open intervals I i := G k(i) ∩ L in L ordered so that |I i | ≤ |I j | if i ≥ j,
where | | denotes the length of an interval. Let 1 ≤ i ≤ j − 1. There are two cases:
(a) if k(i) < k(j) then
dist(I i , I j ) ≥ dist(G k(i) , G k(j) ) ≥ τ (C) diam(G k(j) ) ≥ τ (C)|I j |; (b) if k(j) < k(i) then dist(I i , I j ) ≥ dist(G k(i) , G k(j) ) ≥ τ (C) diam(G k(i) ) ≥ τ (C)|I i | ≥ τ (C)|I j |.
In both cases dist(I i , I j ) ≥ |I j | for all 1 ≤ i ≤ j − 1 so τ (C ∩ L) ≥ τ (C) from the definition of thickness.
We can now obtain a lower bound for the Hausdorff dimension for these sets in terms of thickness using the bound (7) for sets in R.
Proposition 21. Let C 0 ⊆ R d be a proper compact convex set, and let C = C 0 \ ∞ i=1 G i where {G i } i are open convex gaps. Then (10) dim H (C) ≥ d − 1 + log 2 log(2 + 1/τ (C))
where τ (C) is the thickness of C.
Proof. Let L be a straight line that properly intersects C 0 . Combining the relationship between thickness and Hausdorff dimension for subsets of R stated in (7) with Proposition 20, dim H (C ∩ L) ≥ log 2 log(2 + 1/τ (C ∩ L)) ≥ log 2 log(2 + 1/τ (C)) .
This is true for all lines L in a given direction that properly intersect C 0 , so by a standard result relating the Hausdorff dimension of a set to the Hausdorff dimensions of parallel sections, see for example, [7, Corollary 7.10], inequality (10) follows.
Observation 22. When d = 1 Proposition 21 is better than Corollary 19. For d ≥ 2, Corollary 19 gives a better bound than Proposition 21 when τ is large but when τ is small Proposition 21 is better.
Appendix A. Proof of Theorem 18
The proof of Theorem 18 is based on [4, Theorem 5.5] and [22,Theorem 4] and adapted to our particular setting.
Proof. We can assume without loss of generality that the radius of B 0 is ρ. We let x 0 be the center of B 0 , ρ n := β n ρ radii of balls, and E n := ρ n 2 Z d + x 0 centers of balls of the family
E n := B( ρ n 2 z + x 0 , ρ n ) : z ∈ Z d .
We will take Bob's move of the nth turn from E n . We also define D n := 3ρ n Z d + x 0 ⊂ E n , D n := {B(3ρ n z + x 0 , ρ n ) : z ∈ Z d } ⊂ E n . Note that the elements of D n are disjoint (moreover they are at distance ρ n ).
We fix γ ∈ (0, 1), a small number to be determined later (independent of α, β, c and ρ). Let N := γ d α d . We define the function π n : E n+1 → E n , B → π n (B) in the following way:
• When n = jN for all j: we define π n (B) as the element of E n that contains B such that B is as centered as possible inside that element. • When n = jN for some j: If there exists B ∈ D jN containing B, we define π n (B) := B (it is well defined because in that case there is only one element belonging to D jN ). If not, we define the function as before. Intuitively the function π n carries the elements of level n + 1 to its ancestor of level n.
We use the following notation: for m < n and B ∈ E n , π m (B) := π m • π m+1 • · · · • π n−1 (B) ∈ E m . This is to say, we carry B to its ancestor of level m via the functions π. If Bob plays B ∈ E n in the turn n, we consider that in the previous turns m ∈ {0, · · · , n − 1} Bob has played π m (B). Then, we have the following inclusions of movements from the turn n to the turn 0:
B ⊂ π n−1 (B) ⊂ · · · ⊂ π 0 (B).
We defined the function in this way to guarantee that Bob's moves are legal. Alice responds under her winning strategy. If in the turn n Bob plays B ∈ E n , we define Given any ball B, we denote by 1 2 B the ball with the same center as B and the half of the radius.
Note that as β ≤ 1 4 , if B ∈ D jN and B ∈ E jN +1 satisfy B ∩ 1 2 B = ∅, then B ⊂ B, so π jN (B ) = B. It follows that (11) if n > jN, B ⊂ 1 2 B with B ∈ E n and B ∈ D jN , then π jN (B ) = B.
This is true because if we look at the ancestor of B of level jN + 1, since π n chooses the element belonging to E n that contains B such that B is as centered as possible, that element must intersect 1 2 B. We define for every B ∈ D j
φ j (B) := n<j A∈A * n (B) diam(A i,n ) c .
This is a measure of all of Alice's answers to the ancestors of B. Note that φ 0 (B) = 0. Let
D j := {B ∈ D j : φ j (B) ≤ (γρ j ) c }.
We define D j (B) := {B ∈ D j : B ⊂ 1 2 B}. Some useful bounds.
We denote by rad(B) the radius of the ball B.
Observation 23. If B ∈ D jN , we have that rad( 1 2 B) = 1 2 β jN ρ, and rad(B ) = β (j+1)N ρ for every B ∈ D (j+1)N . We can cover 1 2 B with enlarged balls from D (j+1)N (B) (with radii 4ρ (j+1)N √ d). This gives us a lower bound for #D (j+1)N (B):
L d ( 1 2 B) ≤ #D (j+1)N (B)L d (B 4ρ (j+1)N √ d ), so β −N d 1 2 d 4 d √ d d ≤ #D (j+1)N (B). Proposition 24. If α c ≤ 1 K 2 (1 − β d−c ) where K 2 := max{γ −2d , 2γ −d log(γ −d )}, we have that #(D (j+1)N (B) ∩ D (j+1)N ) ≥ β −N d 1 2 d 4 d √ d d − 3 d γ d (1 + 4 d 2) for all B ∈ D jN .
We start by proving two preliminary lemmas:
Lemma 25. a) For all n ∈ N and B ∈ E n we have that
A∈A(B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B ) d ≤ 3 d α c max α d−c , γ −c ρ (j+1)N rad(B ) d−c . b) If B ∈ D jN then n<jN A∈A * n (B) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B) d ≤ 3 d γ c max γ d−c , γ −c ρ (j+1)N rad(B) d−c .
Proof of Lemma 25. Firstly, splitting into the cases x ≤ y and y ≤ x, it is easy to see that
(12) min 1, x c (γy) c (x + 2y) d ≤ 3 d x c max x d−c , y d−c γ c
for all x, y > 0.
Secondly, we will prove that if n ∈ N and B ∈ E n then the claim a) holds. By applying the inequality (12)
to x := diam(A i,n ) rad(B )
and y :=
ρ (j+1)N rad(B )
, summing over all A i,n ∈ A(B ) and using that Alice is playing legally, we have that
A i,n ∈A(B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B ) d ≤ 3 d A i,n ∈A(B ) diam(A i,n ) rad(B ) c max diam(A i,n ) rad(B ) d−c , γ −c ρ (j+1)N rad(B ) d−c ≤ 3 d max max A i,n ∈A(B ) diam(A i,n ) rad(B ) d−c , γ −c ρ (j+1)N rad(B ) d−c A i,n ∈A(B ) diam(A i,n ) rad(B ) c ≤ 3 d α c max α d−c , γ −c ρ (j+1)N rad(B ) d−c .
Finally, we prove the claim b). By applying the inequality (12)
to x := diam(A i,n ) rad(B)
and y :=
ρ (j+1)N rad(B)
, summing over all elements of n<jN A * n (B), and using that, since B ∈ D jN , we have
n<jN A i,n ∈A * n (B) diam(A i,n ) rad(B) c ≤ γ c , and in particular diam(A i,n ) rad(B)
≤ γ for every i and every n < jN , we obtain that:
n<jN A i,n ∈A * n (B) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B) d ≤ 3 d n<jN A i,n ∈A * n (B) diam(A i,n ) rad(B) c max diam(A i,n ) rad(B) d−c , γ −c ρ (j+1)N rad(B) d−c ≤ 3 d n<jN A i,n ∈A * n (B) diam(A i,n ) rad(B) c max γ d−c , γ −c ρ (j+1)N rad(B) d−c ≤ 3 d γ c max γ d−c , γ −c ρ (j+1)N rad(B) d−c .
Now we are ready to prove Proposition 24.
Proof of Proposition 24.
#(D (j+1)N (B) \ D (j+1)N ) ≤ # B ∈ D (j+1)N (B) : φ (j+1)N (B ) (γρ (j+1)N ) c > 1 ≤ B ∈D (j+1)N (B) min 1, φ (j+1)N (B ) (γρ (j+1)N ) c ≤ B ∈D (j+1)N (B) min 1, n<(j+1)N A i,n ∈A * n (B ) diam(A i,n ) c (γρ (j+1)N ) c ≤ B ∈D (j+1)N (B) n<(j+1)N A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c ≤ B ∈D (j+1)N (B) n<jN A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c + B ∈D (j+1)N (B) jN ≤n<(j+1)N A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c .(13)
We have split the sum in (13) into two parts, depending on whether n < jN or jN ≤ n < (j + 1)N .
To get a bound for the left-hand sum of (13) we will use that if n < jN then
(B , A) : B ∈ D (j+1)N (B), A ∈ A * n (B ) ⊂ (B , A) : B ∈ D (j+1)N (B), A ∈ A * n (B), A ∩ B = ∅ .
Since B ∈ D jN the set A * n (B) only makes sense for n < jN . This inclusion holds because of (11), as A(π n (B )) ⊂ A(π n (B)) since B ⊂ B.
So,
B ∈D (j+1)N (B) A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c ≤ A i,n ∈A * n (B) B ∈D (j+1)N (B) B ∩A =∅ min 1, diam(A i,n ) c (γρ (j+1)N ) c = A i,n ∈A * n (B) min 1, diam(A i,n ) c (γρ (j+1)N ) c #{B ∈ D (j+1)N (B), B ∩ A = ∅}.
Now, we will get a bound for the right-hand sum in (13), when jN ≤ n < (j + 1)N .
Recall that B ∈ D jN . First, note that if B ∈ D (j+1)N (B), then B ∈ E (j+1)N , B ⊂ 1 2 B where B ∈ D jN . If we take B := π jN (B ) ∈ E jN , by (11) we have that B = B. For all B ∈ D (j+1)N (B), there exists B ∈ E n with B ⊂ B and B ⊂ 1 2 B (B = B if n = jN ). Hence B ∈D (j+1)N (B) A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c ≤ B ∈En B ⊂B B ∈D (j+1)N (B ) A∈A(B ) A∩B =∅ min 1, diam(A i,n ) c (γρ (j+1)N ) c = B ∈En B ⊂B A∈A(B ) B ∈D (j+1)N (B ) A∩B =∅ min 1, diam(A i,n ) c (γρ (j+1)N ) c = B ∈En B ⊂B A∈A(B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c #{B ∈ D (j+1)N (B ) : A ∩ B = ∅},
where the inequality holds by considering in particular B := π n (B ) ⊂ B. By inequality (13), and what we have noted before,
#(D (j+1)N (B) \ D (j+1)N ) ≤ B ∈D (j+1)N (B) n<jN A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c + B ∈D (j+1)N (B) (j+1)N −1 n=jN A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c ≤ n<jN A i,n ∈A * n (B) min 1, diam(A i,n ) c (γρ (j+1)N ) c #{B ∈ D (j+1)N (B) : B ∩ A = ∅} + (j+1)N −1 n=jN B ∈En B ⊂B A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c #{B ∈ D (j+1)N (B ) : B ∩ A = ∅} ≤ rad(B) ρ (j+1)N d n<jN A i,n ∈A * n (B) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B) d + (j+1)N −1 n=jN B ∈En B ⊂B rad(B ) ρ (j+1)N d A i,n ∈A * n (B ) min 1, diam(A i,n ) c (γρ (j+1)N ) c diam(A i,n ) + 2ρ (j+1)N rad(B ) d ,(14)
where in the first term of the last inequality we use that B ∈ D jN (so φ jN (B) ≤ (γρ jN ) c ), in the second term that Alice is playing legally (i.e.: i diam(A i,m ) c ≤ (αρ m ) c ), and in both terms that: for every B ∈ n E n and every A i,n , since the elements of
D (j+1)N are disjoint, L d (B ) = C d ρ d (j+1)N , and if moreover B ∩A i,n = ∅ then B ⊂ N (A i,n , 2ρ (j+1)N ) (the 2ρ (j+1)N -neighborhood of A i,n ), which is contained in a ball of radius diam(A i,n ) + 2ρ (j+1)N . Therefore, #{B ∈ D (j+1)N (B ) : B ∩ A i,n = ∅}C d ρ d (j+1)N = L d B ∈D (j+1)N (B ) B ∩A i,n =∅ B ≤ L d N (A i,n , 2ρ (j+1)N ) ≤ C d (diam(A i,n ) + 2ρ (j+1)N ) d ,
in other words,
#{B ∈ D (j+1)N (B ) : B ∩ A i,n = ∅} ≤ (diam(A i,n ) + 2ρ (j+1)N ) d ρ d (j+1)N .
By inequality (14), using claim b) from Lemma 25 to bound the first term, and claim a) from Lemma 25 to bound the second one, we obtain:
#(D (j+1)N (B) \ D (j+1)N ) ≤ rad(B) ρ (j+1)N
To continue the estimates, we will use that
rad(B) ρ (j+1)N = β jN ρ β (j+1)N ρ = β −N .
To bound the second term in (15) we write n = (j + 1)N − k for some k ∈ {1, · · · , N }. We know that B := B(3ρ jN z +x 0 , ρ jN ) for some z ∈ Z d , and recall that E n := {B( ρn 2 z + x 0 , ρ n ) : z ∈ Z d }. So, rad(B ) ρ (j+1)N = β −k and #{B ∈ E n :
B ⊂ B} ≤ #{ ρ n 2 z + x 0 ∈ B : z ∈ Z d } = # z ∈ Z d ∩ B 6 β N −k z, 2 1 β N −k − 1 ≤ 4 1 β N −k − 1 + 1 d ≤ 4 d 1 β d(N −k) .
Combining with (15)
≤ β −N d 3 d γ c max{γ d−c , γ −c β N (d−c) } + 1≤k≤N 4 d β −d(N −k) β dk 3 d α c max{α d−c , γ −c β k(d−c) } ≤ β −N d 3 d max{γ d , β N (d−c) } + 4 d N α d + α c γ −c 1≤k≤N β k(d−c) ,(16)
where in the last inequality we have used that if a n , b n ≥ 0 then n max{a n , b n } ≤ n a n + n b n . Provided we can establish the following claims:
(i) N α d ≤ γ d , (ii) α c γ −c k∈N 0 β k(d−c) ≤ γ d , (iii) β N (d−c) ≤ γ d ,γ c (1 − β d−c ) ≤ 1 γ c K 2 ≤ γ 2d−c < γ d ,
for the second claim. (iii) Continuing, since α c ≤ α c 1 1−β d−c ≤ 1 K 2 ≤ γ 2d , then 1 ≤ γ −(2d−c) ≤ ( γ α ) c , so γ/α ≥ 1, and thus
(17) N ≥ 1 2 γ d α −d .
On the other hand, using the hypotheses, c ∈ (0, d) and α ∈ (0, 1),
(18) α d ≤ α c ≤ 1 K 2 (1 − β d−c ) ≤ 1 K 2 | log(β d−c )| = 1 K 2 (d − c)| log(β)|,
where in the last inequality we have used that d − c ∈ (0, 1), β ∈ (0, 1 4 ], z := β d−c ∈ (0, 1), and f (z) := log( 1 z ) + z + 1 is a positive function on (0, 1), so 1 − z ≤ log( 1 z ). Then, (19) N α d K 2 ≤ N (d − c)| log(β)|.
By inequalities (17) and (19) and the definition of K 2 ,
N (d − c)| log(β)| ≥ N α d K 2 ≥ γ d 2 K 2 ≥ | log(γ d )|
which is equivalent to claim (iii). This concludes the proof of Proposition 24.
Conclusion of the proof.
For each γ ∈ (0, 1) we proceed as follows: By definition, B 0 ∈ D 0 . Moreover, φ 0 (B 0 ) := 0 < (γρ) c , so B 0 ∈ D 0 . We will construct a Cantor set F as the intersection of a sequence of unions of closed sets:
• B 0 := {B 0 } ⊂ D 0 .
legal move for Alice we know that i A i,m ∈ A(B m )). So x ∈ A ∈ A(B m ) for some m, and as x ∈ B m we have x ∈ A ∩ B m . Since A * m (B n ) = A(B m ) for every n > m (because π m (B n ) = B m ), then φ j (B jN ) ≥ (diam A) c for every j such that jN > m (because (diam A) c is just one term in the sum of the definition of φ j (B jN ) when A ∈ A * m (B n )). On the other hand, since B jN ∈ D j , then φ j (B jN ) ≤ (γρ jN ) c . Putting everything together, diam A ≤ γρ jN for all j such that jN > m. Letting j → ∞, we get diam A = 0, a contradiction. So x ∈ S, that is F ⊂ S.
Finally, using (18), and that K 1 /K 2 < 1,
K 1 α d | log(β)| ≤ (d − c)K 1 K 2 < d, so d − K 1 α d | log(β)| > 0 if α c ≤ 1 K 2 (1 − β d−c ).
This concludes the proof.
15 diam(B) 16 diam(C).
Figure 1 .
1An example of gaps G 1 and G 2 which intersect but are not linked and which might be parts of the complements of compact sets C 1 and C 2 which satisfy the hypotheses of Theorem 10 but not of Proposition 9.
•
On the m-th turn, Bob plays a closed ball B m := B[x m , ρ m ], satisfying ρ 0 ≥ ρ, and ρ m ≥ βρ m−1 and B m ⊆ B m−1 for every m ∈ N. • On the m-th turn Alice responds by choosing and erasing a finite or countably infinite collection A m of open sets. Alice's collection must satisfy i (diam A i,m ) c ≤ (αρ m ) c if c > 0, or diam A 1,m ≤ αρ m if c = 0 (in the case c = 0 Alice can erase just one set).
Proposition 14 (
14Invariance under similarities). Let f : R d → R d be a similarity satisfying dist(f (x), f (y)) = λdist(x, y) for all x, y ∈ R d . Then a set S is (α, β, c, ρ)-winning if and only if the set f (S) is (α, β, c, λρ)-winning.
Proposition 16 .
16Let C ⊂ R d be compact with unbounded complement E and write S := C ∪E. If τ := τ (C) > 0, then S is 1 τ β , β, 0, β diam(C) 2 -winning for every β ∈ (0, 1).
A
(B) as Alice's answer (each A ∈ A(B) is a countable collection of sets A := {A i,n } i , and a legal movement as an answer for B, i.e.: i diam(A i,n ) c ≤ (αρ n ) c ). Let A * m (B) := {A ∈ A(π m (B)) : B ∩ A = ∅} be Alice's answer (this is a list of sets) to the ancestor of B of level m < n.
γ c max γ d−c , γ −c ρ (j+1)N rad(B) α c max α d−c , γ −c ρ (j+1)N rad(B ) d−c
then #(D (j+1)N (B) \ D (j+1)N ) ≤ β −N d 3 d γ d (1 + 4 d 2); hence, by Observation 23,#(D (j+1)N (B) ∩ D (j+1)N ) ≥ β −N d 1 2 d 4 d √ d d − 3 d γ d (1 + 4 d 2) ,as required. Let us prove (i)-(iii): (i) This holds by the definition of N . (ii) Take K 2 := max{γ −2d , 2γ −d log(γ −d )}. By hypothesis and by using c ∈ (0, d), β ∈ (0, 1 4 ], we have α c
d γ c max γ d−c , γ −c ρ (j+1)N rad(B) d−c
AcknowledgementsAlexia Yavicoli was financially supported by the Swiss National Science Foundation, grant n • P2SKP2 184047.• Given a collection B j ⊂ D jN we construct the next level of sets B j+1 ⊂ D (j+1)N by replacing each element of B ∈ B j by M := β −N dBy a standard argument (see e.g. [7, Example 4.6]),where we have used(17). This last inequality holds for every γ ∈ (0, 1). We can take, for example, γ ∈ (0, 1)It remains to prove that F ⊂ S ∩ B 0 , since thenClearly F ⊂ B 0 , by definition of F . We need to show that F ⊂ S. Let x ∈ F . For every j ∈ N there exists a unique B jN ∈ B j containing x. By definition of B j+1 we have that B (j+1) ⊂ 1 2 B jN . By(11), π jN (B (j+1)N ) = B jN . The sequence (B jN ) j can be extended in a unique way to a sequence (B n ) n satisfying B n ∈ E n and B n := π n (B n+1 ) for all n. We interpret this sequence as Bob's moves, to which Alice responds according to her winning strategy.Thus, for each x ∈ F we construct a sequence (B n ) n as before, where x is the only element of n B n (so x = x ∞ is the outcome of the game). We will show that x ∈ S by contradiction. Otherwise, suppose that x / ∈ S where S is an (α, β, c, ρ)-winning set. Then, x ∈ m∈N 0 i A i,m , where i (diam A i,m ) c ≤ (αρ m ) c = (αβ m ρ) c (since it is a
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| [] |
[
"ROOTS OF EHRHART POLYNOMIALS OF SMOOTH FANO POLYTOPES",
"ROOTS OF EHRHART POLYNOMIALS OF SMOOTH FANO POLYTOPES"
] | [
"Gábor Hegedüs ",
"Alexander M Kasprzyk "
] | [] | [] | V. Golyshev conjectured that for any smooth polytope P with dim(P ) ≤ 5 the roots z ∈ C of the Ehrhart polynomial for P have real part equal to −1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six. | 10.1007/s00454-010-9275-y | [
"https://arxiv.org/pdf/1004.3817v1.pdf"
] | 21,179,830 | 1004.3817 | cc41f8706e50f21ea00cb815e4383e8b0389212f |
ROOTS OF EHRHART POLYNOMIALS OF SMOOTH FANO POLYTOPES
21 Apr 2010
Gábor Hegedüs
Alexander M Kasprzyk
ROOTS OF EHRHART POLYNOMIALS OF SMOOTH FANO POLYTOPES
21 Apr 2010arXiv:1004.3817v1 [math.CO]
V. Golyshev conjectured that for any smooth polytope P with dim(P ) ≤ 5 the roots z ∈ C of the Ehrhart polynomial for P have real part equal to −1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.
Introduction
Let P be a d-dimensional convex lattice polytope in R d . Let L P (m) := mP ∩ Z d denote the number of lattice points in P dilated by a factor of m ∈ Z ≥0 . In general the function L P is a polynomial of degree d, called the Ehrhart polynomial [Ehr67].
The roots of Ehrhart polynomials have recently been the subject of much study (for example [BHW07, BD08, HHO10, Pfe07]), with a significant portion of this work being based on exhaustive computer calculations using the known classifications of polytopes. It has been conjectured in [BDLD + 05] that if z ∈ C is a root of L P , then the real part Re(z) is bounded by −d ≤ Re(z) ≤ d − 1; Braun has shown [Bra08] that z lies inside the disc centred at −1/2 of radius d(d − 1/2). Definition 1.1. A convex lattice polytope P is called reflexive if the dual polytope
P ∨ := {u ∈ R d | u, v ≤ 1 for all v ∈ P }
is also a lattice polytope.
There are many interesting and well-known characterisations of reflexive polytopes (for example [HK10, Theorem 3.5]). They are of particular relevance to toric geometry: reflexive polytopes correspond to Gorenstein toric Fano varieties (see [Bat94]) and have been classified up to dimension four.
Any reflexive polytope P satisfies
(1.1) L P (m) = L ∂P (m) + L P (m − 1) for all m ∈ Z >0 ,
where ∂P denotes the boundary of P . As a consequence, Macdonald's Reciprocity Theorem [Mac71] tells us that L P (−m − 1) = (−1) d L P (m). In particular we observe that the roots of L P are symmetrically distributed with respect to the line Re(z) = −1/2. Clear any smooth polytope is simplicial and reflexive. Smooth polytopes are in bijective correspondence with non-singular toric Fano varieties, and have been classified up to dimension eight [Øbr07].
V. Golyshev conjectured in [Gol09,§5] that, for any smooth polytope P of dimension d ≤ 5, the roots z ∈ C of L P satisfy Re(z) = −1/2 (the "canonical line hypothesis"). Notice that it is not required that z / ∈ R. We prove Golyshev's conjecture without resorting to the known classifications -see Sections 2 and 3 below.
Theorem 1.5 (Golyshev). Let P be a smooth polytope of dimension d ≤ 5. If z ∈ C is a root of L P (m) then Re(z) = −1/2.
Explicit descriptions of the roots are given in Corollaries 2.6 and 3.8. We summarise them in the following theorem.
Theorem 1.6. Let P be a smooth d-dimensional polytope, and suppose that z = −1/2 + βi ∈ C is a root of L P . If d = 2 then
β 2 = − 1 4 + 2 f 0 .
If d = 3 then β = 0 or
β 2 = − 1 4 + 6 f 0 − 2 .
If d = 4 then
β 2 = − 17 4 + 3b 2 b 2 − 2f 0 ± 1 − 12(f 0 + 2) b 2 − 2f 0 + 36f 2 0 (b 2 − 2f 0 ) 2 .
If d = 5 then β = 0 or
β 2 = − 5 4 + 10(f 0 − 2) 6 + b 2 − 4f 0 ± 1 − 20(f 0 + 4) 6 + b 2 − 4f 0 + 100(f 0 − 2) 2 (6 + b 2 − 4f 0 ) 2 .
The following example demonstrates that we cannot extend Theorem 1.5 to dimension 6.
Example 1.7. There exist exactly four smooth polytopes in dimension six having roots z of the Ehrhart polynomial such that Re(z) = −1/2; in each case z / ∈ R. The second polytope has roots where Re(z) > 0, and where Re(z) < −1. This demonstrates that the more general "canonical strip hypothesis" does not hold in dimension six.
Dimensions Two and Three
One of the fundamental pieces of numerical data associated with a polytope is the f -vector, which enumerates the number of faces of P . We begin by deriving an expression for the Ehrhart polynomial of a smooth polytope in terms of its f -vector.
≤ i ≤ d − 1. The f -vector of P is the sequence (f −1 , f 0 , . . . , f d ).
Lemma 2.2. Let P be a d-dimensional smooth polytope. Then
L P (m) = d−1 i=−1 f i m i + 1 and L ∂P (m) = d−1 i=0 f i m − 1 i . Proof. Clearly L ∂P (m) = f 0 + F (mF ) • ∩ Z d ,
where the sum is taken over all i-dimensional faces F of P , i > 0, and Q • denotes the (relative) interior of Q. Since P is smooth, F ∩ Z d forms part of a basis for the underlying lattice Z d for any face F . Hence
L ∂P (m) = d−1 i=0 f i m − 1 i .
To calculate L P (m) we make use of (1.1):
L P (m) = 1 + m k=1 L ∂P (k) = 1 + m k=1 d−1 i=0 f i k − 1 i = 1 + d−1 i=0 f i m k=1 k − 1 i = 1 + d−1 i=0 f i m i + 1 = d−1 i=−1 f i m i + 1 .
The f -vectors of low-dimensional smooth polytopes were calculated in [HK10, Theorem 4.2]. As a consequence we obtain the following formulae for the Ehrhart polynomial:
Corollary 2.3. Let P be a d-dimensional smooth polytope. Define b 2 := ∂(2P ) ∩ Z d . If d = 2 then L P (m) = 1 + 1 2 f 0 m + 1 2 f 0 m 2 . If d = 3 then L P (m) = 1 + 1 6 (f 0 + 10)m + 1 2 (f 0 − 2)m 2 + 1 3 (f 0 − 2)m 3 If d = 4 then L P (m) = 1 + 1 12 (8f 0 − b 2 )m + 1 24 (14f 0 − b 2 )m 2 − 1 12 (2f 0 − b 2 )m 3 − 1 24 (2f 0 − b 2 )m 4 . If d = 5 then L P (m) = 1 + 1 60 (14f 0 − b 2 + 94)m + 1 24 (16f 0 − b 2 − 30)m 2 + 1 3 (f 0 − 2)m 3 − 1 24 (4f 0 − b 2 − 6)m 4 − 1 60 (4f 0 − b 2 − 6)m 5 .
Casagrande provides sharp bounds on the number of vertices f 0 of a smooth polytope in terms of the dimension: Cas06]). Let P be a d-dimensional smooth polytope. Then
Theorem 2.4 ([f 0 ≤ 3d, if d is even; 3d − 1, if d is odd.
We now prove Theorem 1.5 without resorting to the classifications in dimensions 2 and 3.
Proposition 2.5. Let P be a smooth polytope of dimension two or three. If z ∈ C is a root of L P (m) then Re(z) = −1/2.
Proof. d = 2: By Corollary 2.3 we know that
L P (m) = 1 + 1 2 f 0 m + 1 2 f 0 m 2 .
Let α + βi ∈ C be a root of L P , where α, β ∈ R. Assume that β = 0. By considering the imaginary part we obtain β(1 + 2α) = 0, hence α = −1/2 as required. The real part simplifies to
β 2 = 2 f 0 − 1 4 .
Theorem 2.4 tells us that this is always positive, thus we obtain both roots of L P . d = 3: In this case Corollary 2.3 tells us that
L P (m) = 1 + 1 6 (f 0 + 10)m + 1 2 (f 0 − 2)m 2 + 1 3 (f 0 − 2)m 3 ,
giving real and imaginary parts:
(2.1) 1 + 1 6 (f 0 + 10)α + 1 2 (f 0 − 2)(α 2 − β 2 ) + 1 3 (f 0 − 2)(α 2 − 3β 2 )α = 0, (2.2) 1 6 (f 0 + 10)β + (f 0 − 2)αβ + 1 3 (f 0 − 2)(3α 2 − β 2 )β = 0.
Assume that β = 0. Equation (2.2) gives us
(2.3) (f 0 − 2)β 2 = 1 2 f 0 + 5 + 3(f 0 − 2)α + 3(f 0 − 2)α 2 .
Substituting (2.3) into (2.1) gives 1 12 (2α + 1) 4(f 0 − 2)(2α + 1) 2 + 26 − f 0 .
Clearly α = −1/2 is one possible solution. The discriminant of 4(f 0 − 2)(2α + 1) 2 + 26 − f 0 , regarded as a quadratic in 2α + 1, is 16(f 0 − 2)(f 0 − 26). This is negative when 2 ≤ f 0 ≤ 26, and by Theorem 2.4 this covers all possible values of f 0 . Hence α = −1/2 is the only solution.
The values for β are determined by (2.3):
β 2 = 26 − f 0 4f 0 − 8 .
If we allow β = 0 then (2.1) becomes
1 24 (2α + 1) (f 0 − 2)(2α + 1) 2 + 26 − f 0 .
Once more the discriminant of the quadratic component tells us that the only solution is when α = −1/2.
The proof of Proposition 2.5 gives us explicit equations for the roots of L P .
Corollary 2.6. Let P be a smooth d-dimensional polytope, and suppose that z = −1/2 + βi ∈ C is a root of L P . If d = 2 then
β 2 = − 1 4 + 2 f 0 . If d = 3 then β = 0 or β 2 = − 1 4 + 6 f 0 − 2 .
Dimensions Four and Five
In order to prove Theorem 1.5 in dimension 4 we require a some additional results. Throughout we write b 2 := ∂(2P ) ∩ Z d , where d is the dimension of P .
5f 0 − 10 ≤ b 2 ≤ 5f 0 .
Lemma 3.2. Let P be a four-dimensional smooth polytope. Then
(b 2 − 8f 0 ) 2 > 24(b 2 − 2f 0 ). Proof. From Lemma 3.1 we have that (b 2 − 8f 0 ) 2 = (b 2 − 16f 0 )b 2 + 64f 2 0 ≥ (10 − 5f 0 )(10 + 11f 0 ) + 64f 2 0 = 9f 2 0 + 60f 0 + 100 = (3f 0 + 10) 2
Clearly 72f 0 < (3f 0 + 10) 2 , and since 24(b 2 − 2f 0 ) ≤ 72f 0 (by Lemma 3.1) we obtain the result.
We shall also make use of the following trivial observation:
Lemma 3.3. Let g(x) := ax 4 + bx 2 + c ∈ R[x]
be a polynomial such that a > 0, b < 0, c > 0 and b 2 − 4ac > 0. Then g has four distinct real roots.
Proposition 3.4. Let P be a four-dimensional smooth polytope. If z ∈ C is a root of L P (m) then Re(z) = −1/2.
Proof. In four dimensions the Ehrhart polynomial simplifies to
L P (m) = 1 + 1 12 (8f 0 − b 2 )m(m + 1) − 1 24 (2f 0 − b 2 )m 2 (m + 1) 2 .
If z = α + iβ is a root of L P then, by considering the real and imaginary parts, we obtain
(3.1) 24+ 12f 0 ((α+ 1)α− β 2 )− (2f 0 − b 2 )α(α+ 1)(α(α+ 1)− 2− 6β 2 )− (2f 0 − b 2 )β 2 (β 2 + 1) = 0, (3.2) 6f 0 − (2f 0 − b 2 ) (α + 1)α − β 2 − 1 (2α + 1)β = 0.
Clearly α = −1/2 is a possible solution to equation (3.2), in which case β satisfies (by (3.1))
(3.3) 16(b 2 − 2f 0 )β 4 + 8(5b 2 − 34f 0 )β 2 + 3(128 + 3b 2 − 22f 0 ) = 0.
This quadratic in β 2 has distinct real solutions if and only if
(b 2 − 8f 0 ) 2 − 24(b 2 − 2f 0 ) > 0.
By Lemma 3.2 we know that this is always true. Now we consider the signs of the coefficients of (3.3). The leading coefficient is equal to 1/2f 2 , and so is positive. The coefficient of β 2 is always negative by Lemma 3.1, and the constant term is positive by Lemma 3.2. Hence, by Lemma 3.3, there are four distinct real solutions to equation (3.1).
We have found four distinct roots when Re(z) = −1/2. Since L P is of degree four, we are done.
Finally we consider dimension five.
42f 0 − 105 ≤ 7b 2 ≤ 52f 0 − 90.
whenever 6 ≤ f 0 ≤ 11. It is enough to prove that, in the given range,
(3.4) b 2 < 2(161f 2 0 + 219f 0 − 662) 23f 0 + 374 . Now b 2 − f 0 = f 1 ≤ f 0 2 , and so b 2 ≤ f 0 (f 0 + 1) 2 .
We shall show that
f 0 (f 0 + 1) 2 < 2(161f 2 0 + 219f 0 − 662) 23f 0 + 374 .
But this is trivial; the cubic
f 0 (f 0 + 1)(23f 0 + 374) − 4(161f 2 0 + 219f 0 − 662) = 23f 3 0 − 247f 2 0 − 502f 0 + 2648
is negative when 6 ≤ f 0 ≤ 11, hence equation (3.4) holds.
Proposition 3.7. Let P be a five-dimensional smooth polytope. If z ∈ C is a root of L P (m) then Re(z) = −1/2.
Proof. Let z = α + iβ ∈ C be a root of L P , where P is a five-dimensional smooth polytope. By Corollary 2.3 we see that α and β must satisfy (2α + 1) (6 + b 2 − 4f 0 ) (α − 1)α(α + 1)(α+2) − 10(α + 1)αβ 2 + 5(β 2 + 1)β 2 + 20(f 0 − 2) (α + 1)α − 3β 2 + 120 = 0,
(3.5) (14f 0 − b 2 + 94)β + 5(16f 0 − b 2 − 30)αβ + 20(f 0 − 2)(3α 2 − β 2 )β− 10(4f 0 − b 2 − 6)(α 2 − β 2 )αβ − (4f 0 − b 2 − 6)(5α 4 − 10α 2 β 2 + β 4 )β = 0. (3.6)
Clearly α = −1/2, β = 0 is always a solution. Suppose that α = −1/2 and β = 0. Equation (3.5) holds, and from (3.6) we obtain
(3.7) 16(6 + b 2 − 4f 0 )β 4 + 40(22 + b 2 − 12f 0 )β 2 + 2134 + 9b 2 − 116f 0 = 0.
This quadratic in β 2 has distinct real solutions if and only if
100(f 0 − 2) 2 + (6 + b 2 − 4f 0 ) 2 > 20(6 + b 2 − 4f 0 )(f 0 + 4),
which holds by Lemma 3.6. As in the four-dimensional case we consider the signs of the coefficients of (3.7). The leading coefficient equals 1/2f 4 and so is positive. The coefficient of β 2 is negative by Lemma 3.5 and the fact that f 0 ≥ 6, and the constant term is positive (again by Lemma 3.5). Thus, by Lemma 3.3, equation (3.7) has four distinct real solutions.
Hence we have found all five roots of L P , and in each case Re(z) = −1/2 as required.
From equations (3.3) and (3.7) we have f 0 5 6 6 7 7 7 8 8 8 8 9 9 9 9 9 10 10 10 11 12 b 2 15 20 21 25 26 27 31 32 33 34 36 38 39 41 42 44 45 50 52 60 Table 1. The possible pairs (f 0 , b 2 ) for the 124 four-dimensional smooth polytopes.
Corollary 3.8. Let P be a smooth d-dimensional polytope, and suppose that z = −1/2 + βi ∈ C is a root of L P . If d = 4 then
β 2 = − 17 4 + 3b 2 b 2 − 2f 0 ± 1 − 12(f 0 + 2) b 2 − 2f 0 + 36f 2 0 (b 2 − 2f 0 ) 2 .
If d = 5 then β = 0 or
β 2 = − 5 4 + 10(f 0 − 2) 6 + b 2 − 4f 0 ± 1 − 20(f 0 + 4) 6 + b 2 − 4f 0 + 100(f 0 − 2) 2 (6 + b 2 − 4f 0 ) 2 .
Concluding Remarks
In four dimensions one can prove Theorem 1.5 without knowing the explicit equation for the Ehrhart polynomial. We require the following result. Alternative proof in dimension four. First we show that condition (i) of Proposition 4.1 is satisfied. Since P is smooth, f 0 = ∂P ∩ Z 4 . It follows from Lemma 3.1 that 15f 0 ≤ 3b 2 + 30. Hence 9f 0 ≤ 3(b 2 − 2f 0 ) + 30. By Theorem 2.4 we have that f 0 ≤ 12, giving us the (very crude) inequality (4.1) 16f 0 < 3(b 2 − 2f 0 ) + 128.
In four dimensions we have that f 3 = b 2 − 2f 0 ([HK10, Theorem 4.2]) and, since P is smooth, f 3 = 24 vol(P ). Substituting into equation (4.1) gives condition (i). That Proposition 4.1 (ii) holds is immediate from Lemma 3.2 and the fact that b 2 − 2f 0 = 24 vol(P ). Theorem 1.6 tells us that in order to compute the roots of the Ehrhart polynomial we need only know f 0 and, in dimensions four and five, b 2 := ∂(2P ) ∩ Z d . Clearly f 0 ≥ d + 1, and Theorem 2.4 provides a sharp upper bound. The values of b 2 can be calculated from Øbro's classification [Øbr07]. The possible pairs (f 0 , b 2 ) are reproduced in Tables 1 and 2. Table 2. The possible pairs (f 0 , b 2 ) for the 866 five-dimensional smooth polytopes.
Definition 2 . 1 .
21Let P be a d-dimensional convex polytope. Define f −1 := 1, f d := 1, and f i equal to the number of i-dimensional faces of P , for any 0
Lemma 3.1 ([HK10, Corollary 4.4]). Let P be a four-dimensional smooth polytope. Then
Lemma 3.5 ([HK10, Corollary 4.4]). Let P be a five-dimensional smooth polytope. Then
Proposition 4.1 ([BHW07, Proposition 1.9]). Let P be a four-dimensional reflexive polytope. Every root z ∈ C of L P (m) has Re(z) = −1/2 if and only if (i) 2 ∂P ∩ Z 4 ≤ 9 vol(P ) + 16, and (ii) ∂P ∩ Z 4 − 4 vol(P ) 2 ≥ 16 vol(P ).
Let P be a d-dimensional convex lattice polytope such that for all roots z of L P , Re(z) = −1/2. Then, up to unimodular translation, P is a reflexive polytope with vol(P ) ≤ 2 d . In each dimension d there exists a reflexive polytope P such that if z ∈ C \ R is a root of L P then Re(z) = −1/2. Definition 1.4. A d-dimensional convex lattice polytope P is called smooth if the vertices of any facet of P form a Z-basis of the ambient lattice Z d .2010 Mathematics Subject Classification. 52B20 (Primary); 52C07, 11H06 (Secondary).
Research supported in part by OTKA grant K77476.
Theorem 1.2 ([BHW07, Proposition 1.8]). Theorem 1.3 ([HHO10, Theorem 0.1]).
The polytopes have IDs 1895, 1930, 4853, and 5817 in the Graded Ring Database 1 . The corresponding Ehrhart polynomialsare:
1 +
31
10
m +
257
60
m 2 +
5
2
m 3 +
19
12
m 4 +
2
5
m 5 +
2
15
m 6 ,
1 +
7
2
m +
175
36
m 2 +
35
12
m 3 +
35
18
m 4 +
7
12
m 5 +
7
36
m 6 ,
1 +
7
2
m +
21
4
m 2 +
15
4
m 3 +
5
2
m 4 +
3
4
m 5 +
1
4
m 6 ,
1 +
31
10
m +
257
60
m 2 +
5
2
m 3 +
19
12
m 4 +
2
5
m 5 +
2
15
m 6 .
http://grdb.lboro.ac.uk/search/toricsmooth?id cmp=in&id=1895,1930,4853,5817
Acknowledgments. The authors wish to express their gratitude to Alessio Corti for alerting them to[Gol09].Lemma 3.6. Let P be a five-dimensional smooth polytope. ThenProof. We begin by observing that the statement is equivalent towhich in turn is equivalent toFrom Lemma 3.5 we have thatwhich is always positive since f 0 ≥ 6. HenceThis is positive for all f 0 ≥ 12.To prove the inequality when f 0 ≤ 11 we consider. We wish to show that − 2 7 (23f 0 + 374)b 2 + 4 7 (161f 2 0 + 219f 0 − 662) > 0
Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. V Victor, Batyrev, J. Algebraic Geom. 33Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric vari- eties, J. Algebraic Geom. 3 (1994), no. 3, 493-535.
Ehrhart polynomial roots and Stanley's non-negativity theorem, Integer points in polyhedra-geometry, number theory, representation theory, algebra, optimization, statistics. Benjamin Braun, Mike Develin, Contemp. Math. 452Amer. Math. SocBenjamin Braun and Mike Develin, Ehrhart polynomial roots and Stanley's non-negativity theorem, Integer points in polyhedra-geometry, number theory, representation theory, algebra, optimization, statistics, Contemp. Math., vol. 452, Amer. Math. Soc., Providence, RI, 2008, pp. 67-78.
Coefficients and roots of Ehrhart polynomials, Integer points in polyhedra-geometry, number theory, algebra, optimization. ] M + 05, J A Beck, M De Loera, J Develin, R P Pfeifle, Stanley, Contemp. Math. 374Amer. Math. Soc+ 05] M. Beck, J. A. De Loera, M. Develin, J. Pfeifle, and R. P. Stanley, Coefficients and roots of Ehrhart polynomials, Integer points in polyhedra-geometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 15-36.
Christian Bey, Martin Henk, Jörg M Wills, Notes on the roots of Ehrhart polynomials. 38Christian Bey, Martin Henk, and Jörg M. Wills, Notes on the roots of Ehrhart polynomials, Discrete Comput. Geom. 38 (2007), no. 1, 81-98.
Norm bounds for Ehrhart polynomial roots. Benjamin Braun, Discrete Comput. Geom. 391-3Benjamin Braun, Norm bounds for Ehrhart polynomial roots, Discrete Comput. Geom. 39 (2008), no. 1-3, 191-193.
The number of vertices of a Fano polytope. Cinzia Casagrande, Ann. Inst. Fourier (Grenoble). 561Cinzia Casagrande, The number of vertices of a Fano polytope, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 1, 121-130.
Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires. Eugène Ehrhart, J. Reine Angew. Math. 227Eugène Ehrhart, Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires, J. Reine Angew. Math. 227 (1967), 25-49.
On the canonical strip, Uspekhi Mat. V V Golyshev, Nauk. 641V. V. Golyshev, On the canonical strip, Uspekhi Mat. Nauk 64 (2009), no. 1(385), 139-140.
Takayuki Hibi, Akihiro Higashitani, Hidefumi Ohsugi, arXiv:1001.4165v1Roots of Ehrhart polynomials of Gorenstein Fano polytopes. math.COTakayuki Hibi, Akihiro Higashitani, and Hidefumi Ohsugi, Roots of Ehrhart polynomials of Goren- stein Fano polytopes, arXiv:1001.4165v1 [math.CO].
Gábor Hegedüs, Alexander M Kasprzyk, arXiv:1002.2815v2The boundary volume of lattice polytopes. math.COGábor Hegedüs and Alexander M. Kasprzyk, The boundary volume of lattice polytopes, arXiv:1002.2815v2 [math.CO].
Polynomials associated with finite cell-complexes. I G Macdonald, J. London Math. Soc. 2I. G. Macdonald, Polynomials associated with finite cell-complexes, J. London Math. Soc. (2) 4 (1971), 181-192.
Mikkel Øbro, arXiv:0704.0049v1An algorithm for the classification of smooth Fano polytopes. math.COMikkel Øbro, An algorithm for the classification of smooth Fano polytopes, arXiv:0704.0049v1 [math.CO], classifications available from http://grdb.lboro.ac.uk/.
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A-4040 Linz, Austria E-mail address: [email protected]. at School of Mathematics and Statistics. University of Sydney, Sydney NSW69Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of SciencesAustralia E-mail address: [email protected] Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria E-mail address: [email protected] School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia E-mail address: [email protected]
| [] |
[
"Standard Deviation-Based Quantization for Deep Neural Networks",
"Standard Deviation-Based Quantization for Deep Neural Networks"
] | [
"Amir Ardakani \nDepartment of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada\n",
"Arash Ardakani \nDepartment of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada\n",
"Brett Meyer \nDepartment of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada\n",
"James J Clark \nDepartment of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada\n",
"Warren J Gross \nDepartment of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada\n"
] | [
"Department of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada",
"Department of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada",
"Department of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada",
"Department of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada",
"Department of Electrical and Computer Engineering\nMcGill University\nMontrealQCCanada"
] | [] | Quantization of deep neural networks is a promising approach that reduces the inference cost, making it feasible to run deep networks on resourcerestricted devices. Inspired by existing methods, we propose a new framework to learn the quantization intervals (discrete values) using the knowledge of the network's weight and activation distributions, i.e., standard deviation. Furthermore, we propose a novel base-2 logarithmic quantization scheme to quantize weights to power-of-two discrete values. Our proposed scheme allows us to replace resource-hungry high-precision multipliers with simple shift-add operations. According to our evaluations, our method outperforms existing work on CIFAR10 and ImageNet datasets and even achieves better accuracy performance with 3-bit weights and activations when compared to the full-precision models. Moreover, our scheme simultaneously prunes the network's parameters and allows us to flexibly adjust the pruning ratio during the quantization process. | null | [
"https://arxiv.org/pdf/2202.12422v1.pdf"
] | 247,154,903 | 2202.12422 | 943d61dad74525df3664099b1b29e3722702781c |
Standard Deviation-Based Quantization for Deep Neural Networks
Amir Ardakani
Department of Electrical and Computer Engineering
McGill University
MontrealQCCanada
Arash Ardakani
Department of Electrical and Computer Engineering
McGill University
MontrealQCCanada
Brett Meyer
Department of Electrical and Computer Engineering
McGill University
MontrealQCCanada
James J Clark
Department of Electrical and Computer Engineering
McGill University
MontrealQCCanada
Warren J Gross
Department of Electrical and Computer Engineering
McGill University
MontrealQCCanada
Standard Deviation-Based Quantization for Deep Neural Networks
Quantization of deep neural networks is a promising approach that reduces the inference cost, making it feasible to run deep networks on resourcerestricted devices. Inspired by existing methods, we propose a new framework to learn the quantization intervals (discrete values) using the knowledge of the network's weight and activation distributions, i.e., standard deviation. Furthermore, we propose a novel base-2 logarithmic quantization scheme to quantize weights to power-of-two discrete values. Our proposed scheme allows us to replace resource-hungry high-precision multipliers with simple shift-add operations. According to our evaluations, our method outperforms existing work on CIFAR10 and ImageNet datasets and even achieves better accuracy performance with 3-bit weights and activations when compared to the full-precision models. Moreover, our scheme simultaneously prunes the network's parameters and allows us to flexibly adjust the pruning ratio during the quantization process.
Introduction
Deep neural networks (DNNs) have shown promising results in various applications including Image classification and language processing tasks. However, deployment of the current state-of-the-art (SOTA) DNNs requires expensive and powerful hardware (e.g., GPUs) to perform costly highprecision multiplications (Courbariaux et al., 2015). Moreover, DNNs are generally over-parameterized and utilize large memories, thereby require heavy data transmission between the computation and memory units (You et al., 2020). When it comes to resource-constrained hardware such as mobile devices and embedded systems, the aforementioned requirements make the deployment of such networks a challenging task. Correspondingly, a large amount of effort has gone into developing algorithms and hardware, aiming to reduce the deployment costs of DNNs, whilst maintaining high accuracy. Efficient architecture design (Gholami et al., 2018;Sandler et al., 2018;Howard et al., 2019), pruning (Han et al., 2015;Liu et al., 2018), stochastic computing (Ardakani et al., 2017;Gross & Gaudet, 2019) and spiking neural networks (Ghosh-Dastidar & Adeli, 2009;Smithson et al., 2016) are examples of such efforts.
Another realm of studies focuses on quantization of DNNs where real values are mapped to discrete integer values (Zhou et al., 2016). Using integer values at inference dramatically reduces the computational and memory costs. In this work, we also focus on the quantization of DNNs. Although most existing works have employed uniform quantization methods where discrete levels have the same step size (Choi et al., 2018;Esser et al., 2019;Jung et al., 2019), DNNs can be quantized to non-uniform discrete integers as well Miyashita et al., 2016;Li et al., 2021;Zhou et al., 2017).
In general, the quantization methods mainly fall into three categories. The first category includes the methods trying to minimize the quantization error where real values are mapped to the discrete levels that accurately represents the data (Li et al., 2016;Cai et al., 2017;Wang et al., 2018;Zhang et al., 2018;Zhao et al., 2020). Most recent works that fall into the first category often aim to achieve accurate representation of data by minimizing the quantization error between the quantizer's input and output. These methods require to constantly monitor weights and activations values (online) or to use the structural characterization of data from the full-precision network (offline) for minimizing the quantization error. For instance, in the HWGQ scheme (Cai et al., 2017), the quantization error is minimized using Lloyd's algorithm (Lloyd, 1982) and the statistical structure (distribution) of the activations obtained from the full-precision network to fit the quantizer to the data. The main drawback of this method is its offline optimization and that the distribution of the full-precision activations is not necessarily the same as the quantized network. Contrary to HWGQ, LLSQ method uses an online algorithm to minimize the quantization error during the training phase (Zhao et al., 2020). LQ-Net is another online method , where a non-uniform quantization scheme was developed that transforms the quantization error to a linear regression problem with a closed form solution. The performance of HWGQ, LLSQ and LQ-Net has been outperformed by other methods (i.e., the second category), empirically showing that minimizing the quantization error is not the best approach to obtain quantized networks.
In the second category, the quantization is treated as an optimization problem that can be simultaneously optimized with the loss function during the training phase. More specifically, the quantizers are redesigned with learnable parameters that can be then used to find the optimal quantization intervals. In this approach, the goal is to find the optimizer that can produce the best accuracy performance, whereas in the first category, the goal is to find the optimizer that represents the data more accurately (less quantization error). PACT (Choi et al., 2018), QIL (Jung et al., 2019) and LSQ (Esser et al., 2019) are the well-known quantization methods from this category.
Finally, the third category mainly focuses on the training process of the quantized networks (Zhou et al., 2017;Zhuang et al., 2018). This includes techniques and tricks that are used during the training of the quantized network to improve its performance accuracy. For instance, it has been shown that a progressive training framework with the help of knowledge distillation significantly improves the performance of the DoReFa-Net (Zhou et al., 2016) quantization method on various tasks (Zhuang et al., 2018).
In this work, we aim to close the gap between these methods by exploiting the statistical characterization of weights and activations distribution during the quantization process. In other words, we propose a new quantization scheme that utilizes the standard deviation of weights/activations distribution during the training phase to find the optimal optimizer that produces the best accuracy performance using task loss and back-propagation. We also explain how standard deviation contributes in improving our quantization method in Section 3. Furthermore, in Section 4, we propose two training techniques which are employed in our proposed quantization method to further improve the performance of the quantized networks.
Moreover, unlike the existing methods that have only focused on one type of quantization, that is either uniform or non-uniform quantization, we propose a quantization framework that is capable of both uniform and non-uniform quantization. Our non-uniform quantization method uses power-of-two discrete levels. We show that the proposed non-uniform quantization method outperforms the SOTA results (Li et al., 2021) while having 10× faster convergence.
Quantizers with the discrete intervals that include zero level, inherently benefit from pruning as values that fall into this level during the quantization are set to 0 and can be pruned. This valuable property makes quantization methods more desirable over other cost reduction methods. However, there are only a few work investigating the relation between prun-ing and quantization, and their impact on each other (Jung et al., 2019;Han et al., 2015). In this work, not only we study the pruning property of quantization in more depth, we also show that our method allows us to flexibly adjust the pruning ratio during the quantization process. For instance, we show that up to 40% of weights in a 3-bit ResNet-18 model can be pruned at the cost of less than 1% accuracy loss when compared to its full-precision counterpart.
Standard Deviation-Based Quantization
To make the connection between the existing quantization methods, we try to solve the following question: "How much of the information is important in a quantized network?". To answer this question, we need to look at the statistical characterization of weights and activations more carefully. It has been shown that the output of the convolution/fully-connected layers has a bell-shaped distribution, especially if followed by a batch-normalization layer (Ioffe & Szegedy, 2015). Besides the activations, the weights of a network regularized with L2-regularization method have a bell-shaped distribution as well. The width of a bell-shaped distribution is defined by its standard deviation. For instance, in a Gaussian distribution, the values within three standard deviation (σ) from the mean (µ) account for 99.73% of the entire data. However, not all of these values are as important as the others in a neural network. Here we propose a standard deviation-based clipping function with a learnable parameter (α) which is capable of determining how much of these values is actually important. The backbone of our proposed clipping function is the PACT (Choi et al., 2018) method. In our proposed function, we integrate the standard deviation of the weights/activations in to the PACT such that
y(x) = x |x| < ασ sign(x) · ασ |x| ≥ ασ ,(1)
where α and σ are the quantizer's learnable parameter and the standard deviation of the weights/activations, respectively. For the activations, we can integrate the ReLU function with our clipping function, i.e.,
y(x) = 0 x ≤ 0 x 0 < x < ασ ασ x ≥ ασ .(2)
The proposed clipping function simply discards the "outliers" that are ασ far from the mean of the weights/activations distribution. Note that in our clipping function, the mean value of weights is considered to be zero all the time. This is because in our experiments, the mean value of weights was always close to zero and did not impact the results when it was omitted. The mean of the activations, however, is always zero. Note that in Equation (2), we use ReLU function to assign zeros to all negative values. Consequently, we should not calculate σ directly from the activations input. Instead, we calculate the σ by removing all negative values from the inputs and then we apply a horizontal mirroring on the remaining values. It is evident that after this step, the mean value of activations distribution is always zero.
Unlike the standard deviation of the weights distribution, the standard deviation of activations can have a significantly different value whenever a new batch of data is introduced to the network. Therefore, we need to average σ for activations distribution during the training phase. Inspired by the batchnormalization technique, we keep track of σ by employing the exponential running average with the momentum of 0.001,
σ new = (1 − momentum) ×σ + momentum × σ t ,(3)
whereσ is the running average and σ t is the standard deviation of the current batch. It is worth mentioning that the momentum value of 0.001 was empirically obtained from our experiments.
The output of the clipping function can be then quantized to L P + L N + 1 discrete levels (integer values) using b bits such that
y d = clip( y · L P ασ , −L N , L P ),(4)
where function a rounds a to its nearest integer value and y d ∈ N. Variable L is determined by the quantizer bit-width (b) and the data type (signed/Unsigned). For unsigned data (activations) L N = 0 and L P = 2 b − 1 and for signed data (weights) L P = L N = 2 b−1 − 1. It is worth mentioning that our method quantizes signed values symmetrically. For instance, quantization of signed values using 2 bits results in three discrete values (i.e., ternary discrete values).
Finally, the quantization process is completed by multiplying the quantizer scale value with y d such that
y q = y d · ασ L P ,(5)
where y q ∈ R. We would like to emphasize that the multiplication of the quantizer scale with y d can simply be integrated with the batch normalization layer without any additional costs. As shown in Figure 1, our proposed quantization method automatically prunes the network as well.
In fact, any values that falls into the pruning area of the quantizer is pruned (set to zero) during the quantization. From Equation (4), we can observe that
if |y| < ασ 2L P , then y d = 0,
which shows a direct relation between the pruning ratio and the clipping threshold of the quantizer. In other words, higher clipping thresholds result in greater pruning ratios.
Optimizing α with Backpropagation
Following the same principles introduced in PACT and using Straight-Through Estimator (ETS) (Bengio et al., 2013), we can derive the gradient for the quantizer parameter α by
g α = σ · g y x ≥ ασ 0 otherwise ,(6)
where g y is the incoming gradient from the front layers. Furthermore, the gradient of the activations input g x can be computed using
g x = g y 0 < x < ασ 0 otherwise .(7)
We can add a gradient scale s and weight decay λα to g α and rewrite Equation (6) as
g α = sσ · g y + λα x ≥ ασ 0 otherwise .(8)
The gradient scale and weight decay were added to g α to prevent α from exploding/vanishing and to control the pruning ratio.
Base-2 Logarithmic Quantization
We can use the clipping function described in Equation (1) to quantize weights to power-of-two discrete values, i.e., y p2 ∈ {0, ±2 k }, where k ∈ Z + 0 . With this new powerof-two quantized weights, we can replace the expensive multiplications with simple shift and addition/subtraction operations. To quantize weights to power-of-two discrete values, we first modify Equation (4) to produce discrete integers (y int ∈ N) such that
y int = log 2 (|y| · L p2 ασ ) ,(9)
where L p2 = 2 2 b−1 −2 . We then transform y int to discrete power-of-two integers including zero (y p2 ∈ {0, ±2 k }) us-
ing y p2 = clip(sign(y) · 2 yint , −L p2 , L p2 ) y int ≥ 0 0 otherwise .
(10) For instance, using 3 bits (b = 3) we can quantize weights to seven discrete values, i.e., y 3−bit p2 ∈ {0, ±1, ±2, ±4}. Similarly, we can use Equation (5) to multiply the quantizer scale with y p2 to complete the quantization process. Finally, we can use the same method described in Equation (8) to optimize the quantizer parameter α.
Contributing Factors
The main intuition behind our heuristic approach to parametize the quantizers with statistical structure of the weights/activations is to include a new feature, i.e., the standard deviation of the data distribution during the quantization process. Unlike the previous parameterization methods, the loss optimization process (backpropagation) of weights and the quantizer parameter α of our proposed method, relies on both the input samples and the statistical structure of weights/activations from each layer. For instance, the standard deviation shows how dense one distribution is. In a highly dense distribution (small σ), where a huge portion of data resides inside the pruning area, clearly we do not want the outliers to move the clipping threshold further from the zero. In this example, the small value of σ does not allow the gradients of the outliers to significantly shift the clipping threshold, according the Equation (8). In addition to the aforementioned explanation, we believe that there are three other contributing factors to why our method works: inclusivity, adaptive gradient scale factor and faster convergence.
Inclusivity: Unlike PACT where the gradients of the quantizer parameter only rely on the outliers, the gradients in our method are sensitive to the entire data. More specifically, we introduce the standard deviation of weights and activations distribution as a new feature that appears in the task loss and is used to scale the quantizer parameter.
Adaptive gradient scale factor: The impact of the gradient scale has been investigated in LSQ. It was shown that large gradient scales (e.g., 1) produce weak accuracy results. Looking at Equation (8), the standard deviation in our method can also be interpreted as an adaptive scaling factor for the quantizer parameter α. In most of our experiments, the gradient scale was initially set to 1 and then adjusted to a proper value that achieves the best accuracy performance. In fact, the role of standard deviation as the adaptive gradient scale enables us to trade-off between the accuracy performance and the pruning ratio by treating the gradient scale as a hyper-parameter. Using a small gradient scale forces α to converge to smaller values and consequently, less data is pruned. On the other hand, setting the gradient scale to larger values allows us to prune more data, though at the cost of loosing accuracy performance. We investigate this property in Section 5.5.
Faster convergence: In the previously-proposed methods, the clipping threshold of weights and activations solely relies on the quantizer parameter α. Consequently, if the distribution changes rapidly, it may take a longer time for the clipping threshold to catch up with the new distribution of data. In our proposed quantizer, the clipping threshold relies on both the α and the standard deviation of weights/activations distribution. As a result, even without updating α, the clipping threshold can follow the new distribution effortlessly as illustrated in Figure 2.
Quantization Techniques
In this section, we propose two quantization techniques that can be employed to further improve the performance of our quantization method.
Improved Progressive Training
Arguably, any neural network quantized to extremely low bit-widths suffers from accuracy loss due to poor data representation. However, the inaccurate representation of the weights/activations is not the only factor, negatively impacting neural networks. In fact, the gradient vanishing problem as the result of intense pruning during the quantization process is a major issue causing significant performance degradation (Li et al., 2021). Furthermore, intense pruning reduces the learning capacity of neural networks since a huge portion of the weights are removed (set to zero). Previous studies tried to address this issue by proposing different progressive training methods (Zhou et al., 2017;Zhuang et al., 2018). A recent method, for instance, proposed to quantize higher values of weights first while keeping the lower values in full-precision (Zhou et al., 2017). This method allows the gradient to back propagate through the weights with lower values which would have been pruned due to quantization. In another method, it has been shown that a progressive training (e.g., training 2-bit width networks using the trained parameters from the 3-bit width networks) improves the training capacity and the accuracy performance of the quantized networks. However, by just transferring the learned parameters, we are changing the learned quantization intervals according to Equation (4). Consequently, a huge portion of the network parameters are pruned during the transformation from higher bit-width networks to lower bit-width networks due to the larger pruning area as illustrated in Figure 3. By changing the quantization intervals, the learning capacity of the quantized network will be reduced even more, not only because of the poor representation of data due to low bit-width quantization, but also as a result of the pruning.
To address this, we propose to re-scale the quantizer parameter α so that the lower bit-width network starts with the same quantization intervals of the higher bit-width network by applying
α b = α b+n × L b L b+n ,(11)
where α b and L b are the quantizer parameter and the discretization level of the quantized networks with b bits, respectively.
Our proposed progressive training method offers two advantages: improving the training capacity of the quantized network and preventing the quantizer from pruning the parameters further. Consequently, our method reduces the impact of gradient vanishing problem. More importantly, this allows us to limit the search space for the quantizer parameter α, forcing the network to find optimal intervals near Weight Value Density Figure 4. Distribution of a convolution layer from ResNet-18 model trained with our quantization method using 3 bits. The spikes in the distributions are the transition boundaries of the 3-bit quantizer.
the ones found in networks with higher bit-widths. This can be done by using smaller gradient scale values s in Equation (8) which allows the weight decay to prevent the quantizer parameter α from getting larger.
Two-Phase Training
It has been shown that a huge portion of the weights resides near the quantization intervals boundaries (transition boundaries) (Jung et al., 2019). Our experimental result also shows the same patterns across different networks as illustrated in Figure 4. Based on these observations, we suspected that jointly optimizing the network parameters along with the quantizer's could negatively impact the performance of the network since a small change in quantization intervals can possibly result in a dramatic change in the quantized weights, especially for 2-bit quantized networks where there are fewer discrete levels. In addition, it is more difficult for weights near the transition boundaries to converge to the optimal quantization intervals when the transition boundaries are constantly fluctuating. Furthermore, while we do not see the same pattern for the quantized activations (because of the batch normalization), this fluctuation could still hurt the quantization of the activation but with less severity. Furthermore, the gradients of the quantized networks with learnable parameter are affected by that parameter when passing through the layers. Looking at Equation (7), it is evident that both α and σ are controlling the gradients passing to the previous layers. Since α and σ are both getting updated continuously during the training, we suspected that this might introduce some noise to the gradients. To investigate this hypothesis, we retrained our quantized networks keeping α and σ frozen to their optimal value from the first training. Interestingly, our empirical results show constant improvement across different networks and datasets, which highlights the importance of our proposed two-phase training method when training parameterized quantized networks. , DSQ (Gong et al., 2019) and PACT (Choi et al., 2018)
Experiments
To validate our proposed quantization method, we conduct several experiments on CIFAR10 (Krizhevsky et al., 2009) and ImageNet (Russakovsky et al., 2015) datasets on various architecture. Furthermore, we present several ablation studies to evaluate the impact of the hyper-parameters and the proposed quantization techniques on the performance of the quantized networks. In all experiments, we use the network's weight decay value for the quantizer's weight decay. Furthermore, we use the same data augmentation proposed in (He et al., 2016) for both CIFAR10 and ImagNet datasets.
ResNet-20 on CIFAR10
We demonstrate the effectiveness of our proposed method by quantizing ResNet-20 model (He et al., 2016) on CIFAR10 dataset. To have a fair comparison with PACT, we conduct two different experiments based on the quantization method applied on weights. In the first experiment, we quantize ResNet-20 model progressively with re-scaling of the clipping threshold and applying the proposed two-phase training technique. In this experiment, both weights and activations are quantized using our proposed quantization scheme. The gradient scale value (s) is set to {1, 1, 0.1, 0.01} for the {5, 4, 3, 2}-bit quantized ResNet-20, respectively. It is worth mentioning that the gradient scale values were adjusted for each bit-width after performing hyper-parameter search (see Section 5.6).
In the second experiment, we employ the same weight quantization method used in PACT. We also quantize ResNet-20 from scratch, and provide the accuracy results separately in Table 1. Finally, we employ the progressive method without re-scaling and our proposed two-phase training technique step by step to demonstrate their impact on the performance of the quantized network. Similar to PACT, we do not quantize the first layer and the last layer of ResNet-20. In this experiment, the gradient scale used to quantize activations is set to 1 regardless the bit-widths.
From Table 1, we observe that our quantization method achieves a higher accuracy compared to PACT even when only the activations are quantized with our method. Furthermore, we achieve a higher accuracy when our two-phase training technique is applied over the progressive training method, showing its effectiveness in reducing the gradient noise. In addition, we achieve the best performance when both weights and activations are quantized using our method across all the quantization bit-widths.
SmallVGG on CIFAR10
We also evaluate our method on CIFAR10 dataset by quantizing the SmallVGG network (Louizos et al., 2018) under three different setups: Setup-1 We use our quantization method to quantize both weights and activations of SmallVGG network with 2 bits. In this setup, all layers except the first convolution layer and the fully-connected layer are quantized. The network's parameters are initialized with the full-precision parameters. We train the network for 300 epochs using gradient scale of 0.001. As shown in Table 2, our method outperforms the state-of-the-art and even achieves a higher accuracy performance compared to the full-precision model. Setup-2 In this setup, we quantize activations to 2 bits (similar to the first experiment), but we binarize weights (i.e., quantize with 1 bit) using the sign function as described in (Courbariaux et al., 2016). Interestingly, our method still manages to outperform SOTA and full-precision while using binarized weights and 2-bit activations during inference. Setup-3 We repeat the same experiment described in setup-1 to quantize all layers of SmallVGG network. It should be noted that only the weights of the first layer are quantized while its inputs are kept in full-precision. Similar to the first experiment, our method outperformed SOTA and fullprecision models. The results from these three experiments show quantization can have a regularization effect as long as the capacity of the quantized network is not decreased due to quantization. Furthermore, we can conclude that Small-lVGG network is over-parameterized for small datasets like CIFAR10. , HWGQ (Cai et al., 2017), LLSQ (Zhao et al., 2020) and RQST (Louizos et al., 2018).
Method All layers Accuracy @ precision (A/W)
AlexNet on ImageNet
We test our quantization method on ImageNet dataset using modified AlexNet (Krizhevsky et al., 2012) where a batch normalization layer was added after convolution and fully-connected layers except for the last layer. We use the progressive training method with re-scaling when training the 3-bit and the 2-bit networks. The quantized network is optimized with cosine annealing scheduler with the initial learning rate of 0.001 for 70 epochs. Following the common practice from previous methods, all layers are quantized except for the first layer and the last layer. As shown in Table 3, our proposed method outperforms the previously-proposed quantization methods.
When training the 2-bit AlexNet with the gradient scale of 1 (i.e., s = 1), we observed that the training loss starts to increase after several epochs and the network converges to a poor local minima due to extreme pruning, even with re-scaling of the clipping threshold. To address this issue, we changed s to 0.01 to force the 2-bit AlexNet use the intervals found from the 3-bit network with less weight pruning. Furthermore, we trained the 2-bit AlexNet with the intervals obtained from the 4-bit network with s = 0.01. Finally, we used our double training method for the 2-bit AlexNet which increased the accuracy by 0.2%. As shown in Table 4, the 2-bit AlexNet with 3-bit intervals has the best performance accuracy while the one trained with 4-bit (Jung et al., 2019), LQ-Net and TSQ . intervals also outperforms the QIL method.
ResNet on ImageNet
We evaluate our proposed base-2 logarithmic quantization method described in Section 2.2 on ImageNet dataset using ResNet-18 and ResNet-50 models (He et al., 2016). The under-test models are specifically chosen to fairly compare the accuracy performance of our method with SOTA shiftadd methods. We employ previously described two-phase training technique. Both models are initialized with the pre-trained parameters from the Pytorch model zoo (Paszke et al., 2019). We train the models for 70 epochs with cosine learning scheduler. The results summarized in Table 5 show that our method outperforms the SOTA results for both models. In addition to having higher accuracy performance, our method has several advantages over the Sign-Sparse-Shift (Li et al., 2021) method. First, the full-precision parameters of the pre-trained models can be used to initialize the quantized models. Consequently, the quantized models can converge to an acceptable accuracy much faster than the ones initialized randomly. In fact, the Sign-Sparse-Shift method requires 200 epochs to reach the results provided in Table 5, whereas our method achieves a higher accuracy after only 20 epochs (10× faster convergence). Furthermore, the Sign-Sparse-Shift method requires 4 times the number of parameters of its full-precision model during the training phase, which significantly increases the training hardware memory footprint and utilization. Contrary, our method only adds a small number of parameters and registers i.e., α and the standard deviation of the weights.
Pruning Ratio and Accuracy Trade-off
As mentioned previously, pruning is an inherit outcome of quantization. A well-pruned network can significantly reduce the size of the memory. However, too much pruning has a negative impact on the model learning capacity. In our quantization method, we can control the pruning ratio indirectly by adjusting the gradient scale (s in Equation (8)). We evaluate this property by training the same 3-bit quantized ResNet-18 model from previous experiment (Section 5.4) Table 6 show that we can achieve slightly better accuracy performances at the cost of loosing a significant amount of pruning ratio. More specifically, we can achieve 1.12% higher accuracy if we accept 18.64% less pruning. It is up to the users to decide whether the benefits of networks with higher pruning ratios outweigh their slight drop in accuracy. It should be noted that this trade-off can only be considered in a situation where several gradient scale values result in a good convergence and accuracy performance. Furthermore, the clipping thresholds and the pruning rates of each layers for the quantized networks with the gradient scale values of 1 and 0.001 are shown in Figure 5 to better demonstrate how the gradient scale affects the pruning ratio. As expected, the quantized network with the smaller gradient scale value has smaller clipping thresholds. Furthermore, higher clipping thresholds result in grater pruning rates.
Progressive Training and Re-scaling
We evaluate the effectiveness of our proposed progressive training, i.e., the re-scaling of the clipping threshold for the weights quantizer, by training ResNet-20 on CIFAR10 dataset. In this experiment, ResNet-20 model is quantized to 2 and 3 bits. The 2-bit quantized model is initialized and re-scaled from the 3-bit and 4-bit models, and the 3bit model is initialized and re-scaled using the 4-bit model. From Table 7, we observe that the 2-bit model yields a higher accuracy when the clipping threshold is re-scaled from the 3-bit model. Interestingly, both the 2-bit and the 3-bit models produce comparable accuracy even when the quantizer parameter (α) is not updated (the gradient scale is 0). In other words, we can use the same quantization intervals, found from the higher-bit networks, in the lowerbit networks and still achieve acceptable performance. This observation shows the importance and effectiveness of our proposed re-scaling. Furthermore, from the columns where the gradient scale is 0.1 and 0.01, we observe that updating α with a smaller gradient scale does not necessarily achieve a better performance all the time.
Conclusion
We proposed a new quantization method that takes advantage of the knowledge of weights and activations distribution during the quantization process. Using the standard deviation of weights and activation, our proposed method outperforms the previous works on various image classification tasks. Furthermore, we proposed two training techniques to further improve the performance of our quantization scheme. We introduced a two-phase training technique to address the gradient noise and fluctuation of the quantizer's transition boundaries caused by the joint optimization of the network's and the quantizer's parameters. In addition, we presented a novel re-scaling technique to improve the training of quantized networks by reducing the impact of gradient vanishing problem. Finally, we proposed a novel non-uniform quantization framework (base-2 logarithmic quantization) where weights are quantized to power-of-two discrete intervals to replace expensive multipliers with simple shift-add operations. The proposed base-2 logarithmic quantization method converges 10× faster that the SOTA method and outperforms existing work. Our proposed quantization method provides the flexibility to trade-off between accuracy and network size by controlling the pruning ratio. In future work, we plan to investigate the compatibility of our method with transformers.
Figure 1 .
1An example of a 3-bit quantizer for signed values.
Figure 2 .
2Clipping thresholds of two set of data with standard deviation of 0.5 and 0.25. The quantizer parameter α is fixed (α = 2) for both data distributions.
Figure 3 .
3Quantization levels (intervals) of a 3-and 2-bit quantizers with the same α. The blue line shows the intervals of the 3-bit quantizer, whereas the red line shows the intervals for the 2-bit quantizer. The pruning area of the 2-bit quantizer (rectangle with the shade of red) is 3 times wider than the 3-bit quantizer (rectangle with the shade of blue).
Figure 5 .
5The clipping thresholds and the pruning rates of the quantized ResNet-18 weights on ImageNet dataset with various gradient scale factors. Each model was trained for 20 epochs. The results provided in
Table 1 .
1Quantization accuracy performance on CIFAR10 dataset with ResNet-20 model (FP accuracy: 91.74%). Methods included in this table are LQ-Net
Table 2 .
2Comparisonwith SOTA methods on CIFAR10 dataset us-
ing SmallVGG network (FP accuracy: 93.66%). Methods included
in this table are LQ-Net
Table 3 .
3Comparionwith the existing methods on AlexNet (FP
accuracy: 61.8%). Methods included in this table are QIL
Table 4 .
4Top-1 accuracy of 2-bit AlexNet under different training
setup.
Setup
Gradient scale Top-1 Acc. (%)
No re-scaling
1
54.4
Re-scaled
from 4-bit
0.01
58.76
Re-scaled
from 3-bit
0.01
59.01
Re-scaled
from 3-bit +
two-phase training
0.01
59.24
Table 5 .
5Comparison between existing shift-add networks on Ima-geNet dataset. Methods included in this table are DeepShift (Elhoushi et al., 2021), INQ(Zhou et al., 2017) and Sign-Sparse-Shift (S 3 )(Li et al., 2021).Model
Method
Width
Top-1/Top-5
Acc. (%)
ResNet-18
FP
32
69.76/89.08
DeepShift
5
69.56/89.17
INQ
3
68.08/88.36
S 3
3
69.82/89.23
Ours
3
70.04/89.14
INQ
4
68.89/89.01
S 3
4
70.47/89.93
Ours
4
70.70/89.62
ResNet-50
FP
32
76.13/92.86
INQ
5
74.81/92.45
DeepShift
5
76.33/93.05
S 3
3
75.75/92.80
Ours
3
76.41/93.01
Table 6 .Table 7 .
67Top-1 accuracy and pruning ratio for various gradient scale factors. Top-1 Acc. (%) 68.92 69.59 69.70 70.04 Pruning ratio (%) 40.21 29.74 24.97 21.57 Accuracy performance of ResNet-20 on CIFAR10, quantized using 2 and 3 bits with different clipping threshold initialization and gradient scale values.Gradient scale
1
0.1
0.01
0.001
Bit-width
Clipping
Threshold
Initialization
Gradient scale
1
0.1
0.01
0
2
From 3
88.06 90.34 90.69 90.12
2
From 4
87.55 90.29 90.11 89.57
3
From 4
92.14 92.23 92.07 92.10
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| [] |
[
"Dissipation-assisted operator evolution method for capturing hydrodynamic transport",
"Dissipation-assisted operator evolution method for capturing hydrodynamic transport"
] | [
"Tibor Rakovszky \nDepartment of Physics\nTechnische Universität München\nJames-Franck-Straße 1T42, D-85748GarchingGermany\n",
"C W Von Keyserlingk \nSchool of Physics & Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUK\n",
"Frank Pollmann \nDepartment of Physics\nTechnische Universität München\nJames-Franck-Straße 1T42, D-85748GarchingGermany\n\nMunich Center for Quantum Science and Technology (MCQST)\nSchellingstr. 4D-80799MünchenGermany\n"
] | [
"Department of Physics\nTechnische Universität München\nJames-Franck-Straße 1T42, D-85748GarchingGermany",
"School of Physics & Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUK",
"Department of Physics\nTechnische Universität München\nJames-Franck-Straße 1T42, D-85748GarchingGermany",
"Munich Center for Quantum Science and Technology (MCQST)\nSchellingstr. 4D-80799MünchenGermany"
] | [] | We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.Introduction.-Despite their complexity, thermalizing quantum many-body systems often exhibit universal hydrodynamical features in their low-frequency, longwavelength limit[1][2][3][4][5][6][7][8]. Although these features are routinely measured in transport experiments, quantitatively connecting them to the underlying microscopic dynamics, e.g., deriving the transport coefficients from first principles in generic interacting quantum systems, is notoriously difficult in practice[2,[9][10][11][12][13]. It usually requires evaluating dynamical correlations, and existing methods are typically restricted to small systems or short times, often leading to unreliable results[4,[14][15][16][17]. While methods have been proposed to circumvent these issues in certain cases[18][19][20][21][22][23][24][25][26][27], calculating transport properties reliably in generic models remains a challenging task.The purpose of this paper is to introduce a numerical method for calculating transport properties from first principles in a controlled manner, while avoiding finite size and time restrictions. We achieve this by focusing on the Heisenberg picture dynamics of conserved densities. Motivated by recent results on operator spreading [28-31], we introduce an artificial dissipation that removes operators based on their spatial support. As a consequence, the time-evolved operator may be stored more compactly using standard tensor network techniques. The resulting dynamics depends on the specifics of the dissipative procedure, but in the limit of weak dissipation, the different methods all appear to converge. This allows us to estimate the physical result (here, a spin or energy diffusion constant) through extrapolation.Numerical method.-We work with one-dimensional lattice models, labeling sites by j = 1, . . . , L. Consider the local density, q j , of some conserved quantity Q = j q j (e.g., charge or energy). We are interested in dynamical correlations of these densities, q i (0)q j (t) eq , evaluated in some equilibrium state. We focus on infinite temperature, so that . . . eq ≡ Tr[. . .]/N , with N the Hilbert space dimension. Here q j (t) is evolved unitarily in the Heisenberg picture, with a Hamiltonian H | 10.1103/physrevb.105.075131 | [
"https://arxiv.org/pdf/2004.05177v1.pdf"
] | 215,745,740 | 2004.05177 | cb728aa9260a1850cd741a06acf47fe0dacbce84 |
Dissipation-assisted operator evolution method for capturing hydrodynamic transport
Tibor Rakovszky
Department of Physics
Technische Universität München
James-Franck-Straße 1T42, D-85748GarchingGermany
C W Von Keyserlingk
School of Physics & Astronomy
University of Birmingham
B15 2TTBirminghamUK
Frank Pollmann
Department of Physics
Technische Universität München
James-Franck-Straße 1T42, D-85748GarchingGermany
Munich Center for Quantum Science and Technology (MCQST)
Schellingstr. 4D-80799MünchenGermany
Dissipation-assisted operator evolution method for capturing hydrodynamic transport
We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.Introduction.-Despite their complexity, thermalizing quantum many-body systems often exhibit universal hydrodynamical features in their low-frequency, longwavelength limit[1][2][3][4][5][6][7][8]. Although these features are routinely measured in transport experiments, quantitatively connecting them to the underlying microscopic dynamics, e.g., deriving the transport coefficients from first principles in generic interacting quantum systems, is notoriously difficult in practice[2,[9][10][11][12][13]. It usually requires evaluating dynamical correlations, and existing methods are typically restricted to small systems or short times, often leading to unreliable results[4,[14][15][16][17]. While methods have been proposed to circumvent these issues in certain cases[18][19][20][21][22][23][24][25][26][27], calculating transport properties reliably in generic models remains a challenging task.The purpose of this paper is to introduce a numerical method for calculating transport properties from first principles in a controlled manner, while avoiding finite size and time restrictions. We achieve this by focusing on the Heisenberg picture dynamics of conserved densities. Motivated by recent results on operator spreading [28-31], we introduce an artificial dissipation that removes operators based on their spatial support. As a consequence, the time-evolved operator may be stored more compactly using standard tensor network techniques. The resulting dynamics depends on the specifics of the dissipative procedure, but in the limit of weak dissipation, the different methods all appear to converge. This allows us to estimate the physical result (here, a spin or energy diffusion constant) through extrapolation.Numerical method.-We work with one-dimensional lattice models, labeling sites by j = 1, . . . , L. Consider the local density, q j , of some conserved quantity Q = j q j (e.g., charge or energy). We are interested in dynamical correlations of these densities, q i (0)q j (t) eq , evaluated in some equilibrium state. We focus on infinite temperature, so that . . . eq ≡ Tr[. . .]/N , with N the Hilbert space dimension. Here q j (t) is evolved unitarily in the Heisenberg picture, with a Hamiltonian H
We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.
Introduction.-Despite their complexity, thermalizing quantum many-body systems often exhibit universal hydrodynamical features in their low-frequency, longwavelength limit [1][2][3][4][5][6][7][8]. Although these features are routinely measured in transport experiments, quantitatively connecting them to the underlying microscopic dynamics, e.g., deriving the transport coefficients from first principles in generic interacting quantum systems, is notoriously difficult in practice [2,[9][10][11][12][13]. It usually requires evaluating dynamical correlations, and existing methods are typically restricted to small systems or short times, often leading to unreliable results [4,[14][15][16][17]. While methods have been proposed to circumvent these issues in certain cases [18][19][20][21][22][23][24][25][26][27], calculating transport properties reliably in generic models remains a challenging task.
The purpose of this paper is to introduce a numerical method for calculating transport properties from first principles in a controlled manner, while avoiding finite size and time restrictions. We achieve this by focusing on the Heisenberg picture dynamics of conserved densities. Motivated by recent results on operator spreading [28][29][30][31], we introduce an artificial dissipation that removes operators based on their spatial support. As a consequence, the time-evolved operator may be stored more compactly using standard tensor network techniques. The resulting dynamics depends on the specifics of the dissipative procedure, but in the limit of weak dissipation, the different methods all appear to converge. This allows us to estimate the physical result (here, a spin or energy diffusion constant) through extrapolation.
Numerical method.-We work with one-dimensional lattice models, labeling sites by j = 1, . . . , L. Consider the local density, q j , of some conserved quantity Q = j q j (e.g., charge or energy). We are interested in dynamical correlations of these densities, q i (0)q j (t) eq , evaluated in some equilibrium state. We focus on infinite temperature, so that . . . eq ≡ Tr[. . .]/N , with N the Hilbert space dimension. Here q j (t) is evolved unitarily in the Heisenberg picture, with a Hamiltonian H (2) as a sum over paths in operator space. For ∆t → 0, γ → ∞, paths that leave the ≤ * subspace are discarded immediately. Making ∆t finite, we keep paths that wander off from this subspace but return before the next integer multiple of ∆t. Finally, when ∆t, γ are both finite, all paths are kept, but the weight of those that spend time outside the 'slow' subspace is gradually reduced. (b): The operator (MPO) qj can be reinterpreted as a state (MPS) |qj on a doubled Hilbert space. (c): One period of the DAOE as a tensor network. |qj is evolved with the TEBD algorithm up to time ∆t. Then the dissipator D * ,γ is applied as a bond dimension * + 1 MPO.
that conserves Q, [H, Q] = 0. Transport properties can be extracted from such correlations, as we detail below.
In what follows, we shall find it useful to think of operators as vectors in an enlarged Hilbert space of size N 2 . In a matrix product operator (MPO) representation, this is equivalent to combining the two physical legs into a single leg, turning it into a matrix product state (MPS), as illustrated by Fig. 1(b). We use the notation |q j for the 'vectorized' operator, and introduce an inner product on this space as A|B ≡ A † B eq . The Heisenberg equation of motion can be rewritten as ∂ t |q j = i[H, q j ] ≡ iL|q j , which defines the Liouvillian superoperator, L. This is solved by |q j (t) = e iLt |q j . The key point is that we do not need the full evolution e iLt -we are only interested in its matrix elements in the 'slow' subspace, spanned by the conserved densities: q i |e iLt |q j = q i q j (t) eq . This projected evolution is generically no longer unitary.
We wish to approximate this non-unitary evolution by gradually taking into account the effect of the 'bath', meaning all the remaining operators that we are not projecting onto. We will do this in a more general way, where we include not only conserved densities, but all sufficiently local operators in the slow subspace. To be concrete, let us imagine a spin-1/2 chain. Then a basis of all 4 L operators is given by Pauli strings, products of the four Pauli matrices 1 1, X, Y, Z. To each Pauli string S, we can associate a length S , which is simply the number of non-trivial Paulis occurring in it. For example, 1 1, X j , Z i Y j have lengths = 0, 1, 2, respectively. We can then define a dissipation superoperator that decreases the weight of all strings longer than some cutoff length * as
D * ,γ [S] = S if S ≤ * e −γ( S − * ) S otherwise.(1)
The cutoff length * is introduced to ensure that the physically most relevant operators, such as conserved densities, are not affected by the dissipator.
We are now in a position to describe our proposed method. We define a modified time evolution, by applying the dissipator with period ∆t. That is, for time t ∈ [N, N + 1)∆t (for N ∈ N), we consider the time evolved local density defined by |q j (t) = e iL(t−N ∆t) D * ,γ e iL∆t N |q j ;
(2)
we dub this dissipation-assisted operator evolution (DAOE). Eq. (2) is clearly very different from the true, unitarily evolved operator |q j (t) . However, we propose that the dissipative evolution can be made to correctly capture the correlations with other slow operators, particularly conserved densities, q i |q j (t) ≈ q i |q j (t) . Intuitively, ∆t and 1/γ both play a similar role, limiting the amount of time an operator is allowed to spend outside the ≤ * subspace. While at ∆t → 0, γ → ∞ the dynamics is projected down to this subspace [32], making either ∆t or γ finite allows the operators to go outside, but only for a limited amount of time (in fact, when γ is small, results depend on the ratio γ/∆t only [33]). One can think of this as summing up certain contributions in a path-integral representation of the propagator q i |e iLt |q j , as illustrated in Fig. 1(a).
Unitary evolution is recovered by taking either γ → 0, ∆t → ∞ or * → ∞. In practice, we shall find it most useful to take the first option, keeping * and ∆t fixed while approaching the unitary limit through decreasing γ. We expect DAOE to achieve a good approximation to q i |q j (t) for the following reasons. The correlators considered are affected by the dissipation through 'backflow' processes [30], wherein a long Pauli string in q j (t ) at time t < t develops a component on a short operator such as q i by time t. Random circuit calculations [30,31] suggest that the combination of entropic effects and dephasing makes the backflow contributions to the diffusion constant decay exponentially with * . We leave a detailed discussion of this to future work [34].
The spirit of our approximation is closely related to the well-known memory matrix formalism (MMF) [2, 9-12, 35, 36]. The 'short' ( ≤ * ) and 'long' ( > * ) operators play the role of the 'slow' and 'fast' subspaces in the MMF. The backflow processes mentioned above, where a short operator grows to a long string and then back, are then similar to the kind of memory effects that MMF delineates. The dissipation acts as a cutoff on the memory time, more strongly suppressing processes where the operator is 'long' for a greater duration. This is expected to be a good approximation if these processes are indeed 'fast'. However, our method is directly applicable to generic lattice models in 1D, unlike the MMF, where one either perturbs around some fine-tuned point [37,38], or makes some uncontrolled approximations to simplify the dynamics within the fast subspace [2,9,12].
To reap the benefits of the dissipation, we represent |q j (t) as an MPS. The unitary part of the evolution can then be done with standard MPS techniques; for the nearest-neighbor Hamiltonians studied below, the timeevolving block decimation (TEBD) algorithm [15,16,39,40] provides an efficient solution. In this language, the superoperator D * ,γ becomes a matrix product operator (MPO) [15,40,41]. One can then straightforwardly evaluate Eq. (2), as illustrated in Fig. 1(c). As we will show, this can be done accurately with relatively low bond dimension, even for large systems and long times, provided that the dissipation is sufficiently strong. D * ,γ in fact has an exact MPO representation with bond dimension * +1. We label the local basis 'states' by n = 1 1, X, Y, Z (generalization to higher spin is straightforward). We then write the local MPO tensor, W nn ab , as a matrix acting on the virtual indices a, b = 0, 1, . . . , * . They read W 1 11 1 ab = δ a=b and W XX ab = W Y Y ab = W ZZ ab = δ a=b−1 + e −γ δ a=b= * , all others being zero. The MPO is contracted with the vector v L = (1, 0, . . . , 0) on the left, and v R = (1, . . . , 1, 1) on the right. It is easy to check that this reproduces the desired result.
The main limitation in the MPS representation of |q j is the operator entanglement [42][43][44][45][46][47], S vN [q j (t)], defined as the half-chain von Neumann entropy of the normalized state |q j (t) / q j (t)|q j (t) . For generic unitary dynamics, it tends to increase linearly [48,49]
, S vN [q j (t)] ∝ t.
In this case, the bond dimension χ needed for a faithful MPO/MPS representation grows exponentially with t, cutting short the times one can simulate [15,16]. We find that applying the dissipator decreases the operator entanglement, and this effect always becomes dominant at long times (see Fig. 2(a)). This key observation means that we can calculate |q j (t) with high precision, up to very long times, with a finite χ.
Results.-We use our method to calculate the dynamical correlations between the central site i = L+1 2 (we take L odd) and all other positions,
C j (t) ≡ Tr[q jq L+1 2 (t)]/N .
We normalize these such that j C j (0) = 1. One can characterize the spreading of correlations by the mean-square displacement (MSD),
d 2 (t) ≡ j C j (t)j 2 − j C j (t)j 2 .
(
In the strongly interacting, non-integrable systems we study, high-temperature transport of conserved quantities is expected to be diffusive [2,3,50,51], which manifests in a linear growth of the MSD at long times, d 2 (t) ∝ t. This suggests defining a time-dependent diffusion constant [4,[52][53][54][55] as 2D(t) ≡ ∂ t d 2 (t). The physical diffusion constant is then D ≡ lim t→∞ D(t) (assuming L → ∞ first). Further information about the frequency-and wavevector-dependence of the conductivity can be obtained by looking at space-time dependence of C j (t) [4,6,56]. Our approach is as follows. We calculate D(t) for the dissipative evolution, and then approach the unitary dynamics by decreasing γ, while keeping ∆t and * fixed. We decrease γ until we observe signs of convergence, allowing us to extrapolate the results for D back to γ → 0. We can estimate the accuracy of this extrapolation by comparing different values of * . As stated above, the value of ∆t is in principle irrelevant, as one finds a scaling collapse as a function of γ/∆t for small γ. However, in practice, ∆t should be small enough so that one can follow the full dynamics up to ∆t with the given bond dimension. It is also numerically more efficient not to make ∆t too small, in order to reduce the number of MPO-to-MPS multiplications we need to perform. We find that ∆t ≈ 1 (in units of microscopic couplings) works well. We investigate two Hamiltonians which we expect to be generic; further results on discrete circuit models are presented in [33].
Energy transport in the Ising chain. We first consider the Ising model in a tilted field,
H = j h j ≡ j g x X j + g z Z j + Z j−1 Z j + Z j Z j+1 2 .(4)
We fix g x = 1.4 and g z = 0.9045. At these values, we expect the model to be strongly chaotic [57,58], and hard to simulate exactly, due to fast entanglement growth. Here, h j is the energy associated to site j. This is the only local conserved density in the model, and its correlations capture energy (or heat) transport [58]. We therefore take q j ≡ h j in this case, and evolve h L+1
2
, as an MPO, according to Eq. (2). We perform the unitary part of the dynamics with TEBD, using a small Trotter time-step 0.01. We take large enough systems (L = 51) such that finite size effects are negligible at the times we study. Fig. 2(a) confirms that the dissipation limits the operator entanglement growth, so that the entropy S vN [h j (t)] peaks and then decreases. The time and height of the peak increase as γ gets smaller, but for any non-zero γ, dissipation dominates at long times. Moreover, we find that after the peak, S vN approaches 1 in units of ln 2, indicating that the operator is increasingly dominated by the local densities,h L+1
2 (t) ≈ j C j (t)h j .
We benchmark our method by comparing it to exact results on small systems, calculated using the canonical typicality approach [14,59,60], for up to L = 21 sites. In this case, finite-size effects limit the times one can reach to t ≈ 10. We compare these to the dissipative method for a particular set of parameters, * = 2, ∆t = 0.25, γ = 0.03, which we expect to be close to being converged to the physical diffusion constant (see below). The results for the MSD are presented in Fig. 2(b). The curve from the dissipative evolution follows the exact results, and then continues to grow linearly to much longer times, well beyond the reach of exact numerics. This is despite the fact that at these times, the dissipation already had a large effect (as measured, for example, by the decay of S vN ), andh L+1 2 (t) is far from the true time-evolved operator. Note that the dissipation is essential in allowing us to reach long times; for the same bond dimension (χ = 512), TEBD without dissipation starts deviating from the exact results around times t ≈ 7 − 8 due to truncation errors.
Having established the potential of the DAOE method, we now embark on the strategy outlined above, approaching the unitary limit by decreasing γ gradually from γ = ∞. For each set of parameters, we calculate a time-dependent diffusion constant D * ,∆t (t; γ). In the limit γ → 0 one would recover the physical result, lim γ→0 D * ,∆t (t; γ) = D(t), for any * and ∆t. In practice, we are limited to some minimal γ we can simulate with a certain bond dimension, while avoiding truncation errors. However, as we show, one can extrapolate from the data to get an estimate for the diffusion constant at γ = 0. Estimates for different * then allow us to check the accuracy of this extrapolation.
The results are shown in Fig. 3(a,c), for ∆t = 1 and * = 2, 3, 4. D(t) saturates to a γ-dependent constant. When γ is made sufficiently small, we find that the results converge. The last few data points are well fit by a straight line, which allows us to extrapolate D back to γ = 0. The extrapolated results for different choices of * all agree to within ≈ 1% error, supporting our conclusions that we indeed reached the physical diffusion constant (in this case, D ≈ 1.40). This constitutes strong evidence that our method can successfully capture transport coefficients to a high precision.
Spin transport in the XX ladder. Next, we study a spin-1/2 model on a two-leg ladder. We denote by j = 1, . . . , L the rungs of the ladder, and use a = 1, 2 for the two legs. Pauli operators on a given site are specified as X j,a , etc. The Hamiltonian then reads
H = L j=1 a=1,2 (X j,a X j+1.a + Y j,a Y j+1.a ) + L j=1 (X j,1 X j,2 + Y j,1 Y j,2 ) .(5)
Besides energy, this model also conserves the spin z component, j,a Z j,a . We examine the transport of the corresponding local conserved density q j = Z j ≡ (Z j,1 + Z j,2 )/2 along the chain. We take a system of L = 41 rungs, which is large enough to avoid finite-size effects, up to the times (t ≈ 20) that we simulate.
Spin transport in this model has been studied in a number of previous works, finding clear evidence of diffusive behavior with a diffusion constant D ≈ 0.95 [23,61,62]. Here we show that our method reproduces this result on much larger systems. We perform the same analysis as in the Ising model, comparing D for different γ and extrapolating back to γ = 0; the results are shown in Fig. 3(b,d). We find that the extrapolated results are all within the range D ≈ 0.96 − 0.98 (even for * = 1, where energyconservation is violated). The fact that these values are all very close to one another, and to the previous result, strongly supports the validity of our method.
Conclusions.-We introduced a controlled numerical method for computing transport properties in strongly interacting quantum systems at high temperatures. Our method is based on neglecting 'backflow' from complicated to simple operators. We provided a simple implementation of this method, using matrix product states, which allowed us to calculate dynamical correlations without finite-size or finite-time limitations. We demonstrated the utility of this approach on two spin models, showing that it can be used to estimate diffusion constants with high precision. An interesting open question is whether the method could be further improved by using ideas from Refs. 24 and 27.
There are a variety of physical problems that would be interesting to explore with this method, such as transport in 1D quantum magnets [63][64][65][66], disordered models [67][68][69][70][71] or long-range interacting [72] systems, where existing methods are even more limited. There might also be applications in quantum chemistry, where tensor network methods are becoming increasingly important [73][74][75][76][77]. A natural extension of our method is to finite temperatures. We expect it to work well at high temperatures, where the thermal density matrix is dominated by short operators [78][79][80][81][82][83], while it presumably breaks down as the low-temperature limit is approached. Precisely when and how this happens is itself an interesting question. Supplementary Material for "Dissipation-assisted operator evolution method for capturing hydrodynamic transport"
I. ADDITIONAL DATA FOR THE ISING CHAIN AND XX LADDER MODELS
A. Convergence with bond dimension
In the main text, we showed that the dissipation leads to a decay of the operator entanglement at long times. Crucially, this makes the maximal operator entanglement encountered during the evolution independent of system size, depending only on the parameters of the dissipation. As we argued, we can therefore capture the diffusive spreading of correlations up to arbitrarily long times, without significant finite-size or truncation effects. Here we show explicitly how the curves for D(t) converge as we increase the bond dimension χ.
The results are shown in Fig. 4 for the tilted field Ising model. We fix parameters * = 4, ∆t = 1, γ = 0.2 (same as in Fig. 2(b)) and compare results for different bond dimensions χ = 32, 64, 128, 256, 512. As the operator entanglement peaks and decreases (see Fig. 2(a)), the truncation error of the unitary TEBD time step also starts decreasing. While for small χ, the truncation errors encountered around the peak time are already significant, they decrease (roughly linearly) with χ. This also shows up in the results for the time-dependent diffusion constant, D(t). While at small χ the truncation effects are clearly visible, the curves quickly converge as χ is increased.
Another way of testing the effects of truncation is by looking at whether the conservation law (in this case, of energy) is satisfied. We consider the correlations C j (t) and normalize them such that j C j (0) = 1. The exact dissipative dynamics would maintain this normalization at all subsequent times due to energy conservation (assuming * is larger than the support of the terms in the Hamiltonian, in this case * ≥ 2). This is crucial for correctly capturing transport properties. We find that the errors in the conservation law, as measured by 1 − j C j (t) quickly decrease as χ becomes larger. We conclude that it is possible to simulate the dissipative dynamics (2) up to long times, with a bond dimension that is independent of total system size.
B. Scaling collapse as a function of γ/∆t
Here, we justify our claim in the main text that when γ is sufficiently small, the results (in particular, estimates of D) are functions of the ratio γ/∆t only. This can be seen by utilizing the Baker-Campbell-Hausdorff formula to rewrite the evolution operator (2) where t = N ∆t and we have introduced the logarithm of the dissipator, acting on a Pauli string as
K * [S] = 0 if S ≤ * ( S − * )S otherwise.(7)
In the second equality of Eq. (6) we assumed γ 1 to drop higher order terms that scale as γ 2 ∆t. We also assume that ∆t is at most an O(1) quantity, so that terms that scale as γ∆t 2 are of the same order as γ∆t.
Eq. (6) shows that the dynamics only depends on the ratio γ/∆t, and not on the individual value of γ and ∆t, up to times t ≈ 1/γ. As such, it does not directly constrain the diffusion constant, which is extracted from the long-time limit. However, in practice we find that D(t) saturates to a constant at a finite time t sat . While t sat itself depends on γ and ∆t (as well as on the Hamiltonian), we find that this dependence is relatively weak; in particular, t sat should converge to a finite, O(1) value as γ → 0. Therefore, estimate of D should also depend only on the ratio γ/∆t, provided that we are in the regime where γt sat 1.
Testing this expectation on the Ising chain (4), we find that it works remarkably well, even for γ ≈ 1 (we also find that it works increasingly well as * gets larger). This is shown in Fig. 5. Figs. 5(a,b) show that curves with identical ratio γ/∆t are the same at early times, and, moreover, their late time saturation values are also close to one another, provided that we are in a regime with sufficiently small γ. Consequently, the estimates for D show a scaling collapse when data for the same * but different ∆t, are plotted as a function of γ/∆t, see Fig. 5(c).
C. Operator weights
In the main text, we noted that the operator von Neumann entropy of the dissipatively evolving local density approaches 1 (in units of ln 2) at long times. Our interpretation was that this points to a long-time behavior where the evolving operator is increasingly dominated by its diffusive, 'conserved' part,q 0 (t) ≈ j C j (t)q j . We now further support this by calculating the weight of various operators inq 0 (in this section we use a different notation from the main text, with 0 denoting the center site).
To define what we mean by the weight of an operator, let us expandq 0 in the basis of Pauli strings,q 0 = S c S (t)S; the weight of the Pauli string is then the squared coefficient, |c S | 2 . The total weight on operators with length is given by the following quantity:
W (t) ≡ S S = |c S (t)| 2 .(8)
For unitary evolution one would have a conserved total weight, S |c S (t)| 2 = W (t) = 1. During evolution, the weight gets redistributed from short operators to an essentially random superposition of long ones, such that at time t The dissipator fundamentally changes this picture, as it removes operator weight from long strings. This reverses the effect of the unitary dynamics, making the contribution of short operators dominant at long times, which leads to the observed decay in the entanglement. While short operators, with ≤ * , are not affected directly by the dissipator, their weight also decreases as they get converted into longer strings which are subsequently dissipated. However, due to the hydrodynamic nature of transport, we find that the weight associated to local densities, |C j | 2 ≡ |c qj | 2 decreases parametrically more slowly than those of non-conserved operators, so that they dominate at long times.
To show this, we consider the XX ladder (5) and consider the evolution of the spin density,Z 0 (t). Calculating operator weights for this object, we find that the weight on local densities decays as W =1 ∼ t −1/2 , as expected from the diffusive nature of spin transport [30,31]. Considering larger , we find two things. First, for > * , the weight decreases exponentially with , as expected from the form of the dissipator. More importantly, however, we find that the weights for > 1 decay parametrically faster in time, W >1 ∼ t −3/2 (even when 1 < ≤ * ); this is shown in Fig. . This is consistent with our earlier prediction, thatZ 0 (t) is dominated by the local densities at long times.
The t −3/2 power law can be explained using the operator spreading picture developed in Refs. 30 and 31. In this picture, we rewrite the time-evolved local density at position x as q x (t)
q x (t) = q D x (t) + q B x (t)
where q D x (t) ≡ y C(x, y, t)Z y is the diffusive part of the operator and we assume that C(x, y, t) ≡ q y |q x (t) is well approximated by an unbiased diffusion kernel. q B
x (t) contains the contributions from all remaining Pauli strings, and is dominated by those with length = 2v B t, with v B the operator butterfly velocity [28,29]. The unitary dynamics leads to a conversion of weight from the diffusive to the ballistic part, whose local rate is given by 'current' squared, |∂ y C(x, y, t)| 2 . In this way, at each time step q D x sources new ballistically growing operators which thereafter form part of q B
x . This picture can be used to deduce the behaviour of W as a function of time. According to the above picture, operators of support would correspond to terms in q B which have been ballistically growing for a time interval t − τ = /(2v B ). The weight of such terms is therefore expected to be
dy (∂ y C(x, y, τ )) 2 ∼ D t − 2v B −3/2 .
This shows that the weight on length operators at time t 2vB goes as (Dt) −3/2 .
II. SPIN DIFFUSION IN FLOQUET CIRCUITS
We now complement the results shown for energy-conserving, Hamiltonian dynamics in the main text, with data on time-periodic models. We construct these as circuits of local unitary gates, with a 'brick-wall' structure and consequently, a strict light cone. This structure is illustrated in Fig. 7(a). We use the same two-site unitary u in each gate, such that the system has translation invariance in space (with unit cells composed of two sites) and in time (by two layers of the circuit). We want our circuit to conserve the total spin-z component. For a spin-1/2 chain, such a circuit is fully parametrized by three numbers, and it corresponds to a Trotterized version of an XXZ chain with a staggered magnetic field u = e −i(Jxy(S x 1 S x 2 +S y 1 S y 2 )+JzzS z 1 S z 2 +g(S z 1 −S z 2 )) ,
where we have now used spin operator S α instead of Pauli matrices (the two differ by a factor of 2), and the subscripts refer to the two sites on which the gate acts. We choose irrational values of the three couplings, J xy = 2 √ 7, J zz = 2 √ 5, g = 2 √ 3. We apply our dissipative evolution method for this circuit model, applying the dissipator after every second layer of the circuit (i.e., one Floquet period). We extract spin diffusion constant in the same way as in the main text. The results for the spin-1/2 circuit are plotted in Fig. 7(b). We find that the convergence to γ = 0 is less clear than in the Hamiltonian models we studied in the main text. In particular, for * = 1, 2 we observe a strong non-monotonicity with γ, while * = 3, 4 do appear to converge linearly to compatible values of D. Nevertheless, we note that the variations in D are all relatively small.
Our interpretation is that the apparent lack of convergence in Fig. 7(b) is not related to the Floquet circuit nature of our model; rather, it has to do with the fact that it is close to an integrable point. It was recently shown that for g = 0, the model in Eq. (9) is integrable; this is closely related to the integrability of the XXZ Hamiltonian. In the latter case, a staggered field is known to break integrability [60,86,87], so we expect that for generic g our circuit is also non-integrable. However, we believe that the nearby integrable point is responsible for the non-trivial behavior we observe (for example some almost-conserved operator of length = 3 could explain why the * ≤ 2 curves have a qualitatively different behavior from * ≥ 3).
To test this intuition, we also consider the spin-1 version of the same model. That is, we use the same definition of the two-site gate as in Eq. (9), but with S α 1,2 standing for spin-1 operators. The results for this case are shown in Fig. 7(c). While we find that getting to smaller γ becomes quite difficult in this case, due to a quick initial growth of operator entanglement, so that our results are not as precisely converged as for the models presented in the main text, we find no evidence of strong non-monotonicities in the regime we can simulate. This reinforces our belief that the peculiar behavior exhibited by the spin-1/2 model is tied to the presence of nearby integrable points.
FIG. 1 .
1Dissipation-assisted operator evolution (DAOE) method. (a) Sketch of the non-unitary evolution
FIG. 2 .
2Testing DAOE on the Ising model(4). (a) shows how the dissipation (for * = 2, ∆t = 0.25) suppresses operator entanglement (measured in units of ln 2). (b) shows that the MSD (3) is correctly captured to long times by the DAOE (same * , ∆t; γ = 0.03, using bond dimensions χ = 512), by comparing to exact results on small chains (L = 9, 13, 17, 21).
FIG. 3 .
3Estimating the diffusion constant. (a,c) show data for the Ising chain (4) and (b,d) for the XX ladder (5). We fix ∆t = 1 and use bond dimensions up to χ = 768. In (c) and (d) we show results for the time-dependent diffusion constant at a fixed * = 3 for varying γ, showing clear signs of convergence. In (a,b) we show the the estimate for D (taken as the average of D(t) in the interval t ∈ [15, 20]). Data for the weakest dissipations is well fit by a linear extrapolation, and results for different * give consistent estimates for the physical diffusion constant. In (b) and (d), the and dotted line represent the estimate D = 0.95 from Ref. 61.
FIG. 4 .
4Convergence of results with bond dimension χ in the Ising chain (4) for dissipation parameters * = 4, ∆t = 1, γ = 0.2. (a): Truncation error per TEBD step, summed over all bonds in the chain (L = 51 sites). (b) Convergence of results for D(t) (see main text for definition). (c) Errors in the energy conservation, as measured by the sum of the coefficients of local energy density terms Cj(t).
FIG. 5 .
5as D * ,γ e iL∆t N ≡ e −K * γ e iL∆t N = e −K * γ+iL∆t+O(γ∆t) N = e −K * N γ+iLN ∆t+O(γN ∆t) = e t(iL−K * γ ∆t )+O(γt) , Scaling collapse as a function of γ/∆t. (a,b): comparison of time-dependent diffusion constants for two curves with different ∆t but the same ratio γ/∆t. When γ is sufficiently small, the results remain close to each other even at long times. (c): Estimates of D ≡ limt→∞D(t), comparing ∆t = 1 and ∆t = 1/4. In the small γ regime, relevant for extrapolation, the curves with the same * collapse when plotted as function of γ/∆t.
FIG. 6 .
6Total weight on strings of size as a function of time. The majority of the remaining (not yet dissipated) weight is on 1-site strings as decays as t −1/2 . The weight of longer strings decays as t −3/2 . Data shown for ∆t = 1, * = 5 with γ = 0.05 (left) and γ = 0.25 (right).
the operator is dominated by strings of length ∼ v B t, with v B the butterfly velocity. This leads to the linear growth of operator entanglement with time.
FIG. 7 .
7Diffusion constants in Floquet curcits. (a) The circuits have a brick-wall structure, updating even/odd bonds in turn. Every gate is given by the same Sz-conserving two-site unitary u. (b,c) Estimates of the spin diffusion constant for the circuit defined by Eq. (9), for spin-1/2 and spin-1 chains.
for stimulating discussions, and in particular Ehud Altman for his talk at the KITP Conference "Novel Approaches to Quantum Dynamics" that in part inspired our work.CvK is supported by a Birmingham Fellowship. FP is funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No. 771537). FP acknowledges the support of the DFG Research Unit FOR 1807 through grants no. PO 1370/2-1, TRR80, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2111-390814868. This work was initiated at KITP where TR, CvK, and FP were supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 (KITP) during the "Dynamics of Quantum Information" program. TR further acknowledges the hospitality of KITP as part of the graduate fellowship program of the Fall of 2019, during which some of this work was performed.
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| [] |
[
"Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian",
"Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian"
] | [
"Takafumi Amaba \nFukuoka University\n8-19-1 Nanakuma, Jônan-ku814-0180FukuokaJapan\n",
"Roland Friedrich \nFaculty of Mathematics\nSaarland University\nD-66123SaarbrueckenGermany\n"
] | [
"Fukuoka University\n8-19-1 Nanakuma, Jônan-ku814-0180FukuokaJapan",
"Faculty of Mathematics\nSaarland University\nD-66123SaarbrueckenGermany"
] | [] | We introduce the notion of controlled Loewner-Kufarev equations and discuss aspects of the algebraic nature of the equation embedded into the (Sato)-Segal-Wilson Grassmannian. Further, we relate it to conformal field theory (CFT) and free probability. | 10.3842/sigma.2020.108 | [
"https://arxiv.org/pdf/1809.00534v2.pdf"
] | 119,312,227 | 1809.00534 | 2ad9f43cc8b605f6be9e3abe204fe0e1a8f93a89 |
Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian
Sep 2018
Takafumi Amaba
Fukuoka University
8-19-1 Nanakuma, Jônan-ku814-0180FukuokaJapan
Roland Friedrich
Faculty of Mathematics
Saarland University
D-66123SaarbrueckenGermany
Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian
17Sep 2018
We introduce the notion of controlled Loewner-Kufarev equations and discuss aspects of the algebraic nature of the equation embedded into the (Sato)-Segal-Wilson Grassmannian. Further, we relate it to conformal field theory (CFT) and free probability.
Introduction
C. Loewner [24] and P. P. Kufarev [25] initiated a theory which was then further extended by C. Pommerenke [31], which shows that given any continuously increasing family of simply connected domains containing the origin in the complex plane, the inverses of the Riemann mappings associated to the domains are described by a partial differential equation, the so-called Loewner- (Kufarev) equation
∂ ∂t f t (z) = zf ′ t (z)p(t, z)
(see Section 2.2 for details). More recently, I. Markina and A. Vasil'ev [27,29] considered the so-called alternate Loewner-Kufarev equation, which describes not necessarily increasing chains of domains.
In this paper, we further generalise it and introduce a class of controlled Loewner-Kufarev equations, df t (z) = zf ′ t (z){dx 0 (t) + dξ(x, z) t }, f 0 (z) ≡ z ∈ D 0 , where D 0 is the unit disc in the complex plane centred at zero, x 0 , x 1 , x 2 , · · · are given functions which will be called the driving functions, x = (x 1 , x 2 , · · · ) and ξ(x, z) t = ∞ n=1 x n (t)z n . In writing down the controlled Loewner-Kufarev equation, the right hand side becomes − ∞ n=0 (L n f )(z)dx n , where L n = −z n+1 ∂/(∂z) for n ∈ Z, the L n are exactly the generators of the non-negative part of the Witt algebra, or put differently, central charge zero Virasoro algebra. Therefore, we are going to consider an extension of [9], where the second author established and studied the role of Lie vector fields, boundary variations and the Witt algebra in connection with the Loewner-Kufarev equation.
So, let us start first with the classical work of A. A. Kirillov and D. V. Yuriev [17] / G.B. Segal and G. Wilson [34] / N. Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada [16] which will be also fundamental and basic in the present context, in particular in understanding the appearance of the Virasoro algebra with nontrivial central charge.
A. A. Kirillov and D. V. Yuriev [17], constructed a highest weight representation of the Virasoro algebra, where the representation space is given by the space of all holomorphic sections of an analytic line bundle over the orientation-preserving diffeomorphism group Diff + S 1 of the unit circle S 1 (modulo rotations). They also gave an embedding of (Diff + S 1 )/S 1 into the infinite dimensional Grassmannian. In fact, this embedding is an example of a construction of solutions to the KdV hierarchy found by I.Krichever [21], which we address in Section 3.2.
If we embed a univalent function on the unit disc D 0 into the infinite dimensional Grassmannian, by the methods of Kirillov-Yuriev [17], Krichever [21], or Segal-Wilson [34], then one needs to track the Faber polynomials and Grunsky coefficients associated to the univalent function. In general, it is not easy to calculate them from the definition.
One of our main results is, however, the following. R. Friedrich [9] proposed to lift the embedded Loewner-Kufarev equation to the determinant line bundle over the Sato-Segal-Wilson Grassmannian Gr(H), as a natural extension of the "Virasoro Uniformisation" approach by M. Kontsevich [18] / R. Friedrich and J. Kalkkinen [8] to construct generalised stochastic / Schramm-Loewner evolutions [33] on arbitrary Riemann surfaces, which would also yield a connection with conformal field theory in the spirit of [34,16]. Let us also mention the work of B. Doyon [3], who uses conformal loop ensembles (CLE), which can be nicely related with the content of the present article.
In [29], I. Markina and A. Vasil'ev established basic parts of this program, by considering embedded solutions to the Loewner-Kufarev equation into the Segal-Wilson Grassmannian and related the dynamics therein with the representation of the Virasoro algebra, as discussed by Kirillov-Yuriev [17].
Further, they considered the tau-function associated to the embedded solution as a lift to the determinant line bundle.
As observed and briefly discussed in [18,8], the generator of the stochastic Loewner equation is hypo-elliptic.
This observation was more recently worked out and detailed by J. Dubédat [4,5]. Also, I. Markina, I. Prokhorov and A. Vasil'ev [26] observed and discussed the sub-Riemannian nature of the coefficients of univalent functions. This connects nicely with the general theory of hypo-elliptic flows, as explained e.g. in the book by F. Baudoin [1], and which led to propose a connection of the global geometry of (stochastic) Loewner-Kufarev equations with rough paths. Now, in the theory of rough paths (see e.g., the introduction in [23]), one of the central objects of consideration is the following controlled differential equation:
(1.1) dY t = ϕ(Y t )dX t ,
where X t is a continuous path in a normed space V , called the input of (1.1). On the other hand, the path Y t is called the output of (1.1). When we deal with this equation, an important object is the signature of the input X t , with values in the (extended) tensor algebra associated with V and which is written in the following form:
S(X) s,t := (1, X 1 s,t , X 2 s,t , · · · , X n s,t , · · · ), s t. If X t has finite variation with respect to t, then each X n s,t is the nth iterated integral of X t over the interval [s, t]. With this object, a combination of the Magnus expansion and the Chen-Strichartz expansion theorem (see e.g., [1,Section 1.3]) tells us that the output Y t is given as the result of the action of S(X) 0,t applied to Y 0 .
Heuristically, we may say that a 'group element' S(X) in some big 'group' acts on some element in the (extended) tensor algebra T ((V )) which gives the output Y t , or it might be better to say that the vector field ϕ defines how the 'group element' acts on the algebra. In this spirit, we would like to describe such a picture in the context of controlled Loewner-Kufarev equations.
For this, we extract the algebraic structure of the controlled Loewner-Kufarev equation. If we regard the driving functions x 0 , x 1 , x 2 , · · · just as letters in an alphabet then it turns out that explicit expressions for the associated Grunsky coefficients are given by the algebra of formal power series, where the space of coefficients is given by words over this alphabet. It is worth mentioning that the action of the words over this alphabet will be actually given by the negative part of the Witt generators. Thus the action of the signature encodes many actions of such negative generators. This can be used to derive a formula for f t (z) as the signature 'applied' to the initial data f 0 (z) ≡ z (see Theorem 3.8).
However, the story so far lets us ask how the signature associated with the driving functions describes the corresponding tau-function rather than the f t itself.
Theorem 1.2 (see Theorem 3.9). Along the solution of the controlled Loewner-Kufarev equation, the associated tau-function can be written as the determinant of a quadratic form of the signature.
Let us now summarise the structure of the paper. In Section 2, we formulate solutions f t (z) to controlled Loewner-Kufarev equations. We add also a brief review of the classical Loewner-Kufarev equation, and then explain how the classical one is recovered by the controlled Loewner-Kufarev equation. We track the variation of the Taylor-coefficients of f t and also the Faber polynomials and Grunsky coefficients. In Section 3, we first recall briefly basics of the Segal-Wilson Grassmannian and Krichever's construction. After that, we describe how a univalent function on D 0 is embedded into the Grassmannian. We extract the algebraic structure of the controlled Loewner-Kufarev equations in order to obtain Theorem 3.9. Finally, in Section 4, we establish and briefly discuss connections of the material considered so far with CFT and free probability. All fundamental things about Faber polynomials and Grunsky coefficients are wrapped up in Appendix A. Several proofs of propositions are also relegated to Appendix A, because they tend to be rather long (in particular, the proof of Proposition 2.9 will be very long) as we intended not to interrupt the flow of the principal story.
Controlled Loewner-Kufarev equation
2.1. Definition of solutions to controlled Loewner-Kufarev equations. Given functions x 1 , x 2 , · · · : [0, T ] → C, we will write x := (x 1 , x 2 , · · · ) and
ξ(x, z) t := ∞ n=1
x n (t)z n for z ∈ C if it converges. If A : [0, T ] → C is of bounded variation, we write dA or A(dt) (when emphasising the coordinate t on [0, T ]) the associated complex-valued Lebesgue-Stieltjes measure on [0, T ], and the total variation measure will be denoted by |dA|.
Definition 2.1. Let T > 0. Suppose that x 0 : [0, T ] → R, x 1 , x 2 , · · · : [0, T ] → C are continuous and of bounded variations, and x 0 (0) = 0. Let f t : D 0 → C be conformal mappings for 0 t T . We say {f t } 0 t T is a solution to
df t (z) = zf ′ t (z){dx 0 (t) + dξ(x, z) t }, f 0 (z) ≡ z ∈ D 0 (2.1) if (1) f 0 (z) ≡ z for z ∈ D 0 ;
(2) ∞ n=1 n [0,T ] |dx n |(t)z n converges uniformly in z on each compact set in D 0 ;
(3) for each compact set K ⊂ D 0 , the mapping [0, T ] ∋ t → f ′ t | K ∈ C(K)
is continuous with respect to the uniform norm on K;
(4) it holds that
f t (z) − z = t 0 zf ′ s (z) dx 0 (s) + dξ(x, z) s , (t, z) ∈ [0, T ] × D 0 .
In the sequel, we refer to equation (2.1) as a controlled Loewner-Kufarev equation (with driving paths x 0 and x := (x 1 , x 2 , · · · )).
Remark 2.1.
(a) We do not put the univalency condition into the notion of solution. However, the condition (1) and (3)
(2), µ([s, t]) := ∞ k=1 [s,t] φ(k, u)r k dx k (u) defines a finite signed measure µ. We will denote it by µ(dt) =: ∞ k=1 φ(k, t)r k dx k (t). One can check that the signed measure N k=1 φ(k, t)r k dx k (t) converges to ∞ k=1 φ(k, t)r k dx k (t) weakly as N → ∞, and hence we may write for each z ∈ D 0 , dξ(x, z) t = ∞ k=1 z k dx k (t).
Remark 2.2. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then the following holds:
(a) For each w ∈ C and a compact set K ⊂ D 0 , the mapping t → sup z∈K |f ′ t (z) − w| is continuous since we have
sup z∈K |f ′ t (z) − w| − sup z∈K |f ′ s (z) − w| sup z∈K |f ′ t (z) − f ′ s (z)|
and the condition in Definition 2.1-(3).
(b) The mapping t → f t is continuous with respect to the uniform norm on each compact set in D 0 . In fact, by using Definition 2.1-(4), we have for each compact
set K ⊂ D 0 , sup z∈K |f t (z) − f s (z)| sup 0 u T f ′ u | K ∞ t s |dx 0 (u)| + ∞ n=1 |z 0 | n |dx n (u)| ,
where z 0 ∈ K is such that |z 0 | = sup z∈K |z|. By Definition 2.1-(3) (or the above remark (a)), we find that sup 0 u T f ′ u | K ∞ < +∞. Therefore, by using the dominated convergence theorem with Definition 2.1-(2), the right-hand side goes to zero as |t − s| → 0. (1) f t is analytic and univalent on D 0 for each 0 t T ;
(2) f t (z) = e t z + a 2 (t)z 2 + · · · for z ∈ D 0 ;
( T , the function f t (z) = e t z + · · · is analytic in |z| < r 0 , the mapping [0, T ] ∋ t → f t (z) is absolutely continuous for each |z| < r 0 , and |f t (z)| K 0 e t for all |z| < r 0 and t ∈ [0, T ].
3) f s (D 0 ) ⊂ f t (D 0 ) for each 0 s < t T .(ii) Re{p(t, z)} > 0 for all (t, z) ∈ [0, T ] × D 0 and ∂ ∂t f t (z) = zf ′ t (z)p(t, z) (2.2)
for all |z| < r 0 and for almost all t ∈ [0, T ].
According to the terminology in [2] we call the equation (2.2) the Loewner-Kufarev equation (if we regard p(t, z) as given and f t (z) as unknown).
Because of the equation (2.2), it holds that p(t, 0) = lim z→0 ( ∂ ∂t f t (z))/(zf ′ t (z)) = 1, and hence the 'Herglotz Representation Theorem' applies, which permits to conclude that, for every t ∈ [0, T ], there exists a probability measure ν t on S 1 = ∂D 0 (which is naturally identified with [0, 2π] as measurable spaces, and then the induced probability measure is still denoted by ν t ) such that
p(t, z) = 2π 0 e iθ + z e iθ − z ν t (dθ) for z ∈ D 0 .
Substituting this into (2.2), the Loewner-Kufarev equation becomes
∂f t ∂t (z) = zf ′ t (z) 2π 0 e iθ + z e iθ − z ν t (dθ). (2.3)
Assuming that ν t (dθ) =: ν t (θ)dθ, is sufficiently regular, we write the Fourier series of ν t (θ) as
ν t (θ) = 1 2π a 0 (t) + ∞ k=1 a k (t) cos(kθ) + b k (t) sin(kθ) .
We temporally introduce the notations x 0 (t) := t 0 a 0 (s)ds and
u k (t) := t 0 a k (s)ds, v k (t) := − t 0 b k (s)ds
for k = 1, 2, · · · . By using the relations 1 2π
2π 0 e iθ + z e iθ − z cos(kθ)dθ = z k , 1 2π 2π 0 e iθ + z e iθ − z sin(kθ)dθ = −iz k ,
for k = 1, 2, · · · and |z| < 1, equation (2.3) assumes the following form:
∂f t ∂t (z) = zf ′ t (z) ẋ 0 (t) + ∞ k=1 u k (t) + iv k (t) z k .
This can be rewritten as the following controlled differential equation
df t (z) = zf ′ t (z){dx 0 (t) + dξ(x, z) t } (2.4) where x k (t) = u k (t) + iv k (t) for k ≥ 1, and ξ(x, z) t = ∞ k=1 x k (t)z k .
If we omit the condition Re{p(t, z)} > 0, that is, we allow the real part of p(t, z) to have arbitrary signs, then the equation (2.2) is called alternate Loewner-Kufarev equation, as considered by I. Markina and A. Vasil'ev [27]. Intuitively, this describes evolutions of conformal mappings whose images of D 0 are not necessary increasing, i.e. not strict subordinations. It appears that the general theory with respect to the existence and uniqueness of solutions is not yet fully developed. However, our controlled Loewner-Kufarev equations (2.1) deal with this alternate case because we have not assumed that p(t, z) := d dt (x 0 (t) + ξ(x, z) t ) has a positive real part. Remark 2.3. Readers focusing on radial Loewner equations might feel puzzled by the heuristic assumption that the Radon-Nikodym density νt(dθ) dθ = ν t (θ) exists, because the radial Loewner equation describes the case ν t (dθ) = δ e iw(t) (dθ) where w(t) is a continuous path in R, so that there does not exist a Radon-Nikodym density. However, several explicit examples of Loewner-Kufarev equations within this setting, are presented with simulations in Sola [35].
Taylor coefficients along the controlled Loewner
-Kufarev equation. Sup- pose that x 0 : [0, +∞) → R, x 1 , x 2 , · · · : [0, +∞) → C are continuous and of bounded variation. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). We parametrise f t as f t (z) = C(t) z + c 1 (t)z 2 + c 2 (t)z 3 + c 3 (t)z 4 + · · · , (2.5) with the convention c 0 (t) ≡ 1.
The dynamics of the coefficients (c 1 , c 2 , · · · ) has been previously studied by Vasil'ev and his co-authors [13,26,27,28]. The (stochastic/Schramm)-Loewner (equation/evolution) (SLE) case is discussed by Friedrich [9]. In our framework, we obtain similarly the following.
Proposition 2.6. Let {f t } 0≤t≤T be a solution to the controlled Loewner-Kufarev equation (2.1) with the parametrisation (2.5). Then we have
dC(t) = C(t)dx 0 (t) (2.6) and dc 1 (t) = dx 1 (t) + c 1 (t)dx 0 (t), dc 2 (t) = dx 2 (t) + 2c 1 (t)dx 1 (t) + 2c 2 (t)dx 0 (t), . . . dc n (t) = dx n (t) + n−1 k=1 (k + 1)c k (t)dx n−k (t) + nc n (t)dx 0 (t),
for n ≥ 2,
(2.7)
with the initial conditions C(0) = 1 and c 1 (0) = c 2 (0) = · · · = 0. In particular, C = {C(t)} 0≤t≤T takes its values in R.
Since then f ′ t (0) = C(t) = e x 0 (t)−x 0 (0) = 0, we obtain Corollary 2.7. Let {f t } 0≤t≤T be a solution to the controlled Loewner-Kufarev equation (2.1). Then f t is univalent in a neighbourhood of 0 for each 0 t T .
By applying variation of constants to (2.7), we obtain the following recurence relation
c n (t) = e nx 0 (t) t 0 e −nx 0 (s) dx n (s) + n−1 k=1 (k + 1)e nx 0 (t) t 0 e −nx 0 (s) c k (s)dx n−k (s)
for n 2, and we get
Theorem 2.8. Let {f t } 0≤t≤T be a solution to the controlled Loewner-Kufarev equation (2.1). Then for each n ∈ N, the coefficient c n in (2.5) is given by c n (t) = n p=1 i 1 ,··· ,ip∈N: i 1 +···+ip=n w(n) i 1 ,··· ,ip e nx 0 (t) × 0 s 1 <s 2 <···<sp t e −i 1 x 0 (s 1 ) dx i 1 (s 1 )e −i 2 x 0 (s 2 ) dx i 2 (s 2 ) · · · e −ipx 0 (sp) dx ip (s p ) where w(n) i 1 ,··· ,ip := (n − i 1 ) + 1 (n − (i 1 + i 2 )) + 1 · · · (n − (i 1 + i 2 + · · · + i p−1 )) + 1 , and where n = i 1 + · · · + i p .
The proof is similar to the one of Proposition 2.12 below and hence omitted.
Variation of Grunsky coefficients induced by a Loewner-Kufarev equation.
Here
(i) U ⊂ D 0 ; (ii) f t | U is univalent for each t ∈ [0, T ]; (iii) V := ∩ 0 t T f t (U)
is an open neighbourhood of the origin;
(iv) For each ζ ∈ V , [0, T ] ∋ t → f −1 t (ζ)
is continuous and of bounded variation;
(v) For each ζ ∈ V , with f −1 (t, ζ) := f −1 t (ζ) and df −1 t (ζ) := f −1 (dt, ζ), we have df −1 t (ζ) = −f −1 t (ζ) dx 0 (t) + ∞ k=1 (f −1 t (ζ)) k dx k (t)
as Lebesgue-Stieltjes measures on [0, T ].
The proof of Proposition 2.9 is based on classical techniques to prove inverse function theorems however a little bit involved. Therefore we shall postpone it to Appendix A.2.
Let {f t } 0≤t≤T be a solution to the controlled Loewner-Kufarev equation (2.1). Because of Corollary 2.7, associated to f t (z) are Faber polynomials and Grunsky coefficients (see Appendix A-Definition A.1) which will be denoted by Q n (t, w) and b −n,−m (t) respectively. Proposition 2.10.
(i) (Variation of Faber polynomials): We have for each n ∈ N,
dQ n (t, w) = ndx n (t) + n n k=1 Q k (t, w)dx n−k (t).
(ii) (Variation of Grunsky coefficients): For each n, m ∈ N,
db −n,−m (t) = −dx n+m (t) + k,l∈Z 0 ; k+l=m−1 (k + 1)b −n,−(k+1) (t)dx l (t) + k,l∈Z 0 ; k+l=n−1 (k + 1)b −m,−(k+1) dx l (t), (2.8)
with the initial conditions b −n,−m (0) = 0 for all n, m ∈ N.
Proof. (i) Let n ∈ N. Let U and V be as in Proposition 2.9. Then f −1 t (ζ), ζ ∈ V satisfies the equation
df −1 t (ζ) = −f −1 t (ζ) dx 0 (t) + ∞ k=1 (f −1 t (ζ)) k dx k (t) .
Let X 0 ⊂ V be an open disc centred at 0. By using Cauchy's integral formula, we have for w ∈ X 0 ,
dQ n (t, w) = 1 2πi ∂X 0 dζ ζ d(f −1 t (ζ)) −n 1 − ζw −1 = 1 2πi ∂X 0 (−n) (f −1 t (ζ)) −n−1 1 − ζw −1 (−f −1 t (ζ)) ∞ k=0 (f −1 t (ζ)) k dx k (t) dζ ζ = n k=0 n 2πi ∂X 0 (f −1 t (ζ)) −n+k 1 − ζw −1 dζ ζ dx k (t) = ndx n (t) 2πi ∂X 0 1 1 − ζw −1 dζ ζ + n n−1 k=0 Q n−k (t, w)dx k (t).
Noting that the orientation of ∂X 0 is anti-clockwise, we get
1 2πi ∂X 0 1 1 − ζw −1 dζ ζ = 1,
and hence the result.
(ii) Since f t (z) satisfies the controlled Loewner-Kufarev equation, by putting p(dt, z) :
= dx 0 (t) + dξ(x, z) t , we have dQ n (t, f t (z)) = Q n (dt, f t (z)) + Q ′ n (t, f t (z))df t (z) = Q n (dt, f t (z)) + Q ′ n (t, f t (z)) zf ′ t (z)p(dt, z) = Q n (dt, f t (z)) + z ∂ z Q n (t, f t (z)) p(dt, z), so that dQ n (t, f t (z)) = Q n (dt, f t (z)) + z ∂ z Q n (t, f t (z)) p(dt, z). (2.9) Recalling that Q n (t, f t (z)) = z −n + n ∞ m=1 b −n,−m (t)z m , (LHS of (2.9)) 1 = (LHS of (2.9)) = n ∞ m=1 z m db −n,−m (t). (2.10)
On the other hand, by Proposition 2.10-(i), we have
dQ n (t, f t (z)) = ndx n (t) + n n k=1 Q k (t, f t (z))dx n−k (t) = ndx n (t) + n n k=1 z −k + k ∞ m=1 b −k,−m (t)z m dx n−k (t) = ndx n (t) + n n k=1 z −k dx n−k (t) + n ∞ m=1 n k=1 kb −k,−m (t)dx n−k (t) z m so that dQ n (t, f t (z)) 1 = n ∞ m=1 n k=1 kb −k,−m dx n−k (t) z m . (2.11) We further have z ∂ z Q n (t, f t (z)) p(dt, z) = z − nz −n−1 + n ∞ k=1 kb −n,−k z k−1 dx 0 (t) + ∞ l=1 dx l (t)z l = n − dx 0 (t)z −n − ∞ m=1−n dx m+n (t)z m + ∞ m=1 mb −n,−m (t)dx 0 (t)z m + ∞ m=2 k,l 1; k+l=m kb −n,−k (t)dx l (t)z m ,
from which we conclude
z ∂ z Q n (t, f t (z)) p(dt, z) 1 = n ∞ m=1 − dx n+m (t) + k 1, l 0; k+l=m kb −n,−k (t)dx l (t) z m .
(2.12)
Combining (2.11) and (2.12), we obtain
(RHS of (2.9)) 1 = n ∞ m=1 − dx n+m (t) + k,l∈Z 0 ; k+l=m−1 (k + 1)b −n,−(k+1) (t)dx l (t) + k,l∈Z 0 ; k+l=n−1 (k + 1)b −m,−(k+1) dx l (t) z m ,
and then by comparing with (2.10), we get the result. The initial condition is obviously derived from f 0 (z) ≡ z.
To derive an explicit formula for general b −n,−m (t), we shall introduce some notation.
Definition 2.11. Let p, q ∈ N.
(1) A bijection σ : {1, 2, · · · , p + q} → {1, 2, · · · , p + q} is called a (p, q)-shuffle if it holds that σ(1) < σ(2) < · · · < σ(p) and σ(p + 1) < σ(p + 2) < · · · < σ(p + q).
(2) Suppose that x 1 , x 2 , · · · , x p+q : [0, T ] → C are continuous and of bounded variations. Then for each 0 t T , we set
(x 1 · · · x p ) ¡ (x p+1 · · · x p+q ) (t) :=: 0 sq ··· s 1 tp ··· t 1 t dx 1 (t 1 ) · · · dx p (t p ) ¡ dx p+1 (s 1 ) · · · dx p+q (s q ) := σ: (p, q)-shuffle t 0 dx σ(1) (t 1 ) t 1 0 dx σ(2) (t 2 ) · · · t p−1 0 dx σ(p) (t p ) × tp 0 dx σ(p+1) (s 1 ) s 1 0 dx σ(p+2) (s 2 ) · · · s q−1 0 dx σ(p+q) (s q ).
The general formula for the Grunsky-coefficients along the controlled Loewner-Kufarev equation (2.1) is given by the following. The proof is given in Section A.3. Proposition 2.12. For n, m ∈ N and t 0,
b −m,−n (t) = −e (n+m)x 0 (t) t 0 e −(n+m)x 0 (s) dx m+n (s) − n+m−2 k=2 1 i<m; 1 j<n: i+j=k m−i p=1 n−j q=1 i 1 ,··· ,ip∈N: i 1 +···+ip=m−i j 1 ,··· ,jq∈N: j 1 +···+jq=n−j w(i, j) i 1 ,··· ,ip;j 1 ,··· ,jq × e (m+n)x 0 (t) 0 uq ··· u 1 sq ··· s 1 t e −i 1 x 0 (s 1 ) dx i 1 (s 1 ) · · · e −ipx 0 (sp) dx ip (s p ) ¡ e −j 1 x 0 (u 1 ) dx j 1 (u 1 ) · · · e −jqx 0 (uq) dx jq (u q ) uq 0 e −kx 0 (s) dx k (s) − n+m−1 k=m+1 n+m−k q=1 j 1 ,··· ,jq∈N: j 1 +···+jq=n+m−k w(k − m) ∅;j 1 ,··· ,jq × e (m+n)x 0 (t) 0 sq ··· s 1 t e −j 1 x 0 (s 1 ) dx j 1 (s 1 ) · · · e −jqx 0 (sq) dx jq (s q ) sq 0 e −kx 0 (s) dx k (s) − n+m−1 k=n+1 m+n−k p=1 i 1 ,··· ,ip∈N: i 1 +···+ip=m+n−k w(k − n) i 1 ,··· ,ip;∅ × e (m+n)x 0 (t) 0 up ··· u 1 t e −i 1 x 0 (u 1 ) dx i 1 (u 1 ) · · · e −ipx 0 (up) dx ip (u p ) up 0 e −kx 0 (u) dx k (u), (2.13)
where, for m = i 1 + · · · + i p + r and n = j 1 + · · · + j q + s, we have put
w(r) i 1 ,··· ,ip;∅ = (m − i 1 )(m − (i 1 + i 2 )) · · · (m − (i 1 + i 2 + · · · + i p )), w(s) ∅;j 1 ,··· ,jq = (n − j 1 )(n − (j 1 + j 2 )) · · · (n − (j 1 + j 2 + · · · + j q )),
and w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq := w(r) i 1 ,··· ,ip;∅ w(s) ∅;j 1 ,··· ,jq . (2) The orthogonal projection pr − : W → H − is compact. The Fredholm index of the orthogonal projection pr + : W → H + is called the virtual dimension of W . For d ∈ Z, we set
Gr( ∞ 2 + d, ∞) := {W ∈ Gr : the virtual dimension of W is d} and Gr( ∞ 2 , ∞) := Gr( ∞ 2 + 0, ∞).
If we take W = H + , then the corresponding projections are given by pr + = id H + and pr − = 0 which are Fredholm and compact operators, respectively. Therefore we have H + ∈ Gr( ∞ 2 , ∞). Definition 3.2 ([34, Section 5]). We denote by Γ + the set of all continuous functions g :
S 1 → C * such that g(z) = e ∞ k=1 t k z k , z ∈ S 1 for some t = (t 1 , t 2 , t 3 , · · · ).
The set Γ + acts on H by pointwise multiplication. In particular, Γ + forms a group. This action induces the action of Γ + on Gr:
Γ + ×Gr ∋ (g, W ) → gW ∈ Gr (see [34, Lemma 2.2 and Proposition 2.3]), where gW = {gf : f ∈ W }. For any g = e ∞ k=1 t k z k ∈ Γ + , the action of g on H is of the form g = a b 0 d along H = H + ⊕ H − , where a : H + → H + is invertible and b : H − → H + is of trace class (see [34, Proposi- tion 2.3]). Let U be the set of all W ∈ Gr( ∞ 2 , ∞) such that the orthogonal projection W → H + is an isomorphism. Then, associated to each W ∈ U is the tau-function τ W (t) of W , where t = (t 1 , t 2 , · · · )
, satisfying the following:
Proposition 3.3 ([34, Proposition 3.3]). Let W ∈ U. For g = e ∞ n=1 tnz n ∈ Γ + , we have τ W (t) = det(1 + a −1 bA), where t = (t 1 , t 2 , t 3 , · · · ), g −1 = a b 0 d along H = H + ⊕ H −
and A : H + → H − is the linear operator such that graph(A) = W .
3.2.
Krichever's construction. In connection with algebraic geometry and infinitedimensional integrable systems, a fundamental observation / construction of Krichever [19,20,21] states that, associated to each non-singular algebraic curve with some additional data (which are called algebro-geometric data) is a solution of the KdV equation. This construction has been developed further, and after a remark by Mumford [30], it was formalised by Segal-Wilson [34] as follows.
In this context, an algebro-geometric datum (X, L, x ∞ , z, ϕ) consists of a complete irreducible complex algebraic curve X, a rank-one torsion-free coherent sheaf L over X, a non-singular point x ∞ ∈ X, a closed neighbourhood X ∞ of x ∞ , a local parameter 1/z : X ∞ → D 0 ⊂ C which sends x ∞ to 0, and ϕ : L| X∞ → D 0 × C is a trivialisation of L| X∞ . Each section of L| X∞ is identified with a complex function on D 0 under ϕ. Let X 0 := X \(the interior of X ∞ ) and then the closed sets X 0 and X ∞ cover X, and X 0 ∩X ∞ is identified with S 1 under z.
Given this algebro-geometric datum, one can associate a closed subspace W ⊂ H consisting of all analytic functions S 1 → C which, under the above identification, extend to a holomorphic section of L on an open neighbourhood of X 0 . More explicitly, one can write
W = the second component of ϕ • s • (1/z) −1 | S 1 :
s is a holomorphic section on a neighbourhood of X 0
H ,
where (1/z) −1 : D 0 → X ∞ is the inverse function of 1/z. It is known that W ∈ Gr (see [34, Proposition 6.1]) and further, in the class where X is a compact Riemann surface (then L is automatically a complex line bundle, hence a maximal torsion-free sheaf), this correspondence (X, L, x ∞ , z, ϕ) → W ∈ Gr is one-to-one (see [34, Proposition 6.2]).
3.3. The appearance of Faber polynomials and Grunsky coefficients. Let f : D 0 → C be a univalent function such that f (0) = 0 and f (D 0 ) is bounded by a Jordan curve. We set β : C → C by β(w) := 1/w. For a subset A ⊂ C, we shall write A −1 := β(A). Let D ∞ := C \ D 0 . We obtain an algebro-geometric datum (X, L, x ∞ , z, ϕ) by setting X = C, L = C × C, x ∞ := ∞,
X ∞ := f (D 0 ) −1 , z := β • f −1 • β −1 : X ∞ → D ∞ ,and ϕ = (1/z) × id C . Accordingly we have X 0 = C \ (f (D 0 ) −1 )
. Here we note that z extends continuously to X ∞ by virtue of the Caratheodory's extension theorem for z.
Therefore we can embed f into the Grassmannian by assigning a Hilbert space W = W f as above. In this case, we have C \ (f (D 0 ) −1 ), and hence
W f = F • (1/z) −1 | S 1 : F is a holomorphic function on a neighbourhood of C \ (f (D 0 ) −1 ) H .
In order to start this paper's main calculation, let us specify this more explicitly.
h(g −1 (ξ)) ξ − z dξ, z ∈ C \ g(D ∞ ) the Faber transform of φ (with respect to g). When the boundary ∂(C \ g(D ∞ )) is analytic, it is known that h ∈ O(D 0 ) iff Fh ∈ O(C \ g(D ∞ )) (see [14, Theorem 1]) and F : O(D 0 ) → O(C \ g(D ∞ )) is bijective.
In our case, we put
g := (1/z) −1 = β • f • β −1 : D ∞ → f (D 0 ) −1 ,
and then we can describe O(X 0 ) by O( D ∞ ) through the transformation
F • (β −1 ) * = (β −1 ) * • Ad β * (F) : O( D ∞ ) → O(X 0 ) where Ad β * (F) := β * • F • (β −1 ) * : O( D ∞ ) → O( C \ f (D 0 )). For each h(η) = ∞ k=0 a k η −k ∈ O( D ∞ ), a direct calculation shows that (3.1) (Ad β * (F)h)(w) = 1 2πi ∂f (D 0 ) h(f −1 (ζ)) 1 − ζw −1 dζ ζ , w ∈ C \ f (D 0 ).
As a result, (Ad β * (F)φ)(w) is a power series in 1/w. Actually, in view of the Cauchy integral formula
1 2πi S 1 ζ n 1 − ζη −1 dζ ζ = η n if n 0, 0 if n 1, η ∈ D ∞ where S 1 = ∂D 0 , we have (Ad β * (F)φ)(w) = n k=0 a k 2πi ∂X 0 (f −1 (ζ)) −k 1 − ζw −1 dζ ζ = n k=0 a k [(f −1 (w)) −k ] 0
where [(f −1 (w)) −k ] 0 denotes the constant-part plus the principal-part of the Laurent series for (f −1 (w)) −k = (1/f −1 (w)) k . In particular, every element in O( C \ f (D 0 )) can be written as a series of 1 and Q k (w) := [(f −1 (w)) −k ] 0 for k 1. Each Q k is called the k-th Faber polynomial associated to the domain C \ f (D 0 ) (or just simply f ). By definition, Q k (w) is a polynomial of degree n in 1/w.
Then we conclude that [(β −1 ) * • Ad β * (h)] • (1/z) −1 = [Ad β * (h)] • f • β −1 and hence W f = span {1} ∪ {Q n • f • (1/z)| S 1 } n 1 H ,
where z is the identity map on D ∞ . We note that W f = H + in the case of f (z) ≡ z.
Remark 3.1. The Faber polynomials appeared first (with a different formalism, but equivalent to our presentation) in the context of approximations of functions in one complex variable by analytic functions (see [7] and [6]). Since then, they also play an important role in the theory of univalent functions (see [32]). We introduced the Faber polynomials in a slightly non-standard way in order to have them in a form which is suitable for embedding univalent functions into the Grassmannian by using Faber polynomials.
3.4. Action of words. Let X = {x 1 , x 2 , x 3 , · · · } be a countable set of letters. The free monoid X * on X is the set of all words in the letters X, including the empty word ∅. We denote by
C X := w∈X * Cw = C ⊕ ∞ n=1
C X n the free associative and unital C-algebra on X. The unit of this algebra is the empty word which we will denote by 1 := ∅. The set C X n stands for ⊕ w Cw where the summation is taken over all words w of length n.
(x ip · · · x i 2 x i 1 ) s,t := s u 1 <u 2 <···<up t e −i 1 x 0 (u 1 ) dx i 1 (u 1 )e −i 2 x 0 (u 2 ) dx i 2 (u 2 ) · · · e −ipx 0 (up) dx ip (u p ).
The action of naturally extends to C X [[z]] and then we call S(ξ(x, z)) s,t := S(ξ(x, z)) s,t the signature of ξ(x, z).
Definition 3.6. We define a C-bilinear map
T : C X ((z −1 )) × C X → C X ((z −1 ))
by extending C-bilinearly the pairing T (f, 1) := f,
T (f, x ip · · · x i 1 ) := (L −i 1 · · · L −ip f )x ip · · · x i 1
for f ∈ C X ((z −1 )), p 1 and i 1 , · · · , i p ∈ N, where L k := −z k+1 ∂/(∂z) for k −1. Here, ∂/(∂z) is a formal derivation on C X ((z −1 )).
For f ∈ C X ((z −1 )) and x ∈ C X , T (f, x) will be denoted by f. z x in the sequel. The following is clear by definition: Proposition 3.7. T defines an action of the C-algebra C X on C X ((z −1 )) from the right.
The right action T can be extended to the right action
(3.2) C X ((w −1 )) × C X [[z]] → C X ((w −1 ))[[z]],
under which, the image of (f, z n x ip · · · x i 1 ) is mapped to z n (f. w x ip · · · x i 1 ) =: f. w (z n x ip · · · x i 1 ). Note that now the notation f. w S(x) makes sense.
e x 0 (t) z 1 − zw w −1 . w S(ξ(x, e x 0 (t) )) 0,t .
Proof. By setting w(n) i 1 ,··· ,ip := (n − i 1 ) + 1 (n − (i 1 + i 2 )) + 1 · · · n − (i 1 + i 2 + · · · + i p−1 ) + 1 ,
where n = i 1 + · · · + i p , we have w −1 . w 1 = w −1 , w −1 . w x ip · · · x i 1 = w(n) i 1 ,··· ,ip x ip · · · x i 1 w −(i 1 +···+ip+1) .
Therefore Res w=0 ∞ m=0 z m w m (w −1 . w 1) = 1 = (the empty word), and
Res w=0 ∞ m=0 z m w m (w −1 . w x ip · · · x i 1 ) = z (i 1 +···+ip) w(n) i 1 ,··· ,ip x ip · · · x i 1 .
Hence we get
Res w=0 e x 0 (t) z 1 − zw w −1 . w S(ξ(x, e x 0 (t) )) = e x 0 (t) z + ∞ n=1 e (n+1)x 0 (t) z n+1 n p=1 i 1 ,··· ,ip∈N:
i 1 +···+ip=n w(n) i 1 ,··· ,ip x ip · · · x i 1 .
Now, in view of Theorem 2.8, we obtain the result.
By tensoring the right action (3.2) it gives rise to
C X ((w −1 )) ⊗ C X ((u −1 )) × C X [[z]] ⊗ C X [[z]] → C X ((w −1 )) ⊗ C X ((u −1 )),
under which, the image of (f ⊗ g, x ⊗ y) will be denoted by (f. w x) ⊗ (g. u y) in the sequel. We recall that the tau-function corresponding to W ∈ Gr, is given by
τ (t) = det(w + ) = det(1 + a −1 bA) (see [34,h ′ (u) w − z ∞ r,s=1
e (r+s)x 0 (t) x r+s (w −r . w S(ξ(x, e x 0 (t) ))) ¡ (u −s . u S(ξ(x, e x 0 (t) ))) 0,t .
The proof can be found in Section A.4
Connections with Conformal Field Theory and Free Probability
In this last section, we would like to briefly discuss aspects of the relation the material developed so far has with free probability and conform field theory. Let us first recall D. V. Voiculescu's [38] basic definitions and notions.
A pair (A, ϕ) consisting of an unital algebra A and a linear functional ϕ : A → C with ϕ(1 A ) = 1, is called a non-commutative probability space, and then any element in A is called a random variable.
For any random variable a ∈ A with µ a (1 A ) := 1, the C-linear function µ a : C[[X]] → C defined by µ a (X n ) := ϕ(a n ), n ∈ N is called the distribution of a. More generally, any element in the set Σ := {µ : C[X] → C| µ is linear and µ(1) = 1} is called a distribution. We denote by Σ × 1 := {µ ∈ Σ : µ(X) = 1} the family of distributions of mean one. Then, D. V. Voiculescu [38] introduced two operations ⊞ and ⊠ as follows.
For each µ ∈ Σ,
R µ (z) := G −1 (z) − 1 z ∈ C[[z]]
is called the R-transform of µ, where G −1 (z) is the compositional inverse of the Cauchy transform G(z) := ∞ n=0 µ(X n )z −(n+1) of µ. Then R µ (z) := zR µ (z) ∈ zC[[z]] is called the (modified) R-transform of µ. For two distributions µ, ν ∈ Σ, a new distribution µ ⊞ ν ∈ Σ can be defined by the following formula R µ⊞ν (z) = R ν (z) + R ν (z) (then we also have R µ⊞ν (z) = R ν (z) + R ν (z)) and it is called the additive free convolution of µ and ν ([38, Definition 3.1.1 and Remark 3.
1.2]).
On the other hand, for any distribution µ ∈ Σ × ,
S µ (z) := 1 + z z M −1 µ (z) ∈ C[[z]] × is called the S-transform of µ, where M −1 µ (z)
is the compositional inverse of the associated moment series of µ: M µ (z) := ∞ n=1 µ(X n )z n . Then for any two distribution µ, ν ∈ Σ × , a new distribution µ ⊠ ν ∈ Σ × can be determined by the relation
S µ⊠ν (z) = S ν (z) · S ν (z)
which is called the multiplicative free convolution of µ and ν ([38, Definition 3.6.1 and Remark 3.6.2]).
These two sets (Σ, ⊞) and (Σ × , ⊠) with operations form commutative groups ([38, Theorem 3.2.3 and Theorem 3.6.3]). Moreover, it is a result by R. Friedrich and J. McKay [11] that there exists a group isomorphism EXP : Σ → Σ × such that the following diagram commutes.
Σ ×
1 S-transform / / LOG Λ(C) z d dz log Σ R-transform / / EXP O O zC[[z]]
Here, we use that LOG = EXP −1 . Given a Hilbert space H, the pair of (B (F (H)), τ Ω ) of the unital algebra of all bounded linear operators on the full Fock space
l * (e)(x 1 ⊗ · · · ⊗ x n ) := e, x 1 if n = 1, e, x 1 H x 2 ⊗ · · · ⊗ x n if n 2.
If one fixes a unit vector e and writes l := l(e) and l * := l * (e), then the pair ( E 1 , τ 1 ) of the unital C-algebra E 1 := C[[l, l * ]] and τ 1 : E 1 ∋ α(l, l * ) → α(0, 0) ∈ C is also a non-commutative probability space. Now, U. Haagerup [12] showed the following: For any distribution ν ∈ Σ × , there exists a unique f ∈ C[[z]] × such that the distribution (with respect to τ 1 ) of the random variable a := (1 + l)f (l * ) ∈ E 1 coincides with ν. Moreover, we have then S µa (z) = 1/f (z).
If one restricts this result to ν ∈ Σ × 1 , then it follows from results by R. Friedrich and J. McKay [11], that the following diagram commutes: In 2017, R. Friedrich * , pointed out that any distribution of mean one corresponds to a tau-function of the KP hierarchy, extending results obtained with J. McKay [10]. This can be seen as follows: A sequence K = {K n (x 1 , · · · , x n )} n∈N of polynomials K n (x 1 , · · · , x n ) ∈ C[x 1 , · · · , x n ] determines a mapping
{a ∈ E 1 : τ 1 (a) = 1} taking τ 1 -distribution Λ(C) (1 + l) ×(evaluation at z = l * ) (Haagerup) o o Σ × 1 id 2 2 S-transform (Voiculescu) / / Λ(C) inversion O O z d dz log Σ R-transform (Voiculescu) / / EXP (Friedrich-McKay) O O zC[[z]]K : Λ(C) ∋ 1 + ∞ k=1 p k z k → ∞ n=0 K k (p 1 , · · · , p k )z k ∈ C[[z]],
where Λ(A) := 1 + zC[[z]] and K 0 := 1. The sequence K = {K n (x 1 , · · · , x n )} n∈N is called a (Hirzebruch) multiplicative sequence (or m-sequence over C) if the above mapping is a homomorphism between the monoid Λ(A) (where the product is defined by pointwise multiplication) and the ring of formal power series C[[z]] (but considered as a monoid). A result by T. Katsura, Y. Shimizu and K. Ueno [15] says that, if one defines b n ∈ C as the coefficient of x n in the polynomial K n (x 1 , · · · , x n ), then
τ T (t) := exp ∞ k=1 (−1) k+1 b k t k ,
where t = (t 1 , t 2 , · · · ), is a tau-function of the KP hierarchy ([15, Theorem 5.3, 1)]). Furthermore, the corresponding wave operator belongs to the class 1 + ∂ −1
t 1 C[[∂ −1 t 1 ]] (i.e.
, all the coefficients are constants, and hence the corresponding Lax operator is given by L = ∂ t 1 ). In particular, in [10,Theorem 4.3] this was related to the infinite Lie group (Σ 1 , ⊠).
Summarising the above, we have the following isomorphisms of commutative groups and their embedding into the universal Grassmannian UGM: (ii) We have
Σ ×log w − f (z) w = log f (z) f ′ (0)z − ∞ n=1 Q n (w) n z n (A.2)
for z ∈ D 0 and w ∈ C \ f (D 0 ).
(iii) Q n (w) is a polynomial of degree n in 1/w such that
Q n (f (z)) = z −n + n ∞ m=1 b −n,−m z m .
(iv) Q n (w) is a polynomial of degree n in 1/w such that (ii) For each ζ ∈ ∩ 0 t T f t (D), the mapping [0, T ] ∋ t → f −1 t (ζ) ∈ D is continuous and of bounded variation.
Q n (f (z)) − z −n → 0 as z → 0.
Proof. (i) Consider the mapping
ϕ : [0, T ] × D 0 ∋ (t, z) → (t, f t (z)) ∈ [0, T ] × C.
We will show that, there exists an open neighbourhood D of 0 ∈ C such that ϕ| [0,T ]×D is injective. We divide the proof into two parts.
(i-a) We first prepare the necessary estimates. Define g :
[0, T ] × D 0 → [0, T ] × C by g(t, z) := ϕ(t, z) − (t, f ′ t (0)z) = (0, f t (z) − f ′ t (0)z). Then, for each (t, z), (s, w) ∈ [0, T ] × D 0 , by writing z − w = |z − w|e iθ and µ(du, w) := dx 0 (u) + dξ(x, w) u , we have g(t, z) − g(s, w) = {g(t, z) − g(t, w)} + {g(t, w) − g(s, w)} = (0, I + II + III),
where, by writing z = w + |z − w|e iθ and η = w + ae iθ for 0 a |z − w|,
I := f t (z) − f s (z), II := (f ′ s (0) − f ′ t (0))z, III := f s (z) − f ′ s (0)z − f s (w) − f ′ s (0)w = |z−w| 0 (f ′ s (η) − f ′ s (0))da.
By Proposition 2.6, we have f ′ t (0) = e x 0 (t) . Let r ∈ (0, 1 2 ] be arbitrary. It is straightforward to get the following estimates: If 0 s t T and z, w ∈ rD 0 , then
|I| c 1 (r) t s |dx 0 (u)| + ∞ n=1 2 −n |dx n (u)| , |II| c 2 (r) t s |dx 0 (u)| and |III| c 3 (r)|z − w|, where c 1 (r) := r sup u∈[0,T ]×( 1 2 D 0 ) |f ′ u (z)|, c 2 (r) := r exp( x 0 ∞ ) and c 3 (r) := sup (u,η)∈[0,T ]×(rD 0 ) |f ′ u (η) − f ′ u (0)|.
We also note the following inequality: For any z, w ∈ rD 0 and t, s ∈ [0, T ],
|z − w| e x 0 ∞ |f ′ t (0)z − f ′ s (0)w| + r|f ′ t (0) − f ′ s (0)| ,
which follows from f ′ t (0) = e x 0 (t) and the trivial identity
z − w = 1 f ′ 1 f ′ t (0) w(f ′ s (0) − f ′ t (0)). (i-b) Define h : R × C → [0, +∞) by h(t, z) := |z| + |t| + sup 0 u T min{u+|t|,T } u |dx 0 (u)| + ∞ k=1 2 −k |dx k (u)| .
Since the measure µ(ds, a) := |dx 0 (s)| + ∞ n=1 a n |dx n (s)| on [0, T ] doesn't have atoms for each a ∈ [0, 1), h is continuous by virtue of Lemma A.3. It is easy to see that for each (t, z), (s, w) ∈ R × C,
(1) h(t, z) = 0 ⇐⇒ (t, z) = (0, 0);
(2) h(t + s, z + w) h(t, z) + h(s, w). Also note that for each (t, z), (s, w) ∈ [0, T ] × D 0 , we have
(A.3) h((t, z) − (s, w)) |z − w| + h(ϕ(t, z) − ϕ(s, w)).
Let (t, z), (s, w) ∈ [0, T ]×rD 0 be arbitrary with s t. By using the relation (t, f ′ t (0)z) = ϕ(t, z) − g(t, z) and estimates in (a), we obtain |z − w|
e x 0 ∞ h ϕ(t, z) − g(t, z) − ϕ(s, w) − g(s, w) + c 2 (r) t s |dx 0 (u)| e x 0 ∞ h ϕ(t, z) − ϕ(s, w) + h g(t, z) − g(s, w) + c 2 (r) t s |dx 0 (u)| e x 0 ∞ h ϕ(t, z) − ϕ(s, w) + 2e x 0 ∞ c 4 (r)h (t, z) − (s, w) ,
where c 4 (r) := c 1 (r) + c 2 (r) + c 3 (r). Then, by using (A.3), we get
|z − w| e x 0 ∞ (1 + 2c 4 (r))h ϕ(t, z) − ϕ(s, w) + 2e x 0 ∞ c 4 (r)|z − w|.
Then, by taking r ∈ (0, 1 2 ] small so that 2e x 0 ∞ c 4 (r) < 1 2 and putting D := rD 0 , we reached: For every (t, z), (s, w) ∈ [0, T ] × D, it holds that
1 2 |z − w| e x 0 ∞ (1 + 2c 4 (r))h ϕ(t, z) − ϕ(s, w) , (A.4)
which implies that ϕ| [0,T ]×D is injective.
(ii) Let D be as in the proof of (i). If ζ ∈ ∩ 0 t T f t (D), then for each t ∈ [0, T ], we have (t, ζ) ∈ ϕ([0, T ] × D), and then ϕ −1 (t, ζ) = (t, f −1 t (ζ)). Now the assertion follows by (A.4).
Definition A.5. Let X and Y be nonempty subsets of C.
(1) For each x ∈ C, we set d(x, Y ) := inf{|x − y| : y ∈ Y }.
(2) The Hausdorff distance between X and Y is defined by
d H (X, Y ) := inf{ε > 0 : X ⊂ Y ε and Y ⊂ X ε },
where X ε := ∪ x∈X {z ∈ C : |z − x| < ε}.
Definition A.6. Let γ ⊂ C be a Jordan curve and δ > 0. We call
T δ (γ) := {z ∈ C : d(z, γ) < δ} a tubular δ-neighbourhood of γ if for every ε ∈ (0, δ],(1)
there is a homeomorphism between T ε (γ) and (−1, 1) × S 1 , under which γ ∼ = {0} × S 1 ; (2) the boundary ∂T ε (γ) is homeomorphic to a disjoint union of two circles.
Remark A.1.
(a) If the Jordan curve γ is C ∞ , there exists a tubular δ-neighbourhood of γ for sufficiently small δ > 0 (Tubular neighbourhood theorem). (b) For two Jordan curves γ 1 , γ 2 ⊂ C, it is easy to see that
d H (γ 1 , γ 2 ) inf homeomorphisms φ i :S 1 →γ i ; i=1,2 φ 1 − φ 2 ∞ . (A.5) Furthermore, |f t (r 0 (ε)z) − f t (r 0 (ε ′ )w)| |f t (r 0 (ε)z) − f s (r 0 (ε ′ )w)| + |f s (r 0 (ε ′ )w) − f t (r 0 (ε ′ )w)| |f t (r 0 (ε)z) − f s (r 0 (ε ′ )w)| + sup (u,ζ)∈J ′ ×(2r)D 0 |f ′ u (ζ)| t s µ(du, 2 −1 ).
Thus we obtain
|εG(t, ε, z) − ε ′ G(s, ε ′ , w)| |G(t, ε, z)| + |ε| 4 i=1 c i (2c −1 0 ) 4 + sup (u,ζ)∈J ′ ×(2r)D 0 |f ′ u (ζ)| 2 × |f t (r 0 (ε)) − f s (r 0 (ε ′ ))| + t s µ(du, 2 −1 ) ,
which implies that h(g(t, ε, z) − g(s, ε ′ , w))
c 5 |G(t, ε, z)| + |ε| 4 i=1 c i h (t, f t (r 0 (ε)z)) − (s, f s (r 0 (ε ′ )w)) ,
where c 5 := (2c −1 0 )(4 + sup (u,ζ)∈J ′ ×(2r)D 0 |f ′ u (ζ)|) 2 . Now, take an interval J ⊂ J ′ containing t 0 , which is open in [0, T ], r ′′ > 0 so that sup (u,ε,z)∈J×(−r ′′ ,r ′′ )×B r ′′ (z 0 ) |G(t, ε, z)| (4c 5 ) −1 and then, we put r ′′′ := min{r ′′ , (4c 5
4 i=1 c i ) −1 } and U := J × (−r ′′′ , r ′′′ ) × (∂(rD 0 ) ∩ B r ′′′ (z 0 )).
Then for each (t, ε, z), (s, ε ′ , w) ∈ U, it holds that
h(g(t, ε, z) − g(s, ε ′ , w)) 1 2 h (t, f t (r 0 (ε)z)) − (s, f s (r 0 (ε ′ )w)) .
On the other hand, since it holds that (t,
f t (r 0 (ε)z)) − (s, f s (r 0 (ε ′ )w)) = {ϕ(t, ε, z) − g(t, ε, z)} − {ϕ(s, ε ′ , w) − g(s, ε ′ , w)}, we have h (t, f t (r 0 (ε)z)) − (s, f s (r 0 (ε ′ )w)) h ϕ(t, ε, z) − ϕ(s, ε ′ , w) + h g(t, ε, z) − g(s, ε ′ , w) h ϕ(t, ε, z) − ϕ(s, ε ′ , w) + 1 2 h (t, f t (r 0 (ε)z)) − (s, f s (r 0 (ε ′ )w)) ,
which proves (A.6).
(i-b) By compactness of ∂(rD 0 ) and (a), there exist an interval I, containing t 0 which is open in [0, T ], m ∈ N, z 1 , z 2 , · · · , z m ∈ ∂(rD 0 ) and r 1 , r 2 , · · · , r m ∈ (0, r) such that ∪ m i=1 B r i (z i ) ⊃ ∂(rD 0 ) and, with by putting anew r ′ := min{r i : i = 1, 2, · · · , m} and V ′ i :=
I × (−r ′ , r ′ ) × [B r i (z i ) ∩ ∂(rD 0 )], ϕ| V ′ i is injective for each i = 1, 2, · · · , m. Take δ ′ > 0 so that T δ ′ (∂(rD 0 )) ⊂ ∪ m i=1 B r i (z i )
. Take further δ ′′ ∈ (0, δ ′ ) such that for any z, w ∈ ∂(rD 0 ), |z − w| < δ ′′ implies z, w ∈ B r i (z i ) for some i. By the way, note that we have (A.4), which implies that the mapping from a compact set, {(t, ζ) : t ∈ I, ζ ∈ f t (∂(rD 0 ))} ∋ (t, ζ) → (t, f −1 t (ζ)) ∈ I × ∂(rD 0 ), is continuous, in particular, uniformly continuous. Therefore, there exists δ ∈ (0, min{δ ′′ /2, r ′ }) such that for any t ∈ I and z, w ∈ ∂(rD 0 ), |f t (z) − f t (w)| < 2δ implies |z − w| < δ ′′ , so that z, w ∈ B r i (z i ) for some i. Then ϕ is injective on V := I × (−δ, δ) × ∂(rD 0 ). In fact, if (t, ε, z), (s, ε ′ , w) ∈ V and ϕ(t, ε, z) = ϕ(s, ε ′ , w), then t = s and f t (z) + εn(t, z) = f s (w) + εn(s, w) =: ζ. Then we have |f t (z) − f s (w)| |f t (z) − ζ| + |ζ − f s (w)| = 2|ε| < 2δ. Therefore (t, ε, z), (s, ε ′ , w) ∈ V i for some i, on which ϕ is injective. Hence (t, ε, z) = (s, ε ′ , w). Now we show that ϕ is a homeomorphism between V and the open set
V := (t, ζ) ∈ I × C : d(ζ, f t (∂(rD 0 ))) < δ .
(the openness of V follows from the continuity t → f t (∂(rD 0 )) which is because of Definition 2.1-(3) and (A.5)) Suppose that (t, ζ) ∈ V. It is clear that t ∈ I. By compactness of f t (∂(rD)), we can take w ∈ ∂(rD) such that δ > min z∈∂(rD 0 ) |f t (z) − ζ| = |f t (w) − ζ| =: ε. Note that w ∈ ∂(rD) is uniquely determined, otherwise it would contradict the the injectivity of ϕ| V . Hence, by writing w = re iθ 0 , we have d dθ θ=θ 0 f t (re iθ ) = iwf t (w) and 0 = d dθ θ=θ 0 |f t (re iθ ) − ζ| 2 = −2Re(iwf t (w)f t (w) − ζ) = iwf t (w), f t (w) − ζ R 2 , the segment joining ζ and f t (w) is perpendicular to f t (∂(rD 0 )) at f t (w). Therefore, it holds that ζ = f t (w) + (κε)n(t, w) where κ = −1 or +1. Then we have δ > d(ζ, f t (∂(rD 0 ))) = |f t (w) − ζ| = ε.
Hence (t, κε, w) ∈ V and we established a mapping ψ : V ∋ (t, ζ) → (t, κε, w) ∈ V . By the construction, we have ϕ| V • ψ = id V , which shows ϕ| V is surjective, in particular, it follows that ϕ(V ) ⊂ V.
We know so far that ϕ| V : V → V is injective and surjective. It remains to prove the continuity of (ϕ| V ) −1 , but this is clear from (A.6) and the fact that h is continuous.
(i-c) According to the notation in (b), we find that (−δ, δ) × (∂(rD 0 )) ∼ = T δ (f t (∂(rD 0 ))) for each t ∈ I, and hence the assertion (1) is proved.
(2) Let I, δ and r be as in (c). By the condition in Definition 2.1-(3), we can take an interval I ′ ⊂ I containing t 0 , which is open in [0, T ] so that for any s, t ∈ I ′ , sup s,t∈I ′ f t | ∂(rD 0 ) − f s | ∂(rD 0 ) ∞ < δ. Now we fix t ∈ I, arbitrarily. Then for each s ∈ I and z ∈ ∂(rD 0 ), we have
d(f s (z), f t (∂(rD 0 ))) f t | ∂(rD 0 ) − f s | ∂(rD 0 ) ∞ < δ,
which implies f s (∂(rD 0 )) ⊂ T δ (f t (∂(rD 0 ))).
Let γ 1 , γ 2 ⊂ C be two Jordan curves. According to the Jordan curve theorem, we can decompose C \ γ i into a disjoint union
C \ γ i = C i ∪ C ′ i , for i = 1, 2, where C i is the bounded connected component of C\γ i and C ′
i is the unbounded connected component of C \ γ i . By this notation, we have:
Lemma A.8. Suppose that both of γ 1 and γ 2 are surrounding the origin, and there exists δ > 0 such that
(i) T δ (γ i ) is the tubular δ-neighbourhood of γ i for i = 1, 2; (ii) γ 1 ⊂ T δ (γ 2 ) and γ 2 ⊂ T δ (γ 1 ). Then it holds that d H (C ′ 1 , C ′ 2 ) d H (γ 1 , γ 2 )
. Proof. Let d H (γ 1 , γ 2 ) =: ε 0 < δ and ε ∈ (ε 0 , δ) be arbitrary. For each subset A ⊂ C, we write A ε := {z ∈ C : d(z, A) < ε}.
Since T ε (γ i ) is a tubular neighbourhood of γ i for each i = 1, 2, we can decompose
∂T ε (γ i ) = η i ∪ η ′ i , i = 1, 2 where η i ⊂ C i and η ′ i ⊂ C ′ i are homeomorphic to S 1 . Since γ 2 ⊂ T ε (γ 1 ), we have η 1 ∩γ 2 = ∅, so that η 1 ⊂ C 2 ∪ C ′ 2 .
Since C 2 and C ′ 2 are connected and η 1 is homeomorphic to S 1 , it holds that η 1 ⊂ C 2 or η 1 ⊂ C ′ 2 . Similarly, we have η 2 ⊂ C 1 or η 2 ⊂ C ′ 1 . Totally, we may have the following cases:
(1) η 1 ⊂ C 2 and η 2 ⊂ C 1 .
(2) η 1 ⊂ C 2 and η 2 ⊂ C ′ 1 .
(3) η 1 ⊂ C ′ 2 and η 2 ⊂ C 1 . (4) η 1 ⊂ C ′ 2 and η 2 ⊂ C ′ 1 .
Here we shall observe that η 2 ⊂ C ′ 1 does not occur. In fact, suppose that this is the case. Since η 2 ∩ γ 1 = ∅, we have only two cases: C 1 ⊂ (inner domain of η 2 ) or C 1 ∩ (inner domain of η 2 ) = ∅. The first case can not occur since then contradicts to γ 1 ⊂ T ε (γ 2 ). The second case are also impossible because γ 1 and γ 2 surround the origin. Similarly, η 1 ⊂ C ′ 2 does not occur. Therefore, (1) is the only case. It is clear that C ′ 1 ∩ (C ′ 2 ) ε is nonempty and open in C ′ 1 . We shall show that it is also closed in C ′ 1 . Suppose that z n ∈ C ′ 1 ∩ (C ′ 2 ) ε , n ∈ N converges to a point z ∈ C ′ 1 . By (1), we have ∂(C ′ 2 ) ε = η 2 ⊂ C 1 , so that it must be z ∈ (C ′ 2 ) ε .
Therefore C ′ 1 ∩ (C ′ 2 ) ε is closed in C ′ 1 . Since C ′ 1 is connected, we have C ′ 1 ∩ (C ′ 2 ) ε = C ′ 1 , namely, C ′ 1 ⊂ (C ′ 2 ) ε . Similarly, we get C ′ 2 ⊂ (C ′ 1 ) ε , and hence the result.
Henceforth, by taking anew x 1 , x 2 , · · · as non-commutative indeterminates, and b −m,−n 's as polynomials in x i 's, we shall consider the following equation:
b −m,−n = −x n+m + (n − 1)b −m,−(n−1) x 1 + · · · + 2b −m,−2 x n−2 + b −m,−1 x n−1 + (m − 1)b −(m−1),−n x 1 + · · · + 2b −2,−n x m−2 + b −1,−n x m−1 , (A.9)
(roughly speaking, the polynomial b −m,−n means e −(m+n)x 0 (t) b −m,−n (t) and 'applying the indeterminate x k from the right' means 'applying t 0 e −kx 0 (s) dx k (s)× to functions of s') and then we shall make some observations about the equation (A.9) and introduce some notations: If we apply (A.9) to b −m,−n , we get (a) The terms (n−k)b −m,−(n−k) x k and (m−k)b −(m−k),−n x k for each k. We shall denote these phenomena by
b −m,−n (n − k)x k × → b −m,−(n−k) and b −m,−n (m − k)x k × ↓ b −(m−k),−n
respectively (Note that the multiplication by the x * 's must sit just right to the next b * , * 's).
(b) The term −x n+m , to which we can not apply (A.9) anymore. This means, consider the situation that we apply (A.9) iteratively to b * , * 's appeared in the previous stage. Suppose we have the term b −m,−n at some stage. Then chasing the term multiplied by −x * which arose from the first term on the right-hand side in (A.9), makes us get out of the loop of iterations. We shall denote this phenomenon by Let k ∈ N be such that 2 k n + m. We shall find the term of the form x k (...) in the polynomial expression of b −m,−n in the x i 's. For this, we shall fix i ∈ {1, · · · , m} and j ∈ {1, · · · , n} such that i + j = k. Suppose that p, q ∈ N and i 1 , · · · , i p , j 1 , · · · , j q ∈ N satisfy i 1 + · · · + i p = m − i and j 1 + · · · + j q = n − j. We then put a r := m − (i 1 + · · · + i r ) for r = 1, · · · , p and c s := n − (j 1 + · · · + j s ) for s = 1, · · · , q. Note that a p = i and c q = j. According to this notation, we can divide the situation into the following three cases:
where w i 1 ,··· ,ip;j 1 ,··· ,jq = a 1 a 2 · · · a p = (m − i 1 )(m − (i 1 + i 2 )) · · · (m − (i 1 + i 2 + · · · + i p )).
(3) If there exist such q and (j 1 , · · · , j q ) but not for p and (i 1 , · · · , i p ) (then we have i = m), then the diagram which we can have is the following: Hence we have a single path from b −m,−n to the 'end' in the above diagram. This path produces the term −w j 1 ,··· ,jq x k (x jq · · · x j 2 x j 1 ) where w j 1 ,··· ,jq = c 1 c 2 · · · c q = (n − j 1 )(n − (j 1 + j 2 )) · · · (n − (j 1 + j 2 + · · · + j q )). Now by reinterpreting it in the language of paths x k (t)'s, we conclude the result.
A.4. Proof of Theorem 3.9. Since {u n } n 1 forms a basis of H + , it is enough to show n Res w=0, u=0
u n−1 w − z ∞ r,s=1
e (r+s)x 0 (t) x r+s (w −r . w S(ξ(x, e x 0 (t) ))) ¡ (u −s . u S(ξ(x, e x 0 (t) ))) t = n x i 1 x i 2 · · · x ip , we have w −r . w S(ξ(x, e x 0 (t) )) ¡ u −s . u S(ξ(x, e x 0 (t) )) = (w −r . w 1) ¡ (u −s . u 1) + F r,s (w, u) + G r,s (w, u) + H r,s (w, u),
where F r,s (w, u) := w −r . w S(ξ(x, e x 0 (t) )) − 1 ¡ u −s . u S(ξ(x, e x 0 (t) )) − 1
= ∞ m ′ =1 ∞ n ′ =1 m ′ p=1
n ′ q=1 i 1 ,··· ,ip∈N: i 1 +···+ip=m ′ j 1 ,··· ,jq∈N: j 1 +···+jq=n ′ e (m ′ +n ′ )x 0 (t) × (w −r . w x i 1 x i 2 · · · x ip ) ¡ (u −s . u x j 1 x j 2 · · · x jq ), G r,s (w, u) := (w −r . w 1) ¡ u −s . u S(ξ(x, e x 0 (t) )) − 1 = ∞ n ′ =1 e n ′ x 0 (t) n ′ q=1 j 1 ,··· ,jq∈N: j 1 +···+jq=n ′ (w −r . w 1) ¡ (u −s . u x j 1 x j 2 · · · x jq ), H r,s (w, u) := w −r . w S(ξ(x, e x 0 (t) )) − 1 ¡ (u −s . u 1)
= ∞ m ′ =1
e m ′ x 0 (t) m ′ p=1 i 1 ,··· ,ip∈N: i 1 +···+ip=m ′ (w −r . w x i 1 x i 2 · · · x ip ) ¡ (u −s . u 1).
Since w −r . w 1 = w −r , we get (w −r . w 1) ¡ (u −s . u 1) = w −r u −s . Then, by using
1 w − z = − ∞ m=1
z −m w m−1 for |z| > |w|, Let us set w(r) i 1 ,··· ,ip;∅ := r(i p + r)(i p + i p−1 + r) · · · (i p + i p−1 + · · · + i 2 + r)
= m − (i 1 + · · · + i p ) · · · m − (i 1 + i 2 ) (m − i 1 ) where m = i 1 + · · · + i p + r, w(s) ∅;j 1 ,··· ,jq := s(j q + s)(j q + j q−1 + s) · · · (j q + j q−1 + · · · + j 2 + s) = n − (j 1 + · · · + j q ) · · · n − (j 1 + j 2 ) (n − j 1 ) where n = j 1 + · · · + j q + s, and w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq := w(r) i 1 ,··· ,ip;∅ w(s) ∅;j 1 ,··· ,jq . For F r+s (w, u), we first observe that
w −r . w x ip · · · x i 2 x i 1 = x ip · · · x i 2 x i 1 L −i 1 L −i 2 · · · L −ip w −r
= w(r) i 1 ,··· ,ip;∅ x ip · · · x i 2 x i 1 w −(i 1 +i 2 +···+ip+r) , and similarly u −s . u x jq · · · x j 2 x j 1 = w(s) ∅;j 1 ,··· ,jq x jq · · · x j 2 x j 1 u −(j 1 +j 2 +···+jp+s) .
Therefore we have Res w=0; u=0
u n−1 w − z x r+s (w −r . w x i 1 x i 2 · · · x ip ) ¡ (u −s . u x i 1 x i 2 · · · x ip ) = −1 {1 n−s=j 1 +···+jq} ∞ m=1 z −m 1 {1 m−r=i 1 +···+ip}
× w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq x r+s [(x ip · · · x i 2 x i 1 ) ¡ (x jq · · · x j 2 x j 1 )], so that Res w=0; u=0
u n−1 w − z x r+s F r,s (w, u) = − ∞ m=1 z −m ∞ m ′ =1 ∞ n ′ =1 e (m ′ +n ′ )x 0 (t)× x r+s [(x ip · · · x i 2 x i 1 ) ¡ (x jq · · · x j 2 x j 1 )].
Theorem 1. 1
1(see Proposition 2.10 and Proposition 2.12). The Faber polynomials and Grunsky coefficients associated to solutions of the controlled Loewner-Kufarev equation satisfy linear differential equations, and the Grunsky coefficients can be explicitly calculated.
assure that f t , 0 t T are univalent on some open neighbourhood of 0 for some T > 0. (b) Under the conditions of Definition 2.1-(2), the function t → ξ(x, z) t is continuous and of finite variation for each z ∈ D 0 . (c) Let φ : N × [0, T ] → R be a bounded measurable function and r ∈ [0, 1). Under the conditions of Definition 2.1-
Definition 2 . 2 .
22We say {f t } 0 t T is a univalent solution to the controlled Loewner-Kufarev equation if it is a solution to (2.1) and f t is a univalent function on D 0 for each 0 t T .
is a simply connected domain (i.e., open, connected and simply connected) for each t ∈ [0, T ]; (3) (Continuity in the sense of Carathéodory, under the conditions (1) and (2)): For each t ∈ [0, T ] and any sequence 0 t n ↑ t, ∪ ∞ n=1 Ω(t n ) = Ω(t). For the following Definition 2.4, cf. specifically [31, Chapter 6, Section 6.1, pp. 156-157; Chapter 2, Section 2.1, p. 35 and Lemma 2.1]. Definition 2.4 ([31]). Let f t : D 0 → C be given for 0 t T . Then {f t } 0 t T is called a Loewner chain if
The above chains {Ω(t)} and {f t } are known to be in one-to-one correspondence via the relation Ω(τ ) = f t (D 0 ), where t = log f ′ τ (0) is a time-reparametrisation to satisfy Definition 2.4-(2) (see [31, Chapter 6, Section 6.1]).
Theorem 2.5 ([31, Theorem 6.2]). Let f t : D 0 → C be given for 0 t T . Then {f t } 0 t T is a Loewner chain if and only if there exist constants r 0 , K 0 > 0 and a function p(t, z), analytic in z ∈ D 0 , and measurable in t ∈ [0, T ] such that (i) for each 0 t
we recall that for each holomorphic function f , defined on an open neighbourhood U of the origin, the Faber polynomials and hence Grunsky coefficients are defined once f is univalent on an open neighbourhood V ⊂ U of the origin (see Appendix A-Definition A.1). Proposition 2.9. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then there exists an open neighbourhood U of the origin such that
3 .
3Controlled Loewner-Kufarev equation embedded into the Segal-Wilson Grassmannian 3.1. Segal-Wilson Grassmannian. Let H := L 2 (S 1 , C) be the Hilbert space which consists of all square-integrable complex functions on the unit circle S 1 . It decomposes orthogonally into H = H + ⊕ H − , where H + and H − are the closure of span{z k : k 0} and span{z k : k < 0}, respectively.
Definition 3. 1
1(G. Segal and G. Wilson [34, Section 2]). The Segal-Wilson Grassmannian Gr := Gr(H) is the set of all closed subspace W of H satisfying the following:(1) The orthogonal projection pr + : W → H + is Fredholm;
For a closed subset V in C, we denote by O(V ) the space of all holomorphic functions defined on an open neighbourhood of V . For a univalent function g : D ∞ → C with g(∞) = ∞ and for each h ∈ O(D 0 ), we call (Fh)(z) := 1 2πi ∂(C\g(D∞))
. 5 .
5z k ∈ C X [[z]]and a distinguished element S(ξ(x, z)) ∈ C X [[z]] by S(ξ(x, z)) Let x 0 : [0, +∞) → R and x 1 , x 2 , · · · : [0, +∞) → C be continuous and of bounded variation. For 0 s t, we define [ 1] s,t := 1 and
Theorem 3. 8 .
8Let {f t } t 0 be a solution to the Loewner-Kufarev equation. Then f t (z) = Res
Proposition 3.3 and p. 50-51]) up to a multiplicative constant, where w + : e ξ(t,z) W → H + is the orthogonal projection, e ξ(t,z) : H → H is the multiplication operator by e ξ(t,z) with matrix representation e −ξ(t,z) = a b 0 d along H = H + ⊕ H − , and A : H + → H − is such that graph(A) = W . Given a bounded univalent function f : D 0 → C with f (0) = 0, we denote by A f : H + → H − the linear map such that graph(A f ) = W f .
Theorem 3 . 9 .
39Let {f t } 0 t T be a univalent solution to the Loewner-Kufarev equation such that f t (D 0 ) is bounded for every t ∈ [0, T ]. Then for each h ∈ H + and |z| > 1
F
(⊗ is understood in the sense of ⊗ C ) of H, and the vacuum expectationτ Ω : B(F (H)) ∋ a → a := a(Ω), Ω A ∈ Cis a non-commutative probability space. For each unit vector e ∈ H, we define l(e), l * (e) ∈ B(F (H)) by l(e)(x) := e ⊗ x, x ∈ F (H) and l * (e)(Ω) := 0
From this point of view, the solution f t (z) to the Loewner-Kufarev equation lives in zC[[z]] in the above diagram. This, via the diagram, is realised as a random variable of the free probability space ( E 1 , τ 1 ). This would imply that we would have a Bosonic expression of the Loewner-Kufarev equation. Thus, the Loewner-Kufarev equation acts on the Bosonic Fock space. Here we shall recall that we have the positive part of the Witt generators in the expression of the Loewner-Kufarev equation. One reasonable questions might be, what is the meaning of these Witt generators in this Bosonic expression.
b
pointed out by the second author, any distribution of mean one is embeddable into the universal Grassmannian. Now, it would be interesting to interpret the structure of the convolution semigroup (or flow) in (Σ × 1 , ⊠) as a submanifold of UGM. A further question is, how the central limit theorem in Σ × 1 is described in the Grassmannian in relation with the tau-function of the KdV hierarchy associated to △ = L 2 = ∂ 2t 1 ? In this picture, the solution to the Loewner-Kufarev equation lives in Σ × 1 and thus can be again embedded into the Grassmannian. So, how does the positive part of the Witt generators in the Loewner-Kufarev equation act in this context? Appendix A. A.1. Elements of the theory of Faber polynomials and Grunsky coefficients. Definition A.1. Let f : D 0 → C be a univalent function with f (0) = 0. * Talk delivered at the conference "Stochastic Analysis and Related Topics 2017", Ritsumeikan University (BKC), 18. October 2017 (i) For each n ∈ N, Q n (w) := [(f −1 (w)) −n ] 0 (the principal plus constant part of the Laurent series for (1/f −1 (w)) n ) is called the n-th Faber polynomial associated with the domain C \ f (D 0 ) (or just simply f ). z, ζ) = (0, 0), are called the Grunsky coefficients of f . This definition still works for any analytic functions f defined on an open neighbourhood of the origin (not only D 0 ) of the form f (z) = a 1 z + a 2 z 2 + · · · with a 1 = 0, and then the {Q n } n 1 and (b −m,−n ) m,n 1 are also called the Faber polynomials and Grunsky coefficients of f respectively. Also, the following Proposition A.2 extends obviously to this situation if we replace D 0 by 'some open neighbourhood of the origin'. The Grunsky coefficients are, by definition, symmetric: b −m,−n = b −n,−m for n, m ∈ Z 0 . −m,0 z m for z ∈ D 0 . Proposition A.2. Let f : D 0 → C be a univalent function with f (0) = 0 and n ∈ N. Then the following are equivalent: (i) Q n (w) is the n-th Faber polynomial associated to the domain C \ f (D 0 ), i.e., Q n (w) = [(1/f −1 (w)) n ] 0 , the principal plus constant part of the Laurent series for (1/f −1 (w)) n .
.
Let µ be a finite measure on [0, T ] without atoms. Then the nondecreasing function h : [0, T ] → [0, +∞) t ∈ [0, T ] is uniformly continuous. Proof. Let ε > 0 be arbitrary. Since R ∋ a → min{a, T } and φ : [0, T ] ∋ a → a 0 dµ are uniformly continuous, there exists δ > 0 such that, for each a, b ∈ [0, T ] with |a − b| < δ, it holds that | min{a,T } 0 dµ − min{b,T } 0 dµ| < ε/2. Now, let s, t ∈ [0, T ] be arbitrary with |t − s| < δ. By the continuity of φ, we can take a, b ∈ [0, T ] such that h(t) Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then there exists an open neighbourhood D of 0 ∈ C such that (i) f t | D is univalent for each t ∈ [0, T ];
..
Note that the multiplication by x * 's must be from the left. Hence in particular, to get the term of the form x k (...) in the polynomial expression of b −m,−n in the x i 's, we have to escape the loop by passing to the phenomena i, j ∈ N with i + j = k.(c) If we have b −1,−1 , applying (A.9) does not produce b * , * 's. Namely we must have Again, the multiplication by x 2 must be from the left. In particular, b −m,−n does not contain the term x 1 (...) and hence b −m,−n is a linear combination of x k (...)'s for k 2, though the factor (...) may involve x 1 .
−n,−m (t) are the Grunsky coefficients associated with f t . According to the decomposition S(ξ(x, e x 0 (t)
ee
(r+s)x 0 (t) x r+s (w −r . w 1) ¡ (u −s . (r+s)x 0 (t) x r+s w −r u −s = − ∞ m=1 e (m+n)x 0 (t) x m+n .
1
{1 n−s=j 1 +···+jq} × w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq x r+s [(x ip · · · x i 2 x i 1 ) ¡ (x jq · · · x j 2 x j 1 )] = −1 {1 n−s}∞ m=1 z −m e ((m−r)+(n−s))x 0 (t) 1 {1 m−1 ,··· ,ip∈N: i 1 +···+ip=m−r j 1 ,··· ,jq∈N: j 1 +···+jq=n−s w(r, s) i 1 ,··· ,ip;j 1 ,··· ,jq
t (0) (f ′ t (0)z − f ′ s (0)w) +
Acknowledgment. Both authors thank Theo Sturm for the hospitality he granted to T.A. at the University of Bonn. T.A. thanks Roland Speicher for the hospitality offered him in Saarbrücken. R.F. thanks Fukuoka University and the MPI in Bonn for its hospitality, Roland Speicher for his support from which this paper substantially profited.Lemma A.7. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then for each t 0 ∈ [0, T ], we can take r ′ ∈ (0, 1) satisfying the following property: For each r ∈ (0, r ′ ), there exist δ > 0 and an interval I ∋ t 0 which is open in [0, T ] such that, by putting D := rD 0 , we have for any s, t ∈ I, (i) the tubular δ-neighbourhood T δ (f t (∂D)) exists;(ii) f s (∂D) ⊂ T δ (f t (∂D)).Proof. Let t 0 ∈ [0, T ] be arbitrary. By Proposition A.4, its proof and the fact that f ′ t 0 (0) > 0, we can take r ∈ (0,1 4) and an interval J ′ containing t 0 , which is open in [0, T ], so that f t | (2rD 0 ) is injective for every t ∈ J ′ , (A.4) holds on J ′ × (2rD 0 ) andWe putwhich is the outer normal unit vector of f t (∂(rD 0 )) at f t (z). We then consider the mapping ϕ :Since the measure µ(ds, a) := |dx 0 (s)| + ∞ n=1 na n |dx n (s)| on [0, T ] doesn't have atoms, h is continuous (see Lemma A.3) and it is easy to see that for each (t, z), (s, w) ∈ [0, T ] × C,(1) h(t, z) = 0 ⇐⇒ (t, z) = (0, 0);(2) h(t + s, z + w) h(t, z) + h(s, w). We shall divide the proof into three parts.(i-a) Firstly, we show that for each z 0 ∈ ∂(rD 0 ), there exists an open neighbourhood U ⊂ [0, T ] × (−r, r) × ∂(rD 0 ) of (t 0 , 0, z 0 ) such that, for any (t, ε, z), (s, ε ′ , w) ∈ U, it holds thatwhere r 0 (ε) := 1 + ε r|f ′ t 0 (z 0 )| , and hence ϕ| U is injective. For this, we further set g :For (t, ε, z), (s, ε ′ , z ′ ) ∈ [0, T ] × (−r, r) × ∂(rD 0 ), we havewhere, for the last term, we havewhere I := n(t, z) − n(t, w), II := n(t, w) − n(s, w),Suppose that s, t ∈ J ′ with s t, ε, ε ′ ∈ (−r ′ , r ′ ) and z, w ∈ ∂(rD 0 ) ∩ B r/2 (z 0 ). Then, by writing z = w + |z − w|e iθ and ζ = w + ae iθ for a ∈ [0, 1], we haveFor the second term, we have.Note that |w|We shall note also that, as measures on [0, T ], it holds thatnw n dx n (u).By using this, we have |II| c 2 t s µ(du, 2 −1 ), whereFor the third term, we haveFor the fourth term, we haveCombining all the above, we obtainedWe also note that for (ε, z),Moreover, for any (ε, z), (ε ′ , w) ∈ (−r ′ , r ′ ) × ∂(rD 0 ), by noting that |r 0 (ε)z|, |r 0 (ε ′ )w| < 2r, we obtainLemma A.9. Let {f t } 0 t T be a solution to the controlled Loewner-Kufarev equation (2.1). Then there exists r ′ ∈ (0, 1) such that for any r ∈ (0, r ′ ), the open set U := rD 0 satisfies the following.(i) f t | U is univalent for each t ∈ [0, T ];is an open neighbourhood of the origin.Proof. Let r ′ ∈ (0, 1) be as in Lemma A.7. Take r ′′ ∈ (0, r ′ ) small so that D := r ′′ D 0 satisfies Lemma A.4-(i, ii). Now, for each r ∈ (0, r ′′ ), put U := rD 0 . Then (i) is clearly satisfied.(ii) Let x ∈ ∩ t∈[0,T ] f t (U) and t 0 ∈ [0, T ] be arbitrary. Clearly we have x ∈ f t 0 (U). By the Open Mapping Theorem, f t 0 (D) is an open set, which implies that d(x, C\f t 0 (U)) > 0.By Lemma A.7 and Lemma A.8, we find that t → C \ f t (U) is continuous with respect to the Hausdorff distance. Since thenBefore entering the proof of Proposition 2.9, it would be better to recall Remark 2.1-(c).Proof of Proposition 2.9. Take D in Lemma A.4 and U in Lemma A.9 so that U ⊂ D. Then (i), (ii), (iii) and (iv) are now clear. We shall turn to (v). We have(see e.g.,[25,Theorem 1.1]). For the last term, we have by Definition 2.1-(4) that for each t ∈ [0, T ] and ζ ∈ V ,Since f −1 s (ζ) ∈ D and f s is univalent on D, it must be f ′ s (f −1 s (ζ)) = 0. Now combining the above, we get the result. (1) If there exist such p, q, (i 1 , · · · , i p ) and (j 1 , · · · , j q ), then we can consider the following diagram:During the loop of iterations of (A.9), we have p+q p = p+q q -paths from b −m,−n to the 'end' in the above diagram, each of which produces terms) · · · (n − (j 1 + j 2 + · · · + j q )), (note that w i 1 ,··· ,ip;j 1 ,··· ,jq depends only on i 1 , · · · , i p and j 1 , · · · , j q but not on the choice of paths in the diagram) and (...) is a monomial consisting of x ip , x i p−1 , · · · , x i 1 and x jq , x j q−1 · · · , x j 1 , which is interlacing according to a riffle shuffle permutation (Note that we should distinguish, for example x i 1 x j 1 and x j 1 x i 1 even if i 1 = j 1 ). Hence all paths produce in total −w i 1 ,··· ,ip;j 1 ,··· ,jq x k (x ip x i p−1 · · · x i 1 ) ¡ (x jq x j q−1 · · · x j 1 ) .(2) If there exist such p and (i 1 , · · · , i p ) but not for q and (j 1 , · · · , j q ) (then we have j = n), then the diagram which we can have is the following:Hence we have a single path from b −m,−n to the 'end' in the above diagram. This path produces the termHence we reached Res w=0; u=0Similarly, we find that i 1 ,··· ,ip∈N: i 1 +···+ip=m+n−k w(k − n) i 1 ,··· ,ip;∅ x k (x ip · · · x i 1 ). Now, in view of Theorem 2.13, we obtain (A.10), and hence the result.
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| [] |
[
"CLASS-SPECIFIC POISSON DENOISING BY PATCH-BASED IMPORTANCE SAMPLING",
"CLASS-SPECIFIC POISSON DENOISING BY PATCH-BASED IMPORTANCE SAMPLING"
] | [
"Milad Niknejad \nInstituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal\n",
"José M Bioucas-Dias \nInstituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal\n",
"Mário A T Figueiredo \nInstituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal\n"
] | [
"Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal",
"Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal",
"Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa\nPortugal"
] | [] | In this paper, we address the problem of recovering images degraded by Poisson noise, where the image is known to belong to a specific class. In the proposed method, a dataset of clean patches from images of the class of interest is clustered using multivariate Gaussian distributions. In order to recover the noisy image, each noisy patch is assigned to one of these distributions, and the corresponding minimum mean squared error (MMSE) estimate is obtained. We propose to use a self-normalized importance sampling approach, which is a method of the Monte-Carlo family, for the both determining the most likely distribution and approximating the MMSE estimate of the clean patch. Experimental results shows that our proposed method outperforms other methods for Poisson denoising at a low SNR regime. | 10.1109/icip.2017.8296481 | [
"https://arxiv.org/pdf/1706.02867v1.pdf"
] | 3,452,274 | 1706.02867 | 0116ff6ebbaa270b6079caee90bbf5e36c4d657b |
CLASS-SPECIFIC POISSON DENOISING BY PATCH-BASED IMPORTANCE SAMPLING
Milad Niknejad
Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa
Portugal
José M Bioucas-Dias
Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa
Portugal
Mário A T Figueiredo
Instituto de Telecomunicacoes Instituto Superior Tecnico Universidade de Lisboa
Portugal
CLASS-SPECIFIC POISSON DENOISING BY PATCH-BASED IMPORTANCE SAMPLING
Index Terms-Image denoisingPoisson noiseclass- specific datasetimportance sampling
In this paper, we address the problem of recovering images degraded by Poisson noise, where the image is known to belong to a specific class. In the proposed method, a dataset of clean patches from images of the class of interest is clustered using multivariate Gaussian distributions. In order to recover the noisy image, each noisy patch is assigned to one of these distributions, and the corresponding minimum mean squared error (MMSE) estimate is obtained. We propose to use a self-normalized importance sampling approach, which is a method of the Monte-Carlo family, for the both determining the most likely distribution and approximating the MMSE estimate of the clean patch. Experimental results shows that our proposed method outperforms other methods for Poisson denoising at a low SNR regime.
INTRODUCTION
Recovering noisy images is a fundamental problem in image processing and computer vision. In many image denoising applications, especially in low intensity (photon-limited) images, the noise follows a Poisson distribution. Photonlimited acquisition scenarios are often found in astronomical and medical imaging, and in many other areas.
The classical formulation for Poisson denoising considers the task of recovering an underlying image, the pixels of which are stacked in a vector x ∈ R N + , from Poissonian observations y ∈ N N 0 , i.e.
P(y|x) = N j=1 e −x [j] x y [j] [j] y [j] ! .(1)
A common approach to Poisson densoising proceeds as follows: first, a variance stabilization transform (VST) is
The research leading to these results has been funded by the European Union's 7-th Framework Programme (FP7-PEOPLE-2013-ITN), grant agreement 607290 (SpaRTaN), and from Fundação para a Ciência e Tecnologia (FCT), grants UID/EEA/5008/2013 and PTDC/EEI-PRO/0426/2014. applied (e.g., the Anscombe transform [1]), which approximately turns the Poisson distribution into a Gaussian of unit variance. Then, one of the many existing methods for Gaussian denoising is applied to the transformed image. Finally an inverse transformation yields the image estimate [2,3]. However, this approach is inaccurate for low intensity (low SNR) images, thus many methods denoise the Poissonian images without any transformation [4]. Our method belongs to this second category.
In many applications, the image to be recovered is known to belong to a certain class, such as text, face, or fingerprints. Knowing this information can help to learn a statistical prior that is better adapted to that specific class than a generic prior, and this prior can be learned from a dataset of clean images from that class. Recently this class-specific approach has been addressed in [5,6], for denoising and deblurring under Gaussian noise.
Some patch-based methods for recovering images degraded by Gaussian noises use multivariate priors for image restoration to obtain maximum a posteriori (MAP) or minimum mean squared error (MMSE) estimates of the clean image. Most of these methods fit a mixture multivariate Gaussian distributions to patches, either from an external dataset [7], or from the noisy image itself [8,9]. Although for Poisson denoising, patch-based methods have been proposed [4,10], there is a lack of methods that use an explicit prior on image patches, due to the non-Gaussian forward model, which makes it difficult to compute the corresponding MAP or MMSE estimates. On the other hand, the fitted Gaussian mixture prior is an approximation of the true distribution that characterizes the image patches.
In this paper, we propose a class-adapted method to recover images degraded by Poisson noise using a dataset of clean images of the same class. In our method, a mixture of multivariate Gaussian distribution is fitted to the external dataset of clean image patches. For each noisy patch of the observed image, we use the self-normalized importance sampling [11], which is an approach from the family of Monte-Carlo methods, to both choose the proper prior distribution and approximate the MMSE estimation of each patch. Using importance sampling allows us to approximate the MMSE estimate using the true distribution of image patches, if the sam-ples are the patches of clean images in the dataset. Furthermore, our proposed method can be extended to other forward models with known noise distribution, and different prior densities of clean patches.
In the following sections, we first briefly review the selfnormalizing importance sampling approach. Then, we describe the proposed method for MMSE Poisson image denoising based on self-normalizing importance sampling. Section 4 reports experimental results.
SELF-NORMALIZED IMPORTANCE SAMPLING
Our approach uses a technique from the family of Monte-Carlo methods, called self-normalized importance sampling (SNIS) [11,12], which we will now briefly review. Assume that the objective is to compute (or approximate)
E[f (X)], the expectation of a function f : R d → R p of a random variable X ∈ R d .
Denoting the support of the random variable X as R ⊂ R d , the expectation is given by
E[f (x)] = R f (x)p(x)dx.(2)
Consider also that only an un-normalized versionp(x) = c p(x) of the probability density function p(x) of X is known, where c is unknown. Letq(x) = d q(x) be another unnormalized density, where the normalizing constant d is also unknown, but from which samples can be more efficiently obtained than from p(x).
SNIS produces an estimate of E[f (X)] given bŷ
E n [f (x)] = n j=1 f (x j )w(x j ) n j=1 w(x j ) ,(3)
where
w(x j ) =p (xj ) q(xj )
, and x 1 , ..., x n are n independent and identically distributed samples from distribution q(x) [11,12]. This approximation can be shown to converges to the true value of E[f (x)] as n goes to infinity.
PROPOSED METHOD
Prior Learning
The first step of the proposed method is to fit a set of K multivariate Gaussians to the patches in the external dataset of clean images. We adopt the so-called classification EM (CEM) algorithm [13], rather than a standard EM algorithm as in [8,6,7], for reasons explained below. The CEM algorithm works by alternating between the following two steps (after being initialized by standard K-means clustering):
1. From the set of patches assigned to each cluster k ∈ {1, ..., K}, denoted X k , obtain estimates of the mean µ k and covariance matrix Σ k of the corresponding Gaussian density, which are simply the sample mean and the sample covariance of the patches in X k .
2. Assign each patch to the cluster under which it has the highest likelihood, that is, patch
x j is assigned to X k if k = arg max m N (x j ; µ m , Σ m ),
where, as usual, N (x; µ, Σ) denotes a Gaussian density of mean µ and covariance Σ, computed at x.
In the denoising step, each noisy patch will be assigned to one of these clusters. The reason for this kind of clustering is that in the simple importance sampling approach, for a fixed number of samples n, the MSE of the estimator is proportional to the variance of samples being averaged [12]. Similarly in the multivariate case, for a fixed n, it has been shown that the MSE of the estimator of a particular entry in the vector decreases as the variance of the samples of that entry, given the noisy patch, decreases [14]. Consequently, since in practice we use a limited number of patches from the external dataset, by clustering this way we expect to reduce the estimator variance, without increasing too much the number of samples.
It should be noted that the above procedure needs to be applied once for a given dataset of class-specific images.
Image denoising
In the denoising step, each patch is assigned to one of the clusters obtained in the learning stage. However, we only have noisy patches, thus the assignment is not trivial. If the noise was Gaussian, the assignments could be made in closed-form, but this is not the case with Poisson observations. Our main contribution is a new method, based on SNIS, to simultaneously determine the cluster and estimate the clean patch.
Define the random variable k i ∈ {1, ..., K}, which indicates the cluster to which the i-th patch belongs, and denote the corresponding distribution as p(x i |k i ). Having a set of learned cluster distributions, our objective is to solve the following simultaneous classification and MMSE estimation problem, given a noisy patch y i :
(x i ,k i ) = arg min (u,k) R m + u − x 2 2 p(x|y i , k) dx,(4)
where m is the number of pixels in each patch. In other words, we seek the estimate and the cluster that yield the minimum MSE. As shown below, we solve the above problem by the alternating minimization approach, but first we need to address the problem of how to approximate this integral. First, using Bayes rule and the fact that p(y i |x, k) = p(y i |x) (i.e., given the clean patch, the noisy one does not depend on the cluster), the integral in (4) can be written as
E[ x − u 2 2 |y i , k] = R m + u − x 2 2 p(y i |x) p(x|k) p(y i |k) dx. (5)
Then, using SNIS, the above integral can be approximated bŷ
E n [ x − u 2 2 |y i , k] = n j=1 u − x j 2 2 p(y i |x j ) n j=1 p(y i |x j )(6)
where the x j , for j = 1, ..., n are samples from the distribution p(x|k). Exploiting the self-normalized importance sampling formula (3), in (6) we consideredp(x) = p(y i |x) p(x|k), with the unknown constant c = 1/p(y i |k) and q(x) = p(x|k).
The minimization with respect to k in (4), while u is fixed to bex i , can then be approximated aŝ
k i = arg min kÊ n2 [ x −x i 2 2 |y i , k].(7)
Note that for computing (7), n 2 samples from each distribution p(x|k) are randomly extracted. The next step towards an alternating minimization method is a way to minimize E n [ x − u 2 2 |y i , k] with respect to u, for a given k =k i . This is the well-known MMSE estimation criterion, which is given by the posterior expectation E(x|y i ,k i ). Computing this expectation cannot be done in closed-form, under the Poisson observation model and Gaussian prior p(x|k) = N (x; µ k , Σ k ). To tackle this difficulty, we resort again to SNIS,
x i =Ê n1 [x|y i ,k i ] = n1 j=1 x j p(y i |x j ) n1 j=1 p(y i |x j )(8)
where x 1 , ..., x n1 are samples drawn from the distribution p(x|k). The approximation in (8) has been used before for patch-based denoising [15], but without noticing its connection to SNIS.
Since the multivariate Gaussian distributions fitted to the set of patches is only a crude approximation of the true distribution of the patches, samples from p(x|k) may yield poor results in the SNIS approximation. Instead of sampling from p(x|k), we directly sample form the set patches in the dataset assigned to that k-th cluster, X k , which turns out to yield better results.
We initialize our algorithm by u = y i . Our algorithm then alternates between (8) and (7), to assign each patch to a cluster and obtain an MMSE estimate of the patch. The procedure of iterative clustering and denoising has been used in some well-known patch-based denoising methods [16,7]. The patches are returned to the original position in the image and are averaged in the overlapping pixels.
Note that our method can be applied to any noise model with a known distribution. In this paper, we only consider Poissonian noise and we leave other possible noise distributions for future work. For the Poisson case, the conditional distribution of a noisy patch y, given a clean patch x ∈ R m + , is as shown in (1), with N = m.
• Initialization: cluster the training patches into K clusters, X
k = {µ (i) k , Σ (i)
k } as the sample mean and the sample covariance matrix of X (i) k -For each patch x j , assign it to a cluster, i.e., put x j into X
(i) kj , wherê k j = arg max k N (x j ; µ (i) k , Σ (i) k )
• Output: {X
PRACTICAL CONSIDERATIONS AND EXPERIMENTAL RESULTS
The reported results in this section were obtained on the Gore dataset of face images [17] and on the dataset of text images used in [5]. For each dataset, 5 images are randomly chosen as test images and the rest are chosen for training. For the face images, we extracted 95 × 10 3 patches from the training data, whereas from the text dataset, 75 × 10 3 patches were extracted. The described method is computationally expensive, if it is applied to all patches in the dataset. However, the key to reduce the computational complexity is to limit the number of patch samples used for estimating the clean patches and determining the cluster i.e. n 1 and n 2 . Our results in this section show that for a very limited number of samples, we obtain acceptable results, which outperforms other Poisson denoising methods for the tested datasets. For determining the cluster, we set n 2 = 30 which is overall 600 patches for all k = 20 clusters and is less than 1% of the samples in each external datasets. The number of samples, n 1 , used for denoising each patch was set to 300. So, overall for each noisy patch we processed 900 patches which is in computational complexity roughly similar to an internal non-local denoising with the patches constrained in 30 × 30 window. Unlike the original non-local means, which only the central pixel of each patch is denoised, in our method the whole patch is denoised by (8), the patches are then returned to the original position in the image and are averaged in the overlapping pixels. In order to further reduce the computational complexity, we extract the patches from the noisy image every 2 pixels along the row and the column of the image.
In Table 1, the average result of PSNR of the 5 tested images for the face dataset are compared for different methods. The results of our method result from two iterations of the alternating minimization approach. We found that increasing the number of iteration to more than two does not noticeably increase the quality of obtained image estimate, while it obviously increases the computational cost. The reason may lie in the fact that the discrete variables k i computed in (7) stabilize after a couple of iterations.
In Fig. (2), an example of denoising results for the face images with the peak value of 10 using the dataset in [17] is illustrated. It can be seen that our method also improves noticeably the visuals quality of image.
An example of denoising text images is shown in Fig. (3). It can be seen that the general image denoising methods fail to reconstruct the true image properly. However our classspecific method outperforms by a relatively high margin.
CONCLUSION
We proposed a method for class-specific denoising of images degraded by Poisson noise. We proposed to use a method from the Monte-Carlo family, called self-normalized importance sampling, in order to determine the prior and approximate the MMSE estimation. The sampling approach allowed us to approximate the MMSE using the true underlying priors rather than fitted multivariate Gaussian distributions. Our results showed that our method outperforms other methods in both PSNR and visual quality.
------------
K
, via the k-means algorithm • Main loop: for i = 1, . . . , I -Estimate the parameter of Gaussian distributions Θ (i)
Fig. 1 :
1Learning the prior from a class-specific dataset.
Fig. 2 :Fig. 3 :
23An example of denoising of a face image using the Gore dataset ; (a) Noisy image (Peak value=10); (b) Nonlocal PCA (PSNR=22.60); (c) VST+BM3D (PSNR=24.79); (d) Poisson non-local means (PSNR=24.55); (e) Proposed (First iteration) (PSNR=25.88); (f) Proposed (Second iteration) (PSNR=26.40). Comparison of denoising of general methods with our class-specific one for recovering the text images; (a) Original image (b) Noisy image (Peak value=2) (c) Non-local PCA (PSNR=14.95) (d) VST+BM3D (PSNR=14.55); (e) Proposed (First iteration) (PSNR=17.21); (f) Proposed (Second iteration) (PSNR=18.64). 6. REFERENCES [1] F. Anscombe, "The transformation of Poisson, binomial and negative-binomial data," Biometrika, vol. 35, no. 3/4, pp. 246-254, 1948.
Table 1 :
1Denoising PSNR results (in dB) for different peak values of face images in the Gore face database[17]. The results are averaged over 5 test images.2
5
10
15
NL-PCA [4]
19.69 22.87 23.80 25.01
VST+BM3D [3]
20.80 23.15 24.79 25.41
Poisson NL-means [18] 21.12 23.41 24.73 25.32
P4IP [19]
20.03 23.78 24.88 25.84
Our method
21.31 23.95 25.78 27.40
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| [] |
[
"Magnification of Subwavelength Field Distributions at Microwave Frequencies Using a Wire Medium Slab Operating in the Canalization Regime",
"Magnification of Subwavelength Field Distributions at Microwave Frequencies Using a Wire Medium Slab Operating in the Canalization Regime"
] | [
"Pekka Ikonen \nRadio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland\n",
"Pavel Belov \nQueen Mary College\nUniversity of London\nMile End RoadE1 4NSLondonUnited Kingdom\n",
"Constantin Simovski \nRadio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland\n",
"Yang Hao \nQueen Mary College\nUniversity of London\nMile End RoadE1 4NSLondonUnited Kingdom\n",
"Sergei Tretyakov \nRadio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland\n"
] | [
"Radio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland",
"Queen Mary College\nUniversity of London\nMile End RoadE1 4NSLondonUnited Kingdom",
"Radio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland",
"Queen Mary College\nUniversity of London\nMile End RoadE1 4NSLondonUnited Kingdom",
"Radio Laboratory/SMARAD\nTKK Helsinki University of Technology\nP.O. Box 3000FI-02015TKKFinland"
] | [] | Magnification of subwavelength field distributions using a wire medium slab operating in the canalization regime is demonstrated using numerical simulations. The magnifying slab is implemented by radially enlarging the distance between adjacent wires, and the operational frequency is tuned to coincide with the Fabry-Perot resonance condition. The near-field distribution of a complex-shaped source is canalized over an electrical distance corresponding roughly to 3λ, and the distribution details are magnified by a factor of three. The operation of the slab is studied at several frequencies deviating from the Fabry-Perot resonance. | 10.1063/1.2767996 | [
"https://arxiv.org/pdf/0705.3183v1.pdf"
] | 119,583,426 | 0705.3183 | db7cd6ffd73cc2a5bf5005ebf678250a1c7a8395 |
Magnification of Subwavelength Field Distributions at Microwave Frequencies Using a Wire Medium Slab Operating in the Canalization Regime
22 May 2007
Pekka Ikonen
Radio Laboratory/SMARAD
TKK Helsinki University of Technology
P.O. Box 3000FI-02015TKKFinland
Pavel Belov
Queen Mary College
University of London
Mile End RoadE1 4NSLondonUnited Kingdom
Constantin Simovski
Radio Laboratory/SMARAD
TKK Helsinki University of Technology
P.O. Box 3000FI-02015TKKFinland
Yang Hao
Queen Mary College
University of London
Mile End RoadE1 4NSLondonUnited Kingdom
Sergei Tretyakov
Radio Laboratory/SMARAD
TKK Helsinki University of Technology
P.O. Box 3000FI-02015TKKFinland
Magnification of Subwavelength Field Distributions at Microwave Frequencies Using a Wire Medium Slab Operating in the Canalization Regime
22 May 2007arXiv:0705.3183v1 [physics.class-ph]
Magnification of subwavelength field distributions using a wire medium slab operating in the canalization regime is demonstrated using numerical simulations. The magnifying slab is implemented by radially enlarging the distance between adjacent wires, and the operational frequency is tuned to coincide with the Fabry-Perot resonance condition. The near-field distribution of a complex-shaped source is canalized over an electrical distance corresponding roughly to 3λ, and the distribution details are magnified by a factor of three. The operation of the slab is studied at several frequencies deviating from the Fabry-Perot resonance.
Canalization of subwavelength images using electromagnetic crystals was proposed in [1].
Later, two successful experiments were carried out to demonstrate the canalization of TEpolarized (transverse electric field with respect to the slab interface) [2] and TM-polarized (transverse magnetic field) waves [3] at microwave frequencies. The slab used in the former experiment was based on capacitively loaded wires aligned parallel to the slab interfaces, whereas in the latter experiment unloaded wires aligned perpendicular to the slab interfaces were used. The canalization slab considered in [3] utilizes so called wire-medium TEM-modes (transverse electromagnetic modes) [4,5] to transport the details of the source distribution across the slab. Recently, the limitations of subwavelength imaging using such slabs were analytically studied in [6], and an experimental study aimed to verification of the analytical findings is available in [7]. Followed by the first studies conducted at microwave frequencies, authors of [8] proposed to implement the canalization regime in the THz range using stacks of uniaxially positioned alternating dielectric layers.
Recently, motivated mainly by the limitations in the optical microscopy, there has been a growing interest in structures that are able to magnify subwavelength field distributions in the visible range [9,10,11,12]. This means that the details of the source distribution are retained while transferring the distribution over a certain distance, and at the same time the distribution is linearly magnified or enlarged. Essentially, the structures used to demonstrate such an effect in the visible are based on stacks of two different alternating dielectric layers arranged uniaxially in cartesian or cylindrical geometries, and one of the layers is implemented as a sheet of plasmonic metal (see also [8]). In the microwave regime Alitalo and co-authors demonstrated experimentally in [13] simultaneous enhancement and magnification of evanescent fields (distributions) using double cylindrical polariton-resonant structures. In this Letter we demonstrate with full-wave simulations the magnification phenomenon at microwave frequencies using a modified version of the structure considered in [3,7]. The proposed magnifying slab utilizes the canalization phenomenon, thus, it is capable of magnifying distributions comprising any TM-polarized incident wave (propagating or evanescent) with any transverse component of the wave vector [1].
A schematic illustration of the proposed slab is depicted in Fig. 1. The slab consists of metal wires (assumed in the simulations to be perfect electric conductors) whose separation is radially enlarged. The wire ends corresponding to the source interface (input interface of the slab) lie on the surface of a sphere having radius 500 mm, and the wire ends corresponding to the canalized field interface (output interface) lie on the surface of a sphere having radius 1500 mm. Altogether 21 × 21 wires comprise the slab, and the separation between the wire ends at the input interface is approximately a = 10 mm.
When the operational frequency (with a fixed slab thickness) is tuned to the Fabry-Perot resonance, the source-field distortion due to reflections is minimized, and the distribution details are transferred across the slab by the wire-medium TEM-modes. Please note that in theory the Fabry-Perot resonance condition holds for any (including complex) incidence angle [3]. When the Fabry-Perot resonance condition is met there is no need, e.g., to alter the radius of wires (essentially, to maintain a uniform transmission-line characteristic impedance), and this significantly eases possible future implementations of the slab. The structure presented in Fig. 1 is expected to yield, in addition to the canalization effect, a magnification of the source distribution by a factor of three.
The following simulations have been performed using a commercial method-of-moments solver FEKO. The source is a piece of wire forming letter "M" (to reflect "magnification"), and it is fed by three current sources, Fig. 1. The distance between the source-wire plane, and the wire end located in the middle of the slab input interface is 13 mm. The source field distribution is scanned over a planar surface that covers the slab input interface, and is located at a distance 8 mm behind the source-wire plane. The canalized field distribution is scanned over a spherical surface that covers the slab output interface, and is located at a 15 mm distance in front of the wire ends. The two scanned regions are in the following referred to as the "source region" and "canalized field region", respectively, and they are schematically depicted in Fig. 1.
According to the theory [3,7], only the normal (with respect to the slab interfaces) component of electric field of a TM-polarized wave is completely (in the ideal case) restored at the slab output interface. The other two field components contain also contributions from TE-polarized waves, and will in this implementation be reproduced with distortion. For this reason we present below simulation results only for the following electric field components: in the source region we present the component perpendicular to the source-region plane, and in the canalized field region we present the radial component of electric field. The results indicate that the realized frequency corresponds roughly to 910 MHz, and at this frequency the electrical length of the wires is 3.03λ. The small deviation from the theoretical Fabry-Perot condition is most likely caused by the radially enlarging characteristic dimension of the slab. As predicted, at the realized operational frequency the source distribution is not affected by reflections, and the details of the distribution are canalized and simultaneously magnified across the slab. The letter "M" (radial electric field component) is accurately reproduced in the canalized field region, and the characteristic size of the distribution is magnified by a factor of three. Additional simulations (not shown) indicate that when the canalized field distribution is scanned very close to the output interface, the re-radiation of the field by the wire ends is clearly visible. When the canalized field region is located at a distance corresponding to half of the lattice period at the output interface, this interference vanishes.
Simulations performed at frequencies deviating from the predicted Fabry-Perot resonance indicate the following (see Fig. 2): as the frequency decreases below the predicted Fabry-Perot resonance, strong interference appears already after a very small frequency shift.
Authors of [7] speculated (in connection with experimental results for the "regular" canal- ization slab) that such interference is mainly caused by strongly excited surface waves (also the theoretical analysis presented in [6] leads to the same conclusion). Clearly, the degradation of the canalization effect is not so rapid when the frequency increases from the predicted Fabry-Perot resonance. The theoretical analysis dealing with regular canalization structures [6] predicts in this case that the obtainable resolution slightly decreases with increasing frequency. However, the experimental results presented in [7] indicate that such a degradation is barely visible. Indeed, the meander-line source distribution considered in [7] is strongly affected by reflections from the slab input interface, nevertheless, the modified (distorted) source distribution is still well canalized at frequencies above the Fabry-Perot resonance. In the simulation scenario considered here the reflections from the input interface only very moderately affect the details of the source distribution in the frequency range 910...930
MHz. However, in the canalized field region clear interference caused by the slab edges is observed as the frequency deviates from the Fabry-Perot resonance. Additional simulations (not shown) performed in the frequency range 930...1000 MHz indicate that the source-field disturbance caused by the reflections (with our particular source) is rather moderate over the entire frequency range. The form of the canalized letter "M"' can be somehow recognized up to 960 MHz, at higher frequencies the form is unrecognizable. It is important to observe that the shape, and the technique used to feed the source rather strongly dictates how noticeable is the degradation of the initial source distribution due to reflections from the slab interface (this fact is more extensively discussed in [7]). Evidently, due to a smaller amount of subwavelength details, the letter "M" is more tolerant to reflections as compared, e.g., to the meander line considered in [7].
In conclusion, in this Letter we have demonstrated simultaneous canalization and magnification of subwavelength field distributions in the microwave regime using a wire medium slab. The proposed structure consists of an array of metal wires, and the separation between adjacent wires is radially enlarged. In the example simulations we have canalized the near field distribution of a complex-shaped source over an electrical distance corresponding roughly to 3λ, and at the same time magnified the characteristic size of the distribution by a factor of three. Simulation results showing the operation of the slab at several frequencies deviating from the operational frequency have been presented and analyzed. In addition to the magnification effect the proposed slab could be utilized in the opposite way: electrically large source distributions can be decreased by simply placing the source in front of the larger slab interface. In this case the source distribution is canalized across the slab, and the characteristic dimensions of the distribution are simultaneously decreased by a certain factor.
FIG. 1 :
1Schematic illustration of the proposed magnifying slab (dimensions are in millimeters). The radius of wires r 0 = 1 mm. The white letter "M" denotes the wire used as the source. The feed locations (current sources) are depicted as the black dots in the insert.
Fig. 2
2depicts the simulated results. We have performed the simulations at several frequencies in the vicinity of 900 MHz to identify the frequency that corresponds to the Fabry-Perot resonance (the electrical thickness of the slab in this frequency range is roughly 3λ).
FIG. 2 :
2Simulated field distributions at different frequencies. a) f = 900 MHz, b) f = 910 MHz, c) f = 920 MHz, d) f = 930 MHz. The relative sizes of the source distribution and the canalized-field distribution are in scale.
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| [] |
[
"THE END OF AMNESIA: A NEW METHOD FOR MEASURING THE METALLICITY OF TYPE IA SUPERNOVA PROGENITORS USING MANGANESE LINES IN SUPERNOVA REMNANTS",
"THE END OF AMNESIA: A NEW METHOD FOR MEASURING THE METALLICITY OF TYPE IA SUPERNOVA PROGENITORS USING MANGANESE LINES IN SUPERNOVA REMNANTS"
] | [
"Carles Badenes ",
"Eduardo Bravo ",
"John P Hughes "
] | [] | [] | We propose a new method to measure the metallicity of Type Ia supernova progenitors using Mn and Cr lines in the X-ray spectra of young supernova remnants. We show that the Mn to Cr mass ratio in Type Ia supernova ejecta is tightly correlated with the initial metallicity of the progenitor, as determined by the neutron excess of the white dwarf material before thermonuclear runaway. We use this correlation, together with the flux of the Cr and Mn Kα X-ray lines in the Tycho supernova remnant recently detected by Suzaku(Tamagawa et al. 2008)to derive a metallicity of log(Z) = −1.32 +0.67 −0.33 for the progenitor of this supernova, which corresponds to log(Z/Z ⊙ ) = 0.60 +0.31 −0.60 according to the latest determination of the solar metallicity byAsplund et al. (2005). The uncertainty in the measurement is large, but metallicities much smaller than the solar value can be confidently discarded. We discuss the implications of this result for future research on Type Ia supernova progenitors. | 10.1086/589832 | [
"https://arxiv.org/pdf/0805.3344v1.pdf"
] | 14,124,537 | 0805.3344 | 618d6b559337d23bb8da4a630fef50f262c25ecb |
THE END OF AMNESIA: A NEW METHOD FOR MEASURING THE METALLICITY OF TYPE IA SUPERNOVA PROGENITORS USING MANGANESE LINES IN SUPERNOVA REMNANTS
21 May 2008 May 21, 2008
Carles Badenes
Eduardo Bravo
John P Hughes
THE END OF AMNESIA: A NEW METHOD FOR MEASURING THE METALLICITY OF TYPE IA SUPERNOVA PROGENITORS USING MANGANESE LINES IN SUPERNOVA REMNANTS
21 May 2008 May 21, 2008Draft Version May 21, 2008Draft Version Preprint typeset using L A T E X style emulateapj v. 10/09/06Subject headings: stars:binaries:close -supernova remnants -supernovae:general -X-rays:ISM
We propose a new method to measure the metallicity of Type Ia supernova progenitors using Mn and Cr lines in the X-ray spectra of young supernova remnants. We show that the Mn to Cr mass ratio in Type Ia supernova ejecta is tightly correlated with the initial metallicity of the progenitor, as determined by the neutron excess of the white dwarf material before thermonuclear runaway. We use this correlation, together with the flux of the Cr and Mn Kα X-ray lines in the Tycho supernova remnant recently detected by Suzaku(Tamagawa et al. 2008)to derive a metallicity of log(Z) = −1.32 +0.67 −0.33 for the progenitor of this supernova, which corresponds to log(Z/Z ⊙ ) = 0.60 +0.31 −0.60 according to the latest determination of the solar metallicity byAsplund et al. (2005). The uncertainty in the measurement is large, but metallicities much smaller than the solar value can be confidently discarded. We discuss the implications of this result for future research on Type Ia supernova progenitors.
INTRODUCTION
Despite decades of continuing effort, the nature of the progenitor systems of Type Ia supernovae (SNe) remains unknown. The most widely accepted theoretical scenarios involve the thermonuclear explosion of a C+O white dwarf (WD) destabilized by accretion of material from a binary companion, either another WD (double degenerate systems, DDs) or a normal star (single degenerate systems, SDs). Observations of supernova rates in distant galaxies suggest at least two different ways to produce Type Ia SNe (Scannapieco & Bildsten 2005): a 'prompt' channel associated with young stellar populations and a 'delayed' channel associated with old stellar populations. At present, it is not clear what relationship, if any, these channels have with the theoretical scenarios, or even if the proposed scenarios can yield Type Ia SNe at the observed rate (Maoz 2008). Most attempts to constrain the fundamental properties of the progenitors have been unsuccessful, including direct identification in optical pre-explosion images (Maoz & Mannucci 2008), searches for H stripped from the binary companion in SN spectra (Leonard 2007), and searches for prompt emission from circumstellar interaction (Panagia et al. 2006;Hughes et al. 2007). Very recently, the possible identification of the progenitor in a pre-explosion X-ray image of SN 2007on by has been questioned by post-explosion images (Roelofs et al. 2008). Campaigns to look for the surviving companion star to the exploded WD in the nearby Tycho supernova remnant (SNR), known to be of Type Ia origin, have also produced controversial results Ihara et al. 2007). This lack of clues to the identity of the progenitors in the observations of Type Ia SNe is sometimes referred to as 'stellar amnesia'.
One of the most important constraints on the age and nature of the progenitor systems is their metallicity Z. During stellar evolution, the C, N, and O that act as catalysts for the CNO cycle pile up into 14 N, which is converted to 22 Ne in the hydrostatic He burning phase through the reactions 14 N(α,γ) 18 F(β + ,ν e ) 18 O(α,γ) 22 Ne. The β + decay of 18 F increases the neutron excess of the WD material, defined as η = 1 − 2 Z A /A (where Z A is the atomic number and A is the mass number), resulting in a linear scaling of η with metallicity: η = 0.101 × Z (Timmes et al. 2003). This can have an observable impact on Type Ia supernova spectra (Lentz et al. 2000), but the predicted effects are hard to detect in practice (in one case, SN 2005bl, a progenitor of subsolar metallicity has been suggested by Taubenberger et al. 2008). In this Letter, we describe a new direct method to measure the metallicity of Type Ia SN progenitors using Mn and Cr lines from shocked ejecta in the X-ray spectra of SNRs. 2.1. Mn/Cr vs. Z Nuclear burning in Type Ia SNe happens in different regimes depending on the peak temperature reached by the WD material. From higher to lower temperatures, the regimes are: nuclear statistical equilibrium (NSE), incomplete Si burning, incomplete O burning and incomplete C-Ne burning (Woosley 1986). During the explosion itself, electron captures are too slow to change the original value of η except in the innermost ∼ 0.2 M ⊙ of ejecta, where nuclear burning happens at higher densities, leading to neutron-rich NSE (Brachwitz et al. 2000). Outside this region, an imbalance between free neutrons and protons does not affect the production of the most abundant species, with the possible exception of 56 Ni at supersolar metallicities (Timmes et al. 2003
M MS =3 O • M MS =6 O • Fig. 1.
-Mn to Cr mass ratio as a function of progenitor metallicity for Type Ia (Badenes et al. 2003(Badenes et al. , 2005, this work) and core collapse (Woosley & Weaver 1995) SN models, represented with red circles and blue triangles, respectively. The red and blue shaded regions encompass all the models in each class. The dashed line represents a power-law fit to the Type Ia models. The orange and green plots illustrate the impact of weak interactions during the simmering phase for very subluminous Type Ia SNe (∆m 15 = 1.8) in WDs with main sequence progenitor masses of 3 M ⊙ and 6 M ⊙ , respectively (see § 2.2). The horizontal solid and dotted lines represent the estimated M M n /M Cr ratio in the Tycho SNR from the Suzaku observations (0.74 ± 0.47, shaded in gray), which corresponds to log(Z) = −1.32 +0.67 −0.33 (see § 4).
neutrons are very efficient at storing the neutron excess. The most abundant among these nuclei is 55 Mn, which is produced during incomplete Si burning as 55 Co. When normalized to another product of incomplete Si burning whose nucleosynthesis is insensitive to the neutron excess (Cr is the ideal choice), the yield of Mn becomes an excellent tracer of the progenitor metallicity. This diagnostic can be quantified with Type Ia SN explosion models. We have used the model grid from Badenes et al. (2003Badenes et al. ( , 2005, complemented with other models calculated with the code described in Bravo et al. (1996). Physical inputs and initial conditions are as in Badenes et al. (2003), except in the cases noted below. We consider examples of several explosion mechanisms: deflagrations, delayed detonations, and pulsating delayed detonations. For the present work, we have recalculated the nucleosynthesis of four delayed detonation models (DDTa, DDTc, DDTe, and DDTf) at different metallicities by altering the value of X( 22 Ne) in the preexplosion WD from the canonical 0.01 (corresponding to Z = 9.0 × 10 −3 ). We have also calculated one deflagration model starting from a central density twice as large as the other models, and one delayed detonation model with the C to O mass ratio M C /M O = 1/3 in the WD instead of the usual 1. In spite of these fundamental differences in explosion mechanisms and initial conditions, we find a very tight correlation between the Mn to Cr mass ratio outside the neutron-rich NSE region in the SN ejecta, M Mn /M Cr , and the progenitor metallicity Z (see Figure 1). A power-law fit yields the following relation (with a correlation coefficient r 2 = 0.9975):
M Mn /M Cr = 5.3 × Z 0.65(1)
Removal of the inner 0.2 M ⊙ of neutron-rich NSE material from this relation is justified because (1) there is observational evidence that this material does not mix outwards during the explosion Mazzali et al. 2007) or its aftermath (Fesen et al. 2007); and (2) the dynamical ages of most ejecta-dominated Type Ia SNRs are too small for the reverse shock to have reached so deep into the ejecta. For illustration, we have plotted in Figure 1 the core collapse SN models from Woosley & Weaver (1995), which also show some correlation between M Mn /M Cr and Z, albeit with a much larger spread and a much shallower slope.
M MS =6 M O • M MS =5 M O • M MS =3 M O • ∆m 15 =1.8 ∆m 15 =1.6 ∆m 15 =1.4
The impact of C simmering
In order for Eq. 1 to hold, the value of η set by the progenitor metallicity must remain unchanged between the formation of the WD and the SN explosion. This should be true for DD progenitors if the final merger and runaway happen on dynamical timescales. In slowly accreting WDs, η can be modified through electron captures during the so-called 'simmering' phase of non-explosive C burning that takes place in the ∼ 1000 yr prior to the explosion (Piro & Bildsten 2008). The impact of this additional neutronization on the M Mn /M Cr ratio will depend on the extent of the convective region over which the neutronized material is mixed (M conv ), and its overlap with the portion of the WD that will undergo incomplete Si burning where Mn and Cr are synthesized. Some previous studies (Kuhlen et al. 2006;Piro & Bildsten 2008;Chamulak et al. 2008) have found large values for M conv by assuming an homogeneous chemical composition for the WD, effectively applying the Schwarzschild criterion for convection. We will adopt the hypothesis that the convective region is limited by the Ledoux criterion to the C-depleted core of the WD created during hydrostatic He-shell burning, M core (Höflich & Stein 2002). In the absence of a self-consistent picture for convection inside WDs, this must remain an open issue (see Piro & Chang 2008, for a discussion), but we believe that our hypothesis is reasonable given the existence of other processes that can reduce the extent of M conv and limit the mixing of neutronized material. These include the presence of Urca shells (Stein & Wheeler 2006) and the long thermal diffusion time scale for buoyant bubbles larger than ∼ 100 m (Garcia-Senz & Woosley 1995), which prevents them from mixing completely with their surroundings during convection.
According to Mazzali et al. (2007), Si-rich ejecta in Type Ia SNe lie between Lagrangian mass coordinates 1.05 M ⊙ and (1.55 − 0.69 × ∆m 15 ) M ⊙ , where ∆m 15 is the light curve width parameter (defined as the decline in blue magnitude 15 days after maximum), which acts as a proxy for SN brightness. In this context, the extent of the overlap between M conv and the Si-rich ejecta depends on the main sequence mass M MS and initial metallicity of the WD progenitor (which determine M core , Domínguez et al. 2001) and the SN brightness (see Figure 2). This overlap is only significant for subluminous (∆m 15 ≥ 1.6) Type Ia SNe originated by progenitors with either large M MS or low Z, or both. Chamulak et al. (2008) find an upper limit for the increase of η during the simmering phase of ∆η = 0.0015, which is comparable to the value of η in solar material. The impact of simmering on the M Mn /M Cr ratio can then be estimated by mixing material with M Mn /M Cr = 0.3 (appropriate for the value of Z ⊙ derived by Asplund et al. 2005) into the incomplete Si burning region, in a proportion equivalent to the extent of the overlap shown in Figure 2. The green and orange plots in Figure 1 are two examples of such 'simmeringmodified' models for very subluminous (∆m 15 = 1.8) Type Ia SNe, illustrating our conclusion that C simmering will only modify the M Mn /M Cr ratio in the SN ejecta for very subluminous SNe, and then only in cases where M MS is large, or Z is low, or both.
MEASURING THE M M N /M CR RATIO IN SNRS
The work in this Letter is motivated by the recent Suzaku detection of Mn and Cr in the X-ray spectrum of the Tycho SN reported by Tamagawa et al. (2008) (see Figure 3). This observational result opens the possibility of studying the M Mn /M Cr ratio in Type Ia SN ejecta, which cannot be done using optical SN spectra due to the 2.7 yr half-life of 55 Fe in the de- Public data bases for X-ray astronomy do not usually include lines from trace elements like Mn and Cr, but the value of E Mn /E Cr can be estimated by interpolation along the atomic number sequence from elements with available data (Hwang et al. 2000). We have used the ATOMDB data base (Smith et al. 2001) to retrieve Kα line emissivities for Si,S,Ar,Ca,Fe,and Ni (Z A = 14,16,18,20,26,28) as a function of ionization timescale n e t and electron temperature kT , the two variables that control the line emission of a plasma in nonequilibrium ionization. Then we performed a spline interpolation to obtain E Mn /E Cr , which we plot in Figure 4, together with the region of the (n e t, kT ) parameter space that is populated by young Type Ia SNRs (see the Appendix in Badenes et al. 2007). The uncertainty in the value of E Mn /E Cr comes from the variation within this region and the error introduced by the interpolation itself, which we have estimated by comparing the true and interpolated values for the E Ar /E S ratio. For the larger region of the (n e t, kT ) plane plotted in Figure 4, we find E Mn /E Cr = 0.69±0.32 ( E Mn /E Cr = 0.66±0.26 for the smaller region appropriate for Tycho). −0.33 ). The large error bar in this result is dominated by the statistical uncertainties in the X-ray fluxes from the faint Mn and Cr lines, and it will be reduced in an upcoming, approved deeper Suzaku observation. At present, we find a strong indication for a supersolar metallic-ity (log(Z/Z ⊙ ) = 0.45 +0.31 −0.60 with the solar value from Grevesse & Sauval (1998); log(Z/Z ⊙ ) = 0.60 +0.31 −0.60 with the newer value from Asplund et al. (2005)). Our results are compatible with a solar metallicity, but subsolar values can be discarded with confidence. Given this measurement and the fact that Tycho's SN was probably either normal or slightly overluminous (Badenes et al. 2006;, we do not have to concern ourselves with the impact of neutronization during C simmering in this particular case. The Tycho SNR is 59 pc above the Galactic plane at a Galactocentric radius of 9.4 kpc (assuming a distance of 2.4 kpc, Smith et al. 1991). A supersolar metallicity is higher than average for this location, but well within the spread of measured [Fe/H] values (Nordström et al. 2004). This combination of Galactrocentric radius and metallicity suggests a young progenitor age (a few Gyr or less), which would make Tycho a candidate for the 'prompt' channel of Type Ia SNe, but the scatter in the age-metallicity relations and the uncertainties in our measurement are too large to be more specific on this point.
In this Letter, we have proposed a new method to measure the metallicity of Type Ia SN progenitors using Mn and Cr lines in the X-ray spectra of their SNRs. We have applied it to the Tycho SNR and obtained a strong indication that its progenitor had a solar or supersolar metallicity. The main strength of our method is its simplicity: it is based on well-known nuclear physics and observable parameters that are easy to measure. Detection of Mn and Cr lines in other young Type Ia SNRs in the Galaxy should be possible with Suzaku, Chandra, and XMM-Newton observations. A few additional SNRs in the Magellanic Clouds might be reached with deep exposures, bringing the total of potential targets to perhaps a dozen. This number would increase sig-nificantly with the inclusion of SNRs in nearby galaxies like M31 and M33, which may be accessible to next generation X-ray observatories like Constellation-X and XEUS. Even with a small sample of objects, we should be able to use this method to verify and complement the indirect metallicity studies of extragalactic Type Ia SNe (Gallagher et al. 2005;Prieto et al. 2008) and test claims about the metallicity dependence of the Type Ia SN rate (Kobayashi et al. 1998). Measurements or upper limits below a certain threshold (M Mn /M Cr 0.1) would also provide interesting constraints on the extent of the convective region in accreting WDs. We conclude by noting that Mn and Cr lines have already been detected by ASCA in the Galactic SNR W49B (Hwang et al. 2000). The measured line flux ratio also suggests a solar or supersolar progenitor metallicity, but both the age and the SN type of W49B are controversial (Hwang et al. 2000;Badenes et al. 2007). We defer a detailed discussion of this object and the application of our method to other Galactic SNRs to a forthcoming publication.
2 .
2THE M M N /M CR RATIO AS A TRACER OF PROGENITOR METALLICITY
Fig. 2 .
2-Upper limits to Mconv during the simmering phase (left axis, black plots) and its overlap with the Si rich region of ejecta (right axis, colored plots) as a function of Z. The black plots represent quadratic interpolations for different M M S : 3 M ⊙ (dashed), 5 M ⊙ (dotted), and 6 M ⊙ (solid), with the symbols indicating the values from Domínguez et al. (2001) (for M M S = 6 M ⊙ at Z = 0.02, we have taken the average between M M S = 7 M ⊙ and M M S = 5 M ⊙ ). The colored plots represent the overlap (in %) between Mconv and the Si-rich region of ejecta, for very subluminous (∆m 15 = 1.8, red), mildly subluminous (∆m 15 = 1.6, orange), and normal, but faint (∆m 15 = 1.4, blue) Type Ia SNe.
Fig. 3 .
3-X-ray spectrum of the Tycho SNR observed by Suzaku in the vicinity of the Fe Kα line. The Kα lines of Cr and Mn are detected at > 10σ and 7σ confidence levels, with fluxes of 2.45 +0.48 −0.42 × 10 −5 and 1.13 +0.38 −0.45 × 10 −5 photons cm −2 s −1 respectively(Tamagawa et al. 2008).
Fig. 4 .
4-Interpolated Mn to Cr specific Kα emissivity ratio as a function of net and kT . The dotted box encompasses the values of net and kT found in the Si-rich ejecta of six young Type Ia SNRs by Badenes et al. (2007): 9.49 ≤ log(net) ≤ 11.94 cm −3 s; 1.0 ≤ kT ≤ 10.0 keV. The striped area corresponds to the region of parameter space appropriate for the Tycho SNR, 10.23 ≤ log(net) ≤ 10.99 cm −3 s; 1.0 ≤ kT ≤ 10.0 keV. cay chain 55 Co→ 55 Fe→ 55 Mn. Since Mn and Cr are synthesized together in the explosion and have very similar electronic structures, it is possible to estimate their mass ratio from the line flux ratio: M Mn /M Cr = 1.057 × (F Mn /F Cr )/(E Mn /E Cr ), where 1.057 is the ratio of atomic masses, F Mn /F Cr is the line flux ratio, and E Mn /E Cr is the ratio of specific emissivities per ion.
4 .
4DISCUSSION AND CONCLUSIONS: THE METALLICITY OF TYCHO'S PROGENITOR From the Suzaku observation, F Mn /F Cr = 0.46 ± 0.21, which gives M Mn /M Cr = 0.74 ± 0.47. This mass ratio translates into a metallicity of Z = 0.048 +0.051 −0.036 for the progenitor of the Tycho SNR (log(Z) = −1.32 +0.67
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Diagonal 647, Barcelona 08028, Spain; and Institut d'Estudis Espacials de Catalunya, Campus UAB, Facultat de Ciències, Bellaterra, Barcelona 08193, Spain; [email protected] Department of Astrophysical Sciences, Princeton Univer-
sity.
Peyton Hall, Ivy Lane, Princeton NJ 08544-1001;
[email protected]
2 Chandra Fellow
3 4 Department of Physics and Astronomy, Rutgers Univer-
sity.
136 Frelinghuysen Rd., Piscataway NJ 08854-8019;
[email protected]
). However, some trace nuclei with unequal numbers of protons and10 -4
10 -3
10 -2
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Z
0.01
0.10
1.00
M
Mn /M
Cr
10 -4
10 -3
10 -2
10 -1
Z
0.01
0.10
1.00
M
Mn /M
Cr
Tycho SNR
Explosion models:
Type Ia
Core Collapse
M
Mn /M
Cr
=5.3xZ 0.65
Simmering models
(for ∆m
15 =1.8):
The authors are grateful to Inese Ivans, Bruce Draine, and Jim Stone for discussions. We are also happy to acknowledge the work of the Suzaku Science Working Group and the members of the Suzaku Tycho team. Support for this work was provided by NASA through Chandra Postdoctoral Fellowship Award Number PF6-70046 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. EB is supported by grants AYA2007-66256 and AYA2005-08013-C03-01. JPH is partially supported by NASA grant NNG05GP87G.
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| [] |
[
"LIE-ALGEBRAIC CURVATURE CONDITIONS PRESERVED BY THE HERMITIAN CURVATURE FLOW",
"LIE-ALGEBRAIC CURVATURE CONDITIONS PRESERVED BY THE HERMITIAN CURVATURE FLOW"
] | [
"Yury Ustinovskiy "
] | [] | [] | The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold (M, g, J) preserves many natural curvature positivity conditions. Following Wilking[26], for an Ad GL(T 1,0 M )-invariant subset S ⊂ End(T 1,0 M ) and a nice function F : End(T 1,0 M ) → R we construct a convex set of curvature operators C(S, F ), which is invariant under the HCF. Varying S and F , we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds M with the trivial canonical bundle on the universal cover M. | 10.1007/s00208-020-01965-y | [
"https://arxiv.org/pdf/1710.06035v1.pdf"
] | 54,546,735 | 1710.06035 | f87cc2b50cd8e525652fe417aa2147970524d17e |
LIE-ALGEBRAIC CURVATURE CONDITIONS PRESERVED BY THE HERMITIAN CURVATURE FLOW
17 Oct 2017
Yury Ustinovskiy
LIE-ALGEBRAIC CURVATURE CONDITIONS PRESERVED BY THE HERMITIAN CURVATURE FLOW
17 Oct 2017
The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold (M, g, J) preserves many natural curvature positivity conditions. Following Wilking[26], for an Ad GL(T 1,0 M )-invariant subset S ⊂ End(T 1,0 M ) and a nice function F : End(T 1,0 M ) → R we construct a convex set of curvature operators C(S, F ), which is invariant under the HCF. Varying S and F , we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds M with the trivial canonical bundle on the universal cover M.
Introduction
In the last decades the Ricci flow has been successfully used in many classification and uniformization problems in Riemannian and Kähler geometry. The direction started in the 1980's with Hamilton's original papers on the classification of three/four-dimensional manifolds admitting metrics with positive Ricci curvature/positive curvature operator [11,12]. The main idea in Hamilton's approach is to control positivity of the curvature tensor along the Ricci flow, using a form of the parabolic maximum principle for tensors. Following this route one can prove the pinching of the curvature tensor towards a constant curvature tensor. In 2000's this program was used by Brendle and Schoen [5,4] to classify manifolds with (weakly) 1/4-pinched sectional curvature and by Böhm and Wilking [3] to describe manifolds admitting positive curvature operator in all dimensions. In the Kähler setting, Chen, Song and Tian [6] gave a Ricci flow-based proof of the Frankel conjecture [8]. This conjecture states that the existence of a Kähler metric of positive holomorphic bisectional curvature on a complex manifold M implies that M is biholomorphic to a complex projective space; it was solved by Siu and Yau [16] by studying the space of harmonic maps S 2 → M . The authors of [6] proved that any Kähler metrics of positive holomorphic bisectional curvature under the Ricci flow pinches towards the metric of constant curvature.
Unlike the Kähler situation, in a general Hermitian setting there are very few efficient analytic tools. For this reason it is interesting to extend the Ricci flow onto Hermitian manifolds. Unfortunately, the Ricci flow itself is not well-suited for the category of Hermitian manifolds, since, on a general Hermitian manifold (M, g, J), the Ricci tensor Ric(g) is not necessarily J-invariant. Motivated by this problem, Streets ant Tian [20] introduced a family of Hermitian Curvature Flows, generalizing the Ricci flow:
(1) ∂ t g = −S (2) + Q(T ),
where S (2) it the second Chern-Ricci curvature (see formula (2) below) and Q(T ) is an arbitrary type-(1,1) quadratic term in torsion of g. If (M, g, J) is Kähler, then S (2) = Ric(g), T = 0, and all the Hermitian Curvature Flows coincide with the Kähler-Ricci flow. Recently, in a series of papers [19,21,18,17] Streets and Tian investigated a specific member of family (1) with Q(T ) ij = g mn g ps T p im T s jn , which they call the pluriclosed flow. This flow preserves the class of pluriclosed metrics (i.e, g, s.t., ∂∂ω g = 0, where ω g = g(J·, ·)), and in many geometric situations satisfies global existence and convergence results. In this paper we study another member of (1) with Q(T ) ij = −1/2g mn g ps T mpj T nsi . This specific flow was first introduced in [24], where we proved that it preserves Griffiths positivity (non-negativity) of (T M, g). We refer to this flow as the HCF (Hermitian Curvature Flow). The choice of the quadratic term Q(T ) is motivated by a very special evolution equation for the Chern curvature Ω under (1). In [25] we computed the HCF for the induced metrics on all complex homogeneous manifolds. In the present paper, we reinterpret this equation, by setting a space of algebraic curvature tensors and introducing natural operations on this space. It allows us to find a much clearer expression for ∂ t Ω (Proposition 1.10).
Our principle goal in this paper is to prove that the HCF preserves many natural curvature positivity conditions, besides Griffiths positivity. Following Wilking [26], for every Ad GL(T 1,0 M )invariant S ⊂ End(T 1,0 M ) and any nice function (see Definition 3.1) F : End(T 1,0 M ) → R, we define a closed convex set of curvature-type tensors C(S, F ). We prove a modification of Hamilton's maximum principle [12] (Section 2), and adopt arguments of [26] to prove that sets C(S, F ) are preserved by the HCF. Algebraic structure of the evolution equation for Ω plays the crucial role in this proof. We also prove a strong version of the above theorem. Namely, we characterize the set of points, where Ω, corresponding to the evolved metric, hits the boundary of C(S, F ) (Theorem 5.1). Choosing various S and F , we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature (Section 4). The latter has important consequences for the understanding of possible stationary and periodic solutions to the HCF. In particular, we prove:
Theorem 0.2. If a compact complex manifold M admits an HCF-periodic Hermitian metric, then the pull-back of the canonical bundle to the universal cover of M is holomorphically trivial.
We conjecture that, as in the Ricci flow case, the HCF admits only stationary periodic solutions. There exist other Hermitian generalizations of the Ricci flow. Among them we mention the Chern-Ricci flow, introduced ans investigated by Gill [9]. Its long-time existence and convergence properties were also studied by Tosatti and Weinkove [22,23]. Under this flow, the metric form ω g is evolved in the direction of the Chern-Ricci form ρ, which represents (the multiple of) the first Chern class of M .
The rest of the paper is organized as follows. In Section 1 we fix basic notations of Hermitian geometry, set up the space of algebraic Chern curvature tensors, define natural operations on this space and introduce the HCF. We also recall some of the computations of [24]. Next, in Section 2 we formulate and reprove Hamilton's maximum principle for tensors in a slightly more general context. With this generalization Hamilton's maximum principle becomes applicable to the evolution equation for Ω under the HCF. In Section 3 define the convex sets C(S, F ) of curvaturetype tensors and prove that they are preserved under the HCF. This section essentially follows [26]. We provide several examples of the HCF-preserved curvature conditions in Section 4. Finally, we discuss some applications and further questions in Section 5.
Background
This section consists of three parts. First, we provide some background on Hermitian geometry and set up notations, introducing basic objects, such as Chern connection, its torsion and the Chern curvature tensors. In the second part, we define a vector space of algebraic curvature tensors, and present some natural algebraic operations on it. Lastly, we define the Hermitian Curvature Flow (HCF). Using the aforementioned operations we write down the evolution equation for the Chern curvature under the HCF.
1.1. Hermitian geometry. For a complex vector space V , we denote by V the underlying real vector space with a conjugate complex structure. We denote by
Sym 1,1 (V ) the subspace of V ⊗ V spanned over R by all the elements of the form v ⊗ v, v ∈ V .
Let (M, J) be a compact complex manifold with the operator of almost complex structure J : T M → T M . Denote by T M ⊗ C = T 1,0 M ⊕ T 0,1 M the decomposition the complexified tangent bundle into ± √ −1-eigenspaces of J. Any J-invariant Riemannian metric g defines an Hermitian metric on T 1,0 M . We denote this Hermitian metric by the same symbol.
Chern connection on T M is the connection ∇ characterized by the following properties:
(1) ∇g = 0, (2) ∇J = 0, (3) T (X, JY ) = T (JX, Y ) for X, Y ∈ T M , where T (X, Y ) := ∇ X Y − ∇ Y X − [X, Y ]
is the torsion tensor. There exists a unique connection satisfying these properties. After extending ∇ to a C-linear connection on T M ⊗ C, we can substitute property (3) with either of the following properties
(3 1 ) T (ξ, η) = 0 for ξ, η ∈ T 1,0 M , (3 2 ) ∇ ξ η = i ξ ∂η for ξ, η ∈ T 1,0 M , where ∂ : Γ(T 1,0 M ) → Γ(Λ 0,1 ⊗ T 1,0 M ) is the operator of the holomorphic structure.
The Chern curvature Ω is the curvature of the Chern connection, namely,
Ω(X, Y, Z, W ) := g((∇ X ∇ Y − ∇ Y ∇ X − ∇ [X,Y ] )Z, W ).
Extend Ω to a C-linear tensor. Chern curvature tensor has many symmetries: it is antisymmetric and J-invariant in its first and second pairs of arguments, i.e., Ω(X, Y, Z, W ) = Ω(JX, JY, Z, W ) = Ω(X, Y, JZ, JW ),
Ω(X, Y, Z, W ) = −Ω(Y, X, Z, W ) = −Ω(X, Y, W, X).
These symmetries imply that Ω is completely determined by the components
Ω ijkl := Ω(∂/∂z i , ∂/∂z j , ∂/∂z k , ∂/∂z l ),
where {∂/∂z i } is a local frame. If we want to specify explicitly that Ω corresponds to a metric g, we write Ω = Ω g . We also denote the components of the torsion tensor (assuming the Einstein summation convention) as T (∂/∂z i , ∂/∂z j ) := T k ij ∂/∂z k , T ijl := T k ij g kl . If g ij is a coordinate expression for the metric, then the components of the torsion and the curvature tensors are given by
T k ij = g kl (∂ i g jl − ∂ j g il ), Ω ijkl = −∂ i ∂ j g kl + g ps ∂ j g pl ∂ i g ks .
Unlike the Riemannian case, the Chern curvature does not satisfy the classical Bianchi identities, since the Chern connection has torsion. However, in this case, slightly modified identities, involving torsion still hold [13, Ch. III, Thm. 5.3]. Proposition 1.1 (Bianchi identities for the Chern curvature and torsion).
Ω ijkl = Ω kjil + ∇ j T kil , Ω ijkl = Ω ilkj + ∇ i T ljk , ∇ m Ω ijkl = ∇ i Ω mjkl + T p im Ω pjkl , ∇ n Ω ijkl = ∇ j Ω inkl + T s jn Ω iskl , ∇ i T l jk + ∇ k T l ij + ∇ j T l ki =T p ij T l kp + T p jk T l ip + T p ki T l jp .
Presence of the torsion terms in the Bianchi identities for Ω indicates that the four Ricci contractions of the Chern curvature tensor differ from each other (see [14] for the explicit description of the differences between these contractions):
S (1) ij := Ω ijmn g mn , S (2) ij := Ω mnij g mn , S (3) ij := Ω njim g mn , S (4) ij := Ω inmj g mn .
(2)
We call these contractions the Chern-Ricci tensors. All tensors will play certain role below. Symmetries of Ω imply that the first and the second Chern-Ricci tensors define Hermitian products on T 1,0 M . In general this is not the case for S (3) and S (4) , since S
(3) ij = S (3) ji = S (4) ij .
There are also two scalar contractions of Ω: the scalar curvature sc = g ij g kl Ω ijkl = tr g S (1) = tr g S (2) and a quantity which will be referred to as the second scalar curvature and will play important role below: sc = g il g kj Ω ijkl = tr g S (3) = tr g S (4) .
By rising the last two indices of the Chern curvature tensor, we can interpret Ω as a section of End(T 1,0 M ) ⊗ End(T 1,0 M ):
Ω lk ij (e k ⊗ ǫ i ) ⊗ (e l ⊗ ǫ j ), Ω lk ij = Ω ijmn g ml g kn ,
where {e i } is a local frame of T 1,0 M and {ǫ i } is the dual frame. Symmetries of Ω imply that this form is Hermitian, i.e., lies in Sym 1,1 (End(T 1,0 M )).
Remark 1.2.
If Ω ∈ Sym 1,1 (End(T 1,0 M )) is positive definite (resp. semidefinite), then the tangent bundle T 1,0 M is said to be dual-Nakano positive (resp. non-negative), [7]. Dual-Nakano nonnegative metrics exist on all complex homogeneous spaces, see Example 4.2 below.
1.2. Algebra of the space of curvature tensors. Let (V, g) be a complex vector space equipped with an Hermitian inner product. We extend g to all associated tensor powers of V and V . Denote by g = End(V ) the endomorphism Lie algebra of V . Let ·, · tr : g ⊗ g → C be the trace pairing u, v tr := tr(uv).
Definition 1.3. The space of algebraic curvature tensors on V is the vector space Sym 1,1 (g).
Pairing ·, · tr extends to a bilinear form on Sym 1,1 (g) in an obvious way:
v ⊗ v, u ⊗ u tr := |tr(uv)| 2 .
Clearly, Ω ∈ Sym 1,1 (g) is positive (resp. non-negative) if and only if Ω, u ⊗ u tr > 0 (resp. 0) for any nonzero u ∈ g.
Remark 1.4. For V = T 1,0 M the space Sym 1,1 (g) models the space of Chern curvature tensors. Unlike the Riemannian/Kähler setting, where one is interested only in the part of Sym 1,1 (g), satisfying the algebraic Bianchi identity, we consider the whole space Sym 1,1 (g), since the Chern curvature has less symmetries.
There is a natural R-linear adjoint action of g on Sym 1,1 (g):
ad v (u ⊗ u) = [v, u] ⊗ u + u ⊗ [v, u], v, u ∈ g.
For Ω ∈ Sym 1,1 (g), let {v i } be an orthonormal basis of g, diagonalizing Ω:
Ω = i λ i v i ⊗ v i .
We define two important quadratic operations on the space Sym 1,1 (g). Ω 2 : Metric g induces the isomorphism ι g : Ω → R Ω , mapping Ω to the corresponding selfadjoint operator R Ω : g → g.
Define Ω 2 := ι −1 g ((R Ω ) 2 ). In the basis {v i } the square of Ω is given by
Ω 2 := i λ 2 i v i ⊗ v i .
Note that Ω 2 is positive semidefinite, i.e., Ω 2 , u ⊗ u tr 0 for any u ∈ g. Moreover,
Ω 2 , u ⊗ u tr = 0 if and only if u ∈ Ker Ω. Ω # : For v 1 ⊗ w 1 , v 2 ⊗ w 2 ∈ g ⊗ g define (v 1 ⊗ w 1 )#(v 2 ⊗ w 2 ) = [v 1 , v 2 ] ⊗ [w 1 , w 2 ].
This map gives rise to a bilinear operation # : Sym 1,1 (g) ⊗ Sym 1,1 (g) → Sym 1,1 (g). Let Ω # := 1/2(Ω#Ω) be the #-square of Ω. In the basis {v i } the #-square of Ω is given by
Ω # = i<j λ i λ j [v i , v j ] ⊗ [v i , v j ].
Operation Ω # was introduced by Hamilton in [12], while studying the evolution equation for the Riemannian curvature tensor under the Ricci flow. In [25], we used this operation for an arbitrary Lie algebra g to study the HCF on complex homogeneous manifolds G/H. Note that Ω # does not depend on the choice of metric on g. The following proposition provides coordinate expressions for Ω 2 and Ω # .
(g) Ω = Ω lk ij (e k ⊗ ǫ i ) ⊗ (e l ⊗ ǫ j ) we have (Ω # ) lk ij = Ω lk pn Ω np ij − Ω nk pj Ω lp in , (Ω 2 ) lk ij = g mn g ps Ω sk in Ω lp mj . Proof. For e m ⊗ ǫ n , e p ⊗ ǫ s ∈ g we have [e m ⊗ ǫ n , e p ⊗ ǫ s ] = δ n p e m ⊗ ǫ s − δ s m e p ⊗ ǫ n . Therefore Ω#Ω =Ω lk ij Ω dc ab [e k ⊗ ǫ i , e c ⊗ ǫ a ] ⊗ [e l ⊗ ǫ j , e d ⊗ ǫ b ] =Ω lk ij Ω dc ab (δ i c e k ⊗ ǫ a − δ a k e c ⊗ ǫ i ) ⊗ (δ j d e l ⊗ ǫ b − δ b l e d ⊗ ǫ j ).
After expanding the Kronecker δ's we get the expression for Ω # . In coordinates, R Ω is given by
R Ω (e m ⊗ ǫ p ) = Ω lk ij g pj g ml (e k ⊗ ǫ i ). Therefore (R Ω ) 2 (e m ⊗ ǫ p )
= Ω ls nj Ω qk ir g pj g ml g sq g nr (e k ⊗ ǫ i ), which implies the stated formula for Ω 2 .
1.3. Hermitian curvature flow. Let (M, g, J) be an Hermitian manifold. Consider an evolution equation for the Hermitian metric g = g(t).
(3) ∂ t g ij = −S (2) ij − Q ij , where S (2) ij
= g mn Ω mnij is the second Chern-Ricci curvature and Q ij = 1 2 g mn g ps T mpj T nsi is a quadratic torsion term. Flow (3) is a member of the family of Hermitian Curvature Flows, introduced by Streets and Tian in [20]. It is proved in [20] that all these flows are defined by strictly parabolic equations for g, and, hence, admit short-time solutions. The particular flow (3) was first considered by the author in [24] (see also [25]). Further we refer to the flow (3) as the HCF.
With the HCF, the curvature tensor Ω also evolve along a nonlinear heat-type equation. Precise form of this equation was computed in [24]. Before stating it, let us recall the notion of torsiontwisted connections. Definition 1.6 (Torsion-twisted connections). We define ∇ T , ∇ T ♯ to be two torsion-twisted connections on T M given by the identities
∇ T X Y = ∇ X Y − T (X, Y ), ∇ T ♯ X Y = ∇ X Y + g(Y,
T (X, ·)) ♯ , where X, Y are sections of T M , ∇ is the Chern connection and ♯ : T * M → T M is the musical isomorphism induced by g. Equivalently, in the coordinates, for a vector field ξ = ξ p ∂ ∂z p one has
∇ T i ξ p = ∇ i ξ p − T p ij ξ j , ∇ T i ξ p = ∇ i ξ p , ∇ T ♯ i ξ p = ∇ i ξ p , ∇ T ♯ i ξ p = ∇ i ξ p + g ps T isj ξ j .
Both connections preserve the operator of almost complex structure; ∇ T is compatible with the holomorphic structure, i.e., (∇ T ) 0,1 = ∂. Remark 1.7. In general, metric g is not preserved by either of the connections ∇ T , ∇ T ♯ . However, g is parallel with respect to the connection ∇ T ⊗ ∇ T ♯ , i.e., for any vector fields X, Y, The Laplacian of this connection is defined as
Z ∈ Γ(T M ) we have X ·g(Y, Z) = g(∇ T X Y, Z) + g(Y, ∇ T ♯ X Z). In other words, ∇ T ♯ is dual conjugate to ∇ T via g.∆ := 1 2 i ( ∇ ei ∇ ei + ∇ ei ∇ ei ),
where local holomorphic fields {e i } form a unitary frame of T 1,0 m M , m ∈ M . Similarly we define Laplacian ∆ T for connection ∇ T . Definition 1.8 helps to write down the evolution equation for Ω under the HCF. The next proposition is a compilation of Propositions 3.1, 3.4, and 3.9 of [24]. Proposition 1.9. Let g = g(t) be the solution to the HCF (3). Then Ω = Ω ijkl (t) satisfies equation
∂ t Ω ijkl = ∆Ω ijkl + 1 2 g mn g ps ∇ i T mpl ∇ j T nsk + + g mn g ps (Ω ijms Ω pnkl − Ω inks Ω pjml + Ω inpl Ω mjks )− − 1 2 g ps (S(2)
is Ω pjkl + S (2) pj Ω iskl + S (2) ks Ω ijpl + S (2) pl Ω ijks ) − g ps (Q ks Ω ijpl + Q pl Ω ijks )+
+ 1 2 g mn (−∇ n T p mi Ω pjkl − ∇ m T s nj Ω iskl + g ps ∇ n T mpl Ω ijks + g ps ∇ m T nsk Ω ijpl ).(4)
In [24] one was interested in the equation for ∂ t Ω only up to a first order variation of Ω in its arguments. Equation (4) keeps track of the precise expression for this first order variation. In the form (4) the evolution equation for Ω looks unstructured and messy. Using Proposition 1.9 and the the algebraic operations on Sym 1,1 (g), we can considerably simplify it.
Proposition 1.10. Let g = g(t) be the solution to the HCF (3). Then Ω ∈ Sym 1,1 (End(T 1,0 M )), Ω = Ω lk ij (t) satisfies equation (5) ∂ t Ω = ∆ T Ω + Ω 2 + Ω # + D(∇T ) + ad v Ω, where (a) D(∇T ) lk ij = 1/2g mn g ps ∇ i T k mp ∇ j T l ns (b) v b a = − 1 2 S (4)
as g bs .
Proof. The proof of this proposition is a matter of straightforward computations. We will not provide these computations in details, but explain the origin of all the summands in (5).
Recall that Ω lk ij is obtained from Ω ijkl by rising the last two indices. Since the metric g is not preserved by ∇ T and ∇ T ♯ , the derivatives ∇Ω lk ij and ( ∇Ω ijmn )g ml g kn , in general, do not coincide. However, since ∇ T and ∇ T ♯ are g-dual to each other, we have
∇ T Ω lk ij = ( ∇Ω ijmn )g ml g kn .
This explains the presence of the Laplacian ∆ T in (5). Terms quadratic in Ω in equation (4) give rise to summands Ω 2 and Ω # . It can be easily deduced from Proposition 1.5.
Term D(∇T ) trivially comes from the corresponding ∇T -quadratic term in (4). Finally, the summand ad v Ω comes from the summands of (4) linear in Ω plus the terms involving metric derivative: Ω ijmn (∂ t g) ml g kn and Ω ijmn g ml (∂ t g) kn .
Remark 1.11. Modulo terms D(∇T ) and ad v Ω equation (5) coincides with the evolution equation for the Riemannian curvature under the Ricci flow [12]. This is the only member of a general Streets-Tian's family of flows for which we were able to obtain such a nice evolution equation. It would be interesting to find similar expressions for other versions of the HCF.
For a subset of the space of algebraic curvature tensors, X ⊂ Sym 1,1 (End(T 1,0 M )), we write Ω ∈ X (Ω belongs to X), if Ω m ∈ X m at any point m ∈ M . We call such an X a curvature condition and say that Ω satisfies X.
Let g = g(t) be a solution to the HCF on M . Tensor Ω g(t) satisfies a nonlinear heat-type equation (5) and our aim is to find invariant curvature conditions for Ω under this equation. That are subsets X ⊂ Sym 1,1 (End(T 1,0 M )) s.t, Ω g(t) satisfies X for t > 0, provided Ω g(0) satisfies X. A general approach to this kind of problems was developed by Hamilton in his seminal paper [12]. This approach is based on maximum principle for tensors. In the next section we explain how to modify this principle, so that it will become applicable to equation (5).
Hamilton's maximum principle
In this section we prove a modification of Hamilton's maximum principle for tensors. Let us start with recalling this principle in its original form. Let M be a closed smooth manifold with a Riemannian metric g, and let E → M be a vector bundle equipped with a metric h and a metric connection ∇ E . With the use of the Levi-Civita connection we extend ∇ E to a connection on
Λ 1 M ⊗ E. The Laplacian ∆ E : Γ(M, E) → Γ(M, E) is defined as ∆ E s := tr g (∇ E • ∇ E (s)).
Let ϕ(f ) be a smooth vertical vector field on the total space of E. We are interested in the short-time behavior of the solutions to a nonlinear parabolic equation
for f ∈ C ∞ (M × [0, ǫ), R) (6) df dt = ∆ E f + ϕ(f ),
where the background data (h, g, ∇ E , ϕ) is allowed to depend smoothly on t.
Recall that a support functional for a closed convex set Y ∈ R N at a boundary point y ∈ ∂Y is a linear function α : R N → R, such that α, y α, y ′ for any y ′ ∈ Y . The set of support functionals at y ∈ ∂Y forms a nonempty closed convex cone in (R N ) * . The set of support functionals of the unit length (with respect to an underlying metric) will be denoted S y .
Let X ⊂ E be a subset of the total space of E satisfying the following properties (P1) X is closed; (P2) the fiber X m = X ∩ E m over any m ∈ M is convex; (P3) X is invariant under the parallel transport induced by ∇ E ; (P4) For any boundary point f ∈ ∂X m and any support functional α ∈ S f ⊂ E * m at x we have α, ϕ(x) 0.
Assume that the initial data f 0 lies in X, i.e, f 0 (m) ∈ X m for any m ∈ M . Hamilton's maximum principle states that the set X is invariant under the PDE (6), i.e., f (m, t) remains in X m for t > 0, as long, as the equation is solvable. Specifically, we have the following results. The proof of both results is based on the fact that the invariance of X under PDE/ODE is equivalent to the invariance of all the 'half-spaces
df dt = ϕ(f ) remains in X m iff X m satisfies property (P4). {x ∈ E m | α * , x α * , k * } k * • f (m, t) • ρ(m, t) X m' {x ∈ X m | α, x α, f } for all f ∈ ∂X m , α ∈ S f , m ∈ M .
In what follows, we will need these results in a slightly more general setup. Namely, we (a) do not assume that connection ∇ E preserves the bundle metric h; Proof of Theorem 2.1 in a general setup. We will go over the Hamilton's proof of the theorem and point the steps requiring the invariance of h under ∇ E . In each case we provide the necessary modifications to drop this assumption. As in Hamilton's proof, we will use the basic theory of differential inequalities for Lipschitz functions [12, §3].
Denote by | · | the length function induced by h on E and E * . Let f (m, 0) = f 0 (m) be the initial data with f 0 (m) ∈ X m for any m ∈ M . Let f (m, t) be the solution to the PDE (6) on [0, ǫ] and denote by B R = {e ∈ E | |e| R} the disk bundle of radius R in E.
Step 1. Without loss of generality we can assume that X is compact and ϕ is compactly supported. Indeed, for R large enough f (m, t) ∈ B R for any m ∈ M , t ∈ [0, ǫ]. Consider X = X ∩ B 3R and multiply ϕ(f ) by a cutoff function, which is supported on B 3R and equals 1 on B 2R . Clearly, if the solution of a new equation on [0, ǫ] stays in X, then the solution of the initial equation stays in X. From now X is compact. Remark 2.3. Unlike the situation in the original proof, with the above modification the set X does not remain invariant under ∇ E , since h and B R are not preserved by ∇ E . However, we still have the following local invariance property on X ∩ B R (P3 * ) There exists δ = δ(g, h, R) > 0 such that for any path γ(τ ) ∈ M, τ ∈ [0, 1] of length < δ and any s ∈ ∂X γ(0) ∩ B R the ∇ E -parallel transport of s along γ lies in ∂X γ (1) . Moreover, this parallel transport carries support functionals to support functionals.
Step 2. For a fixed m ∈ M , and t ∈ [0, ǫ] define
ρ(m, t) = sup{ α, f (m, t) − k },
where the supremum is taken over k ∈ ∂X m , α ∈ S k (i.e., α is a support functional at k and |α| = 1). Since the domain of this supremum is compact, it is attained at some α = α * , k = k * . By our choice of R, point k * lies in X m ∩ B R . If f (m, t) ∈ X m , then ρ(m, t) equals the distance from f (m, t) to ∂X m (see Figure 1). Otherwise, if f (m, t) ∈ X m , then ρ(m, t) equals the negative distance from f (m, t) to ∂X m . Now define ρ(t) = sup m∈M ρ(m, t).
Function ρ(t) is Lipschitz, and ρ(t) 0 (resp. < 0) if and only f ∈ X (resp. f belongs to the interior of X). Therefore, to prove Theorem 2.1 it is enough to prove that ρ(t) 0, provided ρ(0) 0. We claim that there exists a constant C > 0 such that d ρ dt C| ρ(t)|.
To prove the claim we plug in the definition of ρ and use [12, Lemma 3.5]:
d ρ dt sup d dt α, f (m, t) − k ,
where supremum is taken over m ∈ M , k ∈ ∂X m , α ∈ S k such that the maximum of α, f (m, t)−k is attained, i.e., α, f (m, t) − k = ρ(t). Together with the equation for f this gives
d ρ dt sup{ α, ∆ E f + ϕ(f ) } = sup{ α, ∆ E f + α, ϕ(f ) }.
We claim that both summands could be bounded from above by C| ρ(t)| for some constant C > 0.
Step 2a. Let {e i } be a g-orthonormal frame of T m M . Define γ i (τ ), i = 1, . . . dim M , to be the geodesic path of connection ∇ T M in the direction e i and denote by D i the covariant derivative along γ ′ i . Then
∆ E = dim M i=1 D 2 i .
We extend vectors k ∈ E m , α ∈ E * m along each of the paths γ i by ∇ E -parallel transport. By property (P3 * ), in a small neighborhood of m ∈ M we still have k ∈ ∂X, and α is a support functional at k. Note, however, that in order to get an element in S k over a point m 0 = m, we need to normalize α, since ∇ E does not preserve metric h; so α/|α| ∈ S k . Since point m ∈ M , and the corresponding vectors k ∈ ∂X m , α ∈ S k are chosen in such a way that α, f (m, t) − k attains its maximum -ρ(t), the function Φ i (τ ) := α/|α|, f (γ i (τ ), t) − k is maximal at τ = 0. Therefore
0 = Φ ′ i (0) = (D i |α| −1 ) ρ(t) + D i α, f − k ; 0 Φ ′′ i (0) = (D 2 i |α| −1 ) ρ(t) + 2(D i |α| −1 )D i α, f − k + α, D 2 i f .
With the use of the first equation we can rewrite the inequality as
α, D 2 i f −(D 2 i |α| −1 ) ρ(t) + 2(D i |α| −1 ) 2 ρ(t). Let C ′ be an upper bound for 2(D i |α| −1 ) 2 − D 2 i |α| −1 over m ∈ M , α ∈ {α ∈ E * m | |α| = 1}, e i ∈ {v ∈ T m M | |v| = 1}
. Summing the inequality above over i = 1, . . . , dim M , we deduce the inequality α, ∆ E f C| ρ(t)|,
for C = C ′ dim M , as required.
Remark 2.4. In the original proof, the bundle connection ∇ E preserves h, hence for the ∇ Eparallel extension of α we have |α| ≡ 1, so α ∈ S k . In particular, we could take C = 0.
Step 2b. Recall that α ∈ S k , therefore by property (P4) α, ϕ(k) 0. Hence we have
α, ϕ(f ) = α, ϕ(k) + α, ϕ(f ) − ϕ(k) α, ϕ(f ) − ϕ(k) C|f − k| = C| ρ(t)|,
where C is a generic constant bounding the norm of the derivative of ϕ : E → E.
Step 3. Lipschitz function ρ(t) satisfies initial condition ρ(0) 0 and differential inequality d ρ/dt C| ρ|. By a general result [12], it implies ρ(t) 0 for t 0. This is equivalent to the required invariance: f (m, t) ∈ X m for any m ∈ M , t 0. This proves Theorem 2.1.
With the same reasoning, we can prove a bit more. If ρ(0) < 0, then ρ(t) ρ(0)e −Ct < 0 for t 0. Therefore, if f (m, 0) lies in the interior of X for all m ∈ M , then the same is true for f (m, t), t > 0. So, the interior of X is also preserved by the PDE (6). Theorem 2.1 allows to construct many invariant sets X for certain PDEs of the form (6). In practice, conditions (P1), (P2), (P3) are satisfied automatically for a wide range of subsets X ⊂ E, while (P4) is the most essential and difficult to verify. In the next section we apply these results to the evolution equation of the Chern curvature under the HCF.
Invariant sets of curvature operators
By the philosophy of Hamilton's maximum principle in order to find invariant sets for a heattype equation ∂ t f = ∆f + ϕ(f ) one has to study an ODE ∂ t f = ϕ(f ). Following this idea, in this section we study an ODE on the space of algebraic curvature tensors Sym 1,1 (g) given by the zero-order part of the equation (5), where g = End(V ) as in Section 1.2. We construct a family of invariant sets for this ODE by verifying property (P4).
ODE-invariant sets.
Let Ω ∈ Sym 1,1 (g). Consider an ODE for Ω = Ω(t)
(7) ∂ t Ω = Ω 2 + Ω # + ad v Ω + AA * ,
where (1) v is an element of g;
(2) AA * = i a i ⊗ a i , for some collection of vectors {a i ∈ g}. Both v and A are allowed to depend on time. Following the notations of Section 2, denote ϕ(Ω) := Ω 2 + Ω # + ad v Ω + AA * . We describe a family of convex subsets of Sym 1,1 (g), for which we are aiming to prove invariance under (7), and, eventually, the invariance under the HCF. Let S ⊂ g be a subset invariant under the adjoint action of G = GL(V ) and let F : g → R be a continuous function satisfying the following properties.
(⋆ 1 ) F is Ad G-invariant. Since diagonalizable matrices are dense in g, F can be though of as a symmetric function in the eigenvalues {µ i } of s ∈ g; (⋆ 2 ) For any sequence s i ∈ g and any λ i ց 0, such that λ i s i → s, there exists a finite limit
lim i→∞ F (s i )λ 2 i ,
and its value depends only on s. We denote this limit by F ∞ (s). As the intersection of closed halfspaces, the set C(S, F ) is closed and convex. Since S and F are Ad G-invariant, set C(S, F ) is also invariant under the induced action of G. We claim that C(S, F ) is preserved by ODE (7).
Theorem 3.2. The set C(S, F ) ⊂ Sym 1,1 (g) is closed, convex and satisfies property (P4) for ODE (7). In particular, by Lemma 2.2 set C(S, F ) is invariant under this ODE.
Let us first prove a lemma. In this lemma we do not assume that F is continuous. Its proof follows the lines of [26].
Lemma 3.3.
If Ω ∈ C(S, F ) and u ∈ S is such that Ω, u ⊗ u tr = F (u), then Ω 2 + Ω # + ad v Ω + AA * , u ⊗ u tr 0.
Proof. Recall that by definition of Ω 2 we have Ω 2 , u ⊗ u tr 0. Similarly AA * , u ⊗ u tr = i |tr(a i u)| 2 0. Hence, it remains to prove that Ω # + ad v Ω, u ⊗ u tr 0. We claim, that (C1) ad v Ω, u ⊗ u tr = 0, (C2) Ω # , u ⊗ u tr 0.
Proof of both claims is based on the variation of Ω, u ⊗ u tr in u.
Fix an element x ∈ g and let u(τ ) = exp(τ ad x )u be an orbit of u induced by a 1-parameter subgroup of Ad G. By the invariance of S under the adjoint action, we have u(τ ) ∈ S. Therefore by the definition of C(S, F ), the function Ψ(τ ) := Ω, u(τ ) ⊗ u(τ ) tr is bounded below by F (u) and, by our choice of u, attains this minimum at τ = 0. It implies that Ψ ′ (0) = 0 and Ψ ′′ (0) 0. Specifically, Ω, ad x u ⊗ u + u ⊗ ad x u tr = 0, Ω, ad x u ⊗ ad x u tr + Ω, ad x ad x u ⊗ u tr + Ω, u ⊗ ad x ad x u tr 0. (8) The first identity is equivalent to ad x Ω, u ⊗ u tr = 0 for all x ∈ g, implying the vanishing of (C1). Now let us prove (C2). After summing up the second line of (8) for x and √ −1x we arrive at Ω, ad x u ⊗ ad x u tr 0.
This inequality holds for any x ∈ g, thus Hermitian form Q Ω (x, x) := Ω, ad x u ⊗ ad x u tr = Ω, ad u x ⊗ ad u x tr is positive semidefinite. Let us choose a basis {v i } N i=1 of g such that • {v i } N i=r+1 is a basis of Ker ad u , so Q Ω (v i , ·) = 0 for any r + 1 i N ;
• Q Ω (v i , v j ) = δ ij µ i , µ i 0, for 1 i, j r, or, equivalently, the form Ω, · ⊗ · tr Im adu is diagonalized in the basis {w i = ad u v i } r i=1 ; • {w i = ad u v i } r i=1
is an orthonormal basis of Im ad u with respect to the inner Hermitian product tr(ab * ), a, b ∈ g, where b * := b tr is the transposed conjugate of b in some fixed basis of V . In other words, tr(w i w * j ) = δ ij . Let Ω = N i,j=1 a ij v i ⊗ v j be the expression for Ω in this basis. Then
Ω # = 1 2 N i,j,k,l=1 a ij a kl [v i , v k ] ⊗ [v j , v l ].
Therefore
Ω # , u ⊗ u tr = 1 2 N i,j,k,l=1 a ij a kl tr(u[v i , v k ]) ⊗ tr(u[v j , v l ]) = 1 2 r i,j,k,l=1 a ij a kl tr(v i [u, v k ]) ⊗ tr(v j [u, v l ]) = 1 2 r k,l=1 a kl Q Ω (v k , v l ) = 1 2 r k=1 a kk µ k .
It remains to show that a kk 0:
Q Ω (w * k , w * k ) = r i,j=1 a ij tr([v i , u]w * k )tr([v j , u]w * k ) = r i,j=1 a ij tr(w i w * k )tr(w j w * k ) = a kk ,
hence a kk 0, and Ω # , u ⊗ u tr = k a kk µ k 0, as required.
Proof of Theorem 3.2. Now we prove property (P4) for ODE (7) and convex set C(S, F ). Let S be the closure of S. Clearly C(S, F ) = C(S, F ), so without loss of generality we can assume that S = S. Take a point at the boundary of C(S, F ):
y ∈ ∂C(S, F ).
We want to describe the set of support functionals for C(S, F ) at y. Let α be such a functional and take any w ∈ Sym 1,1 (g) such that α, w > 0. Since α is a support functional, for any θ > 0 we have y + θw ∈ C(S, F ), i.e., there exists s ∈ S (depending on w and θ) such that y + θw, s ⊗ s tr < F (s).
Let θ i ց 0 be a monotonically decreasing sequence of real numbers. Choose s i ∈ S such that the inequality above holds. Since y ∈ C(S, F ), we have
F (s i ) + θ i w, s i ⊗ s i tr y + θ i w, s i ⊗ s i tr < F (s i ).
There are two options.
(1) Some subsequence of |s i | stays bounded. Then after passing to a subsequence we may assume that s i → s ∈ S. In this case we have w, s ⊗ s tr 0, y, s ⊗ s tr = F (s), for some s ∈ S.
(2) |s i | → ∞. Then after passing to a subsequence we may assume that for some λ i ց 0 the sequence λ i s i converges to an element s in the set:
∂ ∞ S := {Y ∈ g | there exists λ i ց 0, s i ∈ S with λ i s i → Y }.
This set is called the boundary of S at infinity. From the definition of F ∞ it is clear that Ω ∈ C(∂ ∞ S, F ∞ ). In this case we have w, s ⊗ s tr 0, y, s ⊗ s tr = F ∞ (s), for some s ∈ ∂ ∞ S.
Inequality w, s ⊗ s tr 0 is valid in both cases for any w such that α, w > 0. Therefore α cannot be separated in Sym 1,1 (g) * by a hyperplane α, w = 0 from the set of functionals
F y := F b y ∪ F ∞ y , where F b y = {− ·, s ⊗ s tr | s ∈ S s.t. y, s ⊗ s tr = F (s)}, F ∞ y = {− ·, s ⊗ s tr | s ∈ ∂ ∞ S s.t. y, s ⊗ s tr = F ∞ (s)}.
Hence, α lies in the convex cone spanned by the elements of F y :
α ∈ Cone(F y ).
Thus, in order to verify property (P4), we need to check that α, ϕ(Ω) 0 for all α in F y . For α ∈ F b y this is exactly the statement of Lemma 3.3. For α ∈ F ∞ y this is the statement of Lemma 3.3 applied to C(∂ ∞ S, F ∞ ) (at this point continuity of F ∞ is not required). This proves property (P4), and, by Lemma 2.2, the invariance of C(S, F ) under ODE (7).
PDE-invariant sets.
With the results of previous subsection we can turn back to PDE (5) satisfied by the Chern curvature tensor Ω under the HCF on (M, g, J). First we note that the term
D(∇T ) lk ij = 1/2g mn g ps ∇ i T k mp ∇ j T l ns
is of the form AA * (see (7)) for
{a (mp) } = {∇ i T k mp e k ⊗ ǫ i | m < p} in some orthonormal basis {e i }.
Hence the zero order part of the PDE for Ω is a specialization of the right hand side of ODE (7). Example 4.1 (Dual-Nakano non-negativity). Recall that the Chern curvature Ω is dual-Nakano non-negative, if it represents a non-negative element in Sym 1,1 (End(T 1,0 M )), see Remark 1.2. Choose S = End(T 1,0 M ). Then the cone C(S, 0) is the set of dual-Nakano non-negative curvature tensors. By Theorem 3.4 this set is preserved by the HCF.
Manifolds admitting Dual-Nakano non-negative hermitian metrics are rather scarce. However, as demonstrates the following example, such metrics exist on all complex homogeneous manifolds.
0 → Λ 1,0 M ι − → g * π − → h * → 0.
The second fundamental form β ∈ Λ 1,0 (M, Hom(Λ 1,0 M, h * )) of this exact sequence is given by
β ξ (α) = π(D ξ (ια)) ∈ h * , ξ ∈ T 1,0 M, α ∈ Λ 1,0 M,
where D is the flat connection on the trivialized vector bundle g * . Second fundamental form β naturally corresponds to a map A β : h → End(T 1,0 M ). Now let g be any Hermitian metric on g. It induces an Hermitian metric on T 1,0 M via projection g → T 1,0 M . By a standard computation [7, §14] its curvature Ω ∈ Sym 1,1 (End(T 1,0 M )) is given by
A * β A β = i A β (h i ) ⊗ A β (h i ),
where {h i } is the orthonormal basis of h. This form is clearly non-negative. Example 4.4 (Griffiths non-negativity). Now we demonstrate preservation of Griffiths nonnegativity under the HCF. It was first proved in [24] by adopting the arguments of Mok [15] and Bando [1], who proved the corresponding statement for the Kähler-Ricci flow. Below we deduce preservation of Griffiths non-negativity as a particular case of Theorem 3.4. Let us recall the definition. In the Kähler setting Griffiths positivity is sometimes referred to as positivity of the holomorphic bisectional curvature. Some authors use this notion in the Hermitian setting as well, see, e.g., [27]. Griffiths positivity implies ampleness of T 1,0 M .
It is easy to see that Chern curvature Ω considered as a section of Sym 1,1 (End(T 1,0 M )) is Griffiths non-negative if and only if Ω ∈ C(S, 0), where
S = {u ∈ End(T 1,0 M ) | rank(u) = 1}.
Therefore Griffiths non-negativity is preserved under the HCF. This curvature condition is a relaxation of Griffiths non-negativity. Kähler manifolds admitting non-negative holomorphic orthogonal bisectional curvature were classified by Gu and Zhang [10].
Chern curvature Ω considered as a section of Sym 1,1 (End(T 1,0 M )) has non-negative holomorphic orthogonal bisectional curvature if and only if Ω ∈ C(S, 0), where S = {u ∈ End(T 1,0 M ) | rank(u) = 1, tr(u) = 0}.
Therefore non-negativity is preserved under the HCF.
(T 1,0 M ) is dual m-non-negative if Ω ∈ C(S m , 0), where S m = {u ∈ g | rank(u) = m}.
It is clear from the definition and Theorem 3.4 that dual m-non-negativity is preserved under the HCF.
Example 4.10 (Lower bounds on the second scalar curvature). It is well-known that under the Ricci flow the lower bound on the scalar curvature is improved, unless the manifold is Ricci-flat. It turns out that the second scalar curvature under the HCF satisfies similar monotonicity. Namely, take S = {Id} ∈ End(T 1,0 M ). Then for any q ∈ R the set C(S, q) is preserved under the HCF. In particular, the infimum of Ω, Id ⊗ Id tr = Ω ji ij = sc is nondecreasing. The same result can be obtained without invoking Hamilton's maximum principle for tensors. Indeed, after contracting equation (5), we get
∂ t sc = ∆ sc + |S (3) | 2 + 1 2 |divT | 2 , where (divT ) jk = ∇ i T i jk .
The zero-order expression on the right-hand side is non-negative and by the standard maximum principle for parabolic equations the quantity inf M sc is nondecreasing in t.
Applications
5.1. Strong maximum principle. Hamilton's maximum principle in the form of Theorem 2.1 is a variant of a weak parabolic maximum principle, i.e., a statement about preservation of a nonstrict inequality along a heat-type flow. In many settings a strong maximum principle is satisfied. That is a statement characterizing solutions f (t) of (6), which meet the boundary of a preserved set X at some t > 0. We describe a version of the strong maximum principle for Theorem 3.4.
Theorem 5.1. Consider an Ad G-invariant S ⊂ g and a nice function F : g → R. Let (M, g, J) be an Hermitian manifold with metric g = g(t), t ∈ [0, τ ) evolved along the HCF. Assume that Ω g(0) satisfies C(S, F ). Then for any t > 0 the set Proof. We may assume that S is an orbit of the Ad G-action on g, otherwise, we decompose S into separate orbits and prove the result for each orbit independently. Then S M = ∪ m S m defines a smooth fiber bundle S M → M .
N (t, m) := {s ∈ S | Ω g(t) m , s ⊗ s tr = F (s)}, m ∈ M, is invariant under the ∇ T -parallel transport. Moreover, if s 0 ∈ N (t, m), then
The idea is to treat N as the zero set of the function Φ : S M × [0, τ ) → R, Φ(s, t) = Ω g(t) , s ⊗ s tr − F (s) and to prove certain differential inequality for Φ, which makes it possible to apply Proposition 4 of [4].
First, note that by assumption, Φ(s, 0) is non-negative on S M , and by Theorem 3.4 the same holds for t > 0. By the evolution equation for Ω, (9), function Φ : S M → R satisfies equation
(10) ∂ t Φ(s, t) = ∆ h Φ(s, t) + Ω 2 , s ⊗ s tr + Ω # , s ⊗ s tr + i<j | ∇T ij , s tr | 2 + ad v Ω, s ⊗ s tr ,
where ∆ h is the horizontal Laplacian of the connection ∇ T . It is defined as follows. Let {X i } be a g(t)-orthonormal collection of vector fields in a neighbourhood of m ∈ M . Denote Y i := ∇ Xi X i , and let { X i }, { Y i } be the ∇ T -horizontal lifts of these vector fields to S M . Then ∆ h Φ(s, t) :=
i ( X i · X i ·Φ − Y i ·Φ).
The second and the fourth summands on the right hand side of (10) are always non-negative. Using the same bounds for the remaining terms, as in [26, Th. A.1], we conclude, that on a small relatively compact coordinate neighbourhood of s ∈ S M , for a positive constant K we have
i X i · X i ·Φ −K inf{Hess(Φ)(a, a) |a| 1} + K|grad(Φ)|.
At this point, we can apply [4,Prop. 4] and conclude that the zero set of Φ is invariant under the flow generated by the vector fields X i . Since these fields span the ∇ T -horizontal subspaces on the fiber bundle S M → M , we obtain that zeros of Φ are invariant under the ∇ T -parallel transport. Now, let us prove (a) and (b). Let s 0 ∈ S M be a zero of Φ(s 0 , t 0 ) for some t 0 > 0. Hence function Φ(s 0 , t 0 ) attains a local minimum at s = s 0 , t = t 0 , so ∂ t Φ(s 0 , t 0 ) = 0. As we have proved in Theorem 3.4, all summands on right hand side of (10) are non-negative, therefore they must vanish. In particular: (a) Ω 2 , s 0 ⊗ s 0 tr = 0, so s 0 ∈ Ker Ω (see definition of Ω 2 ); (b) ∇T ij , s 0 tr = 0. Theorem 5.1 implies a more familiar version of maximum principle.
Corollary 5.2. Consider an Ad G-invariant subset S ⊂ g and a nice function F : g → R. Let (M, g, J) be an Hermitian manifold with metric g = g(t), t ∈ [0, τ ) evolved along the HCF. Assume that Ω g(0) satisfies C(S, F ), and that there exists m 0 ∈ M such that Ω g(0) satisfies strict inequalities, defining C(S, F ) and C(∂ ∞ S, F ∞ ), at m 0 :
Ω g(0)
m0 , s ⊗ s tr > F (s), for any s ∈ S, Ω g(0) m0 , s ⊗ s tr > F ∞ (s), for any s ∈ ∂ ∞ S.
(11)
Then for any t ∈ (0, τ ), inequalities (11) hold everywhere on M .
boundary of C(S, µ|trs| 2 ) (here we are using closedness and scale-invariance of S). Therefore, in notations of Theorem 5.1, N (t, m) = ∅. By Theorem 5.1 (a), it can happen only if µ = 0.
A similar statement is valid for other one-parametric monotonic family of functions F (s).
For S = {λ Id | λ ∈ R} Proposition 5.3 gives the monotonicity of the lower bound for sc, see Example 4.10. Of course, this monotonicity can be deduced by applying the usual parabolic maximum principle to the equation ∂ t sc = ∆ sc + |S (3) | 2 + 1 2 |div T | 2 .
Note that inf M sc is strictly increasing, unless sc ≡ 0, and S (3) = 0, div T = 0. This monotonicity with the vanishing of S (3) and div T = 0 have important consequences for the understanding of periodic and stationary solutions to the HCF. In the Kähler setting the metrics fixed by the Kähler-Ricci flow are tautologically the Ricci flat (Calabi-Yau) metrics. At first glance the situation with the HCF is much more subtle, since vanishing of the term S (2) +Q in (3) does not have any clear cohomological interpretation. Surprisingly, we still can conclude that c 1 (M ) = 0.
More generally, assume that g = g(t) is a periodic solution to the HCF (3) on some time interval [0, t max ), i.e., g(0) = g(T ) for some 0 < T < t max . Hence inf M sc is constant on [0, T ], and by the discussion above it follows div T = 0, S (3) = 0. We claim that the vanishing of S (3) and div T imply that c 1 (M ) = 0 in H 2 (M, C). Indeed, for the first Chern-Ricci form ρ := √ −1S (1) ks dz k ∧ dz s , we have [ρ] = 2πc 1 (M ). Now the claim follows from a simple lemma.
Lemma 5.4. Differential 2-forms ρ and
ρ T := √ −1 2 (div T ) jk dz j ∧ dz k + √ −1S(3)
ks dz k ∧ dz s are cohomologous.
Proof. Let α = √ −1T p kp dz k = ∂ * ω be the (1, 0)-part of the Lee form of (M, g, J). Then, by a standard formula relating the exterior and the covariant derivatives, we get
∂α = − √ −1∇ s T p kp dz k ∧ dz s ∂α = √ −1(∇ i T p kp + 1 2 T r ik T p rp )dz i ∧ dz k = √ −1 2 (∇ i T p kp − ∇ k T p ip + T r ik T p rp )dz i ∧ dz k = = √ −1 2 ∇ p T p ki dz i ∧ dz k = − √ −1 2 (div T ) ik dz i ∧ dz k ,
where in the last line we used the differential Bianchi identity.
∇ i T l jk + ∇ k T l ij + ∇ j T l ki = T p ij T l kp + T p jk T l ip + T p ki T l jp . Using now the first Bianchi identity Ω m ksm + ∇ s T m km = Ω m msk , we conclude ρ − ρ T = dα.
Form ρ T has a clear geometric interpretation. It is the curvature form of the connection ∇ T induced on the anticanonical bundle −K M . A priori ∇ T does not preserve any metric, so its curvature form ρ T might contain a part of type (2, 0). Since ∇ T is compatible with the holomorphic structure, i.e., (∇ T ) 0,1 = ∂, we can use ∇ T -parallel transport to construct a nowhere vanishing holomorphic section of π * K M , where π : M → M is the universal cover. This proves the following result.
Theorem 5.5. If a compact complex manifold M admits an HCF-periodic Hermitian metric, then the pull-back of the canonical bundle to the universal cover of M is holomorphically trivial.
The statement of the above theorem is stronger then just vanishing of c 1 (M ) ∈ H 2 (M, C). For example, Calabi-Eckman complex structures on S 3 ×S 3 have holomorphically non-trivial canonical bundle, while c 1 = 0, see, e.g., discussion in [2, §2].
It is still an open question, whether the HCF (3) admits non-trivial, i.e., not stationary, periodic solutions.
Problem 5.6. Is is true that if g = g(t) is a periodic solution to the HCF on (M, g, J), then g(t) is a stationary solution, i.e., g ≡ g(0)? Theorem 5.5 motivates us to formulate the following problem. This problem is a non-Kähler version of Calabi's conjecture. Namely, if the underlying manifold (M, J) admits a Kähler metric ω, then by Yau's theorem, there exists a unique Kähler metric ω ϕ = ω + i∂∂ϕ, such that Ric(ω ϕ ) = 0. Of course, in this case the torsion vanishes, and all four Chern-Ricci curvatures coincide and equal zero.
Theorem 0 . 1 .
01For any Ad GL(T 1,0 M )-invariant subset S ⊂ End(T 1,0 M ) and any nice function F : End(T 1,0 M ) → R, the curvature condition C(S, F ) is preserved by the HCF.
Proposition 1 . 5 .
15Let {e m } be a basis of V and {ǫ m } be the dual basis. For an element Ω ∈ Sym 1,1
Definition 1. 8 (
8Torsion-twisted connection on the space of curvature tensors). Torsion-twisted connection ∇ on the space Λ 1,0 M ⊗ Λ 0,1 M ⊗ Λ 1,0 M ⊗ Λ 0,1 M acts as ∇ T on the first two factors and as ∇ T ♯ on the last two factors.
Theorem 2. 1 .
1If for every fiber E m , m ∈ M , the solutions of the ODEdf dt = ϕ(f ) remain in X m ⊂ E m ,then the solutions of the PDE (6) also remain in X. Lemma 2.2. For a closed convex subset X m ⊂ E m the solution of the ODE
Figure 1 .
1Definition of ρ(m, t).
(b) in the definition of ∆ E , allow to use any (not necessarily the Levi-Civita) connection ∇ T M on T M to extend ∇ E to a connection on Λ 1 M ⊗ E. Necessity of this generalization comes from the presence of a non-metric Laplacian ∆ T in equation (5). These modifications do not affect neither the setup nor the original proof of Lemma 2.2, since it depends only on the properties of X in each individual fiber X m . Hence, only the proof of Theorem 2.1 requires modifications.
Definition 3. 1 .
1Continuous function F : g → R satisfying properties (⋆ 1 ) and (⋆ 2 ) is called nice. There is a function F ∞ : g → R attached to any nice function.Examples of nice F are F (s) = a|trs| 2 + b with the corresponding limit F ∞ (s) = a|trs| 2 and F (s) = µi∈spec(s) |µ i | with F ∞ (s) ≡ 0. In most of the examples below the only reasonable choice is F ≡ 0.Given a tuple (S, F ) we define a subset of Sym 1,1 (g):C(S, F ) := {Ω ∈ Sym 1,1 (g) | Ω, s ⊗ s tr F (s) for all s ∈ S}.
( 9 )
9∂ t Ω = ∆ T Ω + Ω 2 + Ω # + D(∇T ) + ad v Ω Let V = C dim M . For every m ∈ M choose a linear isomorphism V ≃ T 1,0 m M , with the corresponding isomorphism g ≃ End(T 1,0 m M ). Then any G = GL(V )-invariant S ⊂ g canonically corresponds to S m ⊂ End(T 1,0 m ). The set C(S, F ) := ∪ m∈M C(S m , F ) ⊂ Sym 1,1 (End(T 1,0 M ))is closed, convex and satisfies (P4). It also satisfies property (P3), since C(S, F ) ⊂ End(T 1,0 M ) is invariant under the adjoint action of GL(T 1,0 M ), and, therefore is invariant under the action of the holonomy group any connection compatible with J. Hence, we can apply Theorem 2.1 and conclude that C(S, F ) is invariant under (9), proving our main result.
Theorem 3 . 4 .
34For any Ad G-invariant subset S ⊂ g and any nice F : g → R the curvature condition C(S, F ) is preserved by the HCF (3).4. ExamplesLet us provide some specific examples of curvature conditions preserved by the HCF. In most of the examples F ≡ 0.
Example 4. 2 (
2Dual-Nakano non-negative metrics on complex homogeneous manifolds). Let M = G/H be a complex homogeneous manifold acted on by a complex Lie group G. Denote by g, h the corresponding Lie algebras. There is an exact sequence of holomorphic vector bundles
Remark 4. 3 .
3In Example 4.2 for any Hermitian metric g on g we considered the induced metric on M = G/H. It is proved in[25] that the set of induced metrics is invariant under the HCF.
Definition 4 . 5 .
45Chern curvature tensor Ω = Ω ijkl , considered as a section of Λ 1,0 M ⊗ Λ 0,1 M ⊗ Λ 1,0 M ⊗ Λ 0,1 M is said to be Griffiths non-negative, if for any ξ, η ∈ T 1,0 M Ω(ξ, ξ, η, η) 0.
Example 4 . 6 (
46Nonnegativity of holomorphic orthogonal bisectional curvature).
Definition 4 . 7 .
47Chern curvature tensor Ω = Ω ijkl , considered as a section of Λ 1,0 M ⊗ Λ 0,1 M ⊗ Λ 1,0 M ⊗ Λ 0,1 M is said to have non-negative holomorphic orthogonal bisectional curvature, if for any ξ, η ∈ T 1,0 M s.t g(ξ, η) = 0 Ω(ξ, ξ, η, η) 0.
Example 4. 8 (
8Dual m-non-negativity). This curvature positivity notion interpolates between Griffiths non-negativity (m = 1) and dual-Nakano non-negativity (m = dim M ).
Definition 4. 9 .
9Take a number 1 m dim M . Chern curvature tensor Ω ∈ Sym 1,1
a) s 0 belongs to the kernel of Ωg(t) m , · ⊗ · tr , in particular, F (s 0 ) = 0; (b) ∇T (ξ, η), s 0 tr = (s 0 ) i j ∇ i T j kl ξ k η l = 0 for any ξ, η ∈ T 1,0 m M .This theorem is an extension of Brendle and Schoen's [4] strong maximum principle, which was originally proved for the isotropic curvature evolved under the Ricci flow. A general argument in the context of the Ricci flow was given by Wilking [26, A.1]. The same proof works for the HCF with minor modifications. In [24, Th. 5.2] this argument was used in the case of Griffiths positivity.
Problem 5 . 7 .
57Let (M, J) be a compact complex manifold with a trivial canonical bundle. Does M admit an HCF-stationary metric? By Theorem 5.5, such a metric necessarily will have S (3) = 0, div T = 0.
Acknowledgements I would like to thank my advisor Gang Tian for introducing me a circle of problems concerning Hermitian Curvature Flows, and for constant support in my research. I also wish to thank Jeffrey Streets for valuable discussions.
We claim that Ω m lies in the interior of C(S, F ) if and only if Ω m satisfies inequalities. Proof. We claim that Ω m lies in the interior of C(S, F ) if and only if Ω m satisfies inequalities (11).
Pick t ε > 0 such that Ω g(t) m0 still lies in the interior of C(S, F ) for t ∈ (0, t ε ). Fix any t ∈ (0, t ε ). By Theorem 5.1 the set N (t, m) for (S, F ) is invariant under ∇ T -parallel transport. · , S ⊗ S Tr , S ∈ S ∪ ∂ ∞ S, (tε) , s ⊗ s tr > F (s) for any s ∈ S everywhere on M . Applying the same reasoning to (∂ ∞ S, F ∞ ), we conclude that Ω g(tε) , s ⊗ s tr > F ∞ (s) for any s ∈ ∂ ∞. S everywhere on M . Hence, Ω g(t) , lies in the interior of C(S, F ). By the proof of Hamilton's maximum principle, Ω remains in the interior of the convex set C(S, F ) for all t ∈ (0, τIndeed, if Ω lies on the boundary of C(S, F ), then, following the proof of Theorem 3.2, we can find a support functional of the form ·, s ⊗ s tr , s ∈ S ∪ ∂ ∞ S, such that in (11) we have equality. Conversely, if for some s ∈ S ∪∂ ∞ S we have equality in (11), then there is y ∈ Sym 1,1 (End(T 1,0 m M )) arbitrary close to Ω m , such that y ∈ C(S, F ). So, Ω m ∈ ∂C(S, F ). Pick t ε > 0 such that Ω g(t) m0 still lies in the interior of C(S, F ) for t ∈ (0, t ε ). Fix any t ∈ (0, t ε ). By Theorem 5.1 the set N (t, m) for (S, F ) is invariant under ∇ T -parallel transport. At the same time, N (t, m 0 ) is empty, therefore for any m ∈ M the set N (t, m) is empty as well, i.e., Ω g(tε) , s ⊗ s tr > F (s) for any s ∈ S everywhere on M . Applying the same reasoning to (∂ ∞ S, F ∞ ), we conclude that Ω g(tε) , s ⊗ s tr > F ∞ (s) for any s ∈ ∂ ∞ S everywhere on M . Hence, Ω g(t) , lies in the interior of C(S, F ). By the proof of Hamilton's maximum principle, Ω remains in the interior of the convex set C(S, F ) for all t ∈ (0, τ ).
Let S ⊂ g be a closed scale-invariant, Ad G-invariant subset. Define µ. M , G , J ) ; } ∈ R ∪ {±∞}, Monotonicity under HCF. Theorem 3.4 allows to produce many monotonic quantities for the HCF on. = max{µ ∈ R | Ω ∈ C(S, F )S, g= −∞F (s) = µ|trs| 22. Monotonicity under HCF. Theorem 3.4 allows to produce many monotonic quantities for the HCF on (M, g, J). Let S ⊂ g be a closed scale-invariant, Ad G-invariant subset. Define µ(S, g) := max{µ ∈ R | Ω ∈ C(S, F ), F (s) = µ|trs| 2 } ∈ R ∪ {±∞}, where max{∅} := −∞.
Then for any S ⊂ g as above the quantity µ(S, g(t)) is non-decreasing along the HCF. Moreover, µ(S, g(t)) > µ(S, g(0)) for t > 0. M , G , J ) , Proposition 5.3. Let g = g(t) be the solution to the HCF on. unless µ(S, g(0)) ∈ {−∞, 0, +∞}Proposition 5.3. Let g = g(t) be the solution to the HCF on (M, g, J). Then for any S ⊂ g as above the quantity µ(S, g(t)) is non-decreasing along the HCF. Moreover, µ(S, g(t)) > µ(S, g(0)) for t > 0, unless µ(S, g(0)) ∈ {−∞, 0, +∞}.
If Ω g(0) satisfies C(S, µ|trs| 2 ), then by Theorem 3.4 Ω g(t) also satisfies C(S, µ|trs| 2 ). Hence µ(S, g(t)) µ(S, g(0)). Now assume µ(S, g(0)) ∈ {−∞, 0, +∞}. Proof, but µ(S, g(t)) = µ(S, g(0)) = µ. Therefore for anyProof. If Ω g(0) satisfies C(S, µ|trs| 2 ), then by Theorem 3.4 Ω g(t) also satisfies C(S, µ|trs| 2 ). Hence µ(S, g(t)) µ(S, g(0)). Now assume µ(S, g(0)) ∈ {−∞, 0, +∞}, but µ(S, g(t)) = µ(S, g(0)) = µ. Therefore for any
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E-mail address: yuryu@math. princeton.eduE-mail address: [email protected]
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"Advance Access Publication Date: DD Month YYYY Original Paper Sequence Alignment Shouji: A Fast and Efficient Pre-Alignment Filter for Sequence Alignment",
"Advance Access Publication Date: DD Month YYYY Original Paper Sequence Alignment Shouji: A Fast and Efficient Pre-Alignment Filter for Sequence Alignment"
] | [
"Mohammed Alser \nComputer Science Department\nETH Zürich\n8092ZürichSwitzerland\n\nInstitute for Computer Engineering\nTechnische Universität Dresden\nCfAED, Germany\n\nComputer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey\n",
"Hasan Hassan \nComputer Science Department\nETH Zürich\n8092ZürichSwitzerland\n",
"Akash Kumar \nInstitute for Computer Engineering\nTechnische Universität Dresden\nCfAED, Germany\n",
"Onur Mutlu [email protected] \nComputer Science Department\nETH Zürich\n8092ZürichSwitzerland\n\nComputer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey\n",
"Can Alkan [email protected] \nComputer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey\n"
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"Computer Science Department\nETH Zürich\n8092ZürichSwitzerland",
"Institute for Computer Engineering\nTechnische Universität Dresden\nCfAED, Germany",
"Computer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey",
"Computer Science Department\nETH Zürich\n8092ZürichSwitzerland",
"Institute for Computer Engineering\nTechnische Universität Dresden\nCfAED, Germany",
"Computer Science Department\nETH Zürich\n8092ZürichSwitzerland",
"Computer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey",
"Computer Engineering Department\nBilkent University\n06800Bilkent, AnkaraTurkey"
] | [
"Bioinformatics, YYYY"
] | Motivation:The ability to generate massive amounts of sequencing data continues to overwhelm the processing capability of existing algorithms and compute infrastructures. In this work, we explore the use of hardware/software co-design and hardware acceleration to significantly reduce the execution time of short sequence alignment, a crucial step in analyzing sequenced genomes. We introduce Shouji, a highly-parallel and accurate pre-alignment filter that remarkably reduces the need for computationally-costly dynamic programming algorithms. The first key idea of our proposed pre-alignment filter is to provide high filtering accuracy by correctly detecting all common subsequences shared between two given sequences. The second key idea is to design a hardware accelerator that adopts modern FPGA (Field-Programmable Gate Array) architectures to further boost the performance of our algorithm. Results: Shouji significantly improves the accuracy of pre-alignment filtering by up to two orders of magnitude compared to the state-of-the-art pre-alignment filters, GateKeeper and SHD. Our FPGAbased accelerator is up to three orders of magnitude faster than the equivalent CPU implementation of Shouji. Using a single FPGA chip, we benchmark the benefits of integrating Shouji with five state-ofthe-art sequence aligners, designed for different computing platforms. The addition of Shouji as a prealignment step reduces the execution time of the five state-of-the-art sequence aligners by up to 18.8x. Shouji can be adapted for any bioinformatics pipeline that performs sequence alignment for verification. Unlike most existing methods that aim to accelerate sequence alignment, Shouji does not sacrifice any of the aligner capabilities, as it does not modify or replace the alignment step. Availability: https://github.com/CMU-SAFARI/Shouji | 10.1093/bioinformatics/btz234 | [
"https://arxiv.org/pdf/1809.07858v3.pdf"
] | 85,529,605 | 1809.07858 | 70adcb8a75e25b871081162ddbc65ce226560528 |
Advance Access Publication Date: DD Month YYYY Original Paper Sequence Alignment Shouji: A Fast and Efficient Pre-Alignment Filter for Sequence Alignment
Mohammed Alser
Computer Science Department
ETH Zürich
8092ZürichSwitzerland
Institute for Computer Engineering
Technische Universität Dresden
CfAED, Germany
Computer Engineering Department
Bilkent University
06800Bilkent, AnkaraTurkey
Hasan Hassan
Computer Science Department
ETH Zürich
8092ZürichSwitzerland
Akash Kumar
Institute for Computer Engineering
Technische Universität Dresden
CfAED, Germany
Onur Mutlu [email protected]
Computer Science Department
ETH Zürich
8092ZürichSwitzerland
Computer Engineering Department
Bilkent University
06800Bilkent, AnkaraTurkey
Can Alkan [email protected]
Computer Engineering Department
Bilkent University
06800Bilkent, AnkaraTurkey
Advance Access Publication Date: DD Month YYYY Original Paper Sequence Alignment Shouji: A Fast and Efficient Pre-Alignment Filter for Sequence Alignment
Bioinformatics, YYYY
10.1093/bioinformatics/xxxxxReceived on XXXXX; revised on XXXXX; accepted on XXXXX*To whom correspondence should be addressed. Associate Editor: XXXXXXX Supplementary information: Supplementary data are available at Bioinformatics online.
Motivation:The ability to generate massive amounts of sequencing data continues to overwhelm the processing capability of existing algorithms and compute infrastructures. In this work, we explore the use of hardware/software co-design and hardware acceleration to significantly reduce the execution time of short sequence alignment, a crucial step in analyzing sequenced genomes. We introduce Shouji, a highly-parallel and accurate pre-alignment filter that remarkably reduces the need for computationally-costly dynamic programming algorithms. The first key idea of our proposed pre-alignment filter is to provide high filtering accuracy by correctly detecting all common subsequences shared between two given sequences. The second key idea is to design a hardware accelerator that adopts modern FPGA (Field-Programmable Gate Array) architectures to further boost the performance of our algorithm. Results: Shouji significantly improves the accuracy of pre-alignment filtering by up to two orders of magnitude compared to the state-of-the-art pre-alignment filters, GateKeeper and SHD. Our FPGAbased accelerator is up to three orders of magnitude faster than the equivalent CPU implementation of Shouji. Using a single FPGA chip, we benchmark the benefits of integrating Shouji with five state-ofthe-art sequence aligners, designed for different computing platforms. The addition of Shouji as a prealignment step reduces the execution time of the five state-of-the-art sequence aligners by up to 18.8x. Shouji can be adapted for any bioinformatics pipeline that performs sequence alignment for verification. Unlike most existing methods that aim to accelerate sequence alignment, Shouji does not sacrifice any of the aligner capabilities, as it does not modify or replace the alignment step. Availability: https://github.com/CMU-SAFARI/Shouji
Introduction
One of the most fundamental computational steps in most bioinformatics analyses is the detection of the differences/similarities between two genomic sequences. Edit distance and pairwise alignment are two approaches to achieve this step, formulated as approximate string matching (Navarro, 2001). Edit distance approach is a measure of how much two sequences differ. It calculates the minimum number of edits needed to convert a sequence into the other. The higher the edit distance, the more different the sequences from one another. Commonly-allowed edit operations include deletion, insertion, and substitution of characters in one or both sequences. Pairwise alignment is a measure of how much the sequences are alike. It calculates the alignment that is an ordered list of characters representing possible edit operations and matches required to change one of the two given sequences into the other. As any two sequences can have several different arrangements of the edit operations and matches (and hence different alignments), the alignment algorithm usually involves a backtracking step. This step finds the alignment that has the highest alignment score (called optimal alignment). The alignment score is the sum of the scores of all edits and matches along the alignment implied by a user-defined scoring function. The edit distance and pairwise alignment approaches are non-additive measures (Calude et al., 2002). This means that if we divide the sequence pair into two consecutive subsequence pairs, the edit distance of the entire sequence pair is not necessarily equivalent to the sum of the edit distances of the shorter pairs. Instead, we need to examine all possible prefixes of the two input sequences and keep track of the pairs of prefixes that provide an optimal solution. Enumerating all possible prefixes is necessary for tolerating edits that result from both sequencing errors (Fox et al., 2014) and genetic variations (McKernan et al., 2009). Therefore, the edit distance and pairwise alignment approaches are typically implemented as dynamic programming algorithms to avoid re-examining the same prefixes many times. These implementations, such as Levenshtein distance (Levenshtein, 1966), Smith-Waterman (Smith and Waterman, 1981), and Needleman-Wunsch (Needleman and Wunsch, 1970), are inefficient as they have quadratic time and space complexity (i.e., O(m 2 ) for a sequence length of m). Many attempts were made to boost the performance of existing sequence aligners. Despite more than three decades of attempts, the fastest known edit distance algorithm (Masek and Paterson, 1980) has a running time of O(m 2 /log 2 m) for sequences of length m, which is still nearly quadratic (Backurs and Indyk, 2017). Therefore, more recent works tend to follow one of two key new directions to boost the performance of sequence alignment and edit distance implementations: (1) Accelerating the dynamic programming algorithms using hardware accelerators.
(2) Developing filtering heuristics that reduce the need for the dynamic programming algorithms, given an edit distance threshold. Hardware accelerators include multi-core and SIMD (single instruction multiple data) capable central processing units (CPUs), graphics processing units (GPUs), and field-programmable gate arrays (FPGAs). The classical dynamic programming algorithms are typically accelerated by computing only the necessary regions (i.e., diagonal vectors) of the dynamic programming matrix rather than the entire matrix, as proposed in Ukkonen's banded algorithm (Ukkonen, 1985). The number of the diagonal bands required for computing the dynamic programming matrix is 2E+1, where E is a user-defined edit distance threshold. The banded algorithm is still beneficial even with its recent sequential implementations as in Edlib (Šošić and Šikić, 2017). The Edlib algorithm is implemented in C for standard CPUs and it calculates the banded Levenshtein distance. Parasail (Daily, 2016) exploits both Ukkonen's banded algorithm and SIMD-capable CPUs to compute a banded alignment for a sequence pair with a user-defined scoring function. SIMD instructions offer significant parallelism to the matrix computation by executing the same vector operation on multiple operands at once. The multi-core architecture of CPUs and GPUs provides the ability to compute alignments of many sequence pairs independently and concurrently (Georganas et al., 2015;Liu and Schmidt, 2015). GSWABE (Liu and Schmidt, 2015) exploits GPUs (Tesla K40) for highly-parallel computation of global alignment with a user-defined scoring function. CUDASW++ 3.0 (Liu et al., 2013) exploits the SIMD capability of both CPUs and GPUs (GTX690) to accelerate the computation of the Smith-Waterman algorithm with a user-defined scoring function. CUDASW++ 3.0 provides only the optimal score, not the optimal alignment (i.e., no backtracking step). Other designs, for instance FPGASW (Fei et al., 2018), exploit the very large number of hardware execution units in FPGAs (Xilinx VC707) to form a linear systolic array (Kung, 1982). Each execution unit in the systolic array is responsible for computing the value of a single entry of the dynamic programming matrix. The systolic array computes a single vector of the matrix at a time. The data dependencies between the entries restrict the systolic array to computing the vectors sequentially (e.g., top-to-bottom, left-to-right, or in an anti-diagonal manner). FPGA accelerators seem to yield the highest performance gain compared to the other hardware accelerators (Banerjee et al., 2018;Chen et al., 2016;Fei et al., 2018;Waidyasooriya and Hariyama, 2015). However, many of these efforts either simplify the scoring function, or only take into account accelerating the computation of the dynamic programming matrix without providing the optimal alignment as in (Chen et al., 2014;Liu et al., 2013;Nishimura et al., 2017). Different and more sophisticated scoring functions are typically needed to better quantify the similarity between two sequences (Henikoff and Henikoff, 1992;Wang et al., 2011). The backtracking step required for the optimal alignment computation involves unpredictable and irregular memory access patterns, which poses a difficult challenge for efficient hardware implementation.
Hardware accelerators are becoming increasingly popular for speeding up the computationally-expensive alignment and edit distance algorithms (Al
Pre-alignment filtering heuristics aim to quickly eliminate some of the dissimilar sequences before using the computationally-expensive optimal alignment algorithms. There are a few existing filtering techniques such as the Adjacency Filter (Xin et al., 2013), which is implemented for standard CPUs as part of FastHASH (Xin et al., 2013). SHD is a SIMD-friendly bit-vector filter that provides higher filtering accuracy compared to the Adjacency Filter. GRIM-Filter (Kim et al., 2018) exploits the high memory bandwidth and the logic layer of 3Dstacked memory to perform highly-parallel filtering in the DRAM chip itself. GateKeeper is designed to utilize the large amounts of parallelism offered by FPGA architectures. MAGNET (Alser et al., July 2017) shows a low number of falsely-accepted sequence pairs but its current implementation is much slower than that of SHD or Gate-Keeper. GateKeeper ) provides a high filtering speed but suffers from relatively high number of falsely-accepted sequence pairs.
Our goal in this work is to significantly reduce the time spent on calculating the optimal alignment of short sequences and maintain high filtering accuracy. To this end, we introduce Shouji 1 , a new, fast, and very accurate pre-alignment filter. Shouji is based on two key ideas: (1) A new filtering algorithm that remarkably reduces the need for computationallyexpensive banded optimal alignment by rapidly excluding dissimilar sequences from the optimal alignment calculation. (2) Judicious use of the parallelism-friendly architecture of modern FPGAs to greatly speed up this new filtering algorithm.
The contributions of this paper are as follows:
• We introduce Shouji, a highly-parallel and highly-accurate prealignment filter, which uses a sliding search window approach to quickly identify dissimilar sequences without the need for computationally-expensive alignment algorithms. We overcome the implementation limitations of MAGNET (Alser et al., July 2017). We build two hardware accelerator designs that adopt modern FPGA architectures to boost the performance of both Shouji and MAGNET.
• We provide a comprehensive analysis of the run time and space complexity of Shouji and MAGNET algorithms. Shouji and MAGNET are asymptomatically inexpensive and run in linear time with respect to the sequence length and the edit distance threshold.
• We demonstrate that Shouji and MAGNET significantly improve the accuracy of pre-alignment filtering by up to two and four orders of magnitude, respectively, compared to GateKeeper and SHD.
• We demonstrate that our FPGA implementations of Shouji and MAGNET are two to three orders of magnitude faster than their CPU implementations. We demonstrate that integrating Shouji with five state-of-the-art aligners reduces the execution time of the sequence aligner by up to 18.8x.
METHODS
Overview
Our goal is to quickly reject dissimilar sequences with high accuracy such that we reduce the need for the computationally-costly alignment step. To this end, we propose the Shouji algorithm to achieve highly-accurate filtering. Then, we accelerate Shouji by taking advantage of the parallelism of FPGAs to achieve fast filtering operations. The key filtering strategy of Shouji is inspired by the pigeonhole principle, which states that if E items are distributed into E+1 boxes, then one or more boxes would remain empty. In the context of pre-alignment filtering, this principle provides the following key observation: if two sequences differ by E edits, then the two sequences should share at least a single common subsequence (i.e., free of edits) and at most E+1 non-overlapping common subsequences, where E is the edit distance threshold. With the existence of at most E edits, the total length of these non-overlapping common subsequences should not be less than m-E, where m is the sequence length. Shouji employs the pigeonhole principle to decide whether or not two sequences are potentially similar. Shouji finds all the non-overlapping subsequences that exist in both sequences. If the total length of these common subsequences is less than m-E, then there exist more edits than the allowed edit distance threshold, and hence Shouji rejects the two given sequences. Otherwise, Shouji accepts the two sequences. Next, we discuss the details of Shouji.
Shouji Pre-alignment Filter
Shouji identifies the dissimilar sequences, without calculating the optimal alignment, in three main steps.
(1) The first step is to construct what we call a neighborhood map that visualizes the pairwise matches and mismatches between two sequences given an edit distance threshold of E characters.
(2) The second step is to find all the non-overlapping common subsequences in the neighborhood map using a sliding search window approach.
(3) The last step is to accept or reject the given sequence pairs based on the length of the found matches. If the length of the found matches is small, then Shouji rejects the input sequence pair.
Building the Neighborhood Map
We present in Fig. 1 an example of a neighborhood map for two sequences, where a pattern P differs from a text T by three edits. The entry N [i, j] is set to zero if the i th character of the pattern matches the j th character of the text. Otherwise, it is set to one. The way we build our neighborhood map ensures that computing each of its entries is independent of every other, and thus the entire map can be computed all at once in a parallel fashion. Hence, our neighborhood map is well suited for highly-parallel computing platforms Seshadri et al., 2017). Note that in sequence alignment algorithms, computing each entry of the dynamic programming matrix depends on the values of the immediate left, upper left, and upper entries of its own. Different from "dot plot" or "dot matrix" (visual representation of the similarities between two closely similar genomic sequences) that is used in FASTA/FASTP (Lipman and Pearson, 1985), our neighborhood map computes only necessary diagonals near the main diagonal of the matrix (e.g., seven diagonals shown in Fig. 1).
Identifying the Diagonally-Consecutive Matches
The key goal of this step is to accurately find all the non-overlapping common subsequences shared between a pair of sequences. The accuracy of finding these subsequences is crucial for the overall filtering accuracy, as the filtering decision is made solely based on total subsequence length. With the existence of E edits, there are at most E+1 non-overlapping common subsequences (based on the pigeonhole principle) shared between a pair of sequences. Each non-overlapping common subsequence is represented as a streak of diagonally-consecutive zeros in the neighborhood map (as highlighted in yellow in Fig. 1). These streaks of diagonally-consecutive zeros are distributed along the diagonals of the neighborhood map without any prior information about their length or number. One way of finding these common subsequences is to use a brute-force approach, j 1 2 3 4 5 6 7 8 9 10 11 12 which examines all the streaks of diagonally-consecutive zeros that start at the first column and selects the streak that has the largest number of zeros as the first common subsequences. It then iterates over the remaining part of the neighborhood map to find the other common subsequences. However, this brute-force approach is infeasible for highly-optimized hardware implementation as the search space is unknown at design time. Shouji overcomes this issue by dividing the neighborhood map into equalsize parts. We call each part a search window. Limiting the size of the search space from the entire neighborhood map to a search window has three key benefits. (1) It helps to provide a scalable architecture that can be implemented for any sequence length and edit distance threshold. (2) Downsizing the search space into a reasonably small sub-matrix with a known dimension at design time limits the number of all possible permutations of each bit-vector to 2 n , where n is the search window size. This reduces the size of the look-up tables (LUTs) required for an FPGA implementation and simplifies the overall design. (3) Each search window is considered as a smaller sub-problem that can be solved independently and rapidly with high parallelism. Shouji uses a search window of 4 columns wide, as we illustrate in Fig. 1. We need m search windows for processing two sequences, each of which is of length m characters. Each search window overlaps with its next neighboring search window by 3 columns. This ensures covering the entire neighborhood map and finding all the common subsequences regardless of their starting location. We select the width of each search window to be 4 columns to guarantee finding the shortest possible common subsequence, which is a single match located between two mismatches (i.e., '101'). However, we observe that the bit pattern '101' is not always necessarily a part of the correct alignment (or the common subsequences). For example, the bit pattern '101' exists once as a part of the correct alignment in Fig.1, but it also appears five times in other different locations that are not included in the correct alignment. To improve the accuracy of finding the diagonally-consecutive matches, we increase the length of the diagonal vector to be examined to four bits. We also experimentally evaluate different search window sizes in Supplementary Materials, Section 6.1. We find that a search window size of 4 columns provides the highest filtering accuracy without falsely-rejecting similar sequences.
G G T G C A G A G C T C G G T G A G A G
Shouji finds the diagonally-consecutive matches that are part of the common subsequences in the neighborhood map in two main steps.
Step 1: For each search window, Shouji finds a 4-bit diagonal vector that has the largest number of zeros. Shouji greedily considers this vector as a part of the common subsequence as it has the least possible number of edits (i.e., 1's). Finding always the maximum number of matches is necessary to avoid overestimating the actual number of edits and eventually preserving all similar sequences. Shouji achieves this step by comparing the 4 bits of each of the 2E+1 diagonal vectors within a search window and selects the 4-bit vector that has the largest number of zeros. In the case where two 4-bit subsequences have the same number of zeros, Shouji breaks the ties by selecting the first one that has a leading zero. Then, Shouji slides the search window by a single column (i.e., step size = 1 column) towards the last bottom right entry of the neighborhood map and repeats the previous computations. Thus, Shouji performs "Step 1" m times using m search windows, where m is the sequence length.
Step 2: The last step is to gather the results found for each search window (i.e., 4-bit vector that has the largest number of zeros) and construct back all the diagonally-consecutive matches. For this purpose, Shouji maintains a Shouji bit-vector of length m that stores all the zeros found in the neighborhood map as we illustrate in Fig. 1. For each sliding search window, Shouji examines if the selected 4-bit vector maximizes the number of zeros in the Shouji bit-vector at the same corresponding location. If so, Shouji stores the selected 4-bit vector in the Shouji bit-vector at the same corresponding location. This is neces-sary to avoid overestimating the number of edits between two given sequences. The common subsequences are represented as streaks of consecutive zeros in the Shouji bit-vector.
Filtering out Dissimilar Sequences
The last step of Shouji is to calculate the total number of edits (i.e., ones) in the Shouji bit-vector. Shouji examines if the total number of ones in the Shouji bit-vector is greater than E. If so, Shouji excludes the two sequences from the optimal alignment calculation. Otherwise, Shouji considers the two sequences similar within the allowed edit distance threshold and allows their optimal alignment to be computed using optimal alignment algorithms. The Shouji bit-vector represents the differences between two sequences along the entire length of the sequence, m. However, Shouji is not limited to end-to-end edit distance calculation. Shouji is also able to provide edit distance calculation in local and glocal (semi-global) fashion. For example, achieving local edit distance calculation requires ignoring the ones that are located at the two ends of the Shouji bit-vector. We present an example of local edit distance between two sequences of different length in Supplementary Materials, Section 8. Achieving glocal edit distance calculation requires excluding the ones that are located at one of the two ends of the Shouji bit-vector from the total count of the ones in the Shouji bit-vector. This is important for correct pre-alignment filtering for global, local, and glocal alignment algorithms. We provide the pseudocode of Shouji and discuss its computational complexity in Supplementary Materials, Section 6.2. We also present two examples of applying the Shouji filtering algorithm in Supplementary Materials, Section 8.
Accelerator Architecture
Our second aim is to substantially accelerate Shouji, by leveraging the parallelism of FPGAs. In this section, we present our hardware accelerator that is designed to exploit the large amounts of parallelism offered by modern FPGA architectures (Aluru and Jammula, 2014;Herbordt et al., 2007;Trimberger, 2015). We then outline the implementation of Shouji to be used in our accelerator design. Fig. 2 shows the hardware architecture of the accelerator. It contains a user-configurable number of filtering units. Each filtering unit provides pre-alignment filtering independently from other units. The workflow of the accelerator starts with transmitting the sequence pair to the FPGA through the fastest communication medium available on the FPGA board (i.e., PCIe). The sequence controller manages and provides the necessary input signals for each filtering unit in the accelerator. Each filtering unit requires two sequences of the same length and an edit distance threshold. The result controller gathers the output result (i.e., a single bit of value '1' for similar sequences and '0' for dissimilar sequences) of each filtering unit and transmits them back to the host side in the same order as their sequences are transmitted to the FPGAs.
The host-FPGA communication is achieved using RIFFA 2.2 (Jacobsen et al., 2015). To make the best use of the available resources in the FPGA chip, our algorithm utilizes the operations that are easily supported on an FPGA, such as bitwise operations, bit shifts, and bit count. To build the neighborhood map on the FPGA, we use the observation that the main diagonal can be implemented using a bitwise XOR operation between the two given sequences. The upper E diagonals can be implemented by gradually shifting the pattern (P) to the right-hand direction and then performing bitwise XOR with the text (T). This allows each character of P to be compared with the right-hand neighbor characters (up to E characters) of its corresponding character of T. The lower E diagonals can be implemented in a way similar to the upper E diagonals, but here the shift oper-ation is performed in the left-hand direction. This ensures that each character of P is compared with the left-hand neighbor characters (up to E characters) of its corresponding character of T.
We also build an efficient hardware architecture for each search window of the Shouji algorithm. It quickly finds the number of zeros in each 4-bit vector using a hardware look-up table that stores the 16 possible permutations of a 4-bit vector along with the number of zeros for each permutation. We present the block diagram of the search window architecture in Supplementary Materials, Section 6.3. Our hardware implementation of the Shouji filtering unit is independent of the specific FPGA-platform as it does not rely on any vendor-specific computing elements (e.g., intellectual property cores). However, each FPGA board has different resources and hardware capabilities that can directly or indirectly affect the performance and the data throughput of the design. The maximum data throughput of the design and the available FPGA resources determine the number of filtering units in the accelerator. Thus, if, for example, the memory bandwidth is saturated, then increasing the number of filtering units would not improve performance.
RESULTS
In this section, we evaluate (1) the filtering accuracy, (2) the FPGA resource utilization, (3) the execution time of Shouji, our hardware implementation of MAGNET (Alser et al., July 2017), GateKeeper , and SHD , (4) the benefits of the pre-alignment filters together with state-of-the-art aligners, and (5) the benefits of Shouji together with state-of-the-art read mappers. As we mention in Section 1, MAGNET leads to a small number of falsely-accepted sequence pairs but suffers from poor performance. We comprehensively explore this algorithm and provide an efficient and fast hardware implementation of MAGNET in Supplementary Materials, Section 7. We run all experiments using a 3.6 GHz Intel i7-3820 CPU with 8 GB RAM. We use a Xilinx Virtex 7 VC709 board (Xilinx, 2014) to implement our accelerator architecture (for both Shouji and MAGNET). We build the FPGA design using Vivado 2015.4 in synthesizable Verilog.
Dataset Description
Our experimental evaluation uses 12 different real datasets. Each dataset contains 30 million real sequence pairs. We obtain three different read sets (ERR240727_1, SRR826460_1, and SRR826471_1) of the whole human genome that include three different read lengths (100 bp, 150 bp, and 250 bp). We download these three read sets from EMBL-ENA (www.ebi.ac.uk/ena). We map each read set to the human reference genome (GRCh37) using the mrFAST (Alkan et al., 2009) mapper. We obtain the human reference genome from the 1000 Genomes Project (Consortium, 2012). For each read set, we use four different maximum numbers of allowed edits using the -e parameter of mrFAST to generate four real datasets. Each dataset contains the sequence pairs that are generated by the mrFAST mapper before the read alignment step. This enables us to measure the effectiveness of the filters using both aligned and unaligned sequences over a wide range of edit distance thresholds. We summarize the details of these 12 datasets in Supplementary Materials, Section 9. For the reader's convenience, when referring to these datasets, we number them from 1 to 12 (e.g., set_1 to set_12). We use Edlib (Šošić and Šikić, 2017) to generate the ground truth edit distance value for each sequence pair.
Filtering Accuracy
We evaluate the accuracy of a pre-alignment filter by computing its false accept rate and false reject rate. We first assess the false accept rate of Shouji, MAGNET (Alser et al., July 2017), SHD , and GateKeeper across different edit distance thresholds and datasets. The false accept rate is the ratio of the number of dissimilar sequences that are falsely-accepted by the filter and the number of dissimilar sequences that are rejected by the optimal sequence alignment algorithm. We aim to minimize the false accept rate to maximize that number of dissimilar sequences that are eliminated. In Fig. 3, we provide the false accept rate of the four filters across our 12 datasets and edit distance thresholds of 0% to 10% of the sequence length (we provide the exact values in Section 10 in Supplementary Materials). Based on Fig. 3, we make four key observations. (1) We observe that Shouji, MAGNET, SHD, and GateKeeper are less accurate in examining the low-edit sequences (i.e., datasets 1, 2, 5, 6, 9, and 10) than the highedit sequences (i.e., datasets 3, 4, 7, 8, 11, and 12). 100% 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Set_1
Set_2 Set_3 Set_4 False Accept Rate
Edit distance threshold (characters) and dataset number
GateKeeper Shouji MAGNET SHD 0% 10% 20% 30% 40% 50% 60% 70% 80% 90%
100% 0 1 3 4 6 7 9 10 12 13 15 0 1 3 4 6 7 9 10 12 13 15 0 1 3 4 6 7 9 10 12 13 15 0 1 3 4 6 7 9 10 12 13 15
Set_5
Set_6 Set_7 Set_8 False Accept Rate (2) SHD and GateKeeper become ineffective for edit distance thresholds of greater than 8% (E=8), 5% (E=7), and 3% (E=7) for sequence lengths of 100, 150, and 250 characters, respectively. This causes them to examine each sequence pair unnecessarily twice (i.e., once by GateKeeper or SHD and once by the alignment algorithm). (3) For high-edit datasets, Shouji provides up to 17.2x, 73x, and 467x (2.4x, 2.7x, and 38x for low-edit datasets) smaller false accept rate compared to GateKeeper and SHD for sequence lengths of 100, 150, and 250 characters, respectively. (4) MAGNET shows up to 1577x, 3550x, and 25552x lower false accept rates for high-edit datasets (3.5x, 14.7x, and 135x for low-edit datasets) compared to GateKeeper and SHD for sequence lengths of 100, 150, and 250 characters, respectively. MAGNET also shows up to 205x, 951x, and 16760x lower false accept rates for high-edit datasets (2.7x, 10x, and 88x for low-edit datasets) over Shouji for sequence lengths of 100, 150, and 250 characters, respectively.
Edit distance threshold (characters) and dataset number
We conclude that Shouji and MAGNET 1) maintain a very low rate of falsely-accepted dissimilar sequences and 2) significantly improve the accuracy of pre-alignment filtering by up to two and four orders of magnitude, respectively, compared to GateKeeper and SHD.
Second, we assess the false reject rates of pre-alignment filters in Supplementary Materials, Section 10. We demonstrate that Shouji, SHD and GateKeeper all have a 0% false reject rate. We also observe that MAGNET falsely-rejects correct sequence pairs, which is unacceptable for a reliable filter. Hence, we conclude that Shouji is the most effective pre-alignment filter, with a low false accept rate and a zero false reject rate.
Data Throughput and Resource Analysis
The operating frequency of our FPGA accelerator is 250 MHz. At this frequency, we observe a data throughput of nearly 3.3 GB/s, which corresponds to ~13.3 billion bases per second. This nearly reaches the peak throughput of 3.64 GB/s provided by the RIFFA (Jacobsen et al., 2015) communication channel that feeds data into the FPGA using Gen3 4-lane PCIe. We examine the FPGA resource utilization of Shouji, MAGNET, and GateKeeper filters. SHD is implemented in C with Intel SSE instructions and cannot be directly implemented on an FPGA. We examine the FPGA resource utilization for two commonly used edit distance thresholds, 2% and 5% of the sequence length, as reported in (Ahmadi et al., 2012;Hatem et al., 2013;Xin et al., 2015). The VC709 FPGA chip contains 433,200 slice LUTs (look-up tables) and 866,400 slice registers (flip-flops). Table 1 lists the FPGA resource utilization for a single filtering unit. We make three main observations. (1) The design for a single MAGNET filtering unit requires about 10.5% and 37.8% of the available LUTs for edit distance thresholds of 2 and 5, respectively. Hence, MAGNET can process 8 and 2 sequence pairs concurrently for edit distance thresholds of 2 and 5, respectively, without violating the timing constraints of our accelerator. (2) The design for a single Shouji filtering unit requires about 15x-21.9x fewer LUTs compared to MAGNET. This enables Shouji to achieve more parallelism over the MAGNET design as it can have 16 filtering units within the same FPGA chip. (3) GateKeeper requires about 26.9x-53x and 1.7x-2.4x fewer LUTs compared to MAGNET and Shouji, respectively. Gate-Keeper can also examine 16 sequence pairs at the same time.
We conclude that the FPGA resource usage is correlated with the filtering accuracy. For example, the least accurate filter, GateKeeper, occupies the least FPGA resources. Yet, Shouji has very low FPGA resource usage.
Filtering Speed
We analyze the execution time of MAGNET and Shouji compared to SHD and GateKeeper . We evaluate Gate-Keeper, MAGNET, and Shouji using a single FPGA chip and run SHD using a single CPU core. SHD supports a sequence length of up to only 128 characters (due to the SIMD register size). To ensure as fair a comparison as possible, we allow SHD to divide the long sequences into batches of 128 characters, examine each batch individually, and then sum up the results. In Table 2, we provide the execution time of the four prealignment filters using 120 million sequence pairs under sequence lengths of 100 and 250 characters. We make four key observations. (1) Shouji's execution time is as low as that of GateKeeper , and 2x-8x lower than that of MAGNET. This observation is in accord with our expectation and can be explained by the fact that MAGNET has more resource overhead that limits the number of filtering units on an FPGA. Yet Shouji is up to two orders of magnitude more accurate than GateKeeper (as we show earlier in Section 3.2). (2) Shouji is up to 28x and 335x faster than SHD using one and 16 filtering units, respectively. (3) MAGNET is up to 28x and 167.5x faster than SHD using one and 8 filtering units, respectively. As we present in Supplementary Materials, Section 12, the hardware-accelerated versions of Shouji and MAGNET provide up to three orders of magnitude of speedup over their functionally-equivalent CPU implementations.
We conclude that Shouji is extremely fast and accurate. Shouji's performance also scales very well over a wide range of both edit distance thresholds and sequence lengths.
Effects of Pre-Alignment Filtering on Sequence Alignment
We analyze the benefits of integrating our proposed pre-alignment filter (and other filters) with state-of-the-art aligners. Table 3 presents the effect of different pre-alignment filters on the overall alignment time. We select five best-performing aligners, each of which is designed for a different type of computing platform. We use a total of 120 million real sequence pairs from our previously-described four datasets (set_1 to set_4) in this analysis. We evaluate the actual execution time of Edlib (Šošić and Šikić, 2017) and are not open-source and not available to us. Therefore, we scale the reported number of computed entries of the dynamic programming matrix in a second (i.e., GCUPS) as follows: 120,000,000 / (GCUPS / 100 2 ). We make three key observations. (1) The execution time of Edlib (Šošić and Šikić, 2017) reduces by up to 18.8x, 16.5x, 13.9x, and 5.2x after the addition of Shouji, MAGNET, GateKeeper, and SHD, respectively, as a pre-alignment filtering step. We also observe a very similar trend for Parasail (Daily, 2016) combined with each of the four prealignment filters.
(2) Aligners designed for FPGAs and GPUs follow a different trend than that we observe in the CPU aligners. We observe that FPGASW ( 14.5x, 14.2x, and 17.9x, respectively. This is up to 1.35x, 1.4x, and 85x more than the effect of MAGNET, GateKeeper, and SHD on the end-to-end alignment time. (3) We observe that if the execution time of the aligner is much larger than that of the pre-alignment filter (which is the case for Edlib, Parasail, and GSWABE for E=5 characters), then MAGNET provides up to 1.3x more end-to-end speedup over Shouji. This is expected as MAGNET produces a smaller false accept rate compared to Shouji. However, unlike MAGNET, Shouji provides a 0% false reject rate. We conclude that among the four pre-alignment filters, Shouji is the best-performing prealignment filter in terms of both speed and accuracy. Integrating Shouji with an aligner leads to strongly positive benefits and reduces the aligner's total execution time by up to 18.8x.
Effects of Pre-Alignment Filtering on the Read Mapper
After confirming the benefits of integrating Shouji with sequence alignment algorithms, we now evaluate the overall benefits of integrating
Shouji with the mrFAST (v. 2.6.1) mapper (Alkan et al., 2009) and BWA-MEM (Li, 2013). Table 4 summarizes the effect of Shouji on the overall mapping time, when all reads from ERR240727_1 (100 bp) are mapped to
GRCh37 with an edit distance threshold of 2% and 5%. We also provide the total execution time breakdown in Table 15 in the Supplementary Materials. We make two observations. (1) The mapping time of mrFAST reduces by a factor of up to 5 after adding Shouji as the pre-alignment step.
(2) Integrating Shouji with BWA-MEM, without optimizing the mapper, shows less benefit than integrating Shouji with mrFAST (up to 1.07x reduction in the overall mapping time). This is due to the fact that BWA-MEM generates a low number of pairs that require verification using the read aligner. We believe by changing the mapper to work better with Shouji, we can achieve larger speedups. We leave this for future work.
DISCUSSION AND FUTURE WORK
We demonstrate that the concept of pre-alignment filtering provides substantial benefits to the existing and future sequence alignment algorithms. Accelerated sequence aligners that offer different strengths and features are frequently introduced. Many of these efforts either simplify the scoring function, or only take into account accelerating the computation of the dynamic programming matrix without supporting the backtracking step. Shouji offers the ability to make the best use of existing aligners without sacrificing any of their capabilities, as it does not modify or replace the alignment step. As such, we hope that it catalyzes the adoption of specialized pre-alignment accelerators in genome sequence analysis. However, the use of specialized hardware chips may discourage users who are not necessarily fluent in FPGAs. This concern can be alleviated in at least two ways. First, the Shouji accelerator can be integrated more closely inside the sequencing machines to perform real-time pre-alignment filtering concurrently with sequencing (Lindner et al., 2016). This allows a significant reduction in total genome analysis time. Second, cloud computing offers access to a large number of advanced FPGA chips that can be used concurrently via a simple user-friendly interface. However, such a scenario requires the development of privacy-preserving pre-alignment filters due to privacy and legal concerns (Salinas and Li, 2017). Our next efforts will focus on exploring privacy-preserving real-time pre-alignment filtering.
Another potential target of our research is to explore the possibility of accelerating optimal alignment calculations for longer sequences (few tens of thousands of characters) (Senol et al., 2018) using pre-alignment filtering. Longer sequences pose two challenges. First, we need to transfer more data to the FPGA chip to be able process a single pair of sequences which is mainly limited by the data transfer rate of the communication link (i.e., PCIe). Second, typical edit distance threshold used for sequence alignment is 5% of the sequence length. For considerably long sequences, edit distance threshold is around few hundreds of characters. For a large edit distance threshold, each character of a given sequence is compared to a large number of neighboring characters of the other given sequence. This makes the short matches (e.g., a single zero or two consecutive zeros) to occur more frequently in the diagonal vectors, which would negatively affect the accuracy of Shouji. We will investigate this effect and explore new prealignment filtering approaches for the sequencing data produced by thirdgeneration sequence machines.
CONCLUSION
In this work, we propose Shouji, a highly-parallel and accurate pre-alignment filtering algorithm accelerated on a specialized hardware platform.
The key idea of Shouji is to rapidly and accurately eliminate dissimilar sequences without calculating banded optimal alignment. Our hardwareaccelerated version of Shouji provides, on average, three orders of magnitude speedup over its functionally-equivalent CPU implementation. Shouji improves the accuracy of pre-alignment filtering by up to two orders of magnitude compared to the best-performing existing pre-alignment filter, GateKeeper. The addition of Shouji as a pre-alignment step significantly reduces the alignment time of state-of-the-art aligners by up to 18.8x, leading to the fastest alignment mechanism that we know of.
References
Ahmadi In Fig. 4, we experimentally evaluate the effect of different window sizes on the false accept rate of Shouji. We observe that as we increase the window size, the rate of dissimilar sequences that are accepted by Shouji decreases. This is because individual matches (i.e., single zeros) are usually useless and they are not necessarily part of the common subsequences. As we increase the search window size, we are ignoring these individual matches and instead we only look for longer streaks of consecutive zeros. We also observe that a window size of 4 columns provides the lowest false accept rate (i.e., the highest accuracy).
Fig. 4: The effect of the window size on the rate of the falsely-accepted sequences (i.e., dissimilar sequences that are considered as similar ones by Shouji filter). We observe that a window width of 4 columns provides the highest accuracy. We also observe that as window size increases beyond 4 columns, more similar sequences are rejected by
Shouji, which should be avoided.
The Shouji Algorithm and Its Analysis
We provide the Shouji algorithm along with analysis of its computational complexity (asymptotic run time and space complexity). Shouji divides the problem of finding the common subsequences into at most m subproblems, as described in Algorithm 1 (line 9). Each subproblem examines each of the 2E+1 bit-vectors and finds the 4-bit subsequence that has the largest number of zeros within the sliding window (line 13 to line 23). Once found, Shouji also compares the found subsequence with its corresponding subsequence in the Shouji bit-vector and stores the subsequence that has more zeros in the Shouji bit-vector (line 24). Now, let c be a constant representing the run time of examining a subsequence of 4 bits long. Then, the time complexity of the Shouji algorithm is as follows:
TShouji(m) = c.m .(2E+2)
(2) This demonstrates that the Shouji algorithm runs in linear time with respect to the sequence length and edit distance threshold. The Shouji algorithm maintains 2E+1 diagonal bit-vectors and an additional auxiliary bit-vector (i.e., the Shouji bit-vector) for each two given sequences. The space complexity of the Shouji algorithm is as follows:
DShouji(m) = m .(2E+2)
(3) Hence, the Shouji algorithm requires linear space with respect to the sequence length and edit distance threshold. Next, we describe the hardware implementation details of the Shouji filter.
Hardware Implementation
We present the FPGA chip layout for our hardware accelerator in Fig. 5. As we illustrated in the main manuscript, Section 2.3, we implement the first step of our Shouji algorithm, building neighborhood map, using shift registers and bitwise XOR operations. The second step of the Shouji algorithm is identifying the diagonally-consecutive matches. This key step involves finding the 4-bit vector that has the largest number of zeros. For each search window, there are 2E+1 diagonal bit-vectors and an additional Shouji bit-vector. To enable the computation to be performed in a parallel fashion, we build 2E+2 counters. As presented in Fig. 5, each counter counts the number of zeros in a single bit-vector. The counter takes four bits as input and generates three bits that represent the number of zeros within the window. Each counter requires three 4-input LUTs, as each LUT has a single output signal. In total, we need 6E+6 4-input LUTs to build a single search window. All bits of the counter output are generated at the same time, as the propagation delay through an FPGA look-up table is independent of the implemented function (Xilinx, November 17, 2014). The comparator is responsible for selecting the 4-bit subsequence that maximizes the number of consecutive matches based on the output of each counter and the Shouji bitvector. Finally, the selected 4-bit subsequence is then stored in the Shouji bit-vector at the same corresponding location. Step 3: Filtering out Dissimilar Sequences
MAGNET Filter
First, we provide the MAGNET (Alser et al., July 2017) algorithm and describe its main filtering mechanism. Second, we analyze the computational complexity of the MAGNET algorithm. Third, we provide details about the hardware implementation of the MAGNET algorithm.
Overview
MAGNET (Alser et al., July 2017) is another filter that uses a divide-and-conquer technique to find all the E+1 common subsequences, if any, and sum up their length. By calculating their total length, we can estimate the total number of edits between the two given sequences. If the total length of the E+1 common subsequences is less than m-E, then there exist more common subsequences than E+1 that are associated with more edits than allowed. If so, then MAGNET excludes the two given sequences from optimal alignment calculation. We present the algorithm of MAGNET in Algorithm 3.
Algorithm 3: MAGNET Comments
Input: text (T), pattern (P), edit distance threshold (E). Output: 1 (Similar/Alignment is needed) / 0 (Dissimilar/Alignment is not needed EXEN(N, 1, m, E, MAGNET, 1);
Step 2 -Step 4 10: // Function CZ() returns number of zeros 11: if CZ(MAGNET) ≥ m-E then return 1; else return 0;
Step 5: Filtering out Dissimilar Sequences Step 1
0's Counter
Step 2 Step 3 subsequence based on its length (Algorithm 4, lines 2-11). MAGNET evaluates if the length of the global longest subsequence is less than ⌈( − )/( + 1)⌉, then the two sequences contain more edits than allowed, which cause the common subsequences to be shorter (i.e., each edit results in dividing the sequence pair into more common subsequences). If so, then the two sequences are rejected (Algorithm 4, lines 12-13). Otherwise, MAGNET stores the length of the global longest subsequence to be used towards calculating the total length of all E+1 common subsequences. The lower bound equality occurs when all edits are equispaced and all E+1 subsequences are of the same length.
(3) Encapsulation. The next step is essential to preserve the original edit (or edits) that causes a single common sequence to be divided into smaller subsequences. MAGNET penalizes the found subsequence by two edits (one for each side). This is achieved by excluding from the search space of all bit-vectors the indices of the found subsequence in addition to the index of the surrounding single bit from both left and right sides (Algorithm 4, lines [14][15][16][17]. (4) Divide-and-Conquer Recursion. In order to locate the other E non-overlapping subsequences, MAGNET applies a divide-and-conquer technique where we decompose the problem of finding the non-overlapping common subsequences into two subproblems. While the first subproblem focuses on finding the next long subsequence that is located on the right-hand side of the previously found subsequence in the first extraction step (Algorithm 4, line 15), the second subproblem focuses on the other side of the found subsequence (Algorithm 4, line 17). Each subproblem is solved by recursively repeating all the three steps mentioned above, but without evaluating again the length of the longest subsequence. MAGNET applies two early termination methods that aim to reduce the execution time of the filter. The first method is evaluating the length of the longest subsequence in the first recursion call (Algorithm 4, lines 12-13). The second method is limiting the number of the subsequences to be found to at most E+1, regardless of their actual number for the given sequence pair (Algorithm 4, line 1).
(5) Filtering out Dissimilar Sequences. Once after the termination, if the total length of all found common subsequences is less than m-E, then the two sequences are rejected. Otherwise, they are considered to be similar and the alignment can be measured using sophisticated alignment algorithms.
Algorithm 4: EXEN function Comments
Analysis of the MAGNET Algorithm
We analyze the asymptotic run time and space complexity of the MAGNET algorithm. MAGNET applies a divide-and-conquer technique that divides the problem of finding the common subsequences into two subproblems in each recursion call. In the first recursion call, the extracted common subsequence is of length at least = ⌈( − ) ( + 1) ⁄ ⌉ bases. This reduces the problem of finding the common subsequences from m to at most m-a, which is further divided into two subproblems: a left subproblem and a right subproblem. For the sake of simplicity, we assume that the size of the left and the right subproblems decreases by a factor of b and c, respectively, as follows: m = + 2 + / + / (4) The addition of 2 bases is for the encapsulation bits added at each recursion call. Now, let TMAGNET(m) be the time complexity of MAGNET algorithm, for identifying non-overlapping subsequences. If it takes O(km) time to find the global longest subsequence and divide the problem into two subproblems, where k = 2E+1 is the number of bit-vectors, we get the following recurrence equation:
TMAGNET(m) = TMAGNET(m/b) + TMAGNET(m/c) + O(km)
(5) Given that the early termination condition of MAGNET algorithm restricts the recursion depth as follows:
Recursion tree depth = ⌈ 6 ( + 1)⌉ − 1 (6) Solving the recurrence in (5) using (4) and (6)
where f is a fractional number satisfies the following range: 1≤f<2. This in turn demonstrates that the MAGNET algorithm runs in linear time with respect to the sequence length and edit distance threshold and hence it is computationally inexpensive. The space complexity of the MAGNET algorithm is as follows:
DMAGNET(m) = DMAGNET(m/b) + DMAGNET(m/c) + (km+m)
≈ ( + ) (8) Hence, MAGNET algorithm requires linear space with respect to the read length and edit distance threshold. Next, we describe the hardware implementation details of MAGNET filter.
Hardware Implementation
We outline the challenges that are encountered in implementing the MAGNET filter to be used in our accelerator design. Implementing the MAGNET algorithm on an FPGA is more challenging than implementing the Shouji algorithm due to the random location and variable length of each of the E+1 common subsequences. Verilog-2011 imposes two challenges on our architecture as it does not support variable-size partial selection and indexing of a group of bits from a vector (McNamara, 2001). In particular, the first challenge lies in excluding the extracted common subsequence along with its encapsulation bits from the search space of the next recursion call. The second challenge lies in dividing the problem into two subproblems, each of which has an unknown size at design time. To address these limitations and tackle the two design challenges, we keep the problem size fixed at each recursion call. We exclude the longest found subsequence from the search space by amending all bits of all 2E+1 bit-vectors that are located within the indices (locations) of the encapsulation bits to '1's. This ensures that we exclude the longest found subsequence and its corresponding location in all other bitvectors during the subsequent recursion calls. We build the MAGNET accelerator using the same FPGA board as that used for Shouji for a fair comparison.
Examples of Applying the Shouji and MAGNET algorithms
In this section, we provide three examples of applying the Shouji and MAGNET filtering algorithms to different sequence pairs. In Fig. 6, we set the edit distance threshold to 4 in these examples. The diagonal vectors of the neighborhood map are horizontally presented in the same order of the diagonal vectors for a better illustration. In the first two examples ( Fig. 6(a) and Fig. 6(b)), we observe that MAGNET is highly accurate in providing the exact location of the edits in the MAGNET bit-vector. This is due to two main reasons. First, MAGNET finds the exact length of each common subsequence by performing multiple individual iteration for each common subsequence. Second, it manually encapsulates each found longest subsequence of consecutive zeros by ones, which ensures to maintain the edits in the MAGNET bit-vector. On the contrary, Shouji uses overlapping search windows to detect segments of consecutive zeros. If two segments of consecutive zeros are overlapped within a single search window, then the edit between the two segments is sometimes eliminated by the overlapping zeros of the two segments as shown in Fig. 6(a). Pairwise alignment can be performed as a global alignment, where two sequences of the same length are aligned end-to-end, or a local alignment, where subsequences of the two given sequences are aligned. It can also be performed as a semi-global alignment (called glocal), where the entirety of one sequence is aligned towards one of the ends of the other sequence. To ensure correct pre-alignment filtering and avoid rejecting a correct alignment, prealignment filter needs to consider counting the number of edits in a similar way to that of optimal alignment algorithm. This means that if the optimal alignment algorithm performs local alignment, then the pre-alignment filter should also perform local edit distance calculation. This can be achieved by not considering the leading and trailing edits in the total count of edits between two given sequences. Fig 6(a) and Fig. 6(b) show examples of global prealignment filtering. Fig 6(c) shows an example of local pre-alignment filtering, where the two given sequences have different lengths. While Shouji is conceptually able to perform local pre-alignment and glocal pre-alignment filtering, such support is not currently implemented in our public release of Shouji (https://github.com/CMU-SAFARI/Shouji). The current implementation of Shouji performs only global pre-alignment filtering that requires the text and reference sequences to be of the same length.
9
Dataset Description Table 5 provides the configuration used for the -e parameter of mrFAST (Alkan et al., 2009) for each of the 12 datasets. We use Edlib (Šošić and Šikić, 2017) to assess the number of similar (i.e., having edits fewer than or equal to the edit distance threshold) and dissimilar (i.e., having more edits than the edit distance threshold) pairs for each of the 12 datasets across different user-defined edit distance thresholds. We provide these details for set 1, set 2, set 3, and set 4 in Table 6. We provide the same details for set 5, set 6, set 7, and set 8 in Table 7 and for set 9, set 10, set 11, and set 12 in Table 8. ,345,842 28,654,158 441,927 29,558,073 44,565 29,955,435 18 29,999,982 2 3,266,455 26,733,545 1,073,808 28,926,192 108,979 29,891,021 24 29,999,976 3 5,595,596 24,404,404 2,053,181 27,946,819 206,903 29,793,097 27 29,999,973 4 7,825,272 22,174,728 3,235,057 26,764,943 334,712 29,665,288 29 29,999,971 5 9,821,308 20,178,692 4,481,341 25,518,659 490,670 29,509,330 34 29,999,966 6 11,650,490 18,349,510 5,756,432 24,243,568 675,357 29,324,643 83 29,999,917 7 13,407,801 16,592,199 7,091,373 22,908,627 891,447 29,108,553 177 29,999,823 8 15,152,501 14,847,499 8,531,811 21,468,189 1,151,447 28,848,553 333 29,999,667 9 16,894,680 13,105,320 10,102,726 19,897,274 1,469,996 28,530,004 711 29,999,289 10 18,610,897 11,389,103 11,807,488 18,192,512 1,868,827 28,131,173 1,627 29,998,373 ,821,308 20,178,692 11,195,124 0 11,195,124 0 2,916,838 0 6,452,280 0 6 11,650,490 18,349,510 13,781,651 0 13,781,651 0 3,406,303 4 9,373,309 0 7 13,407,801 16,592,199 14,283,519 0 14,283,519 0 4,026,433 19 11,113,616 0 8 15,152,501 14,847,499 13,814,295 0 13,814,295 0 4,745,672 27 11,990,529
Evaluating the Number of Falsely-Accepted and Falsely-Rejected Pairs Using Single End and Paired End Reads
We assess the accuracy of Shouji using both single end and paired end reads. We first map 3' reads from ERR240727.fastq (i.e., reads from ERR240727_2.fastq) to the human reference genome (GRCh37) using mrFAST (Alkan et al., 2009) with an edit distance threshold of 2. We then use the first 30 million read-reference pairs that are produced by mrFAST before performing alignment to examine the filtering accuracy of Shouji. In Table 12, we show the number of falsely-accepted and falsely-rejected pairs of Shouji using these 30 million pairs over different edit distance thresholds. Generating the read-reference pairs in this way allows us to examine the filtering accuracy of Shouji using both aligned (i.e., pairs that have edits no more than the allowed edit distance threshold) and unaligned (i.e., pairs that have edits more than the allowed edit distance threshold) pairs. We use the same method to generate set_1 from ERR240727_1.fastq, as we describe in Section 3.1 in the main manuscript. We observe that the accuracy of Shouji using 3' reads from ERR240727.fastq remains almost the same as that of Shouji when we use 5' reads from ERR240727.fastq (which we show in Table 9 when we use set_1). Next, we map both 5' reads and 3' reads from ERR240727.fastq to the human reference genome using the mrFAST mapper in paired end mode. We then use the first 30 million read-reference pairs that are produced by mrFAST before performing alignment to examine the filtering accuracy of Shouji. In Table 13, we show the number of falsely-accepted and falsely-rejected pairs of Shouji using these 30 million pairs. We observe the results are similar when using paired end reads as when using single end reads. Based on Table 12 and Table 13, we conclude that the evaluation of our pre-alignment filter does not depend on the paired end sequencing or paired end reads. Similarly with any dynamic programming sequence alignment algorithm, Shouji always examines a single reference segment with a single read individually and independently from the way this pair is generated. The read mapper is responsible for generating the read-reference pairs that must be verified using a dynamic programming sequence alignment algorithm. Shouji examines these pairs (before using the computationally-expensive sequence alignment algorithms) regardless of the algorithm (e.g., single end read mapping or paired end read mapping) used to generate these pairs.
FPGA Acceleration of Shouji and MAGNET
We analyze the benefits of accelerating the CPU implementation of our pre-alignment filters Shouji and MAGNET using FPGA hardware. As we show in Table 14, our hardware accelerators are two to three orders of magnitude faster than the equivalent CPU implementations of Shouji and MAGNET.
Execution time breakdown of Read Mapping combined with Shouji
We provide the total runtime breakdown of mrFAST (v. 2.6.1) (Alkan et al., 2009) and BWA-MEM (Li, 2013) with Shouji as a pre-alignment filter. We break down the execution time of read mapping with Shouji into 1) read-reference pair generation time, 2) Shouji filtering time, 3) Shouji pre-processing time, 4) Shouji transfer time, and 5) dynamic programming alignment time. The sum of these five runtime values provides the total execution time of read mapping with Shouji as a pre-alignment filter (8 th column of Table 15 entitled total execution time). We provide the total execution time breakdown of mrFAST (v. 2.6.1 that includes FastHASH (Xin et al., 2013)) (Alkan et al., 2009) and BWA-MEM (Li, 2013) with Shouji compared to the baseline (i.e., the last column of Table 15 represents the runtime of mrFAST and BWA-MEM without Shouji) in Table 15. We map all reads from ERR240727_1 (100 bp) to GRCh37 with an edit distance threshold of 2% and 5%. Based on Table 15, we make the following key observation: the dynamic programming alignment time drops by a factor of 4-24 (the 7 th column of Table 15 compared with the 10 th column of Table 15) after integrating Shouji with read mapping as a pre-alignment step. We conclude that the ability of Shouji to accelerate read mapping scales very well over a wide range of edit distance threshold values.
Kawam et al., 2017; Aluru and Jammula, 2014; Ng et al., 2017; Sandes et al., 2016).
Fig. 1 :
1Neighborhood map (N) and the Shouji bit-vector, for text T = GGTGCAGAGCTC, and pattern P = GGTGAGAGTTGT for E=3. The three common subsequences (i.e., GGTG, AGAG, and T) are highlighted in yellow. We use a search window of size 4 columns (two examples of which are highlighted in red) with a step size of a single column. Shouji searches diagonally within each search window for the 4-bit vector that has the largest number of zeros. Once found, Shouji examines if the found 4-bit vector maximizes the number of zeros at the corresponding location of the 4-bit vector in the Shouji bit-vector. If so, then Shouji stores this 4-bit vector in the Shouji bit-vector at its corresponding location.
Fig. 2 :
2Overview of our hardware accelerator architecture. The filtering units can be replicated as many times as possible based on the resources available on the FPGA.
Fig. 3 :
3The false accept rate of Shouji, MAGNET, SHD and GateKeeper across 12 real datasets. We use a wide range of edit distance thresholds (0%-10% of the sequence length) for sequence lengths of (a) 100, (b) 150, and (c) 250.
Parasail (Daily, 2016) on our machine. However, FPGASW (Fei et al., 2018), CUDASW++ 3.0 (Liu et al., 2013), and GSWABE (Liu and Schmidt, 2015)
Fei et al., 2018), CUDASW++ 3.0 (Liu et al., 2013), and GSWABE (Liu and Schmidt, 2015) are faster alone than with SHD (Xin et al., 2015) incorporated as the pre-alignment filtering step. Shouji, MAGNET, and GateKeeper (Alser et al., 2017) still significantly reduce the overall execution time of both FPGA and GPU based aligners. Shouji reduces the overall alignment time of FPGASW (Fei et al., 2018), CUDASW++ 3.0 (Liu et al., 2013), and GSWABE (Liu and Schmidt, 2015) by factors of up to
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(2013) Accelerating read mapping with FastHASH, BMC genomics, 14, S13. the Effect of Different Window Sizes on the Accuracy of the Shouji Algorithm.
Algorithm 2 :
2CZ (count zeros) function Function: CZ() counts the number of occurrences of zeros. Input: bit-vector D. Output: number of occurrences of zeros. 1: count ← 0; 2: for i ← 1 to length(D) do 3: if D[i] == 0 then 4: count ← count + 1; 5: return count; Fig. 5: FPGA chip layout for Shouji and block diagram of the search window scheme implemented in a Xilinx VC709 FPGA for a single filtering unit.
Finding the common subsequences involves four main steps. (1) Building the neighborhood map. Similar to Shouji, MAGNET starts with building the 2E+1 diagonal bit-vectors of the neighborhood map for the two given sequences (Algorithm 3, lines 2-6).(2) Extraction. Each diagonal bit-vector nominates its local longest subsequence of consecutive zeros. Among all nominated subsequences, a single subsequence is selected as a global longestm m m Text . . . . . .
by applying the recursion-tree method provides a loose upper-bound to the time complexity as follows: TMAGNET(m) = ( ) .
Fig. 6 :
6Examples of applying the Shouji and MAGNET filtering algorithms to three different sequence pairs, where the edit distance threshold is set to 4. We present the content of the neighborhood map along with the Shouji and MAGNET bit-vectors. In (a) and (b), we apply Shouji and MAGNET algorithms starting from the leftmost column towards the rightmost column (end-to-end) to perform global pre-alignment filtering. In (c), we ignore the ones that are located at the two ends of the final bit-vector to perform local prealignment filtering.
15 :
15Total execution time breakdown (in seconds) of mrFAST and BWA-MEM with and without Shouji, for an edit distance threshold of 2% and 5%. The green shaded columns represent the processing time spent by each step of the original read mapper (without Shouji). The orange and blue shaded columns represent the processing time spent by each step of the accelerated read mapper (with the addition of Shouji as a pre-alignment step). The orange shaded columns represent the processing time spent by Shouji on the FPGA board and the host CPU.
The neighborhood map, N, is a binary m by m matrix, where m is the sequence length. Given a text sequence T[1…m], a pattern sequence P[1…m], and an edit distance threshold E, the neighborhood map represents the comparison result of the i th character of P with the j th character of T, where i and j satisfy 1 ≤ i ≤ m and i-E ≤ j ≤ i+E. The entry N[i, j] of the neighborhood map can be calculated as follows:[ , ] = (
0,
if [ ] = [ ]
1,
if [ ] ≠ [ ]
Shouji bit-vector:. . . .Last bottom
right entry
Search window # 1
Search window # 2
Search window # 3
Search window # 4
Search window # 5
Search window # 6
Search window # 7
Search window # 8
Table 1 :
1FPGA resource usage for a single filtering unit of Shouji, MAGNET, and GateKeeper, for a sequence length of 100 and under different edit distance thresholds. We highlight the best value in each column.Filter
E
Single Filtering Unit
Max. No. of
Filtering Units
Slice LUT
Slice Register
Shouji
2
0.69%
0.01%
16
5
1.72%
0.01%
16
MAGNET
2
10.50%
0.8%
8
5
37.80%
2.30%
2
GateKeeper
2
0.39%
0.01%
16
5
0.71%
0.01%
16
Table 2 :
2Execution time (in seconds) of FPGA-based GateKeeper, MAGNET, Shouji, and CPU-based SHD under different edit distance thresholds and sequence lengths. We use set_1 to set_4 for a sequence length of 100 and set_9 to set_12 for a sequence length of 250. We provide the performance results for both a single filtering unit and the maximum number of filtering units (in parentheses). Execution time, in seconds, for a single filtering unit. b Execution time, in seconds, for maximum filtering units. c The number of filtering units. d Theoretical results based on the resource utilization and data throughput.E
GateKeeper
MAGNET
Shouji
SHD
Sequence Length = 100
2 2.89 a (0.18 b , 16 c ) 2.89 (0.36, 8)
2.89 (0.18, 16)
60.33
5
2.89 (0.18, 16)
2.89 (1.45, 2)
2.89 (0.18, 16)
67.92
Sequence Length = 250
5
5.78 (0.72, 8)
5.78 (2.89 d , 2)
5.78 (0.72 d , 8) 141.09
15
5.78 (0.72, 8)
5.78 (5.78 d , 1)
5.78 (0.72 d , 8) 163.82
a
Table 3 :
3End-to-end execution time (in seconds) for several state-ofthe-art sequence alignment algorithms, with and without pre-alignment filters (Shouji, MAGNET, GateKeeper, and SHD) and across different edit distance thresholds.E
Edlib
w/ Shouji w/ MAGNET w/ GateKeeper w/ SHD
2
506.66
26.86
30.69
36.39
96.54
5
632.95
147.20
106.80
208.77
276.51
E Parasail
w/ Shouji w/ MAGNET w/ GateKeeper w/ SHD
2
1310.96
69.21
78.83
93.87
154.02
5
2044.58
475.08
341.77
673.99
741.73
E FPGASW
w/ Shouji w/ MAGNET w/ GateKeeper w/ SHD
2
11.33
0.78
1.04
0.99
61.14
5
11.33
2.81
3.34
3.91
71.65
E CUDASW++ 3.0 w/ Shouji w/ MAGNET w/ GateKeeper w/ SHD
2
10.08
0.71
0.96
0.90
61.05
5
10.08
2.52
3.13
3.50
71.24
E GSWABE w/ Shouji w/ MAGNET w/ GateKeeper w/ SHD
2
61.86
3.44
4.06
4.60
64.75
5
61.86
14.55
11.75
20.57
88.31
Table 4 :
4Overall mrFAST and BWA-MEM mapping time (in sec-
onds) with and without Shouji, for an edit distance threshold of 2%
and 5%.
E
# pairs to be
verified
# pairs rejected
by Shouji
map. time
w/o Shouji
mapping time
w/ Shouji
mrFAST
2
40,859,970
30,679,795
242.1s 195.4s (1.2x)
5
874,403,170
764,688,027
2532s 504.6s (5.0x)
BWA-MEM
2
653,543
585,036
668.1s 626.9s (1.07x)
2 *
8,209,193
7,847,125
670.1s 625.8s (1.07x)
5
660,901
593,247
695.1s 655.8s (1.06x)
5 *
8,542,937
8,186,550
696.1s 652.7s (1.07x)
* We configure BWA-MEM to report all secondary alignments using -a.
Table 5 :
5Benchmark illumina-like datasets (read-reference pairs). We map each read set to the human reference genome in order to generate
four datasets using different mappers' edit distance thresholds (using the -e parameter).
Accession no.
ERR240727_1
SRR826460_1
SRR826471_1
Sequence Length
100
150
250
HTS
Illumina HiSeq 2000
Illumina HiSeq 2000
Illumina HiSeq 2000
Dataset
Set_1 Set_2 Set_3 Set_4
Set_5 Set_6 Set_7 Set_8
Set_9 Set_10 Set_11 Set_12
mrFAST -e
2
3
5
40
4
6
10
70
8
12
15
100
Amount of Edits
Low-edit
High-edit
Low-edit
High-edit
Low-edit
High-edit
Table 6 :
6Details of our first four datasets (set 1, set 2, set 3, and set 4). We use Edlib to benchmark the accepted (i.e., aligned) pairs and the
rejected (i.e., unaligned) pairs for edit distance thresholds of E=0 up to E=10 edits.
Dataset
Set_1
Set_2
Set_3
Set_4
E
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
0
381,901
29,618,099
124,531
29,875,469
11,989
29,988,011
11
29,999,989
1
1
Table 7 :
7Details of our second four datasets (set_5, set_6, set_7, and set_8). We report the accepted and the rejected pairs for edit distance
thresholds of E=0 up to E=15 edits.
Dataset
Set_5
Set_6
Set_7
Set_8
E
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
0
1,440,497
28,559,503
248,920
29,751,080
444
29,999,556
201
29,999,799
1
1,868,909
28,131,091
324,056
29,675,944
695
29,999,305
327
29,999,673
3
2,734,841
27,265,159
481,724
29,518,276
927
29,999,073
444
29,999,556
4
3,457,975
26,542,025
612,747
29,387,253
994
29,999,006
475
29,999,525
6
5,320,713
24,679,287
991,606
29,008,394
1,097
29,998,903
529
29,999,471
7
6,261,628
23,738,372
1,226,695
28,773,305
1,136
29,998,864
546
29,999,454
9
7,916,882
22,083,118
1,740,067
28,259,933
1,221
29,998,779
587
29,999,413
10
8,658,021
21,341,979
2,009,835
27,990,165
1,274
29,998,726
612
29,999,388
12
10,131,849
19,868,151
2,591,299
27,408,701
1,701
29,998,299
710
29,999,290
13
10,917,472
19,082,528
2,923,699
27,076,301
2,146
29,997,854
796
29,999,204
15
12,646,165
17,353,835
3,730,089
26,269,911
3,921
29,996,079
1,153
29,998,847
Table 8 :
8Details of our third four datasets (set_9, set_10, set_11, and set_12). We report the accepted and the rejected pairs for edit distance thresholds of E=0 up to E=25 edits.Dataset
Set_9
Set_10
Set_11
Set_12
E
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
Accepted
Rejected
0
707,517
29,292,483
43,565
29,956,435
4,389
29,995,611
49
29,999,951
2
1,462,242
28,537,758
88,141
29,911,859
8,970
29,991,030
163
29,999,837
5
1,973,835
28,026,165
119,100
29,880,900
12,420
29,987,580
301
29,999,699
7
2,361,418
27,638,582
145,290
29,854,710
15,405
29,984,595
375
29,999,625
10
3,183,271
26,816,729
205,536
29,794,464
22,014
29,977,986
472
29,999,528
12
3,862,776
26,137,224
257,360
29,742,640
27,817
29,972,183
520
29,999,480
15
4,915,346
25,084,654
346,809
29,653,191
37,710
29,962,290
575
29,999,425
17
5,550,869
24,449,131
409,978
29,590,022
44,225
29,955,775
623
29,999,377
20
6,404,832
23,595,168
507,177
29,492,823
54,650
29,945,350
718
29,999,282
22
6,959,616
23,040,384
572,769
29,427,231
62,255
29,937,745
842
29,999,158
25
7,857,750
22,142,250
673,254
29,326,746
74,761
29,925,239
1,133
29,998,867
Table 9 :
9Details of evaluating the number of falsely-accepted sequence pairs (FA) and falsely-rejected sequence pairs (FR) of Shouji,
MAGNET, GateKeeper, and SHD using four datasets, set_1, set_2, set_3, and set_4, with a read length of 100 bp.
Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
381,901
29,618,099
10 0
0 0
963,941 0
0 0
1
1,345,842
28,654,158
783,185 0
783,185 0
800,099 0
333,320 0
2
3,266,455
26,733,545
2,704,128 0
2,704,128 0
1,876,518 0
1,283,004 0
3
5,595,596
24,404,404
5,237,529 0
5,237,529 0
2,428,301 0
2,674,876 0
4
7,825,272
22,174,728
8,231,507 0
8,231,507 0
2,662,902
1
4,399,886 0
5
9
Table 10 :
10Details of evaluating the number of falsely-accepted sequence pairs (FA) and falsely-rejected sequence pairs (FR) of Shouji, MAGNET, GateKeeper, and SHD using four datasets, set_5, set_6, set_7, and set_8, with a read length of 150 bp.Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
1,440,497 28,559,503
0
0
0
0
428,412
0
0 0
1
1,868,909 28,131,091
173,573
0
173,573
0
156,891
0
113,519
0
3
2,734,841 27,265,159
2,080,279
0 2,080,279
0
725,873
0 1,539,365
0
4
3,457,975 26,542,025
4,023,762
0 4,023,762
0 1,064,344
0 3,042,831
0
6
5,320,713 24,679,287
9,258,602
0 9,258,602
0 1,430,272
0 6,025,592
0
7
6,261,628 23,738,372
12,481,853
0 12,481,853
0 1,532,024
2
8,219,336
0
9
7,916,882 22,083,118
22,076,837
0 22,076,837
0 1,874,734 20 14,568,337
0
10 8,658,021 21,341,979
21,341,979
0 21,341,979
0 2,194,275 10 16,920,389
0
12 10,131,849 19,868,151
19,868,151
0 19,868,151
0 3,294,672 42 18,270,597
0
13 10,917,472 19,082,528
19,082,528
0 19,082,528
0 4,066,617 46 18,095,207
0
15 12,646,165 17,353,835
17,353,835
0 17,353,835
0 5,810,797 62 16,993,568
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
248,920 29,751,080
0
0
0
0
75,136
0
0 0
1
324,056 29,675,944
31,406
0
31,406
0
28,456
0
20,294
0
3
481,724 29,518,276
440,577
0
440,577
0
131,460
0
309,015
0
4
612,747 29,387,253
1,023,901
0 1,023,901
0
199,248
0
718,847
0
6
991,606 29,008,394
4,165,422
0 4,165,422
0
334,729
0 2,222,934
0
7
1,226,695 28,773,305
7,137,889
0 7,137,889
0
405,052
0 3,762,706
0
9
1,740,067 28,259,933
28,215,257
0 28,215,257
0
600,124
0 10,299,935
0
10 2,009,835 27,990,165
27,990,165
0 27,990,165
0
753,866
2 13,826,393
0
12 2,591,299 27,408,701
27,408,701
0 27,408,701
0 1,336,246 10 17,542,652
0
13 2,923,699 27,076,301
27,076,301
0 27,076,301
0 1,835,774 19 18,371,563
0
15 3,730,089 26,269,911
26,269,911
0 26,269,911
0 3,354,276 33 19,528,254
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
444 29,999,556
0
0
0
0
251
0
0 0
1
695 29,999,305
104
0
104
0
77
0
94
0
3
927 29,999,073
191
0
191
0
68
0
180
0
4
994 29,999,006
643
0
643
0
53
0
421
0
6
1,097 29,998,903
47,924
0
47,924
0
57
0
19,097
0
7
1,136 29,998,864
175,481
0
175,481
0
74
0
70,540
0
9
1,221 29,998,779
29,595,345
0 29,595,345
0
461
0
857,547
0
10
1,274 29,998,726
29,998,726
0 29,998,726
0
1,017
0 1,829,338
0
12
1,701 29,998,299
29,998,299
0 29,998,299
0
4,218
0 4,893,299
0
13
2,146 29,997,854
29,997,854
0 29,997,854
0
8,620
0 6,955,205
0
15
3,921 29,996,079
29,996,079
0 29,996,079
0
31,783
0 12,854,488
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
201 29,999,799
0
0
0
0
126
0
0 0
1
327 29,999,673
58
0
58
0
42
0
43
0
3
444 29,999,556
90
0
90
0
35
0
83
0
4
475 29,999,525
267
0
267
0
28
0
137
0
6
529 29,999,471
18,110
0
18,110
0
25
0
6,259
0
7
546 29,999,454
79,418
0
79,418
0
27
0
27,092
0
9
587 29,999,413
29,698,666
0 29,698,666
0
108
0
404,742
0
10
612 29,999,388
29,999,388
0 29,999,388
0
231
0
935,486
0
12
710 29,999,290
29,999,290
0 29,999,290
0
965
0 2,514,950
0
13
796 29,999,204
29,999,204
0 29,999,204
0
2,018
0 3,693,298
0
15
1,153 29,998,847
29,998,847
0 29,998,847
0
8,448
0 8,034,737
0
Pre-alignment Filter
Edlib
SHD
GateKeeper
MAGNET
Shouji
Set_8
Set_7
Set_6
E
Read Aligner
Set_5
Table 11 :
11Details of evaluating the number of falsely-accepted sequence pairs (FA) and falsely-rejected sequence pairs (FR) of Shouji, MAGNET, GateKeeper, and SHD using four datasets, set_9, set_10, set_11, and set_12, with a read length of 250 bp.Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
707,517 29,292,483
0
0
0
0
479,104
0
0 0
2 1,462,242 28,537,758
238,368
0
238,368
0
143,066
0
174,366
0
5 1,973,835 28,026,165
1,546,126
0 1,546,126
0
226,864
0 1,071,218
0
7 2,361,418 27,638,582
3,933,916
0 3,933,916
0
347,819
1
2,775,419
0
10 3,183,271 26,816,729
26,816,729
0 26,816,729
0
624,927
1
6,669,084
0
12 3,862,776 26,137,224
26,137,224
0 26,137,224
0
825,468
9 11,147,373
0
15 4,915,346 25,084,654
25,084,654
0 25,084,654
0 1,066,633 14 18,406,823
0
17 5,550,869 24,449,131
24,449,131
0 24,449,131
0 1,235,999 23 20,971,826
0
20 6,404,832 23,595,168
23,595,168
0 23,595,168
0 1,695,351 35 22,223,170
0
22 6,959,616 23,040,384
23,040,384
0 23,040,384
0 2,241,984 42 22,271,215
0
25 7,857,750 22,142,250
22,142,250
0 22,142,250
0 3,514,515 54 21,849,454
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
43,565 29,956,435
0
0
0
0
28,540
0
0 0
2
88,141 29,911,859
13,092
0
13,092
0
8,367
0
11,238
0
5
119,100 29,880,900
113,106
0
113,106
0
14,685
0
77,095
0
7
145,290 29,854,710
364,611
0
364,611
0
24,919
0
227,073
0
10
205,536 29,794,464
29,794,464
0 29,794,464
0
45,768
0
782,844
0
12
257,360 29,742,640
29,742,640
0 29,742,640
0
63,557
2
2,195,021
0
15
346,809 29,653,191
29,653,191
0 29,653,191
0
92,443
1
7,573,911
0
17
409,978 29,590,022
29,590,022
0 29,590,022
0
116,740
1 11,603,069
0
20
507,177 29,492,823
29,492,823
0 29,492,823
0
165,502
2 16,075,487
0
22
572,769 29,427,231
29,427,231
0 29,427,231
0
217,274
6 19,167,498
0
25
673,254 29,326,746
29,326,746
0 29,326,746
0
376,323
7 24,778,497
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
4,389 29,995,611
0
0
0
0
2,933
0
0 0
2
8,970 29,991,030
1,405
0
1,405
0
890
0
1,173
0
5
12,420 29,987,580
12,185
0
12,185
0
1,704
0
8,489
0
7
15,405 29,984,595
41,555
0
41,555
0
2,644
0
24,946
0
10
22,014 29,977,986
29,977,986
0 29,977,986
0
4,759
0
145,053
0
12
27,817 29,972,183
29,972,183
0 29,972,183
0
6,729
1
833,703
0
15
37,710 29,962,290
29,962,290
0 29,962,290
0
9,498
0 5,088,387
0
17
44,225 29,955,775
29,955,775
0 29,955,775
0
12,134
0 9,832,285
0
20
54,650 29,945,350
29,945,350
0 29,945,350
0
18,366
0 16,815,067
0
22
62,255 29,937,745
29,937,745
0 29,937,745
0
25,411
2 20,798,178
0
25
74,761 29,925,239
29,925,239
0 29,925,239
0
44,377
1 26,094,659
0
E Accepted
Rejected
FA
FR
FA
FR
FA
FR
FA
FR
0
49 29,999,951
0
0
0
0
53
0
0 0
2
163 29,999,837
71
0
71
0
44
0
55
0
5
301 29,999,699
249
0
249
0
49
0
161
0
7
375 29,999,625
698
0
698
0
48
0
212
0
10
472 29,999,528
29,999,528
0 29,999,528
0
42
0
5,627
0
12
520 29,999,480
29,999,480
0 29,999,480
0
45
0
64,225
0
15
575 29,999,425
29,999,425
0 29,999,425
0
82
0
775,314
0
17
623 29,999,377
29,999,377
0 29,999,377
0
175
0 2,052,498
0
20
718 29,999,282
29,999,282
0 29,999,282
0
417
0 5,679,869
0
22
842 29,999,158
29,999,158
0 29,999,158
0
593
0 10,277,297
0
25
1,133 29,998,867
29,998,867
0 29,998,867
0
1,174
0 19,676,652
0
SHD
GateKeeper
MAGNET
Shouji
E
Read Aligner
Pre-alignment Filter
Edlib
Set_12
Set_11
Set_10
Set_9
Table 12 :
12Number of falsely-accepted and falsely-rejected sequence pairs of Shouji using single end reads from ERR240727_2.fastq mapped to the human reference genome. We use Edlib(Šošić and Šikić, 2017) to generate the ground truth edit distance value for each sequence pair.
Table 13 :
13Number of falsely-accepted and falsely-rejected sequence pairs of Shouji using paired end reads from ERR240727.fastq mapped to the human reference genome. We use Edlib(Šošić and Šikić, 2017) to generate the ground truth edit distance value for each sequence pair.Aligned
Unaligned
Aligned
Unaligned
Falsely-Accepted
Falsely-Rejected
0
206,252
29,793,748
206,252
29,793,748
0
0
1
1,359,165
28,640,835
1,680,722
28,319,278
321,557
0
2
3,308,445
26,691,555
4,562,146
25,437,854
1,253,701
0
3
5,673,028
24,326,972
8,290,885
21,709,115
2,617,857
0
4
7,929,996
22,070,004
12,171,061
17,828,939
4,241,065
0
5
9,920,919
20,079,081
16,051,171
13,948,829
6,130,252
0
6
11,710,868
18,289,132
20,532,091
9,467,909
8,821,223
0
7
13,409,936
16,590,064
23,845,857
6,154,143
10,435,921
0
8
15,078,030
14,921,970
26,405,117
3,594,883
11,327,087
0
9
16,727,424
13,272,576
27,901,872
2,098,128
11,174,448
0
10
18,339,408
11,660,592
28,680,484
1,319,516
10,341,076
0
E
Edlib baseline
Shouji
Aligned
Unaligned
Aligned
Unaligned
Falsely-Accepted
Falsely-Rejected
0
0
30,000,000
0
30,000,000
0
0
1
373,921
29,626,079
453,808
29,546,192
79,887
0
2
1,318,319
28,681,681
1,947,127
28,052,873
628,808
0
3
3,207,952
26,792,048
5,224,261
24,775,739
2,016,309
0
4
5,500,950
24,499,050
9,227,434
20,772,566
3,726,484
0
5
7,709,237
22,290,763
13,305,866
16,694,134
5,596,629
0
6
9,698,512
20,301,488
18,208,145
11,791,855
8,509,633
0
7
11,529,693
18,470,307
22,281,600
7,718,400
10,751,907
0
8
13,293,029
16,706,971
25,736,052
4,263,948
12,443,023
0
9
15,041,936
14,958,064
27,833,759
2,166,241
12,791,823
0
10
16,782,466
13,217,534
28,890,050
1,109,950
12,107,584
0
E
Edlib baseline
Shouji
Table 14 :
14Execution time (in seconds) of the CPU implementations of Shouji and MAGNET filters and that of their hardware-accelerated versions (using a single filtering unit). Estimated based on the resource utilization and data throughputE
Shouji-CPU
Shouji-FPGA
Speedup
MAGNET-CPU
MAGNET-FPGA
Speedup
Sequence Length = 100
2
474.27
2.89
164.11x
632.02
2.89
218.69x
5
1,305.15
2.89
451.61x
1,641.57
2.89
568.02x
Sequence Length = 250
2
1,689.09
2.89*
584.46x
5,567.62
2.89*
1,926.51x
5
6,096.61
2.89*
2,109.55x
14,328.28
2.89*
4,957.88x
*
Table
Named after a traditional Japanese door that is designed to slide open http://www.aisf.or.jp/~jaanus/deta/s/shouji.htm.
AcknowledgmentsWe thank Tuan Duy Anh Nguyen for his valuable comments on the hardware design. Conflict of Interest: none declared.
Using our 12 low-edit and high-edit datasets for three different sequence lengths, we observe that Shouji, SHD, and GateKeeper do not filter out correct sequence pairs; hence, they provide a 0% false reject rate. The reason is the way we find the common subsequences. We always look for the subsequences that have the largest number of zeros, such that we maximize the number of matches and minimize the number of edits that cause the division of one long common sequence into shorter subsequences. However, this is not the case for MAGNET. We observe that MAGNET provides a very low false reject rate of less than 0.00045% for an edit distance threshold of at least 4% of the sequence length. Magnet Shouji, Shd (xin, Evaluating the Number of Falsely-Accepted Sequence Pairs and Falsely-Rejected Sequence Pairs We evaluate the number of falsely-accepted pairs and falsely-rejected pairs for. Shouji always examines whether or not the selected 4-bit segment that has the largest number of zeros decreases the number of edits in the Shouji bit-vector before considering the 4-bit segment to be part of the common subsequences. In Fig. 7, we show an example of where MAGNET falsely considers two given sequences as dissimilar ones. while they differ by less than the edit distance threshold. This example shows that MAGNET's greedy approach of finding the common subsequences fails in finding the two common subsequences that are highlighted in blue. Instead, MAGNET finds another four shorter subsequences that result in increasing the number of mismatches in the MAGNET bit-vectorEvaluating the Number of Falsely-Accepted Sequence Pairs and Falsely-Rejected Sequence Pairs We evaluate the number of falsely-accepted pairs and falsely-rejected pairs for Shouji, MAGNET, SHD (Xin et al., 2015), and GateKeeper (Alser et al., 2017). We list the number of falsely-accepted and falsely-rejected sequences in Table 9, Table 10, and Table 11 for read lengths of 100 bp, 150 bp, and 250 bp, respectively. The false reject rate is the ratio of the number of similar sequences that are rejected (falsely-rejected pairs) by the filter and the number of similar sequences that are accepted by the optimal sequence alignment algorithm. The false reject rate should always be equal to 0%. Using our 12 low-edit and high-edit datasets for three different sequence lengths, we observe that Shouji, SHD, and GateKeeper do not filter out correct sequence pairs; hence, they provide a 0% false reject rate. The reason is the way we find the common subsequences. We always look for the subsequences that have the largest number of zeros, such that we maximize the number of matches and minimize the number of edits that cause the division of one long common sequence into shorter subsequences. However, this is not the case for MAGNET. We observe that MAGNET provides a very low false reject rate of less than 0.00045% for an edit distance threshold of at least 4% of the sequence length. This is due in large part to the greedy choice of always selecting the longest common subsequence regardless of its contribution to the total number of edits. On the contrary, Shouji always examines whether or not the selected 4- bit segment that has the largest number of zeros decreases the number of edits in the Shouji bit-vector before considering the 4-bit segment to be part of the common subsequences. In Fig. 7, we show an example of where MAGNET falsely considers two given sequences as dissimilar ones, while they differ by less than the edit distance threshold. This example shows that MAGNET's greedy approach of finding the common subsequences fails in finding the two common subsequences that are highlighted in blue. Instead, MAGNET finds another four shorter subsequences that result in increasing the number of mismatches in the MAGNET bit-vector.
An example of a falsely-rejected sequence pair using the MAGNET algorithm for an edit distance threshold of 6. The random zeros (highlighted in red) confuse the MAGNET filter, causing it to select shorter segments of random zeros instead of a longer common subsequences. 7highlighted in blueFig. 7: An example of a falsely-rejected sequence pair using the MAGNET algorithm for an edit distance threshold of 6. The random zeros (highlighted in red) confuse the MAGNET filter, causing it to select shorter segments of random zeros instead of a longer common subsequences (highlighted in blue).
Configurations In Table 16, we list the software packages that we cover in our performance evaluation, including their version numbers and function calls used. Table 16: Read aligners and pre-alignment filters used in our performance evaluations. Parasail Edlib, Bwa-Mem Shd, Edlib, Parasail, SHD, mrFAST, and BWA-MEM Configurations In Table 16, we list the software packages that we cover in our performance evaluation, including their version numbers and function calls used. Table 16: Read aligners and pre-alignment filters used in our performance evaluations. Edlib: November 5 2017
. Accepted =0. else Accepted =0;
. EDLIB_CIGAR_STANDARD. char* cigar = edlibAlignmentToCigar(resultEdlib.alignment, resultEdlib.alignmentLengthchar* cigar = edlibAlignmentToCigar(resultEdlib.alignment, resultEdlib.alignmentLength, EDLIB_CIGAR_STANDARD);
. free(cigar. free(cigar);
function = parasail_lookup_function. Parasail, Parasail: January 7 2018 function = parasail_lookup_function("nw_banded");
result = function. RefSeq, ReadLength, ReadSeq, ReadLength,10, 1, ErrorThreshold,¶sail_blosum62result = function(RefSeq, ReadLength, ReadSeq, ReadLength,10, 1, ErrorThreshold,¶sail_blosum62);
Readlength Refseq, Readseq, Readlength, Query, (result)==1){ parasail_traceback_generic. Target:", ¶sail_blosum62, result, '|', ':', '.', 50, 14, 0)if(parasail_result_is_trace(result)==1){ parasail_traceback_generic(RefSeq, ReadLength, ReadSeq, ReadLength, "Query:", "Target:", ¶sail_blosum62, result, '|', ':', '.', 50, 14, 0);
. } } Shd, compiled using g++-4.9} } SHD: November 7 2017, compiled using g++-4.9
for (k=1;k<=1+ (ReadLength/128);k++) totalEdits= totalEdits + (bit_vec_filter_sse1(read_t, ref_t. for (k=1;k<=1+ (ReadLength/128);k++) totalEdits= totalEdits + (bit_vec_filter_sse1(read_t, ref_t, length, ErrorThreshold));
/mrfast-2.6.1.0/mrfast --search human_g1k_v37.fasta --seq. mrFAST: November 29 2017 ./mrfast-2.6.1.0/mrfast --search human_g1k_v37.fasta --seq ../ERR240727_1_100bp.fastq -e 2
Extracting read-reference pairs: 1-Add the following to line 1786 of. The human reference genome can be. 2-Extract reference segment: for (n = 0; n < 100; n++) printf("%d. _msf_refGen[n + genLoc + _msf_refGenOffset -1 -leftSeqLengthThe human reference genome can be downloaded from: ftp://ftp.ncbi.nlm.nih.gov/1000genomes/ftp/technical/reference/human_g1k_v37.fasta.gz Extracting read-reference pairs: 1-Add the following to line 1786 of https://github.com/BilkentCompGen/mrfast/blob/master/MrFAST.c 2-Extract reference segment: for (n = 0; n < 100; n++) printf("%d", _msf_refGen[n + genLoc + _msf_refGenOffset -1 -leftSeqLength]);
3-Extract read sequence: printf("\t%s\n. _tmpSeq3-Extract read sequence: printf("\t%s\n", _tmpSeq);
. Bwa-Mem, BWA-MEM: November 25 2018
Report all secondary alignments: ./bwa mem -a -w 3 ../human_g1k_v37.fasta. Desktop/Filters_29_11_2016/ERR240727_1_100bp.fastq Extracting read-reference pairs: 1-Add the following code between line 166 and line. 2-Extract reference segment: for (i = 0; i < rlen; ++i) putchar. int)rseq[i]]); putchar('\t'/human_g1k_v37.fasta ../../../Desktop/Filters_29_11_2016/ERR240727_1_100bp.fastq Report all secondary alignments: ./bwa mem -a -w 3 ../human_g1k_v37.fasta ../../../Desktop/Filters_29_11_2016/ERR240727_1_100bp.fastq Extracting read-reference pairs: 1-Add the following code between line 166 and line 167 of https://github.com/lh3/bwa/blob/master/bwa.c 2-Extract reference segment: for (i = 0; i < rlen; ++i) putchar("ACGTN"[(int)rseq[i]]); putchar('\t');
Extract read sequence: for (i = 0; i < l_query; ++i) putchar("ACGTN. int)query[i]]); putchar('\n'3-Extract read sequence: for (i = 0; i < l_query; ++i) putchar("ACGTN"[(int)query[i]]); putchar('\n');
Personalized copy number and segmental duplication maps using next-generation sequencing. C Alkan, J M Kidd, T Marques-Bonet, G Aksay, F Antonacci, F Hormozdiari, J O Kitzman, C Baker, M Malig, O Mutlu, Nature genetics. 41Alkan, C., Kidd, J. M., Marques-Bonet, T., Aksay, G., Antonacci, F., Hormozdiari, F., Kitzman, J. O., Baker, C., Malig, M. and Mutlu, O. (2009) Personalized copy number and segmental duplication maps using next-generation sequencing, Nature genetics, 41, 1061-1067.
GateKeeper: a new hardware architecture for accelerating pre-alignment in DNA short read mapping. M Alser, H Hassan, H Xin, O Ergin, O Mutlu, C Alkan, Bioinformatics. 33Alser, M., Hassan, H., Xin, H., Ergin, O., Mutlu, O. and Alkan, C. (2017) GateKeeper: a new hardware architecture for accelerating pre-alignment in DNA short read mapping, Bioinformatics, 33, 3355-3363.
Magnet: Understanding and improving the accuracy of genome pre-alignment filtering. M Alser, O Mutlu, C Alkan, Transactions on Internet Research. 13Alser, M., Mutlu, O. and Alkan, C. (July 2017) Magnet: Understanding and improving the accuracy of genome pre-alignment filtering, Transactions on Internet Research 13.
Aligning sequence reads, clone sequences and assembly contigs with BWA-MEM. H Li, arXiv:1303.3997arXiv preprintLi, H. (2013) Aligning sequence reads, clone sequences and assembly contigs with BWA-MEM, arXiv preprint arXiv:1303.3997.
IEEE Standard Verilog Hardware Description Language. M Mcnamara, The Institute of Electrical and Electronics Engineers, Inc. IEEE StdMcNamara, M. (2001) IEEE Standard Verilog Hardware Description Language. The Institute of Electrical and Electronics Engineers, Inc. IEEE Std, 1364-2001.
Edlib: a C/C++ library for fast, exact sequence alignment using edit distance. M Šošić, M Šikić, Bioinformatics. 33Šošić, M. and Šikić, M. (2017) Edlib: a C/C++ library for fast, exact sequence alignment using edit distance, Bioinformatics, 33, 1394-1395.
Xilinx, Series FPGAs Configurable Logic Block User Guide. Xilinx. Xilinx (November 17, 2014) 7 Series FPGAs Configurable Logic Block User Guide. Xilinx.
Shifted Hamming Distance: A Fast and Accurate SIMD-Friendly Filter to Accelerate Alignment Verification in Read Mapping. H Xin, J Greth, J Emmons, G Pekhimenko, C Kingsford, C Alkan, O Mutlu, Bioinformatics. 31Xin, H., Greth, J., Emmons, J., Pekhimenko, G., Kingsford, C., Alkan, C. and Mutlu, O. (2015) Shifted Hamming Distance: A Fast and Accurate SIMD- Friendly Filter to Accelerate Alignment Verification in Read Mapping, Bioinformatics, 31, 1553-1560.
Accelerating read mapping with FastHASH. H Xin, D Lee, F Hormozdiari, S Yedkar, O Mutlu, C Alkan, BMC genomics. 1413Xin, H., Lee, D., Hormozdiari, F., Yedkar, S., Mutlu, O. and Alkan, C. (2013) Accelerating read mapping with FastHASH, BMC genomics, 14, S13.
| [
"https://github.com/CMU-SAFARI/Shouji",
"https://github.com/CMU-SAFARI/Shouji).",
"https://github.com/BilkentCompGen/mrfast/blob/master/MrFAST.c",
"https://github.com/lh3/bwa/blob/master/bwa.c"
] |
[
"HERMITE-HADAMARD AND SIMPSON-LIKE TYPE INEQUALITIES FOR DIFFERENTIABLE HARMONICALLY CONVEX FUNCTIONṠ IMDATİŞCAN",
"HERMITE-HADAMARD AND SIMPSON-LIKE TYPE INEQUALITIES FOR DIFFERENTIABLE HARMONICALLY CONVEX FUNCTIONṠ IMDATİŞCAN"
] | [
"\nDepartment of Mathematics\nFaculty of Arts and Sciences\nGiresun University\n28100GiresunTurkey\n"
] | [
"Department of Mathematics\nFaculty of Arts and Sciences\nGiresun University\n28100GiresunTurkey"
] | [] | In this paper, a new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like type for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given.2000 Mathematics Subject Classification. Primary 26A51; Secondary 26D15. | 10.1155/2014/346305 | [
"https://arxiv.org/pdf/1310.4851v1.pdf"
] | 119,616,510 | 1310.4851 | 7e3cb2f9697554938862552df6b4a3e07f16f148 |
HERMITE-HADAMARD AND SIMPSON-LIKE TYPE INEQUALITIES FOR DIFFERENTIABLE HARMONICALLY CONVEX FUNCTIONṠ IMDATİŞCAN
17 Oct 2013
Department of Mathematics
Faculty of Arts and Sciences
Giresun University
28100GiresunTurkey
HERMITE-HADAMARD AND SIMPSON-LIKE TYPE INEQUALITIES FOR DIFFERENTIABLE HARMONICALLY CONVEX FUNCTIONṠ IMDATİŞCAN
17 Oct 2013and phrases Harmonically convex function, Hermite-Hadamard type inequality 1
In this paper, a new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like type for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given.2000 Mathematics Subject Classification. Primary 26A51; Secondary 26D15.
Introduction
Let f : I ⊂ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following inequality
(1.1) f a + b 2 ≤ 1 b − a b a f (x)dx ≤ f (a) + f (b) 2
holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping f . Both inequalities hold in the reversed direction if f is concave.
Following inequality is well known in the literature as Simpson inequality:
Theorem 1. Let f : [a, b] → R be a four times continuously differentiable mapping on (a, b) and f (4) ∞ = sup x∈(a,b) f (4) (x) < ∞. Then the following inequality holds:
1 3 f (a) + f (b) 2 + 2f a + b 2 − 1 b − a b a f (x)dx ≤ 1 2880 f (4) ∞ (b − a) 4 .
For some results which generalize, improve and extend the Hermite-Hadamard and Simpson inequalities, we refer the reader to the recent papers (see [1,2,3,4,6,7,8] ).
In [5], the author introduced the concept of harmonically convex functions and established some results connected with the right-hand side of new inequalities similar to the inequality (1.1) for these classes of functions. Some applications to special means of positive real numbers are also given. Definition 1. Let I ⊂ R\ {0} be a real interval. A function f : I → R is said to be harmonically convex, if
(1.2) f xy tx + (1 − t)y ≤ tf (y) + (1 − t)f (x)
for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.2) is reversed, then f is said to be harmonically concave.
The following result of the Hermite-Hadamard type holds.
Theorem 2. Let f : I ⊂ R\ {0} → R be a harmonically convex function and a, b ∈ I with a < b. If f ∈ L[a, b] then the following inequalities hold
(1.3) f 2ab a + b ≤ ab b − a b a f (x) x 2 dx ≤ f (a) + f (b) 2 .
The above inequalities are sharp.
Some results connected with the right part of (1.3) was given in [5] as follows:
Theorem 3. Let f : I ⊂ (0, ∞) → R be a differentiable function on I • , a, b ∈ I with a < b, and f ′ ∈ L[a, b]. If |f ′ | q is harmonically convex on [a, b] for q ≥ 1, then f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx (1.4) ≤ ab (b − a) 2 λ 1− 1 q 1 λ 2 |f ′ (a)| q + λ 3 |f ′ (b)| q 1 q , where λ 1 = 1 ab − 2 (b − a) 2 ln (a + b) 2 4ab , λ 2 = −1 b (b − a) + 3a + b (b − a) 3 ln (a + b) 2 4ab , λ 3 = 1 a (b − a) − 3b + a (b − a) 3 ln (a + b) 2 4ab = λ 1 − λ 2 . Theorem 4. Let f : I ⊂ (0, ∞) → R be a differentiable function on I • , a, b ∈ I with a < b, and f ′ ∈ L[a, b]. If |f ′ | q is harmonically convex on [a, b] for q > 1, 1 p + 1 q = 1, then f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx (1.5) ≤ ab (b − a) 2 1 p + 1 1 p µ 1 |f ′ (a)| q + µ 2 |f ′ (b)| q 1 q , where µ 1 = a 2−2q + b 1−2q [(b − a) (1 − 2q) − a] 2 (b − a) 2 (1 − q) (1 − 2q) , µ 2 = b 2−2q − a 1−2q [(b − a) (1 − 2q) + b] 2 (b − a) 2 (1 − q) (1 − 2q) .
In this paper, we shall give some general integral inequalities connected with the left and right parts of (1.3), as a result of this, we shall obtained some new midpoint, trapezoid and Simpson like-type inequalities for differentiable harmonically convex functions.
Main results
In order to prove our main resuls we need the following lemma:
Lemma 1. Let f : I ⊂ R\ {0} → R be a differentiable function on I • and a, b ∈ I with a < b. If f ′ ∈ L[a, b] then for λ ∈ [0, 1] we have the equality (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx = ab (b − a) 2 1/2 0 λ − 2t A 2 t f ′ ab A t dt + 1 1/2 2 − λ − 2t A 2 t f ′ ab A t dt , where A t = tb + (1 − t)a.
Proof. It suffices to note that
I 1 = ab (b − a) 1 0 λ − 2t A 2 t f ′ ab A t dt = (2t − λ) f ab A t 1/2 0 − 2 1/2 0 f ab A t dt = (1 − λ) f 2ab a + b + λf (b) − 2 1/2 0 f ab A t dt.
Setting x = ab At and dx = −ab(b−a)
A 2 t dt, which gives
I 1 = (1 − λ) f 2ab a + b + λf (b) − 2ab b − a b 2ab/(a+b) f (x) x 2 dx. 4İMDATİŞCAN
Similarly, we can show that
I 2 = ab (b − a) 1 1/2 2 − λ − 2t A 2 t f ′ ab A t dt = λf (a) + (1 − λ) f 2ab a + b − 2ab b − a 2ab/(a+b) a f (x) x 2 dx.
Thus,
I 1 + I 2 2 = (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx
which is required.
Theorem 5. Let f : I ⊂ (0, ∞) → R be a differentiable function on I • , a, b ∈ I with a < b, and f ′ ∈ L[a, b]. If |f ′ | q is harmonically convex on [a, b]
for q ≥ 1 and then we have the following inequality for λ ∈ [0, 1]
(2.1) (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 C 1− 1 q 1 (λ; a, b) C 2 (λ; a, b) |f ′ (a)| q + C 3 (λ; a, b) |f ′ (b)| q 1 q +C 1− 1 q 1 (λ; b, a) C 3 (λ; b, a) |f ′ (a)| q + C 2 (λ; b, a) |f ′ (b)| q 1 q , where C 1 (λ; u, ϑ) = 1 (ϑ − u) 2 × −4 + [λ (ϑ − u) + 2u] (3u + ϑ) u (u + ϑ) + 2 ln 2u (u + ϑ) (2u + λ (ϑ − u)) 2 , C 2 (λ; u, ϑ) = 1 (ϑ − u) 3 × [λ (ϑ − u) + 4u] ln [λ (ϑ − u) + 2u] 2 2u (u + ϑ) − [λ (ϑ − u) + 2u] (5u + 3ϑ) u + ϑ + 7u + ϑ ,
and
C 3 (λ; u, ϑ) = C 1 (λ; u, ϑ) − C 2 (λ; u, ϑ), u, ϑ > 0. Proof. Let A t = tb + (1 − t)a.
From Lemma 1 and using the power mean inequality, we have
(1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 1/2 0 |λ − 2t| A 2 t f ′ ab A t dt + 1 1/2 |2 − λ − 2t| A 2 t f ′ ab A t dt ≤ ab (b − a) 2 1/2 0 |λ − 2t| A 2 t dt 1− 1 q 1/2 0 |λ − 2t| A 2 t f ′ ab A t q dt 1 q + 1 1/2 |2 − λ − 2t| A 2 t dt 1− 1 q 1 1/2 |2 − λ − 2t| A 2 t f ′ ab A t q dt 1 q . Hence, by harmonically convexity of |f ′ | q on [a, b], we have (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × 1/2 0 |λ − 2t| A 2 t dt 1− 1 q 1/2 0 |λ − 2t| t |f ′ (a)| q + (1 − t) |f ′ (b)| q A 2 t dt 1 q + 1 1/2 |2 − λ − 2t| A 2 t dt 1− 1 q 1 1/2 |2 − λ − 2t| t |f ′ (a)| q + (1 − t) |f ′ (b)| q A 2 t dt 1 q ≤ ab (b − a) 2 C 1− 1 q 1 (λ; a, b) C 2 (λ; a, b) |f ′ (a)| q + C 3 (λ; a, b) |f ′ (b)| q 1 q C 3 (λ; b, a) |f ′ (a)| q + C 2 (λ; b, a) |f ′ (b)| q 1 q .
It is easily check that
1/2 0 |λ − 2t| A 2 t dt = C 1 (λ; a, b) = 1 (b − a) 2 × −4 + [λ (b − a) + 2a] (3a + b) a (a + b) + 2 ln 2a (a + b) (2a + λ (b − a)) 2 , 6İMDATİŞCAN 1/2 0 |λ − 2t| t A 2 t dt = C 2 (λ; a, b) = 1 (b − a) 3 × [λ (b − a) + 4a] ln [λ (b − a) + 2a] 2 2a (a + b) − [λ (b − a) + 2a] (5a + 3b) a + b + 7a + b , 1/2 0 |λ − 2t| (1 − t) A 2 t dt = C 3 (λ; a, b) = C 1 (λ; a, b) − C 2 (λ; a, b), 1 1/2 |2 − λ − 2t| A 2 t dt = C 1 (λ; b, a), 1 1/2 |2 − λ − 2t| (1 − t) A 2 t dt = C 2 (λ; b, a),and 1 1/2 |2 − λ − 2t| t A 2 t dt = C 3 (λ; b, a) = C 1 (λ; b, a) − C 2 (λ; b, a).
This concludes the proof.
Corollary 1. Under the assumptions Theorem 5 with λ = 0, we have
f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1− 1 q 1 (0; a, b) C 2 (0; a, b) |f ′ (a)| q + C 3 (0; a, b) |f ′ (b)| q 1 q C 1− 1 q 1 (0; b, a) C 3 (0; b, a) |f ′ (a)| q + C 2 (0; b, a) |f ′ (b)| q 1 q , where C 1 (0; u, ϑ) = 2 (ϑ − u) 2 ln u + ϑ 2u − ϑ − u u + ϑ , C 2 (0; u, ϑ) = 1 (ϑ − u) 3 (3u + ϑ) (ϑ − u) u + ϑ + 4u ln 2u u + ϑ , C 3 (0; u, ϑ) = 1 (ϑ − u) 2 2 (u + ϑ) ϑ − u ln u + ϑ 2u − u + 3ϑ u + ϑ , u, ϑ > 0.
Corollary 2. Under the assumptions Theorem 5 with λ = 1, we have
f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1− 1 q 1 (1; a, b) C 2 (1; a, b) |f ′ (a)| q + C 3 (1; a, b) |f ′ (b)| q 1 q C 1− 1 q 1 (1; b, a) C 3 (1; b, a) |f ′ (a)| q + C 2 (1; b, a) |f ′ (b)| q 1 q , where C 1 (1; u, ϑ) = 1 (ϑ − u) 2 ϑ − u u + 2 ln 2u u + ϑ , C 2 (1; u, ϑ) = 1 (ϑ − u) 3 (3u + ϑ) ln u + ϑ 2u − 2(ϑ − u) , C 3 (1; u, ϑ) = 1 (ϑ − u) 2 u + ϑ u − u + 3ϑ ϑ − u ln u + ϑ 2u , u, ϑ > 0.
Corollary 3. Under the assumptions Theorem 5 with λ = 1/3, we have
1 3 f (a) + f (b) 2 + 2f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1− 1 q 1 (1/3; a, b) C 2 (1/3; a, b) |f ′ (a)| q + C 3 (1/3; a, b) |f ′ (b)| q 1 q C 1− 1 q 1 (1/3; b, a) C 3 (1/3; b, a) |f ′ (a)| q + C 2 (1/3; b, a) |f ′ (b)| q 1 q , where C 1 (1/3; u, ϑ) = 1 (ϑ − u) 2 (ϑ − u) (ϑ − 3u) 3u (u + ϑ) + 2 ln 18u (u + ϑ) (5u + ϑ) 2 , C 2 (1/3; u, ϑ) = 1 (ϑ − u) 3 11u + ϑ 3 ln (5u + ϑ) 2 18u (u + ϑ) + 4u(ϑ − u) 3 (u + ϑ) , C 3 (1/3; u, ϑ) = 1 (ϑ − u) 2 ϑ 2 − 4uϑ − u 2 3u (u + ϑ) + 5u + 7ϑ 3 (ϑ − u) ln 18u (u + ϑ) (5u + ϑ) 2 , u, ϑ > 0.
Theorem 6. Let f : I ⊂ (0, ∞) → R be a differentiable function on I • , a, b ∈ I with a < b, and f ′ ∈ L[a, b]. If |f ′ | q is harmonically convex on [a, b] for q > 1 and then we have the following inequality for λ ∈ [0, 1]
(2.2) (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1 p 1 (λ, p; a, b) f ′ 2ab a+b q + |f ′ (b)| q 4 1 q + C 1 p 1 (λ, p; b, a) f ′ 2ab a+b q + |f ′ (a)| q 4 1 q ,
where C 1 (λ, p; u, ϑ) = Proof. Let A t = tb + (1 − t)a. Using Lemma 1 and Hölder's integral inequality, we deduce
(1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 1/2 0 |λ − 2t| A 2 t f ′ ab A t dt + 1 1/2 |2 − λ − 2t| A 2 t f ′ ab A t dt ≤ ab (b − a) 2 1/2 0 |λ − 2t| p A 2p t dt 1 p 1/2 0 f ′ ab A t q dt 1 q (2.3) + 1 1/2 |2 − λ − 2t| p A 2p t dt 1 p 1 1/2 f ′ ab A t q dt 1 q .
Using the harmonically convexity of |f ′ | q , we obtain the following inequalities from inequality (1.3):
1/2 0 f ′ ab A t q dt ≤ 1 2 2ab b − a b 2ab a+b |f ′ (x)| q x 2 dx ≤ f ′ 2ab a+b q + |f ′ (b)| q 4 (2.4) and 1 1/2 f ′ ab A t q dt ≤ 1 2 2ab b − a 2ab a+b a |f ′ (x)| q x 2 dx ≤ f ′ 2ab a+b q + |f ′ (a)|f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1 p 1 (0, p; a, b) f ′ 2ab a+b q + |f ′ (b)| q 4 1 q + C 1 p 1 (0, p; b, a) f ′ 2ab a+b q + |f ′ (a)| q 4 1 q .
Corollary 5. Under the assumptions Theorem 6 with λ = 1, we have
f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1 p 1 (1, p; a, b) f ′ 2ab a+b q + |f ′ (b)| q 4 1 q + C 1 p 1 (1, p; b, a) f ′ 2ab a+b q + |f ′ (a)| q 4 1 q .
Corollary 6. Under the assumptions Theorem 6 with λ = 1/3, we have
1 3 f (a) + f (b) 2 + 2f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 × C 1 p 1 ( 1 3 , p; a, b) f ′ 2ab a+b q + |f ′ (b)| q 4 1 q + C 1 p 1 ( 1 3 , p; b, a) f ′ 2ab a+b q + |f ′ (a)| q 4 1 q
Theorem 7. Let f : I ⊂ (0, ∞) → R be a differentiable function on I • , a, b ∈ I with a < b, and f ′ ∈ L[a, b]. If |f ′ | q is harmonically convex on [a, b] for q > 1 and then we have the following inequality for λ ∈ [0, 1]
(2.6) (1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 4 × C 1/p 4 (λ, p) (1 − q) (1 − 2q) (b − a) 2 1/q C 5 (q; a, b) |f ′ (a)| q + C 6 (q; a, b) |f ′ (b)| q 1 q + C 6 (q; b, a) |f ′ (a)| q + C 5 (q; b, a) |f ′ (b)| q 1 q , where C 4 (λ, p) = λ p+1 + (1 − λ) p+1 p + 1 , C 5 (q; u, ϑ) = u + ϑ 2 1−2q ϑ − 3u 2 − q(ϑ − u) + u 2−2q , C 6 (q; u, ϑ) = u + ϑ 2 1−2q 3ϑ − u 2 − q(ϑ − u) + u 1−2q [u − 2ϑ + 2q(ϑ − u)]
, u, ϑ > 0 and 1/p + 1/q = 1.
10İMDATİŞCAN
Proof. Let A t = tb + (1 − t)a. Using Lemma 1 and Hölder's integral inequality, we deduce
(1 − λ) f 2ab a + b + λ f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 2 1/2 0 |λ − 2t| A 2 t f ′ ab A t dt + 1 1/2 |2 − λ − 2t| A 2 t f ′ ab A t dt ≤ ab (b − a) 2 1/2 0 |λ − 2t| p dt 1 p 1/2 0 1 A 2q t f ′ ab A t q dt 1 q (2.7) + 1 1/2 |2 − λ − 2t| p dt 1 p 1 1/2 1 A 2q t f ′ ab A t q dt 1 q .
Using the harmonically convexity of |f ′ | q , we obtain
1/2 0 1 A 2q t f ′ ab A t q dt ≤ 1/2 0 t |f ′ (a)| q + (1 − t) |f ′ (b)| q A 2q t dt = 1 2 (1 − q) (1 − 2q) (b − a) 2 × a + b 2 1−2q b − 3a 2 − q(b − a) + a 2−2q |f ′ (a)| q (2.8) + a + b 2 1−2q 3b − a 2 − q(b − a) + a 1−2q [a − 2b + 2q(b − a)] |f ′ (b)| q and 1 1/2 1 A 2q t f ′ ab A t q dt ≤ 1 1/2 t |f ′ (a)| q + (1 − t) |f ′ (b)| q A 2q t dt = 1 2 (1 − q) (1 − 2q) (b − a) 2 × b 1−2q [b − 2a − 2q(b − a)] + a + b 2 1−2q 3a − b 2 + q(b − a) |f ′ (a)| q (2.9) + a + b 2 1−2q a − 3b 2 + q(b − a) + b 2−2q |f ′ (b)| q Further, we have (2.10) 1/2 0 |λ − 2t| p dt = 1 1/2 |2 − λ − 2t| p dt = λ p+1 + (1 − λ) p+1 2 (p + 1)
A combination of (2.7)-(2.10) gives the required inequality (2.6).
Corollary 7.
Under the assumptions Theorem 7 with λ = 0, we have
f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 4 (p + 1) 1/p × 1 (1 − q) (1 − 2q) (b − a) 2 1/q C 5 (q; a, b) |f ′ (a)| q + C 6 (q; a, b) |f ′ (b)| q 1 q + C 6 (q; b, a) |f ′ (a)| q + C 5 (q; b, a) |f ′ (b)| q 1 q .
Corollary 8. Under the assumptions Theorem 7 with λ = 1, we have
f (a) + f (b) 2 − ab b − a b a f (x) x 2 dx ≤ ab (b − a) 4 (p + 1) 1/p × 1 (1 − q) (1 − 2q) (b − a) 2 1/q C 5 (q; a, b) |f ′ (a)| q + C 6 (q; a, b) |f ′ (b)| q 1 q + C 6 (q; b, a) |f ′ (a)| q + C 5 (q; b, a) |f ′ (b)| q 1 q .
Corollary 9. Under the assumptions Theorem 7 with λ = 1/3, we have
1 3 f (a) + f (b) 2 + 2f 2ab a + b − ab b − a b a f (x) x 2 dx ≤ ab (b − a)
4 (3 p+1 (p + 1))
1/p × 1 + 2 p+1 (1 − q) (1 − 2q) (b − a) 2 1/q C 5 (q; a, b) |f ′ (a)| q + C 6 (q; a, b) |f ′ (b)| q 1 q + C 6 (q; b, a) |f ′ (a)| q + C 5 (q; b, a) |f ′ (b)| q 1 q .
Some applications for special means
Let us recall the following special means of two nonnegative number a, b with b > a :
(1) The arithmetic mean
A = A (a, b) := a + b 2 .
(2) The geometric mean
G = G (a, b) := √ ab.
(3) The harmonic mean
H = H (a, b) := 2ab a + b .
(4) The Logarithmic mean
L = L (a, b) := b − a ln b − ln a .
12İMDATİŞCAN
(5) The p-Logarithmic mean
L p = L p (a, b) := b p+1 − a p+1 (p + 1)(b − a) 1 p , p ∈ R\ {−1, 0} .
(6) the Identric mean
I = I (a, b) = 1 e b b a a 1 b−a .
These means are often used in numerical approximation and in other areas. However, the following simple relationships are known in the literature:
H ≤ G ≤ L ≤ I ≤ A.
It is also known that L p is monotonically increasing over p ∈ R, denoting L 0 = I and L −1 = L. Proposition 1. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality
(1 − λ) H + λA − G 2 L ≤ ab (b − a) 2 {C 1 (λ; a, b) + C 1 (λ; b, a)} ,
where C 1 is defined as in Theorem 5.
Proof. The assertion follows from the inequality (2.1) in Theorem 5, for f :
(0, ∞) → R, f (x) = x.
Proposition 2. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality
(1 − λ) H + λA − G 2 L ≤ ab (b − a) 2 1+1/q C 1 p 1 (λ, p; a, b) + C 1 p 1 (λ, p; b, a) ,
where q > 1, 1/p + 1/q = 1 and C 1 is defined as in Theorem 6.
Proof. The assertion follows from the inequality (2.2) in Theorem 6, for f : (0, ∞) → R, f (x) = x.
Proposition 3. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality
(1 − λ) H + λA − G 2 L ≤ ab (b − a) C 1/p 4 (λ, p) 4 (1 − q) (1 − 2q) (b − a) 2 1/q × (C 5 (q; a, b) + C 6 (q; a, b)) 1 q + (C 6 (q; b, a) + C 5 (q; b, a)) 1 q ,
where q > 1, 1/p + 1/q = 1 and C 4, C 5 and C 6 are defined as in Theorem 7.
Proof. The assertion follows from the inequality (2.6) in Theorem 7, for f :
(0, ∞) → R, f (x) = x.
Proposition 4. Let 0 < a < b, λ ∈ [0, 1] and q ≥ 1. Then we have the following inequality
(1 − λ) H 2 + λA(a 2 , b 2 ) − G 2 ≤ ab (b − a) C 1− 1 q 1 (λ; a, b) [C 2 (λ; a, b)a q + C 3 (λ; a, b)b q ] 1 q +C 1− 1 q 1 (λ; b, a) [C 3 (λ; b, a)a q + C 2 (λ; b, a)b q ] 1 q ,
where C 1 , C 2 and C 3 are defined as in Theorem 5.
Proof. The assertion follows from the inequality (2.1) in Theorem 5, for f :
(0, ∞) → R, f (x) = x 2 .
Proposition 5. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality
(1 − λ) H 2 + λA(a 2 , b 2 ) − G 2 ≤ ab (b − a) 2 1/q × C 1 p 1 (λ, p; a, b)A 1 q (H q , b q ) + C 1 p 1 (λ, p; b, a)A 1 q (a q , H q ) ,
where q > 1 and 1/p + 1/q = 1.
Proof. The assertion follows from the inequality (2.2) in Theorem 6, for f :
(0, ∞) → R, f (x) = x 2 .
Proposition 6. Let 0 < a < b and λ ∈ [0, 1]. Then we have the following inequality
(1 − λ) H 2 + λA(a 2 , b 2 ) − G 2 ≤ ab (b − a) C 1/p 4 (λ, p) 2 (1 − q) (1 − 2q) (b − a) 2
1/q × (C 5 (q; a, b)a q + C 6 (q; a, b)b q ) 1 q + (C 6 (q; b, a)a q + C 5 (q; b, a)b q ) 1 q , where q > 1, 1/p + 1/q = 1 and C 4, C 5 and C 6 are defined as in Theorem 7.
Proof. The assertion follows from the inequality (2.6) in Theorem 7, for f : (0, ∞) → R, f (x) = x 2 . where q > 1, 1/p + 1/q = 1 and C 1 is defined as in Theorem 6.
Proof. The assertion follows from the inequality (2.2) in Theorem 6, for f : (0, ∞) → R, f (x) = x n+2 , n ∈ (−1, ∞) \ {0} . C 1/p 4 (λ, p) (1 − q) (1 − 2q) (b − a) 2 1/q C 5 (q; a, b)G 2q (a, A(1, ln a)) + C 6 (q; a, b)G 2q (b, A(1, ln b)) 1 q + C 6 (q; b, a)G 2q (a, A(1, ln a)) + C 5 (q; b, a)G 2q (b, A(1, ln b)) 1 q , where q > 1, 1/p + 1/q = 1 and C 5 and C 6 are defined as in Theorem 7.
Proof. The assertion follows from the inequality (2.6) in Theorem 7, for f : (0, ∞) → R, f (x) = x 2 ln x.
+ (1 − t)a)2p dt, u, ϑ > 0. and 1/p + 1/q = 1.
Corollary 4 .
4Under the assumptions Theorem 6 with λ = 0, we have
Proposition 7 .
7Let 0 < a < b, n ∈ (−1, ∞) \ {0}, λ ∈ [0, 1] and q ≥ 1. Then we have the following inequality(1 − λ) H n+2 + λA(a n+2 , b n+2 ) − G 2 .; a, b) C 2 (λ; a, b)a (n+1)q + C 3 (λ; a, b)b ; b, a) C 3 (λ; b, a)a (n+1)q + C 2 (λ; b, a)b (n+1)q 1 q ,where C 1 , C 2 and C 3 are defined as in Theorem 5.Proof. The assertion follows from the inequality (2.1) in Theorem 5, for f :(0, ∞) → R, f (x) = x n+2 , n ∈ (−1, ∞) \ {0} .Proposition 8. Let 0 < a < b and n ∈ (−1, ∞) \ {0} . Then we have the following inequality(1 − λ) H n+2 + λA(a n+2 , b n+2 ) − G 2 .L n n ≤ ab (b − a)
Proposition 9. Let 0 < a < b, λ ∈ [0, 1]. and n ∈ (−1, ∞) \ {0} . Then we have the following inequalitywhere q > 1, 1/p + 1/q = 1 and C 5 and C 6 are defined as in Theorem 7.Proof. The assertion follows from the inequality (2.6) in Theorem 7, for f :Proposition 10. Let 0 < a < b, λ ∈ [0, 1] and q ≥ 1. Then we have the following inequalitywhere C 1 , C 2 and C 3 are defined as in Theorem 5.Proof. The assertion follows from the inequality (2.1) in Theorem 5, for f : (0, ∞) → R, f (x) = x 2 ln x.Proposition 11. Let 0 < a < b, and λ ∈ [0, 1]. Then we have the following inequality(H, A(1, ln H)A(1, ln a)) ,where q > 1, 1/p + 1/q = 1 and C 1 is defined as in Theorem 6.Proof. The assertion follows from the inequality (2.2) in Theorem 6, for f : (0, ∞) → R, f (x) = x 2 ln x.Proposition 12. Let 0 < a < b, and λ ∈ [0, 1]. Then we have the following inequality(1 − λ) H 2 ln H + λA a 2 ln a, b 2 ln b − G 2 ln I ≤ ab (b − a)
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Some Inequalities for Differentiable Convex and Concave Mappings. G.-S Yang, D.-Y Hwnag, K.-L Tseng, Comput. Math. Appl. 47G.-S. Yang, D.-Y. Hwnag and K.-L. Tseng, Some Inequalities for Differentiable Convex and Concave Mappings, Comput. Math. Appl. 47 (2004) 207-216
| [] |
[
"CONSTRUCTION OF A GROUP OF AUTOMORPHISMS FOR AN INFINITE FAMILY OF GARSIDE GROUPS",
"CONSTRUCTION OF A GROUP OF AUTOMORPHISMS FOR AN INFINITE FAMILY OF GARSIDE GROUPS"
] | [
"Fabienne Chouraqui "
] | [] | [] | The structure groups of non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation provide an infinite family of Garside groups with many interesting properties. Given a non-degenerate symmetric solution, we construct for its structure group a group of automorphisms. Moreover, we show this group of automorphisms admits a subgroup that preserves the Garside properties of the structure group. In some cases, we could also prove the group of automorphisms obtained is an outer automorphism group. | null | [
"https://arxiv.org/pdf/1411.1189v1.pdf"
] | 119,572,985 | 1411.1189 | 520a448a3c5f1c44723e35633e2f4bdaef9aefd5 |
CONSTRUCTION OF A GROUP OF AUTOMORPHISMS FOR AN INFINITE FAMILY OF GARSIDE GROUPS
5 Nov 2014
Fabienne Chouraqui
CONSTRUCTION OF A GROUP OF AUTOMORPHISMS FOR AN INFINITE FAMILY OF GARSIDE GROUPS
5 Nov 2014AMS Subject Classification: 16T25, 20F36 Keywords: Group of automorphismsOuter automorphism groupSet-theoretical solution of the quantum Yang-Baxter equationGarside groupsbijective cocycle
The structure groups of non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation provide an infinite family of Garside groups with many interesting properties. Given a non-degenerate symmetric solution, we construct for its structure group a group of automorphisms. Moreover, we show this group of automorphisms admits a subgroup that preserves the Garside properties of the structure group. In some cases, we could also prove the group of automorphisms obtained is an outer automorphism group.
Introduction
The quantum Yang-Baxter equation is an equation in the field of mathematical physics and it lies in the foundation of the theory of quantum groups. Let R : V ⊗ V → V ⊗ V be a linear operator, where V is a vector space. The quantum Yang-Baxter equation is the equality R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗V ⊗V , where R ij means R acting on the ith and jth components. V. Drinfeld suggested the study of a set-theoretical solution, that is a pair (X, S) for which V is the vector space spanned by the set X and R is the linear operator induced by the mapping S : X × X → X × X. To each non-degenerate and symmetric set-theoretical solution (X, S) of the quantum Yang-Baxter equation, P.Etingof, T.Schedler and A.Soloviev associate a group G(X, S) called the structure group. Furthermore, they show that each non-degenerate symmetric solution (X, S) is in one-to-one correspondence with a quadruple (G, X, ρ, π), where G is a group, X is a set, ρ is a left action of G on X, and π is a bijective 1-cocycle of G with coefficients in Z X , where Z X is the free abelian group generated by X [14].
In this paper, given a non-degenerate symmetric solution (X, S) with | X |= n and structure group G(X, S), we use the bijective 1-cocycle π : G(X, S) → Z X and the fact that Aut(Z X ) = GL n (Z) to construct a group of automorphisms of G(X, S). Indeed, given σ ∈ GL n (Z), we define a bijection ϕ : G(X, S) → G(X, S) by ϕ = π −1 • σ • π, such that the following diagram is commutative:
G(X, S) Z X G(X, S) Z X ϕ π π σ
We find a necessary and sufficient condition on σ ensuring that ϕ = π −1 • σ • π is a homomorphism of the group G(X, S). Note that depending on the context, we consider σ ∈ Aut(Z X ) = GL n (Z) as a map or as matrix and we write the elements in Z X multiplicatively. We prove the following result and we refer to Theorem 3.6, where another equivalent condition is given and more generally to Section 3.
Theorem 1. Let (X, S) be a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation, with | X |= n. Let (G(X, S), X, •, π) be the corresponding quadruple, where G(X, S) is the structure group, • denotes the left action of G(X, S) on X by permutations extended to Z X and π : G(X, S) → Z X is the bijective 1-cocycle defined by π(x) = t x , x ∈ X and π(a 1 a 2 ) = (a −1 2 •π(a 1 )) π(a 2 ), a 1 , a 2 ∈ G(X, S). Let ℑ π to be the following set:
ℑ π = {σ ∈ GL n (Z) | A j σ = σ (f −1 j ), ∀1 ≤ j ≤ n} where (f −1 j )
and A j are the permutation matrices corresponding to the action of x j and (ϕ(x j )) −1 respectively. We define Φ π to be the following set:
(0.1) Φ π = {ϕ : G(X, S) → G(X, S), ϕ = π −1 • σ • π | σ ∈ ℑ π }
Then ℑ π ≤ GL n (Z) and Φ π ≤ Aut(G(X, S)).
There exists a strong connection between set-theoretical solutions and Garside groups [6,7], [17], [11]. Indeed, there is a one-to-one correspondence between nondegenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation and Garside groups with a particular presentation. The structure group G(X, S) of (X, S) is a Garside group that admits a Garside element of length n =| X | which is the lcm of X with respect to left and right divisibility [6]. Furthermore, G(X, S) admits a normal free abelian subgroup N of rank n, such that the finite quotient group G(X, S)/N plays the role the finite Coxeter groups play for finitetype Artin groups [8], [10]. The subgroup N is freely generated by the so-called frozen elements θ 1 , θ 2 , ..θ n . Whenever ϕ is an automorphism of the group G(X, S), ϕ ∈ Φ π , a question that arises naturally is whether ϕ preserves the Garside structure of G(X, S). The answer is negative in general, as ϕ ∈ Φ π is not necessarily an automorphism of the underlying monoid. Nevertheless, we show that Φ π admits a subgroup of automorphisms that preserves the Garside structure of G(X, S) and N ; the automorphisms in this subgroup are induced by permutation matrices in ℑ π . Theorem 2. Let (X, S) be a non-degenerate symmetric solution, with structure group G(X, S) and | X |= n. Let ∆ be the Garside element of length n and Div(∆) be the set of left (and right) divisors of ∆. Let N be the free abelian normal subgroup of G(X, S) generated by the frozen elements θ 1 , θ 2 , ..θ n . Let σ be a permutation matrix in ℑ π and let ϕ ∈ Aut(G(X, S)) be induced by σ. Then ϕ is a lengthpreserving isomorphism of the bounded lattice Div(∆) w.r to right and left divisibility. Furthermore, ϕ(N ) = N .
To some extent, we could extend the results of Theorem 2 to the subgroup of Φ π induced by the generalized permutation matrices in ℑ π . We refer to Proposition 4.6. Indecomposable solutions are the "building blocks" of the solutions and there is a parallel between the role of the indecomposable solutions and the irreducible components of a finite-type Artin group. But, while a finite-type Artin group is the direct product of its irreducible components, this is not necessarily the case for the structure group of a decomposable solution. For indecomposable solutions, the groups ℑ π and Φ π satisfy some properties that are very specific to them and in particular for the (unique) indecomposable solutions with | X |= 2, 3, we show the group Φ π is an outer automorphism group. More precisely, we prove: Theorem 3. Let (X, S) be an indecomposable non-degenerate symmetric solution with structure group G(X, S). Assume | X |= 2, 3. Then (i) ℑ π is a finite group of generalized permutation matrices in GL n (Z). (ii) Φ π has a normal subgroup of index 2 that preserves the Garside structure.
(iii) Φ π = Out(G) for | X |= 2 and Φ π ≤ Out(G) for | X |= 3.
The paper is organised as it follows. In Section 1, we introduce the structure group of a set-theoretical solution of the quantum Yang-Baxter equation, the bijective 1cocycle and its properties. In Section 2, we recall some basic material on Garside theory and we present the Garside structure specific to the structure group of a settheoretical solution of the quantum Yang-Baxter equation. In Section 3, we describe the construction of the group of automorphisms Φ π , we prove Theorem 1 and we show the invariance of Φ π under some particular changes of bijective 1-cocycle. In Section 4, we show the existence of a group of automorphisms that preserves the Garside structure of the structure group and we prove Theorem 2. In Section 5, we consider the special case of indecomposable solutions, we prove some properties of ℑ π and Φ π that are specific to general indecomposable solutions and we prove Theorem 3. In Section 6, we present an application of our construction to the Coxeter-like quotient group G(X, S)/N . At the end of the paper, we add an appendix, in which we give for completion the proofs of the properties of the bijective 1-cocycle that appear in Section 1.2.
Acknowledgment. I am very grateful to Arye Juhasz and Yuval Ginosar for fruitful discussions.
1. Set-theoretical solutions of the quantum Yang-Baxter equation 1.1. Background on set-theoretical solutions of the QYBE. We follow [14] and refer to [14], [20] and [2,3,4,18,19] for more details. Fix a finite dimensional vector space V over the field R. The Quantum Yang-Baxter Equation on V is the equality
R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗ V ⊗ V , where R : V ⊗ V → V ⊗ V
is a linear operator and R ij means R acting on the ith and jth components. A set-theoretical solution of this equation is a pair (X, S) such that X is a basis for V and S : X × X → X × X is a bijective map. The map S is defined by S(x, y) = (g x (y), f y (x)), where f x , g x : X → X are functions for all x, y ∈ X. The pair (X, S) is non-degenerate if for any x ∈ X, the functions f x and g x are bijections. It is involutive if S • S = Id X , and braided if S 12 S 23 S 12 = S 23 S 12 S 23 , where the map S ii+1 means S acting on the i-th and (i + 1)-th components of X 3 . It is said to be symmetric if it is involutive and braided. Let α : X × X → X × X be defined by α(x, y) = (y, x), and let R = α • S, then R satisfies the QYBE if and only if (X, S) is braided [14]. The solution (X, S) is the trivial solution if g x = f x = Id X , ∀x ∈ X. Definition 1.1. Let (X, S) be a non-degenerate symmetric set-theoretical solution. The structure group of (X, S) is presented by Gp X | xy = g x (y)f y (x) ; x, y ∈ X .
For each x ∈ X, there are unique y, z ∈ X such that S(x, y) = (x, y) and S(z, x) = (z, x), since g x , f x are bijective and S is involutive. So, the presentation of G(X, S) contains n(n−1) 2 defining relations, where | X |= n. Note that the structure group of the trivial solution is the free abelian group generated by X.
Example 1.2. Let X = {x 1 , x 2 , x 3 , x 4 }, and S : X × X → X × X be defined by S(x i , x j ) = (x g i (j) , x f j (i) )
where g i and f j are permutations on {1, 2, 3, 4} as follows: 3,4). The solution (X, S) is non-degenerate and symmetric and its structure group G(X, S) has the following defining relations:
g 1 = (2, 3), g 2 = (1, 4), g 3 = (1, 2, 4, 3), g 4 = (1, 3, 4, 2); f 1 = (2, 4), f 2 = (1, 3), f 3 = (1, 4, 3, 2), f 4 = (1, 2,x 1 x 2 = x 2 3 ; x 1 x 3 = x 2 x 4 ; x 2 x 1 = x 2 4 ; x 2 x 3 = x 3 x 1 ; x 1 x 4 = x 4 x 2 ; x 3 x 2 = x 4 x 1
There are four trivial relations:
x 2 1 = x 2 1 , x 2 2 = x 2 2 , x 3 x 4 = x 3 x 4 , x 4 x 3 = x 4 x 3 . Definition 1.3. [14] Let (X, S) be a non-degenerate symmetric solution. (i) A subset Y of X is an invariant subset if S(Y × Y ) ⊆ Y × Y . (ii) An invariant subset Y is non-degenerate if (Y, S | Y ×Y ) is non-degenerate and symmetric. (iii) The solution (X, S) is decomposable if X is the union of two non-empty disjoint non-degenerate invariant subsets. Otherwise, (X, S) is indecomposable.
Two solutions (X, S) and (X ′ , S ′ ) are equivalent solutions if there exists a bijection α : X → X ′ such that S ′ • α × α = α × α • S. There exists a unique indecomposable solution (X, S) (up to equivalence) for | X |= p, p a prime. Indeed, in this case, (X, S) is a permutation solution, that is all the functions f i are equal to a p-cycle f and all the functions g i are equal to f −1 . A classification of non-degenerate and symmetric solutions with X up to 8 elements, considering their decomposability and other properties is given in [14]. P.Etingof, T.Schedler and A.Soloviev show that non-degenerate symmetric solutions, up to equivalence, are in one-to-one correspondence with quadruples (G, X, ρ, π), where G is a group, X is a set, ρ is a left action of G on X, and π is a bijective 1-cocycle of G with coefficients in Z X , where Z X is the free abelian group spanned by X. Indeed, they show that the group G(X, S) is naturally a subgroup of Sym(X) ⋉Z X , such that the 1-cocycle defined by the projection G(X, S) → Z X is bijective. More precisely they define and prove the following facts:
Theorem 1.4. [14]
Let (X, S) be a non-degenerate symmetric solution of the quantum Yang-Baxter equation and G(X, S) be its structure group. Let Sym(X) be the group of permutations of X and Z X be the free abelian group spanned by X. Let the map φ f : G(X, S) → Sym(X) ⋉Z X be defined by φ f (x) = f −1
x t x , where x ∈ X and t x is the generator of Z X corresponding to x. Then (i) The assignment x → f −1
x is a left action of G(X, S) on X. (ii) Let a ∈ G(X, S) and w = t m 1 1 t m 2 2 ..t mn n ∈ Z X . Assume a acts on X via the permutation f . Then a acts on Z X in the following way: 2) ..t mn f (n) , where • denotes the extension of the left action of G(X, S)
a • t x = t f (x) and a • w = t m 1 f (1) t m 2 f(on X defined in (i) to Z X . (iii)
The map φ f is a monomorphism.
(iv) The map π : G(X, S) → Z X , defined by π(g) = w if φ f (g) = α w, with α ∈ Sym(X) and w ∈ Z X , is a bijective 1-cocycle satisfying π(a 1 a 2 ) = (a −1 2 •π(a 1 )) π(a 2 ). The product in Sym(X) ⋉Z X is defined by:
f −1 x t x f −1 y t y = f −1 x f −1 y t fy(x) t y [14]. 1.2.
Computation rules for the 1-cocycle and its inverse. We assume (X, S) is a non-degenerate symmetric set-theoretical solution with X finite of cardinality n and structure group G(X, S). We denote by Z n the free abelian group generated by X written multiplicatively. Here, we present some technical preliminary results, in particular we give some computation rules for the 1-cocycle π and its inverse π −1 that are implicit in [14] and can be easily derived from [10]. The proofs of the lemmas of this section appear in the appendix. With no loss of generality, we assume π(1) = 1.
Lemma 1.5. [14] Let T : X → X be the map defined by T (x) = f −1 x (x). Then the map T is invertible and T −1 (y) = g −1 y (y). Furthermore, f −1 x • T = T • g x , ∀x ∈ X. Note that inductively T m (x) = f −1 T m−1 (x) T m−1 (x) = f −1 T m−1 (x) f −1 T m−2 (x) ..f −1 T (x) f −1 x (x).
Lemma 1.6. Let π : G(X, S) → Z n be the bijective 1-cocycle defined in Prop. 1.4(iv). Let x i , x j ∈ X. Let t i , t j be generators of Z n corresponding to x i and x j respectively. Let u, v ∈ Z n . Then
(i) π(x −1 i ) = t −1 f −1 i (i) (ii) π −1 (t j ) = x j . (iii) π −1 (t i t j ) = x f −1 j (i) x j . (iv) π −1 (uv) = π −1 (π −1 (v) • u) π −1 (v)
Definition 1.7. Let (X, S) be a non-degenerate symmetric solution, with X finite. (i) Let x, y ∈ X. We say (X, S) satisfies the condition C, if f x • f y = Id X and g x • g y = Id X , whenever S(x, y) = (x, y). The elements xy and yx are called frozen elements of length 2 [8].
(ii) We say (X, S) is of class m, if f x f T (x) f T 2 (x) ... f T m−1 (x) = Id X , for all x ∈ X.
Remark 1.8. In Defn. 1.7(ii), we use the terminology from [10]. The formulation is not the same as Defn. 2.2 [10], but it is equivalent to it. Note that satisfying condition (C) is equivalent to being of class 2, since f x • f y = Id X implies g x • g y = Id X from Lemma 1.5 and whenever S(x, y) = (x, y), y = g −1
x (x) = f −1 x (x) and the condition C can be rewritten as f
x f g −1 x (x) = f x f T (x) = Id X .
Note also that being of class m implies T m (x) = x, for all x ∈ X. Example 1.9. In Example 1.2, T (x 1 ) = x 1 , T (x 2 ) = x 2 , T (x 3 ) = x 4 , T (x 4 ) = x 3 , since we have the four following trivial relations:
x 2 1 = x 2 1 , x 2 2 = x 2 2 , x 3 x 4 = x 3 x 4 , x 4 x 3 = x 4 x 3 .
The solution satisfies condition (C), as f 2 1 = f 2 2 = f 3 f 4 = Id X . In [8], the present author and E.Godelle show that if (X, S) is a non-degenerate symmetric set-theoretical solution, with | X |= n, that satisfies the condition C, then there is a short exact sequence 1 → N → G(X, S) → W → 1, where N is a normal free abelian subgroup of G(X, S) and W is a finite group of order 2 n . Moreover, N is freely generated by the n frozen elements of (X, S) of length two and W is a Coxeter-like group, that is a finite quotient that plays the role that Coxeter groups play for the finite-type Artin groups [8]. P.Dehornoy extends our result and using another terminology he shows that the condition C may be relaxed [10]. Indeed, he shows that if (X, S) is of class m, then for each x ∈ X there is a chain of trivial relations of the form xy 1 = xy 1 , y 1 y 2 = y 1 y 2 , y 2 y 3 = y 2 y 3 , .., y m−1 x = y m−1 x, y i ∈ X. Here, we call the element y 1 y 2 y 3 ..y m−1 x the frozen element of length m ending with x. The subgroup N , generated by the n frozen elements of length m, y 1 y 2 y 3 ..y m−1 x, xy 1 y 2 y 3 ..y m−1 , .., y 2 y 3 ..y m−1 xy 1 , is normal, free abelian of rank n and the group W defined by G(X, S)/N is finite of order m n and plays the role that Coxeter groups play for finite-type Artin groups [10]. Example 1.10. In Example 1.2, the normal subgroup N is generated by the four frozen elements of length two: x 2 1 , x 2 2 , x 3 x 4 , x 4 x 3 ; a presentation of the Coxeterlike group W is obtained by adding the following four relations:
x 2 1 = 1, x 2 2 = 1, x 3 x 4 = 1, x 4 x 3 = 1 to the presentation of G(X, S).
The following lemma is useful in the characterisation of the frozen element of length m ending with x ∈ X, where m is the class of the solution.
Lemma 1.11. Assume (X, S) is of class m. Let x ∈ X. (i) S(x, T −1 (x)) = (x, T −1 (x)) and S(T (x), x) = (T (x), x). (ii) T m−1 (x) T m−2 (x) ... T (x)
x is the frozen of length m ending with x.
In the following lemma, we give some computation rules for π −1 and show the frozen elements act trivially on Z n . It means that the normal subgroup N is contained in the kernel of the left action of G(X, S) on X with the map x → f −1
x . Lemma 1.12. Assume (X, S) is of class m. Let x ∈ X. Assume π(x) = t.
(i) π −1 (t −1 ) = (T m−1 (x)) −1 = (T −1 (x)) −1 = (g −1 x (x)) −1 (ii) For ℓ > 0, π −1 (t ℓ ) = T ℓ−1 (x) T ℓ−2 (x) ... T (x) x. (iii) π −1 (t m ) = θ, where θ is the frozen element of length m ending with x. (iv) π −1 (t ±m ) • w = w, for all w ∈ Z n . (v) π −1 (t −m ) = θ −1 . (vi) π −1 (t ±m i 1 ..t ±m i k ) • w = w, for all w ∈ Z n .
The connection between Garside groups and the QYBE
There exists a strong connection between set-theoretical solutions and Garside groups [6,7,11,17]. Indeed, in [6], we show there is a one-to-one correspondence between non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation and Garside groups with a particular presentation (see Thm. 2.4). We first begin with some basic background on Garside groups in Section 2.1 and then in Section 2.2 we present some prior results relating Garside groups and settheoretical solutions of the quantum Yang-Baxter equation.
2.1. Background on Garside monoids and Garside groups. Here, we recall some basic material on Garside theory, and refer to [11], [13] for more details. We start with some preliminaries. If M is a monoid generated by a set X, and if g ∈ M is the image of the word w by the canonical morphism from the free monoid on X onto M , then we say w represents g. A monoid M is cancellative if for every e, f, g, h in M , the equality ef g = ehg implies f = h. The element f is a right divisor (resp. a left divisor ) of g if there is an element h in M such that g = hf (resp. g = f h). The element f is a a least common multiple w.r to right-divisibility (left lcm) of g and h in M if g and h are right divisors of f and additionally if there is an element f ′ such that g and h are right divisors of f ′ , then f is right divisor of f ′ . We denote it by f = g ∨ R h. The complement at left of g on h is defined to be an element c ∈ M such that g ∨ R h = cg, whenever g ∨ R h exists. We denote it by c = g \ h. The right lcm and complement at right are defined similarly and denoted respectively by ∨ L and\. A monoid M is left noetherian (resp. right noetherian) if every sequence (g n ) n∈N of elements of M such that g n+1 is a left divisor (resp. a right divisor) of g n stabilizes.
It is noetherian if it is both left and right noetherian. An element ∆ is balanced if it has the same set of right and left divisors. In this case, we denote by Div(∆) its set of divisors. If M is a cancellative and noetherian monoid, then left and right divisibilities are partial orders on M . (
2) A Garside element of M is a balanced element ∆ with Div(∆) generating M . (3) A monoid M is a Garside monoid if M is locally Garside with a Garside element ∆ satisfying Div(∆) is finite. A group is a Garside group if it is the group of fractions of a Garside monoid.
Garside groups have been first introduced in [12]. The seminal examples are the braid groups and more generally the Artin groups of finite type. Recall that an element g = 1 in a monoid is called an atom if the equality g = f h implies f = 1 or h = 1. From the defining properties of a Garside monoid, if M is a Garside monoid, then M is generated by its set of atoms, and every atom divides every Garside element. There is no invertible element, except the trivial one, and any two elements in M have a left (resp. right) gcd and a left (resp. right) lcm; in particular, M satisfies the Ore's conditions, so it embeds in its (left and right) group of fractions [9]. The left and right gcd of two Garside elements are Garside elements and coincide; therefore, by the noetherianity property there exists a unique minimal Garside element for both left and right divisibilities. This element ∆ will be called the Garside element of the monoid.
Definition 2.2. [23] Let G be a Garside group with Garside element ∆. (i) An element g ∈ G is said to be periodic if g q is conjugate to ∆ p , where p ∈ Z, q ∈ Z ≥1
, p and q are prime, and q is the smallest positive integer such that g q is conjugate to a power of ∆. (ii) An element g ∈ G is said to be primitive if it is not a non-trivial power of another element, that is if g = h k with k ∈ Z, then k = ±1. [26]There exists a finite-time algorithm that, given an element g ∈ G and an integer k ≥ 1, decides whether there exists h ∈ G such that h k = g and then finds such an element h if it exists.
Theorem 2.3. Let G be a Garside group. (i)
(ii) [23, Prop. 5.2] There exists a finite-time algorithm that makes a list of primitive periodic elements, such that each primitive periodic element of G is conjugate to either exactly one element in the list or its inverse.
2.2.
Description of the Garside structure specific to the structure group. In the following theorem, we describe the one-to-one correspondence between nondegenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation and Garside groups with a particular presentation.
Theorem 2.4. [6, Thm.1] (i) Assume that Mon X | ℜ is a Garside monoid such that:
(a) the cardinality of ℜ is n(n − 1)/2, where n is the cardinality of X and each side of a relation in ℜ has length 2 and (b) if the word x i x j appears in ℜ, then it appears only once.
Then, there exists a function S : X × X → X × X such that (X, S) is a nondegenerate symmetric set-theoretical solution and Gp X | ℜ is its structure group.
(ii) For every non-degenerate symmetric set-theoretical solution (X, S), the structure group G(X, S) is a Garside group, whose Garside monoid is as above.
Moreover, the Garside structure of the structure groups has interesting additional properties described below.
Proposition 2.5. [6, 7] (i) Let x, y ∈ X, x = y. Then x \ y = f −1 x (y). (ii) The Garside element ∆ is the lcm of X for both left and right divisibilities. (iii) Let s belong to M . Then,
s belongs to Div(∆) ⇐⇒ ∃X ℓ ⊆ X such that s is the lcm w.r to left-divisibility of X ℓ ⇐⇒ ∃X r ⊆ X such that s is the lcm w.r to right-divisibility of X r .
(iv) If s belongs to Div(∆) then the subsets X ℓ and X r defined in Point (ii) are unique and have the same cardinality.
Note that for s ∈ Div(∆), X ℓ = X r if and only if s is balanced. In [8], the present author and E.Godelle show that if (X, S) is a non-degenerate symmetric set-theoretical solution, with | X |= n, that satisfies the condition C, then there is a normal free abelian subgroup N of G(X, S) such that G(X, S)/N is a finite quotient group that plays the role that Coxeter groups play for the finite-type Artin groups (Coxeter-like group), that is there is a bijection between the elements in G(X, S)/N and Div(∆) [8]. P.Dehornoy extends our result and shows that if (X, S) is of class m, then there is a bijection between the elements in G(X, S)/N and Div(∆ m−1 ) [10].
M.Picantin defines the notion of a ∆-pure Garside group. A Garside monoid M is ∆-pure if for every x, y in X, it holds that ∆
x = ∆ y , where ∆ x = ∨ L {b \ x; b ∈ M }.
He shows if G is the group of fraction of a ∆-pure Garside monoid with exponent e, then the centre of G is infinite cyclic generated by ∆ e , where e is the order of the conjugation automorphism by ∆ [25]. We show the structure group of (X, S) is ∆-pure Garside if and only if (X, S) is indecomposable [6]. So, if (X, S) is indecomposable, Z(G) = ∆ e and if it is decomposable, Z(G) is the direct product of cyclic groups.
Example 2.6. The solution (X, S) given in Ex.1.2 is an indecomposable solution. So, the monoid with the same presentation as G(X, S) is a ∆-pure Garside monoid and Z(G(X, S)) = ∆ , where ∆ = (
x 1 x 3 ) 2 = (x 2 x 4 ) 2 = (x 3 x 2 ) 2 = (x 4 x 1 ) 2
is the left and right lcm of X. For G 2 and G 3 , the structure groups of the (unique) indecomposable solutions for | X |= 2 and | X |= 3 respectively, the centre is infinite cyclic generated by the Garside element ∆. We have Z(
G 2 ) = ∆ 2 , where ∆ 2 = x 2 1 = x 2 2 and Z(G 3 ) = ∆ 3 , where ∆ 3 = x 3 1 = x 3 2 = x 3 3 .
Using that, we recover the result Inn(G 2 ) = Z 2 * Z 2 [22].
3. Construction of a group of automorphisms of the structure group 3.1. Description of the construction of the group of automorphisms Φ π . Let (X, S) be a non-degenerate symmetric solution of the quantum Yang-Baxter equation and G(X, S) be its structure group. Assume | X |= n. Let π : G(X, S) → Z n , be the bijective 1-cocycle defined in Prop. 1.4. Let σ ∈ GL n (Z) = Aut(Z n ). Depending on the context, σ denotes a map in Aut(Z n ) or a matrix in GL n (Z). We define a map ϕ : G(X, S) → G(X, S) in the the following way:
(3.1) ϕ : G(X, S) → G(X, S), ϕ(a) = π −1 • σ • π(a)
That is, ϕ is defined such that the following diagram is commutative:
G(X, S) Z n G(X, S) Z n ϕ π π σ
As ϕ is a bijection of G(X, S), the question whether ϕ is an automorphism of G(X, S) reduces to the question whether ϕ is a homomorphism of G(X, S). More generally, an epimorphism of G(X, S) is necessarily an automorphism of G(X, S), as G(X, S) is Hopfian. But, a monomorphism of G(X, S) may not be an automorphism of G(X, S), as G(X, S) is co-Hopfian. We give the proof for completion.
Proposition 3.1. The group G(X, S) is residually finite, so G(X, S) is Hopfian. Furthermore, G(X, S) is co-Hopfian.
Proof. The structure group G(X, S) = Z n ⋊H, where Z n is the free abelian group of rank n and H is a subgroup of S n the symmetric group of n elements [14]. Both Z n and H are residually finite, so G(X, S) is residually finite [21]. This implies further that G(X, S) is Hopfian [24,1]. Let σ : Z n → Z n , such that det(σ) = 0, ±1. Then σ is a monomorphism that is not surjective. Let ϕ : G(X, S) → G(X, S) be defined by ϕ = π −1 •σ •π. Assume ϕ is a homomorphism of G(X, S). So, ϕ is a monomorphism that is not necessarily surjective. That is, G(X, S) is co-Hopfian.
We recall some notation from Section 1. The function S is defined by S(x i , x j ) = (x g i (j) , x f j (i) ), where f i , g i : X → X are bijections. The bijective 1-cocycle π : G(X, S) → Z n is defined by π(x i ) = t i , where x i ∈ X, t i is the generator of Z n corresponding to x i and π satisfies π(a 1 a 2 ) = (a −1 2 •π(a 1 ))π(a 2 ), for a 1 , a 2 ∈ G(X, S), where • is the extension of the left action x → f −1
x of G(X, S) to Z n . Given σ ∈ GL n (Z), we denote by (w i ) the i-th column of σ and by w i the element in Z n representing (w i ) written multiplicatively, that is
w i = σ(t i ) = t σ 1,i 1 ..t σ n,i n = j=n j=1 t σ j,i j Example 3.2. We consider Ex. 1.2. Let σ ∈ GL n (Z) be: −4 −3 −4 0 1 1 2 0 2 1 1 0 0 0 0 −1
We compute the function ϕ = π −1 •σ•π, using Lemma 1.6 and Lemma 1.12:
ϕ(x 1 ) = π −1 (w 1 ) = π −1 (t 2 t 2 3 t −4 1 ) = x 2 (x 4 x 3 )(x −4 1 ), ϕ(x 2 ) = π −1 (w 2 ) = π −1 (t 3 t 2 t −1 1 t −2 1 ) = x 2 x 4 x −1 g −1 1 (1) (x −2 1 ) = x 2 x 4 (x −3 1 ), ϕ(x 3 ) = π −1 (w 3 ) = π −1 (t 3 t 2 2 t −4 1 ) = x 3 (x 2 2 )(x −4 1 ) and ϕ(x 4 ) = π −1 (t −1 4 ) = x −1 g −1 4 (4) = x −1 3 . From Prop. 1.4 and Lemma 1.6, the element ϕ(x 1 ) = x 2 (x 4 x 3 )(x −4 1 ) acts on the left on Z n via the permutation f −1 2 f −1 4 f −1 3 f 4 1 = f −1 2 .
The permutations corresponding to π −1 (w 2 ), π −1 (w 3 ) and π −1 (w 4 ) are respectively f 3 , f −1 3 and f 3 . Proposition 3.3. Let σ ∈ GL n (Z) and let ϕ be the bijection defined by σ. Let 1 ≤ k ≤ n. Let (w k ) denote the k-th column of σ and let w k be the element in Z n representing (w k ). Then
(3.2) ϕ ∈ Aut(G(X, S)) ⇔ (π −1 (w j )) −1 • w i = w f j (i) , ∀1 ≤ i, j ≤ n
That is, ϕ is a homomorphism of groups if and only if σ(x −1 j •t i ) = (ϕ(x j )) −1 •σ(t i ). Proof. The defining relations in G(X, S) have the form:
x i x j = x g i (j) x f j (i)) , with x i , x j ∈ X. So, ϕ is a homomorphism of G(X, S) if and only if ϕ(x i x j ) = ϕ(x i ) ϕ(x j ), ∀x i , x j ∈ X. The map π : G(X, S) → Z n is bijective, so ϕ(x i x j ) = ϕ(x i ) ϕ(x j ) in G(X, S) if and only if π(ϕ(x i x j )) = π(ϕ(x i ) ϕ(x j )) in Z n . From commutativity of the diagram 3.1, π(ϕ(x i x j )) = σ • π(x i x j ). From the definition of π, π(x i x j ) = t f j (i) t j , so π(ϕ(x i x j )) = σ • π(x i x j ) = σ(t f j (i) t j ) = σ(t f j (i) ) σ(t j ) = w f j (i) w j .
On the other hand, π(ϕ(x i ) ϕ(x j )) = π(π −1 (w i ) π −1 (w j )) = ( (π −1 (w j )) −1 • π(π −1 (w i )) ) π(π −1 (w j )) = ((π −1 (w j )) −1 • w i ) w j , using first the definition of ϕ and next the property of π. So, ϕ is a homomorphism of
G(X, S) if and only if w f j (i) w j = ((π −1 (w j )) −1 • w i ) w j , for all 1 ≤ i, j ≤ n, that is if and only if (π −1 (w j )) −1 • w i = w f j (i) , for all 1 ≤ i, j ≤ n. We have (π −1 (w j )) −1 • w i = (ϕ(x j )) −1 • σ(t i ) and w f j (i) = σ(t f j (i) ) = σ(x −1 j • t i ), so ϕ is a homomorphism of G(X, S) if and only if σ(x −1 j • t i ) = (ϕ(x j )) −1 • σ(t i ).
Example 3.4. In Ex.3.2, we find that the element (π −1 (w 1 )) −1 acts on the left on Z n via the permutation
f 2 . So, (π −1 (w 1 )) −1 • w 3 is equal to t −4 f 2 (1) t 2 f 2 (2) t f 2 (3) = t 1 t 2 2 t −4
3 , which does not represent any column of σ. So, from Prop. 3.3, ϕ / ∈ Aut(G(X, S)).
We use the following notation. We denote by (f j ) the permutation matrix corresponding to f j , by f ′ j the permutation corresponding to the action of (π −1 (w j )) −1 and by A j the corresponding permutation matrix. We show that the condition for ϕ to be a homomorphism of G(X, S) can be written in a very simple form. To simplify the notation in the proof, we define the permutation matrix as a representing matrix, that is for the permutation (1, 2, 3), its permutation matrix is 0 0 1 1 0 0 0 1 0 Proposition 3.5. Let σ ∈ GL n (Z) and let ϕ be the bijection defined by σ. Then
(3.3) ϕ ∈ Aut(G(X, S)) ⇔ A j σ = σ (f −1 j ), ∀1 ≤ j ≤ n where (f j )
is the permutation matrix corresponding to f j and A j is the permutation matrix corresponding to the action of (π −1 (w j )) −1 .
Proof. From Prop. 3.3, ϕ ∈ Aut(G(X, S)) if and only if (π −1 (w j )) −1 • w i = w f j (i) ,
for all 1 ≤ i, j ≤ n. We fix arbitrarily 1 ≤ j ≤ n. We show that (π −1 (w j )) −1 • w i and w f j (i) are elements in Z n representing the i-th column in the matrices A j σ and σ (f −1 j ) respectively. The i-th column in the matrix σ (f −1 j ) is (w f j (i) ), as multiplying on the right by (f −1 j ) permutes the columns of σ with respect to f j . Assume the i-th
column (w i ) is represented by w i = t m 1 1 t m 2 2 ..t mn n in Z n . The i-th column in the matrix A j σ is equal to A j (w i ) which is represented by the element t m 1 f ′ j (1) t m 2 f ′ j (2) ..t mn f ′ j (n) , as A j permutes the rows of σ with respect to the permutation f ′ j . From Theorem 1.4(ii), t m 1 f ′ j (1) t m 2 f ′ j (2) ..t mn f ′ j (n) is equal to (π −1 (w j )) −1 • w i , so we have (π −1 (w j )) −1 • w i = w f j (i) ,
for all 1 ≤ i, j ≤ n, if and only if the equality of matrices A j σ = σ (f −1 j ) holds for all 1 ≤ j ≤ n.
We prove the set of matrices in GL n (Z) that satisfy Eqn. 3.2 (and Eqn. 3.3) is a subgroup of GL n (Z) and the induced set of automorphisms of G(X, S) is a subgroup of Aut(G(X, S)) that preserves the free abelian subgroup N of G(X, S) (see Sec.1.2). Theorem 3.6. Let (X, S) be a non-degenerate symmetric set-theoretical solution of class m, with | X |= n. Let G(X, S) be its structure group. Let θ 1 , .., θ n be the n frozen elements of length m ending with x 1 , .., x n respectively. Let N be the free abelian subgroup of G(X, S) generated by θ 1 , .., θ n . Let π : G(X, S) → Z n be the bijective 1-cocycle defined in Prop. 1.4. We define ℑ π to be the following set:
ℑ π = {σ ∈ GL n (Z) | (π −1 (w j )) −1 • w i = w f j (i) , ∀1 ≤ j ≤ n} (3.4) = {σ ∈ GL n (Z) | A j σ = σ (f −1 j ), ∀1 ≤ j ≤ n} (3.5)
where w j is the element in Z n representing the j-th column of σ, (f −1 j ) and A j are the permutation matrices corresponding to the action of x j and (π −1 (w j )) −1 respectively. We define Φ π to be the following set:
Φ π = {ϕ : G(X, S) → G(X, S), ϕ = π −1 • σ • π | σ ∈ ℑ π } Then (i) ℑ π ≤ GL n (Z) (ii) Φ π ≤ Aut(G(X, S)). (iii) Φ π (N ) ⊆ N .
Proof. (i), (ii) The trivial automorphism belongs to Φ π , since I n×n ∈ ℑ π , as A j = (f −1 j ), for all 1 ≤ j ≤ n. Let ϕ ∈ Φ π . From Prop. 3.5 and Prop. 3.3, ϕ ∈ Aut(G(X, S)). So, Φ π ⊆ Aut(G(X, S)). We have ℑ π is a subgroup of GL n (Z) and Φ π is a subgroup of Aut(G(X, S)), as both are closed under composition and inverses. Indeed, if ϕ 1 = π −1 • σ 1 • π and ϕ 2 = π −1 • σ 2 • π belong to Φ π , then ϕ 1 • ϕ 2 ∈ Aut(G(X, S)). As, ϕ 1 • ϕ 2 = π −1 • σ 1 σ 2 • π, we have σ 1 σ 2 ∈ ℑ π and ϕ 1 • ϕ 2 ∈ Φ π from Prop. 3.3. In the same way, we prove closure under inverses.
(iii) Let ϕ ∈ Φ π , with ϕ = π −1 •σ•π. Assume σ(t i ) = t m 1 1 ..t mn n , where 1 ≤ i ≤ n. We show ϕ(θ i ) ∈ N . We have ϕ(θ i ) = π −1 •σ•π(θ i ) = π −1 •σ(t m i ) = π −1 ((t m 1 1 ..t mn n ) m ) = π −1 ((t m 1 ) m 1 ..(t m n ) mn ) = (π −1 (t m 1 )) m 1 ..(π −1 (t m n )) mn = θ m 1 1 .
.θ mn n ∈ N , using first Lemma 1.12(iii), next the definition of σ and the commutativity in Z n and at last Lemma 1.6(iv) and Lemma 1.12(v), (vi).
Note that if the solution is trivial, G(X, S) = Z n , we recover Φ π = ℑ π = GL n (Z). From the proof of Thm. 3.6, we have that the correspondence σ ϕ satisfies σ 1 • σ 2 ϕ 1 • ϕ 2 and σ −1 ϕ −1 . T. Gateva-Ivanova and M. Van den Bergh show the structure group G(X, S) of a non-degenerate symmetric solution (X, S) with | X |= n is a Bieberbach group of rank n. Indeed, they show G(X, S) acts freely on R n by isometries with fundamental domain [0, 1] n and the action is defined using the I-structure [18] (see also [19,20]). P. Etingof uses the 1-cocycle π to define a free action of G(X, S) on R n by isometries to show the classifying space of G(X, S) is a compact manifold of dimension n [15]. The study of the automorphisms of a Bieberbach group has been initiated in [5] and A. Szczepanski uses the tools developed there to study the outer automorphism group of a Bieberbach group. In [27], he studies the question when Out(G), the outer automorphism group of a Bieberbach group G, is finite. He shows that for any given n ≥ 3, there exists a Bieberbach group of rank n with finite outer automorphism group and there exists also a Bieberbach group of rank n with infinite outer automorphism group (quasiisometric to GL k (Z) with k ≤ n − 1 ). For all n ≥ 3, we do not know to which category the structure groups belong, since, for each n, the number of structure groups is less than the number of Bieberbach groups of rank n.
In what follows, we consider computational issues and we give some tools that permit an effective construction of the groups ℑ π and Φ π . In the following lemma, we give for each column (w j ) a characterisation of the set of possible permutations for A j . This set is clearly finite and the lemma gives strong restrictions on it.
Lemma 3.7. Let σ ∈ GL n (Z). Let w j ∈ Z n represent the j-th column of σ. Let f ′ j be the permutation corresponding to the action of (π −1 (w j )) −1 and A j the corresponding permutation matrix. If A j σ = σ (f −1 j ), then (i) f ′ j has the same parity as f j .
(ii) f ′ j has the same number of fixed points as f j .
(iii) f ′ j ∈ f 1 , f 2 , .., f n . (iv) Assume A i σ = σ (f −1 i ), 1 ≤ i ≤ n. Then A i = A j if and only if f i = f j . Proof. (i), (ii) From A j σ = σ (f −1 j )
, it results that A j and (f −1 j ) are similar matrices, so det(A j ) = det(f −1 j ) and trace(A j ) = trace(f −1 j ). As (f j ) is a permutation matrix, it is an orthogonal matrix, so det(f −1 j ) = det(f j ) and trace(f −1 j ) = trace(f j ). Moreover, det(f j ) and trace(f j ) represent respectively the parity and the number of fixed points of the permutation f j . The same holds for A j . So, f ′ j has the same parity and the same number of fixed points as f j . (iii) By induction on the length of w j . If w j = t ǫ k , ǫ = ±1, then π −1 (t ǫ k ) is equal to x k or x −1
g −1 k (k) , from lemma 1.6. So, the corresponding permutation f ′ j is f −1 k or f g −1 k (k) , that is f ′ j ∈ f 1 , f 2 ,
.., f n . Using Lemma 1.6 (iv) we have inductively f ′ j ∈ f 1 , f 2 , .., f n . (iv) results from cancellation by σ. Example 3.8. We consider Ex.1.2. Assume σ ∈ GL n (Z) has first column (w 1 ) = (−4, 2, 8, 6) t , then the element (π −1 (w 1 )) −1 acts trivially on Z n (as the corresponding permutation is Id X ), so A 1 = I 4×4 . From Lemma 3.7, σ cannot be a candidate, as f 1 is an odd permutation and Id X an even one.
The following proposition gives a very efficient tool to compute the permutation corresponding to the action of the element (π −1 (w j )) −1 , where w j represents the j-th column on σ: the computations can be done modulo the class of the solution, and whenever the solution is of class 2 only the parity of the numbers occurring in σ is important.
i 1 t m ′ i 2 i 2 ..t m ′ in in , where m ′ i ≡ m i (modm)
and m ′ i k ≡ 0 (modm). Then (π −1 (w j )) −1 • w = (π −1 (w ′ j )) −1 • w, for all w ∈ Z n . That is, the action of (π −1 (w j )) −1 is equivalent to the action of (π −1 (w ′ j )) −1 .
Proof. In Z n , w j = t m ′ 1 i 1 t m ′ 2 i 2 ..t m ′ n in t k 1 m 1 t k 2 m 2 ..t knm n , m ′ i k ≡ m i , m ′ i k ≡ 0 (modm). So, π −1 (w j ) • w = (π −1 (π −1 (t k 1 m 1 t k 2 m 2 ..t knm n ) • t m ′ 1 1 t m ′ 2 2 ..t m ′ n n ) π −1 (t k 1 m 1 t k 2 m 2 ..t knm n )) • w = π −1 (t m ′ 1 1 t m ′ℑ π = {I 4×4 , −I 4×4 , 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 , 0 −1 0 0 −1 0 0 0 0 0 0 −1 0 0 −1 0 }
Note that the automorphism ϕ induced by the matrix −I ∈ ℑ π satisfies ϕ(
x 1 ) = x −1 1 , ϕ(x 2 ) = x −1 2 , ϕ(x 3 ) = x −1 4 and ϕ(x 4 ) = x −1 3 , using Lemma 1.12(i). Let G 2 = x 2 , x 3 | x 2 2 = x 2 3
be the structure group of the unique non-trivial (indecomposable) solution for | X |= 2. For G 2 , we find
ℑ π = {σ 1 = I 2×2 , σ 2 = −I 2×2 , σ 3 = 0 1 1 0 , σ 4 = 0 −1 −1 0 } Accordingly, Φ π = Out(G 2 ) = Z 2 × Z 2 [22]. Let G 3 = x 1 , x 2 , x 3 | x 1 x 3 = x 2 2 , x 2 x 1 = x 2 3 , x 3 x 2 = x 2 1
, the structure group of the unique indecomposable solution for | X |= 3 (of class 3), the group ℑ π is:
ℑ π = {σ 1 = I 3×3 , σ 2 = 0 0 1 1 0 0 0 1 0 , σ 3 = 0 1 0 0 0 1 1 0 0 , σ 4 = −1 0 0 0 0 −1 0 −1 0 , σ 5 = 0 0 −1 0 −1 0 −1 0 0 , σ 6 = 0 −1 0 −1 0 0 0 0 −1 }
And accordingly, Φ π = Z 3 ⋊ Z 2 . In section 5, we show Φ π ≤ Out(G(X, S)). We present a decomposable solution (X,
S) with X = {x 1 , x 2 , x 3 }, f i = g i = (2, 3) 1 ≤ i ≤ 3, and structure group x 1 , x 2 , x 3 | x 1 x 3 = x 2 x 1 , x 1 x 2 = x 3 x 1 , x 2 2 = x 2 3 ≃ Z⋉G 2 . Then ℑ π = { a b b c d e c e d : e = d ± 1, a(d + e) − 2bc = ±1; a, c, d, e ∈ Z, b ∈ 2Z}
is an infinite subgroup of GL n (Z). Furthermore, Φ π ∩ Inn(G) = Id, as the inner automorphism obtained from conjugation by x 1 can be represented by the matrix in ℑ π satisfying a = e = 1 and b = c = d = 0.
3.2.
Invariance of the group Φ π under the change of bijective 1-cocycle.
Consider the one-to-one correspondence between non-degenerate symmetric solutions with quadruples (see Sec. 1.1). If (X, S) and (X ′ , S ′ ) are different solutions, then they are in correspondence with different quadruples (G(X, S), X, ρ, π) and (G(X ′ , S ′ ), X ′ , ρ ′ , π ′ ), which, using the construction from Sec. 3.1, give the groups Φ π and Φ π ′ , respectively. In this context, a question that arises naturally is: if G(X, S) ≃ G(X ′ , S ′ ) then what is the relation between the groups Φ π and Φ π ′ , as the bijective 1-cocycles π and π ′ may be different. We show that under some additional conditions, G(X, S) ≃ G(X ′ , S ′ ) if and only if (X, S) and (X ′ , S ′ ) are equivalent solutions (Lemma 3.11) and in such a case we show Φ π = Φ π ′ , that is Φ π is invariant under change of bijective 1-cocycle.
Lemma 3.11. Let (X, S) and (X ′ , S ′ ) be non-degenerate symmetric solutions, with structure groups G(X, S) and G(X ′ , S ′ ) respectively. If (X, S) and (X ′ , S ′ ) are equivalent solutions, then G(X, S) ≃ G(X ′ , S ′ ). Conversely, if G(X, S) ≃ α G(X ′ , S ′ ) and α(X) = X ′ , then (X, S) and (X ′ , S ′ ) are equivalent solutions.
Proof. We recall that G(X, S) is generated by X subject to the defining relations x y = S 1 (x, y) S 2 (x, y) for all x, y ∈ X, where S 1 (x, y) and S 2 (x, y) denote the first and second component of S(x, y). If (X, S) and (X ′ , S ′ ) are equivalent solutions, then G(X ′ , S ′ ) is generated by X ′ subject to the defining relations α(x) α(y) = α(S 1 (x, y)) α(S 2 (x, y)), for all x, y ∈ X. So, G(X, S) ≃ G(X ′ , S ′ ). Conversely, if G(X, S) ≃ α G(X ′ , S ′ ) and α(X) = X ′ , then α(x) α(y) = α(S 1 (x, y)) α(S 2 (x, y)), for all x, y ∈ X, are relations in G(X ′ , S ′ ). As, α(x) α(y) = S ′ 1 (α(x) , α(y))S ′ 2 (α(x) , α(y)) holds also in G(X ′ , S ′ ), we have (X, S) and (X ′ , S ′ ) are equivalent solutions.
Proposition 3.12. Let (X, S) and (X ′ , S ′ ) be non-degenerate symmetric solutions, with structure groups G(X, S), G(X ′ , S ′ ), bijective 1-cocycles π : G(X, S) → Z n , π ′ : G(X ′ , S ′ ) → Z n , and groups of automorphisms Φ π , Φ π ′ , respectively. Assume (X, S) and (X ′ , S ′ ) are equivalent solutions and π(x i ) = π ′ (x i ) = t i , with t i , 1 ≤ i ≤ n, a generator of Z n . Assume we have one of the following three situations:
Z n G(X, S) ≃ G(X ′ , S ′ ) Z n Z n G(X, S) ≃ G(X ′ , S ′ ) Z n σ π −1 π −1 ϕ π ′ π ′ σ ′ Figure 3.1. G(X, S) ≃ G(X ′ , S ′ ) Z n G(X, S) ≃ G(X ′ , S ′ ) Z n ϕ π ′ π σ Figure 3.2. G(X, S) ≃ G(X ′ , S ′ ) Z n G(X, S) ≃ G(X ′ , S ′ ) Z n ϕ π π ′ σ Figure 3.3.
Then, in all the cases, the set of automorphisms obtained is Φ π = Φ ′ π . Proof. Assume the case of Fig. 3.1. Let ϕ ∈ Φ π defined by σ ∈ ℑ π . We define the bijection σ ′ : Z n → Z n by σ ′ = π ′ • ϕ • π ′−1 . Since ϕ ∈ Φ π , ϕ ∈ Aut(G(X, S)). So, from Prop.3.3 applied to π ′ , σ ′ ∈ ℑ π ′ . As, ϕ = π ′−1 • σ ′ • π ′ , we have ϕ ∈ Φ π ′ , that is Φ π ⊆ Φ π ′ . The reverse inclusion is obtained in the same way. Assume the case of Fig. 3.2 or Fig. 3.3 That is, ϕ is defined by ϕ = π −1 •σ •π ′ or ϕ = π ′−1 •σ •π. In the case of Fig. 3.2, we obtain the group of automorphisms Φ π , as π ′ (x i ) = π(x i ) = t i . In the case of Fig. 3.3, we obtain the group of automorphisms Φ ′ π , as π(x i ) = π ′ (x i ) = t i . From the above, we have Φ π = Φ ′ π .
A group of automorphisms that preserves the Garside structure
We show the set of automorphisms of G(X, S) induced by permutation matrices in ℑ π is a subgroup of Aut(G(X, S)) that preserves the Garside structure of G(X, S), and more generally the subgroup of automorphisms induced by generalized permutation matrices in ℑ π preserves the properties of an extended Garside structure of G(X, S). We begin with the description of the subgroup of generalized permutation matrices in ℑ π . 4.1. Properties of the subgroup of generalized permutation matrices in ℑ π . We recall that in a generalized permutation matrix there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the non-zero entry must be 1, for a generalized permutation matrix in GL n (Z) the nonzero entry is ±1. We use the following notation. Let σ be a generalized permutation matrix in ℑ π . Let w j be the element in Z n representing the j-th column (w j ), where w j = σ(t j ). If σ i j ,j = ±1, we define the permutationσ byσ(j) = i j and we write w j = t ǫ j σ(j) , where ǫ j = ±1. Lemma 4.1. Let (X, S) be a non-degenerate symmetric solution, with structure group G(X, S). Let ℑ π be the subgroup of GL n (Z) defined in Theorem 3.6. Let σ ∈ ℑ π . Let w j and w k be elements in Z n representing the j-th column (w j ), and the k-th column (w k ) of σ. Assume x j , x k ∈ X belong to the same indecomposable component of (X, S). Then (i) (w j ) is a permutation of the rows of (w k ).
(ii) If additionally, σ is a generalized permutation matrix with w j = t
ǫ j i j , w k = t ǫ k i k and ǫ j , ǫ k = ±1, then ǫ j = ǫ k .
Proof. (i) Since x j , x k ∈ X belong to the same indecomposable component of (X, S), there exists a permutationf j ∈ f 1 , .., f n ,f j = f j 1 f j 2 ..f j l , such that j =f j (k). The proof is by induction on l. If l = 1, thenf j = f j 1 and w j = w f j 1 (k) . From Eqn. 3.4, (π −1 (w j 1 )) −1 • w k = w f j 1 (k) . So, (w j ) is a permutation of the rows of (w k ) according to the action of (π −1 (w j 1 )) −1 . Assume l > 1. We have
((π −1 (w j 1 )) −1 ..(π −1 (w j l )) −1 ) • w k = ((π −1 (w j 1 )) −1 ..(π −1 (w j l−1 ) −1 ) • w f j l (k) = w f j 1 f j 2 .
.f j l (k) = w j , using the action of G(X, S) and l times Eqn. 3.4. So, (w j ) is a permutation of the rows of (w k ) according to the action of (π −1 (w j 1 )) −1 ..(π −1 (w j l )) −1 . (ii) From (i), (w j ) is a permutation of the rows of (w k ), so (w j ) and (w k ) have elements with the same sign.
We show that if σ is a generalized permutation matrix in GL n (Z) , the condition for σ to belong to ℑ π or equivalently for ϕ to belong to Aut(G(X, S)) described in Eqn.3.2 has a simple form.
Proposition 4.2. Let (X, S) be a non-degenerate symmetric solution, with structure group G(X, S). Let σ be a generalized permutation matrix in GL n (Z) defined by S)) if and only if σ ∈ ℑ π if and only if Eqn. 3.2 holds, that is (π −1 (w j )) −1 • w i = w f j (i) for all 1 ≤ i, j ≤ n. We show Eqn. 3.2 is equivalent to the condition cited in each case. Generally, we have (π −1 (w j )) −1
σ(t j ) = t ǫ j σ(j) , where ǫ j = ±1, 1 ≤ j ≤ n. Let ϕ be defined by σ. Then (i) If ǫ j = 1, ∀1 ≤ j ≤ n, then ϕ ∈ Aut(G(X, S)) ⇔σ • f j •σ −1 = fσ (j) , ∀1 ≤ j ≤ n. (ii) If (X, S) is of class 2, then ϕ ∈ Aut(G(X, S)) ⇔σ •f j •σ −1 = fσ (j) , ∀1 ≤ j ≤ n. (iii) If (X, S) is of class m > 2, then ϕ ∈ Aut(G(X, S)) if and only if σ • f j •σ −1 = fσ (j) , ǫ j = 1 σ • f j •σ −1 = f −1 g −1 σ(j)σ (j) , ǫ j = −1 Proof. From Prop. 3.3, ϕ ∈ Aut(G(X,•w i = w f j (i) ⇔ (π −1 (w j )) −1 • σ(t i ) = σ(t f j (i) ) ⇔ (π −1 (w j )) −1 • t ǫσ (i) = t ǫσ •f j (i)
, from the definition of σ and from lemma 4.1(ii), ǫ i = ǫ f j (i) = ǫ. Now, we need to compute (π −1 (w j )) −1 in each case. (i) If ǫ j = 1, then w j = tσ (j) and (π −1 (w j )) −1 = x −1 σ(j) . So, (π −1 (w j )) −1 • t ǫσ
(i) = t ǫσ •f j (i) , ⇔ t ǫ fσ (j) •σ(i) = t ǫσ •f j (i) ⇔ fσ (j) •σ(i) =σ • f j (i). (ii), (iii) If ǫ j = −1, then w j = t −1
σ(j) and π −1 (w j ) = x −1 g −1 σ(j)σ (j) , from lemma 1.12(i). Letσ be a permutation matrix in ℑ π and let ϕ ∈ Aut(G(X, S)) be induced byσ. Then ϕ is a length-preserving isomorphism of the bounded lattice Div(∆) w.r to right-divisibility and for all s, t ∈ Div(∆):
So, (π −1 (w j )) −1 = x g −1 σ(j)σ (j) and (π −1 (w j )) −1 • t ǫσ (i) = t ǫσ •f j (i) , ⇔ f −1 g −1 σ(j)σ (j) •σ(i) = σ • f j (i) ⇔σ • f j •σ −1 = f −1 g −1 σ(j)σ (j) . If (X, S) is of class 2, then f −1 g −1 σ(j)σ (j) = fσ (j) (see remark 1.8) and we have σ ∈ ℑ π ⇔σ • f j •σ −1 = fσ (j) , ∀1 ≤ j ≤ n.(i) ϕ(s ∨ R t) = ϕ(s) ∨ R ϕ(t). (ii) ϕ(s ∧ R t) = ϕ(s) ∧ R ϕ(t) (iii) ϕ(Div(∆)) = Div(∆), with ϕ(∆) = ∆ and ϕ(1) = 1. (iv) ϕ(N ) = N ,
where N is the free abelian subgroup of rank n of G(X, S) generated by the frozen elements. Furthermore, ϕ induces an automorphism of the finite quotient group G(X, S)/N and a permutation of the elements in Div(∆).
In order to prove Thm. 4.4, we show in the following proposition, that a bijective function ϕ induced by a permutation matrixσ is an automorphism of G(X, S) if and only ifσ preserves the left complement in some sense. We use the following notation: if x j \ x k = x m andσ(m) = m ′ , then we writeσ(j \ k) = m ′ Proposition 4.5. Let (X, S) be a non-degenerate symmetric solution, with structure group G(X, S). Let σ be a generalized permutation matrix in GL n (Z) defined by σ(t j ) = t ǫ j σ(j) , where ǫ j = ±1, 1 ≤ j ≤ n. Let ϕ be defined by σ. If σ =σ or (X, S) is of class 2, then (i) ϕ ∈ Aut(G(X, S)) if and only ifσ(j \ k) =σ(j) \σ(k) for all 1 ≤ j, k ≤ n.
(ii) If ϕ ∈ Aut(G(X, S)), thenσ belongs to C Sn (T ) the centralizer of T in S n .
Proof. (i) From Prop. 4.2, under these assumptions, ϕ ∈ Aut(G(X, S)) ⇔σ • f −1 j • σ −1 = f −1 σ(j) , ∀1 ≤ j ≤ n. Let 1 ≤ k ≤ n. We evaluate f −1 σ(j) andσ •f −1 j •σ −1 onσ(k). We have f −1 σ(j)σ (k) =σ(j) \σ(k) andσ • f −1 j •σ −1 (σ(k)) =σ • f −1 j (k) =σ(j \ k), from Prop.2.5(i). So, ϕ ∈ Aut(G(X, S)) if and only ifσ(j \ k) =σ(j) \σ(k) for all 1 ≤ j, k ≤ n. (ii) Assume ϕ ∈ Aut(G(X, S)). From Prop. 4.2(i),(ii),σ • f −1 j = f −1 σ(j) •σ, ∀1 ≤ j ≤ n. In particular, we haveσ • f −1 j (j) = f −1 σ(j) •σ(j)
, that isσ(T (j)) = T (σ(j)). Proof of Theorem 4.4 Proof. (i), (ii), (iii) By definition
x i ∨ R x j = (x i \ x j ) x i . So, ϕ(x i ∨ R x j ) = ϕ(x i \ x j )ϕ(x i ) = (ϕ(x i ) \ ϕ(x j )) ϕ(x i ) = ϕ(x i ) ∨ R ϕ(x j )
, using first ϕ ∈ Aut(G(X, S)), next Prop. 4.5 and the fact that ϕ(x i ) = x j if and only ifσ(i) = j (asσ is a permutation matrix) and at last the definition of the left lcm. Inductively we obtain
( * ) ϕ(x i 1 ∨ R .. ∨ R x i k ) = ϕ(x i 1 ) ∨ R .. ∨ R ϕ(x i k ). From Prop. 2.5, if s ∈ Div(∆)
then there exist a subset of X, X r (s) = {x i 1 , .., x i k } such that s is the left lcm of X r (s). So, from ( * ), ϕ(s) ∈ Div(∆) and has the same length as s. In the same way, t ∈ Div(∆) is the left lcm of X r (t). Clearly, s ∨ R t is the left lcm of X r (s) ∪ X r (t) and s ∧ R t is the left gcd of (X r (s) ∩ X r (t)) ∪ {1}. So, using ( * ), we have (ii) and (iii). Furthermore, ϕ(∆) = ∆ and ϕ(1) = 1. (iv) From Thm. 3.6, ϕ(N ) ⊆ N and using the fact that σ is a permutation matrix in the proof, we have ϕ(θ i ) = θσ (i) , where θ i is the frozen ending with x i . So, ϕ(N ) = N . Moreover, ϕ induces an automorphismφ of the finite quotient group G(X, S)/N defined byφ(N a) = N ϕ(a).
We now consider the group of automorphisms of G(X, S) induced by the generalized permutation matrices in ℑ π . We show this group preserves a Garside structure which is not exactly the original one. Indeed, we define an extended Garside structure that takes into account generators with negative sign. This is done in the following way. Let X = {x 1 , .., x n }. We define X − = {x −1 1 , .., x −1 n } and X = X ∪ X − . For X − , we define left complement and left lcm in the following way: If
x i x j = x k x l is a defining relation in G(X, S), we define x −1 i ∨ R x −1 k = x −1 j x −1 i = x −1 l x −1 k , x −1 i \ x −1 k = x −1 j and x −1 k \ x −1 i = x −1 l .
In the same way, we define right complement and lcm. Obviously, complement and lcm are defined for every pair of elements in X − and are unique. The element ∆ −1 is the left and right lcm of X − , using Prop. 2.5. We now extend the definition of left and right lcm to X in the following way. If x i x j = x k x l is a defining relation in G(X, S), then x −1 i x k = x j x −1 l and x −1 k x i = x l x −1 j hold in G(X, S) and we can define the following elements:
x −1 i ∨ L x j , x −1 k ∨ L x l , x −1 l ∨ R x k and x −1 j ∨ R x i .
Note that we use the same symbols for the complement and lcm in each set of elements. If for every pair of elements in X and for every pair of elements in X − , left and right lcm are defined and unique, this is not the case for mixed pairs with an element from X and another one from X − . Proposition 4.6. Let (X, S) be a non-degenerate symmetric solution, with structure group G(X, S). Let σ be a generalized permutation matrix in ℑ π and let ϕ ∈ Φ π be induced by σ. Let x i , x j ∈ X. Then ϕ satisfies:
(i) ϕ(x i ∨ R x j ) = ϕ(x i ) ∨ R ϕ(x j ), and ϕ(x i \ x j ) = ϕ(x i ) \ ϕ(x j ), when defined. (ii) ϕ(x i ∨ L x j ) = ϕ(x i ) ∨ L ϕ(x j ) and ϕ(x i\ x j ) = ϕ(x i )\ϕ(x j ), when defined. (iii) ϕ(N ) = N ,
where N is the free abelian subgroup of rank n of G(X, S) generated by the frozen elements. Furthermore, ϕ induces an automorphism of the finite quotient group G(X, S)/N .
Proof. (i), (ii) Assume x i x j = x k x l is a defining relation in G(X, S), with x i , x j , x k , x l ∈ X, so x i ∨ L x k = x i x j = x k x l .
The elements x i and x l belong to the same irreducible component and the same holds for x j and x k . We have ϕ(x i ∨ L x k ) = ϕ(x i )ϕ(x j ) = ϕ(x k )ϕ(x l ), since ϕ belongs to Aut(G(X, S)). Moreover, as σ is a generalized permutation matrix, ϕ(x i ), ϕ(x j ), ϕ(x k ), ϕ(x l ) ∈ X, with ϕ(x i ), ϕ(x l ) sharing the same sign and ϕ(x j ), ϕ(x k ) sharing the same sign, from Lemma 4.1(ii). So, the equation ϕ(x i )ϕ(x j ) = ϕ(x k )ϕ(x l ) is necessarily derived from a defining relation in G(X, S) and from our definition of ∨, \, ∨ R , \ L in X, we have (i), (ii), (iii), (iv).
(v) From Thm. 3.6, ϕ(N ) ⊆ N . From the proof of Thm. 3.6 and Lemma 1.12(v), as σ is a generalized permutation matrix, we have ϕ(θ i ) = π −1 (t −ṁ σ(i) ) = θ −1 σ(i) , where θ i is the frozen ending with x i . So, ϕ(N ) = N and ϕ induces an automorphism of the finite quotient group G(X, S)/N .
5.
Special case of indecomposable solutions with Φ π ≤ Out(G(X, S))
In this section, we consider the special case of indecomposable solutions. In particular, for indecomposable solutions with n = 2, 3 we show Φ π = Out(G(X, S)) for | X |= 2 and Φ π ≤ Out(G(X, S)) for | X |= 3. For | X |= 3, we do not know whether the inclusion Φ π ≤ Out(G(X, S)) is an equality. Another question that arises naturally is whether it is possible to prove Φ π ≤ Out(G(X, S)) for general indecomposable solutions or Φ π ∩ Out(G(X, S)) is not trivial.
5.1.
Properties of ℑ π for general indecomposable solutions. We show that some properties are specific to indecomposable solutions.
Proposition 5.1. Let (X, S) be an indecomposable non-degenerate symmetric solution. Let G(X, S) be its structure group. Let σ ∈ ℑ π and ϕ ∈ Φ π defined by σ. Let w j be the element in Z n representing the j-th column (w j ) of σ. Assume Proof. (i) From Lemma 4.1(i), if x j and x k belong to the same indecomposable component of (X, S), then (w j ) is a permutation of the rows of (w k ). The solution (X, S) is indecomposable, so each (w j ) is obtained by permuting the rows of (w 1 ) according to some permutation (that depends on j). (ii) From (i), in each column of σ the same numbers m 1 , .., m n appear (in a different order). The sum of each column is then m 1 + m 2 + .. + m n , so the number m 1 + m 2 + .. + m n is an integer eigenvalue of the matrix σ, where det(σ) = ±1. (iii) From (ii), m 1 + m 2 + .. + m n = ±1. Assume first m 1 + m 2 + .. + m n = 1. So, σ(t 1 t 2 ..t n ) = σ(t 1 )σ(t 2 )..σ(t n ) = w 1 w 2 ..w n = t m 1 +m 2 +..+mn 1 ..t m 1 +m 2 +..+mn n = t 1 t 2 ..t n , using (i) and the assumption. That means t 1 t 2 ..t n is a fixed element of σ. It holds that π(∆) = t 1 t 2 ..t n . Indeed, from Prop.2.5(i), ∆ is the right and left lcm of X and it has length n, so for all 1 ≤ i ≤ n, t i appears in π(∆) and π(∆) has length n also. So, π(∆) = t 1 t 2 ..t n . We have then σ(t 1 t 2 ..t n ) = σ • π(∆) = π(ϕ(∆)) = t 1 t 2 ..t n , from the commutativity of the diagram. Since π is bijective and π(∆) = π(ϕ(∆)) = t 1 t 2 ..t n , we have ϕ(∆) = ∆. Assume next m 1 + m 2 + .. + m n = −1. So, as before,
w 1 = t m 1 1 t m 2 2 ..σ(t 1 t 2 ..t n ) = t −1 1 t −1 2 ..t −1 n . It holds that π(∆ −1 ) = t −1 1 t −1 2 ..t −1 n .
As before, for all 1 ≤ i ≤ n, π(x −1 i ) appears in π(∆ −1 ). From lemma 1.6, π(x −1 i ) = t −1 T (i) and T is bijective, so for all 1 ≤ i ≤ n, t −1 i appears in π(∆ −1 ). As π(∆ −1 ) has length n, π(∆ −1 ) = t −1 1 t −1 2 ..t −1 n . Using the same argument as before, ϕ(∆) = ∆ −1 .
It seems the assumption added in Prop. 5.1 (iii) holds for all indecomposable solutions. As the following example illustrates, if the solution is decomposable, ϕ(∆) = ∆ ±1 does not necessarily hold.
Example 5.2. Consider the structure group G(X, S) = Z ⋉ G 2 from Ex. 3.10 and take the following σ ∈ ℑ π : 1 2 2 1 3 2 1 2 3 . Then the induced automorphism
ϕ satisfies ϕ(x 1 ) = x 1 x 2 x 2 , ϕ(x 2 ) = x 2 (x 3 x 2 )(x 2 x 3 )(x 1 x 1 ) = x 2 θ 1 θ 2 θ 3 , ϕ(x 3 ) = x 3 (x 2 x 3 )(x 3 x 2 )(x 1 x 1 ) = x 3 θ 1 θ 2 θ 3 , and ϕ(x 1 x 2 2 ) = x 2 2 ∆ 5 , where ∆ = x 1 x 2 2 .
The image x 2 2 ∆ 5 of the central element ∆ is also a central element, as Z(G) = x 1 × x 2 2 . In this case, we have ϕ(N ) = N , as ϕ(θ 1 ) = θ 1 θ 2 θ 3 , ϕ(θ 2 ) = θ 2 1 θ 2 2 θ 3 3 and ϕ(θ 3 ) = θ 2 1 θ 3 2 θ 2 3 . 5.2. For indecomposable solutions with n = 2, 3, we show Φ π ≤ Out(G(X, S)). We consider the special case of indecomposable solutions with n = 2, 3 and first we show that ℑ π is a group of generalized permutation matrices. This means that for these solutions, the extended Garside structure is preserved by Φ π (from Prop. 4.6) and the Garside structure is preserved by its subgroup of permutation matrices (from Thm. 4.4).
Proposition 5.3. Let (X, S) be an indecomposable non-degenerate symmetric solution, with | X |= n = 2, 3. Let G(X, S) be its structure group. Then ℑ π is a group of generalized permutation matrices in GL n (Z).
Proof. The proof is a case-by-case checking, using Prop. 5.1(i). For n = 2, the unique indecomposable solution has structure group G 2 = x 1 , x 2 | x 2 1 = x 2 2 . The set of matrices satisfying Eqn.3.3 has the following form:
{ a b b a | a, b ∈ Z, a + b = ±1}
In order to belong to GL n (Z), a matrix in this set satisfies a 2 − b 2 = ±1, that is a = 0 and b = ±1 or b = 0 and a = ±1. This gives the set ℑ π of generalized permutation matrices described in Ex.3.10. For n = 3, the unique indecomposable solution has structure group
G 3 = x 1 , x 2 , x 3 | x 1 x 3 = x 2 2 , x 2 x 1 = x 2 3 , x 3 x 2 = x 2 1 .
The set of matrices satisfying Eqn.3.3 is a subset of one of the two following sets of matrices:
{ a b c b c a c a b | a, b, c ∈ Z, a+b+c = ±1}; { a b c c a b b c a | a, b, c ∈ Z, a+b+c = ±1}
In order to belong to GL n (Z), a matrix in one of these two sets satisfies (a + b + c)(a 2 − a(b + c) + (b 2 + c 2 − bc) = ±1. Using very basic tools from number theory, this gives the set ℑ π of generalized permutation matrices described in Ex.3.10.
For n = 4, there are five indecomposable solutions, three of them are of class 4 and the two other of class 2. One of the two indecomposable solutions of class 2 is described in Ex. 1.2 and the corresponding group ℑ π is also a group of generalized permutation matrices in GL n (Z) as described in Ex.3.10. We conjecture this is the case for all the five indecomposable solutions. More generally, it would be interesting to understand the group ℑ π for indecomposable solutions.
Theorem 5.4. Let (X, S) be an indecomposable non-degenerate symmetric solution, with | X |= n = 2, 3. Let G(X, S) be its structure group. Let Φ π be the group of automorphisms of G(X, S) induced by the group of generalized permutation matrices ℑ π from Prop. 5.3. Then, for n = 2, Φ π = Out(G(X, S)) and for n = 3, Φ π ≤ Out(G(X, S)).
Proof. For n = 2, the group Φ π ≃ Z 2 × Z 2 , with Φ π = {ϕ 1 = Id, ϕ 2 :
x 1 → x −1 2 x 2 → x −1 1 , ϕ 3 : x 1 → x 2 x 2 → x 1 , ϕ 4 : x 1 → x −1 1 x 2 → x −1 2 }
Each automorphism ϕ i in Φ π corresponds to a matrix σ i in ℑ π (see Ex.3.10). As G 2 = Z * 2Z Z, using the work of A. Karrass, A. Pietrowski and D. Solitar on automorphisms of a free product with an amalgamated subgroup [22], Φ π = Out(G 2 ). For n = 3, the group Φ π ≃ Z 3 ⋊ Z 2 , with Φ π = {ϕ 1 = Id, ϕ 2 :
x 1 → x 2 x 2 → x 3 x 3 → x 1 , ϕ 3 : x 1 → x 3 x 2 → x 1 x 3 → x 2 , ϕ 4 : x 1 → x −1 2 x 2 → x −1 1 x 3 → x −1 3 , ϕ 5 : x 1 → x −1 1 x 2 → x −1 3 x 3 → x −1 2 , ϕ 6 : x 1 → x −1 3 x 2 → x −1 2 x 3 → x −1 1 }
Each automorphism ϕ i in Φ π corresponds to a matrix σ i in ℑ π (see Ex.3.10). We show Φ π ∩ Inn(G 3 ) = {Id}. Assume that there is ϕ ∈ Φ π ∩ Inn(G 3 ). That is, there exists an element a ∈ G such that ϕ(x i ) = ax i a −1 , for all 1 ≤ i ≤ n. Since each ϕ ∈ Φ π has finite order and Z(G 3 ) = ∆ (see Ex. 2.6), we have that a is a periodic element, that is there exists p ∈ Z and q ∈ Z ≥1 such that g q = ∆ p (see Defn. , we find the following set of roots of ∆: {x 1 , x 2 , x 3 , x 2 1 , x 2 2 , x 2 3 , ∆}. If a is equal to one of these roots of ∆, then ϕ fixes necessarily one of the generators. As none of the automorphisms in Φ π except Id fixes a generator, we have ϕ = Id.
This result holds also for the indecomposable solution with n = 4 presented in Ex. 1.2 using a case by case computation. We conclude this section with the following observation. If we consider the unique indecomposable solution for | X |= 5, which is a permutation solution with f = (1, 2, 3, 4, 5) and g = (1, 5, 4, 3, 2), then the group of matrices ℑ π is not necessarily a group of generalized permutation matrices. As an example, the circulant matrix with first column (1, −1, 1, 0, 0) t belongs to ℑ π . The existence of matrices that are not necessarily generalized permutation matrices in ℑ π illustrates the difficulty in understanding the connection between Φ π and Out(G(X, S)) for general solutions.
6. Application of our results to the quotient group G(X, S)/N 6.1. Definition of a bijective 1-cocycleπ from the quotient group G(X, S)/N . Let (X, S) be a non-degenerate, symmetric solution with structure group G(X, S). Let π be the bijective 1-cocycle defined in section 1.2. Assume (X, S) is of class m. Let N be the free abelian group generated by the n frozen elements of length m, θ 1 , .., θ n , and W be the finite quotient group G(X, S)/N . We show that the normal subgroup N acts trivially on Z n and therefore there exists a bijective 1-cocyclẽ π : W → Z n /(π(N )) that is induced by π. Lemma 6.1. Let w ∈ Z n . Then N •w = w. Moreover, the mapπ : W → Z n /(π(N )) defined byπ(aN ) = π(a) (π(N )) is a bijective 1-cocycle.
Proof. From Lemma 1.12 (iii), (iv), θ i • w = w for all 1 ≤ i ≤ n. So, N • w = w. A direct computation showsπ(aN ) = π(a) (π(N )) is a bijective 1-cocycle.
Remark 6.2. Assuming m = 2, that is the solution (X, S) satisfies condition C, W is a subgroup of S n ⋉ Diag(n, Z) that is a Weyl (or finite Coxeter) group of type C n and we give another equivalent definition of the bijective 1-cocycleπ induced by π. Indeed, let τ : W → S n ⋉ Diag(n, Z) be defined by τ (x i N ) = (f −1 i , I (i) ), where Diag(n, Z) is the subgroup of diagonal matrices in GL n (Z) and I (i) is given by I j,j = 1, for j = i. A direct checking shows τ : W → S n ⋉ Diag(n, Z) is a monomorphim. The bijective 1-cocycleπ induced by π is given byπ : W → Diag(n, Z) and defined byπ(aN ) = A, whenever τ (aN ) = (f, A).
6.2.
A group of automorphisms of the quotient group G(X, S)/N . Using the bijective 1-cocycleπ : W → Z n /(π(N )) and the construction of Section 3.1 applied to W , we find a subgroup of automorphisms Φπ of W . A question that arises naturally is whetherφ : W → W belongs to Φπ, wheneverφ is an automorphism of W induced by ϕ ∈ Φ π . Proposition 6.3. Let ϕ ∈ Φ π . Assume ϕ(N ) = N . Letφ : W → W be defined bŷ ϕ(N a) = N ϕ(a). Thenφ belongs to Φπ.
Proof. Assume ϕ(N ) = N . From the definition of ϕ, π • ϕ = σ • π, so σ(π(N )) = π(N ) andσ : Z n /π(N ) → Z n /π(N ) is well-defined. Furthermore,σ inducesφ such that the following diagram is commutative andφ belongs to Φπ:
G Z n W Z n /π(N ) G Z n W Z n /π(N )
π ϕ σ π ϕ π πσ Appendix: Proofs of Lemmas 1.6, 1.11, 1.12
Lemma. 1.6 Let π : G(X, S) → Z n be the bijective 1-cocycle defined in Prop. 1.4(iv). Let x i , x j ∈ X. Let t i , t j be generators of Z n corresponding to x i and x j respectively. Let u, v ∈ Z n . Then
Definition 2.1. (1) A locally Garside monoid M is a cancellative noetherian monoid such that any two elements in M have a lcm for left (and right) divisibility if and only if they have a common multiple for left (and right) divisibility.
Proposition 3 . 9 .
39Assume (X, S) is of class m. Let σ ∈ GL n (Z). Let (w j ) denote the j-th column of σ, with representing element w j = t m 1 1 t m 2 2 ..t mn n in Z n , m 1 , .., m n ∈ Z. Let denote w ′
• w, using first Lemma 1.6(iv) and then Lemma 1.12(vi). We conclude Sec. 3.1 with some examples of groups of automorphisms obtained.
Example 3.10. In Ex. 1.2, we find
Example 4. 3 .
3Consider in Ex. 3.10 the group G 3 of class 3 with f 1 = f 2 = f 3 = (1, 2, 3) and the matrix σ 4 ∈ ℑ π , withσ 4 = (2, 3). For 1 ≤ i ≤ 3,σ 4 • f i •σ , so σ 4 satisfies Prop. 4.2(iii), as expected. 4.2. Preservation of the Garside structure by groups of automorphisms.Our first purpose is to show Thm. 4.4 which states that the group of automorphisms induced by the permutation matrices in ℑ π preserves entirely the Garside structure.
Theorem 4 . 4 .
44Let (X, S) be a non-degenerate symmetric solution of class m, with structure group G(X, S).
t mn n , with m 1 , .., m n ∈ Z. Then (i) Each column of σ is a permutation of the rows of the first column. (ii) m 1 + m 2 + .. + m n = ±1 (iii) If additionally, the sum of each row is m 1 + m 2 + .. + m n , then ϕ(∆) = ∆ ±1 , for all ϕ ∈ Φ π .
2.2). From Theorem 2.3, there exists a finite-time algorithm that finds all the primitive periodic elements. Applying the algorithm described in [23, Prop. 5.2] (or in [26, Prop.2.13])
Proof. (i) and (ii) result from the definition of π and the assumption that π(1) = 1.using the definition of π and then the fact that • is an action on Z n ., using first the definition of S and then the non-degeneracy and involutivity of S. Using the same arguments,T m−1 (x) ) −1 = (t T m (x) ) −1 = t −1 , using first Lemma 1.6(i), next the definition of T and at last since the class of the solution is m,x, using first Lemma1.6(iii), (ii) and then the induction assumption for t f −1x (x) and the fact thatx and from lemma 1.11(ii), this is the frozen element of length m ending with x.On the other hand, π −1 (t −m t m ) = π −1 (π −1 (t m ) • t −m ) π −1 (t m ), from Lemma 1.6(iv). As π −1 (t m ) acts trivially on Z n , π −1 (t −m t m ) • w = π −1 (t −m ) • w. So, π −1 (t −m ) • w = w.(v) From (iv), π −1 (t −m t m ) = π −1 (π −1 (t m )•t −m ) π −1 (t m ) = π −1 (t −m ) π −1 (t m ) = 1,
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| [] |
[
"On the Effect of the Large Magellanic Cloud on the Orbital Poles of Milky Way Satellite Galaxies",
"On the Effect of the Large Magellanic Cloud on the Orbital Poles of Milky Way Satellite Galaxies"
] | [
"Marcel S Pawlowski \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany\n",
"Pierre-Antoine Oria \nUMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance\n",
"Salvatore Taibi \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany\n",
"Benoit Famaey \nUMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance\n",
"Rodrigo Ibata \nUMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance\n"
] | [
"Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany",
"UMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance",
"Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D-14482PotsdamGermany",
"UMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance",
"UMR 7550\nUniversité de Strasbourg\nCNRS\nObservatoire astronomique de Strasbourg\nF-67000StrasbourgFrance"
] | [] | The reflex motion and distortion of the Milky Way (MW) halo caused by the infall of a massive Large Magellanic Cloud (LMC) has been demonstrated to result in an excess of orbital poles of dark matter halo particles towards the LMC orbital pole. This was suggested to help explain the observed preference of MW satellite galaxies to co-orbit along the Vast Polar Structure (VPOS). We test this idea by correcting the positions and velocities of the MW satellites for the Galactocentric-distancedependent shifts inferred from a LMC-infall simulation. While this should substantially reduce the observed clustering of orbital poles if it were mainly caused by the LMC, we instead find that the strong clustering remains preserved. We confirm the initial study's main result with our simulation of an MW-LMC-like interaction, and use it to identify two reasons why this scenario is unable to explain the VPOS: (1) the orbital pole density enhancement in our simulation is very mild (∼ 10% within 50-250 kpc) compared to the observed enhancement (∼220-300%), and (2) it is very sensitive to the specific angular momenta (AM) of the simulation particles, with higher AM particles being affected the least. Particles in simulated dark matter halos tend to follow more radial orbits (lower AM), so their orbital poles are more easily affected by small offsets in position and velocity caused by an LMC infall than objects with more tangential velocity (higher AM), such as the observed dwarf galaxies surrounding the MW. The origin of the VPOS thus remains unexplained. | null | [
"https://arxiv.org/pdf/2111.05358v2.pdf"
] | 243,938,749 | 2111.05358 | 4be4a32616f59e958a87779b25ce3458629f678b |
On the Effect of the Large Magellanic Cloud on the Orbital Poles of Milky Way Satellite Galaxies
May 4, 2022
Marcel S Pawlowski
Leibniz-Institut für Astrophysik Potsdam (AIP)
An der Sternwarte 16D-14482PotsdamGermany
Pierre-Antoine Oria
UMR 7550
Université de Strasbourg
CNRS
Observatoire astronomique de Strasbourg
F-67000StrasbourgFrance
Salvatore Taibi
Leibniz-Institut für Astrophysik Potsdam (AIP)
An der Sternwarte 16D-14482PotsdamGermany
Benoit Famaey
UMR 7550
Université de Strasbourg
CNRS
Observatoire astronomique de Strasbourg
F-67000StrasbourgFrance
Rodrigo Ibata
UMR 7550
Université de Strasbourg
CNRS
Observatoire astronomique de Strasbourg
F-67000StrasbourgFrance
On the Effect of the Large Magellanic Cloud on the Orbital Poles of Milky Way Satellite Galaxies
May 4, 2022Draft version Typeset using L A T E X twocolumn style in AASTeX631Dark matter (353) -Milky Way dark matter halo (1049) -Milky Way dynamics (1051) -Dwarf galaxies (416) -Orbits (1184)
The reflex motion and distortion of the Milky Way (MW) halo caused by the infall of a massive Large Magellanic Cloud (LMC) has been demonstrated to result in an excess of orbital poles of dark matter halo particles towards the LMC orbital pole. This was suggested to help explain the observed preference of MW satellite galaxies to co-orbit along the Vast Polar Structure (VPOS). We test this idea by correcting the positions and velocities of the MW satellites for the Galactocentric-distancedependent shifts inferred from a LMC-infall simulation. While this should substantially reduce the observed clustering of orbital poles if it were mainly caused by the LMC, we instead find that the strong clustering remains preserved. We confirm the initial study's main result with our simulation of an MW-LMC-like interaction, and use it to identify two reasons why this scenario is unable to explain the VPOS: (1) the orbital pole density enhancement in our simulation is very mild (∼ 10% within 50-250 kpc) compared to the observed enhancement (∼220-300%), and (2) it is very sensitive to the specific angular momenta (AM) of the simulation particles, with higher AM particles being affected the least. Particles in simulated dark matter halos tend to follow more radial orbits (lower AM), so their orbital poles are more easily affected by small offsets in position and velocity caused by an LMC infall than objects with more tangential velocity (higher AM), such as the observed dwarf galaxies surrounding the MW. The origin of the VPOS thus remains unexplained.
INTRODUCTION
The phase-space correlation of satellite galaxies around their hosts is currently one of the most pressing challenges for our understanding of galaxy formation (Bullock & Boylan-Kolchin 2017;Pawlowski 2018). In short, a large fraction of dwarf satellite galaxies of the Milky Way (MW) and M31 are distributed within flattened and kinematically coherent structures that are at odds with the expected phase-space distribution of their associated dark matter sub-halos (Kroupa et al. 2005;Ibata et al. 2013). In the MW, this structure is called the Vast Polar Structure (VPOS, Pawlowski et al. 2012): it is perpendicular to the galactic disc, has a root-meansquare thickness of only ∼25 kpc for a 10 times larger spatial extension, and its normal vector is aligned with the orbital poles of 50% to 75% of the galaxies spatially located in it (Li et al. 2021).
Similar planar structures have been found outside of the Local Group (e.g. Müller et al. 2018Müller et al. , 2021Paudel et al. 2021;Martínez-Delgado et al. 2021;Heesters et al. 2021), and are calling for an explanation. Many theoretical scenarios for the formation of these satellite-plane structures have been proposed over the years (for a review, see Pawlowski 2018), but thus far all have failed to clearly reproduce the extreme phase-space distributions observed around real galaxies. Recently, a new possible explanation for the VPOS around the MW has been put forward by Garavito-Camargo et al. (2021) (hereafter GC21). It partly relies on the old observation by Lynden-Bell (1976) and Kunkel & Demers (1976) that the Large Magellanic Cloud (LMC) and the Magellanic Stream, align with and orbit along the VPOS.
In recent years, an array of evidence has accumulated for a more massive LMC than initially thought (Kallivayalil et al. 2013;Peñarrubia et al. 2016; 2019; Erkal & Belokurov 2020;Petersen & Peñarrubia 2021), with a total mass at infall of at least 1.5 × 10 11 M . Such a massive LMC should necessarily cause a shift in the orbital barycenter of the MW-LMC system compared to the center of the MW's disk, which could influence the kinematics of satellite galaxies. GC21 demonstrated with a numerical simulation of a massive LMC-analog interacting with a MW-like host that the resulting reflex motion and offset in overall center-of-mass conspire to induce an overdensity of orbital poles in the dark matter particles that constitute the dark halo of the MW model. The direction of this overdensity aligns with the orbital pole of their LMC analog. As the orbital pole of the observed LMC is well aligned with the observed VPOS and its associated overdensity of satellite galaxy orbital poles, GC21 suggest that this LMC-induced effect might help explain the presence of the VPOS.
We here go beyond the original study and test whether this proposed mechanism can have a sufficient effect on the inferred orbital poles of the observed MW satellite system.
EFFECT ON MILKY WAY SATELLITES
GC21 argue that the orbital pole enhancement in their simulation can be reproduced by shifting the positions and velocities of simulation particles in the initial conditions. Effectively, this means that for different radial shells, different reference centers-of-mass (COM) are adopted, mimicking the perturbation to the MW's dark matter halo by the infall of a massive LMC. They have demonstrated that this approach accounts for both the degree and direction of the orbital pole overdensity of dark matter particles in their simulation, making a convincing argument for the validity of their model. We adopt this model but invert the approach to estimate to what degree the infall of the LMC can be responsible for the observed clustering of satellite orbital poles. If adding shifts to an isotropic initial distribution reproduces the LMC's influence, then subtracting those shifts from the observed MW dwarf galaxies, which are exposed to the LMC's influence, should substantially reduce the observed orbital pole overdensity if its main cause is the gravitational influence of the LMC. We note that this approach does only take the reflex motion and COM offset into account, but not any orbital pole enhancement due to the wake of the LMC or its direct torque effects on MW satellites galaxies. As recently shown analytically by Rozier et al. (2022), the latter effect from the dark matter wake itself on the response of stars or satellites is a minor one. However, these additional effects will all be included in a self-consistent manner in the numerical simulations presented and analysed from Sect. 3 onwards.
In Fig. 1 we reproduce the COM shifts (as obtained from figures 4 and 6 in GC21). We subtract the respective distance-dependent values from the Cartesian positions and velocities of the observed MW satellites, for which we adopt the dataset by Battaglia et al. (2022) who provide proper motions based on the early-third data release (EDR3) of Gaia. Here and in the following we focus on satellites in the Galactocentric distance range of 50 to 250 kpc, which results in a sample of 31 dwarfs. For more nearby objects no effect due to the LMC is expected according to GC21, while more distant objects have only weakly constrained orbital poles due to the large proper motion uncertainties.
Panel b of Fig. 1 plots the most-likely positions and velocities of the MW satellites. Also shown are the shifted positions and velocities. For most dwarfs the shifts are minor. The overall orbital directions do not appear to be substantially altered; especially the objects co-orbiting closely along the VPOS (red) keep their preferred orbital sense (clockwise). This impression is confirmed by Fig. 2, which shows the relative change in position r and velocity v, the relative change in the norm of the specific angular momentum h, and the difference ∆θ pole in the direction of the resulting orbital pole (direction of the angular momentum) before and after applying the shifts of each dwarf galaxy, respectively. The positions shift by no more than 10% of the Galactocentric distance of each dwarf, most velocities change by no more than 20%, though for some the velocity change can reach up to 50% of their full 3D velocity. Note, however, that a complete reversal of the velocity requires a change of 200%. The specific angular momenta change by no more than 30%. Most importantly, the direction of the orbital poles of the majority of MW dwarfs changes by no more than ∼ 5 • . Only 7 of the 31 considered dwarfs display ∆θ pole > 10 • . Of these, only one is strongly co-orbiting and one strongly counter-orbiting relative to the VPOS. The mean (me-dian) change on orbital pole direction for the MW dwarfs is 3 • (6 • ).
The impact on the distribution of orbital poles is illustrated in Fig. 3. To account for measurement uncertainties in distances, velocities, and proper motions, we draw 5000 realizations for each dwarf. The distributions without and with shifts are virtually indistinguishable. In both cases a clear overdensity of orbital poles close to the center of the figures is apparent. Overall, the number of MW dwarfs with most likely orbital poles within 36.87 • (10% of the area of the sphere around the VPOS normal direction, see e.g. Fritz et al. 2018;Li et al. 2021) is 12 for the observed orbital poles out of the 31 dwarfs (ignoring two counter-orbiting dwarfs) 1 . This corresponds to an orbital pole enhancement in the 10% VPOS region of almost 300% over the expected isotropic share of 3.1 orbital poles out of 31 dwarfs. After applying the shifts, all 12 orbital poles remain within the VPOS region, and the median alignment angle θ VPOS with the VPOS changes from 59 • to 61 • .
It has been demonstrated that the LMC has likely brought along a number of dwarf galaxies as satellites of its own (Erkal & Belokurov 2020;Patel et al. 2020;Battaglia et al. 2022). These would not constitute independent objects, and could boost the VPOS orbital enhancement given that they should follow orbits similar to the LMC. The exact strength of the observed orbital pole enhancement thus depends on which dwarfs were satellites of the LMC before infall. By considering only objects beyond 50 kpc in this work, we already ensure that many likely LMC satellites, specifically Carina II, Carina III, Hydrus I and Reticulum II, are not part of our sample. However, of the likely LMC satellites identified by Battaglia et al. (2022), Phoenix II and Horologium I are part of our sample of 31 dwarfs. In addition, Battaglia et al. (2022) report inconclusive findings on the LMC-association of Horologium II, which we thus consider as a potential past LMC satellite. Note that we must not exclude dwarf galaxies that had a recent interaction but were unlikly to have been brought in as LMC satellites, because such interactions with the LMC are exactly what is being studied here and in particular in the following simulation-based investigation. An example for this is Grus II, which Battaglia et al. (2022) report to not originate with the Magellanic system, but to merely have interacted with the LMC recently. We furthermore do not consider the classical MW satellites Carina or Fornax as likely past LMC satellites, because both were excluded as such by Patel et al. (2020), and also Battaglia et al. (2022) report this as unlikely. Excluding the two likely LMC satellites with orbital poles within the VPOS region from our sample of 31 dwarfs results in an orbital pole enhancement of 250%, while also excluding the possible past LMC satellite Grus II results in an enhancement of 220%. The enhancement thus remains substantial even if accounting for dwarfs brought in as LMC satellites.
We note that this is only a first-order estimate of the influence of the LMC. More accurate results require full orbit modelling under the influence of the time-varying potential sourced by the MW and LMC. Recently, Correa Magnus & Vasiliev (2021) have presented such a work. They report that re-winding the satellite orbits under the influence of a massive LMC, and then forwardintegrating them again without an LMC until present time, results in no significant change in the orbital pole distribution and preserves its strong non-uniformity. We are thus confident that the LMC's impact on the orbital pole clustering observed for the MW satellites is indeed very minor.
This begs the question as to why the model of GC21 appears to be insufficient to account for the observed orbital pole clustering. Our hypothesis is that the reason lies in different orbital properties between typical dark matter particles in a halo, on which their work is based, and the MW dwarf galaxies. GC21 used a slightly radially biased setup for their dark matter halo particle velocities. If an orbit is strongly radial, then a minor shift in position or velocity of the reference point for angular momentum calculations has much greater impact on the orbital pole direction than if the orbit were more circular. While dark matter particles can be on highly radial orbits, the observed MW dwarf galaxies have been found to follow more tangentially biased orbits (Cautun & Frenk 2017;Riley et al. 2019;Hammer et al. 2021). This is not entirely unexpected in a ΛCDM framework. Diemand et al. (2004) report for galaxy cluster mass simulations that dark matter particles in the inner regions of halos are on slightly more radial orbits than subhalos, and that subhalos typically have higher velocities than dark matter particles. The presence of a baryonic central galaxy seems to further strengthen this difference, because the additional potential in the inner region of a halo results in stronger tidal forces that even more efficiently destroy subhalos on radial orbits. Using the hydrodynamical FIRE simulations, Garrison-Kimmel et al. (2017) showed that this leads to more tangentially biased subhalo velocities when compared to dark-matteronly simulations. They also showed that the surviving subhalos in the hydrodynamical runs have considerably The most-likely orbital poles are shown as open circles, and the orbital poles from 5000 Monte-Carlo realizations drawing from the measurement uncertainties as dots. The colors follow those in Fig. 1, with strongly co-orbiting dwarfs in red and counterorbiting ones in green. If the LMC's influence were the main reason for the observed overdensity of orbital poles towards the VPOS normal vector (green cross), then the lower panels should show a much reduced overdensity. They do not. Most orbital pole directions are barely affected by the shifts.
higher tangential velocities within 100 kpc than those in dark-matter-only runs, which implies higher angular momenta. This effect has been confirmed by Kelley et al. (2019) for the PhatELVIS simulations which contain an analytically grown MW-like disk potential, and by Riley et al. (2019) who find that satellites in the Auriga simulations are on more tangentially biased orbits, probably because the Auriga halos contain rather massive baryonic disks that destroy subhalos on radial orbits.
OUR SIMULATION
To test our hypothesis, we run our own simulation of a MW-LMC-like interaction. The N -body simulation is run with Gyrfalcon (Dehnen 2000), and we use Agama (Vasiliev 2019) to generate initial conditions. For the MW, we use the Model 1 of Binney & Tremaine (2008) as given in Agama, which includes a stellar bulge, a thin disk, a thick disk, and a dark matter (DM) halo. We make two changes: (i) we use the same halo mass of 1.2 × 10 12 M as GC21, and (ii) in order to have a spherical halo, we remove the z-axis flattening coefficient of its potential. We generate 2 × 10 5 particles for the stars and 8 × 10 5 for the dark matter halo. For the LMC, we generate 10 5 particles based on the spheroid potential of Agama for a total mass of 1.8 × 10 11 M , following the fiducial model of GC21. The initial conditions of our simulated MW halo model deliberately adopt isotropic orbits, in order to facilitate our aim to examine the effect of the infall of the LMC model on the halo particles as a function of their specific angular momenta. The anisotropy parameter calculated using all dark matter halo particles thus is very close to
zero: β = 1 − σ 2 θ +σ 2 φ 2σ 2 r = 0.0083,
where σ are the velocity dispersions of the particles in the three spherical coordinate components (r, θ, φ).
Initially we place the LMC at (x, y, z) = (12, 215, 130) kpc and (v x , v y , v z ) = (12, 13, −77) km s −1 in the MW-centred reference frame, a slight alteration of the initial conditions of GC21. We run the simulation for 2 Gyr. For each snapshot, we use the snapcenter function of Nemo (Teuben 1995) to obtain the positions and velocities of the COM of the MW based on its stars, then we shift the positions and velocities of all particles by these values in order to make the MW the center of phase space. In this reference frame, the coordinates of the COM of the LMC af-
RESULTS
We follow the evolution of individual DM particles in the simulation from the initial to final snapshot, selecting all particles at a distance of 50 to 250 kpc from the MW analog center in the final snapshot. We divide them in three bins of low (h ini < 5×10 3 kpc km s −1 , N = 49, 884), intermediate (5 × 10 3 ≤ h ini < 10 4 kpc km s −1 , N = 109, 905) and high (10 4 kpc km s −1 ≤ h ini , N = 232, 533) initial specific angular momenta. While the two bins with lower angular momentum constitute only 41% of all particles in the considered distance range, we will show in the following that they contribute the bulk to the orbital pole overdensity around the VPOS direction. Fig. 4 plots the specific angular momenta of the observed MW dwarfs against their Galactocentric distance. It compares to particles in our simulation, selected to reside in the considered distance range at the end of the simulation. The LMC affects the angular momenta of the particles, spreading them beyond their initial bin boundaries, though overall they preserve their ranking in specific angular momentum (note the density scale is logarithmic). The observed MW dwarfs agree best with the final particle distribution of the high-angularmomentum bin. As expected, the MW dwarfs have relatively high specific angular momenta compared to dark matter particles in a MW-like halo. To more quantitatively judge this visual impression, we calculate the likelihood ratios of the high angular momentum bin to the low and the intermediate angular momentum bins, respectively.
The likelihoods are estimated using the maps of angular momenta and distances in Fig. 4 as follows: for each observed satellite, we identify the Galactocentric distance d bin (of 10 kpc width) it falls in. We then calculate the weighted contribution of this observed satellite's angular momentum to each of the angular momen-tum bins (of 2500 kpc km s −1 width) at this d. This is done by adopting the most-likely measured value of h for this satellite and assuming a normal distribution with a width as given by the uncertainties on h (see error bars in Fig. 4). Since h can only be positive we cut off these distributions at h = 0 and re-normalize them accordingly. To avoid being dominated by a few high-h outliers, we also do not consider satellites or particles with h > 5×10 4 kpc km s −1 . The likelihood for one individual satellite is then the weighted contribution of simulation particles in these bins, normalized to the total number of simulation particles in the considered set (i.e. either low, intermediate, or high h ini ). Finally, the individual likelihoods of all observed satellites are multiplied to obtain the overall likelihood. We find that the likelihood ratio of high-to-low h ini is 1.3 × 10 22 , while that of high-to-intermediate h ini is 4.1 × 10 8 . This strongly confirms the impression that the set of observed Milky Way satellite galaxies match best with the high initial angular momentum bin.
To test whether the particles most affected by the LMC-effect indeed have more radial, eccentric orbits, we have also calculated the anisotropy parameter β for bins in initial angular momentum h ini and distance d, see panel (b) in Fig. 4. As expected, the high-angular momentum particles are dominated by tangentially biased orbits (β < 0, blue in the plot), while the lowand intermediate-angular momentum particles are characterized by radially biased orbits (β > 0, red in the plot).
Orbital Pole Enhancement
The enhancement of the density of orbital pole directions for the particles in our simulation are plotted in Fig. 5. We follow the method of GC21, and calculate for each bin in the Healpix map how many orbital poles in the particle sample contribute to it, subtract the expected number for an isotropic distribution, and divide by the latter. Despite the lower resolution of the maps due to our smaller particle numbers, we clearly confirm the results of GC21. The infall of an LMC-like object onto a MW-like host results in an enhancement of orbital poles of dark matter particles in the general direction of the VPOS.
We also calculate how many more orbital poles than in an isotropic distribution contribute to the VPOS region. We assume the VPOS normal points to Galactic coordinates (l, b) VPOS = (169.3 • , −2.8 • ) to be consistent with previous works (e.g. Fritz et al. 2018;Li et al. 2021;Garavito-Camargo et al. 2021). Also following these previous works, we consider a region of 36.87 • around this direction, which corresponds to 10% of the area on the sphere. The excess of the number of particle orbital poles in this area in the final simulation snapshot over the corresponding number using the same set of particles in the initial setup, divided by the latter, is the relative VPOS enhancement. Within the adopted distance range of 50 to 250 kpc, we find an overall VPOS enhancement of 9%. This means that there are 1.09 times the number of orbital poles within the VPOS region than expected from isotropy. As the region constitutes 10% of the sphere, for 31 dwarf galaxies 3 should typically be found in the region if they are isotropically distributed. The LMC effect would then suggest an enhancement of about one quarter of an additional orbital pole. This is hardly sufficient to account for the observed cluster-ing of 12 (or more within their uncertainties) out of 31 orbital poles aligned with the VPOS.
In the other panels of Fig. 5 we separate the DM particles by their initial angular momentum. The resulting maps of orbital pole enhancement strongly confirm our suspicion that mainly particles with small angular momenta are affected by the LMC's influence. The low angular momentum bin shows the strongest VPOS enhancement of 35%. The pole density distribution also displays a slight s-shape that further confirms the results reported by GC21. Particles with intermediate specific angular momenta still display some enhancement in the orbital pole density, but to a much reduced degree (VPOS enhancement of 9%). Once only high angular momentum particles are considered, the map mainly consists of noise, with no substantial enhancement of orbital poles towards the VPOS direction (VPOS enhancement of 3%).
Change in Orbital Pole Directions
We now investigate how the orbital poles change between the initial and the final simulation snapshot to assess how much the LMC-like infall has affected the orbital direction of each individual particle. Panel (a) of Fig. 6 plots the angle between the initial and the final direction of orbital poles, ∆θ pole . The vast majority of simulation particles (note the logarithmic color scale) overlap with the region covered by the observed MW dwarfs. There is a clear tendency for particles with higher initial specific angular momentum to have smaller changes in their orbital pole direction. Panel (b) of Fig. 6 confirms this. Low angular momentum particles display an extended tail to high ∆θ pole , some reaching as high as ∆θ pole = 170 • and thus almost flip their orbital direction. Their mean (median) change in orbital pole direction is 43.9 • (29.2 • ). The particles with intermediate angular momentum show smaller changes of 19.7 • (12.8 • ). The vast majority of particles with high initial angular momentum, however, change their orbital pole direction by less than 20 • , and correspondingly their mean (median) ∆θ pole are only 12.5 • (9.4 • ).
Change in Alignment with VPOS
Changing the direction of the orbital poles does not necessarily imply an enhancement of poles towards the Figure 7. The enhancement in the alignment of orbital poles with the VPOS normal direction of particles in the simulation depends on the specific angular momentum they had initially. Panel (a) plots the Cosines of the angle between the VPOS normal vector and the particle orbital poles as measured in the initial (θ ini VPOS ) and final (θ obs VPOS ) simulation snapshot. The corresponding angles for the observed (and shifted) MW dwarf positions and velocities are also shown as colored symbols. Panel (b) compares histograms of θ obs VPOS for three different bins in specific angular momentum for the particles in the simulation, with the alignment of orbital poles of the observed set of MW dwarfs. Only the low angular momentum (red dotted line) particles show some enhancement towards the VPOS normal (at cos(θ obs VPOS ) = 1). The high angular momentum particles (blue solid line) follow a flat distribution as expected for isotropically distributed orbital poles. The observed MW dwarfs have specific angular momenta that are comparable to the high angular momentum bin, but they display a very pronounced overdensity of orbital poles close to the VPOS normal, which is more than an order of magnitude higher than even the enhancement shown by the low angular momentum particles.
VPOS. To quantify this, we measure the angle θ VPOS between each particle's orbital pole and the normal vec-tor to the VPOS. We again do this for both the initial snapshot which should be comparable to the "corrected" positions after shifting the observed MW dwarf positions and velocities (θ ini VPOS ), and the final one which is comparable to the observed situation (θ obs VPOS ). Panel (a) of Fig. 7 compares these two quantities 2 . If the clustering of observed MW orbital poles close to the VPOS were substantially affected by the LMC infall, θ ini VPOS should show a much wider distribution than θ obs VPOS , and in particular the dwarfs with closely aligned poles (red) should scatter appreciably towards lower cos(θ ini VPOS ). Yet, the distributions largely follow the diagonal in the figure, suggesting that the overall changes are small.
The situation is even clearer when looking at histograms of θ obs VPOS (panel (b) in Fig. 7). An isotropic distribution of directions on the sphere, such as the DM particle orbital poles in our initial simulation snapshot, gives a flat histrogram with probability density 0.5. Clearly, the simulation particles in the final snapshot mostly follow this distribution, in particular for the high angular momentum bin. While an isotropic distribution should have a mean (
Dependency on Distance from Host
These results are not strongly dependent on the exact radial range considered. Fig. 8 plots the VPOS enhancement as a function of radius, split into the three angular momentum bins. The high initial angular momentum particles always show the least VPOS enhancement. In particular, they show no more than a 10% enhancement within 200 kpc from the host, the distance range covered by most MW satellites with reliably measured orbital poles. We also note that our overall VPOS enhancement is very comparable to that reported by GC21 for their fiducial model (see their Fig. 15), upon which we modelled our initial conditions. This confirms that our results are indeed comparable with their study.
It is to be expected that differences in initial conditions and model parameters, such as in the anisotropy parameter of the dark matter halo particles, the exact phase along the LMC-infall orbit at which the model is evaluated, or the mass ratio of the LMC and MW analog, all influence the degree of inferred orbital pole enhancement. For example, a likely source of differences ompared to GC21 is our choice of dark matter halo setup. While we deliberately chose one with isotropic orbits, the GC21 model implies somewhat more radially biased orbits. Since we find that particles on more radial orbits show stronger enhancements, a different mix of orbits will lead to differences in the overall degree of orbital pole enhancement. Furthermore, the simulations of GC21 and ours do not share the exact same orbits of the LMC analogs and are likely analyzed as slightly different phases of their orbits, which can be expected to propagate into further differences in the exact degree of orbital pole enhancement. Finally, we note that our simulation has a much lower particle resolution than that of GC21. While the overall dynamics and effect of the LMC is certainly captured with our approach, it is possible that the lower resolution does not capture all the resonances of the dark matter halo. For example, while our simulation does show a wake induced by the LMC, its detailed properties and behaviour might not yet be fully converged at our resolution. An increased resolution could thus potentially change the detailed results slightly. It is reassuring how closely the overall degree of enhancement of the two studies agree despite these effects, with reported orbital pole enhancements in the range of 9 to 15% in the two simulations.
CONCLUSIONS
GC21 have demonstrated that the infall of a massive, LMC-like galaxy onto a MW-like host results in an enhancement of orbital poles of dark matter particles in the direction of the orbital plane of the interaction. For the MW, this direction aligns with the VPOS. This in turn led GC21 to propose that the infall of a massive LMC could have changed the orbital pole directions of the observed MW satellite galaxies, and that this could help explain the observed clustering or satellite orbital poles towards the VPOS normal.
We confirm the overall results of GC21 with our own simulation of a LMC-MW like interaction. However, our results strongly disagree with their suggestion that this can help explain the origin of the VPOS. We find that accounting for the radius-dependent shifts in position and velocity of the MW halo center-of-mass induced by the LMC infall does not remove nor appreciably weaken the clustering of orbital poles towards the VPOS normal. This is likely due to the small relative COM shifts on the positions and velocities of the MW satellites. While the orbital pole direction of objects on more radial orbits (low angular momenta) can be substantially changed by even a small perturbation to their position or velocity, objects on less radial orbits (higher angular momenta) are much less affected by the same shifts. The observed MW satellites, as well as dark matter sub-halos in hydrodynamical cosmological simulations, have less radial orbits and higher tangential velocities (and thus higher specific angular momenta) than many dark matter halo particles or subhalos in dark-matter-only simulations (Diemand et al. 2004;Cautun & Frenk 2017;Garrison-Kimmel et al. 2017;Li et al. 2020). We hypothesize that the observed MW satellites have too high angular momenta to be sufficiently affected by the LMC infall to explain their strong clustering of orbital poles.
Our numerical simulation of an LMC-MW like interaction confirms this, while -beyond the mere shift in positiona and the reflex motion -also accounting for the additional effects of direct torques by the LMC and its induced wake. Within the distance range of 50 to 250 kpc, which is most relevant for the observed MW satel-lites, the simulation predicts an orbital pole enhancement of 9% in the VPOS direction. In the same area of sky, 12 out of 31 MW satellite orbital poles are found, corresponding to an enhancement of 300% over the expected isotropic share of three orbital poles. Even excluding the two likely and one possible LMC satellites among these, which could artificially boost the enhancement because their orbits would be aligned with that of the LMC, results in an enhancement of 220%. The observed signal is thus substantially larger than the one predicted to be induced by the LMC. This discrepancy increases further to almost two orders of magnitude (3% vs. 220-300%) if only particles with angular momenta as high as those of the MW satellites are considered. Particles that initially had very low angular momenta show the strongest enhancement in orbital pole density towards the VPOS normal. However, these angular momenta are inconsistent with the observed dwarf galaxies, as based on Gaia-EDR3 proper motions (Battaglia et al. 2022). Restricting the analysis to particles of comparably high angular momenta, the orbital pole enhancement all but vanishes. The LMC-like infall is therefore unable to induce a pronounced enhancement of orbital poles in the VPOS direction for simulation particles with angular momenta comparable to those inferred for the observed MW dwarfs.
In summary, the LMC-induced overdensity of orbital poles is (i) only present for particles with low specific angular momenta, while the effect becomes fully negligible towards the higher specific angular momenta of the observed MW satellites, and (ii) is still too small an effect to account for the strong degree of orbital pole clustering among the observed MW satellites even if assuming the impact on them were as strong as on particles with low specific angular momenta. The LMC's infall onto the MW therefore does not suffice by a large margin to explain the observed VPOS.
If the LMC cannot explain the VPOS, its orbital alignment with the VPOS despite its very recent infall nevertheless remains a puzzling coincidence (Pawlowski et al. 2013). While alternative approaches have proposed some potential explanatory mechanisms (Lynden-Bell 1976;Yang & Hammer 2010;Bílek et al. 2018), at least within a classical ΛCDM framework this alignment appears to be nothing but a chance event.
DATA AVAILABILITY
The initial conditions and final snapshot of the simulation described in Section 3 can be obtained at the following address: https://seafile.unistra.fr/d/f0ff2f519f4e42b792a0/.
Figure 1 .
1Effect of an LMC-induced shift in the COM positions and velocities on the observed MW dwarfs. Panel (a) shows the shifts in the center-of-mass position ∆ri and velocity ∆vi for simulation particles in spherical shells of distance r around their host galaxy center, in three Cartesian coordinates i ∈ {x, y, z}, with x measured along the Galactic Center-Sun direction, y in the direction of Galactic rotation at the Sun, and z towards the North Galactic Pole. The results reported by GC21 are shown (solid lines) and compared to our final simulation snapshot (dashed lines). Panel (b) shows the positions and velocities (as vectors) of the MW satellites. They are color-coded by the observed alignment angle θ obs VPOS of their orbital pole directions with the VPOS normal (red is co-orbiting). Smaller, fainter symbols indicate the position after applying the GC21 shifts from panel (a) in reverse. The y − z plane is displayed, in which the majority of the shift takes place and which aligns well with the VPOS. The position and orientation of the edge-on MW is indicated in green.
Figure 2 .
2Panels (a) to (c) show the relative effect that the COM shifts have (seeFig. 1) on the position and velocity, angular momentum, and direction of orbital pole, respectively. Here and throughout the following figures the MW dwarfs are color-coded as in panel (b) ofFig. 1.
poles after correcting for LMC-induced reflex motion and COM shift
Figure 3 .
3Directions of orbital poles for the observed MW satellites calculated relative to the MW center (upper panel a), and after shifting the dwarf galaxy positions and velocities as in Fig. 1 to correct for the effect of the LMC (lower panel b).
ter 2
2Gyr are (x, y, z) = (−1.87, −31.42, −17.07) kpc and (v x , v y , v z ) = (−48.61, −207.98, 150.63) km s −1 . For comparison, the present day values retained by Vasiliev et al. (2021) are (x, y, z) = (−0.6, −41.3, −27.1) kpc and (v x , v y , v z ) = (−63.9, −213.8, 206.6) km s −1 .
Figure 4 .
4Specific angular momenta of MW dwarfs (symbols) and dark matter particles in our simulation (density maps, with logarithmic scale). Panel (a) shows the initial specific angular momentum hini (and the one inferred after shifting the observed MW dwarf positions and velocities), while panel (b) shows the velocity anisotropy parameter β for all particles contributing to the corresponding bins in panel (a). Panels (c) to (e) show the angular momentum h of the same set of particles in the final simulation snapshot (and the angular momenta of the MW satellites relative to today's MW center), split into low, intermediate, and high initial angular momentum bins (as indicated by the green lines in panel a). Particles are selected from the distance range of 50 to 250 kpc in the final snapshot. The observed MW dwarfs best agree with the simulation particles in the high angular momentum bin.
Figure 5 .
5d ≤ 250 kpc, 5000 ≤ h ini ≤ 10000 kpc km s −1 , d ≤ 250 kpc, 10000 ≤ h ini ≤ ∞ kpc km s −1 , All-sky maps in Galactic Coordinates of the orbital pole enhancement for dark matter particles in our simulation. Panel (a) contains all particles within a distance range of 50 to 250 kpc, while the other panels only show subsets based on the particles' initial specific angular momenta. The green cross marks the VPOS normal vector direction, and the green circle indicates the 10% area on the sphere where the relative VPOS enhancement is measured. While panel (a) confirms the overall findings of GC21 that an LMC-like infall enhances the density of orbital poles towards the VPOS normal, low angular momentum particles (panel b) are most affected and show a pronounced overdensity, while high angular momentum particles (panel d)which are comparable to the observed MW dwarfs -show only a very slight overdensity.
Figure 6 .
6The change in orbital pole direction of particles in the simulation depends on the specific angular momentum they had initially. Panel (a) shows that particles with smaller initial specific angular momentum hini show larger changes in the direction of their orbital poles as measured by the angle ∆θ pole between their orbital pole in the initial and final simulation snapshot. Also shown are the observed MW dwarfs. Panel (b) shows the distribution of ∆θ pole for three specific angular momentum bins. Low angular momentum particles (red dotted line) are affected most, while high angular momentum particles (blue solid line) show the least change in their orbital pole directions. Poles with VPOS (50 ≤ d ≤ 250 kpc) MW satellites hini < 5 × 10 3 kpc km s −1 5 × 10 3 kpc km s −1 ≤ hini < 10 4 kpc km s −1 10 4 kpc km s −1 ≤ hini (b)
median) alignment angle of 90 • , the low, intermediate, and high specific angular momentum bins show only mildly smaller angles, namely 85.8 • (84.6 • ), 88.0 • (87.2 • ), and 89.4 • (89.1 • ) respectively. The corresponding values for the observed MW dwarfs are 63.0 • (58.5 • ). Only the low angular momentum bin shows some clear excess towards cos(θ obs VPOS ) = 1, i.e. around the VPOS direction. However, this excess is neither comparable in strength to the distribution of observed dwarf galaxy orbital poles, nor do the observed MW satellites have such low specific angular momenta.
Figure 8 .
8The enhancement of orbital pole density of dark matter particles in the simulation over that expected for isotropy in the 10% region around the VPOS normal vector, considering only particles with Galactic latitude |l| > 20 • . The enhancement is measured as a function of distance from the host center d, in radial shells of 50 kpc width. While low angular momentum particles can show very pronounced enhancements at larger distances, the higher-angular momentum particles, which are more comparable to the observed MW satellites, show a substantially lower enhancement and also dominate the sample of particles at larger distances. Also plotted are the VPOS enhancements as reported in GC21 and extracted from their figure 15. Dotted grey lines of decreasing width indicate enhancements of 0, 10, and 25%.
MSP and ST acknowledge funding of a Leibniz-Junior Research Group (project number J94/2020) via the Leibniz Competition, MSP also thanks the Klaus Tschira Stiftung and German Scholars Organization for support via a KT Boost Fund. MSP and BF acknowledge support from the Partenariat Hubert Curien (PHC) for PROCOPE project 44677UE and the Deutscher Akademischer Austauschdienst for PPP grant 57512596 funded by the Bundesministerium fur Bildung und Forschung. PAO, BF and RI acknowledge funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 834148) and from the Agence Nationale de la Recherche (ANR projects ANR-18-CE31-0006 and ANR-19-CE31-0017). This research has made use of NASA's Astrophysics Data System.
Note that many of the others remain consistent with aligning with the VPOS within their considerable proper motion uncertainties; their orbital poles are only very weakly constrained.
The Cosines are plotted as they ensure that isotropically distributed directions result in a flat distribution.
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| [] |
[
"SPARSE REGULAR RANDOM GRAPHS: SPECTRAL DENSITY AND EIGENVECTORS",
"SPARSE REGULAR RANDOM GRAPHS: SPECTRAL DENSITY AND EIGENVECTORS"
] | [
"Ioana Dumitriu ",
"Soumik Pal "
] | [] | [] | We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.Edelman[21]. Recently, more sophisticated probability tools have been generalized and applied to random matrix theory (e.g. the Lindeberg Principle, by Chatterjee [16] and Tao and Vu [60,61]).Most of the focus in universality research has been on proving, under progressively weaker assumptions on the entry distribution, the following:-convergence of the ESD to the semicircle or Marčenko-Pastur laws and establishing rates of convergence in various ways (large deviations, concentration estimates, central limit theorems). For a comprehensive treatment of the subject, see the books by Bai and Silverstein [5] and by Anderson, Guionnet, and Zeitouni [2]; -fluctuations of the spectrum at the edge (the famous Tracy-Widom laws [62, 63, 64]) for general Wigner matrices, settled by Tao and Vu [61] and Erdős, Ramirez, Schlein, Tao, Vu, and Yau [27]; -universality of correlation functions in the bulk, under various assumptions (Tao and Vu [60], Erdős, Ramirez, Schlein, Tao, Vu, and Yau [27], Erdős, Péché, Ramirez, Schlein, and Yau [26], Erdős, Yau, and Yin [28]; -partial or complete delocalization of the eigenvectors (by Erdős, Schlein, and Yau [24], [25], and Tao and Vu [60], [61]). The aim of said research has been to show that the spectral statistics agree in the large n limit to the spectral statistics of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), depending on whether the matrices are real symmetric/positive-definite, or complex hermitian/positive definite. The Gaussian and Wishart ensembles are some of the most studied and best understood random matrix models; for an easy introduction to classical random matrix theory, see the books by Mehta [49] and Muirhead[53].Parallel to these developments, in combinatorics and discrete mathematics, there has always been an interest in studying spectral properties of deterministic and random graphs. There are two matrices of interest in spectral graph theory: the adjacency (sometimes called the incidence) matrix, which we already defined, and the Laplacian matrix. These matrices are the same for regular graphs (although, in general, they can be quite different), and the spectrum of the graph is the spectrum of the matrix.Among the properties of random graphs that have been the focus of intense research are connectivity, phase transitions, and the limiting spectral distribution of random graphs, including trees(McKay [45], Feige and Ofek [29], Mirlin and Fyodorov [52], Bauer and Golinelli [9], Semerjian and Gugliandolo [55], Bordenave and Lelarge [13], Bhamidi, Evans, and Sen [11]). Other properties include concentration of eigenvalues (Krivelevich and Sudakov ([41], Alon, Krivelevich, and Vu [1]), the spectral gap (Fűredi and Komlós [34], Friedman [31] and Friedman and Alon [33], Broder and Shamir [15]).Another area of recent interest is the study of quasi-random graphs and expanders. These are nonrandom graphs which display properties one expects to hold with high-probability for certain classes of random graph models. For example, expanders are sparse graphs that have high connectivity properties (e.g., a large spectral gap). These graphs are often regular (for example, the famous Ramanujan graph, described in the seminal articles by Lubotzky, Phillips, and Sarnak [43] and Morgenstern [51]). Random d-regular graphs display the same connectivity properties with very high probability, when d is kept fixed and the order is large; this is in essence the Alon Conjecture, recently settled by Friedman[33]. Thus a study of random regular graphs suggests possible properties of (deterministic) expanders.It is easy for a probability audience to appreciate the importance of studying eigenvalues of the graph (e.g., the spectral gap which determines the mixing properties of a random walk), but eigenvectors of graphs are equally important, especially since they are the solutions of various combinatorial optimization problems. Traditionally, there has been much less work on computing the actual graph eigenvector distributions, with the notable and recent exception of Wishart-like sample covariance matrices (see Bai, Miao, and Pan [4]). Thus, developments in examining properties of the eigenvectors of large random graphs (as in Friedman [32] and Dekel, Lee, and Linial [20]) are relatively new, and motivated by the applications of eigenvectors to engineering and computer science. Such applications include the Google Page-Rank algorithm [14], the Shi-Malik algorithm [57], the Meila-Shi algorithm [50] and other spectral clustering techniques and related segmentation problems (Weiss [67], Pothen, Simon, and Liou [54], etc.).It is probably clear by now that the two fields of research that we have very briefly sketched here (universality studies in random matrix theory and spectra of random graphs) are vast and, by examining the two lists of important problems we have outlined, one can see that there is a certain amount of overlap. | 10.1214/11-aop673 | [
"https://arxiv.org/pdf/0910.5306v5.pdf"
] | 17,811,569 | 0910.5306 | 0fefa4230d0024abd7468d6d7acc10d0a6f44053 |
SPARSE REGULAR RANDOM GRAPHS: SPECTRAL DENSITY AND EIGENVECTORS
Ioana Dumitriu
Soumik Pal
SPARSE REGULAR RANDOM GRAPHS: SPECTRAL DENSITY AND EIGENVECTORS
We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.Edelman[21]. Recently, more sophisticated probability tools have been generalized and applied to random matrix theory (e.g. the Lindeberg Principle, by Chatterjee [16] and Tao and Vu [60,61]).Most of the focus in universality research has been on proving, under progressively weaker assumptions on the entry distribution, the following:-convergence of the ESD to the semicircle or Marčenko-Pastur laws and establishing rates of convergence in various ways (large deviations, concentration estimates, central limit theorems). For a comprehensive treatment of the subject, see the books by Bai and Silverstein [5] and by Anderson, Guionnet, and Zeitouni [2]; -fluctuations of the spectrum at the edge (the famous Tracy-Widom laws [62, 63, 64]) for general Wigner matrices, settled by Tao and Vu [61] and Erdős, Ramirez, Schlein, Tao, Vu, and Yau [27]; -universality of correlation functions in the bulk, under various assumptions (Tao and Vu [60], Erdős, Ramirez, Schlein, Tao, Vu, and Yau [27], Erdős, Péché, Ramirez, Schlein, and Yau [26], Erdős, Yau, and Yin [28]; -partial or complete delocalization of the eigenvectors (by Erdős, Schlein, and Yau [24], [25], and Tao and Vu [60], [61]). The aim of said research has been to show that the spectral statistics agree in the large n limit to the spectral statistics of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), depending on whether the matrices are real symmetric/positive-definite, or complex hermitian/positive definite. The Gaussian and Wishart ensembles are some of the most studied and best understood random matrix models; for an easy introduction to classical random matrix theory, see the books by Mehta [49] and Muirhead[53].Parallel to these developments, in combinatorics and discrete mathematics, there has always been an interest in studying spectral properties of deterministic and random graphs. There are two matrices of interest in spectral graph theory: the adjacency (sometimes called the incidence) matrix, which we already defined, and the Laplacian matrix. These matrices are the same for regular graphs (although, in general, they can be quite different), and the spectrum of the graph is the spectrum of the matrix.Among the properties of random graphs that have been the focus of intense research are connectivity, phase transitions, and the limiting spectral distribution of random graphs, including trees(McKay [45], Feige and Ofek [29], Mirlin and Fyodorov [52], Bauer and Golinelli [9], Semerjian and Gugliandolo [55], Bordenave and Lelarge [13], Bhamidi, Evans, and Sen [11]). Other properties include concentration of eigenvalues (Krivelevich and Sudakov ([41], Alon, Krivelevich, and Vu [1]), the spectral gap (Fűredi and Komlós [34], Friedman [31] and Friedman and Alon [33], Broder and Shamir [15]).Another area of recent interest is the study of quasi-random graphs and expanders. These are nonrandom graphs which display properties one expects to hold with high-probability for certain classes of random graph models. For example, expanders are sparse graphs that have high connectivity properties (e.g., a large spectral gap). These graphs are often regular (for example, the famous Ramanujan graph, described in the seminal articles by Lubotzky, Phillips, and Sarnak [43] and Morgenstern [51]). Random d-regular graphs display the same connectivity properties with very high probability, when d is kept fixed and the order is large; this is in essence the Alon Conjecture, recently settled by Friedman[33]. Thus a study of random regular graphs suggests possible properties of (deterministic) expanders.It is easy for a probability audience to appreciate the importance of studying eigenvalues of the graph (e.g., the spectral gap which determines the mixing properties of a random walk), but eigenvectors of graphs are equally important, especially since they are the solutions of various combinatorial optimization problems. Traditionally, there has been much less work on computing the actual graph eigenvector distributions, with the notable and recent exception of Wishart-like sample covariance matrices (see Bai, Miao, and Pan [4]). Thus, developments in examining properties of the eigenvectors of large random graphs (as in Friedman [32] and Dekel, Lee, and Linial [20]) are relatively new, and motivated by the applications of eigenvectors to engineering and computer science. Such applications include the Google Page-Rank algorithm [14], the Shi-Malik algorithm [57], the Meila-Shi algorithm [50] and other spectral clustering techniques and related segmentation problems (Weiss [67], Pothen, Simon, and Liou [54], etc.).It is probably clear by now that the two fields of research that we have very briefly sketched here (universality studies in random matrix theory and spectra of random graphs) are vast and, by examining the two lists of important problems we have outlined, one can see that there is a certain amount of overlap.
Introduction
Consider the uniform distribution over the space of all labeled simple graphs on n vertices where every vertex has degree d. We denote a graph randomly selected from this distribution by G(n, d); the vertices of G(n, d) will always be labeled by {1, 2, . . . , n}. Now, consider a sequence of such random graphs {G(n, d n ), n ∈ N} which are d n -regular of order n. We assume d n to be slowly growing with n in a manner which will be made more precise later. Consider the adjacency matrix A n of G(n, d n ); the (i, j)th element of A n is one or zero depending on whether there is an edge between vertices i and j in the graph G(n, d n ). The random matrix A n is always symmetric, and it has n real eigenvalues (perhaps not all distinct) and corresponding real eigenspaces. Under appropriate conditions on the growth of the sequence d n , we study the following phenomena as n tends to infinity:
(i) Global semicircle law. We prove the convergence of the empirical spectral distribution (ESD) of the scaled adjacency matrix to the probability measure on [−2, 2] with density
(1) f sc (x) = 1 2π 4 − x 2 , −2 < x < 2.
(ii) Local semicircle law (a.k.a. the semicircle law on short scales). We obtain concentration estimates of the deviation of the number of eigenvalues N I that lie in a small interval I from its predicted number n I f sc (x)dx. The size of I will be taken to be vanishing at an appropriate rate with increasing n. (iii) Delocalization of eigenvector coordinates. We obtain probability estimates of the event that, for some eigenvector, a few of the coordinates are significantly larger in magnitude than the rest. These problems connect two areas of study: Wigner random matrices and spectra of sparse random graphs; since we have already mentioned the latter, we will now talk about the former.
In the last couple of decades there has been an enormous amount of activity in the study of universal properties of random matrices, inspired by their connection to (universal) physical systems. The literature on universality studies in random matrices is vast; we mention here only a few references and ask the reader to look to them for further ones.
For an introduction and motivation to the subject, we recommend Deift's ICM address [18]. Of particular interest are the Wigner matrices (see Bai [3], Soshnikov [58], Bai and Yao [6], Khorunzhy, Khoruzhenko, and Pastur [37], Guionnet and Zeitouni [35], Ben Arous and Peche [10], Tao and Vu [60,61]. Deift and Goiev [19] looked at different potential functions on symmetric, hermitian, and self-dual matrices; Baik and Suidan explored connections to percolation [7] and random walks [8]; β-generalizations of the classical ensembles and universal properties thereof have been explored in Forrester and Baker [30], Johansson [36], Dumitriu and Naturally, this lead to a few papers where the two fields have intersected, despite differences in both the goals and the methodology of each.
A famous such example is McKay's derivation of the limiting empirical spectrum of random d-regular graphs on n vertices, as d is fixed and n grows to infinity [45]. In that case, the empirical spectral distribution converges in probability to what is known as the McKay (or Kesten-McKay) law, which has a density (2) f
d (x) = d 4(d − 1) − x 2 2π(d 2 − x 2 ) , −2 √ d − 1 ≤ x ≤ 2 √ d − 1.
This density had appeared earlier in Kesten's work on random walks on groups [39]. It can be easily verified that as d grows to infinity, if we normalize the variable x in the above by √ d − 1, the resulting density converges to the semicircle law on [−2, 2].
This naturally raised the question of whether the study of "universal" properties could be pushed into the domain of regular random graphs with increasing degree. The rate of growth of the degree sequence plays an important role, since at both extremes (d fixed and d = n − 1) the ESDs do not converge to the semicircle law.
The answer to this question turns out to be difficult. There are a number of major obstacles to developing an applicable universality theory in the spirit of Wigner random matrices to adjacency matrices of random graphs, which are non-Wigner: these matrices are sparse and the entries are not independently distributed.
To see how sparsity affects concentration, consider the question of proper scaling of the adjacency matrix A n . In the Wigner case, the scaling factor is clearly 1/ √ n, which puts all of the eigenvalues in [−2 − , 2 + ], for any positive , with very high probability for a sufficiently large n. One might be tempted then to believe that the proper scaling for adjacency matrices is 1/ √ d n , as this achieves the same kind of finite row-variance as 1/ √ n does in the Wigner case. Unfortunately, it is not known if this scaling will place all the eigenvalues (except the first) in a compact interval, as d n → ∞.
For the regime when d is fixed, the Alon conjecture states that the second largest eigenvalue (in absolute value) λ 2 has an upper bound |λ 2 | ≤ 2 √ d − 1 + , with very high probability. The well-known lower bound holds for every d-regular graph and we cite it from Friedman [31]:
|λ 2 | ≥ 2 √ d − 1 + O (log d/log n).
Unfortunately, when d grows with n, the upper bound is not known to hold outside of a narrow growth regime 1 . Khorunzhy [40] has shown that, for a random matrix model similar to the adjacency matrix of the Erdős-Renýi random graph on n vertices with an expected degree d n log n, with probability one, the spectral norm of the adjacency matrix grows faster than √ d n . Although this does not necessarily affect convergence of the ESD to the semicircle law, it eliminates the possibility of containing all the eigenvalues of the rescaled centralized adjacency matrix within any compact interval.
Our results investigate the extent to which universality can be extended to the slowly growing d n case. Our first result is Theorem 1 stated below. Theorem 1. Let d n satisfy the asymptotic condition
(3) lim n→∞ d n = ∞, d n − 1 = n n , for some n = o(1).
Then the ESD of the matrix (d n − 1) −1/2 A n , where A n denotes the adjacency matrix of G n , converges in distribution to the semicircle law on [−2, 2] which has a density
(4) f sc (x) := 1 2π 4 − x 2 , −2 < x < 2.
The condition on d n , for example, includes the logarithmic regime, d n = (log n) γ for any positive γ; in which case we can define n as γ log log n/ log n.
Our proof of this result (and the following ones) depends crucially on two facts: (i) the "locally tree-like" property, which states that with high probability, most vertices in a random regular graph will have a (increasingly larger) neighborhood which is free of any cycles, and (ii) the fact that d n grows to infinity, which smooths out irregularities as n tends to infinity. Our second result is arguably the most important one in this paper. Theorem 2. Fix δ > 0. Let d n = (log n) γ , where γ > 0. Let η n = (r n − r −1 n )/2 where r n = exp (d −α n ) for some 0 < α < min(1, 1/γ). Then there exists an N large enough such that for all n ≥ N , for any interval I ⊂ R of length |I| ≥ max{2η n , η n /(−δ log δ)},
N I − n I f sc (x)dx < δn |I| with probability at least 1 − o(1/n).
Here N I is the number of eigenvalues of 1 √ dn−1 A n in the interval I, and f sc refers to the density of the semicircle law as in (1).
Remark 1. Note that the shortest length of the interval I that our methods can narrow down to is of length η n , which is roughly about 1/ log n, if d n log n, and 1/d n , if d n log n. For Wigner matrices a far shorter scale can be achieved (effectively poly-log over n in [24]). Such sharp estimates are not to be expected in the graph case, and this again is a consequence of sparsity and lack of concentration estimates.
Remark 2. A close examination of the proof of Theorem 2 reveals that it can be extended to any deterministic sequence of regular graphs of increasing size and degree, as long as the "locally tree-like" property holds at "most" vertices.
Remark 3. Since the submission of this paper, significant progress has been made in proving the local semicircle law for random regular graphs in any kind of growth regime for d n (see Tran,Vu,and Wang [65]). Their methods rely on proving the local semicircle law first for Erdős-Rényi graphs with suitable parameters, and then using a result by McKay and Wormald [47] about the probabililty that an Erdős-Rényi graph is regular. Their result subsumes ours (in the sense that the lower bound on the length of the interval I is smaller) for the case when d n = Ω((log n) 10 ); when d n = o((log n) 10 ), our result is slightly stronger in the same sense.
Although these results are similar up to a point to the Wigner matrix results, our methodology is essentially different. Due to sparsity and lack of concentration, we had to adapt a more combinatorial set of tools (in particular, the tree approximation) as well as tools from linear algebra to the Stieltjes transform approach used in [24] and [60].
Several recent articles have done extensive simulations on eigenvalues and eigenvectors of random graphs, with surprising conclusions. For example, Jakobson et al. [38] carries out a numerical study of fluctuations in the spectrum of regular graphs. Their experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the GOE as we increase the number of vertices. On the eigenvector front, in the article [22] by Elon, the author attempts to characterize the structure of the eigenvectors by suggesting (with numerical observations) that all, except the first, follow approximately a Gaussian distribution. Additionally, the local covariance structure has been conjectured to be given by explicit functions of the Chebyshev polynomials of the second kind. In particular, if two vertices on the graph are at a distance k from each other, it is conjectured that the covariance between the coordinates of any eigenvector at the two vertices decays exponentially in k.
All this empirical data points to universality properties of the adjacency matrices of large, sparse regular graphs; we took here a first step toward proving them.
If the eigenvectors are indeed uniformly distributed over the sphere then they must (with high probability) satisfy delocalization. We give upper bounds on the probability of this phenomenon.
We use the following definition of delocalization, similar to the one used in [24].
Definition 1. Let T be a subset of {1, 2, . . . , n} of size L ≥ 1. Let δ > 0 be some fixed number. We say that a vector v = (v(1), . . . , v(n)) ∈ R n with L 2 norm ||v|| 2 = 1 exhibits (T, δ) localization if
||v | T || 2 2 = j∈T |v(j)| 2 ≥ 1 − δ.
The vector v is said to be (L, δ) localized there exists some set T ⊂ {1, 2, . . . , n} such that |T | = L such that v is (T, δ) localized.
Below is our result on eigenvector delocalization. (i) Let T n ⊆ {1, 2, . . . , n} be a deterministic sequence of sets of size L n = o(η −1 n ). Let Ω 1 (n) be the event that some normalized eigenvector of the matrix A n is (T n , δ) localized. Then, for all sufficiently large n,
P ((Ω 1 (n)) c ) ≥ e −Lnηn/dn 1 − o 1 d n = 1 − o 1 d n .
(ii) Define the sequence
ζ n = 1 4 log n log(d n − 1) − 4, n ≥ 2.
Consider the (random) subset J(n) of all vertices in the graph whose ζ n -neighborhood is free of cycles. Then,
P |J(n)| n ≥ 1 − η n d n = 1 − o 1 n .
Moreover, there exists an n large enough such that the event that T n ⊂ J(n) and some normalized eigenvector is (T n , δ) has probability zero.
Remark 4. More progress has been made on the eigenvector delocalization front since the submission of this paper. In their paper [65], the authors prove that the ∞ norms of all eigenvectors are o(1), regardless of the regime of growth of d n . Very recently, Erdős, Knowles, Yau, and Yin have posted a paper [23] proving the local semicircle law for Erdős-Rényi graphs with pn = Ω(log n) up to a spectral window (an interval I) of size larger than 1/n; from this, they could deduce that the eigenvectors of such Erdős-Rényi graphs are completely delocalized, i.e. that the ∞ norms of the normalized (unit) eigenvectors are at most of order 1/ √ N with high probability. It would be interesting to see if the methods of [65] (of deducing results for random regular graphs from the same results for Erdős-Rényi) can be combined with the theorems of [23] to obtain complete eigenvector delocalization (and, potentially, a much smaller spectral window) for random regular graphs with d n = Ω(log n).
The bounds in Theorem 3 is not sharp. There are severe technical obstacles in producing sharp bounds by adapting the strategy of Wigner matrices. One such example is eigenvalue collision, i.e., the event that A n does not have n distinct eigenvalues. Since A n has discrete entries, this event has a positive probability. However, to the best of our knowledge, no good bound on this probability is known.
The paper is organized as follows. In Section 2 we prove the global convergence to the semicircle law (Theorem 1). This is followed by the proof of the local semicircle law (Theorem 2) in Section 3. The eigenvector delocalization is proved in Section 4. Finally, Appendix A contains an exact calculation of eigenvalues and eigenvectors for the random regular (finite) tree, defined in Section 2.
Global convergence to the semi-circle law
Recall that G n = G(n, d n ) denotes a random d n -regular graph on n vertices whose adjacency matrix is A n . Recall that d n satisfies the asymptotic condition (5) lim n→∞ d n = ∞, d n − 1 = n n , for some n = o(1).
We prove here Theorem 1, namely, that the empirical spectral distribution (ESD) of the adjacency matrix A n converges in probability to the semicircle law on [−2, 2] which we recall from (4).
Our main instrument is to use the moment method. Our arguments depend crucially on the following local approximation of G n by a rooted tree. Consider the deterministic rooted tree S n , which is the infinite regular tree of degree d n with a distinguished vertex marked as the root. For a graph G whose every edge is taken to have unit length, consider the induced metric structure on G. We define the r-neighborhood of the vertex i, to be the subgraph of G whose vertices are at a distance at most r from i and whose edges are all the edges between those vertices. The following lemma makes precise the idea that, except for a vanishing proportion of the vertices, the r-neighborhood of any vertex is isomorphic to the corresponding neighborhood of the root in the tree S n .
Recall that a cycle is a sequence of vertices {i 1 , . . . , i k } of a graph such that i 1 = i k , there is no other repeated vertex, and there is an edge between every successive i j and i j+1 . The length of the cycle is the number of vertices except the initial one. A cycle of length k will be called a k-cycle. Finally, a cycle-free or acyclic graph is a tree. Lemma 4. Fix a positive integer r. Let τ (n) be the subset of vertices of G n which have no cycles in their r-neighborhoods, and let |τ (n)| denote the size of τ (n). Then, under the assumptions of (5), we have
P 1 − |τ (n)| n > n −1/4 = o n −5/4 .
Proof. We use the estimates of McKay, Wormald, and Wysocka [48] on the Poisson approximation to the number of short cycles in regular graphs. Let g(n) be a sequence such that
(6) g(n) ≥ 3, and (d n − 1) 2g(n)−1 = o(n).
For any s ≤ g, let M s denote the number of cycles of length s in the graph G n . It has been shown in [48] that M s is approximately distributed as a Poisson random variable and
E (M s ) = µ s (1 + O (s(s + d)/n)), where µ s = (d − 1) s 2s , and Var (M s ) = µ s + O (s(s + d)/n))µ 2 s .(7)
Consider now the growth of the degree sequence as in (5).
If we choose g(n) such that 2g(n) − 1 = 1/ √ n , it will satisfy (d n − 1) 2g(n)−1 = n √ n = o(n). Additionally g(n) grows to infinity with n, since (n) = o(1).
Consider an s-cycle for some s ≤ g(n). It has exactly s-vertices. Now, the number of vertices whose r neighborhoods fail to be acyclic because of this s-cycle are precisely those vertices which are at a distance of at most (2r − s)/2 from any of the vertices in the s-cycle. The number of such vertices has an easy upper bound of 2(d n − 1) (2r−s)/2 s, for all large enough d n . Thus, the total number of vertices whose r neighborhoods are not acyclic can be bounded above by
(8) N * r = 2r s=3 2s(d n − 1) (2r−s)/2 M s . Also,(9)
n − |τ (n)| ≤ N * r . Taking expectations on both sides of (8), and using formulas (7), we get
EN * r = 2r s=3 2s(d − 1) (2r−s)/2 (d − 1) s 2s (1 + O (s(s + d)/n))
The quantity O(s(s + d)/n) denotes a function h(s, d, n) such that n s(s + d) h(s, d, n)
remains bounded for all choices of s, d, and n. Thus we get
EN * r = (d − 1) r 2r s=3 (d − 1) s/2 + O 1 n 2r s=3 s(s + d)(d − 1) r+s/2 = O (d − 1) 2r .
The last equality is true since, by our assumption on d n , the second term in the sum is o(1). Similarly, we can compute the second moment. By the Cauchy-Schwarz inequality
Var(N * r ) ≤ 2r 2r s=3 4s 2 (d − 1) 2r−s Var(M s ) ≤ 2r 2r s=3 4s 2 (d − 1) 2r−s µ s + O (s(s + d)/n))µ 2 s ≤ 2r 2r s=3 4s 2 (d − 1) 2r−s µ s + 2r 2r s=3 4s 2 (d − 1) 2r−s O (s(s + d)/n) µ 2 s .
Plugging in the value of µ s from (7) we get
Var(N * r ) ≤ 4r 2 (2r + 1)(d − 1) 2r + 2r(d − 1) 2r 2r s=3 (d − 1) s O (s(s + d)/n) .
As before, it thus follows that
2r s=3 (d − 1) s O (s(s + d)/n) = O 2r s=3 (d − 1) s s(s + d)/n O n −1 (2r + d)(d − 1) 2r 2r s=3 = O n −1 (2r + d)(d − 1) 2r r(2r + 1) . Hence Var(N * r ) ≤ 4r 2 (2r + 1)(d − 1) 2r + O r 2 (2r + d)(d − 1) 4r /n .
Note again that, by our assumption, the quantity (
d − 1) 4r /n is o(1).
We now want to use Markov's inequality to bound the tail probability of the quantity 1 − |τ (n)| /n. Fix any > 0. Then, by inequality (9), we get
P 1 − |τ (n)| n > ≤ P (N * r > n ) ≤ 1 n 2 2 E (N * r ) 2 = 1 n 2 2 Var(N * r ) + (E (N * r )) 2 ≤ 1 n 2 2 4r 2 (2r + 1)(d − 1) 2r + r 2 (2r + d)o(1) + O (d − 1) 4r ≤ −2 O (d − 1) 4r n 2 = −2 O n 4r n −2 ,
by our choice of the sequence d n . Choosing = n −1/4 we get
P 1 − |τ (n)| n > n −1/4 ≤ √ nO n 4r n−2 = o n −5/4 , since n = o(1)
. This completes the proof of the Lemma.
Lemma 5. Let {µ i , i = 1, 2, . . .} be a sequence of random probability measures on the real line, defined on the same probability space. Let µ be a non-random continuous probability measure supported on a compact interval I. Suppose there exits a pair of doubly indexed real-valued sequences {a n (r), b n (r), r, n ∈ N} such that the following hold.
(1) For every r = 1, 2, . . ., we have
P ∞ N =1 n≥N a n (r) ≤ x r dµ n (x) ≤ b n (r) = 1.
(2) For every r = 1, 2, . . ., we have lim n→∞ a n (r) = lim
n→∞ b n (r) = x r dµ(x) < ∞.
Then the sequence of measures {µ n } converges to µ in probability.
Proof. Let Ω r be the event
∞ N =1 n≥N a n (r) ≤ x r dµ n (x) ≤ b n (r) .
Then, from condition (1), it follows that
1 − P ∞ r=1 Ω r = P ∞ r=1 Ω c r ≤ ∞ r=1 P (Ω c r ) = 0.
Thus P (∩ ∞ r=1 Ω r ) = 1. Consider any fixed realization of the sequence {µ n } ∈ ∩ ∞ r=1 Ω r . By Helly's Selection Theorem, this sequence has a limit point ν. Thus, there is a subsequence {µ n k } that converges to ν in the topology of weak convergence. Now take r to be a positive integer. We would like to show that
lim n k →∞ x r dµ n k = x r dν.
From the standard theory of weak convergence, it follows that this will be true if the function x r is uniformly integrable under the sequence of measures {µ n k }. However, uniform integrability follows from the following L 2 -boundedness condition: max
n k x 2r dµ n k < max n k b n k (2r) < ∞,
by conditions (1) and (2). In particular, from condition (2) we reach the conclusion
x r dν(x) = x r dµ(x), r = 0, 1, 2, . . .
Since the support of µ is the compact interval I, it follows that the Moment Problem has a unique solution, and hence, ν must be equal to µ. This shows that any limit point of any sequence {µ n } in ∩ ∞ r=1 Ω r is given by µ. By the usual subsequence argument, this shows that µ n converges to µ in the set ∩ ∞ r=1 Ω r , and hence with probability one. This proves the result.
Proof of Theorem 1. Consider the random graph sequence G n = G(n, d n ) as in the statement, and let A n be the adjacency matrix of G n .
Let µ n be the ESD of the matrix (d n − 1) −1/2 A n . Then, for any positive integer r,
x r dµ n (x) = 1 n tr (d n − 1) −r/2 A r n = (d n − 1) −r/2 n n i=1 A r n (i, i).
Here A r n (i, i) is the ith diagonal element of the matrix A r n . Note that A r n (i, i) counts the number of paths of length r that start and end at i. Consider the set of vertices in τ (n), as in Lemma 4, whose r/2 -neighborhood is acyclic. For any i ∈ τ (n), the number of such paths, B r n , is equal to the number of paths of size r that start and end at the root of the tree S n . If i / ∈ τ (n), we use the trivial bound A r n (i, i) ≤ d r n . Thus
(d n − 1) −r/2 |τ (n)| n B r n ≤ (d n − 1) −r/2 n n i=1 A r n (i, i) ≤ (d n − 1) −r/2 B r n + n − |τ (n)| n d r n .
If we define
(10) a n (r) = 1 − n −1/4 (d n − 1) −r/2 B r n , b n (r) = (d n − 1) −r/2 B r n + n −1/4 d r/2 n ,
then from Lemma 4 we get
(11) P a n (r) ≤ x r dµ n (x) ≤ b n (r) ≥ 1 − o n −5/4 .
In particular, by taking complements of the events above, we get
∞ n=1 P a n (r) ≤ x r dµ n (x) ≤ b n (r) c < ∞ n=1 o n −5/4 < ∞.
Now consider a product probability space on which independent copies of our (countably many) random graphs are defined. Applying the Borel-Cantelli Lemma and (11) we get that
P ∞ N =1 ∞ n=N a n (r) ≤ x r dµ n (x) ≤ b n (r) c = 0.
Taking the complements again, we get
P ∞ N =1 ∞ n=N a n (r) ≤ x r dµ n (x) ≤ b n (r) = 1.
This satisfies the condition (1) in Lemma 5.
Once we show the validity of condition (2) for µ equal to the semicircle law, we will be done by Lemma 5. Clearly, by our choice of d, n as in the statement, and the functions a n (r), b n (r) as in (10), this will be true once we establish lim
n→∞ (d n − 1) −r/2 B r n = x r f sc (x)dx.
We only need to verify above for even r, since for odd r, both sides are zero (B r n = 0 since in a tree one cannot return to the root in an odd number of steps, and the moment is zero since f sc is a symmetric density). Now, for an even r, the value of B n (r) has been computed by McKay in [45] (denoted by θ(r) in eq. (15) on [45]). It is given by
B n (r) = 2 √ dn−1 −2 √ dn−1 x r f n (x)dx, where f n (x) is the Kesten-McKay density f n (x) = d n 4(d n − 1) − x 2 2π(d 2 n − x 2 ) , −2 d n − 1 < x < 2 d n − 1. Thus, changing variable to y = (d n − 1) −1/2 x, we get lim n→∞ (d n − 1) −r/2 B r n = lim n→∞ 1 (d n − 1) r/2 2 √ dn−1 −2 √ dn−1 x r d n 4(d n − 1) − x 2 2π(d 2 n − x 2 ) dx = lim n→∞ 2 −2 y r d n √ d n − 1 4 − y 2 2π(d n − 1) 2 ((d n /(d n − 1)) 2 − y 2 /(d n − 1)) (d n − 1) 1/2 dy = 1 2π 2 −2 y r f sc (y)dy.
The last equality follows by the Dominated Convergence Theorem and the fact that lim n→∞ d n = ∞. This completes our proof.
Estimating the rate of convergence of the ESD
This is the longest section of the paper, and it is quite technical, so we provide an outline of the proof. The approach we will use is given by the Stieltjes transform of the adjacency matrix of the graph. To estimate how far the Stieltjes transform of the graph is from the Stieltjes transform of the semicircle, we will use as a stepping stone the resolvent of the adjacency matrix of a finite regular tree, which we will show to be very close to both.
The estimation consists of the following steps:
Step 0. Basic definitions and properties of the quantities involved (Section 3.1).
Step 1. Compute the resolvent of the regular tree, and show that, in a certain growth regime for d n , its (root, root) elements is close to the Stieltjes transform of the semicircle (Section 3.2).
Step 2. Show that, in the same growth regime as before, the (root, root) element of the resolvent of the regular tree is very close to the Stieltjes transform of the regular graph (Section 3.3).
Step 3. Use the estimations from the previous steps to conclude that the Stieltjes transform of the regular graph is close to that of the semicircle, and use the methods of [60] to obtain bounds on the rate of convergence of the ESD (Section 3.4).
Basic Definitions.
Definition 2. For a n × n Hermitian matrix A and a variable z ∈ C for which Im(z) > 0 (thus z is not an eigenvalue of A), define the Stieltjes transform to be the function
s(A; z) := 1 n tr (A − zI n ) −1 := 1 n tr (A − z) −1 .
Here I n is the n × n identity matrix; for convenience, we will drop the identity matrix and use the second notation.
We will also require the notion of Chebyshev orthogonal polynomials of a complex variable. For more details, see the book by Mason and Handscomb [44, p. 14].
For a complex number z, define
w = z + z 2 − 1, z = 1 2 w + w −1 ,
where the square root of a complex number is taken such that the imaginary part is always positive. It can be verified easily that for any r > 1, the set Definition 3. The nth Chebyshev polynomial of the second kind U n is defined as
(13) U n (z) = w n+1 − w −(n+1) w − w −1 , n = 1, 2, . . . ,
with U 0 (z) ≡ 1. It is easy to check that, in addition, U n (z) satisfies the recursion (14) U n (z) = 2zU n−1 (z) − U n−2 (z), n = 1, 2, . . . ,
with the initial conditions U 0 (z) = 1, U −1 (z) = 0.
Remark 5. When r = 1, the above gives us the traditional orthogonal polynomials for the semicircle law on the interval [−1, 1].
We will need the following bound on U n which can be found in [44, eqns. (1.53), (1.55)]:
(15) r n − r −n r + r −1 ≤ |U n−1 (z)| ≤ r n − r −n r − r −1 , z ∈ E r .
Finally, we will need the standard formula for inverses of symmetric block matrices, given below.
Proposition 6. Let A and D be complex symmetric matrices with sizes n × n, respectively, m × m, and let B be an m × n real matrix. Define the (m + n) × (m + n) complex symmetric matrix
M = A B B D ,
where B denotes the transpose of B. Then
(16) M −1 = A −1 + A −1 BF −1 B A −1 −A −1 BF −1 −F −1 B A −1 F −1 , F = D − B A −1 B.
Equivalently, by reversing the roles of the blocks A and D,
M −1 = G −1 −G −1 BD −1 −D −1 B G −1 D −1 + D −1 B G −1 BD −1 , G = A − BD −1 B .(17)
These formulas are easy to verify and their proofs can be found in standard matrix algebra books.
3.2.
Resolvents of regular and almost regular trees. Fix a positive integer d ≥ 2. Let T be a finite ordered rooted tree of depth ζ ∈ N such that every vertex has exactly (d − 1) children. That is, the root has degree (d − 1), and every other vertex, except the leaves, has degree d. Such a tree is almost regular since all vertices, excluding the root and the leaves, have degree d.
In order to define the adjacency matrix of this graph, we must fix a labeling; we will define this labeling recursively down to ζ = 0, in which case all we have is a root vertex which we label 1.
Imagine the tree embedded in the plane. If the depth is zero, the only element is the root, and the adjacency matrix is obvious. If the depth is one, the root has (d − 1) children. Consider each child vertex as a tree of depth zero, order their adjacency matrices H 1 , H 2 , . . . , H d−1 from left to right, and consider a block matrix with these as the diagonal blocks from upper left to bottom right. Finally add a bottom-most row and a rightmost column for the root vertex.
By induction, suppose we have labeled the adjacency matrix for the tree of depth ζ − 1. Consider now the tree of depth ζ. If we remove the root and the edges incident to it, we are left with (d − 1) trees of depth ζ − 1 arranged from left to right. We consider their (d − 1) adjacency matrices and arrange them as diagonal blocks and add the root as the last element.
Denote by H the adjacency matrix thus obtained.
Lemma 7. For any complex number z such that (z) > 0, and recall the nth order Chebyshev polynomial, U n (z). Then the elements of the resolvent of the adjacency matrix H,
(1/ √ d − 1H − z) −1 , have the following properties: (i) 1 √ d − 1 H − z −1 root,root = −1 z+ −1 z+ . . . −1 z ,
where the previous refers to a continued fraction of depth ζ (i.e., ζ + 1 recursions); (ii) the above can also be represented as
(18) ϕ(ζ) = 1 √ d − 1 H − z −1 root,root = − U ζ (z/2) U ζ+1 (z/2) ; (iii) furthermore,(19)ψ(ζ) = 1 √ d − 1 H − z −1 root,leaf = − (d − 1) −ζ/2 U ζ+1 (z/2) ,
where leaf represents any leaf of T;
Proof.
(i) Note that when ζ = 0 (i.e., the tree has only the root vertex), the equality is trivially true. We proceed by induction. Suppose the equality is true until depth ζ − 1. Consider a tree of depth ζ and label the adjacency H matrix as above. Thus
(20) 1 √ d − 1 H − z = 1 √ d−1 H 1 − z 1 √ d−1 H 2 − z u . . . 1 √ d−1 H d−1 − z u −z .
Here u is the column vector representing the children of the root. Notice that u is (d − 1) −1/2 exactly at the (d − 1) coordinates which are the last elements in each of the block matrices H 1 , . . . , H d−1 and zero elsewhere. The vector u is the transpose of u. We now use formula (16) treating the the final element [−z] as one block. Thus if ϕ(ζ) denote the element on the left side of (i) above, we get ϕ(ζ) = F −1 , where
F = −z − 1 d − 1 i∼root ϕ(ζ − 1) = −z − ϕ(ζ − 1).(21)
Here "i ∼ root" refers to the children of the root which are, in their turn, the roots of trees of depth ζ − 1. The formula now follows by induction.
(ii) We will use the three term recurrence formula for continued fractions which we state below. More details can be found in the excellent book by Lorentzen and Waadeland [42, p. 5-6]. Given sequences of complex numbers {a n } and {b n } and a complex argument ω, one can define a continued fraction function with argument ω by defining
S n (ω) = b 0 + a 1 b 1 + a 2 b 2 + .
. . a n b n + ω .
By Lemma 1.1 in [42] we get the existence of complex sequences {A n } and {B n } such that S n (ω) = A n−1 ω + A n B n−1 ω + B n for n = 1, 2, . . .
where (22)
A n = b n A n−1 + a n A n−2 , B n = b n B n−1 + a n B Comparing with the recursions of the Chebyshev polynomials U n given in (14) we get that B n (z) = U n (z/2) , A n (z) = −U n−1 (z/2).
Since clearly ϕ(ζ) = S ζ+1 (0) we get formula (18). This proves part (ii). (iii) Since there is an obvious isomorphism of the tree that can exchange the labeling of leaves, it is enough to consider the leaf labeled 1 in the adjacency matrix. We will express
ψ(ζ) = 1 √ d − 1 H − z −1 1,root .
in terms of ψ(ζ − 1). Let N be the total number of vertices in the tree, and let A be the diagonal block matrix which is the upper left (N − 1) × (N − 1) block in (20). Then from the formula of inverses of block matrices we get
ψ(ζ) = −F −1 (A −1 u) 1 ,
where F is defined in (21). But F −1 is ϕ(ζ) and simplifying the elements of w = A −1 u, we see that w(1) = (d − 1) −1/2 ψ(ζ − 1). In other words, ψ(ζ) = −(d − 1) −1/2 ϕ(ζ)ψ(ζ − 1). We get by induction
ψ(ζ) = (−1) ζ (d − 1) ζ/2 ζ i=0 ϕ(i), ζ = 1, 2, . . . .
Now we substitute formula (18) to obtain (19):
ψ(ζ) = (−1) ζ (d − 1) ζ/2 ζ i=0 − U i (z/2) U i+1 (z/2) = − 1 (d − 1) ζ/2 U 0 (z/2) U ζ+1 (z/2) = − (d − 1) −ζ/2 U ζ+1 (z/2) .
Having now calculated the quantities ϕ and ψ for this "slightly irregular" tree T, let us use them to find the corresponding quantities for the regular one, where the root is adjacent (just like all of the other non-leaf nodes) to precisely d edges (and thus has d children). We consider the same kind of labeling as before.
Lemma 8 below is a variation of Lemma 7 above.
Lemma 8. Let T d denote a d-regular tree of depth ζ ∈ N such that every vertex has degree d. Let H d denote the adjacency matrix of the graph. The entries of its resolvent (1/ √ d − 1H d − z) −1 have the following properties:
(i) (23) ϕ d (ζ) := 1 √ d − 1 H d − z −1 root,root = − U ζ (z/2) U ζ+1 (z/2) − (d − 1) −1 U ζ−1 (z/2) ;
(ii) the above can also be represented as
(24) ψ d (ζ) := 1 √ d − 1 H d − z −1 root,leaf = − (d − 1) −ζ/2 U ζ+1 (z/2) − (d − 1) −1 U ζ−1 (z/2)
where leaf represents any leaf of T d ;
Proof. The proof is identical to that of the last lemma, except that we need to be careful in the first step of the recursion. Since the labeling of the vertices has the same principle as before, we have
(25) 1 √ d − 1 H d − z −1 root,root = 1 −z − d d−1 ϕ(ζ − 1)
, where ϕ(·) has been defined in (21). This reflects the fact that the only change from before is in the number of children on the root (used to be d − 1, now is d).
Substituting the value of ϕ from (18) we get
1 √ d − 1 H d − z −1 root,root = −z + d d − 1 U ζ−1 (z/2) U ζ (z/2) −1 = U ζ (z/2) −zU ζ (z/2) + U ζ−1 (z/2) + (d − 1) −1 U ζ−1 (z/2) = − U ζ (z/2) U ζ+1 (z/2) − (d − 1) −1 U ζ−1 (z/2) .
The final step above follows from the recursion of the Chebyshev polynomials given in (14). This proves (i). The proof of (ii) follows by a similar argument.
Recall that we are ultimately interested in how close the Stieltjes transform of the d-regular graph is to the Stieltjes transform of the semicircle, s(z). Toward this goal, we will need some estimates for the functions defined in Lemma 8. Lemma 9. Consider the functions defined in Lemmas 7 and 8. Then for all z such that z/2 ∈ E r = y ∈ C : y + y 2 − 1 = r , for some r > 1 such that r −ζ < 1/2, one has the following estimates.
(i) Consider ϕ d (ζ) and s(z) = −(z − √ z 2 − 4)/2. We have the following estimate
(26) |ϕ d (ζ) − s(z)| ≤ C 0 2r −2ζ 1 − r −2ζ−2 + 1 d − 1 where C 0 is a constant. (ii)
The following bound on ψ(·) holds:
(27) |ψ(ζ)| ≤ r −ζ−1 (d − 1) ζ/2 2 1 − r −2ζ−4 . (iii) Similarly (28) |ψ d (ζ)| ≤ C 0 r −ζ−1 (d − 1) ζ/2 1 1 − r −2ζ−4 .
Remark 6. The condition r −ζ < 1/2 is a priori more restrictive than necessary for the purposes of Lemma 9; since we will in fact be interested in the case when r −ζ = o(1), this does not matter.
Proof.
(i) Let ω = (z + √ z 2 − 4)/2 = −1/s(z). Note that |ω| = r > 1. Recall from (23) and (13) that
ϕ d (ζ) = ω ζ+1 − ω −ζ−1 ω ζ+2 − ω −ζ−2 − 1 d−1 (ω ζ − ω −ζ ) = 1 ω 1 − ω −ζ−2 1 − ω −2ζ−4 − 1 d−1 (ω −2 − ω −2ζ−2 )
.
Thus
|ϕ d (ζ)| ≤ r −1 1 + r −ζ−2 1 − r −2ζ−4 − (d − 1) −1 (r −2 + r −2ζ−2 )
as long as the right side above is positive. Since r > 1 we get ϕ d (ζ) is bounded by an absolute constant C 0 when r −ζ < 1/2.
We now estimate the quantity ϕ(ζ). By (18) and (13) we get
(29) ϕ(ζ) = − ω ζ+1 − ω −ζ−1 ω ζ+2 − ω −ζ−2 = −ω −1 1 − ω −2ζ−2 1 − ω −2ζ−4 .
Thus
ϕ(ζ) + ω −1 = |ϕ(ζ) − s(z)| = ω −1 1 − 1 − ω −2ζ−2 1 − ω −2ζ−4 = r −1 ω −2ζ−2 − ω −2ζ−4 1 − ω −2ζ−4 = r −2ζ−3 1 − ω −2 1 − ω −2ζ−4 ≤ r −2ζ−3 1 − r −2ζ−4 .
To get to ϕ d , consider the formula (25). Note that
− 1 z − s(z) = − 1 z/2 + √ z 2 − 4/2 = −ω −1 = s(z).
Thus
ϕ d (ζ) − s(z) = − 1 z + d d−1 ϕ(ζ − 1) + 1 z − s(z) Hence |ϕ d (ζ) − s(z)| = s(z) − d d − 1 ϕ(ζ − 1) 1 z − s(z) 1 z + d d−1 ϕ(ζ − 1) = d d − 1 (s(z) − ϕ(ζ − 1)) − s(z) d − 1 ω −1 |ϕ d (ζ)| ≤ C 0 d d − 1 r −2ζ−1 1 − r −2ζ−2 + r −1 d − 1 r −1 .
Since r > 1 this completes the proof. (ii) We use the estimate on the Chebyshev polynomials given in (15) and our assumption on z to get
|ψ(ζ)| = 1 (d − 1) ζ/2 1 U ζ+1 (z/2) ≤ 1 (d − 1) ζ/2 r + r −1 r ζ+2 − r −ζ−2 = r −ζ−1 (d − 1) ζ/2 1 + r −2 1 − r −2ζ−4 .
(iii) This part is similar, since
|ψ d (ζ)| = 1 (d − 1) ζ/2 1 U ζ+1 (z/2) 1 1 − (d − 1) −1 U ζ−1 (z/2)/U ζ+1 (z/2) ≤ r −ζ−1 (d − 1) ζ/2 1 + r −2 1 − r −2ζ−4 1 1 − (d − 1) −1 |U ζ−1 (z/2)/U ζ+1 (z/2)| .
Note that, by way of its definition, the constant C 0 in part (i) is an upper bound on the final term-hence the estimate.
3.3.
From trees to regular graphs. Consider now a (deterministic) d-regular graph G with a distinguished vertex called the root such that, for some ζ ≥ 1, the (ζ + 1)-neighborhood of the root is a tree. That is to say, consider the subgraph consisting of all vertices in G whose distance from the root is at most ζ + 1 and the edges between them; we assume that this subgraph has no cycles. This gives us a natural partition of the graph. We denote the tree subgraph induced by the root and all vertices of distance at most ζ from the root by T d . We denote the "boundary" of this graph, that is, the set of vertices that are at distance exactly ζ from the root, by ∂T d . The subgraph induced by the vertices in the complement of T d will be denoted by T c d , and its own boundary, that is, the set of vertices at distance exactly ζ + 1 from the root, will be denote by ∂T c d . For further clarification, please refer to Figure 1. Note that all edges betwen T d and T c d are between ∂T d and ∂T c d . Additionally we will denote the set of vertices of
T d (respectively T c d ) by V (T d ) (respectively V (T c d )). T c T T c T d . . . . . . ...A = D B B H d .
Here D is the adjacency matrix of T c d , the matrix B records only and all the edges between ∂T d and ∂T c d , and we again use the notation B for the transpose of B.
We will now proceed to estimate how close the Stieltjes transform of the regular graph is to that of the tree.
Lemma 10. Fix a complex number z such that z ∈ E r (see (12)) for some r > 1. Let denote the quantity
= 1 √ d − 1 A − z −1 root,root − 1 √ d − 1 H d − z −1 root,root
We have the following bound
| | ≤ 2C 2 0 1 − r −2ζ−4 r −2ζ−2 (z) . Proof. Define the vector v = B 1 √ d − 1 H d − z −1 e root ,
where e root is the vector that puts mass at the root vertex and zero elsewhere. Then by using the formula for the inverse of block matrices (17) we get
(30) = 1 d − 1 v 1 √ d − 1 D − z − 1 d − 1 B 1 √ d − 1 H d − z −1 B −1 v .
Our first job is to estimate the elements of the vector v. From the definition it is clear that the rows of B (and columns of B ) are labeled by the vertices of T c d . We write B i * to designate the ith row of B. We obtain
v i = B i * 1 √ d − 1 H d − z −1 e root = k∈V (T d ) B ik 1 √ d − 1 H d − z −1 k,root = k∈V (T c d ) B ik 1 √ d − 1 H d − z −1 root,k , by symmetry of H d .(31)
Note that B ik is positive (i.e., 1) if and only if i ∈ ∂T c d , k ∈ ∂T d and i and k have an edge between them.
Thus (i) v i = 0 unless i ∈ ∂T c d ; (ii) when i ∈ ∂T c d ,(32)v i = k∈∂T d ,k∼i 1 √ d − 1 H d − z −1 root,k = ψ d (ζ).
Here ψ d (ζ) has been defined in (24). The fact that there is exactly one k ∈ ∂T d such that i ∼ k follows from our assumption that the (ζ + 1) neighborhood of the root is a tree (see Figure 1); (iii) by counting the number of elements in ∂T c d , we get
(33) v 2 = d(d − 1) ζ |ψ d (ζ)| 2 ≤ C 2 0 dr −2ζ−2 1 − r −2ζ−4 ,
where the final estimate is from (28).
Note that the matrix
1 √ d − 1 D − z − 1 d − 1 B 1 √ d − 1 H d − z −1 B
is precisely the matrix G appearing in (17). In particular, it is the top left block of the matrix (d
− 1) −1/2 A − z −1 .
Hence, by padding the vector v with extra zeros, we get a vectorv such that
= 1 d − 1v 1 √ d − 1 A − z −1v .
Since the matrix A is real symmetric, it has only real eigenvalues. It follows from spectral decomposition of the real, symmetric matrix A, that for any real vector y
(34) y 1 √ d − 1 A − z −1 y ≤ y 2 (z) .
Observe from (32) that v = ψ d (ζ)e, where e is a real vector of ones and zeroes which is one precisely for the labels corresponding to ∂T d .
Combining our previous observations, we get
v z − 1 √ d − 1 D − 1 d − 1 χ −1 v = ψ 2 d (ζ)e 1 √ d − 1 A − z −1 e.
Now using the bound in (33) to (30) we get
| | ≤ ψ 2 d (ζ) d − 1 e 2 (z) = 1 d − 1 v 2 (z) ≤ C 2 0 d d − 1 r −2ζ−2 1 − r −2ζ−4 1 (z) , ≤ 2C 2 0 1 − r −2ζ−4 r −2ζ−2 (z) .
This completes the proof of the lemma.
We now arrive at our main result about deterministic regular graphs. Consider the set-up as in Lemma 10. Now a consider a sequence of graphs G n such that G n is d n -regular. Each G n has a marked vertex called the root such that for some sequence {ζ n }, the ζ n + 1 neighborhood of the root is acyclic in G n .
Lemma 11. Assume that that sequences {d n } and {ζ n } both tend to infinity with n with the following restriction. There exists a sequence r n such that (35) r n = e d −α n , for some 0 < α < 1, and r −ζn
n = o(1/d n ).
Then, if A n denotes the adjacency matrix of the graph G n , for all d n such that d n ≥ (log 2) −1/α , we have
(36) 1 √ d n − 1 A n − z −1 root,root − s(z) = O (1/d n ) ,
where s(z) is the Stieltjes transform of the semicircle law s(z) = −z/2 + √ z 2 − 4/2.
Proof. Under our assumptions certain simplifications are immediate. The constant C 0 appearing in Lemma 9 (and later) can be taken to be an absolute constant. Since r −ζn
n = o(1/d n ), we can choose C 0 large enough in (26) such that (37) |ϕ dn (ζ n ) − s(z n )| ≤ C 0 /d n .
Here z n is any sequence of complex numbers such that z n /2 belongs to the ellipse E rn defined in (12). Note that
ϕ dn (ζ n ) = 1 √ d n − 1 A n − z n −1 root,root .
We now use Lemma 10. Consider now
(z) = r n − r −1 n 2 = 1 2 e d −α n − e −d −α n .
One can easily verify the inequality
(38) e x − e −x 2 ≥ x/2, for all x ≤ log 2.
Applying this inequality above, we get (z) ≥ d −α n /2 for all d n as stated in the lemma. Combining with Lemma 10 we get
| | ≤ 2C 2 0 r −2ζn n 1 − r −2ζn−4 n d α n ≤ C 1 1 d 2 n d α n = o (1/d n ) .
Combining with (37) this completes the proof of the theorem.
We will need the following lemma to show that the previous "tree approximation" result about deterministic regular graphs can be applied to the random graph by choosing almost any vertex as the "root".
Lemma 12. Let d n = (log n) γ , for some positive γ. Let {r n } be as in Lemma 11. Let η n be defined as (r n − r −1 n )/2. For some β > 1, we define the sequence {ζ n , n = 1, 2, . . .} satisfying
(39) ζ n = 1 4 log n log(d n − 1) − β.
Let J(n) be the set of vertices in G n whose ζ n -neighborhoods are acyclic and let |J(n)| denote its size. Let Ω(n) be the event (40) |J(n)|/n > 1 − η n /d n .
One can choose β no larger than 4 such that
(41) P (Ω c (n)) ≤ o 1 n .
Proof. The proof of this result is very similar to the proof of Lemma 4. We again use the estimates of McKay, Wormald, and Wysocka [48] on the Poisson approximation to the number of short cycles. Consider a sequence {ζ n } as in the statement. It is clear from the choice that 4ζ n log(d n − 1) = log n − β log(d n − 1), or (42)
(d n − 1) 4ζn = n (d n − 1) β = o(n) .(43)
Thus, we can take g = 2ζ n in (6).
The argument is essentially the same as the one used in the proof of Lemma 4; rather than repeating it, we choose to only highlight the differences.
The total number of vertices whose ζ n -neighborhoods are not acyclic can be bounded above by
N * ζn = 2ζn s=3 s(d n − 1) (2ζn−s)/2 M s ,
where M s is the number of cycles of length s. Taking expectations and variances above we get
EN * ζn = O (d n − 1) 2ζn and
Var(N * ζn ) ≤ ζ 2 n (2ζ n + 1)(d n − 1) 2ζn + ζ 2 n (2ζ n + d n )o(1) .
We now want to use Chebyshev's inequality on the quantity 1 − |J(n)| /n:
P 1 − |J(n)| n > η n d n ≤ P N * ζn > nη n /d n ≤ d 2 n n 2 η 2 n E N * ζn 2 ≤ 3d 2 n ζ 3 n (d n − 1) 2ζn n 2 η 2 n + 3ζ 3 n d 2 n o(1) n 2 η 2 n + d 2 n nη 2 n O (d n − 1) 4ζn n .
Let us analyze the three terms that appear above. We use the inequality (38) to obtain η n ≥ d −α n /2. Using (42) we get
3d 2 n ζ 3 n (d n − 1) 2ζn n 2 η 2 n ≤ 3d 2 n ζ 3 n √ n (d n − 1) β/2 1 n 2 4d 2α n = poly-log n n 3/2 . 3ζ 3 n d 2 n o(1) n 2 η 2 n ≤ 12ζ 3 n d 2+2α n o(1) n 2 = poly-log n n 2 o(1).
The leading term is the last term on the right which is of the order d 2−β n /(nη 2 n ), because by (42) we get (d n − 1) 4ζn /n = (d n − 1) −β .
We now choose β > 2α + 2 (and thus no larger than 4, since α < 1) such that
d 2−β n η 2 n ≤ 4d 2−β n d 2α n = o(1).
This completes the proof of the lemma.
The Stieltjes transform.
We bring now all the results of the previous sections together. Below is the first theorem of this section.
Theorem 13. Consider a sequence of d n -regular graphs on n vertices where d n = (log n) γ for some γ > 0. Let A n be the adjacency matrix of the graph and consider the Stieltjes transform
s n (z) = tr 1 √ d n − 1 A n − z −1 , (z) > 0.
Let η n = (r n − r −1 n )/2 where r n = exp (d −α n ) for some 0 < α < min(1, 1/γ). Let U n denote all complex numbers z such that (z) > η n . Then there is a large enough constant C > 0 such that the Stieltjes transform of the empirical eigenvalue distribution of the n × n matrix A n satisfies P sup
z∈Un |s n (z) − s(z)| > C/d n ≤ o (1/n) .
Here s(z) refers to the Stieltjes transform of the semicircle law.
Proof of Theorem 13. We condition on the event Ω(n) which, from Lemma 12, happens with a probability of at least 1 − o(1/n).
Consider the set J(n) from the Lemma 12 and write
(44) s n (z) = 1 n k∈J(n) 1 √ d n − 1 A n − z −1 k,k + 1 n k / ∈J(n) 1 √ d n − 1 A n − z −1 k,k . Now, the spectral norm of 1 √ dn−1 A n − z −1
is bounded above by 1/ (z), which in turn is bounded above by η −1 n for all z ∈ U n . Thus, for all z ∈ U n and all k, we have the obvious bound
1 √ dn−1 A n − z −1 k,k ≤ η −1 n . Summing up over all k / ∈ J(n), we get 1 n k / ∈J(n) 1 √ d n − 1 A n − z −1 k,k ≤ n − |J(n)| nη n ≤ 1 d n ,
where the final inequality holds on the event Ω. Thus, we get
s n (z) = 1 n k∈J(n) 1 √ d n − 1 A n − z −1 k,k + O 1 d n .
Now, on the event Ω(n), every k ∈ J(n) can be considered as the root for a d n -regular tree of depth ζ n . We can now apply Lemma 10 and Lemma 11. Note that, technically, the ζ n in Lemma 12 and the ζ used in Lemma 10 differ by at most one; however, this does not affect the following calculations.
We verify the assumptions in Lemma 11:
r −ζn n = exp − 1 4 d −α n log n log(d n − 1) + βd −α n = exp − 1 4 (log n) 1−αγ log(d n − 1) + β (log n) α ≤ C exp − (log n) 1−αγ 4γ log log n exp (−γ log log n) = 1 d n ,
whenever αγ < 1. We now combine our error estimate (36) to see that for any k ∈ J(n), on the event Ω(n), and for all z ∈ U n , we have
(45) 1 √ d n − 1 A n − z −1 (k, k) − s(z) = O 1 d n ,
where the constants in the O(·) above does not depend on k.
Combining this with the decomposition (44) we get
(46) sup z∈Un |s n (z) − s(z)| = O 1 d n .
on the event Ω(n), which holds with probability at least 1 − o(1/n). This completes the proof of the Lemma.
Recall now the setup of Theorem 2. Fix δ > 0. Let d n , η n be as in Theorem 13. Then we will show that there exists an N large enough such that for all n ≥ N , for any interval I ⊂ R of length |I| ≥ max{2η n , η n /(−δ log δ)},
N I − n I f sc (x)dx < δn |I| with probability at least 1 − o(1/n).
Here N I is the number of eigenvalues of W n in the interval I, and f sc refers to the density of the semicircle law as in (1).
Proof of Theorem 2. Theorem 13 leads to Theorem 2 whose proof follows almost identically to Lemma 60 in the article by Tao and Vu [60]. The only major difference between our theorem and Lemma 60 of [60] is the fact that our interval I can lie anywhere on the real line and not restricted to a subset of (−2, 2) as Lemma 60 requires. We provide an outline of the argument but skip the details.
The idea lies in the observation that a good control over the Stieltjes transform near the real line allows one to invert the transform and have an estimate of the empirical spectral density. This is due to the following inversion formula: if G is a continuous distribution on the real line with Stieltjes transform s G , one gets
G[a, b] = lim →0 1 π b a (s G (x + i )) dx.
Fix an interval I ⊆ [−2, 2] such that |I| ≥ 2η. Define the function F (y) = 1 π I η n η 2 n + (y − x) 2 dx. Then it follows that if λ i denotes the ith eigenvalues of the matrix
1 √ dn−1 A n , then 1 n n i=1 F (λ i ) = 1 π I 1 n n i=1 η n η 2 n + (λ i − x) 2 dx = 1 π I (s n (x + iη n )) dx.
Also, if f sc denotes the density of the semicircle law, we get
2 −2 F (y)f sc (y)dy = 1 π I (s(x + iη n )) dx.
Using the approximation between s n and s obtained in Theorem 13 we get
1 n n i=1 F (λ i ) − 2 −2 F (y)f sc (y)dy ≤ 1 π I |s n (x + iη n ) − s(x + iη n )| dx ≤ C |I| d n , with probability 1 − o 1 n .
Choose n large enough such that 1/d n ≤ δ for all subsequent n. Thus
1 n n i=1 F (λ i ) = 2 −2 F(1 n n i=1 F (λ i ) = N n + O η n log |I| η n .
Putting all these together we obtain that with probability 1 − o(1/n) one has
N I − n I f sc dy = O (nδ |I|) + O nη n log |I| η n .
Finally, as observed in Tao & Vu [60], the latter term can be absorbed in the former since |I| ≥ η n /δ log(1/δ). This completes the proof of the Theorem.
Delocalization of eigenvectors
Closely related to approximation of the empirical spectral distribution is the fact that the L 2 -norm of a normalized eigenvector restricted to a large subset of the vertices cannot be small. We follow Definition 1 and prove Theorem 1.
Recall the set-up of Theorem 2. Fix δ > 0. Let T n ⊆ {1, 2, . . . , n} be a sequence of sets of size L n = o(η −1 n ). Let Ω 1 (n) be the event that some unit-norm eigenvector of the matrix A n is (T n , δ) localized, that is,
Ω 1 (n) = {∃ i : ||v i | Tn || 2 ≥ 1 − δ, for some v i such that A n v i = λ i v i } .
Then, for all sufficiently large n, we will show that
P ((Ω 1 (n)) c ) ≥ e −Lnηn/dn 1 − o 1 d n = 1 − o 1 d n .
Proof of Theorem 3. Consider the set J(n), defined in Lemma 12, of vertices whose ζ n -neighborhoods are acyclic. Recall the event Ω(n), whose probability (as soon as n is large enough) is 1 − o(1/n), which is that |J(n)|/n > 1 − η n /d n . We first prove part (i). The event Ω 1 (n) can be decomposed as two disjoint events depending on whether the set T n is a subset of J(n) or not. We first examine the event
Ω 1 (n) = {T n ⊆ J(n)} ∩ Ω 1 (n) ,
for purposes of exclusion.
Assume ω ∈ Ω(n) ∩ Ω 1 (n), and fix i, v i depending on ω and n. The matrix A n has n real eigenvalues and corresponding eigenspaces. The top eigenvalue is d n with corresponding eigenvector v 1 = n −1/2 1, which is completely delocalized. Let us now choose v 2 , v 3 , . . . , v n to be a set of normalized eigenvectors, corresponding respectively to the eigenvalues λ 2 , λ 3 , . . . , λ n of 1 √ dn−1 A n . Fix a subset T n ⊆ {1, 2, . . . , n} of size L n as stated in the theorem.
Consider again the Stieltjes transform of the matrix (d n − 1) −1/2 A n as in the proof of Theorem 13. That is, for z ∈ C, with (z) > 0, consider the matrix
1 √ dn−1 A n − z −1 .
Then by the spectral representation it follows that for any 1 ≤ k ≤ n we get
1 √ d n − 1 A n − z −1 (k, k) = n j=1 v 2 j (k) (λ j − z)
.
Summing over the vertices k ∈ T n , we get
(47) k∈Tn 1 √ d n − 1 A n − z −1 (k, k) = n j=1 k∈Tn v 2 j (k) (λ j − z)
.
Taking the imaginary part on both sides we get
k∈Tn 1 √ d n − 1 A n − z −1 (k, k) = n j=1 k∈Tn v 2 j (k) 1 λ j − z . ≥ k∈Tn v 2 i (k) (λ i − z) −1 ≥ (1 − δ) (λ i − z) −1 .(48)
Now we use the fact that for ω ∈ Ω(n) and for all k ∈ J(n), we get from equation (45) that
sup z∈Un 1 √ d n − 1 A n − z −1 (k, k) − s(z) ≤ C d n ,
for some absolute constant C > 0. Recall that s(z), the Stieltjes transform of the semicircle density, is given by
s(z) = 1 2 −z + z 2 − 4 .
Thus, summing up over all k in T n we get sup z∈Sn k∈Tn
1 √ d n − 1 A n − z −1 (k, k) − L n s(z) ≤ CL n d n .
Combining this estimate with (48) for z such that (z) = λ i and (z) = η n we get
L n (s(z)) + CL n /d n ≥ η −1 n (1 − δ) for all ω ∈ Ω(n) ∩ Ω 1 (n).
Since (s(z)) is bounded and L n = o η −1 n , there is a large enough N such that for all n ≥ N , this inequality will not hold when ω ∈ Ω 1 (n). So we get that P (Ω(n) ∩ Ω 1 (n)) = 0 ,
as soon as n is large enough.
We can now write Ω 1 (n) = ((Ω(n)) c ∩ Ω 1 (n)) ∪ (Ω(n) ∩ Ω 1 (n)) ; the probability of the first of the two events above is bounded by the probability of (Ω(n)) c which is o(1/n). Further, by (49), we get that Ω(n) ∩ Ω 1 (n) = Ω(n) ∩ (Ω 1 (n) \ Ω 1 (n)) .
Note that the last event is equivalent to saying that Ω(n) and Ω 1 (n) happen, and that in addition T n ⊆ J(n).
Putting all of these together, we conclude that
P [Ω 1 (n)] ≤ 1 − e −Lnηn/dn 1 − o 1 d n ,
which completes the proof. Note that part (ii) is proved in (49).
We now use the following well-known formula of determinant of block matrices (akin to (16)):
det
A B C D = det(A) det D − CA −1 B .
We apply this to the matrix z − (d − 1) −1/2 H treating the the final element [z] as one block: Note that, by our labeling, in this case A is a diagonal block matrix, and hence its determinant is a product of the determinants of the individual blocks which are all the same and equal to ∆(z; ζ − 1). Thus we get ∆(z; ζ) = (∆(z; ζ − 1)) d−1 z − u A −1 u .
As shown in Section 3.2, the quantity
z − u A −1 u = z + ϕ(ζ − 1) = z − U ζ−1 (z/2) U ζ (z/2) = zU ζ − U ζ−1 U ζ = U ζ+1 (z/2) U ζ (z/2) .
Here U n is the Chebyshev polynomial of the second kind. Note the reversal of sign from Section 3.2 which is due to current reversal of sign from the resolvent matrix. Hence
∆(z; ζ) = U ζ+1 (z/2) U ζ (z/2) (∆(z; ζ − 1)) d−1 = ζ i=0 U ζ+1−i (z/2) U ζ−i (z/2) (d−1) i
The last term can be verified from the initial conditions of the Chebyshev polynomials. Simplifying a bit more, we get
∆(z; ζ) = U ζ+1 (z/2) ζ i=1 U (d−1) i −(d−1) i−1 ζ+1−i (z/2).
For part (ii), note from above that the i many zeroes of U i appear with multiplicity (d − 1) i − (d − 1) i−1 . Hence the total number of eigenvalues are
ζ + 1 + ζ i=1 (ζ + 2 − i) (d − 1) i − (d − 1) i−1 = ζ i=0 (d − 1) i
which is the total number of vertices of the tree.
The zeros of Chebyshev polynomials of order k can be easily shown to be given by cos jπ k + 1 , j = 1, 2, . . . , k.
Thus an interesting phenomenon transpires in this analysis. If we drop the multiplicities and consider the empirical distribution of the distinct eigenvalues, they are the zeros of the Chebyshev polynomials of increasing order. These zeros are cosine transformations of equidistant points on the unit circle; and hence their empirical distribution converges to the arc-sine law. However, the entire empirical spectral distribution converges (see [13]) to the spectral distribution of the infinite tree which is the semicircle law. The effect of the multiplicities is strong enough to flip the "smile" of the arc-sine law to the "frown" of the semicircle! Also note that the gap of the spectrum from 2 is about twice of π 2 /(ζ + 1) 2 , and does not depend on d.
Some facts about eigenvectors of this tree are also easy to derive and might be also worthwhile to look at. For example, by the spectral theorem, one can write
(53) − U ζ (z/2) U ζ+1 (z/2) = ϕ(z) = 1 √ d − 1 H − z −1 root,root = i P i e root 2 λ i − z .
Here the sum on the right goes over distinct eigenvalues of the adjacency matrix and P i e root refers to the projection of the vector e root on the eigenspace corresponding to λ i . Notice that the above is a meromorphic function of z. From the leftmost expression in (53), it is obvious that the function has poles at (twice) the zeros of U ζ+1 . It follows then that P i e root is zero for all eigenvalues except when λ i is twice of a root of U ζ+1 . However, these roots are simple, as we show in the previous lemma. Hence, P i e root 2 is precise the square of the 'root'-coordinate of the ith eigenvector.
Its value can be easily computed. For any root λ i of U ζ+1 , we get
P i e root 2 = lim z→2λi (z − 2λ i ) U ζ (z/2) U ζ+1 (z/2) = 2U ζ (λ i ) U ζ+1 (λ i ) .
Here U ζ+1 refers to the derivative of the polynomial U ζ+1 . Other coordinates can be similarly derived.
Theorem 3 .
3Assume the set-up of Theorem 2. Fix δ > 0.
( 12 )
12E r := {z : |w| = r} is an ellipse whose foci are at {+1, −1}. when r = 1, this ellipse degenerates to the interval [−1, 1].
− 2 with
2initial values A −1 = 1, A 0 = b 0 , B −1 = 0 and B 0 = 1. In our case we will take each a i = −1 and each b i = z, except b 0 = 0. The recursions in (22) give us A n = zA n−1 − A n−2 , B n = zB n−1 − B n−2 with the initial values A −1 = 1, A 0 = 0, B −1 = 0 and B 0 = 1.
Figure 1 .
1Illustration of T d , T c d , ∂T d , and ∂T c d starting at a vertex root, with maximal tree depth ζ. Note that the neighbors in T c d of the vertices from ∂T c d may, but need not continue the tree-like structure. For this particular tree, d = 4. Let A denote the adjacency matrix of this graph and let H d denote the adjacency matrix of the subgraph T d . Label the vertices of H d as in Lemmas 7 and 8, and write A in the block matrix form
PP
[Ω(n) ∩ (Ω 1 (n) \ Ω 1 (n))] ≤ P [Ω(n) ∩ {T n ⊆ J(n)}]= P [{T n ⊆ J(n)} | Ω(n)] P [Ω({T n ⊆ J(n)} | Ω(n)] .Note now that, given the size j = |J(n)|, any set of j labels chosen from {1, . . . , n} is just as likely as any other, and independent of the set T n . SoP [{T n ⊆ J(n)} | |J(n)| = j] = 1 − P [{T n ⊆ J(n)} | |J(n)| = j] [{T n ⊆ J(n)} | Ω(n)] = P {T n ⊆ J(n)} | |J(n)| n > 1 − η n d n , L n = o(1/η n ) and thus L n /n = o(η n /d n ), it follows that P [{T n ⊆ J(n)} | Ω(n)] ≤ 1 − e −Lnηn/dn 1 − o 1 d n .
y)f sc (y)dy + O (δ |I|) with probability 1 − o (1/n). Now following the bounds in [60, page 60, Proof of Lemma 64] we obtain the bounds2
−2
F (y)f sc (y)dy =
I
f sc dy + O η n log
|I|
η n
More precisely, even under this narrow growth regime the theorem is valid only for Friedman's permutation model. For d fixed, Friedman's model approaches the uniform distribution with increasing n; this is no longer clear once dn grows with n.
AcknowledgementsIt is our pleasure to thank Chris Burdzy, Sourav Chatterjee, Manju Krishnapur, Nati Linial, and Sasha Soshnikov for very useful discussions. Both authors would like to thank Tobias Johnson for a very careful reading of this paper. Ioana is grateful to MSRI for their hospitality during the Fall 2010 quarter, as part of the program Random Matrix Theory, Interacting Particle Systems and Integrable Systems, during which this work was completed.Appendix A. Eigenvalues and eigenvectors of regular treesOur main step in the above proofs was to understand the Stieltjes transforms or the resolvent matrix of a finite tree where every non-leaf vertex has (d − 1) children. It is quite straightforward to compute all the eigenvalues of such a tree. Explicit eigenvalues of the tree give us ideas about spacing distribution of eigenvalues of the random regular graph. Fix a positive integer d ≥ 2.Lemma 14. Let T be a finite ordered rooted tree of depth ζ ∈ N such that every vertex has exactly (d − 1) children. That is, the root has degree (d − 1), and every other vertex, other than the leaves, has degree d. Let H denote the adjacency matrix of this graph.(i) Then, for any complex number z the characteristic polynomial of H is given by(ii) The eigenvalues of the adjacency matrix are given by the following collection. Consider i = 1, 2 . . . , ζ: then twice the zeros of the Chebyshev polynomial U ζ+1−i appears with multiplicity (d−1) i −(d−1) i−1 . For i = 0, the multiplicity is one.Proof. Recall the recursive labeling of vertices as given in Section 3.2.To prove conclusion (i), note that when ζ = 0 (i.e., the tree has only the root vertex), the equality is trivially true. We proceed by induction. Suppose the equality is true until depth ζ − 1. Consider a tree of depth ζ and label the adjacency H matrix as above. ThusHere u is the column vector representing the children of the root. Notice that u is (d − 1) −1/2 exactly at the (d − 1) coordinates which are the last elements in each of the block matrices H 1 , . . . , H d−1 and zero elsewhere. The vector u is the transpose of u.
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98195 E-mail address: [email protected] C-547 Padelford Hall. Seattle, WA; Seattle, WAC-342 Padelford Hall, University of Washington ; University of Washington98195 E-mail address: [email protected] Padelford Hall, University of Washington, Seattle, WA 98195 E-mail address: [email protected] C-547 Padelford Hall, University of Washington, Seattle, WA 98195 E-mail address: [email protected]
| [] |
[
"How the interbank market becomes systemically dangerous: an agent-based network model of financial distress propagation",
"How the interbank market becomes systemically dangerous: an agent-based network model of financial distress propagation"
] | [
"Matteo Serri \nFacoltà di Economia (Dip. MEMOTEF\nUniversità \"Sapienza\"\n00161RomeItaly\n",
"Guido Caldarelli \nIMT School for Advanced Studies\n55100LuccaItaly\n\nIstituto dei Sistemi Complessi (ISC)\nCNR -00185RomeItaly\n\nLondon Institute for Mathematical Sciences -Mayfair W1\nLondonUK\n",
"Giulio Cimini \nIMT School for Advanced Studies\n55100LuccaItaly\n\nIstituto dei Sistemi Complessi (ISC)\nCNR -00185RomeItaly\n"
] | [
"Facoltà di Economia (Dip. MEMOTEF\nUniversità \"Sapienza\"\n00161RomeItaly",
"IMT School for Advanced Studies\n55100LuccaItaly",
"Istituto dei Sistemi Complessi (ISC)\nCNR -00185RomeItaly",
"London Institute for Mathematical Sciences -Mayfair W1\nLondonUK",
"IMT School for Advanced Studies\n55100LuccaItaly",
"Istituto dei Sistemi Complessi (ISC)\nCNR -00185RomeItaly"
] | [] | Assessing the stability of economic systems is a fundamental research focus in economics, that has become increasingly interdisciplinary in the currently troubled economic situation. In particular, much attention has been devoted to the interbank lending market as an important diffusion channel for financial distress during the recent crisis. In this work we study the stability of the interbank market to exogenous shocks using an agentbased network framework. Our model encompasses several ingredients that have been recognized in the literature as pro-cyclical triggers of financial distress in the banking system: credit and liquidity shocks through bilateral exposures, liquidity hoarding due to counterparty creditworthiness deterioration, target leveraging policies and fire-sales spillovers. But we exclude the possibility of central authorities intervention. We implement this framework on a dataset of 183 European banks that were publicly traded between 2004 and 2013. We document the extreme fragility of the interbank lending market up to 2008, when a systemic crisis leads to total depletion of market equity with an increasing speed of market collapse. After the crisis instead the system is more resilient to systemic events in terms of residual market equity. However, the speed at which the crisis breaks out reaches a new maximum in 2011, and never goes back to values observed before 2007. Our analysis points to the key role of the crisis outbreak speed, which sets the maximum delay for central authorities intervention to be effective. | 10.21314/jntf.2017.025 | [
"https://arxiv.org/pdf/1611.04311v1.pdf"
] | 157,558,624 | 1611.04311 | 46c897d8460c6352724a0041737693376e88a625 |
How the interbank market becomes systemically dangerous: an agent-based network model of financial distress propagation
14 Nov 2016
Matteo Serri
Facoltà di Economia (Dip. MEMOTEF
Università "Sapienza"
00161RomeItaly
Guido Caldarelli
IMT School for Advanced Studies
55100LuccaItaly
Istituto dei Sistemi Complessi (ISC)
CNR -00185RomeItaly
London Institute for Mathematical Sciences -Mayfair W1
LondonUK
Giulio Cimini
IMT School for Advanced Studies
55100LuccaItaly
Istituto dei Sistemi Complessi (ISC)
CNR -00185RomeItaly
How the interbank market becomes systemically dangerous: an agent-based network model of financial distress propagation
14 Nov 201610.1038/nphys2588Financial contagionSystemic riskInterbank lending marketAgent-based models
Assessing the stability of economic systems is a fundamental research focus in economics, that has become increasingly interdisciplinary in the currently troubled economic situation. In particular, much attention has been devoted to the interbank lending market as an important diffusion channel for financial distress during the recent crisis. In this work we study the stability of the interbank market to exogenous shocks using an agentbased network framework. Our model encompasses several ingredients that have been recognized in the literature as pro-cyclical triggers of financial distress in the banking system: credit and liquidity shocks through bilateral exposures, liquidity hoarding due to counterparty creditworthiness deterioration, target leveraging policies and fire-sales spillovers. But we exclude the possibility of central authorities intervention. We implement this framework on a dataset of 183 European banks that were publicly traded between 2004 and 2013. We document the extreme fragility of the interbank lending market up to 2008, when a systemic crisis leads to total depletion of market equity with an increasing speed of market collapse. After the crisis instead the system is more resilient to systemic events in terms of residual market equity. However, the speed at which the crisis breaks out reaches a new maximum in 2011, and never goes back to values observed before 2007. Our analysis points to the key role of the crisis outbreak speed, which sets the maximum delay for central authorities intervention to be effective.
I. INTRODUCTION
The financial instability which characterised the last decade made clear to academic and regulators that the economy and the financial system have become so inherently complex that a multidisciplinary effort is needed to disentangle the intertwined set of connections between different actors and institutions (Acemoglu et al., 2015;Beale et al., 2011;Gai et al., 2011;Haubrich and Lo, 2013;Sorkin, 2009). Indeed, the network structure of the financial system is now acknowledged as a potential trigger of instability Chan-Lau et al., 2009), thus the many recent studies on the origin of the crisis investigating the interplay between network topology and contagion processes (an approach originally developed in statistical physics 1 ). Among the various subjects, researchers focused in particular on the interbank lending market, namely the network of financial interlinkages between banks resulting from unsecured overnight loans. This system has been ascribed as one of the principal diffusion channels for financial distress during the 2007/2008 crisis (Bluhm and Krahnen, 2011;Cont et al., 2013;Gabrieli and Co-Pierre, 2014;Georg, 2013;Krause and Giansante, 2012): after the collapse of Lehman Brothers, the interbank market froze causing a severe liquidity drought within the whole financial system (Acharya and Merrouche, 2013;Adrian and Shin, 2008;Berrospide, 2013;Brunnermeier, 2009). Such a black swan was due to the collapse of different liquidity channels, e.g., Asset-Backed-Commercial-Paper and Repo, and to a burst in the spread between long-term and over-night interest rates (or Libor-OIS spread) (Brunnermeier, 2009). As this problem became systemic, central banking authorities intervened with extraordinary monetary policies to restore the solvency of the financial system.
Liquidity issues threaten the stability of the financial system by generating important spillover effects. Different subcategories of problems have been identified in the literature: funding liquidity, market liquidity, flight to liquidityquality, liquidity spirals, liquidity hoarding, market freeze, and assets fire sales. Authors like Acharya and Skeie (2011) and Brunnermeier and Pedersen (2009) have modeled liquidity dynamics using a theoretical approach. Others like Berrospide (2013) and Acharya and Merrouche (2013) resorted to empirical econometric analyses to study the causes of interest rate spreads. Eisenberg and Noe (2001) have been the first to tame the complexity of the problem using a theoretical approach explicitly considering the set of interconnections within the financial system. Their work originated a flourishing line of research aimed at assessing the systemic importance of financial institutions under a network perspective-see, among the others, Barucca et al. (2016); Bluhm and Krahnen (2011);Gabrieli and Co-Pierre (2014); Greenwood et al. (2015); Hausenblas et al. (2015).
A second approach to deal with the complexity of the financial system has been that of using agent-based models (ABM). An ABM is a simulated framework in which several agents interact following optimal selfish strategies, creating spin-off effects such as the emergence of an endogenous trading market (Caporale et al., 2009;Lucas et al., 2014). The use of ABM in economics and finance started in parallel with the development of calculators and computer science. The Santa Fe Institute Artificial Stock Market model in the early 1990s was one of the first ABM developed-and later complemented with market orders (Lux and Marchesi, 1999). Recent advances in ABM for financial stability studies include the work of Fischer and Riedler (2014), who showed the fundamental role of leverage in assessing systemic risk, and of Georg (2013) and Ha laj and Kok (2014), who modeled an emerging interbank market via a stylised trading mechanisms. All these studies agree on the relevance of the topology of interactions for contagion mechanisms. Others studies like Cont et al. (2013), Bookstaber et al. (2014) and Klinger and Teplý (2014) consider an exogenous interbank network (data-driven or simulated) affected by shocks that induce an idiosyncratic response such as the emergence of bank runs and fire sales. The aim is to evaluate systemic risk, and find effective regulatory capital buffer and requirements to prevent cascade failures.
In this work we bring together these two approaches by introducing and ABM which describes the network dynamics of the interbank market. We build on the framework introduced by Chan-Lau et al. (2009) and Krause and Giansante (2012), and later developed by Cimini and Serri (2016). We consider a data-driven network of bilateral exposures between banks (the agents of our model), which use micro-optimal rules to interact with each other and with the rest of the financial system. We explicitly model various categories of spillover effects arising during financial crises, such as fire sales and interest rate jumps due to leverage targeting and liquidity hoarding behavior of banks. The modeled dynamics of pro-cyclical policies then spread financial losses via credit and liquidity interconnections, and may result in cascades of defaults. In our approach, we just assume the existence of events in order to focus on the description of the dynamical evolution of the financial system. Our ABM can thus be used to stress-test the robustness of the financial system to an external shock, which can be either absorbed or cause an avalanche of failures eventually leading to the market freezing.
Note that the use of an ABM allows us to have a complete description of the system dynamics during a crisis (i.e., out of the economic equilibrium), which would be very difficult to obtain by analytical modeling. The ABM presented here also allows to consider a flow of events which is different from what happened in reality during, e.g., the 2007-2008 financial crisis. In particular, we are interested in the scenario characterised by the absence of a lender of last resort like a central bank, whose monetary policies can completely re-design the market. Indeed, our aim is not to reproduce the real dynamics of the crisis, but to define the worst-possible scenario without any quantitative easing nor bail-out program by regulatory institutions. The rest of the paper is organised as follows. Sections II and III report basic assumptions and detailed description of the ABM framework, respectively. Results of our extensive simulation program are discussed in Section IV, and Section V concludes.
II. MODEL ASSUMPTIONS
The main ingredients of the model are the set of connections and the strategies of the agents. In order to define them, we make the choices listed below.
Network definition
• The interbank network is assumed to consist of loans with overnight (ON) duration. Thus, A ij is the overall amount that bank i lends to bank j (i.e., the interbank asset of i towards j), which corresponds to the liquidity L ji ≡ A ij that j borrowed from i (i.e., the interbank liability of j towards i). As contracts are of short duration (ON), we assume that links are continuously placed and immediately resolved and rolled over, so that the same (current) interest rates r > 1 applies to both assets and liabilities. In other words, the market dynamics we consider here is on a time scale longer than that of contracts duration.
• The network is derived from aggregate interbank exposures and obligations: A i = j A ij and L i = j L ij ≡ j A ji . We use the Bureau van Dijk Bankscope database 2 , that contains yearly-aggregated balance sheets information of N = 183 large European banks from 2004 to 2013, and resort to the procedure described in Cimini et al. (2015) which uses the fitness model (Caldarelli et al., 2002) to build an ensemble of interbank networks from such aggregate data.
• For each bank i, the balance sheet equation reads:
E i := A E i − L E i + r j A ij − r j L ij ,(1)
where A E i and L E i are, respectively, the external assets and liabilities of i. A i = A E i + r j A ij and L i = L E i + r j L ij are the total amount of assets and liabilities (external plus interbank) held by i, respectively. For each bank i to be solvent, it must be E i > 0.
Strategies definition
In order to build agent strategies and model dynamics, we take inspiration from the most important facts characterising the crises.
• If hit by a shock, a bank sells assets following a leverage targeting policy in order to reinforce its reputation and expectation of the stakeholders.
• After the shock and during the realignment, worries about creditworthiness may cause a "flight to quality", for which banks withdraw liquidity from the market.
• Liquidity hoarding coupled with a constant liquidity demand triggers an increase of interbank interest rates, and the consequent revaluation of interbank assets and liabilities.
• If a bank defaults, credit and funding shocks propagate through its bilateral exposures like a bank-run contagion on financial interbank contracts.
• Interbank network connections and fire sales spillovers may lead to default cascades, with a consequent increasing of liquidity hoarding and interest rate.
• In extreme conditions, the market freezes triggering exacerbated fire sales.
III. MODEL DYNAMICS
Building on the above definitions and assumptions, we now specify the model dynamics of the interbank market.
Exogenous shock
• At a given time step t = t 0 , bank s is hit by an exogenous shock, so that its external assets A E s decrease by a quantity Φ (Chan-Lau et al., 2009;Krause and Giansante, 2012):
A E s (t 0 + 1) = A E s (t 0 ) − Φ ⇒ E s (t 0 + 1) = E s (t 0 ) − Φ(2)
• At first, bank s tries to realign to its target leverage B s (t 0 ) = A s (t 0 )/E s (t 0 ) by selling assets. To this end, the amount of assets to be sold is given by (Adrian and Shin, 2008;Brunnermeier, 2009):
A s (t 0 + 1) − E s (t 0 + 1)B s (t 0 ) = A s (t 0 ) − Φ − [E s (t 0 ) − Φ] B s (t 0 ) = Φ[B s (t 0 ) − 1].(3)
As bank s has increased worries on its financial situation, it adopts a microprudential policy (Acharya and Merrouche, 2013;Berrospide, 2013) by hoarding the liquidity granted by such sales. This means that its interbank loans are not rolled over for their entire amount. We thus assume that external and interbank assets are sold proportionally to their balance sheet shares f E
s (t 0 ) = A E s (t 0 )/A s (t 0 ) and f I s (t 0 ) = k A sk (t 0 )/A s (t 0 )
, respectively. We further assume that interbank assets rescale proportionally to the contracts size: each loan A sk (t 0 ) decreases of an amount equal to Φ[B s (t 0 )−1]f I s (t 0 ) times the ratio of A sk (t 0 ) itself to the total exposure k A sk (t 0 ) of s. The net result is that part of the total value of the interbank market is lost. The consequent increase of counterparty and roll-over risk perceived in the market causes the interbank interest rate to grow up (Acharya and Skeie, 2011). In particular we assume:
dr/dt = r ln(1 + α) =⇒ r(t 0 + 1) = (1 + α) r(t 0 ) + ε(4)
where the factor α > 0 is as a small quantity which leaves the system stable, and ε is a random variable drawn from N [0, σ]. Thus, interbank assets and liabilities increase as
A jk (t 0 + 1) = [r(t 0 + 1)/r(t 0 )]A jk (t 0 ) and L jk (t 0 + 1) = [r(t 0 + 1)/r(t 0 )]L jk (t 0 )
, respectively and ∀j, k. Note that while the external assets sold by s are turned into liquidity 3 and thus do not cause any change of equity for s, the liquidated interbank assets causes the bank's equity to shrink as they do not get revalued by the new interest rate. Therefore, the target leverage of s is substantially respected, except for the non appreciation of a part of its interbank assets (that is, however, minimal). Overall, the balance sheet of bank s becomes:
E s (t 0 + 1) = A E s (t 0 ) − Φ − Φ[B s (t 0 ) − 1]f E s (t 0 ) − L E s (t 0 ) − Φ[B s (t 0 ) − 1]f E s (t 0 ) − Φ[B s (t 0 ) − 1]f I s (t 0 ) + [r(t 0 + 1)/r(t 0 )] k A sk (t 0 ) 1 − Φ[B s (t 0 ) − 1]f I s (t 0 ) k A sk (t 0 ) − [r(t 0 + 1)/r(t 0 )] k L sk (t 0 ) = (5) = E(t 0 ) − Φ + [α + ε/r(t 0 )] k [A sk (t 0 ) − L sk (t 0 )] − Φ[B s (t 0 ) − 1]f I s (t 0 ) ,
indicating that the equity of s has changed due to the external shock and the revaluation of its interbank contracts, except the part which is not rolled-over.
• Banks {i} that borrow from s receive a funding shock given by the interbank assets s dries up for liquidity hoarding, and replace it with an external liability. Thus their balance sheets becomes:
E i (t 0 + 1) = E i (t 0 ) + [α + ε/r(t 0 )] k [A ik (t 0 ) − L ik (t 0 )] + Φ[B s (t 0 ) − 1] L is (t 0 ) A s (t 0 ) .(6)
• For all other banks {j}:
E j (t 0 + 1) = E j (t 0 ) + [α + ε/r(t 0 )] k [A jk (t 0 ) − L jk (t 0 )] .(7)
These steps are repeated until a first default is triggered.
Cascade Failures
• After some rounds of exogenous shocks, at iteration step t * a given bank u becomes insolvent and defaults, meaning E u (t * ) ≤ 0. Bank u is removed from the system, but this event triggers a cascade of credit and liquidity losses in the interbank market (Chan-Lau et al., 2009;Krause and Giansante, 2012). We use a onestep Debt-Solvency rank dynamics (Cimini and Serri, 2016) to model this process, i.e., we have A) Credit shocks. Bank u cannot meet its obligations, so that each other bank j suffers a loss equal to λA ju (t * ).
Here λ indicates the amount of loss given default. We set λ = 1 to consider only un-collateralized markets.
B) Funding shocks. Banks are unable to replace all the liquidity previously granted by the defaulted institutions but by selling their assets, triggering fire sales (Brunnermeier and Pedersen, 2009). In particular, each bank j is able to replace only a fraction (1−ρ) of the lost funding from u, and its assets trade at a discount: j must sell assets worth [1 + γ(t * )]ρA uj (t * ) in book value terms 4 , corresponding to an overall loss of γ(t * )ρA uj (t * ).
Here we set ρ = 1, meaning that banks actually cannot replace the lost funding from u and are thus forced to entirely replace the corresponding liquidity by assets sales.
Overall, the balance sheet of a bank j connected to u becomes:
E j (t * + 1) = E j (t * ) − λA ju (t * ) − γ(t * )ρA uj (t * ).(8)
If any other bank u ′ fails because of the suffered loss, the procedure above is repeated, until no other bank fails.
• After the default cascade has ended, the net change of equity drives each survived bank to realign to its leverage before the cascade, and to liquidate some of its assets. Thus, the dynamics restart from the exogenous shock phase, even if the shock is endogenous this time. However, now the interbank market has shrunk significantly by a loss ∆E(t
* ) = i E i (t * + 1) − E i (t * ) ≤ 0.
This triggers a sudden interest rate increase, which we model by adding to eq. (4) a source term depending on the ratio of ∆E(t * ) to the exogenous shock Φ:
dr/dt = r ln(1 + α) + δ ln[|∆E(t * )|/Φ] =⇒ r(t * + 1) = (1 + α) r(t * ) + α δ log [α+1] [|∆E(t * )|/Φ] + ε. (9)
Thus, if |∆E(t * )| ≃ Φ the interest rate grows by the same factor α as before, but if |∆E(t * )| ≫ Φ the interest rate blows up. Overall, the equities of each bank s change 5 according to eq. (5) with t 0 = t * , where however Φ is replaced by E s (t * ) − E s (t * + 1), and the interest rate has changed according to eq. (9). This process may trigger again a default cascade. Otherwise, the exogenous shock dynamics continues afterwards.
Market freeze
The market freezes at iteration t = t c when the total relative equity of the market i E i (t c )/ i E i (0) becomes smaller than a critical ratio ǫ c . At this point, interbank assets get totally liquidated. Whenever for a given bank j this liquidation is not enough to repay debts (that is, if χ j (t c ) = k [L jk (t c ) − A jk (t c )] > 0), such bank is forced to fire sale its external assets. However, differently from normal sales due to funding shocks, now the market is frozen and the value of external assets is therefore enormously decreased. Thus bank j must sell a fraction of assets worth (1+Γ c )χ j (t c ), with Γ c ≫ γ(t c ). To evaluate the depricing factor Γ c , we rescale γ(t c ) by the relative wealth of potential buyers of fire-sold assets (Duarte and Eisenbach, 2013) (i.e., their current wealth compared to the initial value):
Γ c = γ(t c ) i E i (0) i {E i (t c ) − χ i (t c )Θ[χ i (t c )]} ,(10)
meaning that if the interbank market shrinks it is more difficult to sell assets and Γ c grows. Note that to compute Γ c , we subtract from the current total equity the total assets that must be fire sold (because these assets cannot be used to acquire other assets). The market freeze condition ends the ABM dynamics of the interbank market.
IV. RESULTS
We now present results of our ABM simulations ( Table I reports the list of parameter values we use). Firstly, we look at the dynamics of a single realization of the system. Figures 1, 2 and 3 report results of the model run on balance sheet data of some representative years: 2004, 2008 and 2013, respectively. These are the extremal years of the dataset at our disposal plus the year of the global financial crisis 6 . These figures show various important quantities characterizing the market at different iteration steps t: interest rate (upper left panels), percentage of total residual 4 Following a common approach (Ellul et al., 2011;Feldhütter, 2012;Greenwood et al., 2015), we assume that fire sales generate a linear impact on prices. Given that Q(t * ) = ρ j =u A uj (t * ) is the total amount of assets to be liquidated, we have no price change when Q(t * ) = 0, and assume that assets value vanishes when Q(t * ) = C ≡ ij A ij (i.e., when the whole market has to be sold). Thus we have a relative assets price change ∆p/p| t * = −Q(t * )/C. To obtain the corresponding γ(t * ), we equate the loss γ(t * )ρA uj (t * ) to the amount sold [1 + γ(t * )ρA uj (t * ) times ∆p/p| t * , obtaining γ(t * ) = [C/Q(t * ) − 1] −1 . 5 We consider a random sequence of survived banks to perform target leveraging sequentially. 1.0 initial interest rate α 10 −3 interest rate increase factor σ 10 −3 variation of the random variable in the interest rate dynamics δ 10 −2 prefactor of the source term for interest rate dynamics ǫc 0.37 critical ratio of residual equity for market freeze equity (upper right panels), percentage of defaulted banks (lower left panels) and the depricing factor γ (lower right panels). The different trajectories refer to different model configurations, in which we systematically hit a given bank s with the exogenous shock Φ. Thus, the line A-max (A-min) refer to s being the biggest (smallest) bank in terms of total assets; the line B-max (B-min) to s being the bank with the highest (lowest) leverage; the line K-max (K-min) to s being the bank with most (less) bilateral contracts in the interbank market. Looking at the figures, the first striking observation is that the ABM dynamics converge to the market freeze condition much faster in 2008 than in the other years. Indeed, the maximum interest rate which can be sustained by the market is also much lower in 2008. Actually, final values of r reached both in years 2004 and 2013 appear to be unreasonably high, meaning that the interbank market is rather stable to the proposed dynamics. The total residual equity in the market and the number of defaults have, as expected, a symmetrical trend, and the sudden drop of residual equity usually marks the transition to the market freeze state where the assets fire sale depreciation γ is maximal. Note however a significant difference between the "stable" years 2004 and 2013. In the first case, the equity drop ends with the total depletion of the market, just as in 2008. In 2013 instead the system can absorb the first systemic crash, falling into a state with non-zero residual equity. Market freeze is not triggered immediately, and even when it occurs it does not zero the value of the market. This points to the effectiveness of the new regulatory requirements on banks balance sheets in place after the crisis. Concerning the difference of system dynamics between the various shock configurations, we see in general that the convergence of the system to the market freeze is faster when the systematically shocked bank is "small" (i.e., owning a few total assets and a few contracts, and typically having high leverage). "Big" and less leveraged banks are indeed more robust to an extensive exogenous shock, but they eventually fail causing the same kind of market transition. The difference in behavior is less evident in 2013, suggesting that balance sheets became more homogeneous because of the new regulation. We further discuss more robust results, which are obtained as averages over 1000 realizations of the ABM dynamics and for distributed exogenous shocks (the bank to be shocked is randomly drawn at each iteration). Figure 4 supports our findings outlined above: up to years 2008-2009, the final equity in the system i E i (t c + 1) is basically zero, whereas, after 2010 we observe a higher resilience of the system, with a residual equity around 30% even after the freezing of the interbank market. Figure 5 shows instead the length of the ABM dynamics, namely the number of iterations t c for the system to converge to the market freeze. This indicator basically quantifies the maximum delay allowed for a regulatory intervention aimed at taming the crisis spiral. Notably, the minimum value of t c is six times smaller in 2008 than in 2012. We see that the system monotonically loses its resilience before the global crisis, and increases it afterwards. Yet, according to our previous analysis, the system converges to its final state in different ways for early and late years of our dataset. In particular, the first drop of total equity can well represent the outbreak of the crisis event. We thus introduce the half-life of the system t 1/2 as the iteration step at which the total equity in the system is halved, i.e., i E(t 1/2 ) = 1 2 i E(0). In the whole range of years considered, this iteration corresponds to the earliest substantial equity drop. As Figure 6 shows, t 1/2 behaves differently from t c : there is an additional minimum in 2011 (the year of the European sovereign bond crisis), and the more resilient markets are now those far before the global financial crisis. Overall, according to our results, year 2008 marked a transition between a regime in which a crisis was hard to be triggered but would lead to a total market crash, and a regime in which a crisis is easier to occur but part of the system is likely to survive it.
V. DISCUSSION
In this work we have designed an agent based model to mimic the dynamics of the interbank lending market during financial crises. The model relies on banks balance sheet as the only data source, and is build on simple assumptions on banks strategic behavior during periods of financial distress. We find that as we get close to the global financial crisis of 2008, the system becomes less stable in terms of time to collapse. This feature persists after the crisis, and another peak of instability is observed in 2011. However, the consequence of a crisis are much more severe-in term of overall losses-before 2009, as afterwards the new regulation made banks balance positions more solid.
Here we focused on the dynamics interbank market because of its crucial role of liquidity provider to the financial system (Allen et al., 2014), and to the economy in general (Gabbi et al., 2015). As this system results from the usually uncollateralized (OTC) bilateral contracts between banks, it is rather sensitive to market movements (Smaga et al., 2016), and it can dry up under exceptional circumstances (Brunnermeier, 2009) becoming a main vehicle for distress spreading in the financial system. The dynamics of the interbank market are driven by leverage targeting and liquidity hoarding behaviors of banks. These selfish strategies do consolidate the individual bank position, but spread financial distress through spin-off effects like interest rate increase and fire sale spillovers, which in turn induce other banks to adopt similar pro-cyclic behavior. Exceptional monetary policies by central banks are usually implemented to sustain the interbank market during periods of deteriorated creditworthiness and distributed distress. However, in order to assess the stability of the system, in our model we did not include the possibility of a regulatory intervention nor of bail-outs. Indeed, by measuring the speed at which the crisis breaks out, we provide a temporal window for implemented anti-cyclical policies to be effective in mitigating the crisis.
6
6Results for other years are reported in the Supplementary Materials, Figures from S1 to S7.
FIG. 1
1Dynamics of a single realization of the ABM built on balance sheet data of year 2004.
FIG. 2 FIG. 3
23Dynamics of a single realization of the ABM built on balance sheet data of year 2008. Dynamics of a single realization of the ABM built on balance sheet data of year 2013.
FIG. 4 FIG. 5 FIG. 6
456Final relative equity in the system after the market freeze, i Ei(tc + 1)/ i Ei(0), averaged over 1000 ABM run on balance sheet data for different years. Final iteration tc of the dynamics (market freeze condition), averaged over 1000 ABM run on balance sheet data for different years. Half-life t 1/2 of the total equity in the market, averaged over 1000 ABM run on balance sheet data for different years.
FIGFIGFIG
. S1 Dynamics of a single realization of the ABM built on balance sheet data of year 2005. . S2 Dynamics of a single realization of the ABM built on balance sheet data of year 2006. FIG. S3 Dynamics of a single realization of the ABM built on balance sheet data of year 2007. FIG. S6 Dynamics of a single realization of the ABM built on balance sheet data of year 2011. . S7 Dynamics of a single realization of the ABM built on balance sheet data of year 2012.
TABLE I
IList of parameter values used in simulations.Symbol Value Description
d
0.1 density of the reconstructed interbank network
λ
1.0 loss given default
ρ
1.0 lost funding fraction to be replaced
Φ
10 8 $ entity of the exogenous shock
r0
Raw Bankscope data are available from Bureau van Dijk: https://bankscope.bvdinfo.com. Refer to(Battiston et al., 2015) for all the details about the handling of missing data.
Models usually assume that external assets do not revalue as the most unbiased assumption that the overall contribution of market fluctuations averages to zero.
ACKNOWLEDGEMENTSThis work was supported by the EU projects GROWTHCOM (FP7-ICT, grant n. 611272), MULTIPLEX (FP7-ICT, grant n. 317532), SIMPOL (FP7 grant n. 610704), DOLFINS (H2020-EU.1.2.2., grant n. 640772) and CoeGSS (EINFRA, no. 676547). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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| [] |
[
"MULTIPLICITIES OF JUMPING POINTS FOR MIXED MULTIPLIER IDEALS",
"MULTIPLICITIES OF JUMPING POINTS FOR MIXED MULTIPLIER IDEALS"
] | [
"Maria Alberich-Carramiñana ",
"Josepàlvarez Montaner ",
"ANDFerran Dachs-Cadefau ",
"Víctor González-Alonso "
] | [] | [] | In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall. arXiv:1807.09839v1 [math.AG] | 10.1007/s13163-019-00309-y | [
"https://arxiv.org/pdf/1807.09839v1.pdf"
] | 119,324,722 | 1807.09839 | cc6b1d621f85605db9f3d987becb69de6c42eea3 |
MULTIPLICITIES OF JUMPING POINTS FOR MIXED MULTIPLIER IDEALS
Maria Alberich-Carramiñana
Josepàlvarez Montaner
ANDFerran Dachs-Cadefau
Víctor González-Alonso
MULTIPLICITIES OF JUMPING POINTS FOR MIXED MULTIPLIER IDEALS
In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall. arXiv:1807.09839v1 [math.AG]
Introduction
Let X be a complex surface with a rational singularity at a point O ∈ X and O X,O its corresponding local ring. Let a ⊆ O X,O be an m-primary ideal where m = m X,O is the maximal ideal of O X,O . Then, for any real exponent c > 0, we may consider its corresponding multiplier ideal J (a c ). Indeed, the multiplier ideals form a discrete nested sequence (1.1) O X,O J (a λ 1 ) J (a λ 2 ) . . . J (a λ i ) . . .
indexed by an increasing sequence of rational numbers 0 < λ 1 < λ 2 < . . . such that J (a λ i ) = J (a c ) J (a λ i+1 ) for any c ∈ [λ i , λ i+1 ). The λ i are the so-called jumping numbers of the ideal a. Ein, Lazarsfeld, Smith and Varolin [12], using the fact that the multiplier ideals are m-primary as well, defined the multiplicity of a point c ∈ R as introduced by Galindo and Montserrat in [14]. Actually, they proved that this series is a rational function when X is smooth and a is simple. In [3] we gave a systematic study of multiplicities and proved the rationality of the Poincaré series for any ideal a in a complex surface with a rational singularity at O.
Whenever we extend to the case of mixed multiplier ideals J (a a a c ) := J (a c 1 1 · · · a cr r ) associated to a tuple of m-primary ideals a a a := (a 1 , . . . , a r ) ⊆ (O X,O ) r and a point c := (c 1 , . . . , c r ) in the positive orthant R r 0 , things become a little bit more trickier. Instead of having a partition of the positive real line into intervals defined by the jumping numbers where the multiplier ideals are constant, we get a partition of the positive real orthant into constancy regions whose boundary is described by the so-called jumping walls. We point out that only a few results on mixed multiplier ideals are available in the literature. Libgober and Mustaţǎ [19] studied properties of the log-canonical wall, i.e. the jumping wall associated to λ λ λ 0 = (0, . . . , 0). Naie in [21] describes a nice property that jumping walls must satisfy. Cassou-Noguès and Libgober study in [10,11] mixed multiplier ideals and jumping walls associated to germs of plane curves, under the analogous notion of ideals of quasi-adjunction and faces of quasi-adjunction (see [18]).
We may define the multiplicity of any given point c c c ∈ R r 0 and we say that it is a jumping point, if and only if m (c c c) > 0. In particular, jumping points lie on jumping walls. The most natural generalization of a Poincaré series for mixed multiplier ideals is to consider a filtration of ideals J (a a a c ) indexed by points over a ray L in R r 0 with direction vector in Q r 0 and set (1.4) where t c c c := t c 1 1 · · · t cr r . We have to mention that the multiplicity is sensitive to small perturbation of a given point. Indeed, if we consider a sequence of mixed multiplier ideals J (a a a c c c 0 ) · · · J (a a a c c c i−1 ) J (a a a c c c i ) J (a a a c c c i+1 ) · · · indexed by jumping points {c c c i } i 0 over a ray L and perturb minimally this ray, for example taking a parallel ray L that is close enough, then the sequence of mixed multiplier ideals indexed by jumping points in L may vary (see Example 4.1) and thus, the corresponding Poicaré series also varies.
The organization of the paper is as follows. In Section 2 we recall all the basics on mixed multiplier ideals. In Section 3 we extend the results of [3] to this setting. Namely, we make a systematic study of the multiplicities of points in the positive orthant. The main result is Theorem 3.4 where we give a precise formula for the multiplicity. We also prove in Theorem 3.9 that the Poincaré series associated to a ray is a rational function. In Section 4 we study the variation of the multiplicity of a jumping point by small perturbations. We prove that this multiplicity does not vary for points in the interior (with the Euclidean topology) of a C-facet (see Proposition 4.2) but it does so in a controlled way at the intersection of C-facets (see Theorem 4.5). In Section 5 we study the exceptional divisors that contribute to the log-canonical wall. Our main result is Theorem 5.9 where we establish (except for a very particular case considered in Proposition 5.7) a one-to-one correspondence between the facets of the log-canonical wall and the exceptional divisors of the so-called Newton nest (see Definition 5.1) generalizing a result of Cassou-Noguès and Libgober [11,Theorem 4.22].
Mixed multiplier ideals
Let X be a complex surface with at most a rational singularity at a point O ∈ X (see Artin [8] and Lipman [20] for details) and m = m X,O be the maximal ideal of the local ring O X,O at O. Given a tuple of m-primary ideals a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r we will consider a common log-resolution, that is, a birational morphism π :
X → X such that X is smooth, a i · O X = O X (−F i ) for some effective Cartier divisors F i , i = 1, . . . , r and r i=1 F i + E is a divisor with simple normal crossings, where E = Exc (π)
is the exceptional locus. Actually, the divisors F i are supported on the exceptional locus since the ideals are m-primary. The fundamental cycle is the unique smallest non-zero effective divisor Z (with exceptional support) such that Z · E i 0 for every i = 1, . . . , r. The fundamental cycle satisfies m · O X = O X (−Z) (see [8,Theorem 4]). We point out that any effective divisor with integer coefficients D is called antinef if D · E i 0 for every i = 1, . . . , r. Indeed, for any effective divisor D there exists a unique minimal antinef divisor D satisfying D D that is called the antinef closure of D. It can be computed using an inductive procedure called unloading (see [22] and [4] for details).
Since the point O has a rational singularity, the exceptional locus E is a tree of smooth rational curves E 1 , . . . , E s . Moreover, the matrix of intersections (E i · E j ) 1 i,j s is negative-definite. For any exceptional component E j , we define the excess of a i at E j as ρ i,j = −F i · E j . We also recall the following notions:
· A component E j of E is a rupture component if it intersects at least three more components of E (different from E j ). · We say that E j is dicritical if ρ i,j > 0 for some i. Such components correspond to
Rees valuations (see [20]).
We define the mixed multiplier ideal at a point c := (c 1 , .., c r ) ∈ R r 0 as 1
(2.1) J (a a a c ) := J (a c 1 1 · · · a cr r ) = π * O X ( K π − c 1 F 1 − · · · − c r F r )
where · denotes the round-up and the relative canonical divisor
K π = s i=1 k j E j
is the Q-divisor on X supported on the exceptional locus E characterized by the property (K π + E i ) · E i = −2 for every exceptional component E j , j = 1, . . . , s. We say that X is log-canonical (resp. log-terminal) at O if k j −1 ( resp. k j > −1) ∀j. The boundary of the region R a a a (c c c) is what we call the jumping wall associated to c c c. One usually refers to the jumping wall of the origin as the log-canonical wall. It follows from the definition of mixed multiplier ideals that the jumping walls must lie on supporting hyperplanes of the form (2.2) V j, : e 1,j z 1 + · · · + e r,j z r = + k j , j = 1, . . . , s for a suitable ∈ Z >0 . Here we assume that the effective divisors F i such that a i · O X = O X (−F i ), for i = 1, . . . , r, are of the form F i = s j=1 e i,j E j . Notice that each supporting hyperplane V j, is associated to an exceptional component E j . Indeed, we may find other exceptional components associated to the same hyperplane, that is, we may find E i and
∈ Z >0 such that V j, = V i, .
It is proved in [4,Theorem 3.3] that the region R a a a (c c c) is (the interior of) a rational convex polytope defined by the inequalities e 1,j z 1 + · · · + e r,j z r < k j + 1 + e c c c j , j = 1, . . . , s corresponding to either rupture or dicritical divisors E j and D c c c = e c c c j E j is the antinef closure of c 1 F 1 + · · · + c r F r − K π .
The intersection of the boundary of a connected component of a constancy region C a a a (c c c) with a supporting hyperplane of R a a a (c c c) is what we call a C-facet of C a a a (c c c). Every facet of a jumping wall decomposes into several C-facets associated to different mixed multiplier ideals.
The main result of [4] is an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range of the positive orthant R r 0 . In particular the set of jumping walls of a a a, that we will denote from now on as JW a a a , is precisely described. The points on the jumping walls, which we will denote with λ λ λ when we want to emphasize this fact, satisfy the property J (a a a c c c ) J a a a λ λ λ for all c c c ∈ {λ λ λ − R r 0 } ∩ B ε (λ λ λ) and ε > 0 small enough. In the sequel, we will refer to these points as the jumping points of the tuple of ideals a a a.
Multiplicities of jumping points
In this section we are going to provide a systematic study of the multiplicity of any point c c c ∈ R r 0 . The results that we present are a natural generalization of the ones we obtained in [3]. Our first goal is to compute explicitly these multiplicities using the theory of jumping divisors in this mixed multiplier ideals setting as considered in [4]. Since we are dealing with any general point, it will be more convenient to consider the notion of maximal jumping divisors as opposed to the minimal jumping divisors, which are only meaningful for jumping points.
r i=1 F i supported on those components E j such that c 1 e 1,j + · · · + c r e r,j − k j ∈ Z >0 .
Equivalently, for a sufficiently small ε > 0,
H c c c = K π − (1 − ε)c 1 F 1 − · · · − (1 − ε)c r F r − K π − c 1 F 1 − · · · − c r F r .
In particular, we have
J (a a a (1−ε)c c c ) = π * O X ( K π − c 1 F 1 − · · · − c r F r + H c c c ) .
The following numerical properties for maximal jumping divisors will be useful for our purposes. We will skip the details of the proof just because it is a natural generalization of [3, Proposition 3.6] and the same proof holds mutatis mutandi. Proposition 3.3. Fix any c c c ∈ R r 0 , and let H c c c be its associated maximal jumping divisor. Then the following inequalities hold:
• ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · E i −1 for all E i H c c c , and • ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H −1 for any connected component H H c c c .
The main result of this section is the following:
c ∈ R r 0 . Then, m (c c c) = ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H c c c + # {connected components of H c c c } .
Proof. Given the short exact sequence
0 −→ O X ( K π − c 1 F 1 − · · · − c r F r ) −→ O X ( K π − c 1 F 1 − · · · − c r F r + H c c c ) −→ −→ O Hc c c ( K π − c 1 F 1 − · · · − c r F r + H c c c ) −→ 0
we have, after pushing it forward to X and applying local vanishing [17] for the case of mixed multiplier ideals
0 −→ J (a a a c c c ) −→ J (a a a (1−ε)c c c ) −→ −→ H 0 (H c c c , O Hc c c ( K π − c 1 F 1 − · · · − c r F r + H c c c )) ⊗ C O −→ 0
for ε small enough. Therefore the multiplicity of c c c is just
m (c c c) = h 0 (H c c c , O Hc c c ( K π − c 1 F 1 − · · · − c r F r + H c c c )) = E i Hc c c h 0 (E i , O E i ( K π − c 1 F 1 − · · · − c r F r + H c c c )) − a Hc c c ,
where in the second equality we have used that H c c c has simple normal crossings, and hence the sections of the line bundle O Hc c c ( K π − c 1 F 1 − · · · − c r F r + H c c c ) correspond to sections over each component that agree on the a Hc c c intersections, where a Hc c c denotes the number of edges of H c c c in the dual graph. Then, since we have
deg O E i ( K π − c 1 F 1 − · · · − c r F r + H c c c ) = ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · E i −1 by Proposition 3.3, we get m (c c c) = E i Hc c c (( K π − c 1 F 1 − · · · − c r F r + H c c c ) · E i + 1) − a Hc c c = ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H c c c + v Hc c c − a Hc c c = ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H c c c + # {connected components of H c c c } .
The above formula can be rephrased as follows
m (c c c) = E i Hc c c E j ∈Adj(E i ) {c 1 e 1,j + · · · + c r e r,j − k j } + c 1 ρ 1,i + · · · + c r ρ r,i − # {connected components of H c c c } .
We may also provide a very simple numerical criterion to detect whether a given point c c c ∈ R r >0 is a jumping point.
( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H ≥ 0. Proof. We have m (c c c) = ( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H c c c + # {connected components of H c c c } = H Hc c c (( K π − c 1 F 1 − · · · − c r F r + H c c c ) · H + 1) ,
where the sum is taken over all the connected components H H c c c . The result follows since we have ( K π − c 1 F 1 − · · · − c r F r + H c ) · H −1 by Proposition 3.3.
3.1.
Poincaré series of mixed multiplier ideals. Given a m-primary ideal a ⊆ O X,O we consider its Poincaré series
(3.1) P a (t) = c∈R 0 m(c) t c .
which was first considered, in the case that X is smooth and a is simple, by Galindo and Montserrat [14] and extended in [3] to the case where X has a rational singularity and a is any m-primary ideal.
For a tuple of m-primary ideals a a a = {a 1 , . . . , a r } ⊆ (O X,O ) r we are going to give a generalization of this series by considering a sequence of mixed multiplier ideals indexed by points in a ray L : c c c 0 + µu in the positive orthant R r 0 with u = (u 1 , . . . , u r ) ∈ Z r 0 , u = 0 and c c c 0 ∈ Q r 0 . Here we are considering, for simplicity, a point c c c 0 belonging to a coordinate hyperplane but not necessarily being the origin and µ ∈ R 0 . Namely, we consider the sequence of mixed multiplier ideals
J (a a a c c c 0 ) J (a a a c c c 1 ) J (a a a c c c 2 ) · · · J (a a a c c c i ) · · ·
where {c c c i } i>0 = L ∩ JW a a a or equivalently {c c c i } i>0 is the set of jumping points of this sequence. Then we define the Poincaré series of a a a alongside the ray L as
(3.2) P a a a (t; L) = c c c∈L m(c c c) t c c c .
where t c c c := t c 1 1 · · · t cr r . Notice that we have where the last equality follows from the fact that we are considering points of the form c c c = c c c 0 + µu with µ ∈ [0, 1). Our goal is to prove that this Poincaré series is rational in the sense that it belongs to the field of fractional functions C(z 1 , . . . , z r ), where the indeterminate z i corresponds to a fractional power t 1/e i for e ∈ N >0 being the least common multiple of the denominators of the coordinates of all jumping points. To do so we need to prove a linear recurrence among the coefficients of the series. A key ingredient will be a periodicity property of the maximal jumping divisor which follows from its definition.
Lemma 3.7. For any c c c ∈ R r >0 and α α α = (α 1 , . . . , α r ) ∈ Z r 0 we have H c c c = H c c c+α α α .
The linear recurrence that the multiplicities satisfy is described in terms of the excesses at dicritical components.
m (c c c + α α α) − m (c c c) = −(α 1 F 1 + · · · + α r F r ) · H c c c = E i Hc c c 1 j r α j ρ j,i .
In the sequel, we will just denote ρ c c c,α α α := E i Hc c c r j=1 α j ρ j,i . The formula for the Poincaré series that we obtain is the following: Consider the tuple of ideals a a a = (a 1 , a 2 ) on a smooth surface X given by:
m(c c c 0 + µu) 1 − t u + ρ c c c 0 +µu,u t u (1 − t u ) 2 t µu
· a 1 = ((x + y) 4 , x 9 (x + y), x 11 , x 6 (x + y) 2 , x 3 (x + y) 3 ), · a 2 = (y 3 , x 7 , x 5 y, x 3 y 2 ).
The dual graph of the log-resolution of a a a is as follows:
E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10
where the blank dots correspond to dicritical divisors and their excesses are represented by broken arrows. The divisors associated with this resolution are
· F 1 = 4E 1 + 4E 2 + 4E 3 + 8E 4 + 12E 5 + 8E 6 + 11E 7 + 20E 8 + 32E 9 + 44E 10 , · F 2 = 3E 1 + 6E 2 + 7E 3 + 14E 4 + 21E 5 + 3E 6 + 3E 7 + 6E 8 + 9E 9 + 12E 10
and the relative canonical divisor is:
· K π = E 1 + 2E 2 + 3E 3 + 6E 4 + 9E 5 + 2E 6 + 3E 7 + 6E 8 + 10E 9 + 14E 10 .
In Figure 2, we present the constancy regions of the corresponding mixed multiplier ideals, those regions are computed using the algorithm in [4]. The chains of mixed multiplier ideals over the parallel rays L : 0, 101 780 + µ(1, 1) and L : 0, 37 390 + µ(1, 1) are given in Table 1. The sets of generators for these ideals are computed using the algorithm in [1] (see also [9] ). In the previous example, we observe that the chains of mixed multiplier ideals differ whenever the corresponding ray crosses the intersection of C-facets. Indeed, the multiplicity of a jumping point at the intersection of C-facets is bigger than the multiplicities of jumping points in its neighborhood. The aim of this section is to provide an explanation to this phenomenon. We start with the fact that the multiplicity does not increase in the interior of C-facets. (y 3 + 2xy 2 + x 2 y, x 2 y 2 + x 3 y, x 5 y, x 2 y 2 + 2x 3 y + x 4 ) Table 1. Chains of mixed multiplier ideals of a a a over the rays L and L . In bold type we present the jumping points with multiplicity 2.
To prove this result it is more convenient to compute the multiplicity of a jumping point by using the so-called minimal jumping divisor instead of the maximal jumping divisor as we did in Section 3. This minimal jumping divisor is closely related to the algorithm developed in [4] to compute the constancy regions of mixed multiplier ideals. We give its definition below but we refer to [4] for details. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of ideals. Given a jumping point λ λ λ = (λ 1 , . . . , λ r ) ∈ R r 0 , its corresponding minimal jumping divisor is the reduced divisor G λ r i=1 F i supported on those components E j for which the point λ λ λ satisfies λ 1 e 1,j + · · · + λ r e r,j = k j + 1 + e
Definition 4.3.(1−ε)λ λ λ j ,
where, for a sufficiently small ε > 0, D (1−ε)λ λ λ = e (1−ε)λ λ λ j E j is the antinef closure of (1 − ε)λ 1 F 1 + · · · + (1 − ε)λ r F r − K π .
Using the same arguments that we used in the proof of Theorem 3.4 we may provide the following formula for the multiplicity of a jumping point in terms of the minimal jumping divisor. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and G λ λ λ the maximal jumping divisor associated to some jumping point λ λ λ ∈ R r >0 . Then, m(λ λ λ) = ( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · G λ λ λ + #{connected components of G λ λ λ } It was proved in [4,Lemma 4.6] that two interior points of a C-facet have the same minimal jumping divisor, which we refer to as the minimal jumping divisor associated to the C-facet. Therefore by applying 4.4, the multiplicity is constant along the interior points of a C-facet and thus proving Proposition 4.2. We point out that two interior points of a C-facet may have different maximal jumping divisor This constancy property for the multiplicities is no longer true when considering jumping points at the intersection of C-facets. However we can control the multiplicity depending on the number of C-facets that contain this jumping point. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and let L : c c c 0 + µu and L : c c c 0 + µu be two parallel rays that are close enough. Let λ λ λ ∈ L be a jumping point and B ε (λ λ λ) be a ball centered at λ λ λ of a sufficiently small radius ε > 0 such that L ∩ JW a a a ∩ B ε (λ λ λ) = {λ λ λ}. If L ∩ JW a a a ∩ B ε (λ λ λ) = {λ λ λ 1 , . . . , λ λ λ n } then m(λ λ λ) = m(λ λ λ 1 ) + m(λ λ λ 2 ) + · · · + m(λ λ λ n ) .
Proposition 4.4.Theorem 4.5.
Proof. Let V 1 , . . . , V k be all the hyperplanes associated to exceptional divisors that contain the jumping point λ λ λ. For each hyperplane V i we consider the divisor H i = j E j where the sum is taken over the exceptional divisors that support the hyperplane V i , i.e. for all E j ≤ H i there exists some j ∈ Z >0 such that the hyperplane V i is of the form e 1,j z 1 + · · · + e r,j z r = j + k j . Notice that, even though it is possible that not all of these hyperplanes support a jumping wall, we have a decomposition of the maximal jumping divisor as H λ λ λ = H 1 + · · · + H k . Let {c 1 , . . . , c k } be the ordered 2 set of points resulting from the intersection of the ray L with the hyperplanes V 1 , . . . , V k . Notice that we have L ∩ JW a a a ∩ B ε (λ λ λ) = {λ λ λ 1 , . . . , λ λ λ n } ⊆ {c 1 , . . . , c k }.
For each c i = (c i,1 , . . . , c i,r ) we may find a point (1−ε )c i := ((1−ε 1 )c i,1 , . . . , (1−ε r )c i,r ) over the ray L that is close enough but smaller than c 1 and a point over the ray L and smaller than λ λ λ that we will denote as (1 − ε)λ λ λ := ((1 − ε 1 )λ 1 , . . . , (1 − ε r )λ r ) satisfying
K π − (1 − ε 1 )λ 1 F 1 − · · · − (1 − ε r )λ r F r = K π − (1 − ε 1 )c i,1 F 1 − · · · − (1 − ε r )c i,r F r .
From the construction of the hyperplanes V i we have:
· K π − λ 1 F 1 − · · · − λ r F r = K π − (1 − ε 1 )λ 1 F 1 − · · · − (1 − ε r )λ r F r + H λ λ λ . · K π − c i,1 F 1 − · · · − c i,r F r = K π − (1 − ε 1 )c i,1 F 1 − · · · − (1 − ε r )c i,r F r + H 1 + · · · + H i . Therefore (4.1) K π − λ 1 F 1 − · · · − λ r F r − K π − c i,1 F 1 − · · · − c i,r F r = H i+1 + · · · H k .
By Theorem 3.4, one has
m(λ λ λ) = ( K π − λ 1 F 1 − · · · − λ r F r + H λ λ λ ) · H λ λ λ + #{connected components of H λ λ λ } .
Thus, we can rewrite this formula as
m(λ λ λ) = k i=1 ( K π − λ 1 F 1 − · · · − λ r F r + H i ) · H i + k i=1 k j=1 j =i H i H j + #{connected components of H λ λ λ } = k i=1 ( K π − c i,1 F 1 − · · · − c i,r F r + H i ) · H i + k i=1 k j>i H i H j + #{connected components of H λ λ λ }
where the last equality follows from Equation 4.1. Now, recall that for any divisor D with exceptional support #{connected components of D} = v D − a D , where v D and a D denote the number of vertices and edges of D in the dual graph.
Since
v H λ λ λ = v H 1 + · · · + v H k and a H λ λ λ = a H 1 + · · · + a H k + k i=1 k j>i H i H j we deduce #{connected components of H λ λ λ } = k i=1 #{connected components of H i } − k i=1 k j>i H i H j .
Therefore
m(λ λ λ) = k i=1 [( K π − c i,1 F 1 − · · · − c i,r F r + H i ) · H i + #{connected components of H i }] = m(c 1 ) + · · · + m(c k ).
The only points with non zero multiplicity are those over a jumping wall, namely the jumping points {λ λ λ 1 , . . . , λ λ λ n } and thus we get the desired result.
Contribution to the log-canonical wall
Let X be a smooth complex surface and a ⊆ O X,O an ideal. A common theme in the study of multiplier ideals is to check which exceptional divisors contribute to the jumping numbers of a. In the case of the log-canonical threshold we know that it is described by the formula lct(a) = min
i k i + 1 e i .
In the case that a is m-primary and simple, this minimum is achieved at the first rupture or dicritical exceptional component, starting from the origin, in the dual graph of the log-resolution of a (see [15], [23]). For non simple ideals we may find some analogous statements in [16], [13], [6], [5].
For the case of mixed multiplier ideals, Cassou-Noguès and Libgober [11,Theorem 4.22] studied the contribution of exceptional divisors to the log-canonical wall for the case where the tuple of ideals corresponds to the branches of a plane curve. In this section we will give a generalization of their result that works for general tuples of m-primary ideals a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r , where X is a complex surface with a rational singularity at O and the points in the log-canonical wall have multiplicity one.
Their result is described in terms of the so-called Newton nest introduced in [11,Definition 4.19]. In order to give a generalization to our setup of the Newton nest we will need to fix some notation. When X has a rational singularity we may have an strict inclusion O X,O J (a a a 0 ) where 0 = (0, . . . , 0) is the origin of the positive orthant R r 0 . Indeed, the mixed multiplier ideal J (a a a 0 ) is described by a divisor D 0 = e 0 j E j which is the antinef closure of −K π that can be computed using the unloading procedure described in [2]. Therefore, the log-canonical wall is supported on hyperplanes of the form e 1,j z 1 + · · · + e r,j z r = k j + 1 + e 0 j , j = 1, . . . , s.
For each point z z z i = (0, . . . , 0, lct(a i ), 0, . . . , 0) in the i-th coordinate axis corresponding to the log-canonical threshold of the ideal a i , i = 1, . . . , r, we consider the reduced divisor G z z z i = E j , where the sum is taken over those exceptional divisors associated to the supporting hyperplanes of the log-canonical wall which contain the point z z z i . Notice that this divisor is contained in the minimal jumping divisor of z z z i , that is G z z z i G z z z i . Definition 5.1. Consider the minimal connected subgraph Γ a a a of the dual graph Γ a a a containing the divisors G z z z i , for i = 1 . . . , r. The Newton nest of Γ a a a is the set of rupture or dicritical divisors belonging to Γ a a a .
Remark 5.2. In the case that X is smooth and the ideals a a a i are simple, this definition coincides with the one given by Cassou-Noguès and Libgober in [11,Definition 4.19] since in this case we have that G z z z i = G z z z i = E j i , where E j i is the rupture divisor in the dual graph Γ a i which is closest to its root.
Cassou-Noguès and Libgober [11,Theorem 4.22] established a one-to-one correspondence between the divisors of the Newton nest and the C-facets of the log-canonical wall in the case where X is smooth and the tuple of ideals correspond to the branches of a plane curve. The only restriction that we are going to impose in our generalization is that the multiplicity of all the points in the log-canonical wall have multiplicity one. This condition is achieved, for example, in the case that X has a log-terminal singularity at O ∈ X. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of simple m-primary ideals and X is a complex surface with a log-terminal singularity. Then, all the points in the logcanonical wall have multiplicity one.
Lemma 5.3.
Proof. From the definition of log-terminal singularity, it follows that the antinef closure of −K π is 0 because all the coefficients of −K π are strictly smaller than one. Therefore the ideal associated to the point 0 0 0 is the whole ring.
Let λ λ λ be a jumping point in the log-canonical wall. All the coefficients of the divisor λ λ λF − K π must be smaller or equal to one so we have λ λ λF − K π Z where Z is the fundamental cycle. Therefore we have m = π * O X (−Z) ⊆ J a a a λ λ λ O X,O . So J a a a λ λ λ = m, and consequently m(λ λ λ) = 1 for all points in the log-canonical wall.
Before stating the main result of this section we will present some properties concerning jumping points of multiplicity one. This is a very restrictive condition on the corresponding minimal jumping divisors. To such purpose we have to introduce some technical notation. Given any exceptional component E i and a reduced divisor D E = Exc(π), we define the set of components adjacent to E i inside D and its number as: Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and λ λ λ a jumping point such that m(λ λ λ) = 1. Then, the minimal jumping divisor G λ λ λ has only one connected component and no rupture or dicritical divisor E i such that a G λ λ λ (E i ) > 1.
Adj D (E i ) = {E j D | E i · E j = 1} and a D (E i ) = #Adj D (E i ) Lemma 5.4.
Proof. Using Proposition 4.4 we have the following formula. m(λ λ λ) = ( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · G λ λ λ + #{connected components of G λ λ λ } .
In the case that m(λ λ λ) = 1 we can deduce that #{connected components of G λ λ λ } = 1 and [4,Proposition 4.13]. Indeed, using again this result we have
( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · G λ λ λ = E i G λ λ λ ( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · E i = 0 since we already had ( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · G λ λ λ 0 by( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · E i = 0 for all E i G λ λ λ .
We may provide a more explicit description of this equation using [4,Lemma 4.11]. Namely we have
( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · E i = = −2 + λ 1 ρ 1,i + · · · + λ r ρ r,i + a G λ λ λ (E i ) + E j ∈Adj E (E i )
{λ 1 e 1,j + · · · + λ r e r,j − k j } .
Thus, if E i is a rupture or dicritical component with a G λ λ λ (E i ) > 1, then we have
( K π − λ 1 F 1 − · · · − λ r F r + G λ λ λ ) · E i > 0
so we get a contradiction and the result follows.
Corollary 5.5. Let λ λ λ ∈ R r 0 be a jumping point not contained in any coordinate hyperplane such that m(λ λ λ) = 1. Then:
i) If λ λ λ is an interior point of a C-facet which does not intersect any other C-facet, the minimal jumping divisor G λ λ λ contains at most two dicritical or rupture divisors. ii) If λ λ λ is an interior point of a C-facet which intersects, at least, another C-facet, the minimal jumping divisor G λ λ λ is a dicritical or rupture divisor. iii) If λ λ λ is at the intersection of two C-facets, the minimal jumping divisor G λ λ λ is connected and contains exactly two dicritical or rupture divisors, which are its two ends.
Proof. Let λ λ λ ∈ R r 0 be a jumping point. By [4,Theorem 4.14], the ends of the connected components of the minimal jumping divisor G λ λ λ over the dual graph are either rupture or dicritical divisors. If we assume m(λ λ λ) = 1, then, using Lemma 5.4, we have that a G λ λ λ (E j ) 1 for any rupture or dicritical divisor E j . Therefore either G λ λ λ is just one exceptional component or it is connected with just two ends which are rupture or dicritical divisors in the dual graph. In particular, i) follows. Now assume that λ λ λ is at the intersection of two C-facets C 1 and C 2 with associated minimal jumping divisors G 1 and G 2 respectively. We have G λ λ λ = G 1 + G 2 + G for some divisor G with exceptional support. Moreover, the jumping points in the interior of C 1 and C 2 have multiplicity 1 so the same properties considered above also apply for G 1 and G 2 . The two C-facets are supported on different hyperplanes with different slope, so G 1 and G 2 do not share any exceptional divisor. By Lemma 5.4, this forces G 1 and G 2 to be just one exceptional component being a rupture or dicritical divisor and the minimal jumping divisor G λ λ λ contains exactly two dicritical or rupture divisors. Thus, ii) and iii) follow.
Remark 5.6. Given a tuple of m-primary ideals a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r we may pick a subfamily a = {a i 1 , . . . , a i k | 1 ≤ i 1 < · · · < i k ≤ r} and, if no confusion arise, we may view it either as a tuple (O X,O ) s or a subtuple of a a a in (O X,O ) r in the obvious way. Notice for example that the Newton nest of a a a is a subset of the Newton nest of a a a. In the case that λ λ λ = (λ 1 , . . . , λ r ) ∈ R r 0 is a jumping point contained in a coordinate hyperplane, we may consider the tuple a a a = (a i | λ i = 0). Corollary 5.5 holds whenever we consider λ λ λ as a jumping point for a a a and thus, a point not in the coordinate hyperplanes of the lower dimensional positive orthant.
Notice that Corollary 5.5 already singles out a very particular case where we may not have our desired one-to-one correspondence. Namely, assume that the log-canonical wall has a unique C-facet with points of multiplicity one. Part i) of Corollary 5.5 says that the Newton nest contains either one or two divisors. Therefore the desired one-toone correspondence fails when we have exactly two divisors and this case can indeed be achieved. Recall that the effective divisors F i such that a i ·O X = O X (−F i ) are of the form F i = s j=1 e i,j E j for i = 1, . . . , r and the relative canonical divisor is K π = s i=1 k j E j . Let E j and E be the divisors in the Newton nest and V j,1 : e 1,j z 1 + · · · + e r,j z r = k j + 1 and V ,1 : e 1, z 1 + · · · + e r, z r = k + 1 be their associated hyperplanes. The numerical conditions for which these hyperplanes support the unique C-facet of the log-canonical wall are e 1, e 1,j = · · · = e r, e r,j = k + 1 k j + 1 .
This result can be reformulated in the following Lemma 5.7. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals where X is a complex surface with a rational singularity at O ∈ X. Assume that all the points in the log-canonical wall have multiplicity one and the Newton nest contains two divisors E j and E . Then, the log-canonical wall has a unique C-facet if and only if e 1, e 1,j = · · · = e r, e r,j = k + 1 k j + 1 .
We illustrate this case with the following Example 5.8. Consider a smooth surface X and a tuple of ideals a a a = {a 1 , a 2 } such that they have a minimal log-resolution with the following vertex ordering with K π = (1,2,4,7,8,16,24,8,16,25,2,4,6,7,14,21).
E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E
The divisors in the Newton nest are E 4 and E 13 and the log-canonical wall only has a unique C-facet whose supporting hyperplane has the following equation: V 4,1 : 72x 1 + 72x 2 = 8, or equivalently, V 13,1 : 63x 1 + 63x 2 = 7 .
The main result of this section is that the one-to-one correspondence established by Cassou-Noguès and Libgober in [11,Theorem 4.22] still holds in our setup except for this very particular case where we may have two divisors in the Newton nest and just a unique C-facet. Theorem 5.9. Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals where X is a complex surface with a rational singularity at O ∈ X. Assume that the log-canonical wall has at least two C-facets and all its points have multiplicity one. Then, there is a one-to-one correspondence between the exceptional divisors in the Newton nest of a a a and the C-facets of the log-canonical wall.
M = (E i · E j ) 1 i,j 6 = −2 1 1 1 0 0 1 −3 0 0 1 1 1 0 −1 0 0 0 1 0 0 −3 0 0 0 1 0 0 −3 0 0 1 0 0 0 −6 .
The fundamental cycle is the divisor Z = (3, 2, 3, 1, 1, 1) and the relative canonical divisor is K π = − 1 2 , −1, 1 2 , − 1 2 , − 2 3 , − 5 6 so the singularity is log-canonical. Then we consider a duple of ideals a a a = (a 1 , a 2 ), with a 1 non singular and a 2 = m given by the divisors F 1 = (15, 6, 15, 9, 2, 1) and F 2 = Z = (3, 2, 3, 1, 1, 1). The log-canonical wall has two C-facets and the corresponding mixed multiplier ideals are different (see Figure 2). In particular we have jumping points on the log-canonical wall with multiplicity bigger that 1. In this case the Newton nest consists of the exceptional divisors E 1 , E 2 and E 4 so we no longer have the bijection given in Theorem 5.9.
m (c) := dim C J (a c−ε ) J (a c ) for ε > 0 small enough. With this definition, it is clear that c is a jumping number if and only if m (c) > 0. A way to encode the information provided by the filtration of ideals (1.1) is by means of its Poincaré series of multiplier ideals All four authors are partially supported by Spanish Ministerio de Economía y Competitividad MTM2015-69135-P. VGA is partially supported by ERC StG 279723 Arithmetic of algebraic surfaces (SURFARI). MAC and JAM are also supported by Generalitat de Catalunya SGR2017-932 project and they are with the Barcelona Graduate School of Mathematics (BGSMath). MAC is also with the Institut de Robòtica i Informàtica Industrial (CSIC-UPC).
Associated to any point c c c ∈ R r 0 , we consider: · The region of c c c:R a a a (c c c) = c c c ∈ R r 0 J a a a c c c ⊇ J (a a a c c c ) · The constancy region of c c c: C a a a (c c c) = c c c ∈ R r 0 J a a a c c c = J (a a a c c c )
Definition 3 . 1 .
31Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals. We define the multiplicity attached to a point c c c ∈ R r 0 as the codimension of J (a a a c c c ) in J a a a (1−ε)c c c for ε > 0 small enough, i.e. m(c c c) := dim C J a a a (1−ε)c c c J (a a a c c c ) .
Definition 3 . 2 .
32Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of ideals. Given any point c c c ∈ R r 0 , we define its maximal jumping divisor as the reduced divisor H c c c
Theorem 3. 4 .
4Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and H c c c the maximal jumping divisor associated to some c c
Corollary 3 . 5 .
35Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and H c c c the maximal jumping divisor associated to some c c c ∈ R r 0 . Then,
Theorem 3. 6 .
6Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and c c c ∈ R r 0 . Then, c c c is a jumping point if and only if m(c c c) > 0 or equivalently, there exists a connected component H H c c c such that
c + ku) t (µ+k)u
Proposition 3. 8 .
8Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and α α α = (α 1 , . . . , α r ) ∈ Z r 0 . Then, m (c c c + α α α) − m (c c c) = E i Hc c c r j=1 α j ρ j,i .Proof. Lemma 3.7 above states that c c c and c c c + α α α have the same maximal jumping divisor, say H c c c . Therefore, by Theorem 3.4, we have
Theorem 3. 9 .
9Let a a a = {a 1 , . . . , a r } ⊆ (O X,O ) r be a tuple of m-primary ideals and let L : c c c 0 + µu be a ray in the positive orthant R r 0 with u ∈ Z 0 , u = 0. The Poincaré series of a a a alongside L can be expressed as
Proof. Given a point c c c = c c c 0 + µu, with µ ∈ [0, 1) we have, using Proposition 3.8, that m(c c c + ku) = m(c c c) + k E i Hc c c r j=1 u j ρ j,i = m(c c c) + kρ c c c,u
c + ku) t c c c+ku = m(c c c) t c c c + (m(c c c) + ρ c c c,u ) t c c c+u + (m(c c c) + 2ρ c c c,u ) t c c c+2u + · · · = m(c c c) 1 − t u + ρ c c c,u t u (1 − t u ) 2 t c c c and the result follows.
Remark 3.10. In the case that L is the i-th axis of the positive orthant R r 0 , in particular if c c c 0 is the origin, we obtain the Poincaré series of the ideal a i .4. Multiplicities of jumping points after small perturbationsLet a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and consider two parallel rays L : c c c 0 + µu and L : c c c 0 + µu as those considered in the previous section that are close enough. Our aim is to compare the sequences of mixed multiplier ideals indexed by points in both rays and see how the multiplicity of a jumping point varies with a small perturbation. To illustrate this phenomenon we start with the following example.
Figure 1 .
1Constancy regions of the mixed multiplier ideals of a a a and the rays L, L , in blue and Bordeaux respectively.
Proposition 4 . 2 .(y 3 +(y 3 +(y 3 (y 3 (y 3 (y 3 (y 3
423333333Let a a a = (a 1 , . . . , a r ) ⊆ (O X,O ) r be a tuple of m-primary ideals and let λ λ λ, λ λ λ be two jumping points in the interior of a C-facet. Then m(λ λ λ) = m(λ λ λ ). xy 2 , x 3 y, x 3 y + x 4 , x 2 y 2 , xy 2 + x xy 2 , x 3 y, x 3 y + x 4 , x 2 y 2 , xy 2 + x + 2xy 2 + x 2 y, x 4 y, xy 2 + x 2 y, x 2 y 2 + 2x 3 y + x 4 , x 3 y + x 4 ) + 2xy 2 + x 2 y, x 4 y, xy 2 + x 2 y, x 2 y 2 + 2x 3 y + x 4 , x 3 y + x 4 ) + 2xy 2 + x 2 y, x 2 y 2 + x 3 y, x 4 y, x 2 y 2 + 2x 3 y + x 4 ) + 2xy 2 + x 2 y, x 2 y 2 + x 3 y, x 4 y, x 2 y 2 + 2x 3 y + x 4 ) + 2xy 2 + x 2 y, x 2 y 2 + x 3 y, x 5 y, x 2 y 2 + 2x 3 y + x 4 )
Figure 2 .
2Constancy regions of the the mixed multiplier ideals of a a a.
10 E 11 E 12 E 13 E 14 E 15 E 16 and the divisors given by the ideals are: · F 1 = (18, 24, 45, 72, 73, 146, 218, 72, 144, 216, 21, 42, 63, 64, 128, 194), and · F 2 = (18, 24, 45, 72, 72, 144, 216, 74, 147, 222, 21, 42, 63, 64, 128, 194).
By an abuse of notation, we will also denote J (a a a c c c ) its stalk at O so we will omit the word "sheaf" if no confusion arises.
The order on the set of points {c 1 , . . . , c k } is given by their distance to the origin. We order the hyperplanes V 1 , . . . , V k accordingly.
Proof. We start with the case r = 2, that is a a a = (a 1 , a 2 ). In order to construct the Newton nest of a a a we start considering two points z z z 1 and z z z 2 in the coordinate axes corresponding to the log-canonical thresholds of the ideals a 1 and a 2 respectively. Then we order the C-facets of the log-canonical wall C 1 , C 2 , . . . , C l in such a way that z z z 1 ∈ C 1 , z z z 2 ∈ C l and each C-facet C i intersects C i−1 and C i+1 . Roughly speaking, we are considering a path from z z z 1 to z z z 2 over the log-canonical wall. By Corollary 5.5, the exceptional divisors associated to the supporting hyperplanes of C 1 , C 2 , . . . , C l are unique, but they are also different because the hyperplanes have different slopes. Therefore, we can order these dicritical or rupture divisors E j 1 , E j 2 , . . . , E j l accordingly to their corresponding C-facets. Moreover, Corollary 5.5 implies that they form a path in the dual graph Γ a a a of the log-resolution of a a a with no dicritical or rupture divisors in between two consecutive E j i 's. Now, let Γ a a a ⊂ Γ a a a be the minimal connected subgraph containing E j 1 and E js . Then we have that E j 2 , . . . , E j s−1 belong to Γ a a a and the result follows because the Newton nest is the set {E j 1 , E j 2 , . . . , E js } by construction.For the case r > 2, that is a a a = (a 1 , . . . , a r ), we have to consider points z z z 1 , . . . , z z z r in the coordinate axes corresponding to the log-canonical thresholds. We are going to pick one of them, say z z z 1 , and a C-facet C 1 containing this point z z z 1 . Let c ∈ C 1 be an interior point with rational coordinates and q := (0, q 2 , . . . , q r ) a point in the coordinate hyperplane {x 1 = 0} with rational coordinates. Notice that q corresponds to the mixed multiplier ideal J (a q 2 2 · · · a qr r ) = J (a dq i 2 · · · a dqr r ) 1 d for some d ∈ Z such that dq i ∈ Z for i = 2, . . . , r.The mixed multiplier ideals appearing in the restriction of the positive orthant R r 0 to the plane containing the points c, q and the origin 0 are the mixed multiplier ideals of the duple a a a = (a 1 , a dq 2 2 · · · a dqr r ) so the facets of the corresponding log-canonical wall are in one-to-one correspondence with the exceptional divisors in the Newton nest of a a a which is contained in the Newton nest of a a a. Just moving the points c and q conveniently allows us to cover the whole log-canonical wall of a a a and the result follows. We point out that a log-resolution of a a a is also a log-resolution of a a a .In the following example we show that with our definition of the Newton nest we may also consider the case of non-simple ideals in a smooth surface which was not considered in[11].= (a 1 , a 2 , a 3) on a smooth surface X:· a 1 = (y 3 , x 6 y, x 8 , x 3 y 2 ), · a 2 = (x (x 2 + x + y)4, (x 2 + x + y) 2 (x 2 − x − y) 2 , x 5 (x 2 − x − y) 2 , x 5 (x 2 + x + y) 2 ,
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Barcelona 08028, Spain E-mail address: Maria. Matemàtiques Deptartament De, [email protected] Institut für Mathematik. 647Universitat Politècnica de CatalunyaAv. DiagonalDeptartament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal 647, Barcelona 08028, Spain E-mail address: [email protected], [email protected] Institut für Mathematik , Martin-Luther-Universität Halle-Wittenberg
Germany E-mail address: [email protected] Institut für Algebraische Geometrie. Halle (SaaleLeibniz Universität HannoverHalle (Saale), Germany E-mail address: [email protected] Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1,
Germany E-mail address: [email protected]. Hannover, deHannover, Germany E-mail address: [email protected]
| [] |
[
"A set of formulas for primes",
"A set of formulas for primes"
] | [
"Simon Plouffe "
] | [] | [] | In 1947, W. H. Mills published a paper describing a formula that gives primes : if A | null | [
"https://arxiv.org/pdf/1901.01849v4.pdf"
] | 119,131,282 | 1901.01849 | a509a4065a5dec9cd93da23d9928055f2bdf8867 |
A set of formulas for primes
April 4, 2022
Simon Plouffe
A set of formulas for primes
April 4, 2022
In 1947, W. H. Mills published a paper describing a formula that gives primes : if A
first one is : if 43.8046877158 … and , then if is the rounded values of , 113, 367, 1607, 10177, 102217, 1827697, 67201679, 6084503671, …. Other exponents can also give primes like 11/10, or 101/100. If is well chosen then it is conjectured that any exponent 1 can also give an arbitrary series of primes. When the exponent is 3/2 it is conjectured that all the primes are within a series of trees. The method for obtaining the formulas is explained. Five formulas are presented and all results are empirical.
Résumé
Introduction
The first type of prime formula to consider is for example, given a real constant 0 and 10, if 7.3327334517988679… then the sequence 73, 733, 7333, 73327, 733273, … is a sequence of primes but fails for obvious reasons after a few terms. If the base is changed to any other fixed size base, taking into account that the average gap between primes is increasing then eventually the process fails to give any more primes.
If we choose a function that grows faster like , we get better results. The best start constant found is 0.2655883729431433908971294536654661294389… giving 19 primes. But fails at 23 beginning at n 3 . Here ⌊ ⌋ . 7 67 829 12391 218723 4455833 102894377 2655883729 75775462379 2368012611049 80440106764817 2951219812933057 116299525867995629 4899240744635092571 219705395187452015923 10449948501874965563651 525445257345556693801913 27848959374722952425334841 1551723179991864497606172809 Again, for the same reasons mentioned earlier, the process fails to go further, no better example was found. The method used is a homemade Monte-Carlo method that uses Simulated Annealing principe du recuit simulé in French .
If a function grows too slowly, eventually the average gap between primes increases and the process ceases to give any more primes. The next step was to consider formulas like Mills or Wright. The question was then : is there a way to get a useful formula that grows just enough to produce primes ? The natural question that comes next is : Can we generate all the primes with one single exponent ? Here is the tree graph of primes with the exponent 3/2.
If
Other ways of generating primes
Three other methods are presented. The first is based on a very high degree polynomial equation. We begin by the known Euler polynomial 41 that produces 40 primes in a row when n goes from 0 to 39. In 2010, François Dress and Bernard Landreau found a 6 degree polynomial that gives 56 primes values. It was obtained after several months of computation. The polynomial is All other attempts failed to go further. If we take another approach to it there is more.
If c is a real positive constant then
1/2
Where . denotes the integer part function. For a fixed k much larger than 2 it is possible to calculate c using the simulated annealing and Monte-Carlo method used before. For example, if 64 then by using the formula Then the value of the new c is calculated by rounding properly
1 2 1
The hypothesis is : for a large k, c is constant and will produce an arbitrary number of primes. When k 64, the method quickly finds 8 primes values but if k 599 then 70 primes are produced. The value of c found is :
1.0000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000001556808232\ 856463140060524522424280772934429200613394089802373785244587288\ 947728630209290312111238140972072983869588571830870498647900351\ 444778386216146076636515297094962921999030102707899655875083040\ 665145525741443694679398437806398341541385226365101056503219529\ 413038357761603719148231623498218255233850813147568998744093447\ 755692241415245289626420200959624751354414736085310934015664680\ 524107510074349656822480162844410900000000000000000000006345174\ 532890027939294089368385587928671551372984708501441585620418908\ 704346095361979911978459546006902805287612053866464609018035122\ 943293118288150470557741929174121124290158771911527759861726269\ 584165754859588606741441445142853262133802425477033893100601759\ 975564856010884639189352574717896440249667645304974867391425986\ 281906375105044471660347105158250104363991834652263421461752834\ 975705269597512556117680247632410969440028501070105083651713662\ 403142442324953222225461614957943694687384105246348009815340760\ 743692542284884699931895728286412542979603098044884198867262644\ 327255465732046886424762403984621514353794473387936916290029911\ 549331529954509625933626714214956674379391784430822602413846681\ 316909794084400273637943424856690207465817148161052758912779700\ 994744869116506018623229585002806564436851770494944385807359855\ 293872240395791047188062073418411994344296873461271952410865026\ 038448423723192039177383879070574428090707165141606873722573750\ 361413628118939617030810117859967828092642486835827304995753842\ 763548295039042203138474167878091109846423833375302339937989596\ 168347847772485744109449547853544023508997274222527142776574434\ 316734525917951119963956001777673781864527502261570125603162371\ 258224293816247417631732394744100802942101015408972089551902455\ 081699086722777589571816095705739686859606854254953133788523318\ 655544571643416019528317460445413625249324735829
When k 1001 I find this value of c that produces 104 prime values.
1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000001259905884979345486636620885377153178983073452601805833\ 23931435872455532981729710651349444158213948818545323640593772754269090815382794373303605\ 31032253065209164543992867389533988645378869293054521535540437640264924717218337701861582\ 63863610200230076170685773794205724815861643935173361519983613116882690284144248101920208\ 06510182210643573796316380910863539606120370866893232496051485086497467990763578114679066\ 43511345081362364422769127194021649235153662464913703294618822166887983870315588441106214\ 81340962305982173042950831271359740413862293006103829962131522304901811854369079285560801\ 56127395594055635262316335676598235139936600519471752446533884127775136800820321897034833\ 69252163149601536619636700000000000000000000000000000000000002856922463537929582277227375\ 34573209459155555553507901662476263229096688827287105130336980209309890445915932025320437\ 91865264980534443344524609828999110761933292589504030787982875567498653229908488978554566\ 67343197183608335096725513262211244819036081998055736478108166949284397076878265689251779\ 50165437169403417121093904025741600510426629665273037772624863935742023884842513100551563\ 13386198931648636209922198705909308260842342176288588385011256962600297234767059688581215\ 42128399100716160760517497248823653252822696639553588275394359123882969533179056493207692\ 39058072976876703918551210565097947101808802571780934485397548675096907719003422433055454\ 36432892896196532241017067958091280226589158192434780052667595497166072344482563094303132\ 07749969618831846158232622183434850148402756911211398935776546073125564968263058674825326\ 78385994696717675249981709965958000987443947212244644763379515406938514672346131121294611\ 46321844369295493249482895087729753260849786693705728092698851465492392173730706849962492\ 31216596166252982070773148562282049797435559272711648811046626031928698550228861837334923\ 47445282121848836355413332570898852324742813132203189317097096931337090249364473376188254\ 80333372330490460254399100556222506183678193972395584173595952332514796665024284845488485\ 64456888046546242546151999688740068991295519427373610371260392861045349960918614305187606\ 01199946505452669414820340092566861924748261672447488843829277513346089163004507571313765\ 06792446848015950986087309825879656484491208770711299508536138483156405005289125879215048\
Which is a record since the polynomial 1 2 will produce 104 primes when n 2..105.
The second formula is based on an observation about primes in arithmetic progression. The idea is to change the base asumption. Instead of an arithmetic progression, I use a geometric progression. If is defined as : The 3 rd formula is based on Fermat primes. Fermat primes are the primes of the form : 2 1. What is known so far is that for n = 1 to 4 we have the primes 2, 5, 17, 65537 and the pattern stops here apparently. Again, by modifying the premise we just have to write ( ) = [ 2 n + 1 2 ] And to search for a suitable real value of r that will produce the first Fermat primes and the ones after (modified Fermat primes). If r = 2 then we get the usual Fermat numbers after which n > 4 fails to produce more primes. But if r 1.0905077… then we get this series of primes:
Conclusions
There are no proofs of all this, just empirical results. In practical terms, we have now a way to generate an arbitrary series of primes with so far a minimal growth function. The formulas are much smaller in growth rate than of the 2 historical results of Mills and Wright. Perhaps there is even a simpler formulation, I did not find anything simpler. In the appendix, the 50'th term of the sequence beginning with 10 961 is given, breaking the record of known series of primes in either a polynomial 46 values or primes in arithmetic progression 26 values . 179769332237633170170088026884\ 50845059260725731988258572967021469957141212554036273433033527170307819182667579620360866\ 32471077573785377675676242507349422625188943366906764576673141528562623002007942016657593\ 90883867097480294726378517706587276267484299500204412355407141006057494372924502257219698\ 352110801927, 323170128131645365815076255135382646669278924751793572195552962545141719652\ 64801403285247181358559850418369614308791215550771771474749468184741737297690057465193509\ 63945905691375542263397996520864769510659365204874119526029393967225952392638861315457919\ 63108282494937753987330958790356284722814028831250184315941475044351819744600571075621618\ 91978078908010514773644027856618091755427013769073948650806222690073031800766002745811262\ 17470632677757168145724097916814902375242298300482723097426477384391005496248708782736397\ 85136638055501928789129239806873288362297236918221684091632681230224992540958690935289708\ 05311667, 1044389317166240834594605377443097823907897351158848968802396223990763915529923\ 35735754205395972104335511348468746703462926848540116993310568418686077261633016448147919\ 18781556374086563714969978706718863915534446770744346798098445623101221088765942162306552\ 14621897440640949470144818060504055261313257506351263985875277343435085681089703739446815\ 48500161594788046362932860178367430582323227819014770251706417135952611655420880255303182\ 64681512789697795902401025093747702623759547554737815750417887995127647950217987243038766\ 77533890003260364907468957019674181713392537979549059382616660583120543334829823484146591\ 47717501906341315211489961755543220463855382535950075700566535477413593697275201691979923\ 38641568982946895521986824611585033511719052239621316679060986921373718762688888837745422\ 83169367906229978113397931970222522988444158095482888380174612676484538739600000996356088\ 45000911940693727397524975428611394844926595732425377211428922912540710834804687928863044\ 33679420861353478682425113923228694593794164457617570785058604598399524014935395382862380\ 98333161261112242273472110203395411585454959169383671956007018359097021603202728284219951\ 102044249080006323598825172316922737869910163299754485996931556116349948454919395698791, 10907490458109667924271757791513592293983166654301156822150591558598181044442382010399174\ 92919511601279745737961889572367787422187043894748772952513879077053306947607149615936661\ 75112967878119912006205571981584069736977530603282183549613013123916787655598082588924149\ 40891379369301309816262380113991728427333406116199557348489554551799814745807267443201348\ 34707865566910864427699247698231429315759272436031068117019906918001830336446962391804107\ 22985340156979073167257435430296713066681084447199038191801425151002140169249568182087355\ 37216931832857484768370735100604211780047308319725147598436806135282678368780265876110257\ 53988182574348514613527544858201515292421197926011282801578254357559923204391580750693084\ 80293384143141225083638688402007089934075995374391110014446085562005376548363427140256578\ 80908388428167581904401650025654343131473880484355345444054888207153825885498689967570953\ 70985204324044035890167854980468740591429549828136949302322295576903613561409781403727774\ 85357093055388039711174843200724936509378775007630503199092851376287125049320867683678108\ 33352369655462724762257881165906356802040669563085578699320870099396037097662944064618881\ 78983789996479926314834866382506382527233365361033178023818660797724861354693574334402892\ 64201680765323750650601736988219125208588373661116687694701583189612511308498971041370105\ 66940709481058594272986767876269978297997496028861741255819903164165855307471882021560488\ 46419624409275268608448955329531354914194090887107690516242871900980302183047175728790288\ 75725800629989849920195442450635678864794991294524049166732285923131275124915709630178011\ 95007750822749047043672754341054932799971741515843977613033068821444197643295712208115235\ 28096036461198603387546097428870083996323823306860603984960807123643767912944085743355903\ 55150134317023096965637947743236134732979702590603391356175620665336125866408430315394702\ 89835851661541255483770242672579803599405016279803061405113768381039638122682649588584888\ 93930854366167295599946845670094424486542939255420837897352516799960987153631076804984709\ 95101269541394419133543181784050825199638913349633307664893542738931263128355685853014270\ 39519093232327765542481347851062836108789141693181881486942394111395782143969968340113780\ 00491584853335395982836075034898639850515917970108430905151231503584598134574477059277161\ 06628882069129509101520519702817443789175537317708652543820801180775690044296871487181349\ 4856853443180640274846511668911244308922936908792273552407473599
En 1947, W. H. Mills publiait un article montrant une formule qui peut donner un nombre arbitraire de nombres premiers. Si A 1.3063778838630806904686144926… alors donne une suite arbitraire de nombres tous premiers. , ici ⌊ ⌋ est le plancher de x. Plus tard en 1951, E. M.Wright en proposait
une autre, si
1.9287800… et
2 alors
⌊ ⌋
2 …
est toujours premier.
Les premiers consécutifs sont uniquement représentés par . La suite de premiers est 3, 13, 16381,…
Le taux de croissance de ces 2 fonctions est assez élevé puisque le 4 ème terme de la suite de Wright a 4932
chiffres décimaux. La croissance de celle de Mills est moins élevée, le 8 ème terme a quand même une taille de
762 chiffres. Une série de formules est présentée ici qui minimise le taux de croissance et qui possède les mêmes
propriétés de fournir une suite de premiers de longueur arbitraire. Si
43.8046877158 … et
alors la suite
: l'arrondi de , est une suite de premiers de longueur arbitraire. Ici l'exposant 5 4
we consider the recurrence 1 that arises in the context of Sylvester sequence. The Sylvester sequence is A000058 of the OEIS catalogue and begins with Now, what if we carefully chooseso that the exponent is smaller, would it work ? Let's try with 11/10 and start with a larger number.If we want a smaller starting value then has to be bigger, I could get a series of primes when2, like this : 2, 3, 7, 43,
1807, 3263443, 10650056950807, , that sequence has the property that
1
1
2
1
3
1
7
1
43
1
1807
1
3263443
⋯
The natural extension that comes next is: can we choose
so that
will always produce primes ?
The answer is yes, when 0
1.6181418093242092 … and by using the ⌊ ⌋ function we get,
2, 3, 7, 43, 1811, 3277913, 10744710357637, …
The sequence and formula are interesting for one reason the growth rate is quite smaller than the one
of Mills and Wright.
A Formula for primes
What if we choose the exponent to be as small as possible? The problem with that last one is that it is
stilll growing too fast, a 14
9.838 … 10
. The size of primes doubles in length at each step.
The simplest found was
43.80468771580293481… and using the x , rounding to the nearest
integer, we get
For example when
100000000000000000000000000000049.31221074776345 … and the
exponent beeing 11/10 then we get the primes :
100000000000000000000000000000049
158489319246111348520210137339236753
524807460249772597364312157022725894401
3908408957924020300919472370957356345933709
70990461585528724931289825118059422005340095813
3438111840350699188044461057631015443312900908952333
489724690004200094265557071425023036671550364178496540501
…
10
57
, where 0
0.5 chosen at random. In this case the exponent is
If we choose
10
543
then we get our formula to be.
If a 0
2.03823915478206876746349086260954825144862477844317361… and the exponent 3/2 then
the sequence of primes is :
2
3
5
11
37
223
3331
192271
84308429
774116799347
681098209317971743
562101323304225290104514179
13326678220145859782825116625722145759009
1538448162271607869601834587431948506238982765193425993274489
Again by using the previous method I can produce this table of values for c The record number of values is 899 with c 999982.6807693608… From which we could conjecture that for sufficiently large c, an arbitrary number of primes can be produced.1/2
Then when c = 2.553854696… , 7 primes are generated : 3, 7, 17, 43, 109, 277 and 709. But what if
c >> 1 ?
see appendix for more prime values and the precise value of r .Description of the algorithm and methodThere are 3 steps 1 First we choose a starting value and exponent preferably a rational fraction for technical reasons . 2 Use Monte-Carlo method with the Simulated Annealing, in plain english we keep only the values that show primes and ignore the rest. Once we have a series of 4-5 primes we are ready for the next step.3 We use a formula for forward calculation and backward. The forward calculation is Forward : Next smallest prime to . is the next prime candidate. This is where e needs to be in rational form in order to solve easily in floating point to high precision using Newton-like methods.2, 3, 5, 17, 257, 65537, 4294967311, 18446744193968636141,
340282371357715587431288126011714099603,
115792092256830257597513487698137234684227436353307972878385071833485576558709, …
Constant c such that
1/2 is prime
Values
Number of primes
produced
2.553854696…
3, 7, 17, 43, 109, 277,
709
7
2027.1671684764912194343956
n=1..97
97
577.181936975247888…
n=1..22
22
593.46526943871…
n=2..48
47
31622.7767185595693…
n=2..388
387
55237.07504296764715433124781528617 n=2..633
632
999982.6807693608…
n=1..899
899
It is easy to find a probable prime up to thousands of digits. Maple has a limit of about 10000 digits on
a Intel core i7 6700K, if I use PFGW I can get a probable prime of 1000000 digits in a matter of
minutes.
Backward : to check if the formula works
Previous prime solve for in
1 . Where
1
. Encyclopedia Of Integer, N J A Sequences, Simon Sloane, Plouffe, Academic PressSan DiegoEncyclopedia of Integer Sequences, N.J.A. Sloane, Simon Plouffe, Academic Press , San Diego 1995.
A prime-representing function. W H Mills, 10.1090/S0002-9904-1947-08849-2Bulletin of the American Mathematical Society. 536604Mills, W. H. (1947), A prime-representing function, Bulletin of the American Mathematical Society 53 (6): 604, doi:10.1090/S0002-9904-1947-08849-2.
A prime-representing function. E M Wright, 10.2307/2306356.JSTOR2306356American Mathematical Monthly. 589E. M. Wright (1951). A prime-representing function. American Mathematical Monthly. 58 (9): 616- 618. doi:10.2307/2306356. JSTOR 2306356.
Oeis The, Online Encyclopedia of Integer Sequences, sequences : sequences A051021, A051254, A016104 and A323176. 0069886880The OEIS, Online Encyclopedia of Integer Sequences, sequences : sequences A051021, A051254, A016104 and A323176, A006988, A006880.
Wikipedia : formulas for primes (effective and non-effective formulas). Wikipedia : formulas for primes (effective and non-effective formulas).
The Wright's fourth prime. Baillie Robert, Baillie Robert, The Wright's fourth prime: https://arxiv.org/pdf/1705.09741.pdf
. : Wikipedia, Le, Simulé, Wikipedia : Le recuit simulé : https://fr.wikipedia.org/wiki/Recuit_simul%C3%A9
László Tóth, A Variation on Mills-Like Prime-Representing Functions. Wikipedia : Simulated AnnealingWikipedia : Simulated Annealing : https://en.wikipedia.org/wiki/Simulated_annealing [9] László Tóth, A Variation on Mills-Like Prime-Representing Functions, ArXiv : https://arxiv.org/pdf/1801.08014.pdf
Makoto Kamada Prime numbers of the form 7. 73Makoto Kamada Prime numbers of the form 7, 73, https://stdkmd.net/nrr/7/73333.htm#prime
Simon : Pi, the primes and the Lambert W function. Plouffe, Montréal at the ACA 2019 (ETSPlouffe, Simon : Pi, the primes and the Lambert W function, conference in July 2019, Montréal at the ACA 2019 (ETS). https://vixra.org/abs/1907.0108
Undersøgelser angaaende Maengden af Primtal under en given Graeense. J P Gram, K. Videnskab. Selsk. Skr. 2Gram, J. P. "Undersøgelser angaaende Maengden af Primtal under en given Graeense." K. Videnskab. Selsk. Skr. 2, 183-308, 1884.
. Kim Walisch, fastest program to compute primesKim Walisch, primecount and primesieve, fastest program to compute primes. https://github.com/kimwalisch/primecount
Matt Visser, Primes and the Lambert W function. Bruce, Ramanujan Notebooks IV124Visser, Matt : Primes and the Lambert W function : https://arxiv.org/abs/1311.2324 [15] Berndt, Bruce, Ramanujan Notebooks IV, page 124.
Autour de la fonction qui compte le nombre de nombres premiers. Pierre ; Dusart, Simon Plouffe, thèse de DoctoratList of primes computed by the primecount programDusart, Pierre, Autour de la fonction qui compte le nombre de nombres premiers, thèse de Doctorat 1998. [17] Plouffe, Simon, List of primes computed by the primecount program : http://plouffe.fr/NEW/list_primes_pi_of_n_100000000000.txt http://plouffe.fr/NEW/list_primes_10000000000.txt
. Jean-Pierre Kahane, Le Inconnu, Kahane, Jean-Pierre, le nombre cet inconnu. : http://ww3.ac- poitiers.fr/math/prof/resso/kah/conf.pdf [19] Skewe's Number on wikipedia https://en.wikipedia.org/wiki/Skewes%27s_number
Fast computation of some asymptotic functional inverses. Bruno Salvy, J. Symbolic Comput. 17Salvy, Bruno, Fast computation of some asymptotic functional inverses, J. Symbolic Comput.17(1994), 227-236
. Michel Mendès-France, Tannenbaum Les nombres premiers, entre l'ordre et le chaosMendès-France, Michel, Tannenbaum Les nombres premiers, entre l'ordre et le chaos.
Pi, the primes and the Lambert W function. Simon Plouffe, Conférence ACA. Montréal ETSPlouffe, Simon , Pi, the primes and the Lambert W function. Conférence ACA 2019, Montréal ETS. https://vixra.org/abs/1907.0108
. Simon Plouffe, Nombre premiers en progression géométriquePlouffe, Simon , Nombre premiers en progression géométrique : http://plouffe.fr/NEW/
| [
"https://github.com/kimwalisch/primecount"
] |
[
"A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem *",
"A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem *"
] | [
"Changhong Lu [email protected]:[email protected] \nSchool of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiP. R. China\n",
"Qingjie Ye \nSchool of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiP. R. China\n"
] | [
"School of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiP. R. China",
"School of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiP. R. China"
] | [] | In this paper, we consider the minimal doubly resolving set problem in Hamming graphs, hypercubes and folded hypercubes. We prove that the minimal doubly resolving set problem in hypercubes is equivalent to the coin weighing problem. Then we answer an open question on the minimal doubly resolving set problem in hypercubes. We disprove a conjecture on the metric dimension problem in folded hypercubes and give some asymptotic results for the metric dimension and the minimal doubly resolving set problems in Hamming graphs and folded hypercubes by establishing connections between these problems. Using the Lindström's method for the coin weighing problem, we give an efficient algorithm for the minimal doubly resolving set problem in hypercubes and report some new upper bounds. We also prove that the minimal doubly resolving set problem is NP-hard even restrict on split graphs, bipartite graphs and co-bipartite graphs. | 10.1016/j.dam.2021.11.016 | [
"https://arxiv.org/pdf/2012.00396v2.pdf"
] | 244,909,475 | 2012.00396 | 5c2d859e5df1e5c8d8f468ad4f6c74b35455a85d |
A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem *
6 Dec 2021
Changhong Lu [email protected]:[email protected]
School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
200241ShanghaiP. R. China
Qingjie Ye
School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
200241ShanghaiP. R. China
A bridge between the minimal doubly resolving set problem in (folded) hypercubes and the coin weighing problem *
6 Dec 2021arXiv:2012.00396v2 [math.CO]Metric dimensionDoubly resolving setCoin weighing problemHypercubeHamming graph
In this paper, we consider the minimal doubly resolving set problem in Hamming graphs, hypercubes and folded hypercubes. We prove that the minimal doubly resolving set problem in hypercubes is equivalent to the coin weighing problem. Then we answer an open question on the minimal doubly resolving set problem in hypercubes. We disprove a conjecture on the metric dimension problem in folded hypercubes and give some asymptotic results for the metric dimension and the minimal doubly resolving set problems in Hamming graphs and folded hypercubes by establishing connections between these problems. Using the Lindström's method for the coin weighing problem, we give an efficient algorithm for the minimal doubly resolving set problem in hypercubes and report some new upper bounds. We also prove that the minimal doubly resolving set problem is NP-hard even restrict on split graphs, bipartite graphs and co-bipartite graphs.
Introduction
Let G be a finite, connected, simple and undirected graph with vertex set V = V (G) and edge set E = E(G). The distance between vertices u and v is denoted by d G (u, v). The Cartesian product of graphs G and H, denoted by G H, where V (G H) = {(g, h) : g ∈ V (G), h ∈ V (H)}, and (g 1 , h 1 )(g 2 , h 2 ) ∈ E(G H) if and only if g 1 = g 2 , h 1 h 2 ∈ E(H) or g 1 g 2 ∈ E(G), h 1 = h 2 . The cartesian product is associative and G 1 G 2 · · · G d is well-defined.
The metric dimension problem was independently defined by Slater [32], Harary and Melter [12]. A vertex subset S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. More formally, a vertex x of G resolves two vertices u and v of G if d G (u, x) = d G (v, x). A vertex subset S is a resolving set of G if every two vertices in G are resolved by some vertex of S. A resolving set S of G with the minimum cardinality is a metric basis of G, and the size of S is the metric dimension of G, denoted by β(G). Theorem 1.1 (Cáceres et al. [5]). For all graphs G and H = K 1 ,
max{β(G), β(H)} ≤ β(G H) ≤ β(G) + Ψ(H) − 1.
The metric dimension arises in many diverse areas, including network discovery and verification [4], the robot navigation [19] and chemistry [7]. Finding the doubly resolving set in graphs is equivalent to locating the source of a diffusion in complex networks [8]. The metric dimension problem and minimal doubly resolving set problem have many interesting theoretical properties which are out of the scope of this paper. The interested reader is referred, e.g. to [3,13,23].
As far as general graphs are concerned, both problems are NP-hard. The proof for the metric dimension problem is given in [19] and for the minimal doubly resolving set problem is given in [24]. Epstein et al. [10] proved that the metric dimension problem is NP-hard even for split graphs, bipartite graphs and co-bipartite graphs. Therefore, some researchers try to design heuristic algorithms to solve the problems. It has been designed the genetic algorithm (GA) to solve the metric dimension problem in [21] and the minimal doubly resolving set problem in [24]. Mladenović et al. [27] designed the variable neighborhood search algorithm (VNS) to solve the metric dimension problem and the minimal doubly resolving set problem. Chartrand et al. [7] and Kratica et al. [24] gave the 0-1 integer linear programming formulations for the metric dimension problem and the minimal doubly resolving set problem respectively.
The Hamming graph H n,q is the Cartesian product of n copies of the complete graph K q with q vertices H n,q = K q K q · · · K q n .
Specifically, the vertex of H n,q is an n-dimensional vector u = (u 1 , . . . , u n ) ∈ {0, 1, . . . , q − 1} n and two vertices are adjacent if they differ in exactly one coordinate (see Figure 1). The operation of addition (subtraction) in V (H n,q ) is defined by the modulo-q addition (subtraction) of the corresponding vector. For example, if x = (0, 0, 1, 1, 2, 2) and y = (1, 2, 2, 1, 0, 2) are two vertices of H 6,3 , then x + y = (1, 2, 0, 2, 2, 1) and x − y = (2, 1, 2, 0, 2, 0). By the definition of H n,q , it is easy to show that d Hn,q (u, v) = d Hn,q (u − v,
− → 0 ) = n i=1 1 ui =vi , where 1 ui =vi = 1 if u i = v i and 1 ui =vi = 0 if u i = v i .
The n-dimensional hypercube Q n , also called n-cube, is a Cartesian product of n copies of K 2 (see Figure 2). Note that Q n = H n,2 . For each u ∈ V (Q n ), we use u to denote its opposite vertex, that is
u = u + − → 1 . It is clear that d Qn (u, v) = n i=1 |u i − v i | and thus d Qn (u, v) = n − d Qn (u, v)
. The metric dimension of the Hamming graph is connected to Mastermind, which is a deductive game for two players, the code setter and the code breaker. The code setter chooses a secret vector s = (s 1 , . . . , s n ) ∈ {0, 1, . . . , q − 1} n . The task of the code breaker is to infer the secret vector by a series of questions, each a vector t = (t 1 , . . . , t n ) ∈ {0, 1, . . . , q − 1} n . The code setter answers with two integers, denoted by a(s, t) = |{i : s i = t i , 1 ≤ i ≤ n}| and b(s, t) = max{a(s, t) :s is a permutation of s}. The original commercial version of the game is n = 4 and q = 6, which was invented by Mordecai Meirowitz. Knuth [20] showed that four questions suffice to determine s in this case. Let g(n, q) be the smallest number such that the code breaker can determine any s by asking g(n, q) questions at once (without waiting for the answers). Chvátal [9] proved that g(n, q) ≤ (4 + 2 log q 2 + o(1))n/ log q n. Kabatianski et al. [16] showed that β(H n,q ) − (q − 1) ≤ g(n, q) ≤ β(H n,q ). Let f (n, q) be the smallest number such that the code breaker can determine any s by asking f (n, q) questions at once without b(s, t) in the answers. Cáceres et al. [5] showed that g(n, q) ≤ f (n, q) = β(H n,q ).
It has been showed that β(H 2,q ) = Ψ(H 2,q ) = ⌊(4q − 2)/3⌋ for all q ≥ 5 by Cáceres et al. [5] and Kratica et al. [22]. Recently, Jiang and Polyanskii [15] gave the following nice theorem. Theorem 1.2 (Jiang and Polyanskii [15]). β(H n,q ) = (2 + o(1))n/ log q n for all q ≥ 2.
We remark that Kabatyanskiȋ and Lebedev [17] proved the above theorem for q = 3, 4. For q = 2, the metric dimension problem in hypercubes is related to the following coin weighing problem.
Given n coins, some of them may be defective. We know the weight g of the good coins in advance and also the weight h = g of the defective coins. If we weigh a subset of coins with a spring scale, then the outcome will tell us precisely the number of defective coins among them. The coin weighing problem is determining the minimum number M (n) of weighings by means of which the good and defective coins can be separated under the assumption that all the family of tested subsets has to be given in advance.
More formally, the binary vector u = (u 1 , . . . , u n ) ∈ {0, 1} n is corresponding to a distribution of defective coins, where u j = 1 if and only if the j-th coin is defective. Similarly, the binary vector x = (x 1 , . . . , x n ) ∈ {0, 1} n is corresponding to a weighing, where x j = 1 if and only if the j-th coin is chosen to weigh. The outcome of a weighing is a scalar product of x and u, that is u · x = n i=1 u i x i . A set of binary vectors S is called a weighing strategy if for every pair of distinct vectors u, v, there exists x ∈ S such that u · x = v · x. The coin weighing problem was proposed for n = 5 by Shapiro [30] and solved by Shapiro and Fine [31]. Erdős and Rényi [11] presented a lower bound and Lindström [25] (independently by Cantor and Mills [6]) presented an upper bound. The lower bound and the upper bound are asymptotically equivalent. Almost all exact values of M (n) are not known yet. [6], Erdős and Rényi [11], Lindström [25]). M (n) = (2 + o(1))n/ log 2 n.
Theorem 1.3 (Cantor and Mills
A surprising connection between the metric dimension problem in hypercubes and the coin weighing problem was given in [29]. Theorem 1.4 (Sebő and Tannier [29]). |β(Q n ) − M (n)| ≤ 1. Theorems 1.3 and 1.4 imply that β(Q n ) = (2 + o(1))n/ log 2 n. Researchers try to get optimal upper bounds of β(Q n ) and Ψ(Q n ) by heuristic algorithms. Besides the genetic algorithm and the variable neighborhood search algorithm as mentioned previously, some specially algorithms are designed for the metric dimension and minimal doubly resolving set problem in hypercubes. For example, Nikolić et al. [28] designed a greedy algorithm and a dynamic programming procedure for the metric dimension of a hypercube by using the symmetry property of resolving sets to reduce the size of the feasible solution set. Hertz [14] designed an IP-based swapping algorithm for the metric dimension and minimal doubly resolving set problem in hypercubes.
The folded hypercube is a graph obtained by merging opposite vertices of a hypercube. A vertex of a folded n-cube F n is denoted by Figure 3). Note that uv ∈ E(Q n ) if and only ifūv ∈ E(Q n ). It is easy to
[u] = {u, u}, where u is a vertex of Q n . [u][v] ∈ E(F n ) if and only if uv ∈ E(Q n ) or uv ∈ E(Q n ) (seeshow that d Fn ([u], [v]) = min{d Qn (u, v), n − d Qn (u, v)}.
Recently, Zhang et al. [33] gave the upper bound of metric dimension of folded n-cube. Theorem 1.5 (Zhang et al. [33]). β(F n ) ≤ n − 1 for all odd n ≥ 5 and β(F n ) ≤ 2n − 4 for all even n ≥ 6.
They raised the following conjecture. Conjecture 1.1 (Zhang et al. [33]). If n ≥ 5 is odd, then β(F n ) = n − 1. The paper is organized as follows. Based on a new concept of doubly distance resolving sets, we reveal the relationship between resolving sets and doubly resolving sets in Section 2. Using the preliminary results in Section 2, we show that Ψ(H n,q ) = (2 + o(1))n/ log q n and Ψ(Q n ) = M (n) + 1 in Section 3. Hence, we construct a bridge between the minimal doubly resolving set problem in hypercubes and the coin weighing problem. Using the result of the coin weighing problem, we prove that Ψ(Q n ) ≤ Ψ(Q n+1 ) ≤ Ψ(Q n ) + 1, which answer an open question in [14]. In Section 4, by exploring the connection of the metric dimension problems between hypercubes and folded hypercubes, we give a shorter proof for Theorem 1.5 and disprove Conjecture 1.1. Some asymptotic results of β(F n ) and Ψ(F n ) are also given. In Section 5, we prove that the minimal doubly resolving set problem is NP-hard for split graphs, bipartite graphs and co-bipartite graphs. In Section 6, using the bridge between the minimal doubly resolving set problem in hypercubes and the coin weighing problem, we explore algorithms and experimental results for the minimal doubly resolving set problem in hypercubes and get some better upper bounds of Ψ(Q n ) when n ≤ 93. We also give precise values of β(F n ) and Ψ(F n ) when n ≤ 9.
Preliminary results
The following lemma is obvious but helpful to identify a doubly resolving set of a graph.
u, v ∈ V (G), there exists x j ∈ S, such that d G (u, x 1 ) − d G (u, x j ) = d G (v, x 1 ) − d G (v, x j ).
Now we introduce a new concept to reveal the relationship between resolving sets and doubly resolving sets of graphs. Let G be a graph of order n ≥ 2. Given a vertex
x ∈ V (G), a vertex subset S of G is a doubly distance resolving set of G on x if every pair of vertices {u, v} with d G (u, x) = d G (v, x) is doubly resolved by some pair of vertices in S ∪ {x}. In other words, S = {x 1 , x 2 , . . . , x m } is a doubly distance resolving set of G on x if and only if d G (u, x) is uniquely determined by the vector (d G (u, x) − d G (u, x 1 ), d G (u, x) − d G (u, x 2 ), . . . , d G (u, x) − d G (u, x m )) for any u ∈ V (G). Lemma 2.2. Let S be a resolving set of G. Let x ∈ S and T be a doubly distance resolving set of G on x. Then S ∪ T is a doubly resolving set of G. Proof. Let u, v be two distinct vertices of G. If d G (u, x) = d G (v, x), then {u, v} can be doubly resolved by the definition of T . If d G (u, x) = d G (v, x), then there is a vertex y such that d G (u, y) = d G (v, y) by the definition of S and we have d G (u, x) − d G (u, y) = d G (v, x) − d G (v, y). It leads that {u, v} is doubly resolved by {x, y}. Therefore, S ∪ T is a doubly resolving set of G.
Let φ(G, x) denote the minimum cardinality of a doubly distance resolving set of G on x and φ(G) = max{φ(G,
x) : x ∈ V (G)}. Theorem 2.3. Let G be a graph of order n ≥ 2. Then φ(G) ≤ Ψ(G) ≤ β(G) + φ(G).
Proof. For every vertex x, since every doubly resolving set is a doubly distance resolving set on x by definition, we have φ(G, x) ≤ Ψ(G) and thus φ(G) ≤ Ψ(G).
Let S be a resolving set of G with |S| = β(G). Let x ∈ S and T be a doubly distance resolving set of G on x with |T | = φ(G, x). By Lemma 2.2, S ∪ T is a doubly resolving set of G and then Ψ(
G) ≤ |S ∪ T | ≤ |S| + |T | = β(G) + φ(G, x) ≤ β(G) + φ(G). A function f : V (G) → V (G) is an automorphism of G if f is bijective such that uv ∈ E(G) if and only if f (u)f (v) ∈ E(G). A graph G is called a vertex-transitive graph if for every pair of vertices {x, y}, there is an automorphism f such that f (x) = y.
Lemma 2.4. Let G be a vertex-transitive graph of order n ≥ 2. Then the following holds:
(a) φ(G) = φ(G, x) for every x ∈ G. (b) For every x ∈ G, there is a minimum (doubly) resolving set S of G such that x ∈ S.
Proof. We just prove (a). The proof of (b) is similar. Let y be the vertex such that φ(G, y) = φ(G). Let S be a minimum doubly distance resolving set of G on x. By the definition of vertex-transitive graph, there is an automorphism f :
V (G) → V (G) such that f (x) = y. Then f (S) = {f (s) : s ∈ S} is a doubly distance resolving set of G on y. It leads that φ(G) = φ(G, y) ≤ |f (S)| = φ(G, x) ≤ φ(G), i.e. φ(G) = φ(G, x).
Hamming graphs and hypercubes
Let x, y be two vertices of H n,q . It is clear that
f (u) = u − x + y is an automorphism of H n,q with f (x) = y.
Then H n,q is a vertex-transitive graph. By the Lemma 2.4(b), we have the following corollary, which was proved for q = 2 on the metric dimension problem in [28].
Corollary 3.1. There is a minimum (doubly) resolving set S of H n,q such that − → 0 ∈ S. Lemma 3.2. For every positive integer n, φ(H n,q ) ≤ min{q − 1, n}.
Proof. Since H n,q is a vertex-transitive graph, it suffices to prove that φ(H n,q ,
− → 0 ) ≤ min{q −1, n} by Lemma 2.4(a). Let S = { − → 1 , . . . , − −− → q − 1}.
We will prove that S is a doubly distance resolving set of H n,q on − → 0 . Let u be a vertex of H n,q and f (u,
c) = n i=1 1 ui=c , where 1 ui=c = 1 if u i = c and 1 ui=c = 0 if u i = c. Let a u,c = d Hn,q (u, − → 0 ) − d Hn,q (u, − → c ). Then we have a u,c = n i=1 1 ui =0 − n i=1 1 ui =c = (n − f (u, 0)) − (n − f (u, c)) = f (u, c) − f (u, 0). Since q−1 c=0 f (u, c) = n, we have q−1 c=1 a u,c = q−1 c=1 f (u, c) − (q − 1)f (u, 0) = q−1 c=1 f (u, c) − (q − 1) n − q−1 c=1 f (u, c) = q q−1 c=1 f (u, c) − (q − 1)n. Then d Hn,q (u, − → 0 ) = q−1 c=1 f (u, c) = q−1 c=1 a u,c + (q − 1)n q .
Therefore, d Hn,q (u, − → 0 ) is uniquely determined by the vector (a u,1 , . . . , a u,q−1 ), i.e. S is a doubly distance resolving set of H n,q on
− → 0 . Now we assume that n ≤ q − 1. Let T = { − → 1 , . . . , − → n }.
Then we will prove that T is a doubly distance resolving set of H n,q on − → 0 . If there is a pair of vertices {u, v} such that d Hn,q (u,
− → 0 ) = d Hn,q (v, − → 0 ) and a u,c =
a v,c for all c ∈ {1, . . . , n}, then, without loss of generality, we assume that
b = d Hn,q (v, − → 0 )−d Hn,q (u, − → 0 ) ≥ 1. We have f (u, 0) = n − d Hn,q (u, − → 0 ) = n − (d Hn,q (v, − → 0 ) − b) = f (v, 0) + b and f (u, c) = a u,c + f (u, 0) = a v,c +f (v, 0)+b = f (v, c)+b for all c ∈ {1, . . . , n}. But it leads that n c=0 f (u, c) = n c=0 (f (v, c)+b) ≥ n+1, a contradiction.
By the Theorem 2.3, we have β(H n,q ) ≤ Ψ(H n,q ) ≤ β(H n,q ) + min{q − 1, n}. Then we immediately get the following theorem by Theorem 1.2.
Theorem 3.3. Ψ(H n,q ) = (2 + o(1))n/ log q n for all q ≥ 2.
Now we focus on the special Hamming graph, that is the hypercube Q n . We have
d Qn (u, x) = n i=1 |u i − x i | = n i=1 u i + x i − 2u i x i = u · − → 1 + x · − → 1 − 2(u · x). Note that u · x = n i=1 u i x i is the inner product of u and v.
Then
d Qn (u, − → 0 ) − d Qn (u, x) = 2(u · x) − x · − → 1 .
Firstly, we prove the equivalence between the minimal doubly resolving set problem in hypercubes and the coin weighing problem. Proof. We first prove that M (n) ≤ Ψ(Q n ) − 1. By Corollary 3.1, let S be a doubly resolving set of Q n such that − → 0 ∈ S and |S| = Ψ(Q n ). Now we need to prove that
S ′ = S\{ − → 0 } is a weighing strategy. Suppose not, then there are two distinct vertices u, v ∈ V (Q n ), such that u · x = v · x for each x ∈ S ′ . It leads that d Qn (u, − → 0 ) − d Qn (u, x) = 2(u · x) − (x · − → 1 ) = 2(v · x) − (x · − → 1 ) = d Qn (v, − → 0 ) − d Qn (v, x)
for each x ∈ S ′ . By Lemma 2.1, S is not a doubly resolving set, a contradiction. Now we prove that Ψ(Q n ) ≤ M (n) + 1. Let S be a weighing strategy such that |S| = M (n). Then we prove that S ′ = S ∪ { − → 0 } is a doubly resolving set. Suppose not, then there are two distinct vertices
u, v ∈ V (Q n ), such that d Qn (u, − → 0 ) − d Qn (u, x) = d Qn (v, − → 0 ) − d Qn (v, x) for each x ∈ S. Then u · x = (d Qn (u, − → 0 ) − d Qn (u, x) + x · − → 1 )/2 = (d Qn (v, − → 0 ) − d Qn (v, x) + x · − → 1 )/2 = v · x, a contradiction.
By Theorem 1.1, it is easy to know that
β(Q n ) ≤ β(Q n+1 ) = β(Q n K 2 ) ≤ β(Q n ) + Ψ(K 2 ) − 1 = β(Q n ) + 1.
But it is an open problem whether Ψ(Q n ) has the similar property (see Hertz [14]). Since we know that the coin weighing problem and minimal doubly resolving set problem in hypercubes are equivalent, it is not difficult to answer this open problem using the result of the coin weighing problem. . . . , x n+1 ) ∈ S} is a weighing strategy for n coins, i.e. M (n) ≤ M (n + 1).
x = (x 1 , . . . , x n+1 ) ∈ S, such that u ′ · x = v ′ · x, where u ′ = (u 1 , . . . , u n , 0) and v ′ = (v 1 , . . . , v n , 0). Let x ′ = (x 1 , . . . , x n ). Then u · x ′ = u ′ · x = v ′ · x = v · x ′ . Thus, S ′ = {x ′ = (x 1 , . . . , x n ) : x = (x 1 ,
Let u = (u 1 , . . . , u n+1 ) and v = (v 1 , . . . , v n+1 ) be two distinct distribution of defective coins. If u n+1 = v n+1 , then u · y = v · y where y = (0, . . . , 0, 1). Now we assume that u n+1 = v n+1 . Let S be a weighing strategy for n coins such that |S| = M (n). Then there exists
x = (x 1 , . . . , x n ) ∈ S, such that u ′ · x = v ′ · x, where u ′ = (u 1 , . . . , u n ) and v ′ = (v 1 , . . . , v n ). Let x ′ = (x 1 , . . . , x n , 0). Then u · x ′ = u ′ · x = v ′ · x = v · x ′ .
Thus, S ′ = y ∪ {x ′ = (x 1 , . . . , x n , 0) : x = (x 1 , . . . , x n ) ∈ S} is a weighing strategy for n + 1 coins, i.e. M (n + 1) ≤ M (n) + 1.
Folded hypercubes
It is clear that f ([u]) = [u − x + y] is an automorphism of F n with f ([x]) = [x − x + y] = [y]
. Then F n is a vertex-transitive graph. In this section, we will use some precise values of β(Q n ) and β(F n ) that have calculated in [2,5] (see Table 1). n 1 2 3 4 5 6 7 8 9 β(Q n ) 1 2 3 4 4 5 6 6 7 β(F n ) -1 3 6 4 8 6 11 - Table 1. β(Q n ) and β(F n ), n ≤ 9
Lemma 4.1. For every integer n ≥ 3, β(F n ) ≥ β(Q n ). Proof. Let f : V (F n ) → V (Q n ) be a function such that f ([x]) = x if x 1 = 0 and f ([x]) = x if x 1 = 1. If S is a vertex set of F n , then f (S) = {f ([x]) : [x] ∈ S}.
It suffices to prove that if S is a resolving set of F n , then f (S) is a resolving set of Q n .
For
every two distinct vertices u, v ∈ V (Q n ) with u = v, since S is a resolving set of F n , there is a vertex [x] ∈ S, such that d Fn ([u], [x]) = d Fn ([v], [x]) with f ([x]) = x. Let d 1 = d Fn ([u], [x]) and d 2 = d Fn ([v], [x]). Then d Qn (u, x) ∈ {d 1 , n − d 1 } and d Qn (v, x) ∈ {d 2 , n − d 2 }. If d Qn (u, x) = d Qn (v, x), since d 1 = d 2 , we have d 1 + d 2 = n. Since max{d 1 , d 2 } ≤ n/2, we have d 1 = d 2 = n/2, a contradiction. Therefore, d Qn (u, x) = d Qn (v, x).
Now we consider the case that u = v. Since F n is a vertex-transitive graph, we can assume that What is more, we prove that equality holds if n ≥ 3 is odd. Proof. By Lemma 4.1, it suffices to prove that β(F n ) ≤ β(Q n ). Let g : V (Q n ) → V (F n ) be a function such that g(x) = [x]. If S is a vertex set of Q n , then g(S) = {[x] : x ∈ S}. It suffices to prove that if S is a resolving set of Q n , then g(S) is a resolving set of F n .
u = − → 0 , v = − → 1 . Note that for each y ∈ V (Q n ), we have d Qn (u, y)+d Qn (v, y) = n. If there is a vertex x ∈ f (S) such that d Qn (u, x) = d Qn (v, x), then we have done. Otherwise, for each x ∈ f (S), we have d Qn (u, x) = d Qn (v, x) = n/2. It leads that n is even. Let S ′ = {[x] : n i=1 x i = n/
For
(u, x) = d Fn ([u], [x]) = d Fn ([v], [x]) = d Qn (v, x). Then d Qn (u, x) − d Qn (v, x) = u · − → 1 − v · − → 1 − (2u · x − 2v · x) = 0.
It shows that u · − → 1 + v · − → 1 is even. Since S is a resolving set of Q n , there is a vertex y ∈ S such that d Qn (u, y) = d Qn (v, y). Let d 1 = d Qn (u, y) and d 2 = d Qn (v, y).
+ d 2 = n. Besides, d 1 + d 2 = 2y · − → 1 + u · − → 1 + v · − → 1 − (2u · y + 2v · y). It leads that u · − → 1 + v · − → 1 is odd, a contradiction.
If n is even, equality does not hold in general, such as β(F 4 ) = 6 = 4 = β(Q 4 ). If n is even, the following lemma provides the upper bound. Proof. For each x ∈ V (Q n ), let x 0 = (x 1 , . . . , x n , 0) and x 1 = (x 1 , . . . , x n , 1) be the two vertices in V (Q n+1 ). It suffices to prove that if S is a resolving set of Q n , then
S ′ = {[x 0 ], [x 1 ] : x ∈ S} is a resolving set of F n+1 . Let [u],
[v] ∈ V (F n ) be two distinct vertices. Without loss of generality, we assume that u n+1 = v n+1 = 0. Let u ′ = (u 1 , . . . , u n ) and v ′ = (v 1 , . . . , v n ). Then since S is a resolving set of Q n , there is a vertex
x ∈ S such that d Qn (u ′ , x) = d Qn (v ′ , x). Let d 1 = d Qn (u ′ , x) and d 2 = d Qn (v ′ , x). Then d Fn+1 ([u], [x 0 ]) = min{d 1 , n + 1 − d 1 }, d Fn+1 ([u], [x 1 ]) = min{d 1 + 1, n − d 1 }, d Fn+1 ([v], [x 0 ]) = min{d 2 , n + 1 − d 2 }, d Fn+1 ([v], [x 1 ]) = min{d 2 + 1, n − d 2 }. If [x 0 ] or [x 1 ] resolves {[u],
[v]}, we have done. Otherwise, we have d 1 + d 2 = n + 1 and (d 1 + 1) + (d 2 + 1) = n + 1, a contradiction.
Recall that β(Q n+1 ) ≤ β(Q n ) + 1 and hence β(Q n+m ) ≤ β(Q n ) + m. Since β(Q 5 ) = 4, β(Q n ) ≤ β(Q 5 ) + n − 5 = n − 1 if n ≥ 5. By Lemma 4.2, β(F n ) = β(Q n ) ≤ n − 1 for odd n ≥ 5. By Lemma 4.3, β(F n ) ≤ 2β(Q n−1 ) ≤ 2n − 4 for even n ≥ 6. This is a shorter proof of Theorem 1.5. Since β(Q 9 ) = 7, β(Q n ) ≤ β(Q 9 ) + n − 9 = n − 2 for odd n ≥ 9. It implies that Conjecture 1.1 is false. Furthermore, we have the following asymptotic result of β(F n ) by Lemmas 4.1-4.3 and β(Q n ) = (2 + o(1))n/ log 2 n. Theorem 4.4. If n is odd, then β(F n ) = (2 + o(1))n/ log 2 n. If n is even, then (2 + o(1))n/ log 2 n ≤ β(F n ) ≤ (4 + o(1))n/ log 2 n. Now we consider the doubly distance resolving set of F n .
Lemma 4.5. If n = 2k + 1 ≥ 3 is odd, then φ(F n ) ≤ (n + 1)/2. Proof. Let x i = (x i 1 , . . . , x i n ) ∈ V (Q n ) such that x i j = 1 j = 2i − 1 or 2i 0 otherwise for 1 ≤ i ≤ k and x k+1 j = 1 j = 2k + 1 0 otherwise. Let S = {[x 1 ], . . . , [x k+1 ]}.
Then it suffices to prove that S is a doubly distance resolving set of F n on [ − → 0 ]. Let [u] be a vertex of F n . Without loss of generality, we assume that d Qn (u,
− → 0 ) ≤ k. Then d Fn ([u], [ − → 0 ]) = n j=1 u j . Let a i = d Fn ([u], [ − → 0 ]) − d Fn ([u], [x i ]).
Then
a i = d Qn (u, − → 0 ) − min{d Qn (u, x i ), d Qn (u, x i )} = max{2u · x i − x i · − → 1 , 2u · x i − x i · − → 1 }.
Therefore
a i = max 2(u 2i−1 + u 2i ) − 2, 2 n j=1 u j − u 2i − u 2i−1 − (n − 2) for 1 ≤ i ≤ k and a k+1 = max 2u 2k+1 − 1, 2 n j=1 u j − u 2k+1 − (n − 1) .
Firstly, if a i is even for every i ≤ k and a k+1 is odd, then a i = 2(u 2i−1 + u 2i ) − 2 for every i ≤ k and a k+1 = 2u 2k+1 − 1. Thus,
n j=1 u j = k+1 i=1 u · x i = k+1 i=1 a i + n /2. Secondly, if a k+1 is even, then 2 n j=1 u j − u 2k+1 − (n − 1) > 2u 2k+1 − 1 ⇒ 2 n j=1 u j > 4u 2k+1 + n − 2 ≥ n − 2 = 2k − 1. Since d Qn (u, − → 0 ) = n j=1 u j ≤ k, we have n j=1 u j = k. Finally, if a i is odd for some i ≤ k, then a i = 2 n j=1 u j − u 2i − u 2i−1 − (n − 2) and 2(u 2i−1 + u 2i ) − 2 < 2 n j=1 u j − u 2i − u 2i−1 − (n − 2) ⇒ u 2i−1 + u 2i < 2 n j=1 u j − (n − 4) 4 ≤ 3 4 .
It leads that u 2i−1 + u 2i = 0. We have n j=1 u j = (a i + n − 2)/2 + (u 2i−1 + u 2i ) = (a i + n − 2)/2. From the above discussion, d Fn ([u], [ − → 0 ]) is uniquely determined by the vector (a 1 , . . . , a k+1 ). Therefore, S is a doubly distance resolving set of F n on [
− → 0 ].
Lemma 4.6. If n = 2k is even, then φ(F n ) ≤ n − 1.
Proof. Let x i = (x i 1 , . . . , x i n ) ∈ V (Q n ) such that for 0 ≤ i ≤ n, x i j = 1 j ≤ i 0 otherwise Let S = {[x 1 ],− → 0 ) ≤ k. Then d Fn ([u], [ − → 0 ]) = n j=1 u j . Let a i = d Fn ([u], [ − → 0 ]) − d Fn ([u], [x i ]), b i = d Qn (u, − → 0 ) − d Qn (u, x i ) = 2u · x i − x i · − → 1 = 2 i j=1 u j − i and c i = d Qn (u, − → 0 ) − d Qn (u, x i ) = 2u · x i − x i · − → 1 = 2 n j=i+1 u j − (n − i).
Then
a i = max{b i , c i } = max 2 i j=1 u j − i, 2 n j=i+1 u j − (n − i) . Let d i = b i − c i . First, since b 0 = c n = 0, we have d 0 ≤ 0 and d n ≥ 0. Second, |d i+1 − d i | ≤ |b i+1 − b i | + |c i+1 − c i | = |2u i+1 − 1| + |1 − 2u i+1 | = 2.
Third, since b i + c i = 2 n j=1 u j − n is even, d i is even. Combining them with the principle of bisection method, there is a k such that d k = 0, i.e. a k = b k = c k . Besides, for each i such that b i = c i , we have a i = max{b i , c i } > (b i + c i )/2 = n j=1 u j − n/2 = (b k + c k )/2 = a k , i.e. a k = min{a i : i ∈ {0, 1, . . . , n}}. Note that a 0 = a n = 0. Therefore, Theorem 4.7. If n ≥ 3 is odd, then Ψ(F n ) ≤ β(F n ) + (n + 1)/2 = n/2 + o(n). If n is even, then Ψ(F n ) ≤ β(F n ) + n − 1 = n + o(n).
n j=1 u j = k j=1 u j + n j=k+1 u j = a k + k 2 + a k + (n − k) 2 = a k + n 2 .
NP-completeness
A split graph is a graph whose vertex set is the disjoint union of a clique C and an independent set I. In other words, every two vertices in C are connected by an edge, while no two vertices in I are connected by an edge. There is no restriction on edges having one end in C and one end in I. Similarly, a bipartite graph is a graph whose vertex set is the disjoint union of two independent sets S 1 and S 2 . A co-bipartite graph is a graph whose vertex set is the disjoint union of two cliques C 1 and C 2 .
In this section, we prove that the minimal doubly resolving set problem is NP-hard even for split graphs, bipartite graphs and co-bipartite graphs. The proof is an extension of the proof for metric dimension problem in [10]. Besides, we omit some details that were mentioned in [10].
The 3-dimensional matching problem is defined as follows. Given three disjoint sets A, B, C such that |A| = |B| = |C| = n, and a set of triples S ⊆ A × B × C, is there a subset S ′ ⊆ S such that each element of A ∪ B ∪ C occurs in exactly one of the triples of S ′ . It is well-known that 3-dimensional matching problem is NP-hard, due to Karp [18].
For each subset S ′ ⊆ S, the cost of S ′ is calculated by c(S ′ ) = |S ′ |+3n−| (a,b,c)∈S ′ {a, b, c}|. Note that if S ′ is a 3-dimensional matching, then c(S ′ ) = n. Let N = 2 12 n and n ′ = nN .
Let A = N i=1 A i , B = N i=1 B i and C = N i=1 C i , where A i , B i , C i are the copies of A, B, C respectively. Let S = N i=1 S i , where S i is the copy of S corresponding to A i , B i , C i . It is clear that S ′ = N i=1 S ′ i ⊆ S is a 3-dimensional matching if S ′ ⊆ S is a 3-dimensional matching and S ′
i is the copy of S ′ corresponding to A i , B i , C i . Furthermore, Epstein et al. [10] proved the following lemma.
Lemma 5.1 (Epstein et al. [10]). There is a 3-dimensional matching S ′ ⊆ S if and only if there is a subset
S ′ ⊆ S such that c(S ′ ) ≤ n ′ + √ n ′ − 1.
Let S = {s 0 , s 1 , . . . , s τ −1 }, v = ⌈log 2 τ ⌉ and K = n ′ + v + 5. Note that K < n ′ + √ n ′ − 4. They construct a graph G whose vertices are partitioned into two sets I = {s A , s B , s C , s D } ∪ S and J = A ∪ B ∪ C ∪ {d 0 , d 1 , . . . , d v−1 }. If u ∈ J and v ∈ I, then {u, v} ∈ E in the seven following cases (see Figure 4):
1. u ∈ A and v = s A . 2. u ∈ B and v = s B . 3. u ∈ C and v = s C . 4. u ∈ A ∪ B ∪ C and v = s D . 5. u ∈ {a, b, c} and v = (a, b, c) ∈ S 6. u = d i and v = s j such that ⌊j/2 i ⌋ mod 2 = 1 7. u = d i and v = s D ,
The set of additional edges of G is defined according to the following cases. For the case of bipartite graphs there are no additional edges. For the case of split graphs, J is a clique and I is an independent set, and for the case of co-bipartite graphs both I and J are cliques. Clearly, the construction of the graph G in all cases can be done in polynomial time. Then they prove the following lemma.
v ∈ A. Since d G (u, u) − d G (v, u) = 0 − 1 < 1 − 1 ≤ d G (u, s A ) − d G (v, s A ), {u, v} is doubly resolved by {u, s A }.
The following situations only happen
s A s B s C s D a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 s 0 s 1 s 2 s 3 s 4 s 5 s 6 d 0 d 1 d 2 I J I Jd G (u, u) − d G (v, u) = 0 − 1 < 1 − 1 = d G (u, s A ) − d G (v,then d G (u, u) − d G (v, u) = 0 − 2 < 2 − 2 = d G (u, s A ) − d G (v, s A ), i.e. {u, v} is doubly resolved by {u, s A }.
Case 3: d G (u, s D ) = 1 and d G (v, s D ) = 2. If u ∈ A ∪ B ∪ C, then without loss of generality, we assume that u ∈ A.
If v ∈ S, then d G (u, s D ) − d G (v, s D ) = 1 − 2 < 2 − 2 ≤ d G (u, s B ) − d G (v, s B ), i.e. {u, v} is doubly resolved by {s D , s B }. If v ∈ {s A , s B , s C }, then d G (u, s D ) − d G (v, s D ) = 1 − 2 < 1 − 0 ≤ d G (u, v) − d G (v, v), i.e. {u, v} is doubly resolved by {s D , v}. Now we assume that u ∈ {d 0 , d 1 , . . . , d v−1 }. If v ∈ S, then d G (u, s D ) − d G (v, s D ) = 1 − 2 < 2 − 2 ≤ d G (u, s A ) − d G (v, s A ), i.e. {u, v} is doubly resolved by {s D , s A }. If v ∈ {s A , s B , s C }, then d G (u, s D ) − d G (v, s D ) = 1 − 2 < 2 − 0 ≤ d G (u, v) − d G (v, v), i.e. {u, v} is doubly resolved by {s D , v}.
Note that by Lemmas 5.1 and 5.3, there is a 3-dimensional matching S ′ ⊆ S if and only if G has a doubly resolving set L such that |L| ≤ K. From the above, we get the following theorem.
Theorem 5.4. Given a value K and a graph G that is a split graph, a bipartite graph or a co-bipartite graph, deciding whether Ψ(G) ≤ K is NP-complete.
Algorithms and experimental results
Lindström's construction of weighing strategy is a very creative method. For the details and the correctness, the interested reader is referred to Section 2.4 in [1].
Let T be a finite set and F ⊆ 2 T a collection of subsets. F is called a (simplicial) complex if A ∈ F and B ⊆ A imply B ∈ F . Recall that M (n) is the minimum number of weighings for n coins. Lindström proved the following theorem. Theorem 6.1 (Lindström [26]). M ( A∈F |A|) ≤ |F | − 1 for every complex F .
For a positive integer m, the binary representation can be written to m = t i=1 2 ki . Then let F m = {k 1 , k 2 , . . . , k t } with F 0 = ∅. For example, 10 = (1010) 2 = 2 1 + 2 3 and F 10 = {1, 3}. It is easy to know that F m = {F 0 , F 1 , . . . , F m−1 } is a complex. Based on Theorems 3.4 and 3.5, we can construct a doubly resolving set of Q n with cardinality |F | for every complex F and n ≤ A∈F |A|. Our algorithm for finding upper bounds of Ψ(Q n ) is given in Algorithm 1. We use β n and Ψ n to denote the upper bounds of β(Q n ) and Ψ(Q n ), respectively. The genetic algorithm (GA), variable neighborhood search (VNS) algorithm and IP-based swapping (IPBS) algorithm were reported in [24], [27] and [14], respectively. Our computing method is Algorithm 1. Due to the limitation of the memory space and computing time, the previous results only compute for n ≤ 22. Except for n = 14, 16, 18, our upper bounds are same with them (see Table 2). Note that conversely their results actually improved the Lindström's upper bounds for coin weighing problem in n = 14, 16, 18, i.e. M (14) ≤ 8, M (16) ≤ 9 and M (18) ≤ 10. In addition, our upper bound of Ψ(Q 28 ) is better than the upper bound of β(Q 28 ) that founded by IPBS (see Table 3). Recall that β(Q n ) ≤ Ψ(Q n ). What is more, when 29 ≤ n ≤ 90, all of our upper bounds of Ψ(Q n ) are not more than the upper bounds of β(Q n ) that is calculated by a dynamic programming (DP) procedure in [28] (see Table 4). Besides, some of our upper bounds of Ψ(Q n ) are even better than their upper bounds of β(Q n ).
n GA VNS IPBS Our Table 3. β n and Ψ n , 23 ≤ n ≤ 28 Table 4. β n and Ψ n , 29 ≤ n ≤ 93
Recall that Chartrand et al. [7] and Kratica et al. [24] have given the 0-1 integer linear programming formulations for the metric dimension problem and the minimal doubly resolving set problem respectively. Using the similar method, we give the 0-1 integer linear programming formulations for computing φ(G, s).
For a doubly distance resolving set S of G on s, let x t (1)
s.t. t∈V (G) A (u,v),(s,t) x t ≥ 1 ∀(u, v) ∈ T (G, s)(2)
x t ∈ {0, 1} ∀t ∈ V (G).
By Lemma 2.1, it is easy to see that each feasible solution of (2)-(3) defines a doubly distance resolving set S of G on s, and vice versa. We use 0-1 linear programming model to compute β(F n ), φ(F n ) and Ψ(F n ) for n ≤ 10 by Gurobi Optimizer (see Table 5). Note that the values of β(F n ) for n ≤ 8 have computed in [2]. n 2 3 4 5 6 7 8 9 10 β(F n ) 1 3 6 4 8 6 11 7 ≤ 14 Ψ(F n ) 2 3 6 5 9 6 11 7 ≤ 14 φ(F n ) 1 1 3 2 5 3 6 3 ≤ 8 Table 5. β(F n ), φ(F n ) and Ψ(F n ), n ≤ 10
Open problems
In Section 3, we proved that Ψ(H n,q ) ≤ β(H n,q ) + q − 1. However, we do not know whether it is best possible. We pose the following question. By the values of β(F n ) and Ψ(F n ) for n ≤ 10 in Table 5, we raise the following conjecture.
Conjecture 7.1. For every integer n ≥ 2, β(F n ) ≤ Ψ(F n ) ≤ β(F n ) + 1.
We observe that β(F 2n ) ≈ 2β(F 2n−1 ), Ψ(F 2n ) ≈ 2Ψ(F 2n−1 ) and φ(F 2n ) ≈ 2φ(F 2n−1 ) when n is small by Table 5. Besides, it seems to remain true when n is large by comparing Lemma 4.2 with Lemma 4.3, as well as comparing Lemma 4.5 with Lemma 4.6. We pose the following question and conjecture that the values are 2.
Question 7.2. Determine the values of
lim n→+∞ β(F 2n ) β(F 2n−1 ) , lim n→+∞ Ψ(F 2n ) Ψ(F 2n−1 ) , lim n→+∞ φ(F 2n ) φ(F 2n−1 )
.
Figure 3 .
3The graph F 3
Lemma 2 . 1 (
21Kratica et al. [24]). Let S = {x 1 , x 2 , . . . , x m } be a doubly resolving set of G. Then for every pair of distinct vertices
Theorem 3 . 4 .
34For every positive integer n, we have Ψ(Q n ) = M (n) + 1.
Theorem 3. 5 .
5For every positive integer n, we have Ψ(Q n ) ≤ Ψ(Q n+1 ) ≤ Ψ(Q n ) + 1.Proof. By Theorem 3.4, we need to prove that M (n) ≤ M (n + 1) ≤ M (n) + 1.Let u = (u 1 , . . . , u n ) and v = (v 1 , . . . , v n ) be two distinct distribution of defective coins. Let S be a weighing strategy for n + 1 coins such that |S| = M (n + 1). Then there exists
2} .
2}Then S ⊆ S ′ . However, {[s], [t]} cannot be resolved by S ′ , where s = (1, 0, 0, . . . , 0) and t = (0, 1, 0, . . . , 0), since d Fn ([s], [x]) = d Fn ([t], [x]) = n/2 − 1 for each [x] ∈ S ′ . Then S ′ is not a resolving set of F n , a contradiction.
Lemma 4 . 2 .
42If n ≥ 3 is odd, then β(F n ) = β(Q n ).
Lemma 4 . 3 .
43For every positive integer n, β(F n+1 ) ≤ 2β(Q n ).
From the above, d Fn ([u], [ − → 0 ]) is uniquely determined by the vector (a 1 , . . . , a k+1 ). Therefore, S is a doubly distance resolving set of F n on [ − → 0 ]. By Theorems 2.3 and 4.4, we have the following theorem.
Lemma 5 . 2 (
52Epstein et al.[10]). (a) If G has a resolving set L such that |L| ≤ K, then there is a subsetS ′ ⊆ S such that c(S ′ ) ≤ K + 3 < n ′ + √ n ′ − 1. (b) If there is a 3-dimensional matching S ′ ⊆ S, then L = S ′ ∪ {s A , s B , s C , s D } ∪ {d 0 , d 1 , . . . , d v−1 } is a resolving set of G. Note that |L| = K − 1.Now let us consider the minimal doubly resolving set problem.Lemma 5.3. (a) If G has a doubly resolving set L such that |L| ≤ K, then there is a subset S ′ ⊆ S such that c(S ′ ) ≤ K + 3 < n ′ + √ n ′ − 1. (b) If there is a 3-dimensional matching S ′ ⊆ S, then L = S ′ ∪ {s A , s B , s C , s D } ∪ {d 0 , d 1 , . . . , d v−1 } is a doubly resolving set of G. Note that |L| = K − 1.Proof. By Lemma 5.2(a), we get (a) immediately since L is also a resolving set. In order to prove (b), by Lemmas 2.2 and 5.2(b), it suffices to prove thatL ′ = {s A , s B , s C } ∪ {d 0 , d 1 , . . . , d v−1 } is a doubly distance resolving set on s D .It is easy to check that for each u ∈ V (G), d G (u, s D ) ≤ 2. In addition, for the case that G is a co-bipartite graph,d G (u, s D ) ≤ 1. Let {u, v} be the pair of vertices with d G (u, s D ) = d G (v, s D ). Then there exist only three possibilities: Case 1: u = s D and d G (v, s D ) = 1. If v ∈ {d 0 , d 1 , . . . , d v−1 }, then {u, v} is doubly resolved by {u, v}.If v ∈ A ∪ B ∪ C, then without loss of generality, we assume that
Figure 4 .
4An example for constructing for the case of bipartite graphs (the instance actually is much larger), where S = {(a 1 , b 1 , c 1 ), (a 1 , b 2 , c 3 ), (a 1 , b 3 , c 2 ), (a 2 , b 1 , c 2 ), (a 2 , b 2 , c 3 ), (a 3 , b 3 , c 1 ), (a 3 , b 3 , c 2 )} when G is a co-bipartite graph. If v ∈ {s A , s B , s C }, then {u, v} is doubly resolved by {u, v}. If v ∈ S, then
s A ), i.e. {u, v} is doubly resolved by {u, s A }. Case 2: u = s D and d G (v, s D ) = 2. If v ∈ {s A , s B , s C }, then {u, v} is doubly resolved by {u, v}. If v ∈ S,
Algorithm 1 :
1Finding upper bounds of Ψ(Q n ). Input: A positive integer m. Output: Upper bounds P (n) of Ψ(Q n ) for n ≤ A∈Fm |A|.
(G, s) = {(u, v) : d G (u, s) = d G (v, s)}. Let A (u,v),(s,t) = 1 if d G (u, s) − d G (u, t) = d G (v, s) − d G (v, t) 0 otherwise.The following 0-1 linear programming model of calculating the value of φ(G, s) can be formulated as: min t∈V (G)
Question 7 . 1 .
71For every positive integer n, determine the smallest positive value f (q), such that Ψ(H n,q ) ≤ β(H n,q ) + f (q).
every x ∈ S and distinct vertices [u], [v] ∈ V (F n ), if [x] resolves [u],[v], we have done. Otherwise, without loss of generality, we assume that d Qn
Then d Fn ([u], [y]) = min{d 1 , n − d 1 } and d Fn ([v], [y]) = min{d 2 , n − d 2 }. If [y] does not resolve [u] and [v], then d 1
be a vertex of F n . Without loss of generality, we assume that d Qn (u,. . . , [x n−1 ]}. It suffices to prove that S is a doubly distance resolving set of F n on [
− →
0 ].
Let [u]
1 2 --2 2 3 --3 3 4 --4 4 4 --4 5 5 --5 66 --
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| [] |
[
"The interior structure of rotating black holes 1. Concise derivation",
"The interior structure of rotating black holes 1. Concise derivation"
] | [
"Andrew J S Hamilton \nJILA\nU. Colorado\nBox 44080309BoulderCOUSA\n\nDept. Astrophysical & Planetary Sciences\nU. Colorado\n80309BoulderCOUSA\n",
"Gavin Polhemus \nJILA\nU. Colorado\nBox 44080309BoulderCOUSA\n"
] | [
"JILA\nU. Colorado\nBox 44080309BoulderCOUSA",
"Dept. Astrophysical & Planetary Sciences\nU. Colorado\n80309BoulderCOUSA",
"JILA\nU. Colorado\nBox 44080309BoulderCOUSA"
] | [] | This paper presents a concise derivation of a new set of solutions for the interior structure of accreting, rotating black holes. The solutions are conformally stationary, axisymmetric, and conformally separable. Hyper-relativistic counter-streaming between freely-falling collisionless ingoing and outgoing streams leads to mass inflation at the inner horizon, followed by collapse. The solutions fail at an exponentially tiny radius, where the rotational motion of the streams becomes comparable to their radial motion. The papers provide a fully nonlinear, dynamical solution for the interior structure of a rotating black hole from just above the inner horizon inward, down to a tiny scale. | 10.1103/physrevd.84.124055 | [
"https://arxiv.org/pdf/1010.1269v3.pdf"
] | 118,417,778 | 1010.1269 | bbc666441075d8879936cbac1b191f2f635b6a87 |
The interior structure of rotating black holes 1. Concise derivation
13 Jan 2012
Andrew J S Hamilton
JILA
U. Colorado
Box 44080309BoulderCOUSA
Dept. Astrophysical & Planetary Sciences
U. Colorado
80309BoulderCOUSA
Gavin Polhemus
JILA
U. Colorado
Box 44080309BoulderCOUSA
The interior structure of rotating black holes 1. Concise derivation
13 Jan 2012(Dated: January 16, 2012)PACS numbers: 04.20.-q
This paper presents a concise derivation of a new set of solutions for the interior structure of accreting, rotating black holes. The solutions are conformally stationary, axisymmetric, and conformally separable. Hyper-relativistic counter-streaming between freely-falling collisionless ingoing and outgoing streams leads to mass inflation at the inner horizon, followed by collapse. The solutions fail at an exponentially tiny radius, where the rotational motion of the streams becomes comparable to their radial motion. The papers provide a fully nonlinear, dynamical solution for the interior structure of a rotating black hole from just above the inner horizon inward, down to a tiny scale.
I. INTRODUCTION
Two companion technical papers [1,2], hereafter Papers 2 and 3, present conformally stationary, axisymmetric, conformally separable solutions for the interior structure of a rotating black hole that accretes a collisionless fluid, undergoes inflation at its inner horizon, and then collapses. Paper 2 deals with uncharged black holes, while Paper 3 generalizes to charged black holes. The purpose of the present paper is to give an abbreviated derivation of the solution for an uncharged black hole, and to summarize the principal features of the solution. A Mathematica notebook containing many details of the calculations is at [3].
The papers consider only classical general relativity, not alternate theories of gravity, nor speculative quantum processes that might occur at the outer horizon.
The Penrose diagram of the analytically extended Kerr [4] geometry, Figure 1, provides a good starting point for understanding where and how the interior Kerr geometry fails. A spherical charged (Reissner-Nordström) black hole has a similar interior structure, with essentially the same Penrose diagram, and much of the literature has focussed on this simpler case. The Kerr geometry, and more generally the Kerr-Newman geometry, has two inner horizons that are gateways to regions of unpredictability, signalled by the presence of timelike singularities. In 1968, Penrose [5] pointed out that an observer passing through the outgoing inner horizon (the Cauchy horizon) of a spherical charged black hole would see the outside Universe infinitely blueshifted, and he suggested that the infinite blueshift would destabilize the inner horizon. The infinite blueshift is plain from the Penrose diagram, Figure 1, which shows that a person passing through the outgoing inner horizon sees the entire future of the outside Universe go by in a finite time. Perturbation theory, much of it expounded in Chandrasekhar's (1983) monograph [6], confirmed that waves from the outside Universe would amplify to a diverging energy density on the outgoing inner horizon of a spherical charged black hole. The result was widely interpreted as indicating the instability of the inner horizon.
It was not until 1990 that the full nonlinear nature of the instability at the inner horizon was eventually clarified by Poisson & Israel [7]. Poisson & Israel showed that if ingoing and outgoing streams are simultaneously present just above the inner horizon of a spherical charged black hole, then cross-flow between the two streams would lead to an exponential growth of the interior mass. They called the instability "mass inflation." Shortly thereafter, Barrabès, Israel & Poisson [8] generalized the arguments to the case of rotating black holes, showing that whenever two null sheets cross, an effective mass parameter defined by the product of the expansions of the null bundles inflates. The inflationary instability in spherical charged black holes was confirmed analytically and numerically in several studies, as reviewed by [9].
The physical reason for the inflationary instability can be seen in the Penrose diagram, Figure 1. In the Black Hole region between the outer and inner horizons, the time coordinate (the one that expresses time-translation symmetry) is spacelike, so that it is possible to go either forward or backward in time. Inside the inner horizon, the time coordinate reverts to being timelike. Ingoing particles want to fall into a region where the time coordinate is progressing forwards, while outgoing particles want to fall into a region where the time coordinate is progressing backwards. The Penrose diagram, Figure 1, shows that indeed there are two distinct ingoing and outgoing inner horizons which disgorge on to two causally separated pieces of spacetime where the time coordinate is pointed in opposite directions. To achieve this causal separation, the ingoing and outgoing streams must exceed the speed of light relative to each other. This is Penrose's infinite blueshift. In reality, if ingoing and outgoing streams are present, then their attempt to exceed the speed of light relative to each other produces a counter-streaming energy and pressure that, however tiny the initial streams may be, inevitably grows to the point that it becomes a significant source of gravity. As expounded by [9], the counter-streaming pressure produces a gravitational force that is in opposite directions for ingoing and outgoing streams, accelerating the streams ever faster through each other, in turn increasing the counter-streaming pressure. The inflationary instability thus grows exponentially.
II. APPROACH
The strategy adopted in the present papers is motivated by two key physical insights. The first insight is that, as shown in §V of Paper 2 [1], collisionless ingoing and outgoing streams falling towards the inner horizon of the Kerr-Newman geometry become highly focussed into twin narrow, intense beams pointed along the ingoing and outgoing principal null directions. The focussing is along these two special directions regardless of the initial distributions of orbital parameters of the streams. This implies that the energy-momentum tensor of the ingoing and outgoing streams takes a simple and predictable form near the inner horizon. (We use the term inner horizon to describe the narrow region where inflation takes place, even though the inner horizon is destroyed by inflation, and therefore does not actually exist.)
As first shown by [10], the Kerr-Newman geometry (and some other electrovac geometries) is Hamilton-Jacobi separable in a tetrad aligned with the principal null directions. The fact that collisionless streams focus near the inner horizon along precisely these principal null directions suggests that the spacetime might continue to be separable in the presence of inflation.
The second insight is that the geometry of a spherical charged black hole undergoing inflation at (just above) its inner The right panel depicts a black hole of the kind considered in the present series of papers, which undergoes inflation just above the inner horizon, then collapses. In the Kerr geometry, surfaces of constant radius are confocal ellipsoids in Boyer-Lindquist coordinates, while surfaces of constant latitude are confocal hyperboloids, with a ring singularity at their focus. In the inflationary geometry, the streaming energy density and pressure, and Weyl curvature, inflate to exponentially huge values at (just above) the inner horizon, which is destroyed. In the conformally separable solutions presented here, the geometry then collapses radially to exponentially tiny size without changing shape.
horizon has a step-function character. The spacetime is well-approximated by the electrovac (Reissner-Nordström) geometry down to just above the inner horizon. Then, in a tiny interval of radius and proper time near the inner horizon, the centre-of-mass counter-streaming energy and pressure, the Weyl curvature, and the interior mass all inflate to exponentially huge values. Counter-intuitively, the smaller the incident ingoing and outgoing streams, the more rapidly quantities exponentiate [9]. In the limit of tiny accretion rate, the geometry tends to a step-function. This suggests that inflationary spacetimes might be found by looking for solutions with a steplike character. The steplike character of the inflationary solutions can be seen in the sharp turns at the inner horizon in the contours of constant radius and latitude in Figure 2.
As an aside, it is worth commenting on the challenges and pitfalls of computing inflationary spacetimes numerically rather than analytically. One major numerical challenge arises from the fact that during inflation physical quantities inflate to exponentially huge values over tiny intervals of distance and time. One potential pitfall is that inflation requires ingoing and outgoing streams that can stream relativistically through each other. A code, or indeed analytic model, that treats the matter as a single fluid with a sound speed less than the speed of light artificially suppresses inflation by disallowing the relativistic counter-streaming that drives it.
III. SUMMARY DERIVATION
This section summarizes the derivation of the solution for an uncharged rotating black hole. Complete details are given in Paper 2.
The solutions presented in this series of papers are conformally stationary, axisymmetric, and conformally separable. Conformally stationarity combines the assumption of conformal time-translation invariance (self-similarity) with an infinitesimal expansion rate. Let x µ ≡ {x, t, y, φ} be coordinates in which t is conformal time, φ is the azimuthal angle, and x and y are radial and angular coordinates. As shown in Appendix A of Paper 2 [1], the line-element may be taken to be
ds 2 = ρ 2 dx 2 ∆ x − ∆ x σ 4 (dt − ω y dφ) 2 + dy 2 ∆ y + ∆ y σ 4 (dφ − ω x dt) 2 ,(1)
where
σ ≡ 1 − ω x ω y .(2)
The determinant of the 2 × 2 submatrix of t-φ metric coefficents defines the radial and angular horizon functions ∆ x and ∆ y :
g tt g φφ − g 2 tφ = − ρ 4 σ 4 ∆ x ∆ y .(3)
Horizons occur when one or other of the horizon functions ∆ x and ∆ y vanish. The focus here is the region near the inner horizon where the radial horizon function ∆ x is negative and tending to zero. The line-element (1) defines a tetrad that is aligned with the principal null directions. Since the radial coordinate x is timelike near the inner horizon, it is convenient to take x as the time coordinate of the tetrad, and to choose the sign of x so that it increases inwards, the direction of advancing time.
The conformal factor ρ is a product of separable (electrovac) ρ s , time-dependent e vt , and inflationary e −ξ factors,
ρ = ρ s e vt−ξ .(4)
Conformal time-translation symmetry is expressed by the fact that the spacetime expands conformally (that is, without changing shape) by factor ρ → e v∆t ρ when the conformal time increases by t → t + ∆t. Conformal stationarity means taking the limit of small expansion rate, or small accretion rate, after calculations are complete,
v → 0 .(5)
This is not the same as stationarity, which sets v to zero at the outset. A feature of inflation is that the smaller the accretion rate, the faster inflation exponentiates. Even in the limit of infinitesimal accretion rate, inflation drives the centre-of-mass streaming density and pressure, and the Weyl curvature, to exponentially huge values. Mathematically, Einstein's equations contain terms of order ∼ v/∆ x that grow large at the inner horizon ∆ x → −0 however small the accretion rate v may be. This paper adopts a collisionless fluid as the source of energy-momentum that ignites and then drives inflation. Because collisionless particles stream hyper-relativistically through each other during inflation, the trajectories of massive freely-falling particles are well-approximated by those of massless particles. Conformal separability posits that the equations of motion of freely-falling massless particles are Hamilton-Jacobi separable, which implies that a conformal Killing tensor exists. As shown in Appendix A of Paper 2, conformal separability requires that
ω x , ∆ x are functions of x only , ω y , ∆ y are functions of y only .(6)
Unlike strict separability (for massive as well massless particles), conformal separability does not impose conditions on the conformal factor ρ.
Conformally separable inflationary solutions are obtained by separating the Einstein equations systematically. Homogeneous solution of the Einstein components G xy , G tφ , and G xx + G tt and G yy − G φφ leads to the usual electrovac solutions for the electrovac conformal factor ρ s and the vierbein coefficients ω x and ω y :
ρ s = ρ 2 x + ρ 2 y , ρ x = g 0 − g 1 ω x (f 0 g 1 + f 1 g 0 )(f 0 + f 1 ω x ) , ρ y = g 1 − g 0 ω y (f 0 g 1 + f 1 g 0 )(f 1 + f 0 ω y ) ,(7)dω x dx = 2 (f 0 + f 1 ω x ) (g 0 − g 1 ω x ) , dω y dy = 2 (f 1 + f 0 ω y ) (g 1 − g 0 ω y ) ,(8)
where f 0 , f 1 , g 0 , and g 1 are constants determined by boundary conditions. Equations (7) and (8) continue to hold throughout inflation and collapse. The inflationary solutions are generic, applying wherever a separable electrovac spacetime has an inner horizon, so the specific choice of constants f 0 , f 1 , g 0 , and g 1 does not affect the argument. The most important Einstein equations, since they lead to equations governing the evolution of the inflationary exponent ξ and the horizon function ∆ x , are those for the Einstein components G xx − G tt and G yy + G φφ . The collisionless source of both these components can be treated as negligible, in the conformally stationary limit of small accretion rate. The angular components are negligible because inflation amplifies the radial, not angular components; and the trace of the collisionless energy-momentum remains small because it depends on the rest mass of the particles, which is unchanged by inflation. Define U x , U y , X x , X y , Y x , and Y y by
U x ≡ − ∂ξ ∂x ∆ x , U y ≡ ∂ξ ∂y ∆ y ,(9a)X x ≡ ∂U x ∂x + 2 U 2 x − v 2 ∆ x , X y ≡ ∂U y ∂y − 2 U 2 y + v 2 ω 2 y ∆ y ,(9b)Y x ≡ d∆ x dx + 3U x − ∆ x d dx ln (f 0 +f 1 ω x ) dω x dx , Y y ≡ d∆ y dy − 3U y − ∆ y d dy ln (f 1 +f 0 ω y ) dω y dy .(9c)
In terms of these quantities, the Einstein components G xx − G tt and G yy + G φφ are
ρ 2 (G xx − G tt ) = 1 σ 2 Y x d ln ω x dx − Y y d ln ω y dy − 2X x + Y x d dx ln f 0 +f 1 ω x ω x + X y − ∂Y y ∂y + Y y d dy ln ω y (f 1 +f 0 ω y ) dω y /dy + U x ∂ ∂x ln σ 2 (f 0 +f 1 ω x ) − U y ∂ ∂y ln (g 1 −g 0 ω y ) σ 2 dω y dy , (10a) ρ 2 (G yy + G φφ ) = 1 σ 2 Y x d ln ω x dx − Y y d ln ω y dy − 2X y − Y y d dy ln f 1 +f 0 ω y ω y + X x + ∂Y x ∂x − Y x d dx ln ω x (f 0 +f 1 ω x ) dω x /dx + U y ∂ ∂y ln σ 2 (f 1 +f 0 ω y ) − U x ∂ ∂x ln (g 0 −g 1 ω x ) σ 2 dω x dx . (10b)
Homogeneous solutions of these equations can be found by supposing that U x , X x , and Y x are all functions of radius x, while U y , X y , and Y y are all functions of radius y, and by separating each of the equations as
1 σ 2 f 0 h 0 +h 2 ω x +f 1 h 1 ω 2 x ω x − f 1 h 1 +h 2 ω y +f 0 h 0 ω 2 y ω y − f 0 h 0 +h 3 ω x ω x + f 1 h 1 +h 3 ω y ω y = 0 ,(11)
for some constants h 0 , h 1 , h 2 , and h 3 . If one attempts to separate equations (10) exactly, then the attempt fails unless U x and U y are identically zero, which is the usual electrovac case. But if U x is taken to be small but finite, then separation succeeds, and inflation emerges. If U x and U y on the second lines of equations (10) are treated as negligibly small, then separating the first lines of each of equations (10) according to the pattern of equation (11) leads to the homogeneous solutions
X x = 0 , X y = 0 , (12a) Y x = (f 0 + f 1 ω x )(h 0 + h 1 ω x ) dω x /dx , Y y = (f 1 + f 0 ω y )(h 1 + h 0 ω y ) dω y /dy . (12b)
If U x = U y = 0, then solution of equations (9c) and (12b) for Y x and Y y , subject to appropriate boundary conditions, yields the radial and angular horizon functions ∆ x and ∆ y of the Kerr line-element. The result is easily generalized to other electrovac spacetimes by admitting appropriate sources for Y x and Y y .
The quantity X y defined by equation (9b) determines the evolution of U y , and the solution X y = 0, equation (12a), then implies that inflation leaves U y unchanged, and sensibly equal to its electrovac value of zero, in the conformally stationary limit. Thus inflation leaves the angular horizon function ∆ y unchanged from its electrovac value.
On the other hand, inflation drives U x away from zero however small it might initially be. In the vicinity of the inner horizon, where ∆ x → −0, the solutions (12) for X x and Y x defined by equations (9b) and (9c) imply the evolution equations
∂U x ∂x + 2 U 2 x − v 2 ∆ x = 0 ,(13a)d∆ x dx + 3U x = ∆ ′ x ,(13b)
where ∆ ′ x ≡ d∆ x /dx| xin is the (positive) derivative of the electrovac horizon function at the inner horizon x = x in . Below, equation (28c), it will be found that the radius x remains frozen at its inner horizon value x in throughout inflation and collapse, so the right hand side of equation (13b), which is the electrovac solution for Y x evaluated at the inner horizon, is constant during inflation and collapse. The evolution equation (13a) for U x involves a term inversely proportional to the horizon function ∆ x , which diverges at the inner horizon ∆ x → −0, driving U x away from zero however small U x might initially be. Equation (13a) leads to instability only at the inner horizon, where ∆ x → −0. At the outer horizon, where ∆ x → 0, solutions of equation (13a) decay rather than grow.
The separation of the Einstein components (10) that leads to the evolution equations (13) was premised on the assumption that the terms proportional to U x and U y on the second lines of equations (10) could be neglected. However, the separation continues to remain valid during inflation and collapse when U x grows huge. The reason for this is that the dominant terms in the Einstein components (10) during inflation and collapse are of order U 2
x /∆ x , coming from the expression (9b) for X x . Thus, once U x ceases to be negligible, the condition for the validity of the separation becomes U x ≪ U 2
x /|∆ x |, or equivalently |∆ x | ≪ U x . Consequently the condition for the validity of the separation of the Einstein components (10) is
either U x ≪ 1 or |∆ x | ≪ U x .(14)
Condition (14) holds from electrovac through inflation and collapse, provided that the accretion rate is small, as conformal stationarity prescribes. The fact that condition (14) suffices is verified in §VIII J of Paper 2, where the Einstein equations are solved to next order in ∆ x /U x , and it is shown that the effect on the evolution of the inflationary exponent ξ and horizon function ∆ x is negligible. The evolution equations (13) are solved in the next section, §IV, but first it is necessary to attend to the other Einstein equations.
In the conformally separable geometry, freely-falling collisionless ingoing and outgoing streams become highly focussed along the principal ingoing and outgoing null directions as they approach the inner horizon. Inflation accelerates the streams even faster along the same null directions, causing the x and t components of the tetrad-frame momenta of freely-falling collisionless streams to grow exponentially. Consequently the collisionless energy-momentum is dominated by its x-t components. The associated components of the Einstein tensor are
ρ 2 G xx + G tt 2 ± G xt = (U x ∓ v) Y x ± v −∆ x − d dx ln dω x dx + X x .(15)
Since X x = 0, and Y x = ∆ ′ x , and the term proportional to d ln(dω x /dx)/dx is sub-dominant (in fact the term disappears when the Einstein components (10) and corresponding evolution equations (13) are solved to next order in ∆ x /U x ; see Paper 2), equation (15) simplifies to
ρ 2 G xx + G tt 2 ± G xt = 1 −∆ x (U x ∓ v)(∆ ′ x ± v) .(16)
The right hand side of equation (15) agrees with 8π times the energy-momentum tensor of two collisionless streams, one ingoing (+) and one outgoing (−),
T kl = N + p + k p + l + N − p − k p − l ,(17)
with densities
N ± = 1 16π (U x ∓ v)(∆ ′ x ± v) ,(18)
and tetrad-frame momenta
p ± k = 1 ρ − 1 √ −∆ x , ∓ 1 √ −∆ x , 0 , 0 .(19)
The tetrad-frame momenta (19) are null vectors pointed along the principal ingoing and outgoing null directions. That equations (18) and (19) describe correctly the behaviour of freely-falling streams can be shown by solving the Hamilton-Jacobi and collisionless Boltzmann equations (see Paper 2), and can be confirmed by checking that the densities and momenta satisfy, to requisite accuracy, covariant number conservation, D k N ± p ± k = 0, and the geodesic equation dp ± k /dλ = 0, where λ is an affine parameter. It might seem somewhat miraculous that the x-t components of the Einstein equations are satisfied with a collisionless source, but it is no coincidence. Einstein's equations enforce covariant energy-momentum conservation, D k T kl = 0. Since the angular components are sub-dominant, only the 3 distinct x-t components of the energy-momentum tensor are important. The 3 components are subject to 2 energymomentum conservation equations, but in the present instance the 2 conservation equations are redundant, so there is effectively 1 conservation equation. However, the energy-momentum conservation equations for freely-falling ingoing and outgoing streams are symmetrically related to each other by v → −v, so conservation of the sum of their energies, as enforced by Einstein, implies conservation of both. The two conservation equations, coupled with solution of the Einstein equation for G xx − G tt , equation (10a), leads to a complete, self-consistent set of equations.
The angular motions of the freely-falling streams are small compared to their radial motions, but not necessarily zero. Next in order of magnitude, after the 3 radial (x-t) components of the energy-momentum tensor, are its 4 off-diagonal radial-angular components. The corresponding components of the Einstein tensor are
ρ 2 (G xy ± G ty ) = − 1 −∆ x ∆ y (U x ∓ v) ∆ y ∂ ln ρ 2 s ∂y − 2U y − −∆ x ∆ y U y ∂ ln ρ 2 s ∂x ± vω y σ 2 dω y dy ,(20a)ρ 2 (G xφ ± G tφ ) = ± 1 −∆ x ∆ y (U x ∓ v) ∆ y σ 2 dω x dx ∓ 2vω y ∓ −∆ x ∆ y U y σ 2 dω y dy ∓ vω y ∂ ln ρ 2 s ∂x .(20b)
Since U y and the terms proportional to v √ −∆ x are negligible, equations (20) simplify to
ρ 2 (G xy ± G ty ) = − 1 −∆ x ∆ y (U x ∓ v)∆ y ∂ ln ρ 2 s ∂y , (21a) ρ 2 (G xφ ± G tφ ) = ± 1 −∆ x ∆ y (U x ∓ v) ∆ y ω ′ x σ 2 ∓ 2vω y ,(21b)
where ω ′ x ≡ dω x /dx| xin , which is effectively constant throughout inflation and collapse, is the derivative of ω x at the inner horizon x = x in , equation (8). The right hand sides of equations (21) agree with 8π times the energy-momentum tensor of ingoing and outgoing streams with the same densities N ± as before, equation (18), but with tetrad-frame momenta p ± k having finite rather than zero angular components:
p ± k = 1 ρ − 1 √ −∆ x , ∓ 1 √ −∆ x , 1 ∆ y ∆ y ∂ ln ρ 2 s /∂y ∆ ′ x ± v , ∓ 1 ∆ y ∆ y ω ′ x /σ 2 ∓ 2vω y ∆ ′ x ± v .(22)
The tetrad-frame momenta (22) satisfy the Hamilton-Jacobi equations with constant Hamilton-Jacobi parameters along the path of the streams. The angular components of the momenta are small compared to the radial components as long as
|∆ x | ≪ 1 .(23)
The momentum (22) is hyper-relativistic, and p ± t = ±p ± x to an excellent approximation so long as condition (23) is true. If the condition (23) is violated, then it signifies that angular motions are becoming important, and the solution is breaking down.
It should be emphasized that, as long as condition (23) holds, the purely radial (x-t) Einstein equations hold regardless of angular motions, and thus the radial solution is unaffected by angular motions. However, if one requires that the sub-dominant radial-angular Einstein equations are also satisfied, then the angular motion of the collisionless streams must be as given by equation (22). One might perhaps have expected that conformally separable solutions would require that the collisionless streams would move exactly along the principal null directions, but equation (22) shows that this is not true.
Again, it might seem remarkable that the radial-angular Einstein equations are satisfied by collisionless ingoing and outgoing streams. And again, this coincidence results from energy-momentum conservation. There are 4 radial-angular Einstein components, subject to 2 energy-momentum conservation equations. The energy-momentum conservation equations for the freely-falling ingoing and outgoing streams are symmetrically related by v → −v, so conservation of their sum implies conservation of both.
The final, sub-sub-dominant, components of the energy-momentum tensor are the 3 purely angular (y-φ) components. The component G yy + G φφ component has already been addressed, equation (10b). The remaining 2 components are
ρ 2 G yy − G φφ 2 ± iG yφ = (U y ∓ ivω y ) − Y y ± ivω y ∆ y − d dy ln dω y dy + X y ∓ iv dω y dy .(24)
Since X y = 0, and U y and v are negligibly small, and there are no denominators of the radial horizon function ∆ x , equation (24) simplifies to
ρ 2 G yy − G φφ 2 ± iG yφ = 0 .(25)
During inflation, the collisionless streams have negligible angular components of energy-momentum because the densities of the accreting streams are negligible, in the conformally stationary limit, and inflation does not amplify angular motions. During collapse, the conformal factor ρ shrinks, and angular motions grow. However, as long as the angular motions are sub-dominant, which is true as long as condition (23) is satisfied, the angular components of the energy-momentum can be neglected consistently: the Einstein equations for the purely radial components, and for the radial-angular components, are unaffected by the angular components (25) (the angular component G yy + G φφ , along with G xx − G tt , equations (10), determined the angular horizon function ∆ y , which inflation leaves unaltered from its electrovac form). Equation (25) requires that the 2×2 angular submatrix of the energy-momentum tensor be isotropic, proportional to the 2 × 2 unit matrix. As discussed in Paper 2, it is possible to arrange the angular energy-momentum to be isotropic by admitting multiple ingoing and outgoing streams, with mean momenta set by equation (22) and isotropic mean squared momenta. Treating the diagonal components of the angular energy-momentum requires taking equations (10) to next order in ∆ x /U x , but this can be done. Eventually however, the angular components do become important, when |∆ x | ∼ 1, and the solution fails. (28), as a function of the inflationary exponent ξ, for parameters v = 0.001, u = 0.002, and ∆ ′ x = 1 (the solutions in this paper apply in the limit of tiny v and u; small finite values are adopted in this plot to avoid numerical overflow). Inflation ignites as the horizon function |∆x| decreases below Ux. Inflation ends as the absolute value of the horizon function goes through a minimum, and the geometry proceeds to collapse. Once |∆x| 1, the angular components of the collisionless streams exceed their radial components, and the solution breaks down, but this happens only after the geometry has collapsed to exponentially tiny scale. (Right) The tetrad-frame radial, radial-angular, and angular collisionless energy-momenta Txx ∝ ρ −2 Ux/|∆x|, Txy ∝ ρ −2 Ux/ |∆x|, and Tyy ∝ ρ −2 Ux. The energy-momenta grow exponentially huge despite their small initial values. Indeed, the smaller the initial energy-momenta, the faster and larger they grow. The dashed line is minus the polar (real) spin-0 component of the Weyl curvature, −C ∝ ρ −2 U 2
x /|∆x|. The axial (imaginary) spin-0 Weyl component is comparable to Tyy.
IV. INFLATION AND COLLAPSE
Denote the initial value of U x , equation (9a), incident on the inner horizon by u, a small parameter of order v,
U x = u initially .(26)
The densities N ± of ingoing and outgoing streams incident on the inner horizon are proportional to u∓v, equation (18). Inflation is driven by counter-streaming between ingoing and outgoing streams, so both streams must be present for inflation to occur, but even the tiniest amount suffices to trigger inflation. Positivity of both ingoing and outgoing densities requires that
u > v > 0 ,(27)
the condition v > 0 coming from the fact that the black hole must expand as it accretes. The case v = 0 is the stationary (or homogeneous) approximation of [11]. The densities N ± , equation (18), are also proportional to ∆ ′
x ∓ v, where ∆ ′
x is the positive derivative of the electrovac horizon function at the inner horizon. Positivity of both ingoing and outgoing densities requires ∆ ′ x to be strictly positive, which excludes extremal black holes, whose inner and outer horizons coincide, and which have ∆ ′ x = 0 at the horizon. Solution of the evolution equations (13) for U x and ∆ x yields
U x = v 2 + (u 2 − v 2 )e 4ξ ,(28a)∆ x = − U 2 x − v 2 u 2 − v 2 3/4 (U x + v)(u − v) (U x − v)(u + v) ∆ ′ x /(4v) ,(28b)
x
− x in = − ∆ x dU x 2(U 2 x − v 2 ) .(28c)
The integral on the right hand side of equation (28c) can be expressed analytically as an incomplete beta function, but the expression is not useful. Physically, equation (28c) says that the radius x is frozen at its inner horizon value x in during inflation and collapse, where U x is growing, while ∆ x remains small. Figure 3 illustrates the evolution of U x and the horizon function ∆ x as a function of the inflationary exponent ξ, for parameters v = 0.001 and u = 0.002. The value v = 0.001, which physically represents the velocity with which a distant observer sees the characteristic radius of the black hole expand, is large compared to a typical astronomical accretion rate, but a large value is needed to avoid numerical overflow. Inflation ignites near the inner horizon as the horizon function |∆ x | drops below u. During inflation, the horizon function |∆ x | decreases exponentially, while U x increases slowly. During inflation, the inflationary exponent ξ in the conformal factor ρ, equation (4), satisfies
ξ ≪ dξ dx ≪ d 2 ξ dx 2 ,(29)
which is the behaviour characteristic of a step-function. The inequalities (29) essentially say that the acceleration d 2 ξ/dx 2 of the inflationary exponent, which is driven by the radial energy-momentum of the collisionless streams, is much larger than the velocity dξ/dx, which in turn is much larger than the distance moved ξ. Inflation ends when the absolute value of the horizon function reaches a minimum, at an exponentially tiny value,
|∆ x | ∼ e −1/v ,(30)
at which point the spacetime collapses. During collapse, the horizon function |∆ x | increases, while the conformal factor ρ ∝ e −ξ shrinks exponentially, no longer satisfying the inequalities (29). The radial coordinate x remains frozen even while the conformal factor ρ is shrinking. That the spacetime collapses rather than leads to a null singularity accords with the conclusion of [9] that the outcome of inflation is collapse when a black hole continues to accrete, as is ensured in the present case by the assumption of conformal time translation invariance (self-similarity).
During collapse the horizon function increases back to of order unity, |∆ x | ∼ 1. At this point the angular motion of the freely-falling ingoing and outgoing streams becomes comparable to their radial motion, and the solution breaks down. This happens when the conformal factor has collapsed to an exponentially tiny value,
ρ ∼ e −1/v .(31)
The right panel of Figure 3 shows the magnitudes of the radial, radial-angular, and angular components T xx , T xy , and T yy of the tetrad-frame energy-momenta of the collisionless streams. During inflation, the radial energy-momentum grows fastest, reaching an exponentially huge value
T xx ∼ e 1/v .(32)
During collapse, the angular energy-momentum grows fastest. The Weyl curvature tensor has only a spin-0 component, which classifies the spacetime as Petrov Type D. The right panel of Figure 3 shows minus the polar (real) part of the spin-0 component of the Weyl curvature.
It is notable that the smaller the accretion rate v, the more rapidly inflation exponentiates, and the larger the energy-momentum and curvature grow, in agreement with the conclusions of [9,12].
V. BOUNDARY CONDITIONS
The solutions are determined by boundary conditions of the collisionless streams incident on the inner horizon. Since the solution above the inner horizon is well-approximated by the Kerr (or other electrovac) solution, the behaviour of gas above the inner horizon does not affect the solution.
The requirement of conformal separability imposes special boundary conditions. The densities N ± of ingoing (+) and outgoing (−) streams incident on the inner horizon are, equation (18), since U x = u initially,
N ± = 1 16π (u ∓ v)(∆ ′ x ± v) .(33)
This is just a constant, independent of angular position on the inner horizon. Thus conformal separability requires that the incident flow of ingoing and outgoing streams be "monopole," independent of latitude. It makes physical sense that conformal separability would require this high degree of symmetry of the accretion flow. As emphasized in §III, because the radial motions of collisionless streams dominate their angular motions during inflation and collapse (up until the angular motions become important, at |∆ x | ∼ 1), the radial Einstein equations (22) at the inner horizon required if conformal separability is imposed to sub-dominant radial-angular order, for an uncharged black hole with spin parameter a = 0.96M•. The thicker contours mark the outer and inner horizons, the latter being destroyed by inflation. Ingoing and outgoing particles fall along the same trajectories in the x-y plane, but have opposite motions in the azimuthal φ coordinate. Trajectories near the equatorial plane change from ingoing to outgoing, or vice versa, inside the outer horizon; the transition is marked by the lines changing from dashed to solid. The angular flow pattern cannot be achieved with collisionless streams that fall from outside the outer horizon. In the equatorial region, outgoing but not ingoing particles can fall from outside the outer horizon, while in the polar region, ingoing but not outgoing particles can fall from outside the outer horizon.
are unaffected by the angular motion, and the boundary condition (33) is all that is needed to ensure conformal separability with sufficient accuracy. However, if it is required that the sub-dominant radial-angular components of the Einstein equations are also satisfied, which is a more stringent constraint on conformal separability, then the tetrad-frame momenta p ± k of the streams must have finite angular components, satisfying equation (22). Figure 4 illustrates the required flow pattern for a black hole of spin parameter a = 0.96M • . The energy per unit mass of infalling particles, which is unspecified by boundary conditions, is chosen here to be E/m = ±1.
As described in Paper 2, the required angular flow pattern cannot be achieved with collisionless streams falling from outside the outer horizon. Streams that fall from outside the outer horizon must necessarily be ingoing at the outer horizon, which eliminates half the phase space available to collisionless streams, making it impossible to satisfy the required conditions on the angular motion of the streams. Thus, if the angular conditions are imposed, then the collisionless ingoing and outgoing streams must be considered as being delivered ad hoc to just above the inner horizon.
VI. CONCLUSION
This paper has presented conformally stationary, axisymmetric, conformally separable solutions for the interior structure of an uncharged rotating black hole that undergoes inflation just above its inner horizon, then collapses. It has long been known that linear perturbations diverge at the inner horizon of the Kerr geometry, and it has been suspected that perturbations would develop nonlinearly similarly to the inflationary instability [7] known to operate in spherical charged black holes [9], and expected to occur also in rotating black holes [8]. The self-consistent nonlinear solutions found here confirm that, at least in the conformally separable special case considered here, the inflationary instability develops in rotating black holes as anticipated.
A feature of the Kerr geometry (and other separable electrovac geometries) is that, as freely-falling ingoing and outgoing particles approach the inner horizon, they become highly focussed along the ingoing and outgoing null directions, regardless of their initial angular motion. However small the accretion rate may be, eventually the energy and pressure of the twin beams of particles counter-streaming hyper-relativistically along the ingoing and outgoing null directions grows large enough to be a source of gravity. The gravity produced by the counter-streaming acts to accelerate ingoing and outgoing stream even faster through each other, leading to an exponential growth in the streaming density and pressure, and in the curvature. This is inflation. The huge gravitational acceleration produced by the counter-streaming is in the inward direction, to smaller radius, but each stream thinks that they are moving in the inward direction, so the streams are accelerated in opposite directions.
Inflation takes place over an extremely short interval of proper time. Inflation is like a bullet fired in the chamber of a gun: an explosion accelerates the bullet, and shortly after the bullet achieves high velocity, but still the bullet has hardly moved (see the inequalities (29)). Inflation does in due course alter the geometry, but in a predictable way: the conformal factor, having been accelerated to huge velocity inward, proceeds to shrink rapidly. The geometry collapses.
During inflation, the ingoing and outgoing streams were accelerated along the principal null directions, without amplifying the angular motion. During collapse, the angular motions grow. At an exponentially tiny scale, the angular motions become comparable to the radial motion, and the solutions considered in this paper break down. What happens then is undetermined.
The existence of conformally separable solutions for the inflationary zones of rotating black holes is not surprising. Ingoing and outgoing streams focus along the principal null directions as they approach the inner horizon, and the streaming energy and pressure generated by the radial beams accelerates the streams along the same null directions. The acceleration depends on the accretion rate. If the densities of ingoing and outgoing streams incident on the inner horizon are uniform, independent of latitude, then inflation accelerates the beams at the same rate at all latitudes. When the geometry begins to collapse, it does so uniformly, preserving conformal separability. What happens in the more general case when the accretion flow on to the inner horizon varies with angular position remains to be seen. But that inflation will occur is physically inevitable.
FIG. 2 :
2Contours of constant radius x and latitude y in an uncharged black hole with spin parameter a = 0.96M•. The thicker contours mark the outer and inner horizons. The left panel depicts a Kerr black hole.
FIG. 3 :
3Evolution of the geometry and energy-momenta from electrovac through inflation and collapse. (Left) The parameter Ux ≡ −∆xdξ/dx and the horizon function ∆x, equations
FIG. 4 :
4Angular flow pattern of freely-falling particles that produces the conditions
FIG. 1: Partial Penrose diagram illustrating why the Kerr geometry is subject to the inflationary instability. Ingoing and outgoing streams just outside the inner horizon must pass through separate ingoing and outgoing inner horizons into causally separated pieces of spacetime where the timelike Kerr time coordinate t goes in opposite directions. To accomplish this, the ingoing and outgoing streams must exceed the speed of light through each other, which physically they cannot do. In reality, hyper-relativistic counter-streaming between the ingoing and outgoing streams ignites and then drives the exponentially growing inflationary instability. The inset shows the direction of coordinate time t in the various regions. Proper time of course always increases upward in a Penrose diagram.
AcknowledgmentsThis work was supported by NSF award AST-0708607.
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| [] |
[
"T-Net: Encoder-Decoder in Encoder-Decoder architecture for the main vessel segmentation in coronary angiography",
"T-Net: Encoder-Decoder in Encoder-Decoder architecture for the main vessel segmentation in coronary angiography"
] | [
"Tae Joon Jun \nSchool of Computing\nKorea Advanced Institute of Science and Technology\n34141DaejeonRepublic of Korea\n",
"Jihoon Kweon \nDivision of Cardiology\nUniversity of Ulsan College of Medicine\nAsan Medical Center\n05505SeoulRepublic of Korea\n",
"Young-Hak Kim [email protected] \nDivision of Cardiology\nUniversity of Ulsan College of Medicine\nAsan Medical Center\n05505SeoulRepublic of Korea\n",
"Daeyoung Kim \nSchool of Computing\nKorea Advanced Institute of Science and Technology\n34141DaejeonRepublic of Korea\n"
] | [
"School of Computing\nKorea Advanced Institute of Science and Technology\n34141DaejeonRepublic of Korea",
"Division of Cardiology\nUniversity of Ulsan College of Medicine\nAsan Medical Center\n05505SeoulRepublic of Korea",
"Division of Cardiology\nUniversity of Ulsan College of Medicine\nAsan Medical Center\n05505SeoulRepublic of Korea",
"School of Computing\nKorea Advanced Institute of Science and Technology\n34141DaejeonRepublic of Korea"
] | [] | In this paper, we proposed T-Net containing a small encoder-decoder inside the encoder-decoder structure (EDiED). T-Net overcomes the limitation that U-Net can only have a single set of the concatenate layer between encoder and decoder block. To be more precise, the U-Net symmetrically forms the concatenate layers, so the low-level feature of the encoder is connected to the latter part of the decoder, and the high-level feature is connected to the beginning of the decoder. T-Net arranges the pooling and up-sampling appropriately during the encoder process, and likewise during the decoding process so that feature-maps of various sizes are obtained in a single block. As a result, all features from the low-level to the high-level extracted from the encoder are delivered from the beginning of the decoder to predict a more accurate mask. We evaluated T-Net for the problem of segmenting three main vessels in coronary angiography images. The experiment consisted of a comparison of U-Net and T-Nets under the same conditions, and an optimized T-Net for the main vessel segmentation.As a result, T-Net recorded a Dice Similarity Coefficient score (DSC ) of 0.815, 0.095 higher than that of U-Net, and the optimized T-Net recorded a DSC of 0.890 which was 0.170 higher than that of U-Net. In addition, we visualized the * Corresponding author weight activation of the convolutional layer of T-Net and U-Net to show that T-Net actually predicts the mask from earlier decoders. Therefore, we expect that T-Net can be effectively applied to other similar medical image segmentation problems. | 10.1016/j.neunet.2020.05.002 | [
"https://arxiv.org/pdf/1905.04197v1.pdf"
] | 150,373,617 | 1905.04197 | 835bca4b69e7e76ea87ece8f0ddd0652076571ff |
T-Net: Encoder-Decoder in Encoder-Decoder architecture for the main vessel segmentation in coronary angiography
May 13, 2019 10 May 2019
Tae Joon Jun
School of Computing
Korea Advanced Institute of Science and Technology
34141DaejeonRepublic of Korea
Jihoon Kweon
Division of Cardiology
University of Ulsan College of Medicine
Asan Medical Center
05505SeoulRepublic of Korea
Young-Hak Kim [email protected]
Division of Cardiology
University of Ulsan College of Medicine
Asan Medical Center
05505SeoulRepublic of Korea
Daeyoung Kim
School of Computing
Korea Advanced Institute of Science and Technology
34141DaejeonRepublic of Korea
T-Net: Encoder-Decoder in Encoder-Decoder architecture for the main vessel segmentation in coronary angiography
May 13, 2019 10 May 2019Preprint submitted to Neural NetworksConvolutional neural networkMain vessel segmentationCoronary angiographyEncoder and decoder
In this paper, we proposed T-Net containing a small encoder-decoder inside the encoder-decoder structure (EDiED). T-Net overcomes the limitation that U-Net can only have a single set of the concatenate layer between encoder and decoder block. To be more precise, the U-Net symmetrically forms the concatenate layers, so the low-level feature of the encoder is connected to the latter part of the decoder, and the high-level feature is connected to the beginning of the decoder. T-Net arranges the pooling and up-sampling appropriately during the encoder process, and likewise during the decoding process so that feature-maps of various sizes are obtained in a single block. As a result, all features from the low-level to the high-level extracted from the encoder are delivered from the beginning of the decoder to predict a more accurate mask. We evaluated T-Net for the problem of segmenting three main vessels in coronary angiography images. The experiment consisted of a comparison of U-Net and T-Nets under the same conditions, and an optimized T-Net for the main vessel segmentation.As a result, T-Net recorded a Dice Similarity Coefficient score (DSC ) of 0.815, 0.095 higher than that of U-Net, and the optimized T-Net recorded a DSC of 0.890 which was 0.170 higher than that of U-Net. In addition, we visualized the * Corresponding author weight activation of the convolutional layer of T-Net and U-Net to show that T-Net actually predicts the mask from earlier decoders. Therefore, we expect that T-Net can be effectively applied to other similar medical image segmentation problems.
Introduction
Semantic segmentation is a typical problem where deep learning technology is actively applied. Compared with classification, semantic segmentation has the advantage of visualizing the characteristics of an image because it can display a concrete region with classes of object. However, while labeling of classification is word or number level, labeling of semantic segmentation requires much time and effort for labeling because it needs to extract specific area from the image. Therefore, the most active area of semantic segmentation problem is medical image analysis. This is because the effect obtained by marking specific regions in the medical image is large even if time and effort are involved. Unlike general images, which have large number of objects to be segmented and their shape vary, medical images are captured with a specific purpose, so the number of classes for segmentation is relatively small and the image shape is fixed. Therefore, various methods for semantic segmentation is proposed to solve specific medical image segmentation problems [1].
Currently, the most popular method for medical image semantic segmentation is the fully convolutional neural network (CNN) structure based on U-Net [2]. The U-Net consists of an encoder part extracting a feature from the original image and a decoder part restoring the feature to a mask image. However, since the size of the feature map continuously decreases during the encoder process, noise is generated while restoring the extracted features in the decoder process. Therefore, to minimize the loss of the original image, U-Net provides a concatenate layer that directly connects the encoder and decoder. However, due to the structural restriction of U-Net, there is only one set of convolution blocks matching the feature-map of the same size in the encoder and decoder, which has a limitation in generating a precise mask. More specifically, the high-level feature extracted from the latter encoder is connected to the beginning of the decoder, and the low-level feature is connected to the decoder near the prediction layer. Particularly, this limitation is fatal in the medical image problem where the number of classes of the object to be segmented is small, but the region of the mask has to be precisely segmented. An example of such a medical image problem is the segmenting main vessels from coronary angiography. In coronary angiography, the number of main vessels to be segmented is relatively small, but masks should be generated in the same form as the main vessel of the original image. In other words, the main vessel is identified among the various blood vessels of similar shape in the image, and the predicted region should be similar to the actual vessel shape in the original image. Therefore, from the low-level feature indicating the shape of vessels to the high-level feature specifying the main vessel, all levels of features should be considered in the mask restoring process.
In this paper, we propose a T-Net that allows various sizes of feature-maps between encoder and decoder, resulting in sophisticated semantic segmentation. The core concept of T-Net is encoder-decoder in encoder-decoder (EDiED) structure. Through EDiED structure, the size of the feature-map is increased by up-sampling in the encoding process, while the size of the feature-map is reduced by pooling in the decoding process. Thus, there are multiple sizes of feature maps in the same block, which allows for a more versatile combination when constructing the concatenated layers of the encoder-decoder. In other words, precise segmentation is possible from the beginning of the decoder by transmitting all levels of features extracted from every encoder block. We evaluated T-Net for the problem of segmenting three types of main vessels in coronary angiography. The three main vessels are the left anterior descending artery (LAD), the left circumflex artery (LCX), and the right coronary artery (RCA). We first compare the performance of U-Net and T-Net under the same conditions and vi-sualize the intermediate convolutional layers to see how the actual weight varies from original to mask image. Then, we fine-tuned the optimized T-Net structure to show the best segmentation performance. As a result, T-Net showed 0.095 higher Dice Similarity Coefficient score (DSC ), 5.71% higher sensitivity, 12.22% higher precision than those of U-Net in the same experiment condition.
The optimized T-Net showed an average of 0.890 DSC, 88.32% sensitivity and 90.50% precision for the three types of main vessels segmentation from coronary angiography. Our T-Net is also expected to be effectively applied to other medical image segmentation problems that require precise segmentation.
The rest of this Chapter is structured as followed. In Section 2, we review the literature for vessel segmentation in coronary angiography and also briefly review CNN-based studies for semantic segmentation. Section 3 describes the basic structure of T-Net and shows examples of various models that can be derived from T-Net. Section 4 describes the optimal T-Net structure for the main vessel segmentation in coronary angiography. Section 5 evaluates the comparison of T-Net and U-Net and the performance of optimized T-Net. Finally, we conclude this study in Section 6 and discusses future plans.
Related Work
Main vessel segmentation in coronary angiography
The World Health Organization (WHO) has announced that cardiovascular diseases (CVDs) are the leading cause of death in today's world [3]. More than 17 million people died of CVDs in 2016 which is an about 31% of all deaths, and more than 75% of these deaths occurred in low-income and middle-income countries [3]. Among CVDs, coronary artery disease (CAD) is the most common cause of death [4] [5]. In 2015, CAD affected 110 million people, resulting in 8.9 million deaths and 15.6% of all deaths, making it the most common cause of death worldwide [4] [5]. The primary imaging method to observe CAD is X-ray angiography, often called coronary angiography. Especially in coronary angiography, it is important to identify the main blood vessels correctly. However, the identification of main vessels currently depends on the manual segmentation from the radiologist, requiring a lot of time and effort. Moreover, it is difficult to identify the vessel clearly because of low contrast, non-uniform illumination, and low signal to noise ratios (SNR) of X-ray angiography [6]. Therefore, studies on automated vessel segmentation are aimed at reducing time and cost by helping relevant experts. Among them, the main vessel segmentation is a difficult problem because it does not identify the whole blood vessels shown in the image but only the main blood vessel is segmented. Figure 1 shows the LAD, LCX, and RCA vessels observed in coronary angiography. There are several studies on blood vessel segmentation in coronary angiography. Near-Esfahani proposed a CNN-based method to classify whole blood vessels in X-ray angiography [6]. Near-Esfahani used CNN to classify the central pixel of each patch after dividing a single image into several small patches.
A total of 44 coronary angiography images of 512 x 512 size were spliced into 26 train-sets and 18 test-sets and the result was 93.5% segmentation accuracy.
Felfelian proposed a method of extracting the ROI of the coronary arteries with a Hessian filter and segmenting the blood vessel by overlapping the ROI with the flux flow measurement result [7]. As a result, the segmentation accuracy was about 96% for a total of 50 x-ray angiography images. Wang proposed a method of vessel segmentation by combining Hessian matrix multi-scale filtering and region growing algorithm [8]. Similarly, M'hiri proposed a vessel segmentation method that combines Hessian-based vesselness information with a random walk formulation [9]. Compared with existing methods such as Frangi's filter and active contour method for 9 angiography images, the AUC was 0.95. In addition, there are many other vessel segmentation approaches, but as above, they are not suitable for the main vessel segmentation problem because they segment the entire vessel in X-ray angiography [10][11] [12]. In other words, a machine learning based method is needed to extract the characteristics and position of the main vessel in order to segment only the main vessel. Recently, Jo proposed the method of segmenting the LAD in coronary angiography, which is the most consistent with our study [13]. Jo automatically selects the appropriate filter through the selective feature mapping (SFM) method to extract the candidate area. Then, LAD vessel segmentation is performed in the candidate area. The CNN model for segmentation is typical U-Net and is compared to U-Nets with backbone CNN using VGGNet [14] or DenseNet [15]. In a total of 1,987 angiography images, 200 images were used as train-set and 1,787 images were used as test-set, and the highest result showed an average of 0.676 DSC.
Although Jo's approach utilizes U-Net and presents a novel SFM method, the segmentation performance is still low.
CNN for semantic segmentation
CNN for semantic segmentation has been developed in two major directions.
One is the direction of the semantic segmentation of objects in a general image.
Generally, it is evaluated in PASCAL VOC [16] and MS COCO [17] dataset, where numbers of classes to segment are 20 and 91 respectively, excluding the background. Therefore, the feature extraction performance of the encoder is of primary importance in this direction. The first proposed CNN-based method is the fully convolutional network (FCN) proposed by Long [18]. FCN is the model that changes the fully connected layer of the well-known classification models such as AlexNet [19], VGGNet [14], and GoogLeNet [20] to a 1x1 convolution and up-sampling the final prediction. However, there is a limitation that the process of restoring the FCN from a very small feature-map to the original mask at a single step is not accurate. Therefore, transposed convolution is proposed to overcome the limitation of FCN [21]. Meanwhile, various studies for improving the performance in the PASCAL VOC dataset have been proposed [22] [23] [24] [25]. The recently proposed DeepLabv3+ [26] takes into account the encoderdecoder architecture in previous version of DeepLabv3 [27].
Another direction is to perform semantic segmentation in medical images.
In fact, creating a mask of semantic segmentation is very costly because it is almost impossible to segment all objects that appear in a wide variety of generic images. This is why the number of classes in ImageNet [28], a dataset for classification, is 1000, while PASCAL VOC and MS COCO are less than 100. However, since medical image segmentation require relatively fewer classes (tumor, vessel, organs, etc.) in typical types of images (MRI, CT, X-ray, etc.), higher performance can be obtained with a fewer number of images. In addition, because segmentation provides explainable information on medical judgment than simply classifying images, studies on medical image segmentation are very active in a wide range of medical fields [1] [29]. The most popular CNN based model in medical segmentation is U-Net, which is also called encoder-decoder architecture. A detailed description of U-Net follows in the next section. Also, there are variations of U-Net for medical image segmentation such as 3D U-Net [30], V-Net [31], H-DenseUNet [32], etc. However, there has been no study related to transmitting various levels of features extracted from encoders by performing multiple pooling and up-sampling in a single block to generate multisized feature maps.
T-Net: encoder-decoder in encoder-decoder architecture
Before describing T-Net in detail, we introduce the basic structure of U-Net and explain the structural limitation. U-Net is a symmetric structure, as its name implies, an encoder that extracts a feature and a decoder that restores a feature to a mask. The difference with FCN is that as the depth of the encoder increases, the number of filters increases as the general classification CNN model, and conversely, the number of filters decreases as the decoder reaches the prediction layer. Figure 2 shows the basic structure of U-Net. Therefore, E i and D i of U-Net have feature-map of the following sizes.
S(E i ) = S(D i ) = h 2 i-1(1)
Because the concatenate layer connects the same sized convoluted layer, the concatenated layer in U-Net is only one-to-one matching of E i and D i . However, as the depth of the encoder becomes deeper, the high-level feature of the original image is extracted, whereas the corresponding decoder block just started restoring. On the other hand, the earliest block of the encoder extracts the low-level feature, but the matching decoder block connected is the block closest to the prediction. In other words, U-Net connects low-level features close to the prediction layer and connects high-level features far to the prediction layer.
This is an inevitable limit for a single set of encoder-decoder architecture.
Encoder-decoder in encoder-decoder (EDiED)
In order to overcome the structural simplicity of U-Net, we propose the encoder-decoder in encoder-decoder (EDiED) architecture. The purpose of EDiED is to ensure that the various levels of features extracted from the encoder are delivered during the training of the decoder. To do this, the unmatched E i and D j need to be concatenated to each other. The Figure 3 shows the simplest T-Net, which is named T3-Net. so the sum of n(P ) and n(U ) must be odd, and n(P ) is one more than n(U ).
Likewise, D i needs to double the size of the feature-map, so n(U ) is one more than n(P ). Therefore, E i that can exist in T3-Net are PUP i and PPU i , and D i are UPU i and UUP i . Assuming that P and U are general stride 2 pooling and up-sampling, the feature-map sizes of E i and D i that can exist in T3-Net are as follows.
S(E i ) = S(PPU i ) = { h 2 i-1 , h 2 i , h 2 i+1 } S(E i ) = S(PUP i ) = { h 2 i-1 , h 2 i } S(D i ) = S(UUP i ) = { h 2 i-3 , h 2 i-2 , h 2 i-1 } S(D i ) = S(UPU i ) = { h 2 i-2 , h 2 i-1 }(2)
From equation 2, T3-Net with E i and D i with PPU i and UUP i respectively
has E i-1 , E i , E i+1 and D i+1 , D i+2 and D i+3 blocks with feature-map size h/2 i .
Thus, when compared to U-Net, T3-Net can add up to nine times more concatenate layers between encoder and decoder. And concatenation of these various combinations improves segmentation performance by transferring various levels of extracted features to the decoder's restore process. Figure 3 shows the architecture of T3-Net where E i and D i are PPU i and UUP i .
Considerations for designing T-Net
T-Net can exist in various forms depending on the number and order of pooling and up-sampling that build up E i and D i . However, even if n(P ) and n(U ) increase, what is needed to obtain various sizes of feature-maps is continuous pooling or up-sampling. In other words, even though the depth of the block can be increased by placing P and U alternately, the sizes of the feature-maps generated are the same. The following equation shows the sizes of the feature-maps when alternating between P and U.
S(PUP i ) = S(PUPUP i ) = S(PUPUP ... PUP i ) = { h 2 i , h 2 i-1 } S(UPU i ) = S(UPUPU i ) = S(UPUPU ... UPU i ) = { h 2 i-2 , h 2 i-1 }(3)
On the other hand, it should also be taken into consideration that if the continuous P or U are long-lasting, the depth of E i can not be deepened. For example, consider T7-Net, where E i is PPPPUUU i . Suppose that the size h of the original image is 256, which we normally deal in CNN classification. In this case, the feature-map sizes of E i are as follows.
S(PPPPUUU i ) = { 256 2 i-1 , 256 2 i , 256 2 i+1 , 256 2 i+2 , 256 2 i+3 }(4)
That is, the size of the smallest feature-map in the third encoder block becomes 4 (256/64). Empirically, it is not recommended that the size of the last featuremap of the encoder be reduced to less than 8. In the semantic segmentation problem, this is because we can extract the high-level feature as we reduce the feature-map, but it will be difficult to restore it to the mask. Also, repeated placement of pooling or up-sampling at short depths has the disadvantage of making the shape of the feature-map too simple before extracting sufficient levels of features. For example, considering the E i of the PPPUU structure in which the convolution and pooling layers are arranged three times, the size of the feature-maps may vary, but by reducing the size of the feature-map by oneeighth in the same block, this structure is insufficient as an encoder. Therefore, we configured only up to T5-Net in the evaluation and set the smallest featuremap size of the encoder not to be smaller than 8. And E i in T5-Net is designed as PPUPU structure instead of PPPUU to avoid more than two continuous pooling layers.
The last point to consider when designing T-Net is that the P and U of the encoder and decoder need not be symmetric. That is, there is no problem in configuring the encoder with PPUPU and the decoder with UUP. However, since the purpose of EDiED is to transfer the various levels of features extracted from the encoder to the decoder, it is recommended that the length of the block constituting the encoder is longer than the decoder. In other words, T53-Net delivers more feature levels to the decoder than T35-Net. Therefore, we added T53-Net as a comparison with U-Net. In T53-Net, the encoder block is composed of PPUPU and the decoder block is composed of UUP. As a result,
the T-Nets we have configured for comparison with U-Net are T3-Net, T5-Net, and T53-Net. The structure of T3-Net is in Figure 3, and the structure of T5-Net is shown in Figure 4. T53-Net is the same as replacing T5-Net decoder with UUP instead of UUPUP in Figure 4.
Optimized T-Net for main vessel segmentation
In this section, we describe an optimized T-Net structure for main vessel segmentation in coronary angiography. Figure 5 shows the detailed structure of optimized T-Net.
The overall process of glaucoma detection is as follows. First, angiography images are augmented for regularization of the model. And we train the augmented images on optimized T-Net by the mini-batch size. The model is improved to minimize the validation loss and finally, the performance of the model is evaluated using the test-set.
Data augmentation
Our data consists of 4,700 grayscale images with a size of 512 x 512. This is a small number compared to general datasets such as ImageNet [28] or MS COCO [17], and without proper augmentation, the model will inevitably face overfitting problem. Fortunately, because coronary angiography is taken in a defined form with a specific purpose, overfitting can be avoided with augmentation even with vary, the brightness is also changed within ±40% at random rates. Figure 6 shows images when each augmentation policy is applied to a single image at a maximum rate. The optimized T-Net consists of five encoder blocks (E i ) and decoder blocks
(D i ), respectively. E i is composed of PPUPU i , and D i is composed of UUPUP i .
Each pooling or up-sampling layer is preceded by a convolutional layer (C ).
That is, the actual PPUPU is CPCPCUCPCU, but it is called PPUPU for convenience. Let P E i k be the k-th pooling or up-sampling layer that appears in E i , and similarly define U E i k , P Di k , U E i k , where i ∈ {1,2,3,4,5} and k ∈ {1,2,3,4,5}. The sizes of the feature-maps constituting E 1 is as follows.
S(E 1 ) = S({P E 1 1 , P E 1 2 , U E 1 3 , P E 1 4 , U E 1 5 }) = {512,
As previously defined, the decoder block becomes D1 near the prediction, and the block close to the convolutional layers of center becomes D5. Therefore, the sizes of feature-maps from D 5 to D 1 are as follows.
Since the S (P
Concatenate layers in optimized T-Net
In optimzized T-Net, concatenate layers are connected immediately after up-sampling layers of U Di 1 and U Di 2 . In case of U D4 1 , the size of featuremap after the up-sampling is 64, and the layers of the encoders having such feature-map size is as follows.
64 = S(E i ) = {S(P E 4 1 ), S(P E 3 2 ), S(P E 3 4 ), S(U E 2 3 ), S(U E 2 5 )}(8)
If there are layers of the same size in the encoder block, we concatenate the preceding layer only in order to reduce GPU memory consumption. Therefore, the layers concatenated with U D4 1 are P E 4 1 , P E 3 2 , and U E 2 3 . Likewise, U D4 2 is concatenated with P E 3 1 , P E 2 2 , and U E 1 3 . Figure 7 shows the concatenate layers between encoder and decoder blocks.
From Figure 7, we can observe that T-Net delivers various levels of features to the decoder. In case of U-Net, the encoder block connected to D 4 is only E 4 ,
M 1 (D 5 ) = {P E 5 1 , P E 4 2 , U E 3 3 }, M 2 (D 5 ) = {P E 4 1 , P E 3 2 , U E 2 3 } M 1 (D 4 ) = {P E 4 1 , P E 3 2 , U E 2 3 }, M 2 (D 4 ) = {P E 3 1 , P E 2 2 , U E 1 3 } M 1 (D 3 ) = {P E 3 1 , P E 2 2 , U E 1 3 }, M 2 (D 3 ) = {P E 2 1 , P E 1 2 } M 1 (D 2 ) = {P E 2 1 , P E 1 2 }, M 2 (D 2 ) = {P E 1 1 }, M 1 (D 1 ) = {P E 1 1 }(9)
M 2 (D i ) does not exist because the size of the feature-map is 1024 and there is no matching encoder layer with the same size.
Short-cut connections in optimized T-Net
In optimized T-Net, the number of filters in the convolutional layer of E 1 starts from 32, and doubles in the next block. Short-cut connection is a concept proposed in ResNet [33], which has the effect of preventing the performance degradation caused by deeply stacking the convolutional layers. Therefore, recent CNN models necessarily include short-cut connections, and they also form short-cut connections that connect all the convolutional layers within a block, such as DenseNet [15]. Short-cut connections result in the addition between layers, requiring the same number of filters as feature-maps of the same size.
In case of optimized T-Net, the second and fourth, third and fifth layers of all E i and D i have the same number of filters as the same feature-map size, respectively. Through the equation 6-8, the first layer in the next block has a feature-map of the same size as the second and fourth layers of the previous block. Therefore, an additional short-cut connection is possible by making only the first convolutional layer equal to the number of filters of the previous block before double the number of filters of the next block. Figure 8 shows how short-cut connections are constructed in three consecutive encoder blocks.
where Ce denotes two convolutional layers in the center and Pr refers the last two convolutional layers before final prediction.
Fine-tuning T-Net for optimization
Convolutional layers of optimized T-Net consist of convolution, batch normalization [34], and activation function. The activation function uses softmax only in the last convolutional layer and the others use ReLU [35]. The weights of the convolution are initialized by He initialization [36]. For up-sampling, transposed convolution [21] and bi-linear up-sampling are the most common methods. As a result of the experiment, there was no significant difference in performance between two. The detailed performance comparison result applying each method in optimized T-Net is explained in the next section. However, we used bi-linear up-sampling because transposed convolution requires number of free parameters.
In general, Dice Similarity Coefficient score (DSC ) is the performance metric
for semantic segmentation problems. The definition of the DSC is described in the result section. The commonly used loss function in deep learning is cross entropy loss (CELoss). However, CELoss has a disadvantage in that loss cannot be easily improved when a large number of background pixels are included, which corresponds to most medical image segmentation problems. Therefore, the following loss function DSCLoss is used to provide a balance between CELoss and DSC.
DSCLoss = 1 + αCELoss − DSC(11)
Where α is a coefficient for adjusting the scale of DSCLoss which set to 0.1 in this paper. More specifically, 1 -DSC attempts to reduce the loss of the pixels corresponding to the vessels, and αCELoss tries to reduce the overall pixel loss, including the background.
In the case of the optimizer function, we used the Adam [37] and set the initial learning rate to 0.0001. In addition, we reduced the learning rate with a factor of 0.5 if the validation loss does not improve for the last 5 epochs. The software and hardware environment for the evaluation are as follows.
Results
We tested on a 64GB server with two NVIDIA Titan X GPUs and an Intel Core i7-6700K CPU. The operating system is Ubuntu 16.04, and the development of the CNN model uses Python-based machine learning libraries including Keras [39], Scikit-learn [40], and TensorFlow [41].
Evaluation setup
The evaluation of this paper proceeds from two perspectives. The first is However, the loss function uses the same DSCLoss as optimized T-Net because convergence is faster than using CELoss. The T-Nets used in the comparison is T3-Net, T5-Net, and T53-Net. Unlike T-Nets, U-Net uses two consecutive convolutional layers to maintain the number of weights in each block similar to that of T3-Net. Figure 9 shows the schematic structure of the four models used for the U-Net and T-Nets comparisons.
The second is to evaluate the optimized T-Net that exhibits the maximum performance of the main vessel segmentation in coronary angiography. First, Opt-Net is T-Net with all the fine-tuning methods described in section 4. Briefly, the convolutional layer is connected with short-cut connections, the loss function is DSCLoss, and uses bi-linear up-sampling. The T-Nets compared to More specifically, the loss function is DSCLoss, data augmentation is applied, the learning rate has a factor of 0.5 with 5 epochs patience, and the maximum epoch is 100.
Evaluation metrics
The evaluation of the main vessel segmentation was based on the following three metrics: Dice Similarity Coefficient score (DSC ), sensitivity (Se), and precision (Pr ). Dice Similarity Coefficient score, also called F1-score, means the harmonic mean of sensitivity and precision. Generally, in the semantic segmentation problem, we use the name DSC rather than F1-score. Sensitivity, Table 1 summarizes the evaluation results of U-Net and T-Nets compared under the same condition explained in the previous section.
Evaluation results of U-Net and T-Nets
The highest overall DSC was 0.815 for T5-Net, 0.08 higher than T53-Net, 0.028 higher than T3-Net and 0.095 higher than U-Net. T5-Net and T53-Net did not show significant performance differences in LAD and RCA, but T5-Net in performance between U-Net and T3-Net, which is the most similar in terms of the number of weights and structurally, demonstrates that various concatenate layers enhance performance. Considering that T3-Net has lower performance than T53-Net or T5-Net, the number of up-sampling layers of the decoder block to which the concatenate layer is connected is also important. That is, T3-Net is concatenated to the first up-sampling layer of the decoder block, but T5-Net and T53-Net are concatenated in the first and second up-sampling layers. This makes it clear that the performance of T53-Net is closer to T5, despite the fact that T53-Net is a half-mixed structure of T5 and T3.
The difference between U-Net and T-Net is more evident when comparing segmentation performance for each vessel. First, the smallest performance gap between T-Net and U-Net is RCA segmentation, and RCA has higher overall performance than LAD or LCX. The highest DSC in the RCA segmentation is 0.890 in T5-Net, 0.037 higher than the lowest DSC from U-Net. On the other hand, in LCX segmentation, T5-Net achieve 0.753 DSC while U-Net shows only 0.565 DSC. This means that LCX segmentation is the most difficult and RCA segmentation is the easiest problem. In other words, T-Net achieves higher performance than the U-Net in a more difficult problem, which is LCX segmentation. Figure 10 show box plots for overall DSC, sensitivity, and precision for U-Net and T-Nets, respectively. Table 1, T53-Net and T5-Net showed similar performance on average, but the box plot shows that T5-Net achieves a fairly higher performance throughout the test-set. This is also the basis for our decision to design the optimized T-Net with T5-Net as the basis. Figure 11 show the validation DSCLoss and validation DSC for each epoch while training U-Net and T-Nets. As with the above performance analysis, we can see that T-Net performs better than U-Net. In other words, we can see that DSCLoss of U-Net converges at a higher value than those of T-Nets. In the comparison between U-Net and T-Nets, the last thing to evaluate is to visualize how the actual weights of each model are trained. Therefore, we visualized how the weights of the convolutional layer that constitutes each decoder block are activated for the same input image. Figure 12 are visualization of weight activation for U-Net, T3-Net, T5-Net, and T53-Net. The overall performance is best for Opt-Net followed by Opt-Net 1 and Opt-
Evaluation results of optimized T-Net for main vessel segmentation
Net 2 .
Opt-Net is more efficient than Opt-Net 1 because the segmentation performance is better and trained with fewer free-parameters. Comparing the total number of free-parameters, Opt-Net has 43,570,372 free-parameters and Opt-Net 1 has 55,883,236, which is about 28.26% more than Opt-Net. Based on the overall DSC, Opt-Net is the best at 0.890 followed by Opt-Net 1 and Opt-Net 2 .
In addition, Opt-Net is always better in all three metrics, with the exception of precision for LAD and sensitivity for RCA. Opt-Net 2 , which removed short-cut connections, shows a 0.025 lower DSC than Opt-Net. That is, adding a shortcut connection has reasonable effect on performance improvement. The DSC of Opt-Net is 0.170 higher than the DSC of U-Net in Table 1. From Table 1 and 2, proposed T-Net performs better than U-Net in the main vessel segmentation problem, and the optimized T-Net shows the highest performance.
The segmentation performance of each main vessel shows the highest DSC in RCA, followed by LAD and LCX. This is in the same order as Overall, we can observe that the validation loss and DSC improved continuously over 100 epochs. However, we have experimentally found that the minimum validation loss is formed between 80 and 100 epochs and is not improved much thereafter. Of course, if we increase the reduction factor of the learning rate and lengthen the patience epoch, the performance can be improved with longer epochs. However, we did not perform an additional evaluation with longer epochs because it took too much time because our server performance was limited. This effect is clearly observed near the 30 epoch of Figure 14. Figure 15 is the visualization of weight activation for Opt-Net. Unlike Figure 12 with U-Net and T-Nets, we included entire blocks including encoder and decoder. We can see the process of predicting the final mask from the first encoder block through the intermediate convolution layer and then through the decoder block. In the first encoder block, low-level features such as the contour of the blood vessel are observed, and the higher-level features other than the outline of the blood vessel are extracted as the next encoder moves. When we reach the center, activation of weights is no longer similar to the original image's shape. In case of U-Net, only the information of the last encoder is concatenated to the first decoder block D 5 . However, since all encoder blocks, except E 1 , are connected with D 5 , T-Net outputs activation close to the mask image from the beginning of restoring process. As a result, subsequent decoder blocks focus on more sophisticated restoration of the mask, resulting in higher performance. In other words, the closer the latter stage of the decoder, the clearer the differences in the activation of main vessels and background.
Conclusion
In this paper, we proposed T-Net containing a small encoder-decoder inside the encoder-decoder structure (EDiED). T-Net overcomes the limitation that U-Net, which is the most popular model, can only have a single set of the concatenate layer between encoder and decoder block. To be more precise, the U-Net symmetrically forms the concatenate layers, so the low-level feature of the encoder is connected to the latter part of the decoder, and the high-level feature is connected to the beginning of the decoder. T-Net arranges the pooling and up-sampling appropriately during the encoder process, and likewise during the decoding process so that feature-maps of various sizes are obtained in a single block. As a result, all features from the low-level to the high-level extracted from the encoder are delivered from the beginning of the decoder to predict a more accurate mask.
Figure 1 :
1Three main vessels with overlapping mask images
Figure 2 :
2Basic structure of U-Net There are various deformation models of U-Net at present, but the basic structure does not deviate much from Figure 2. That is, the encoder gradually reduces the size of the feature-map in order to extract the high-level features, while in the decoder, the size of the feature-map gradually increases to match the size of the mask. Unlike the classification in which the extracted high-level features are directly connected to the prediction layer, the segmentation requires a process of restoring to the prediction layer, which creates a noisy boundary mask different from the shape of the object in the original image. Therefore, in U-Net, there is a concatenate layer that connects feature-maps of the same size in encoder and decoder. Suppose both the width and height of input image are h. Let E i be the i -th block of the encoder, and halve the width and height of the feature-map towards E n . Assume E 1 be a direct convolutional connection to the input image to have a feature-map of size h. Likewise, the i -th block of the decoder is called D i , and for convenience, let a block near the prediction layer be D 1 . And D i doubles the width and height of the feature-map by up-sampling in the direction of D 1 .
Figure 3 :
3Structure of T3-Net with PPU and UUP We named it T3-Net because there are three pooling and up-sampling in the single convolution block. Likewise, if there are 5 pooling and up-sampling, it is T5-Net and if there are 5 times in encoder and 3 times in decoder, it is T53-Net. Depending on the order of pooling and up-sampling, the same T3-Net can be in various forms. We name the block in the order of P (pooling) and U (up-sampling) appearing in the block. That is, if E i of T3-Net is composed of pooling, up-sampling, and pooling, it is called PUP i , and in the case of D i , it is called UPU i . As can be inferred from EDiED, PUP and UPU can be regarded as small-scale encoder-decoders existing in encoder and decoder, respectively. Like U-Net, E i reduces the size of the final feature-map in half,
Figure 4 :
4Structure of T5-Net with PPUPU and UUPUP
Figure 5 :
5Optimized T-Net structure for main vessel segmentation
4
,700 images. The effect of preventing overfitting can also be confirmed through the training loss of the optimized T-Net in the result section. In addition, we train the model by resizing an image in a classification problem, but since the prediction of segmentation is done by pixel levels, resizing is not recommended if GPU memory allows. Therefore, we used the size of the input image as the original 512 x 512. Our image augmentation policy is as follows. First, we zoom-in and zoom-out an image at a random ratio within ±20%. And the height and width of the image are shifted at a random ratio within ±20% of image size 512 x 512. Next, we rotate the angiography image within ±30 • at random rates. Finally, because the brightness of the angiography image can
Figure 6 :
6Vessel images result from data augmentation4.2. T-Net optimization for main vessel segmentationIn this section, we describe the specific structure and fine-tuning parameters of optimized T-Net for main vessel segmentation. First, we set the overall T-Net structure to T5-Net. This is because we compared U-Net, T3-Net, T5-Net, and T53-Net under the same conditions, and as a result, T5-Net showed the best performance. However, the image used in the comparison is resized to 256 x 256 in order to shorten the training time, and the structure of optimized T-Net is advanced from that of T5-Net. The structures of T3-Net, T5-Net, and T53-Net designed for comparison with U-Net are described in detail in the result section.
the sizes of feature-maps from E 2 to E 5 are as follows. S(E 2 ) = {256, 128, 64, 128, 64} S(E 3 ) = {128, 64, 32, 64, 32} S(E 4 ) = {64, 32, 16, 32, 16} S(E 5 ) = {32, 16, 8, 16, 8}
S(D 5
5) = {16, 32, 64, 32, 64} S(D 4 ) = {32, 64, 128, 64, 128} S(D 3 ) = {64, 128, 256, 128, 256} S(D 2 ) = {128, 256, 512, 256, 512} S(D 1 ) = {256, 512, 1024, 512, 1024}
D1 5
5) is 1024, the size of next convolutional layer's feature-maps become 512. Therefore, connecting this convolutional layer to the final convolutional layer with four filters and applying a softmax function to the result, a feature-map of size 512 x 512 x 4 is generated. This represents the background, LAD, LCX, and RCA scores for the 512 x 512 size mask. And each of these scores is compared with the actual class as pixel-by-pixel, and the total loss is used for the gradient calculation of the next epoch. Next, we explain how the concatenate layers are constructed based on the above feature-maps sizes and describe the specific parameters for fine-tuning.
Figure 7 :
7The concatenate layers in optimized T-Net but T-Net connects D 4 with all blocks in the encoder except the E 5 . Assume that the first concatenate layers list for D i is M 1 (D i ) and the second list is M 2 (D i ). Than the lists of concatenate layers in optimized T-Net are as follow.
Figure 8 :
8Short-cut connections in encoder blocks of optimize T-Net Finally, the number of filters and size of feature-maps constituting each block of optimized T-Net is as follow. The number in parentheses is the size of the feature-map. F(E 1 ) = {32(512), 32(256), 32(128), 32(256), 32(128)} F(E 2 ) = {32(256), 64(128), 64(64), 64(128), 64(64)} F(E 3 ) = {64(128), 128(64), 128(32), 128(64), 128(32)} F(E 4 ) = {128(64), 256(32), 256(16), 256(32), 256(16)} F(E 5 ) = {256(32), 512(16), 512(8), 512(16), 512(8)} F(Ce) = {1024(16), 512(16)} F(D 5 ) = {512(16), 512(32), 512(64), 512(32), 512(64)} F(D 4 ) = {256(32), 256(64), 256(128), 256(64), 256(128)} F(D 3 ) = {256(64), 128(128), 128(256), 128(128), 128(256)} F(D 2 ) = {128(128), 64(256), 64(512), 64(256), 64(512)} F(D 1 ) = {64(256), 32(512), 32(1024), 32(512), 32(1024)} F(Pr) = {32(512), 4(512)}
A total of 4,700 coronary angiography images of patients who visited Asan Medical Center were evaluated. All patients participating in the study provided written informed consent and the institutional review board of Asan Medical Center approved the study. Experts with more than five years of experience split the main vessels (LAD, LCA, RCA) from the ostium to the distal site by using The CAAS QCA system (Pie Medical Imaging BV, the Netherlands)[38].Of the 4,700 angiography images, 1,987 (42.3%) were LAD, 1,307 (27.8%) were LCX, and 1,406 (29.9%) were RCA. From the total number of 4,700 angiography images, 3,755 images (80%) were randomly split into train-set and 945 images into (20%) test-set with the similar class distribution. 945 test-set images consisted of 393 LAD, 271 LCX, and 281 RCA images. Validation-set consists of 565 images which correspond to about 15% of the train-set. As a result, 3,190 train-set images consisted of 1,349 LAD, 876 LCX, and 965 RCA images. Likewise, of the 565 validation-set images, 245 images are LAD, 160 images are LCX, and 160 images are RCA.
to compare U-Net and T-Nets under the same conditions. In other words, it is to see how the performance of the main vessel segmentation differs in the most basic U-Net and T-Net structures. Therefore, unlike the optimized T-Net, the size of the image is resized to 256 x 256 to improve the training speed, and the depth of the encoder and decoder is fixed to 3. The learning rate was fixed at 0.0001 regardless of the validation loss, and no short-cut connection was used. Likewise, since there is no reduction in the learning rate, the max epoch is 50, which corresponds to half of the optimized T-Net. That is, the goal of the first evaluation is to compare the performance of pure U-Net with that of T-Nets, to the greatest extent, excluding other performance-enhancing factors.
Figure 9 :
9Schematic structures of U-Net, T3-Net, T5-Net, and T53-Net. Opt-Net are models with removing two methods. Opt-Net 1 uses transposed convolution instead of bi-linear up-sampling and Opt-Net 2 does not have the short-cut connections. Other unspecified methods apply equally to all models.
also known as the true positive rate or recall, measures the percentage of positives that are correctly identified as the main vessel. Precision measures the percentage of positives that are predicted as the main vessel. In the case of accuracy and specificity, it is not generally included in evaluation metrics for semantic segmentation problem because the class corresponding to negative is a background. These metrics are defined with the following three terminologies. However, since there are three types of main vessels, all evaluation metrics are into four different categories. That is, metrics for all types of main vessels (ALL), LAD, LCX, and RCA. • True Positive(TP ): The number of pixels in an angiography image correctly identified as main vessels. • False Positive(FP ): The number of pixels in an angiography image incorrectly identified as main vessels. • False Negative(FN ): The number of pixels in an angiography image incorrectly identified as background Dice similarity coef f icient(DSC) , loss and DSC according to epoch are graphically represented, and visualizations of model weights and predicted mask with an actual mask are included. Detailed descriptions of the graphs and visualizations are provided in the following section.
obtained 0 .
0023 higher DSC in LCX segmentation. Comparing T3-Net and U-Net, T3-Net was 0.067 higher than U-Net based on overall DSC. The difference ALL LAD Method DSC Se(%) Pr (%) DSC Se(%) Pr (%)
Figure 10 :
10Box plots for comparison of U-Net and T-NetsThe solid line in the box represents the median value and the dotted line represents the mean value. The rounded points above and below the box represent outliers of 5% and 95%, respectively. Other expressions follow the definition of the general box plot. In
Figure 11 :
11Validation loss and DSC of U-Net and T-Nets
Figure 12 :
12Visualization of weight activation for U-Net and T-NetsFirst, we set the one of RCA vessel as the input image which shows relatively little performance difference than LAD and LCX vessels. What we want to see inFigure 12is from which decoder starts to segment the mask clearly. The T-Nets begin to show the outline of the mask from the beginning of the decoder(D 3 ), but D 3 in U-Net is almost invisible. In D 2 ,the weights in T-Nets are fairly clear, but those of U-Net are still unclear. And in D 1 , T-Nets activate the weights similar to mask, but U-Net still activates the other vessels of the original image. This means that T-Net accurately predicts the mask from the beginning decoder block, which is possible through various concatenate layers between encoder and decoder blocks. More specifically, all the blocks of the encoder are connected to D 3 , so early prediction can be performed since the low-level to high-level features extracted from the encoder are transmitted to D 3 .
Figure 13 :
13Box plots for optimized T-NetFigure 14 show the loss and DSC of the Opt-Net training process for each train-set and validation-set. The part where the loss and DSC fluctuate in the staircase form is the epoch where the learning rate is halved because there is no improvement in the validation DSC during the last five epochs.
Figure 14 :
14Loss and DSC of optimized T-Net
Figure 15 :
15Visualization of weight activation for Opt-NetFinally, we show the predicted results of LAD, LCX, and RCA segmentation of U-Net, T-Nets, and Opt-Net with actual masks in Figure 16 through 18. Although these examples are not the entire test-set, but are sufficient to show the performance differences between U-Net, T-Nets, and Opt-Net. From the figures, U-Net and T-Net do not differ greatly in the role of the encoder to extract high-level feature such as main vessel position under the same condition. However, there is a performance gap in the restoration of the actual mask due to the different levels of features passed to the decoder
Figure 16 :Figure 17 :Figure 18 :
161718Visualized LAD segmentation prediction results Visualized LCX segmentation prediction results Visualized RCA segmentation prediction results
We evaluated T-Net for the problem of segmenting three main vessels (LAD, LCX, RCA) in coronary angiography images. The experiment consisted of a comparison of U-Net and T-Nets under the same conditions, and an optimized T-Net for the main vessel segmentation. As a result, under the same conditions, T-Net recorded a DSC of 0.815, 0.095 higher than that of U-Net, and the optimized T-Net recorded a DSC of 0.890 which was 0.170 higher than that of U-Net. In addition, we visualized the weight activation of the convolutional layer of T-Net and U-Net to show that T-Net actually predicts the mask from earlier decoders. Therefore, we expect that T-Net can be effectively applied to other similar medical image segmentation problems. Although this paper only introduces a 2-dimensional T-Net structure, the structure of 3-D T-Net is the same. Only the convolutional layer and the pooling layer change from the existing 2-D to 3-D. Therefore, We will apply 3-D T-Net to 3-D medical image segmentation problems in future work.
Table 2
2summarizes the optimized T-Net for three types of main vessel segmentation in coronary angiography.ALL
LAD
Method
DSC
Se(%) Pr (%) DSC
Se(%) Pr (%)
Opt-Net
0.890 88.32 90.50 0.884 86.57
91.17
Opt-Net 1 0.875
86.24
89.91
0.878
85.11
91.56
Opt-Net 2 0.865
87.10
87.13
0.855
84.49
87.88
LCX
RCA
Method
DSC
Se(%) Pr (%) DSC
Se(%) Pr (%)
Opt-Net
0.860 86.68 86.23 0.927 92.50
93.63
Opt-Net 1 0.831
83.10
84.53
0.914
91.10
92.58
Opt-Net 2 0.818
84.74
80.68
0.923 93.04
92.28
Table 2 :
2Evaluation results of optimized T-Net
Table 1 ,
1with the highest DSC of 0.927 for RCA segmentation, which is an excellent segmentation performance. The highest DSC for LCX is 0.860, which is 0.295 higher than U-Net, which has the lowest DSC inTable 1. In the LAD segmentation, Opt-Net's DSC is 0.884, an improvement of 0.151 over U-Net inTable 1.Figure 13show box plots of DSC, sensitivity, and precision for each main vessel and overall vessel. Box plots also show that the segmentation performance of the LCX is lower than that of the other two main vessels. Considering that the number of LCX and RCA images is 1,307 and 1,406 respectively, this difference is not due to class imbalance. This is because unlike RCA, which is relatively easy to segment with no branching, LAD and LCX are more difficult to segment because they are separated from single vessel. Also, because of the small number of LCX images compared to LAD, there is performance difference between two vessels. The RCA segmentation shows high performance in the entire test-set as well as the highest average DSC seen in theTable 2. That is, the inter-quartile range (IQR) is very narrow and is formed near 0.9 DSC, where IQR means the difference between 75-th and 25-th percentiles. The performance of LAD segmentation is intermediate between RCA and LCX and very similar to the distribution of overall performance.
Table 3
3compares the results of the proposed optimized T-Net with the previous main vessel segmentation study. Jo's study only performed LAD vessel segmentation, so no performance comparison is possible in LCX and RCA vessels. FromTable 3, proposed T-Net in the LAD segmentation achieved 0.208 higher DSC than Jo's approach. In addition, Jo used U-Net, which is similar to the U-Net structure used for comparison with T-Net in this paper. Under the same conditions, T5-Net showed 0.071 higher DSC for LAD segmentation than that of U-Net. Therefore, it can be said that T-Net's segmentation performance is better than U-Net in the main vessel segmentation problem.ALL
LAD
Method
DSC
Se(%) Pr (%) DSC
Se(%) Pr (%)
Proposed
0.890 88.32 90.50 0.884 86.57 91.17
T5-Net
0.815
80.93
83.74
0.804
79.77
82.87
U-Net
0.720
75.39
71.52
0.733
76.25
72.78
Jo et al [13]
-
-
-
0.676
60.70
80.00
LCX
RCA
Method
DSC
Se(%) Pr (%) DSC
Se(%) Pr (%)
Proposed
0.860 86.68 86.23 0.927 92.50 93.63
T5-Net
0.753
75.70
77.08
0.890
87.58
91.39
U-Net
0.565
63.86
54.10
0.853
85.27
86.55
Jo et al [13]
-
-
-
-
-
-
Table 3 :
3Comparison results between proposed method with previous study
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"ADVERSARIAL EXAMPLES FOR GENERATIVE MODELS"
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"Jernej Kos \nNational University of Singapore\nUniversity of California\nBerkeley\n",
"Ian Fischer \nNational University of Singapore\nUniversity of California\nBerkeley\n",
"Google Research \nNational University of Singapore\nUniversity of California\nBerkeley\n",
"Dawn Song \nNational University of Singapore\nUniversity of California\nBerkeley\n"
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] | [] | We explore methods of producing adversarial examples on deep generative models such as the variational autoencoder (VAE) and the VAE-GAN. Deep learning architectures are known to be vulnerable to adversarial examples, but previous work has focused on the application of adversarial examples to classification tasks. Deep generative models have recently become popular due to their ability to model input data distributions and generate realistic examples from those distributions. We present three classes of attacks on the VAE and VAE-GAN architectures and demonstrate them against networks trained on MNIST, SVHN and CelebA. Our first attack leverages classification-based adversaries by attaching a classifier to the trained encoder of the target generative model, which can then be used to indirectly manipulate the latent representation. Our second attack directly uses the VAE loss function to generate a target reconstruction image from the adversarial example. Our third attack moves beyond relying on classification or the standard loss for the gradient and directly optimizes against differences in source and target latent representations. We also motivate why an attacker might be interested in deploying such techniques against a target generative network. | 10.1109/spw.2018.00014 | [
"https://arxiv.org/pdf/1702.06832v1.pdf"
] | 6,462,244 | 1702.06832 | 17f3358d219c05f3cb8d68bdfaf6424567d66984 |
ADVERSARIAL EXAMPLES FOR GENERATIVE MODELS
Jernej Kos
National University of Singapore
University of California
Berkeley
Ian Fischer
National University of Singapore
University of California
Berkeley
Google Research
National University of Singapore
University of California
Berkeley
Dawn Song
National University of Singapore
University of California
Berkeley
ADVERSARIAL EXAMPLES FOR GENERATIVE MODELS
We explore methods of producing adversarial examples on deep generative models such as the variational autoencoder (VAE) and the VAE-GAN. Deep learning architectures are known to be vulnerable to adversarial examples, but previous work has focused on the application of adversarial examples to classification tasks. Deep generative models have recently become popular due to their ability to model input data distributions and generate realistic examples from those distributions. We present three classes of attacks on the VAE and VAE-GAN architectures and demonstrate them against networks trained on MNIST, SVHN and CelebA. Our first attack leverages classification-based adversaries by attaching a classifier to the trained encoder of the target generative model, which can then be used to indirectly manipulate the latent representation. Our second attack directly uses the VAE loss function to generate a target reconstruction image from the adversarial example. Our third attack moves beyond relying on classification or the standard loss for the gradient and directly optimizes against differences in source and target latent representations. We also motivate why an attacker might be interested in deploying such techniques against a target generative network.
INTRODUCTION
Adversarial examples have been shown to exist for a variety of deep learning architectures. 1 They are small perturbations of the original inputs, often barely visible to a human observer, but carefully crafted to misguide the network into producing incorrect outputs. Seminal work by Szegedy et al. (2013) and , as well as much recent work, has shown that adversarial examples are abundant and finding them is easy.
Most previous work focuses on the application of adversarial examples to the task of classification, where the deep network assigns classes to input images. The attack adds small adversarial perturbations to the original input image. These perturbations cause the network to change its classification of the input, from the correct class to some other incorrect class (possibly chosen by the attacker). Critically, the perturbed input must still be recognizable to a human observer as belonging to the original input class. 2 Deep generative models, such as Kingma & Welling (2013), learn to generate a variety of outputs, ranging from handwritten digits to faces (Kulkarni et al., 2015), realistic scenes (Oord et al., 2016), videos (Kalchbrenner et al., 2016), 3D objects (Dosovitskiy et al., 2016), and audio (van den Oord et al., 2016). These models learn an approximation of the input data distribution in different ways, and then sample from this distribution to generate previously unseen but plausible outputs.
To the best of our knowledge, no prior work has explored using adversarial inputs to attack generative models. There are two main requirements for such work: describing a plausible scenario in which an attacker might want to attack a generative model; and designing and demonstrating an attack that succeeds against generative models. We address both of these requirements in this work.
One of the most basic applications of generative models is input reconstruction. Given an input image, the model first encodes it into a lower-dimensional latent representation, and then uses that representation to generate a reconstruction of the original input image. Since the latent representation usually has much fewer dimensions than the original input, it can be used as a form of compression. The latent representation can also be used to remove some types of noise from inputs, even when the network has not been explicitly trained for denoising, due to the lower dimensionality of the latent representation restricting what information the trained network is able to represent. Many generative models also allow manipulation of the generated output by sampling different latent values or modifying individual dimensions of the latent vectors without needing to pass through the encoding step.
These properties of input reconstruction generative networks suggest a variety of different attacks that would be enabled by effective adversaries against generative networks. Any attack that targets the compression bottleneck of the latent representation can exploit natural security vulnerabilities in applications built to use that latent representation. Specifically, if the person doing the encoding step is separated from the person doing the decoding step, the attacker may be able to cause the encoding party to believe they have encoded a particular message for the decoding party, but in reality they have encoded a different message of the attacker's choosing. We explore this idea in more detail as it applies to the application of compressing images using a VAE or VAE-GAN architecture.
RELATED WORK AND BACKGROUND
This work focuses on adversaries for variational autoencoders (VAEs, proposed in Kingma & Welling (2013)) and VAE-GANs (VAEs composed with a generative adversarial network, proposed in Larsen et al. (2015)).
RELATED WORK ON ADVERSARIES
Many adversarial attacks on classification models have been described in existing literature Szegedy et al., 2013). These attacks can be untargeted, where the adversary's goal is to cause any misclassification, or the least likely misclassification Kurakin et al., 2016); or they can be targeted, where the attacker desires a specific misclassification. Moosavi-Dezfooli et al. (2016) gives a recent example of a strong targeted adversarial attack. Some adversarial attacks allow for a threat model where the adversary does not have access to the target model (Szegedy et al., 2013;Papernot et al., 2016), but commonly it is assumed that the attacker does have that access, in an online or offline setting Kurakin et al., 2016). 3 Given a classifier f (x) : x ∈ X → y ∈ Y and original inputs x ∈ X , the problem of generating untargeted adversarial examples can be expressed as the following optimization: argmin x * L(x, x * ) s.t. f (x * ) = f (x), where L(·) is a chosen distance measure between examples from the input space (e.g., the L 2 norm). Similarly, generating a targeted adversarial attack on a classifier can be expressed as argmin x * L(x, x * ) s.t. f (x * ) = y t , where y t ∈ Y is some target label chosen by the attacker.
These optimization problems can often be solved with optimizers like L-BFGS or Adam (Kingma & Ba, 2015), as done in Szegedy et al. (2013) and Carlini & Wagner (2016). They can also be approximated with single-step gradient-based techniques like fast gradient sign , fast gradient L 2 (Huang et al., 2015), or fast least likely class (Kurakin et al., 2016); or they can be approximated with iterative variants of those and other gradient-based techniques (Kurakin et al., 2016;Moosavi-Dezfooli et al., 2016).
An interesting variation of this type of attack can be found in Sabour et al. (2015). In that work, they attack the hidden state of the target network directly by taking an input image x and a target image x t and searching for a perturbed variant of x that generates similar hidden state at layer l of the target network to the hidden state at the same layer generated by x t . This approach can also be applied directly to attacking the latent vector of a generative model.
A variant of this attack has also been applied to VAE models in the concurrent work of Tabacof et al. (2016) 4 , which uses the KL divergence between the latent representation of the source and target images to generate the adversarial example. However in their paper, the authors mention that they tried attacking the output directly and that this only managed to make the reconstructions more Receiver z Sender Attacker f enc f dec Figure 1: Depiction of the attack scenario. The VAE is used as a compression scheme to transmit a latent representation of the image from the sender (left) to the receiver (right). The attacker convinces the sender to compress a particular image into its latent vector, which is sent to the receiver, where the decoder reconstructs the latent vector into some other image chosen by the attacker.
blurry. While they do not explain the exact experimental setting, the attack sounds similar to our L VAE attack, which we find very successful. Also, in their paper the authors do not consider the more advanced VAE-GAN models and more complex datasets like CelebA.
BACKGROUND ON VAES AND VAE-GANS
The general architecture of a variational autoencoder consists of three components, as shown in Figure 8. The encoder f enc (x) is a neural network mapping a high-dimensional input representation x into a lower-dimensional (compressed) latent representation z. All possible values of z form a latent space. Similar values in the latent space should produce similar outputs from the decoder in a well-trained VAE. And finally, the decoder/generator f dec (z), which is a neural network mapping the compressed latent representation back to a high-dimensional outputx. Composing these networks allows basic input reconstructionx = f dec (f enc (x)). This composed architecture is used during training to backpropagate errors from the loss function.
The variational autoencoder's loss function L VAE enables the network to learn a latent representation that approximates the intractable posterior distribution p(z|x):
L VAE = −D KL [q(z|x)||p(z)] + E q [log p(x|z)].
(1) q(z|x) is the learned approximation of the posterior distribution p(z|x). p(z) is the prior distribution of the latent representation z. D KL denotes the Kullback-Leibler divergence. E q [log p(x|z)] is the variational lower bound, which in the case of input reconstruction is the cross-entropy H [x,x] between the inputs x and their reconstructionsx. In order to generatex the VAE needs to sample q(z|x) and then compute f dec (z).
For the VAE to be fully differentiable while sampling from q(z|x), the reparametrization trick (Kingma & Welling, 2013) extracts the random sampling step from the network and turns it into an input, ε. VAEs are often parameterized with Gaussian distributions. In this case, f enc (x) outputs the distribution parameters µ and σ 2 . That distribution is then sampled by computing z = µ+ε √ σ 2 where ε ∼ N (0, 1) is the input random sample, which does not depend on any parameters of f enc , and thus does not impact differentiation of the network.
The VAE-GAN architecture of Larsen et al. (2015) has the same f enc and f dec pair as in the VAE. It also adds a discriminator f disc that is used during training, as in standard generative adversarial networks . The loss function of f dec uses the disciminator loss instead of cross-entropy for estimating the reconstruction error.
PROBLEM DEFINITION
We provide a motivating attack scenario for adversaries against generative models, as well as a formal definition of an adversary in the generative setting.
MOTIVATING ATTACK SCENARIO
To motivate the attacks presented below, we describe the attack scenario depicted in Figure 1. In this scenario, there are two parties, the sender and the receiver, who wish to share images with each other over a computer network. In order to conserve bandwidth, they share a VAE trained on the input distribution of interest, which will allow them to send only latent vectors z. Figure 2: Results for the L 2 optimization latent attack (see Section 4.3) on the VAE-GAN, targeting a specific image from the class 0. Shown are the first 12 non-zero images from the test SVHN data set. The columns are, in order: the original image, the reconstruction of the original image, the adversarial example, the predicted class of the adversarial example, the reconstruction of the adversarial example, the predicted class of the reconstructed adversarial example, the reconstruction of the reconstructed adversarial example (see Section 4.5), and the predicted class of that reconstruction.
The attacker's goal is to convince the sender to send an image of the attacker's choosing to the receiver, but the attacker has no direct control over the bytes sent between the two parties. However, the attacker has a copy of the shared VAE. The attacker presents an image x * to the sender which resembles an image x that the sender wants to share with the receiver. For example, the sender wants to share pictures of kittens with the receiver, so the attacker presents a web page to the sender with a picture of a kitten, which is x * . The sender chooses x * and sends its corresponding z to the receiver, who reconstructs it. However, because the attacker controlled the chosen image, when the receiver reconstructs it, instead of getting a faithful reproductionx of x (e.g., a kitten), the receiver sees some other image of the attacker's choosing,x adv , which has a different meaning from x (e.g., a request to send money to the attacker's bank account).
There are other attacks of this general form, where the sender and the receiver may be separated by distance, as in this example, or by time, in the case of storing compressed images to disk for later retrieval. In the time-separated attack, the sender and the receiver may be the same person or multiple different people. In either case, if they are using the insecure channel of the VAE's latent space, the messages they share may be under the control of an attacker. For example, an attacker may be able to fool an automatic surveillance system if the system uses this type of compression to store the video signal before it is processed by other systems. In this case, the subsequent analysis of the video signal could be on compromised data showing what the attacker wants to show.
DEFINING ADVERSARIAL EXAMPLES AGAINST GENERATIVE MODELS
We make the following assumptions about generating adversarial examples on a target generative model, G targ (x) = f dec (f enc (x)). G targ is trained on inputs X that can naturally be labeled with semantically meaningful classes Y, although there may be no such labels at training time, or the labels may not have been used during training. G targ normally succeeds at generating an output x = G targ (x) in class y when presented with an input x from class y. In other words, whatever target output class the attacker is interested in, we assume that G targ successfully captures it in the latent representation such that it can generate examples of that class from the decoder. This target output class does not need to be from the most salient classes in the training dataset. For example, on models trained on MNIST, the attacker may not care about generating different target digits (which are the most salient classes). The attacker may prefer to generate the same input digits in a different style (perhaps to aid forgery). We also assume that the attacker has access to G targ . Finally, the attacker has access to a set of examples from the same distribution as X that have the target label y t the attacker wants to generate. This does not mean that the attacker needs access to the labeled training dataset (which may not exist), or to an appropriate labeled dataset with large numbers of examples labeled for each class y ∈ Y (which may be hard or expensive to collect). The attacks described here may be successful with only a small amount of data labeled for a single target class of interest.
One way to generate such adversaries is by solving the optimization problem
argmin x * L(x, x * ) s.t. ORACLE(G targ (x * )) = y t ,
where ORACLE reliably discriminates between inputs of class y t and inputs of other classes. In practice, a classifier trained by the attacker may server as ORACLE. Other types of adversaries from Section 2.1 can also be used to approximate this optimization in natural ways, some of which we describe in Section 4.
If the attacker only needs to generate one successful attack, the problem of determining if an attack is successful can be solved by manually reviewing the x * andx adv pairs and choosing whichever the attacker considers best. However, if the attacker wants to generate many successful attacks, an automated method of evaluating the success of an attack is necessary. We show in Section 4.5 how to measure the effectiveness of an attack automatically using a classifier trained on z = f enc (x).
ATTACK METHODOLOGY
The attacker would like to construct an adversarially-perturbed input to influence the latent representation in a way that will cause the reconstruction process to reconstruct an output for a different class. We propose three approaches to attacking generative models: a classifier-based attack, where we train a new classifier on top of the latent space z and use that classifier to find adversarial examples in the latent space; an attack using L VAE to target the output directly; and an attack on the latent space, z. All three methods are technically applicable to any generative architecture that relies on a learned latent representation z. Without loss of generality, we focus on the VAE-GAN architecture.
CLASSIFIER ATTACK
By adding a classifier f class to the pre-trained generative model 5 , we can turn the problem of generating adversaries for generative models back into the previously solved problem of generating adversarial examples for classifiers. This approach allows us to apply all of the existing attacks on classifiers in the literature. However, as discussed below, using this classifier tends to produce lower-quality reconstructions from the adversarial examples than the other two attacks due to the inaccuracies of the classifier.
Step 1. The weights of the target generative model are frozen, and a new classifier f class (z) →ŷ is trained on top of f enc using a standard classification loss L classifier such as cross-entropy, as shown in Figure 3. This process is independent of how the original model is trained, but it requires a training corpus pulled from approximately the same input distribution as was used to train G targ , with ground truth labels for at least two classes: y t and yt, the negative class.
Step 2. With the trained classifier, the attacker finds adversarial examples x * using the methods described in Section 4.4.
Using f class to generate adversarial examples does not always result in high-quality reconstructions, as can be seen in the middle column of Figure 5 and in Figure 11. This appears to be due to the fact that f class adds additional noise to the process. For example, f class sometimes confidently misclassifies latent vectors z that represent inputs that are far from the training data distribution, resulting in f dec failing to reconstruct a plausible output from the adversarial example.
L VAE ATTACK
Our second approach generates adversarial perturbations using the VAE loss function. The attacker chooses two inputs, x s (the source) and x t (the target), and uses one of the standard adversarial methods to perturb x s into x * such that its reconstructionx * matches the reconstruction of x t , using the methods described in Section 4.4.
The adversary precomputes the reconstructionx t by evaluating f dec (f enc (x t )) once before performing optimization. In order to use L VAE in an attack, the second term (the reconstruction loss) of L VAE (see Equation 1) is changed so that instead of computing the reconstruction loss between x andx, the loss is computed betweenx * andx t . This means that during each optimization iteration, the adversary needs to computex * , which requires the full f dec (f enc (x * )) to be evaluated.
LATENT ATTACK
Our third approach attacks the latent space of the generative model.
Single latent vector target. This attack is similar to the work of Sabour et al. (2015), in which they use a pair of source image x s and target image x t to generate x * that induces the target network to produce similar activations at some hidden layer l as are produced by x t , while maintaining similarity between x s and x * .
For this attack to work on latent generative models, it is sufficient to compute z t = f enc (x t ) and then use the following loss function to generate adversarial examples from different source images x s , using the methods described in Section 4.4:
L latent = L(z t , f enc (x * )).
(2) L(·) is a distance measure between two vectors. We use the L 2 norm, under the assumption that the latent space is approximately euclidean.
We also explored a variation on the single latent vector target attack, which we describe in Section A.1 in the Appendix.
METHODS FOR SOLVING THE ADVERSARIAL OPTIMIZATION PROBLEM
We can use a number of different methods to generate the adversarial examples. We initially evaluated both the fast gradient sign method and an L 2 optimization method. As the latter produces much better results we focus on the L 2 optimization method, while we include some FGS results in the Appendix. The attack can be used either in targeted mode (where we want a specific class, y t , to be reconstructed) or untargeted mode (where we just want an incorrect class to be reconstructed). In this paper, we focus on the targeted mode of the attacks.
L 2 optimization. The optimization-based approach, explored in Szegedy et al. (2013) and Carlini & Wagner (2016), poses the adversarial generation problem as the following optimization problem:
argmin x * λL(x, x * ) + L(x * , y t ).
(3) As above, L(·) is a distance measure, and L is one of L classifier , L VAE , or L latent . The constant λ is used to balance the two loss contributions. For the L VAE attack, the optimizer must do a full reconstruction at each step of the optimizer. The other two attacks do not need to do reconstructions while the optimizer is running, so they generate adversarial examples much more quickly, as shown in Table 1.
MEASURING ATTACK EFFECTIVENESS
To generate a large number of adversarial examples automatically against a generative model, the attacker needs a way to judge the quality of the adversarial examples. We leverage f class to estimate whether a particular attack was successful. 6
Reconstruction feedback loop. The architecture is the same as shown in Figure 3. We use the generative model to reconstruct the attempted adversarial inputs x * by computing:
x * = f dec (f enc (x * )).(4)
Then, f class is used to compute:ŷ = f class (f enc (x * )).
(5) The input adversarial examples x * are not classified directly, but are first fed to the generative model for reconstruction. This reconstruction loop improves the accuracy of the classifier by 60% on average against the adversarial attacks we examined. The predicted labelŷ after the reconstruction feedback loop is compared with the attack target y t to determine if the adversarial example successfully reconstructed to the target class. If the precision and recall of f class are sufficiently high on y t , f class can be used to filter out most of the failed adversarial examples while keeping most of the good ones.
We derive two metrics from classifier predictions after one reconstruction feedback loop. The first metric is AS ignore−target , the attack success rate ignoring targeting, i.e., without requiring the output class of the adversarial example to match the target class:
AS ignore−target = 1 N N i=1 1ŷi =y i (6)
N is the total number of reconstructed adversarial examples; 1ŷi =y i is 1 whenŷ i , the classification of the reconstruction for image i, does not equal y i , the ground truth classification of the original image, and 0 otherwise. The second metric is AS target , the attack success rate including targeting (i.e., requiring the output class of the adversarial example to match the target class), which we define similarly as:
AS target = 1 N N i=1 1ŷi =y i t .(7)
Both metrics are expected to be higher for more successful attacks. Note that AS target ≤ AS ignore−target . When computing these metrics, we exclude input examples that have the same ground truth class as the target class.
EVALUATION
We evaluate the three attacks on MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011) and CelebA (Liu et al., 2015), using the standard training and validation set splits. The VAE and VAE-GAN architectures are implemented in TensorFlow (Abadi & et al., 2015). We optimized using Adam with learning rate 0.001 and other parameters set to default values for both the generative model and the classifier. For the VAE, we use two architectures: a simple architecture with a single fully-connected hidden layer with 512 units and ReLU activation function; and a convolutional architecture taken from the original VAE-GAN paper Larsen et al. (2015) (but trained with only the VAE loss). We use the same architecture trained with the additional GAN loss for the VAE-GAN model, as described in that work. For both VAE and VAE-GAN we use a 50-dimensional latent representation on MNIST, a 1024-dimensional latent representation on SVHN and 2048-dimensional latent representation on CelebA. Figure 4: Results for the L 2 optimization latent attack on the VAE-GAN, targeting the mean latent vector for 0. Shown are the first 12 non-zero images from the test MNIST data set. The columns are, in order: the original image, the reconstruction of the original image, the adversarial example, the predicted class of the adversarial example, the reconstruction of the adversarial example, the predicted class of the reconstructed adversarial example, the reconstruction of the reconstructed adversarial example (see Section 4.5), and the predicted class of that reconstruction.
In this section we only show results where no sampling from latent space has been performed. Instead we use the mean vector µ as the latent representation z. As sampling can have an effect on the resulting reconstructions, we evaluated it separately. We show the results with different number of samples in Figure 22 in the Appendix. On most examples, the visible change is small and in general the attack is still successful.
MNIST
Both VAE and VAE-GAN by themselves reconstruct the original inputs well as show in Figure 9, although the quality from the VAE-GAN is noticeably better. As a control, we also generate random noise of the same magnitude as used for the adversarial examples (see Figure 13), to show that random noise does not cause the reconstructed noisy images to change in any significant way. Although we ran experiments on both VAEs and VAE-GANs, we only show results for the VAE-GAN as it generates much higher quality reconstructions than the corresponding VAE.
CLASSIFIER ATTACK
We use a simple classifier architecture to help generate attacks on the VAE and VAE-GAN models. The classifier consists of two fully-connected hidden layers with 512 units each, using the ReLU activation function. The output layer is a 10 dimensional softmax. The input to the classifier is the 50 dimensional latent representation produced by the VAE/VAE-GAN encoder. The classifier achieves 98.05% accuracy on the validation set after training for 100 epochs.
To see if there are differences between classes, we generate targeted adversarial examples for each MNIST class and present the results per-class. For the targeted attacks we used the optimization method with lambda 0.001, where Adam-based optimization was performed for 1000 epochs with a learning rate of 0.1. The mean L 2 norm of the difference between original images and generated adversarial examples using the classifier attack is 3.36, while the mean RMSD is 0.120.
Numerical results in Table 2 show that the targeted classifier attack successfully fools the classifier. Classifier accuracy is reduced to 0%, while the matching rate (the ratio between the number of predictions matching the target class and the number of incorrectly classified images) is 100%, which means that all incorrect predictions match the target class. However, what we are interested in (as per the attack definition from Section 3.2) is how the generative model reconstructs the adversarial examples. If we look at the images generated by the VAE-GAN for class 0, shown in Figure 4, the targeted attack is successful on some reconstructed images (e.g. one, four, five, six and nine are reconstructed as zeroes). But even when the classifier accuracy is 0% and matching rate is 100%, an incorrect classification does not always result in a reconstruction to the target class, which shows that the classifier is fooled by an adversarial example more easily than the generative model.
Reconstruction feedback loop. The reconstruction feedback loop described in Section 4.5 can be used to measure how well a targeted attack succeeds in making the generative model change the reconstructed classes. Table 4 in the Appendix shows AS ignore−target and AS target for all source and target class pairs. A higher value signifies a more successful attack for that pair of classes. It is interesting to observe that attacking some source/target pairs is much easier than others (e.g. pair (4, 0) vs. (0, 8)) and that the results are not symmetric over source/target pairs. Also, some pairs do well in AS ignore−target , but do poorly in AS target (e.g., all source digits when targeting 4). As can be seen in Figure 11, the classifier adversarial examples targeting 4 consistently fail to reconstruct into something easily recognizable as a 4. Most of the reconstructions look like 5, but the adversarial example reconstructions of source 5s instead look like 0 or 3.
L VAE ATTACK
For generating adversarial examples using the L VAE attack, we used the optimization method with λ = 1.0, where Adam-based optimization was performed for 1000 epochs with a learning rate of 0.1. The mean L 2 norm of the difference between original images and generated adversarial examples with this approach is 3.68, while the mean RMSD is 0.131.
We show AS ignore−target and AS target of the L VAE attack in Table 5 in the Appendix. Comparing with the numerical evaluation results of the latent attack (below), we can see that both methods achieve similar results on MNIST.
LATENT ATTACK
To generate adversarial examples using the latent attack, we used the optimization method with λ = 1.0, where Adam-based optimization was performed for 1000 epochs with a learning rate of 0.1. The mean L 2 norm of the difference between original images and generated adversarial examples using this approach is 2.96, while the mean RMSD is 0.105. Table 3 shows AS ignore−target and AS target for all source and target class pairs. Comparing with the numerical evaluation results of the classifier attack we can see that the latent attack performs much better. This result remains true when visually comparing the reconstructed images, shown in Figure 5.
We also tried an untargeted version of the latent attack, where we change Equation 2 to maximize the distance in latent space between the encoding of the original image and the encoding of the adversarial example. In this case the loss we are trying to minimize is unbounded, since the L 2 distance can always grow larger, so the attack normally fails to generate a reasonable adversarial example. Additionally, we also experimented with targeting latent representations of specific images from the training set instead of taking the mean, as described in Section 4.3. We show the numerical results in Table 3 and the generated reconstructions in Figure 15 (in the Appendix). It is also interesting to compare the results with L VAE , by choosing the same image as the target. Results for L VAE for the same target images as in Table 3 are shown in Table 6 in the Appendix. The results are identical between the two attacks, which is expected as the target image is the same -only the loss function differs between the methods.
SVHN
The SVHN dataset consists of cropped street number images and is much less clean than MNIST. Due to the way the images have been processed, each image may contain more than one digit; the target digit is roughly in the center. VAE-GAN produces high-quality reconstructions of the original images as shown in Figure 17 in the Appendix.
For the classifier attack, we set λ = 10 −5 after testing a range of values, although we were unable to find an effective value for this attack against SVHN. For the latent and L VAE attacks we set λ = 10.
In Table 10 we show AS ignore−target and AS target for the L 2 optimization latent attack. The evaluation metrics are less strong on SVHN than on MNIST, but it is still straightforward for an attacker to find a successful attack for almost all source/target pairs. Figure 2 supports this evaluation. Visual inspection shows that 11 out of the 12 adversarial examples reconstructed as 0, the target digit. It is worth noting that 2 out of the 12 adversarial examples look like zeros (rows 1 and 11), and two others look like both the original digit and zero, depending on whether the viewer focuses on the light or dark areas of the image (rows 4 and 7). The L 2 optimization latent attack achieves much better results than the L VAE attack (see Table 11 and Figure 6) on SVHN, while both attacks work equally well on MNIST.
CELEBA
The CelebA dataset consists of more than 200,000 cropped faces of celebrities, each annotated with 40 different attributes. For our experiments, we further scale the images to 64x64 and ignore the attribute annotations. VAE-GAN reconstructions of original images after training are shown in Figure 19 in the Appendix.
Since faces don't have natural classes, we only evaluated the latent and L VAE attacks. We tried lambdas ranging from 0.1 to 0.75 for both attacks. Figure 20 shows adversarial examples generated Table 1 shows a comparison of the mean distances between original images and generated adversarial examples for the three different attack methods. The larger the distance between the original image and the adversarial perturbation, the more noticeable the perturbation will tend to be, and the more likely a human observer will no longer recognize the original input, so effective attacks keep these distances small while still achieving their goal. The latent attack consistently gives the best results in our experiments, and the classifier attack performs the worst.
SUMMARY OF DIFFERENT ATTACK METHODS
We also measure the time it takes to generate 1000 adversarial examples using the given attack method. The L VAE attack is by far the slowest of the three, due to the fact that it requires computing full reconstructions at each step of the optimizer when generating the adversarial examples. The other two attacks do not need to run the reconstruction step during optimization of the adversarial examples.
CONCLUSION
We explored generating adversarial examples against generative models such as VAEs and VAE-GANs. These models are also vulnerable to adversaries that convince them to turn inputs into surprisingly different outputs. We have also motivated why an attacker might want to attack generative models. Our work adds further support to the hypothesis that adversarial examples are a general phenomenon for current neural network architectures, given our successful application of adversarial attacks to popular generative models. In this work, we are helping to lay the foundations for understanding how to build more robust networks. Future work will explore defense and robustification in greater depth as well as attacks on generative models trained using natural image datasets such as CIFAR-10 and ImageNet.
ACKNOWLEDGMENTS
A APPENDIX
A.1 MEAN LATENT VECTOR TARGETED ATTACK
A variant of the single latent vector targeted attack described in Section 4.3, that was not explored in previous work to our knowledge is to take the mean latent vector of many target images and use that vector as x t . This variant is more flexible, in that the attacker can choose different latent properties to target without needing to find the ideal input. For example, in MNIST, the attacker may wish to have a particular line thickness or slant in the reconstructed digit, but may not have such an image available. In that case, by choosing some images of the target class with thinner lines or less slant, and some with thicker lines or more slant, the attacker can find a target latent vector that closely matches the desired properties.
In this case, the attack starts by using f enc to produce the target latent vector, z t , from the chosen target images, x (t) .
z t = 1 |x (t) | |x (t) | i=0 f enc (x i (t) ).(8)
In this work, we choose to reconstruct "ideal" MNIST digits by taking the mean latent vector of all of the training digits of each class, and using those vectors as x t . Given a target class y t , a set of examples X and their corresponding ground truth labels y, we create a subset x (t) as follows:
x (t) = {x i |x i ∈ X ∧ y i = y t } .(9)
Both variants of this attack appear to be similarly effective, as shown in Figure 15 and Figure 5. The trade-off between the two in these experiments is between the simplicity of the first attack and the flexibility of the second attack. Figure 9. Careful visual inspection reveals that none of the VAE reconstructions change class, and only two of the VAE-GAN reconstructions change class (a 6 to a 0 in the next-to-last row, and a 9 to a 4 in the last row). Combining FGS with L VAE does not seem to give an effective attack. The adversarial examples successfully fool f class into predicting the target class almost 100% of the time, which makes this attack seem like a strong attack, but the attack actually fails to generate good reconstructions in many cases.
Reconstructions for target classes 0 and 4 can be seen in Figure 4 and Figure 11. . The magnitude of the random noise is the same as for the generated adversarial noise shown in Figure 12. Random noise does not cause the reconstructed images to change in a significant way.
Source
Source
Target 0 Target 1 Target 2 Target 3 Target 4 Target 5 Target 6 Target 7 Target 8 Target Target 0 Target 1 Target 2 Target 3 Target 4 Target 5 Target 6 Target 7 Target 8 Target
While we do not specifically attack their models, viable compression schemes based on deep neural networks have already been proposed in the literature, showing promising resultsToderici et al. (2015;.
Figure 3 :
3The VAE-GAN classifier architecture used to generate classifier-based adversarial examples on the VAE-GAN. The VAE-GAN in the dashed box is the target network and is frozen while training the classifier. The path x → f enc → z → f class →ŷ is used to generate adversarial examples in z, which can then be reconstructed by f dec .
Figure 5 :
5Left: representative adversarial examples with a target class of 0 on the first 100 nonzero images from the MNIST validation set. These were produced using the L 2 optimization latent attack (Section 4.3). Middle: VAE-GAN reconstructions from adversarial examples produced using the L 2 optimization classifier attack on the same set of 100 validation images (those adversaries are not shown, but are qualitatively similiar, see Section 4.1). Right: VAE-GAN reconstructions from the adversarial examples in the left column. Many of the classifier adversarial examples fail to reconstruct as zeros, whereas almost every adversarial example from the latent attack reconstructs as zero.
Figure 6 :
6Left: VAE-GAN reconstructions of adversarial examples generated using the L 2 optimization L VAE attack (single image target). Right: VAE-GAN reconstructions of adversarial examples generated using the L 2 optimization latent attack (single image target). Approximately 85 out of 100 images are convincing zeros for the L 2 latent attack, whereas only about 5 out of 100 could be mistaken for zeros with the L VAE attack.
using the latent attack and a lambda value of 0.5 (L 2 norm between original images and generated adversarial examples 9.78, RMSD 0.088) and the corresponding VAE-GAN reconstructions. Most of the reconstructions reflect the target image very well. We get even better results with the L VAE attack, using a lambda value of 0.75 (L 2 norm between original images and generated adversarial examples 8.98, RMSD 0.081) as shown inFigure 21.
Figure 7 :
7Summary of different attacks on CelebA dataset: reconstructions of original images (top), reconstructions of adversarial examples generated using the latent attack (middle) and L VAE attack (bottom). Target reconstruction is shown on the right. Full results are in the Appendix.
Figure 9 :
9Original Inputs and Reconstructions: The first 100 images from the validation set reconstructed by the VAE (left) and the VAE-GAN (right).
Figure 10 :
10Untargeted FGS L VAE Attack: VAE reconstructions (left) and VAE-GAN reconstructions (right). Note the difference in reconstructions compared to
Figure 11 :
11L 2 Optimization Classifier Attack: Reconstructions of the first 100 adversarial examples targeting 4, demonstrating why the AS target metric is 0 for all source digits.
Figure 12 :
12Untargeted FGS Classifer Attack: Adversarial examples (left) and their reconstructions by the generative model (right) for the first 100 images from the MNIST validation set. Top results are for VAE, while bottom results are for VAE-GAN. Note the difference in quality of the reconstructed adversarial examples.
Figure 13 :
13Original images with random noise added (top) and their reconstructions by VAE (bottom left) and VAE-GAN (bottom right)
Figure 15 :
15L 2 Optimization Latent Attack (single latent vector target): VAE-GAN reconstructions of adversarial examples generated using the latent attack with target classes 0 and 7 using two random targets in latent space per target class. Original examples which already belong to the target class are excluded. The stylistic differences in the reconstructions are clearly visible.
Figure 18 :
18L 2 Optimization Latent Attack (single latent vector target): Nearest neighbors in latent space for generated adversarial examples (target class 0) on the first 100 images from the MNIST (left) and SVHN (right) validation sets.
Figure 19 :
19Original images in the CelebA dataset (left) and their VAE-GAN reconstructions (right).
Figure 20 :
20L 2 Optimization Latent Attack on CelebA Dataset (single latent vector target): Adversarial examples generated for 100 images from the CelebA dataset (left) and their VAE-GAN reconstructions (right).
Figure 21 :Figure 22 :
2122L 2 Optimization L VAE Attack on CelebA Dataset (single image target): Adversarial examples generated for 100 images from the CelebA dataset (left) and their VAE-GAN reconstructions (right). Effect of sampling on adversarial reconstructions. Columns in order: original image, reconstruction of the original image (no sampling, just the mean), reconstruction of the original image (1 sample), reconstruction of the original image (12 samples), reconstruction of the original image (50 samples), adversarial example (latent attack), reconstruction of the adversarial example (no sampling, just the mean), reconstruction of the adversarial example (1 sample), reconstruction of the adversarial example (12 samples), reconstruction of the adversarial example (50 samples).
Table 1: Comparison of mean L 2 norm and RMSD between the original images and the generated adversarial examples for the different attacks. Time to attack is the mean number of seconds it takes to generate 1000 adversarial examples using the given attack method (with the same number of optimization iterations for each attack).MNIST
SVHN
Method
Mean L 2
Mean RMSD
Time to attack
Mean L 2
Mean RMSD
Time to attack
L 2 Optimization Classifier Attack
3.36
0.120
277
1.77
0.032
274
L 2 Optimization L VAE Attack
3.68
0.131
734
2.36
0.043
895
L 2 Optimization Latent Attack
2.96
0.105
236
2.80
0.051
242
This material is in part based upon work supported by the National Science Foundation under Grant No. TWC-1409915. 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.Aaron van den Oord, Nal Kalchbrenner, Oriol Vinyals, Lasse Espeholt, Alex Graves, and Koray Kavukcuoglu. Conditional image generation with pixelcnn decoders. arXiv preprint arXiv:1606.05328, 2016. Nicolas Papernot, Patrick McDaniel, Somesh Jha, Matt Fredrikson, Z Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. In Proceedings of the 1st IEEE European Symposium on Security and Privacy, 2015. George Toderici, Damien Vincent, Nick Johnston, Sung Jin Hwang, David Minnen, Joel Shor, and Michele Covell. Full resolution image compression with recurrent neural networks. arXiv preprint arXiv:1608.05148, 2016. Aäron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew W. Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. CoRR, abs/1609.03499, 2016. URL http://arxiv.org/abs/1609.03499.Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram
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Table 2 :
2L 2 Optimization Classifier Attack on MNIST: f class accuracy on adversarial examples against the VAE-GAN for each target class (middle row) and the matching rate between the predictions f class made and the adversarial target class (bottom row).
Table 3 :
3L 2 Optimization Latent Attack on MNIST (single latent vector target): AS ignore−target (AS target in parentheses) after one reconstruction loop for different source and target class pairs on the VAE-GAN model. The latent representation of a random image from the target class is used to generate the target latent vector. Higher values indicate more successful attacks against the generative model.Source
Target 0
Target 1
Target 2
Target 3
Target 4
Target 5
Target 6
Target 7
Target 8
Target 9
0
-
40.96%
(1.20%)
6.02%
(4.82%)
10.84%
(7.23%)
75.90%
(0.00%)
6.02%
(3.61%)
28.92%
(28.92%)
37.35%
(20.48%)
6.02%
(1.20%)
10.84%
(3.61%)
1
99.20%
(77.60%)
-
7.20%
(5.60%)
1.60%
(1.60%)
85.60%
(0.00%)
8.00%
(5.60%)
28.80%
(28.00%)
8.80%
(7.20%)
3.20%
(1.60%)
69.60%
(0.80%)
2
85.96%
(80.70%)
3.51%
(2.63%)
-
29.82%
(23.68%)
78.95%
(0.00%)
72.81%
(20.18%)
72.81%
(46.49%)
35.09%
(8.77%)
41.23%
(12.28%)
68.42%
(2.63%)
3
93.46%
(83.18%)
26.17%
(12.15%)
27.10%
(16.82%)
-
67.29%
(0.00%)
66.36%
(62.62%)
87.85%
(22.43%)
50.47%
(27.10%)
23.36%
(8.41%)
33.64%
(8.41%)
4
100.00%
(82.73%)
70.00%
(48.18%)
28.18%
(10.91%)
84.55%
(17.27%)
-
66.36%
(31.82%)
95.45%
(71.82%)
62.73%
(37.27%)
20.91%
(0.91%)
51.82%
(44.55%)
5
93.10%
(89.66%)
21.84%
(1.15%)
68.97%
(11.49%)
28.74%
(18.39%)
3.45%
(0.00%)
-
20.69%
(19.54%)
80.46%
(41.38%)
22.99%
(2.30%)
44.83%
(12.64%)
6
29.89%
(28.74%)
44.83%
(1.15%)
24.14%
(3.45%)
59.77%
(11.49%)
77.01%
(0.00%)
10.34%
(8.05%)
-
62.07%
(8.05%)
8.05%
(0.00%)
75.86%
(4.60%)
7
79.80%
(65.66%)
77.78%
(26.26%)
20.20%
(8.08%)
8.08%
(4.04%)
100.00%
(0.00%)
56.57%
(23.23%)
97.98%
(17.17%)
-
38.38%
(1.01%)
17.17%
(10.10%)
8
94.32%
(84.09%)
96.59%
(18.18%)
60.23%
(42.05%)
57.95%
(43.18%)
100.00%
(0.00%)
93.18%
(80.68%)
100.00%
(57.95%)
100.00%
(34.09%)
-
87.50%
(26.14%)
9
98.91%
(79.35%)
97.83%
(33.70%)
26.09%
(1.09%)
17.39%
(2.17%)
100.00%
(0.00%)
22.83%
(21.74%)
100.00%
(30.43%)
47.83%
(43.48%)
31.52%
(4.35%)
-
Table 4 :
4L 2 Optimization Classifier Attack on MNIST: AS ignore−target (AS target in parenthe-
ses) for all source and target class pairs using adversarial examples generated on the VAE-GAN
model. Higher values indicate more successful attacks against the generative model.
Table 5 :
5L 2 Optimization L VAE Attack on MNIST (single image target): AS ignore−target
(AS target in parentheses) for different source and target class pairs using adversarial examples
generated on the VAE-GAN model. Higher values indicate more successful attacks against the
generative model.
Table 6 :
6L 2 Optimization L VAE Attack (mean reconstruction target): AS ignore−target
(AS target in parentheses) for all source and target class pairs using adversarial examples gener-
ated on the VAE-GAN model. The mean reconstruction image for each target class (over all of the
images of that class in the training set) is used as the target reconstruction. Higher values indicate
more successful attacks against the generative model.
Source
Target 0
Target 1
Target 2
Target 3
Target 4
Target 5
Target 6
Target 7
Target 8
Target 9
0
-
40.96%
(10.84%)
65.06%
(65.06%)
53.01%
(46.99%)
62.65%
(54.22%)
36.14%
(36.14%)
59.04%
(59.04%)
46.99%
(46.99%)
13.25%
(12.05%)
44.58%
(27.71%)
1
100.00%
(100.00%)
-
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(100.00%)
100.00%
(96.80%)
2
96.49%
(96.49%)
60.53%
(59.65%)
-
95.61%
(95.61%)
78.07%
(75.44%)
98.25%
(71.05%)
94.74%
(90.35%)
71.05%
(69.30%)
52.63%
(50.88%)
75.44%
(42.98%)
3
100.00%
(100.00%)
87.85%
(66.36%)
90.65%
(90.65%)
-
85.98%
(73.83%)
95.33%
(95.33%)
79.44%
(53.27%)
65.42%
(64.49%)
59.81%
(46.73%)
70.09%
(58.88%)
4
99.09%
(99.09%)
67.27%
(66.36%)
96.36%
(96.36%)
100.00%
(81.82%)
-
100.00%
(98.18%)
93.64%
(93.64%)
98.18%
(95.45%)
97.27%
(92.73%)
39.09%
(39.09%)
5
100.00%
(100.00%)
79.31%
(51.72%)
100.00%
(83.91%)
70.11%
(70.11%)
80.46%
(72.41%)
-
73.56%
(73.56%)
87.36%
(73.56%)
55.17%
(52.87%)
75.86%
(65.52%)
6
97.70%
(97.70%)
68.97%
(50.57%)
96.55%
(96.55%)
95.40%
(71.26%)
73.56%
(73.56%)
87.36%
(77.01%)
-
88.51%
(72.41%)
90.80%
(55.17%)
91.95%
(35.63%)
7
100.00%
(97.98%)
83.84%
(83.84%)
100.00%
(100.00%)
100.00%
(100.00%)
93.94%
(90.91%)
98.99%
(96.97%)
88.89%
(81.82%)
-
100.00%
(86.87%)
50.51%
(50.51%)
8
100.00%
(100.00%)
96.59%
(78.41%)
100.00%
(100.00%)
98.86%
(95.45%)
94.32%
(86.36%)
98.86%
(98.86%)
98.86%
(93.18%)
98.86%
(73.86%)
-
87.50%
(78.41%)
9
100.00%
(100.00%)
100.00%
(76.09%)
100.00%
(100.00%)
98.91%
(96.74%)
100.00%
(100.00%)
100.00%
(98.91%)
97.83%
(97.83%)
98.91%
(98.91%)
97.83%
(94.57%)
-
Table 7 :
7L 2 Optimization Latent Attack (mean latent vector target): AS ignore−target (AS target
in parentheses) for all source and target class pairs using adversarial examples generated on the
VAE-GAN model. The mean latent vector for each target class (over all of the images of that class
in the training set) is used as the target latent vector. Higher values indicate more successful attacks
against the generative model.
Table 8 :
8L 2 Optimization L VAE Attack (mean reconstruction target): AS ignore−target
(AS target in parentheses) for all source and target class pairs using adversarial examples gener-
ated on the VAE-GAN model. The mean image for each target class (over all of the images of that
class in the training set) is used as the target. Higher values indicate more successful attacks against
the generative model.
Table 9 :
9L 2 Optimization L VAE Attack on MNIST (single image target): AS ignore−target (AS target in parentheses) for different source and target class pairs using adversarial examples generated on the VAE-GAN model. Higher values indicate more successful attacks against the generative model.Figure 17: Original Inputs and Reconstructions: The first 100 images from the SVHN validation set (left) reconstructed by VAE-GAN (right).Source
Target 0
Target 1
Target 2
Target 3
Target 4
Target 5
Target 6
Target 7
Target 8
Target 9
0
-
64.29%
(40.00%)
78.57%
(61.43%)
92.86%
(80.00%)
84.29%
(57.14%)
98.57%
(98.57%)
94.29%
(38.57%)
88.57%
(54.29%)
95.71%
(11.43%)
95.71%
(25.71%)
1
76.80%
(70.72%)
-
74.59%
(67.40%)
93.37%
(88.95%)
75.69%
(65.19%)
98.34%
(97.79%)
86.74%
(24.86%)
46.96%
(36.46%)
96.13%
(4.97%)
96.13%
(28.73%)
2
82.93%
(65.85%)
57.93%
(42.68%)
-
90.24%
(86.59%)
53.66%
(46.34%)
99.39%
(98.17%)
82.93%
(14.02%)
71.34%
(57.32%)
71.34%
(6.71%)
24.39%
(23.17%)
3
92.17%
(64.35%)
58.26%
(41.74%)
83.48%
(68.70%)
-
84.35%
(49.57%)
96.52%
(95.65%)
53.91%
(23.48%)
90.43%
(56.52%)
93.04%
(5.22%)
93.91%
(33.91%)
4
74.44%
(55.56%)
47.78%
(43.33%)
70.00%
(61.11%)
86.67%
(77.78%)
-
100.00%
(98.89%)
93.33%
(35.56%)
90.00%
(36.67%)
85.56%
(14.44%)
94.44%
(27.78%)
5
75.31%
(50.62%)
59.26%
(43.21%)
88.89%
(58.02%)
97.53%
(88.89%)
72.84%
(53.09%)
-
37.04%
(18.52%)
80.25%
(41.98%)
32.10%
(6.17%)
92.59%
(30.86%)
6
67.44%
(47.67%)
56.98%
(27.91%)
84.88%
(55.81%)
86.05%
(79.07%)
65.12%
(39.53%)
94.19%
(94.19%)
-
90.70%
(33.72%)
58.14%
(10.47%)
87.21%
(22.09%)
7
87.34%
(63.29%)
55.70%
(48.10%)
79.75%
(74.68%)
92.41%
(79.75%)
69.62%
(41.77%)
97.47%
(89.87%)
93.67%
(18.99%)
-
91.14%
(7.59%)
97.47%
(17.72%)
8
98.33%
(63.33%)
78.33%
(38.33%)
80.00%
(63.33%)
100.00%
(88.33%)
93.33%
(48.33%)
98.33%
(96.67%)
96.67%
(35.00%)
96.67%
(50.00%)
-
95.00%
(31.67%)
9
87.88%
(66.67%)
72.73%
(43.94%)
92.42%
(80.30%)
93.94%
(86.36%)
80.30%
(51.52%)
95.45%
(93.94%)
98.48%
(27.27%)
92.42%
(62.12%)
93.94%
(9.09%)
-
Table 10 :
10L 2 Optimization Latent Attack on SVHN (single latent vector target): AS ignore−target (AS target in parentheses) after one reconstruction loop for different source and target class pairs on the VAE-GAN model. The latent representation of a random image from the target class is used to generate the target latent vector. Higher values indicate more successful attacks against the generative model.Source
Target 0
Target 1
Target 2
Target 3
Target 4
Target 5
Target 6
Target 7
Target 8
Target 9
0
-
30.00%
(12.86%)
32.86%
(5.71%)
34.29%
(5.71%)
28.57%
(0.00%)
30.00%
(1.43%)
30.00%
(5.71%)
30.00%
(0.00%)
30.00%
(1.43%)
31.43%
(0.00%)
1
13.26%
(1.10%)
-
7.73%
(1.66%)
18.78%
(4.97%)
13.26%
(3.31%)
12.15%
(0.00%)
11.60%
(0.55%)
9.94%
(1.10%)
10.50%
(1.10%)
16.02%
(0.55%)
2
23.17%
(0.61%)
13.41%
(3.66%)
-
17.07%
(3.05%)
14.63%
(1.83%)
14.63%
(2.44%)
15.24%
(0.00%)
15.24%
(1.22%)
14.02%
(0.61%)
15.24%
(1.22%)
3
30.43%
(0.87%)
26.09%
(7.83%)
30.43%
(2.61%)
-
30.43%
(0.00%)
29.57%
(6.96%)
27.83%
(0.00%)
27.83%
(1.74%)
28.70%
(2.61%)
33.91%
(6.09%)
4
21.11%
(0.00%)
15.56%
(5.56%)
16.67%
(2.22%)
25.56%
(4.44%)
-
16.67%
(1.11%)
18.89%
(0.00%)
16.67%
(1.11%)
18.89%
(2.22%)
22.22%
(0.00%)
5
32.10%
(0.00%)
28.40%
(3.70%)
27.16%
(3.70%)
32.10%
(8.64%)
24.69%
(2.47%)
-
28.40%
(6.17%)
23.46%
(0.00%)
27.16%
(3.70%)
27.16%
(0.00%)
6
27.91%
(4.65%)
25.58%
(4.65%)
26.74%
(0.00%)
33.72%
(3.49%)
30.23%
(2.33%)
20.93%
(4.65%)
-
31.40%
(0.00%)
24.42%
(3.49%)
32.56%
(0.00%)
7
30.38%
(0.00%)
27.85%
(12.66%)
26.58%
(10.13%)
31.65%
(5.06%)
31.65%
(0.00%)
30.38%
(0.00%)
32.91%
(0.00%)
-
30.38%
(0.00%)
34.18%
(1.27%)
8
40.00%
(5.00%)
35.00%
(0.00%)
33.33%
(3.33%)
43.33%
(6.67%)
40.00%
(3.33%)
35.00%
(1.67%)
41.67%
(11.67%)
38.33%
(0.00%)
-
36.67%
(0.00%)
9
34.85%
(6.06%)
33.33%
(12.12%)
33.33%
(9.09%)
40.91%
(4.55%)
31.82%
(3.03%)
31.82%
(0.00%)
33.33%
(0.00%)
34.85%
(0.00%)
31.82%
(1.52%)
-
Table 11 :
11L 2 Optimization L VAE Attack on SVHN (single image target): AS ignore−target (AS target in parentheses) after one reconstruction loop for different source and target class pairs on the VAE-GAN model. The latent representation of a random image from the target class is used to generate the target latent vector. Higher values indicate more successful attacks against the generative model.
SeePapernot et al. (2015) for an overview of different adversarial threat models. 4 This work was made public shortly after we published our early drafts.
This is similar to the process of semi-supervised learning inKingma et al. (2014), although the goal is different.
Note that f class here is being used in a different manner than when we use it to generate adversarial examples. However, the network itself is identical, so we don't distinguish between the two uses in the notation.
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| [] |
[
"Physical Layer Security-Aware Routing and Performance Tradeoffs in WANETs",
"Physical Layer Security-Aware Routing and Performance Tradeoffs in WANETs"
] | [
"Yang Xu \nState Key Laboratory of Integrated Services Network\nXidian University\n710071XianChina\n",
"Jia Liu [email protected] \nCyber Security Research Center\nNational Institute of Informatics\n101-8430TokyoJapan\n",
"Yulong Shen [email protected] \nState Key Laboratory of Integrated Services Network\nXidian University\n710071XianChina\n",
"Xiaohong Jiang [email protected] \nSchool of Systems Information Science\nFuture University Hakodate\n041-8655HakodateJapan\n",
"Norio Shiratori \nResearch and Development Initiative\nChuo University\n112-8551TokyoJapan\n"
] | [
"State Key Laboratory of Integrated Services Network\nXidian University\n710071XianChina",
"Cyber Security Research Center\nNational Institute of Informatics\n101-8430TokyoJapan",
"State Key Laboratory of Integrated Services Network\nXidian University\n710071XianChina",
"School of Systems Information Science\nFuture University Hakodate\n041-8655HakodateJapan",
"Research and Development Initiative\nChuo University\n112-8551TokyoJapan"
] | [] | The application of physical layer security in wireless ad hoc networks (WANETs) has attracted considerable academic attention recently. However, the available studies mainly focus on the single-hop and two-hop network scenarios, and the price in terms of degradation of communication quality of service (QoS) caused by improving security is largely uninvestigated. As a step to address these issues, this paper explores the physical layer security-aware routing and performance tradeoffs in a multi-hop WANET. Specifically, for any given end-to-end path in a general multi-hop WANET, we first derive its connection outage probability (COP) and secrecy outage probability (SOP) in closed-form, which serve as the performance metrics of communication QoS and transmission security, respectively. Based on the closed-form expressions, we then study the QoS-security tradeoffs to minimize COP (resp. SOP) conditioned on that SOP (resp. COP) is guaranteed. With the help of analysis of a given path, we further propose the routing algorithms which can achieve the optimal performance tradeoffs for any pair of source and destination nodes in a distributed manner. Finally, simulation and numerical results are presented to validate the efficiency of our theoretical analysis, as well as to illustrate the QoS-security tradeoffs and the | 10.1016/j.comnet.2017.05.012 | [
"https://arxiv.org/pdf/1609.02288v2.pdf"
] | 9,645,532 | 1609.02288 | ca675679c4c0344089f446b238bccb9b050c6abc |
Physical Layer Security-Aware Routing and Performance Tradeoffs in WANETs
12 Dec 2016
Yang Xu
State Key Laboratory of Integrated Services Network
Xidian University
710071XianChina
Jia Liu [email protected]
Cyber Security Research Center
National Institute of Informatics
101-8430TokyoJapan
Yulong Shen [email protected]
State Key Laboratory of Integrated Services Network
Xidian University
710071XianChina
Xiaohong Jiang [email protected]
School of Systems Information Science
Future University Hakodate
041-8655HakodateJapan
Norio Shiratori
Research and Development Initiative
Chuo University
112-8551TokyoJapan
Physical Layer Security-Aware Routing and Performance Tradeoffs in WANETs
12 Dec 2016Preprint submitted to XXX December 13, 2016 routing performance.(Norio Shiratori)Ad hoc networksphysical layer securityQoSrouting
The application of physical layer security in wireless ad hoc networks (WANETs) has attracted considerable academic attention recently. However, the available studies mainly focus on the single-hop and two-hop network scenarios, and the price in terms of degradation of communication quality of service (QoS) caused by improving security is largely uninvestigated. As a step to address these issues, this paper explores the physical layer security-aware routing and performance tradeoffs in a multi-hop WANET. Specifically, for any given end-to-end path in a general multi-hop WANET, we first derive its connection outage probability (COP) and secrecy outage probability (SOP) in closed-form, which serve as the performance metrics of communication QoS and transmission security, respectively. Based on the closed-form expressions, we then study the QoS-security tradeoffs to minimize COP (resp. SOP) conditioned on that SOP (resp. COP) is guaranteed. With the help of analysis of a given path, we further propose the routing algorithms which can achieve the optimal performance tradeoffs for any pair of source and destination nodes in a distributed manner. Finally, simulation and numerical results are presented to validate the efficiency of our theoretical analysis, as well as to illustrate the QoS-security tradeoffs and the
Introduction
Background and Related Works
The wireless ad hoc network (WANET) represents a class of self-organizing network architecture, which consists of nodes communicating with each other over peer-to-peer wireless channels without centralized infrastructure [1]. Since WANETs can be flexibly deployed and reconfigured at very low cost, they are highly promising for many critical applications, such as disaster relief, emergency rescue, daily information exchange, traffic off-loading and coverage extension for 5G cellular networks [2,3]. To facilitate the application and commercialization of WANETs, protecting their transmission security is of great significance [4]. However, due to the broadcast nature of wireless channel and the lack of central administration, it is very challenging for the traditional cryptographicbased security techniques [5] to be applied in such a distributed WANET.
As a complementary technique of cryptographic-based methods, physical layer security, an information-theoretic approach which exploits the fundamental characteristics of wireless channel to achieve perfect secrecy, has been extensively studied over the past few decades. Based on the results of Shannon in [6], Wyner first indicated that perfect secrecy is achievable when the conditions of main channel between transmitter and receiver are better than that of wiretap channel between transmitter and eavesdropper [7]. Following this line, many research activities have been devoted to the study of physical layer security under various channel models, such as the broadcast channel [8], Gaussian wiretap channel [9], two-way wiretap channel [10], multi-access channel [11] and MIMO wiretap channel [12]. Meanwhile, diverse approaches for improving physical layer security have been proposed in the literature. The works of [13,14,15,16,17,18,19] demonstrated that the strategies of cooperative jamming and relay selection can be utilized to enhance physical layer security.
The works of [20,21] indicated that physical layer security can also be facilitated by applying coding schemes. Moreover, the combinations of physical layer security with other techniques such as power allocation, signal processing, and cross-layer optimization were explored in [22], [23] and [24], respectively.
Since physical layer security has the advantage of low computational complexity and can be easily implemented in a distributed manner, its application in WANETs has attracted considerable academic attention recently [25,26,27,28,29,30,31]. For a large-scale WANET, Vasudevan et al. [25] investigated the asymptotic behaviors of security-capacity tradeoff as the number of network nodes tends to infinity. The price in terms of performance degradation for ensuring physical layer security in WANETs was explored under the asymptotically large network scenario [26] and single-hop network scenario [27], respectively.
Goeckel et al. [28] indicated that the artificial noise generated by cooperative relays can be utilized to achieve everlasting secrecy in a two-hop WANET.
Koyluoglu et al. [29] with decode-and-forward relaying, and Lee [38] proposed an optimal power allocation strategy for maximizing the secrecy rate in a special multi-hop relay network with single source-destination pair.
It is notable that security usually comes with a cost in terms of performance degradation [26,27], thus the tradeoffs between security and other network performance should be carefully addressed for a practical multi-hop WANET. In [39,40] and our previous work [41], the issue of integrating security and quality of service (QoS) under some network scenarios was investigated. While in this paper, for the first time, we explore the tradeoffs between transmission security and communication QoS in a multi-hop WANET. We consider a general multihop WANET with randomly distributed legitimate nodes and malicious eavesdroppers, and analyze the connection outage probability (COP) and secrecy outage probability (SOP) for a given path. Based on the outage probability analysis, we study the COP and SOP tradeoffs, and further propose the routing algorithms which can achieve the optimal performance with the guaranteed communication QoS and transmission security in the concerned WANET.
The main contributions of this paper are summarized as follows:
• For any given end-to-end path in a general multi-hop WANET where legitimate nodes and malicious eavesdroppers are randomly distributed following the independent Poisson point processes, we derive its COP and SOP in closed-form, which serve as the performance metrics of communication QoS and transmission security, respectively.
• We formulate the QoS-security tradeoffs of a given path as two constrained optimization problems and provide corresponding analysis to obtain the optimal solutions. Based on the results of a given path, we further propose the routing algorithms which can find the optimal path between any pair of source and destination nodes, and allocate the transmission power for the relay node of each hop on the path to achieve the optimal performance.
• We provide extensive simulation and numerical results to validate the efficiency of our theoretical analysis, and to illustrate the QoS-security tradeoffs as well as the performance of proposed routing algorithms.
Paper Organization
The remainder of this paper is organized as follows. Section 2 introduces the preliminaries involved in this paper. The expressions of COP and SOP are derived in Section 3. We explore the tradeoffs between COP and SOP in Section 4 and propose the routing algorithms in Section 5. Finally, Section 6 presents the simulation and numerical results, and Section 7 concludes this paper.
Preliminaries
In this section, we introduce the network model, wireless channel model and performance metrics involved in this study.
Network Model
We consider a general multi-hop WANET which consists of arbitrarily distributed legitimate nodes and malicious eavesdroppers. A K-hop path (route) Π = l 1 , . . . , l K in the network is formed by K links from l 1 to l K , and a link l k ∈ Π connects two legitimate nodes S k and D k on path Π. We assume that each link l k is exposed to a set of eavesdroppers denoted by Φ
Wireless Channel Model
We consider the decode-and-forward (DF) relaying scheme and assume that the wireless channels between any pair of nodes are characterized by the largescale path loss along with the small-scale Rayleigh fading [42]. In addition, we assume that the network is interference limited and thus the noise at the receiver is negligible. More formally, regarding a transmission from Node S to Node D, let x S and x Jj denote the normalized (unit power) symbol stream to be transmitted by S and its j th jammer J j , respectively. P S and P Jj denote the corresponding transmission power, and y D denote the received signal at D.
Then y D can be expressed as:
y D = √ P S h S,D d α/2 S,D x S + Jj∈ΦJ P Jj h Jj,D d α/2 Jj ,D x Jj ,(1)
where d S,D and h S,D (resp. d Jj ,D and h Jj,D ) are the distance and the fading coefficient of wireless channel between S (resp. J j ) and D, α is the path-loss exponent (typically between 2 and 6). Similarly, for an eavesdropper E ∈ Φ E , the signal y E received at E is given by
y E = √ P S h S,E d α/2 S,E x S + Jj ∈ΦJ P Jj h Jj,E d α/2 Jj ,E x Jj ,(2)
where d S,E and h S,E (resp. d Jj ,E and h Jj ,E ) are the distance and the fading coefficient of wiretap link between S (resp. J j ) and E.
It is notable that the Rayleigh fading implies |h X,Y | 2 between any pair of nodes X and Y is exponentially distributed with E{|h X,Y | 2 } = 1, and the wireless channel model adopted here is also widely employed in other physical layer literature [43,30,36].
Performance Metrics
The performance metrics involved in this paper are defined as follows:
Connection Outage Probability: The event of connection outage refers to the case when the signal-to-interference ratio (SIR) at the intended receiver is below a required threshold γ C , such that the message cannot be correctly decoded by the receiver. The connection outage probability (COP) P co is defined as the probability that the event of connection outage happens.
Secrecy Outage Probability: The event of secrecy outage refers to the case when the SIR at one or more eavesdroppers is above a required threshold γ E , such that the message can be decoded by the eavesdropper(s). The secrecy outage probability (SOP) P so is defined as the probability that the event of secrecy outage happens.
Remark 1. COP and SOP are of high significance as COP represents the communication QoS of a network user, while SOP serves as a measure of the transmission security level.
Outage Probabilities Analysis
In this section, we derive the exact expressions of COP and SOP for a given path, which will help us explore the performance tradeoffs in Section 4.
COP Analysis
Regarding the COP of a given path, we have the following lemma.
Lemma 1. For a concerned WANET with a network model and wireless channel model as described in Section 2, the COP of a K-hop path Π = l 1 , . . . , l K is given by
P co (Π) = 1 − exp −A co l k ∈Π d 2 S k ,D k P − 2 α S k ,(3)
where
A co = λ J π (γ C PJ ) 2 α Γ(1 − 2 α )Γ(1 + 2 α ), Γ(·)
is a gamma function, P S k and PJ denote the transmission power of S k and the average transmission power of jammers, respectively.
Proof. We first derive the COP for a link l k on path Π, which is termed as P co (l k ). Since in such a distributed WANET, it is hardly possible for a node to get the exact information about the transmission power of other nodes, but it is likely to estimate their average transmission power, we use PJ as a reasonable approximation of P Jj to facilitate the efficiency of our theoretical analysis.
Based on the wireless channel model of Expression (1) and the definition of COP, P co (l k ) can be determined as:
P co (l k ) = P P S k |h S k ,D k | 2 /d α S k ,D k Jj ∈ΦJ PJ |h Jj,D k | 2 /d α Jj ,D k < γ C ,(4)
which can be further rewritten as:
P co (l k ) = 1 − E ΦJ E hJ j ,D k exp −γ C Jj ∈ΦJ PJ |h Jj ,D k | 2 /d α Jj ,D k P S k /d α S k ,D k = 1 − E ΦJ Jj ∈ΦJ E hJ j ,D k exp −γ C PJ |h Jj ,D k | 2 /d α Jj,D k P S k /d α S k ,D k = 1 − E ΦJ Jj ∈ΦJ ∞ 0 exp − γ C PJ /d α Jj,D P S k /d α S k ,D k + 1 x dx = 1 − E ΦJ Jj ∈ΦJ 1 1 + γC PJ /d α J j ,D k PS k /d α S k ,D k .(5)
Notice that for a homogeneous PPP, the corresponding probability generating functional (PGFL) is given by [44]
E ΦJ Jj ∈ΦJ f (z Jj ) = exp −λ J R 2 1 − f (z Jj )dz Jj = exp −2πλ J ∞ 0 (1 − f (r))rdr ,(6)
where z Jj is the location of J j . By applying PGFL in (5), then P co (l k ) can be expressed as:
P co (l k ) = 1 − exp −2πλ J ∞ 0 1 1 + PS k /d α S k ,D k γC PJ /r α rdr = 1 − exp −A co d 2 S k ,D k P − 2 α S k .(7)
Based on the COP of a link l k , the COP P co (Π) of the K-hop path Π can be finally determined as:
P co (Π) = 1 − l k ∈Π [1 − P co (l k )] = 1 − exp −A co l k ∈Π d 2 S k ,D k P − 2 α S k .
We can see from Formula (3) that P co (Π) is an increasing function of λ J , PJ and γ C , while being a decreasing function of P S k .
SOP Analysis
Regarding the SOP of a given path, we have the following lemma.
Lemma 2. For a concerned WANET with a network model and wireless channel model as described in Section 2, the SOP of a K-hop path Π = l 1 , . . . , l K is given by
P so (Π) = 1 − exp −B so l k ∈Π P 2 α S k ,(8)
where
B so = λ E λ J (γ E PJ ) 2 α Γ(1 − 2 α )Γ(1 + 2 α ) −1 .
Proof. We first derive the SOP for a link l k on path Π, which is termed as P so (l k ). Based on the wireless channel model of Expression (2) and the definition of SOP, P so (l k ) can be determined as:
Pso(l k ) = 1−E Φ J E Φ E E i ∈Φ E 1−P P S k |h S k ,E i | 2 /d α S k ,E i J j ∈Φ J PJ |h J j ,E i | 2 /d α J j ,E i > γ E Φ E , Φ J .(9)
Applying the PGFL technique for the PPP Φ E , then Equation (9) can be re-expressed as:
Pso(l k ) = 1 − E Φ J exp −λ E R 2 P P S k |h S k ,E i | 2 /d α S k ,E i J j ∈Φ J PJ |h J j ,E i | 2 /d α J j ,E i > γ E Φ J dz E i (10) ≤ 1 − exp −λ E R 2 P P S k |h S k ,E i | 2 /d α S k ,E i J j ∈Φ J PJ |h J j ,E i | 2 /d α J j ,E i > γ E dz E i (11) = 1 − exp −λ E R 2 exp −λ J πd 2 S k ,E i γ E PJ P S k 2 α Γ(1 − 2 α )Γ(1 + 2 α ) dz E i (12) = 1 − exp −2πλ E ∞ 0 exp −λ J πr 2 γ E PJ P S k 2 α Γ(1 − 2 α )Γ(1 + 2 α ) rdr = 1 − exp −BsoP 2 α S k ,(13)
where (11) follows from the Jensen's inequality, and (12) follows from the same procedures which transform (4) into (7).
Based on the SOP of a link l k , the SOP P so (Π) of the K-hop path Π can be finally determined as:
P so (Π) = 1 − l k ∈Π [1 − P so (l k )] = 1 − exp −B so l k ∈Π P 2 α S k .
We can see from Formula (8) that P so (Π) is an increasing function of P S k and λ E , while being a decreasing function of γ E , λ J and PJ .
Remark 2. For a given WANET, the network parameters λ J , PJ , γ C , λ E and γ E are usually pre-determined, the controllable parameter is the transmission power of each transmitter. It is worth noting that increasing P S k will lead to a decrease in P co (Π) and an increase in P so (Π), which agrees with the intuition that a larger transmission power can bring about a larger SIR at the intended receiver to gain a lower COP, at the same time it comes with the cost of a higher SOP since there is also a larger SIR at the eavesdroppers. This observation indicates that by adjusting the transmission power of each transmitter on path Π, we can achieve performance tradeoffs between COP and SOP.
Since the performance tradeoffs between COP and SOP exist, a problem of insight is how to optimize (minimize) one outage probability while ensuring that another outage probability is below some pre-specified threshold. This problem is termed as the optimal performance tradeoffs and will be analyzed in the next section.
Optimal Performance Tradeoffs
In this section, we formally define the optimal performance tradeoffs as the problems of secure-based optimal COP (SO-COP) and QoS-based optimal SOP (QO-SOP), and provide corresponding solutions, respectively.
SO-COP: Secure-based Optimal COP
We first analyze how to achieve optimal QoS performance (minimal COP) conditioned on that secure performance is ensured (SOP is below some prespecified threshold), which is termed as the problem SO-COP.
Let β so (0 < β so ¡1) denote the pre-specified constraint on SOP of path Π, then the problem SO-COP can be formally defined as the following optimization issue:
min l k ∈Π,PS k P co (Π)(14)s.t. P so (Π) ≤ β so .(15)
Regarding the problem SO-COP (14)-(15), we have the following theorem.
Theorem 1. For a concerned multi-hop WANET, where the densities of eavesdroppers and jammers are λ E and λ J , respectively, the required SIRs for an intended receiver correctly decoding the message and an eavesdropper successfully intercepting the message are γ C and γ E , respectively. The constraint on transmission security is β so , then the optimal solution (i.e., optimal transmission power) of problem SO-COP is determined as:
P SO-COP S k = − ln(1 − β so ) B so · d S k ,D k l k ∈Π d S k ,D k α/2 l k ∈ Π,(16)
and the optimal achievable COP with the guaranteed SOP is given by
P * co (Π) = 1 − exp λ E π ln(1 − β so ) γ C γ E 2 α l k ∈Π d S k ,D k 2 .(17)
Proof. Let F k = P 2/α S k , then P co (Π) in Formula (3) and P so (Π) in Formula (8) can be re-expressed as:
P co (Π) = 1 − exp −A co l k ∈Π d 2 S k ,D k F k ,(18)P so (Π) = 1 − exp −B so l k ∈Π F k .(19)
Substituting (19) into (15), we have
l k ∈Π F k ≤ − ln(1 − β so ) B so ǫ so .(20)
Notice that P co (Π) in (18) is a decreasing function of F k while the objective in (14) is to minimize P co (Π), so the inequality constraint (20) can be replaced by the equality constraint l k ∈Π F k = ǫ so . Therefore, the problem SO-COP is equivalent to the following optimization issue:
min l k ∈Π,F k l k ∈Π d 2 S k ,D k F k ,(21)s.t. l k ∈Π F k = ǫ so .(22)
To solve the above optimization issue, we apply the method of Lagrange multipliers [45]. Then, we obtain the following K equations:
∂ ∂F k l k ∈Π d 2 S k ,D k F k + θ 1 l k ∈Π F k − ǫ so F * k = 0, l k ∈ Π,(23)
where θ 1 is the Lagrange multiplier, and we have
− d 2 S k ,D k (F * k ) 2 + θ 1 = 0, l k ∈ Π, ⇒F * k = 1 √ θ 1 d S k ,D k , l k ∈ Π.(24)
Substituting (24) into (22), θ 1 can be determined as:
θ 1 = 1 ǫ so l k ∈Π d S k ,D k 2 .(25)
Substituting (25) into (24), we have
F * k = ǫ so d S k ,D k l k ∈Π d S k ,D k , l k ∈ Π.(26)
Thus, the optimal transmission power P SO-COP S k of Node S k is given by
P SO-COP S k = − ln(1 − β so ) B so · d S k ,D k l k ∈Π d S k ,D k α/2
, and the minimum COP P * co (Π) of path Π under the condition that P so (Π) ≤ β so is determined as:
P * co (Π) = 1 − exp −A co l k ∈Π d 2 S k ,D k F * k = 1 − exp −A co 1 ǫ so l k ∈Π d S k ,D k 2 = 1 − exp A co · B so ln(1 − β so ) l k ∈Π d S k ,D k 2 = 1 − exp λ E π ln(1 − β so ) γ C γ E 2 α l k ∈Π d S k ,D k 2 .
We can see from Formula (16) that P SO-COP S k is an increasing function of λ J , PJ , γ E and β so , while being a decreasing function of λ E . We can see from Formula (17) that P * co (Π) is an increasing function of γ C and λ E , while being a decreasing function of γ E and β so .
QO-SOP: QoS-based Optimal SOP
We then analyze how to achieve optimal secure performance (minimal SOP) conditioned on that QoS performance is ensured (COP is below some prespecified threshold), which is termed as the problem QO-SOP.
Let β co (0 < β co ¡1) denote the pre-specified constraint on COP of path Π, then the problem QO-SOP can be formally defined as the following optimization problem:
min l k ∈Π,PS k P so (Π) (27) s.t. P co (Π) ≤ β co .(28)
Regarding the problem QO-SOP (27)-(28), we have the following theorem.
Theorem 2. For a given multi-hop WANET, where the densities of eavesdroppers and jammers are λ E and λ J , respectively, the required SIRs for an intended receiver correctly decoding the message and an eavesdropper successfully intercepting the message are γ C and γ E , respectively. The constraint on communication QoS is β co , then the optimal solution (i.e., optimal transmission power) of problem QO-SOP is determined as:
P QO-COP S k = − A co ln(1 − β co ) · l k ∈Π d S k ,D k · d S k ,D k α/2 , l k ∈ Π,(29)
and the optimal achievable SOP with the guaranteed COP is given by
P * so (Π) = 1 − exp λ E π ln(1 − β co ) γ C γ E 2 α l k ∈Π d S k ,D k 2 .(30)
Proof. Let F k = P 2/α S k , then P co (Π) and P so (Π) can be expressed as (18) and (19), respectively. Substituting (18) into (28) we have
l k ∈Π d 2 S k ,D k F k ≤ − ln(1 − β co ) A co ǫ co .(31)
Notice that P so (Π) in (19) is an increasing function of F k while the objective in (27) is to minimize P so (Π), so the inequality constraint (31) can be replaced by the equality constraint
l k ∈Π d 2 S k ,D k F k = ǫ co .
Therefore, the problem QO-SOP is equivalent to the following optimization issue:
min l k ∈Π,F k l k ∈Π F k ,(32)s.t. l k ∈Π d 2 S k ,D k F k = ǫ co .(33)
Similar to the proof of Theorem 1, we also apply the method of Lagrange multipliers and obtain the following K equations:
∂ ∂F k l k ∈Π F k + θ 2 l k ∈Π d 2 S k ,D k F k − ǫ co = 0, F * k = 0, l k ∈ Π,(34)
where θ 2 is the Lagrange multiplier. Then we have
1 − θ 2 d 2 S k ,D k (F * k ) 2 = 0, l k ∈ Π, ⇒F * k = √ θ 2 d S k ,D k , l k ∈ Π.(35)
Substituting (35) into (33), θ 2 can be determined as:
θ 2 = 1 ǫ co l k ∈Π d S k ,D k 2 .(36)
Substituting (36) into (35), we have
F * k = 1 ǫ co l k ∈Π d S k ,D k d S k ,D k , l k ∈ Π.(37)
Thus, the optimal transmission power P QO-SOP S k of Node S k is given by
P QO-SOP S k = − A co ln(1 − β co ) · l k ∈Π d S k ,D k · d S k ,D k α/2
, and the minimum SOP P * so (Π) of path Π under the condition that P co (Π) ≤ β co is determined as:
P * so (Π) = 1 − exp −B so l k ∈Π F * k = 1 − exp −B so 1 ǫ co l k ∈Π d S k ,D k 2 = 1 − exp B so · A co ln(1 − β so ) l k ∈Π d S k ,D k 2 = 1 − exp λ E π ln(1 − β co ) γ C γ E 2 α l k ∈Π d S k ,D k 2 .
We can see from Formula (29) that P QO-SOP S k is an increasing function of λ J , PJ and γ C , while being a decreasing function of β co . We can see from Formula (30) that P * so (Π) is an increasing function of γ C and λ E , while being a decreasing function of γ E and β co .
Remark 3. It is worth noting that although the jammer-related parameters λ J and PJ influence P co (Π) and P so (Π) (i.e., the COP and SOP performance of a path Π), as well as P SO-COP S k and P QO-SOP S k (i.e., the optimal transmission powers), they do not influence P * co (Π) and P * so (Π) (i.e., the optimal performance tradeoffs). This is due to the reason that the jammers in a WANET have effects on both intended receivers and eavesdroppers, and the effects on two sides will counteract with each other completely for the performance tradeoffs.
Routing Algorithm
In Section 3, we have derived the expressions of outage probabilities for a given path, and in Section 4, we have explored the optimal performance tradeoffs for a given path. Based on the obtained results, in this section, we further investigate the routing problem, i.e., for a pair of source and destination nodes with multiple optional end-to-end paths, how to select the optimal path to achieve the minimum COP under the security constraint or the minimum SOP under the QoS constraint.
Routing Algorithm for SO-COP
We first consider the routing algorithm for SO-COP. Based on Formula (17), the routing problem of finding the optimal path which achieves the minimum COP under the security constraint can be expressed as:
min Π∈S(Π) 1 − exp λ E π ln(1 − β so ) γ C γ E 2 α l k ∈Π d S k ,D k 2 ,(38)
where S(Π) denotes the set of all potential paths connecting the pair of source and destination nodes. Then (38) is equivalent to
min Π∈S(Π) l k ∈Π d S k ,D k .(39)
Expression (39) indicates that the routing problem for SO-COP is equivalent to finding the shortest path connecting the pair of source and destination nodes.
It means that we can assign the link weights d S k ,D k to each potential link l k and then find the path Π * with the minimum total link weights. This problem can be directly solved by applying the standard Dijkstra's algorithm [46], which returns the shortest paths from a source vertex to all other vertexes in a weighted graph.
After finding the shortest path Π * , the routing algorithm for SO-COP should conduct another key procedure to achieve the optimal COP with a guaranteed SOP, i.e., the transmission power allocation for each node on path Π * (except the destination) based on Formula (16). The details of the routing algorithm are summarized in Algorithm 1.
Routing Algorithm for QO-COP
We then consider the routing algorithm for QO-SOP. Based on Formula (30), the routing problem of finding the optimal path which achieves the minimum
1 − exp λ E π ln(1 − β co ) γ C γ E 2 α l k ∈Π d S k ,D k 2 ,(40)
Then (40) is equivalent to
min Π∈S(Π) l k ∈Π d S k ,D k .(41)
Expression (41) indicates that the routing problem for QO-SOP is also equivalent to finding the shortest path connecting the pair of source and destination nodes. Thus, we also apply the standard Dijkstra's algorithm to find the shortest path Π * , and then allocate the transmission power of each node on path Π * (except the destination) based on Formula (29). The details of the routing algorithm are summarized in Algorithm 2.
We can see from Algorithm 1 and Algorithm 2 that the optimal paths for SO-COP and QO-SOP are the same, the transmission powers, however, should Algorithm 2 The routing algorithm for QO-SOP.
Require: Network parameters {λ J , λ E , γ C , γ E , PJ , α} and QoS constraint β co ;
Ensure: The optimal path Π * for QO-SOP, the corresponding transmission power P QO-SOP S k , the achievable SOP P * so (Π * ); 1: Initialization (assign the value d S k ,D k to the link weights for any potential pair of transmitter S k and receiver D k ); 2: Find a shortest path in terms of the link weights between source node and destination node. The standard Dijkstra's algorithm can be used for this procedure; Remark 4. It is worth noting that the routing algorithms for SO-COP and QO-SOP can be easily implemented by the classical on-demand routing protocols, such as AODV [47]. In AODV routing protocol, the Dijkstra's algorithm for finding the shortest path can be implemented by sending route request (RREQ) messages from source node to destination node; while the procedure of transmission power allocation can be implemented by returning the route replay (RREP) message from destination node to source node.
Numerical Results and Discussions
In this section, we first present the Monte Carlo [48] simulation results to validate our theoretical analysis for the outage probabilities in a concerned multi-hop WANET, and then apply our theoretical results to illustrate the performance tradeoffs and the corresponding routing algorithm.
Simulation Settings
We simulate a multi-hop WANET in a 2000×2000 square area. The jammers (resp. eavesdroppers) are distributed at random positions which follow the homogeneous PPP with density λ J (resp. λ E ). Regarding the basic network parameters, we set PJ = 1, γ C = 1, γ E = 1 and α = 4. In each Monte
Carlo simulation for COP and SOP, we consider the example of a fixed path Π = l 1 , . . . , l 5 with five links, where the transmission power P S k and the distance d S k ,D k of each link are set to be the same, respectively. The duration of each task of Monte Carlo simulation is set to 10 7 rounds, and the simulated outage probability is given by
simulated outage probability = 100% × N o 10 7 ,(42)
where N o denotes the number of times that the event of outage occurs in each simulation.
Validation for COP and SOP
We first summarize in Fig. 1 results are obtained based on Formula (42). Similar to Fig. 1, Fig. 2 shows that the simulation results match well with the theoretical ones for all the cases, which indicates that our theoretical analysis is highly efficient in the evaluation of end-to-end SOP of multi-hop WANETs. We can also see from Fig. 2 that SOP increases (thus the transmission security degrades) monotonically as the eavesdropper's density λ E increases, while increasing the jammer's density λ J will lead to a decrease in SOP, indicating that the jammers can be utilized cooperatively to improve the security performance.
d S k ,D k =3 d S k ,D k =4 d S k ,D k =5
Performance Tradeoffs
We show in Fig. 3 the COP-SOP tradeoff with the variation of transmission power, where λ J = 10 −3 , λ E = 10 −3 , and we consider a path Π with five links (i.e., K = 5), each of which has the same distance and same power. For the points from left to right on each curve of Fig. 3 Fig. 3 is that for the same SOP (for example, P so = 50%), the minimum d S k ,D k can lead to the minimum COP (P co is 64%, 83% and 94% under d S k ,D k = {3, 4, 5}, respectively), which indicates that a shorter transmission distance can lead to a better performance tradeoff.
We summarize in Fig. 4 the performance of SO-COP and QO-SOP, where we set K = 5 and d S k ,D k = 5 for 1 ≤ k ≤ K. It is worth noting that the expressions of P * co (Π) and P * so (Π) are almost same, except that β co in (17) is replaced by β so in (30). Thus, we plot Fig. 4(a) to show how P * co (Π) varies with β so and how P * so (Π) varies with β co , simultaneously. We can see from Fig. 4(a) that as β so (resp. β co ) increases, which means the constraint on SOP (resp. COP) declines, P * co (Π) (resp. P * so (Π)) decreases monotonically. Another observation about Fig. 4(a) is that a big gap exists between the curves under λ E = 10 −4 and λ E = 10 −3 , which indicates the eavesdropper's density has a great impact on the network performance. Since we set the distance of each link on path Π is the same in Fig. 4, the corresponding transmission power of each link is also the same. To further illustrate the performance under the network scenario with different link lengths, we consider a path Π which consists of five links, the distance of each link is uniformly distributed on (1, 10). We set λ J = 10 −3 , λ E = 10 −4 , β so = 0.5 and β co = 0.5, and the results of one implementation are summarized in Table 1.
Routing Performance
To illustrate the routing algorithm for SO-COP and QO-SOP, we focus on a 20 × 20 square area and randomly place 20 legitimate nodes following the uniform distribution. We assign the node which is closest to the lower left corner as the source, and assign the node which is closest to the upper right corner as the destination. Notice that the eavesdroppers and jammers are still randomly distributed over the whole network area, and we set the densities as λ E = 10 −4 and λ J = 10 −3 . In order to ensure the end-to-end transmission is formed by multiple hops, we strategically set the maximal transmission range of a single hop as 8.
We plot in Fig. 5 a snapshot of the optimal path for SO-COP and QO-SOP.
For the snapshot of network scenario in Fig. 5, the optimal path Π * with the shortest path length is selected by executing the Dijkstra's algorithm. Based on the distance of each link on path Π * , our proposed routing algorithms allocate the transmission power for each link to achieve the optimal performance tradeoffs. Here we set both β so and β co as 0.4, then the optimal achievable COP and SOP, as well as the corresponding transmission powers are summarized in Table 2.
Conclusion
This paper studied the physical layer security-aware routing and the tradeoffs between communication QoS and transmission security in multi-hop wireless ad hoc networks. Considering a general network scenario where legitimate nodes and malicious eavesdroppers are randomly distributed following an independent homogeneous Poisson point process, we first derived the closed-form expressions of COP and SOP for a given path. Then, we analyzed the QoS-security tradeoffs to obtain the minimum achievable COP (resp. SOP) with a guaranteed SOP (resp. COP) and the corresponding strategies of power allocation for each transmitter on the path. With the help of theoretical analysis of a given path, we finally proposed the Dijkstra-based routing algorithms to find the optimal path between any pair of source and destination nodes which can achieve the optimal QoS-security tradeoffs.
E = {E i , i = 1, 2, . . .}, and the locations of eavesdroppers follow an independent homogeneous Poisson point process (PPP) with density λ E . Furthermore, due to the broadcast nature of wireless channels, simultaneous transmissions among legitimate nodes interfere with each other. Thus, when S k transmits a message to D k , the concurrent transmitting nodes should be regarded as jammers. Let Φ J = {J j , j = 1, 2, . . .} denote the set of jammers and we assume the corresponding locations also follow an independent homogeneous PPP with density λ J .
Algorithm 1
1The routing algorithm for SO-COP. Require: Network parameters {λ J , λ E , γ C , γ E , PJ , α} and security constraint β so ;Ensure: The optimal path Π * for SO-COP, the corresponding transmission power P SO-COP S k , the achievable COP P * co (Π * ); 1: Initialization (assign the value d S k ,D k to the link weights for any potential pair of transmitter S k and receiver D k ); 2: Find a shortest path in terms of the link weights between source node and destination node. The standard Dijkstra's algorithm can be used for this procedure;3: Assign the shortest path to Π * ; 4: Apply Formula (16) to allocate the corresponding transmission power P SO-COP S k for each transmitter on path Π * ; 5: Apply Formula (17) to calculate the secure-based optimal COP P * co (Π * ) of path Π * ; 6: return {Π * , P SO-COP S k , P * co (Π * )}; SOP under the QoS constraint can be expressed as: min Π∈S(Π)
3 :
3Assign the shortest path to Π * ; 4: Apply Formula (29) to allocate the corresponding transmission power P QO-SOP S k for each transmitter on path Π * ; 5: Apply Formula (30) to calculate the QoS-based optimal SOP P * so (Π * ) of path Π * ; 6: return {Π * , P QO-SOP S k , P * so (Π * )}; be allocated according to Formula (16) for SO-COP and Formula (29) for QO-SOP, respectively.
Figure 1 :
1the theoretical and simulation results of COP performance, where we set P S k = 1 and d S k ,D k = {3, 4, 5} for 1 ≤ k ≤ 5. The theoretical curves are plotted according to Formula (3) while the simulated results are obtained based on Formula (42). We can see from Fig. 1 that the simulation results match nicely with the theoretical ones for all the cases, which indicates that our theoretical analysis is highly efficient in the evaluation of end-to-end COP of multi-hop WANETs. Another observation of Fig. 1 is that as the jammer's density λ J and/or the transmission distance d S k ,D k increase, COP increases and thus the communication QoS is degraded. We then summarize in Fig. 2 the theoretical and simulation results of SOP performance, where we set P S k = 1 for 1 ≤ k ≤ 5, and λ J = {10 −3 , 10 −2 }. The theoretical curves are plotted according to Formula (8) while the simulated Performance validation of COP: Pco(Π) versus λ J under different settings of d S k ,D k . K = 5, P S k = 1 and d S k ,D k = {3, 4, 5} for 1 ≤ k ≤ K, PJ = 1, γ C = 1, α = 4.
Figure 2 :
2Performance validation of SOP: Pso(Π) versus λ E under different settings of λ J . K = 5, P S k = 1 for 1 ≤ k ≤ K, λ J = {10 −3 , 10 −2 }, PJ = 1, γ E = 1, α = 4.
Figure 3 :
3COP-SOP tradeoff with the variation of transmission power. K = 5, for the points from left to right on each curve, P S k takes the value from the set {0, 0.001, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2} sequentially, λ J = 10 −3 , λ E = 10 −3 , γ C = 1, γ E = 1, PJ = 1, α = 4.
Fig. 4
4(b) shows how the transmission power P SO-COP S k to achieve SO-COP varies with β so , and Fig. 4(c) shows how the transmission power P QO-SOP S k to achieve QO-SOP varies with β co . We can see that P SO-COP S k increases monotonically as β so increases, while P QO-SOP S k decreases monotonically as β co increases.Moreover,Fig. 4(b) indicates that to deal with the network scenario with denser eavesdroppers, for example, increasing λ E from 10 −4 to 10 −3 , we should diminish the transmission power; whileFig. 4(c)indicates that to deal with the network scenario with denser jammers, for example, increasing λ J from 10 −4 to 10 −3 , we should increase the transmission power.
SOP β so (Constraint on COP β co ) (%) Optimal COP P * co (Π) (Optimal SOP P * so (Π)) (%) λE = 10 −4 λ E = 10 −3(a) P * co (Π) versus βso (P * so (Π) versus βco).
Figure 4 :
4Performance of SO-COP and QO-SOP. K = 5, d S k ,D k = 5 for 1 ≤ k ≤ K, PJ = 1,γ C = 1, γ E = 1, α = 4.
Figure 5 :
5A snapshot of the optimal path for SO-COP and QO-SOP. The optimal path Π * connecting the source and destination are plotted by the black solid line with "⋆", and other legitimate nodes are plotted by red empty circles.
studied the scaling behaviors of WANETs under secrecy constraints. They demonstrated that under the path loss model, a secure rate of Ω( 1 √ n ) is achievable if the density of eavesdropper is below some threshold 1 ; while under the ergodic fading model, a constant secret rate can be achieved for sufficiently large n. For a two-hop relay WANET, Zou et al. [30] explored the cooperative-based relay selection schemes to improve transmission security against eavesdropping attack. Xie and Ulukus [31] considered the single-hop WANET with four fundamental wireless channels, and studied its secure degrees of freedom as well as provided the corresponding achievable schemes. For a detailed survey on physical layer security and its applications in WANETs, Saad et al. [34] proposed a tree-formation game to choose secure paths in multihop WANETs. Later, Sheikholeslami et al. [35] and Ghaderi et al. [36] explored the minimum energy routings which can guarantee the end-to-end outage probability and security for multi-hop WANETs, respectively. More recently, Yao et al. [37] studied the physical layer security-based routing in multi-hop WANETsplease kindly refer to [33] and references therein.
1.2. Motivation and Our Contributions
Although there have been extensive works for studying physical layer secu-
rity in wireless networks, they mainly focus on either the single-hop and two-hop
network scenarios, or the asymptotically large network scenarios, while the re-
search of physical layer security in multi-hop WANETs which fills the significant
gap between those two extremes is largely untouched and thus remains a tech-
nical challenge. By now, some initial results have been reported on the study of
physical layer security in multi-hop WANETs [34, 35, 36, 37, 38]. Specifically,
, the transmission power P S k takes the value from the set {0, 0.001, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2} sequentially. We can see from Fig. 3 that as P S k increases, SOP increases while COP decreases, indicating that the tradeoff between transmission security and communication QoS can be achieved by controlling the transmission power. A further careful observation of
Table 1 :
1the transmission powers of each link and the optimal outage probabilities.The k th link
1
2
3
4
5
d S k ,D k
3.5726 7.8148 7.7836 4.4240 6.1104
P SO-COP
S k
1.7147 8.2046 8.1391 2.6294 5.0160
P QO-SOP
S k
0.5708 2.7314 2.7097 0.8754 1.6699
P *
co (Π), P *
so (Π)
0.3269
Table 2 :
2link length, transmission powers and optimal outage probabilities of the optimal path Π * .The k th link
1
2
3
4
5
d S k ,D k
6.6027 4.6456 5.9676 4.7477 5.3562
P SO-COP
S k
3.7608 1.8617 3.0721 1.9444 2.4748
P QO-SOP
S k
3.0366 1.5033 2.4806 1.5700 1.9983
P *
co (Π * ), P *
so (Π * )
0.3681
n is the number of source-destination pairs and please kindly refer to[32] for the asymptotic notations.
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| [] |
[
"Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians",
"Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians"
] | [
"Eric A Galapon \nTheoretical Physics Group\nNational Institute of Physics\nUniversity of the Philippines Diliman\n1101Quezon City, Philippines\n"
] | [
"Theoretical Physics Group\nNational Institute of Physics\nUniversity of the Philippines Diliman\n1101Quezon City, Philippines"
] | [] | We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. | 10.1098/rspa.2002.0992 | [
"https://export.arxiv.org/pdf/quant-ph/0111061v2.pdf"
] | 14,335,343 | quant-ph/0111061 | d8a151289039ab9ca1c0f4fb9a24c462cd89cea6 |
Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians
Apr 2002
Eric A Galapon
Theoretical Physics Group
National Institute of Physics
University of the Philippines Diliman
1101Quezon City, Philippines
Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians
Apr 2002arXiv:quant-ph/0111061v2 5Time operatorsQuantum canonical pairsPauli's theorem
We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace.
Introduction
Does a self-adjoint operator canonically conjugate with a semibounded Hamiltonian exist? This operator, if it exists, has been referred to as time operator. The general concensus is that no such operator exists (Toller 1997(Toller , 1999Pegg 1998;Atmanspacher & Amann 1998;Giannitrapani 1997;Eisenberg & Horwitz 1997;Delgado & Muga 1997;Blanchard & Jadczyk 1996;Omnes 1994;Holland 1993;Park 1984;Srinivas & Vijayalakshmi 1981;Holevo 1978;Cohen-Tannoudji 1977;Jammer 1974;Olhovsky & Recami 1974;Rosenbaum 1969;Gotfried 1966;Pauli 1926Pauli , 1933Pauli , 1958. This pessimism traces back to the well-known theorem of Pauli (Pauli 1926(Pauli , 1933(Pauli , 1958 which asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the Hamiltonian has an absolutely continous spectrum filling the entire real line. Thus for the generally semibounded and discrete Hamiltonian of quantum mechanics, Pauli's theorem excludes the possibility of developing a quantum theory of time via quantum operators. This conclusion has been corroborated by succeeding attempts to introduce time operators, particularly for the free particle in the real line where quantization of the classical arrival or passage times has led to maximally symmetric, non-self-adjoint time opertor (Muga & Leavens 2000;Egusquiza & Muga 1999;Delgado & Muga 1997;Grot et al 1996;Alcock 1969a,b;Fick & Engelmann 1963, 1964Paul 1962). Thus it has been tacitly assumed that if one attempts to introduce time in standard quantum mechanics (SQM) as an operator canonically conjugate to a semibounded Hamiltonian, discrete or not, one has to expect that the time operator to be generally maximally symmetric without any self-adjoint extension (Jammer 1974). Thus it has been the current thinking that time operators will generally be meaningful only when the axioms of SQM are modified to include POVM-observables, in which case time is a POVM-observable (Egusquiza & Muga 1999;Toller 1999Toller , 1997Giannitrapani 1997;Busch et al 1995aBusch et al ,b, 1994Srinivas & Vijayalakshmi 1981;Holevo 1978;Helstrom 1970;Toller 1999Toller , 1997.
However, in a recent publication, we have explicitly demonstrated that Pauli's theorem does not hold within SQM, and there is no a priori reason to exclude the existence of self-adjoint time operators canonically conjugate to a semibounded Hamiltonian (Galapon 2002). For this reason, it is imperative to look back and investigate the existence of self-adjoint time operators for quantum mechanical systems. In this paper, we prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. By characteristic we mean that the operator is parameter free and is solely constructed from the spectral decomposition of the Hamiltonian, i.e. from the Hamiltonian eigenvectors and eigenvalues alone. Moreover, we will show that each characteristic self-adjoint time operator generates a class of uncountably many self-adjoint time operators canonically conjugate with the same Hamiltonian. Incidentally our results belie earlier claims that no self-adjoint time operators exists for discrete Hamiltonian systems (Pegg 1998;Canata and Ferrari 1991a,b;Jordan 1927).
Our method of proof will follow that of the physicist's intuition: We formally construct an operator with a dimension of time out of the spectral resolution of the Hamiltonian, then show that, under some mild conditions, it can be assigned a dense subspace to lift its formality, then show that it is canonically conjugate with the Hamiltonian in a dense subspace of the Hilbert space, then finally show that the operator in its assigned domain is essentially self-adjoint-thus with a uniquely associated self-adjoint operator.
Characteristic Time Operators for Non-Degenerate Hamiltonians
Let H 1 be a non-degenerate Hamiltonian whose orthonormal eigenkets are | s , and corresponding eigenvalues, ordered in increasing size, are E s , for all s = 1, 2, . . . . We assume that the Hilbert space of the system corresponding to the given Hamiltonian is spanned by the eigenkets of H 1 , i.e.
H 1 = | ϕ = ∞ s=1 ϕ s | s , ∞ s=1 |ϕ s | 2 < ∞ , (2.1) where H 1 is equiped with the standard norm | ϕ = ∞ s=1 |ϕ s | 2 derived from the standard inner product ψ| φ = ∞ s=1 ψ s φ s .
Under this representation, the Hamiltonian H 1 with domain D(H 1 ) is explicitly given by
H 1 = ∞ s=1 E s | s s | ,(2.
2)
To appear in the Proceedings of the Royal Society of London A
D(H 1 ) = ϕ = ∞ s=1 ϕ s | s ∈ H, ∞ s=1 E 2 s |ϕ s | 2 < ∞ . (2.3)
Given the eigenkets | s 's and the eigenvalues E s 's, we construct the following formal symmetric operator,
T 1 = ∞ s,s ′ ≥1 ′ i ω s,s ′ | s s ′ | , (2.4)
where ω s,s ′ = (E s − E s ′ )/ , and the prime indicates a double summation with the s = s ′ contribution excluded from the sum. We note that T 1 has a dimension of time and it is only constructed out of the spectral decomposition of the Hamiltonian without the need of introducing any parameter. In the following we show that if the eigenvalues of the Hamiltonian satisfies the condition
∞ s=1 1 E 2 s < ∞, (2.5)
then the formal operator T 1 can be assigned a dense subspace in H 1 , and it is canonically conjugate with the Hamiltonian in some dense subspace of H 1 , and it is essentially self-adjoint in its assigned domain. For this reason we call T 1 as the characteristic time operator for the non-degenerate Hamiltonian.
(a) T 1 is Densely Definable
Now we prove that if condition (2.5) is satisfied, then T 1 can be assigned the following dense subspace
D 1 = | ϕ = N s=1 ϕ s | s , ϕ s ∈ C, N < ∞ ,
(2.6) as its domain. The subspace D 1 is dense because for every | ψ in H 1 there exists a sequence of vectors in D 1 which converges to | ψ . In particular the sequence of vectors | φ k = k s=1 s| ψ | s , k = 1, 2, . . . in D 1 converges to | ψ . We first show that the formal time operator T 1 is defined in the entire D 1 . That means for every | ϕ in D 1 the vector T 1 | ϕ belongs in the Hilbert space H 1 . Let | ϕ be in D 1 , then
T 1 | ϕ = ∞ s=1 N s ′ =s iϕ s ′ ω s,s ′ | s .
(2.7)
T 1 | ϕ belongs to the Hilbert space H if and only if T 1 | ϕ < ∞, or equivalently,
T 1 | ϕ 2 = ∞ s=1 N s ′ =s ϕ s ′ ω s,s ′ 2 < ∞. (2.8)
To appear in the Proceedings of the Royal Society of London A Let us divide the sum in two parts,
1 2 ∞ s=1 N s ′ =s ϕ s ′ ω s,s ′ 2 = N s=1 N s ′ =s ϕ s ′ (E s − E s ′ ) 2 + ∞ s=N +1 N s ′ =1 ϕ s ′ (E s − E s ′ ) 2 . (2.9)
The first term in equation (2.9) is already finite for finite N so that we need only to show that the second term is finite.
Using the triangle inequality, we get the bound,
N s ′ =1 ϕ s ′ E s − E s ′ ≤ 1 E s N s ′ =1 |ϕ s ′ | 1 − E s ′ /E s (2.10) since E s − E s ′ > 0 for all s > s ′ . We note that (1 − E s ′ /E s ) −1 is bounded within the range N + 1 ≤ s ≤ ∞ for every 1 ≤ s ′ ≤ N . Thus there exists a finite positive constant A N depending on N alone such that N s ′ =1 |ϕ s ′ | 1 − E s ′ /E s < A N · N s ′ =1 |ϕ ′ s | . (2.11)
One such constant is given by
A N = max 1≤s ′ ≤N N +1≤s≤∞ 1 (1 − E s ′ /E s ) < ∞.
(2.12)
Thus we finally arrive at the upperbound
∞ s=N +1 N s ′ =1 ϕ s ′ E s − E s ′ 2 ≤ A 2 N · N s ′ =1 |ϕ s ′ | 2 · ∞ s=N +1 1 E 2 s < ∞ (2.13)
which is finite as the sum ∞ s=1 E −2 s is assumed finite. Since | ϕ is an arbitrary element of D 1 , the bound (2.13) holds for all vectors in D 1 . Therefore T 1 | ϕ belongs to H for all | ϕ in D 1 . Since T 1 is defined in D 1 and D 1 is dense, we define T 1 to be the densely defined operator T 1 : D(T 1 ) ⊆ H → H where D(T 1 ) = D 1 . The formality of T 1 has thus been lifted.
(b) H 1 and T 1 are Canonically Conjugate Now we claim that the pair of operators H 1 and T 1 form a canonical pair in a dense subspace D 1
c of D(H 1 T 1 ) ∩ D(T 1 H 1 ), i.e. (T 1 H 1 − H 1 T 1 ) | ϕ = i | ϕ for all | ϕ in D c . Let us assume for the moment that D(H 1 T 1 ) ∩ D(T 1 H 1 ) is not empty. Then if | ϕ is in D(H 1 T 1 ) ∩ D(T 1 H 1 ), we have (T 1 H 1 − H 1 T 1 ) | ϕ = −i ∞ s=1 N s ′ =s ϕ s ′ | s .
(2.14)
Now if N s=1 ϕ s = 0, (2.15) then N s ′ =s ϕ s ′ = −ϕ s . In which case equation (2.14) reduces to
(T 1 H 1 − H 1 T 1 ) | ϕ = i | ϕ . (2.16)
That is T 1 and H 1 satisfy the canonical commutation relation if and only if there is a subspace of D(H 1 T 1 ) ∩ D(T 1 H 1 ) satisfying equation (2.15). We then have to identify a dense subspace of D(T 1 ) satisfying condition (2.15) and at the same time belonging to D(H 1 T 1 ) ∩ D(T 1 H 1 ). Now we show that the vectors of the following proper subspace of D(T 1 ) satisfy all these requirements,
D 1 c = | ϕ = L−1 j=1 L i=j+1 a i,j (| i − | j ), a i,j ∈ C, for finite even L > 1 , (2.17)
First let's demonstrate that the vectors in D 1 c satisfy equation (2.15). Expanded in the basis | s , the vectors | ϕ in D 1 c assume the form,
| ϕ = L s=1 ϕ s | s = L s=1 s−1 k=1 a s,k − L k=s+1 a k,s | s , (2.18)
provided we set a i,j = 0 when either i or j is outside of its respective range, 1 ≤ j ≤ (L − 1), (j + 1) ≤ i ≤ L; or when both are outside of their ranges; or when i ≤ j. Taking the sum of the coefficients, we have
L s=1 ϕ s = L s=2 s−1 k=1 a s,k − L−1 s=1 L k=s+1 a k,s = 0.
The sum vanishes because rearrangement of the first term in the second line leads to the equality L s=2
s−1 k=1 a s,k = L−1 s=1 L k=s+1 a k,s . Now D 1 c is dense. Let us assume otherwise. Then there exists a vector | ψ in H 1 with | ψ = 0 such that ϕ| ψ = 0 for all | ϕ in D 1 c . Now since | i, j = (| i − | j ) is in D 1
c for every pair i = j, we must have i, j| ψ = 0 for all such pairs of i and j. This implies that ψ i − ψ j = 0 or ψ i = ψ j for all pairs of i and j. This means that if ψ j = 0 for some fixed j, then the rest of the coefficients will have to be zero, which implies that | ψ = 0, contrary to the assumption that | ψ = 0. On the other hand if ψ j = c for some non-vanishing comlex number c and for some j, then ψ i = c for all i; but then | ψ = c ∞ s=1 | s which does not belong to H 1 , contrary to the assumption that | ψ is in H 1 . Thus if i, j| ψ = 0 for all i and j for some | ψ in
H 1 , then | ψ = 0. Thus D 1 c is dense. Now we show that D 1 c is a subspace of D(H 1 T 1 ) ∩ D(T 1 H 1 ). Since D 1 c is a proper subspace of D(T 1 ) and D(T 1 ) is invariant under H 1 (that means H 1 : D 1 c is a subspace of D(T 1 )), T 1 H 1 is defined in the entire D 1 c . On the other hand, D(T 1 ) is not invariant under T 1 so that it is not necessary that T 1 | ϕ is in D(H 1 ) for all | ϕ in D 1 c . Now D(H 1 T 1 ) consists of those | φ in D(T 1 ) such that T 1 | φ is in D(H 1 ). Specifically the domain consists of those | φ in D(T 1 ) such that ∞ s=1 E 2 s N s ′ =s ϕ s ′ ω s,s ′ 2 < ∞.
(
2.19)
To appear in the Proceedings of the Royal Society of London A To find the suitable condition to determine the domain of H 1 T 1 , we split the sum in two parts,
1 2 ∞ s=1 E 2 s N s ′ =s ϕ s ′ ω s,s ′ 2 = N s=1 E 2 s N s ′ =s ϕ s ′ (E s − E s ′ ) 2 + ∞ s=N +1 E 2 s N s ′ =1 ϕ s ′ (E s − E s ′ ) 2 .
(2.20)
For finite N the first term is already finite, so again we need only to concern ourselves with the second term. If the | ϕ 's satisfy the bound
N s ′ =1 ϕ s ′ E s − E s ′ ≤ C E 2 s , (2.21)
for some constant C independent of s, then | ϕ belongs to D(H 1 T 1 ). It is sufficient then to show that the vectors in D 1
c satisfy (2.21) to establish that D 1 c is a proper dense subspace of D(H 1 T 1 ). Now L s ′ =1 ϕ s ′ E s − E s ′ = L s ′ =1 1 (E s − E s ′ ) s ′ −1 k=1 a s ′ ,k − L s ′ +1 a k,s ′ = 1 E 2 s L−1 s ′ =1 L k=s ′ +1 a k,s ′ (E k − E s ′ ) (1 − E k /E s )(1 − E s ′ /E s ) ,
where the second line follows from a rearrangement of the first term. The triangle inequality then dictates that
L s ′ =1 ϕ s ′ E s − E s ′ ≤ 1 E 2 s L−1 s ′ =1 L k=s ′ +1 |a k,s ′ | (E k − E s ′ ) (1 − E k /E s )(1 − E s ′ /E s ) . (2.22) Again (1 − E s ′ /E s ) −1 is bounded within the range (L + 1) ≤ s ≤ ∞ for every 1 ≤ s ′ ≤ L.
Thus there exists a finite positive constant B L depending on L alone such that
L−1 s ′ =1 L k=s ′ +1 |a k,s ′ | (E k − E s ′ ) (1 − E k /E s )(1 − E s ′ /E s ) < B L L−1 s ′ =1 L k=s ′ +1 |a k,s ′ | .
(2.23)
One such B L is explicitly given by
B L = max 1≤s ′ ≤(L−1), (s ′ +1)≤k≤L (L+1)≤s≤∞ (E k − E s ′ ) (1 − E k /E s )(1 − E s ′ /E s ) < ∞. (2.24)
Thus we get the required bound of inequality (2.21)
L s ′ =1 ϕ s ′ E s − E s ′ ≤ 1 E 2 s · B L · L−1 s ′ =1 L k=s ′ +1 |a k,s ′ | , (2.25) which implies that D 1 c is a subspace of D(H 1 T 1 ) ∩ D(T 1 H 1 ).
The operators H 1 and T 1 are then canonically conjugate in the dense subspace D 1 c .
To appear in the Proceedings of the Royal Society of London A
(c) T 1 is Essentially Self-adjoint and It Generates a Class of Essentially Self-adjoint Time Operators
Now we show that T 1 is essentially self-adjoint. Let T 1 * be the adjoint of T 1 . Simon 1972). However, it is sufficient to show that T 1 and its adjoint T 1 * are symmetric. This is so because the symmetry of T 1 , with the fact that T 1 is densely defined, assures the existence of a unique nontrivial adjoint of T 1 , T 1 * ; on the other hand, the symmetry of T 1 * dictates that T 1 * has real eigenvalues only. If T 1 * is symmetric, and if | φ = 0 is in D(T * 1 ) and (T 1 * ± i I) | φ = 0, then | φ is an eigenvector of T 1 * with the eigenvalues ±i which is a contradiction with the assumption that T 1 * is symmetric.
Then T 1 in D(T 1 ) is essentially self-adjoint if | φ is in D(T * 1 ) and (T 1 * ± i I) | φ = 0 imply that | φ = 0 (Reed &First we show that T 1 is symmetric in its assigned domain D(T 1 ). T 1 is symmet- ric if ψ| T 1 ϕ = T 1 ψ| ϕ for all | ϕ , | ψ in D(T 1 ). Let | ϕ = N s=1 ϕ s | s , | ψ = L s=1 ψ s | s in D(T 1 ). Now ψ| T 1 ϕ = L s=1 ψ * s N s ′ =s iϕ s ′ ω s,s ′ = N s ′ =1 L s =s ′ iψ s ω s ′ ,s * ϕ s ′ = T 1 ψ| ϕ
where the rearrangement of the summations are possible because they have finite limits. Thus T 1 is symmetric in its assigned domain. Now let us determine the adjoint, T 1 * , of T 1 . Since T 1 is densely defined and symmetric, it has a unique non-trivial adjoint. The domain D(T * 1 ) of T 1 * consists of those vectors | ψ in H 1 such that there exists a vector | ψ * in H 1 satisfying the condition ψ|
T 1 ϕ = ψ * | ϕ for all | ϕ in D(T 1 ). Let | ψ = ∞ s=1 ψ s | s be in H 1 . Then for all | ϕ in D(T 1 ), ψ| T 1 ϕ = ∞ s=1 ψ s L s ′ =s iϕ s ′ ω s,s ′ = L s ′ =1 ∞ s =s ′ iψ s ω s,s ′ ϕ s ′ = L s ′ =1 ∞ s =s ′ iψ s ω s ′ ,s ϕ s ′ .
(2.26)
Since L is finite, the interchanging of the order of summation in the second line is allowed.
With ψ s ′ = ∞ s ′ =s i ω −1 s,s ′ ψ s ′ , we can rewrite equation (2.26) in the form ψ| T 1 ϕ = L s ′ =1 ψ * s ′ ϕ s ′ ,
To appear in the Proceedings of the Royal Society of London A from which we identify the vector
| ψ * = ∞ s=1 ∞ s ′ =s i ω s,s ′ ψ s ′ | s (2.27)
to satisfy the relation ψ| T 1 ϕ = ψ * | ϕ for all | ϕ in D(T 1 ). Now | ψ is in the domain of T 1 * if and only if | ψ * is in the Hilbert space. Thus the domain D(T * 1 ) of T * 1 must comprise the following subspace of H 1 ,
D(T * 1 ) = | ψ = ∞ s=1 ψ s | s ∈ H 1 , ∞ s=1 ∞ s ′ =s ψ s ′ ω s,s ′ 2 < ∞ .
(2.28)
Finally the adjoint of T 1 is uniquely determined by the definition of the adjoint, | ψ * = T * 1 | ψ . Equation (2.27) then yields the adjoint
T * 1 = ∞ s,s ′ ≥1 ′ i ω s,s | s s ′ | .
(2.29)
As expected from the symmetry of T 1 , we have the extension relation T 1 ⊂ T 1 * . (It may be possible that D(T * 1 ) = D(T 1 ) already for some systems, in which case T 1 is immediately self-adjoint.)
To complete the proof that T 1 is essentially self-adjoint, we now show that its adjoint T * 1 is symmetric. It is sufficient to show that for every | ψ in D(T * 1 ) the
number ψ| T * 1 ψ is real valued. Let | ψ be in D(T * 1 ), then ψ| T * 1 ψ = ∞ s=1 ψ s ∞ s ′ =s iψ s ′ ω s,s ′ . (2.30)
We note that the double sum in equation (2.30) is absolutely convergent. This follows from the fact that the first summation is already absolutely convergent, | T * 1 ψ being uniquely determined by | ψ , and
∞ s=1 ψ s ∞ s ′ =s ψ s ′ ω s,s ′ ≤ ∞ s=1 |ψ s | 2 · ∞ s=1 ∞ s ′ =s ψ s ′ ω s,s ′ 2 < ∞.
It is well known that if a double sum has been proven to be absolutely convergent for any mode of summation, it will be absolutely convergent for all modes of summation, and the sum of the series is independent of summation (Hardy et al 1952).
Thus we can interchange the order of summation in the right hand side of equation (2.30) to give
ψ| T * 1 ψ = ∞ s ′ =1 ψ s ′ ∞ s =s ′ iψ s ω s,s ′ = ∞ s ′ =1 ψ s ′ ∞ s =s ′ iψ s ω s ′ ,s = ψ| T * 1 ψ .
To appear in the Proceedings of the Royal Society of London A Thus T * 1 is symmetric. And T 1 is consequently essentially self-adjoint. Now we show that T 1 generates a class of uncountable essentially self-adjoint time operators conjugate to the same Hamiltonian H 1 . Let α = {α s , s = 1, 2, . . . } be a bounded sequence of real numbers, i.e. |α s | ≤ A < ∞ for all s; these sequences may satisfy some other properties such as ∞ s=1 α 2 s < ∞, or α s = τ for all s for some real τ . Then the operator
T 1,α = T 1 + ∞ s=1 α s | s s | (2.31)
is essentially self-adjoint in D(T 1 ). Moreover, T 1,α is canonically conjugate with H 1 in D 1 c . To prove our assertion, let ∆T 1 = ∞ s=1 | s α s s |. Obviously ∆T 1 is symmetric in D(T 1 ). Thus T 1,α is essentially self-adjoint in D(T 1 ) if there exists some constants p 1 , p 2 ≥ 0 such that ∆T 1 | ϕ 2 ≤ p 1 ϕ| T 1 ϕ + p 2 | ϕ 2 for all | ϕ in D(T 1 ) (Hellwig 1964). For bounded α s , the operator ∆T 1 is bounded; this follows from the fact that ∆T 1 | φ ≤ A | φ . Thus there exists some p 1 , p 2 ≥ 0 such that ∆T 1 | ϕ 2 ≤ ∆T 1 2 | ϕ 2 ≤ p 1 ϕ| T 1 ϕ + p 2 | ϕ 2 , for all | ϕ in D(T 1 ); a pair of such p 1 and p 2 is p 1 = 0 and p 2 = ∆T 1 . Therefore T 1,α is essentially self-adjoint in D(T 1 ). That T 1,α and H 1 are canonical in D 1 c follows from the fact that T 1 are canonical in D 1 c , and ∆T 1 and H 1 commute in D 1 c . We note that the above construction of ∆T 1 does not necessarilly exhaust all possibilities. However, we emphasize that not all symmetric operators in D(T 1 ) (not even those that are essentially self-adjoint or self-adjoint in D(T 1 )) which commute with the Hamiltonian in D 1 c can be added to T 1 to give an essentially self-adjoint time operator in D(T 1 ). If T 1 is bounded and if ∆T 1 is unbounded in D(T 1 ), then no constants p 1 , p 2 ≥ 0 can satisfy ∆T 1 | ϕ 2 ≤ p 1 ϕ| T 1 ϕ + p 2 | ϕ 2 for all | ϕ in D(T 1 ) because for every p 1 and p 2 there will always be a | ψ in D(T 1 ) such that ∆T 1 | ψ 2 ≥ p 1 ψ| T 1 ψ + p 2 | ψ 2 , ∆T 1 being unbounded and the right hand side of the inequality being always bounded for bounded T 1 .
Characteristic Time Operators for M-Degenerate Hamiltonians
Now let us consider the case when the Hamiltonian has a constant degeneracy M , i.e. to every energy eigenvalue E s corresponds to M linearly independent and orthonormal eigenvectors, | s, r , where s, r| s ′ , r ′ = δ ss ′ δ rr ′ , in which r = 1, , . . . , M and s = 1, 2, . . . . We assume that the Hilbert space is spanned by these eigenvectors,
H M = | ϕ = ∞ s=1 M r=1 ϕ s,r | s, r , ∞ n=1 M r=1 |ϕ s,r | 2 < ∞ (3.1)
and H M is equipped with the standard norm | ϕ =
H M = ∞ s=1 M r=1 E s | s, r r, s | , (3.2) D(H M ) = | ϕ = ∞ s=1 M r=1 ϕ s,r | s, r ∈ H, ∞ s=1 M r=1 E 2 s |ϕ s,r | 2 < ∞ . (3.3)
Now given the eigenvectors | s, r and eigenvalues E s of the Hamiltonian, we construct the following formal symmetric operator,
T M = ∞ s,s ′ ≥1 ′ M r,r ′ ≥1 ′ i ω s,s ′ | s, r r ′ , s ′ | ,(3.4)
where ω s,s ′ = (E s − E s ′ )/ , and again the primes indicate that s = s ′ and r = r ′ are excluded from the summation. Similarly, T M has a dimension of time and it is solely constructed out of the spectral decomposition of the Hamiltonian alone. Similarly we will show below that if the eigenvalues of the Hamiltonian satisfies the condition converges to | ψ . We first show that the formal time operator T M is defined in the entire D M ; that is,
∞ s=1 1 E 2 s < ∞,(3.T M | ϕ is in H M for all | ϕ in D M . Let | ϕ be in D M , then T M | ϕ = ∞ s=1 M r=1 N s ′ =s M r ′ =r iϕ s ′ ,r ′ ω s,s ′ | s, r .
(3.7)
T M | ϕ lies in the Hilbert space H if and only if T M | ϕ < ∞, or equivalently,
T M | ϕ 2 = ∞ s=1 M r=1 N s ′ =s M r ′ =r ϕ s ′ ,r ′ ω s,s ′ 2 < ∞. (3.8)
To appear in the Proceedings of the Royal Society of London A We divide the sum in two parts,
1 2 ∞ s=1 M r=1 N s ′ =s M r ′ =r ϕ s ′ ω s,s ′ 2 = N s=1 M r=1 N s ′ =s M r ′ =r ϕ s ′ ,r ′ (E s − E s ′ ) 2 + ∞ s=N +1 M r=1 N s ′ =1 M r ′ =r ϕ s ′ ,r ′ (E s − E s ′ ) 2 . (3.9)
For finite N the first term is already finite, so we need only to show that the second term is finite.
Again appealing to the boundedness of (1−E s ′ /E s ) −1 within the range (N +1) ≤ s ≤ ∞ for all s ′ < N , we have the following bound,
M r=1 N s ′ =1 M r ′ =r ϕ s ′ ,r ′ (E s − E s ′ ) 2 ≤ M r=1 N s ′ =1 M r ′ =r |ϕ s ′ ,r ′ | (E s − E s ′ ) 2 ≤ 1 E 2 s · A 2 N · M r=1 N s ′ =1 M r ′ =r |ϕ s ′ ,r ′ | 2 .
(3.10)
The second term then takes the bound
∞ s=N +1 M r=1 N s ′ =1 M r ′ =r ϕ s ′ ,r ′ (E s − E s ′ ) 2 ≤ A 2 N · M r=1 N s ′ =1 M r ′ =r |ϕ s ′ ,r ′ | 2 · ∞ s=N +1 1 E 2 s ,(3.(T M H M − H M T M ) | ϕ = −i ∞ s=1 M r=1 L s ′ =s M r ′ =r ϕ s ′ ,r ′ | s, r .D M c = | ξ = L−1 j=1 L i=j+1 M k=1 a i,j,k (| i, k − | j, k ) , a i,| ξ in D M c . Since | i, j, k = (| i, k − | j, k ) is in D M
c for all pair of i and j, and for all k = 1, . . . M . Then i, j, k| ψ = 0 implies ψ i,k − ψ j,k = 0 or ψ i,k = ψ j,k . Let us say that for all k there exists some j (not necessarilly the same j for each k) such that ψ j,k = 0, which implies that | ψ = 0, a contradiction with the assumption that | ψ = 0. If on the other hand ψ j,k is equal to some finite constant c k for some j for a particular k, then ψ i,k = c k for all i; but for this case, the vector will be given by | ψ =
T M | φ 2 = ∞ s=1 E 2 s M r=1 L s ′ =s M r ′ =r φ s ′ ,r ′ ω s,s ′ 2 < ∞.
(3.17)
To appear in the Proceedings of the Royal Society of London A Let us divide equation (3.17) in two parts,
1 2 ∞ s=1 E 2 s M r=1 L s ′ =s M r ′ =r φ s ′ ,r ′ ω s,s ′ 2 = L s=1 E 2 s M r=1 L s ′ =s M r ′ =r φ s ′ ,r ′ (E s − E s ′ ) 2 + ∞ s=L+1 E 2 s M r=1 L s ′ =1 M r ′ =r φ s ′ ,r ′ (E s − E s ′ ) 2 .
(3.18)
The first term is already finite for finite L so that equation (3.17) is satisfied as long as the second term is finite.
Now all of D M c is in D(H M T M ) ∩ D(T M H M ) if L s ′ =1 M r ′ =r φ s ′ ,r ′ (E s − E s ′ ) ≤ D E 2 s (3.19)
for some positive finite D independent of s. Now we show that the vectors | φ in D M c satify this condition. For a given ρ,
we have L s ′ =1 φ s ′ ,ρ E s − E s ′ = L−1 i=1 L j=i+1 a j,i,ρ (E j − E i ) (E s − E i )(E s − E j ) (3.20)
for s > L, where we have used the same method of rearrangement to arrive at equation (3.20). Again we appeal to the boundedness of (1 − E s ′ /E s ) −1 within the indicated range to get the bound
L s ′ =1 φ s ′ ,ρ E s − E s ′ ≤ 1 E 2 s L−1 i=1 N j=i+1 |a j,i,ρ | (E j − E i ) (1 − E i /E s )(1 − E j /E s ) ≤ 1 E 2 s · B L · L−1 i=1 N j=i+1 |a j,i,ρ | .
(c) T M is Essentially Self-adjoint and It Generates a Class of Essentially Self-adjoint Time Operators
Finally we prove that T M is essentially self-adjoint in its assinged subspace.
First we show that T M is symmetric in D. Let | ϕ = N s=1 M r=1 ϕ s,r | s, r and | φ = L s=1 M r=1 φ s,r | s, r , then ψ| T M ϕ = L s=1 M r=1 ψ * s,r N s ′ =s M r =r ′ iϕ s ′ ,r ′ ω s,s ′ = N s ′ =1 M r ′ =1 N s =s ′ M r =r ′ iψ s ′ ,r ′ ω s ′ ,s * ϕ s,r = T M ψ| ϕ ,
where the rearrangement is possible because the limits of summations are finite. Thus T M is symmetric.
Since T M is densely defined and symmetric, it is assured that it has a unique adjoint T * M . Let | φ be in D(T * M ). Then for all | ϕ in D(T M ),
φ| T M ϕ = ∞ s=1 M r=1 φ s,r L s ′ =s M r ′ =r iϕ s ′ ,r ′ ω s,s ′ = N s ′ =1 M r ′ =1 ∞ s =s ′ M r =r ′ iφ s,r ω s,s ′ ϕ s ′ r ′ = φ * | ϕ where | φ * = ∞ s=1 M r=1 ∞ s ′ =s M r ′ =r iφ s ′ ,r ′ ω s,s ′
for every σ and ρ. Since | η is asserted to exist, the summation in (3.26) must converge absolutely for all σ and for every ρ, including the limit σ → ∞. Now for two different values of ρ, say k and l, we get the two coupled equations,
T ρ 1 = ∞ s,s ′ ≥1 ′ i ω s,s ′ | s, ρ s ′ , ρ | , D(T ρ 1 ) = | ψ = N s=1 ψ s | s, ρ , N < ∞ .
T ρ 1 has all the properties as those considered in the non-degenerate case; and it is, most importantly, essentially self-adjoint. Thus there exists no | φ = ∞ s=1 φ s | s, ρ in D(T ρ 1 * ) different from the zero vector such that (T ρ M * ± i I) | φ = 0. Had such a vector existed, then its coefficients φ s 's, not all vanishing, would have satisfied the equation
∞ s =σ i ω s,σ φ s = ±i φ σ . (3.30)
However, if T M is not essentially self-adjoint then it is possible that η s,k − η s,l = 0 for some s, which implies that (3.30) has a non-trivial solution since we can choose φ s = η s,k − η s,l , owing from the fact that ∞ σ=1 |(η σ,k − η σ,l )| 2 < ∞. But this is in contradiction with the essential self-adjointness of T 1 ρ . Therefore in order to maintain the equality in equation (3.29) and avoid any contradiction, we must have η s,k = η s,l for all k and l for every s, i.e. η s,1 = η s,2 = · · · = η s,M . Then for all ρ, equation (3.26) reduces to
∞ s =σ i(M − 1) ω s,σ η s,ρ = ±i η σ,ρ .
(3.31) But equation (3.31) is equivalent to equation (3.30) only that the Hamiltonian is scaled to H ρ 1 /(M − 1). Again because | η belongs to the Hilbert space which implies ∞ s=1 |η s,ρ | 2 < ∞, the assertion that T M is not essentially self-adjoint implies that the charateristic time operator corresponding to (3.31) is not essentially self-adjoint, which is a contradiction with our earlier result for non-degenerate Hamiltonians. Thus we must necessarilly have η s,r = 0 for all s and r, implying that | η is the zero vector, contrary to the assumption that it is otherwise. Therefore T M must be essentially self-adjoint in its assigned domain.
As in the non-degenerate case, the characteristic time operator T M generates a class of uncountable essentially self-adjoint operator canonically conjugate with the same Hamilotnian H M . Now let α = {α s,r , s = 1, 2, . . . , 1 ≤ r ≤ M } be a bounded sequence of real numbers in s for every r, i.e. |α s,r | < A < ∞ for all s and r. Then the operator
T M,α = T M + ∞ s=1 M r=1 α s,r | s, r s, r | (3.32)
is essentially self-adjoint in D(T M ). Moreover, T M,α is canonically conjugate with H M in D M c . We can follow the same line of proof as the one we used in the nondegenerate case to prove our assertation.
We note that the argument found in (Pegg 1998) and (Jordan 1927) on the non-existence of self-adjoint time operators for discrete Hamiltonians assumes that the eigenvectors of the Hamiltonian belong to the domain D M c for all M ≥ 1. The eigenvectors, however, lie outside of D M c , so that no contradiction arises.
Conclusion
The One might notice that the above theorem implicitly assumes that the Hamiltonian has no zero eigenvalue. But the above theorem can be extended without difficulty to Hamiltonians with zero eigenvalues. One only needs to modify condition (3) to require that the sum of the reciprocal of the non-vanishing eigenvalues is finite.The construction of the characteristic time operator and the class it generates are the same as in the cases considered here. We mention that the bounded and self-adjoint operator canonically conjugate with the number operator constructed by Garisson andWong (1970), andGalindo (1984) is an example (See also Busch 1995a,b).
To conclude, we give a class of characteristic time operators. Consider the class of characteristic time operators distinguished by the following requirement further imposed upon the eigenvalues of the Hamiltonian, ∞ s,s ′ ≥1 ′ 1 (E s − E s ′ ) 2 < ∞.
(4.1)
We note that when (4.1) is satisfied, the condition ∞ s=1 E −2 s < ∞ is automatically satisfied. Under this condition, the characteristic essentially self-adjoint time operator for both degenerate and non-degenerate Hamiltonians admits a bounded and compact self-adjoint extension. The boundedness of T M for every finite M ≥ 1 follows from the following inequality,
T M | ϕ ≤ M ∞ s,s ′ ≥1 ′ 1 ω 2 s,s ′ 1 2 | ϕ (4.2)
for every | ϕ in D(T M ). The boundedness of T M means that it can be extended in the entire Hilbert space. And since T M is symmetric, it is self-adjoint in the entire H.
The compactness of T M for every finite M ≥ 1 can be shown in the configuration space representation in which T M assumes the form of a Fredholm integral operator with square integrable kernel. That is for every ϕ(q) = q| ϕ in the domain of T M in configuration space representation, we have (T M ϕ)(q) = Ω q| T M |q ′ ϕ(q ′ ) dσ(q ′ ), (4.3)
where the respective kernels for non-degenerate and M-degenerate Hamiltonians are given by
q| T 1 |q ′ = ∞ s,s ′ ≥1 ′ i ω s,s ′ ϕ s (q)ϕ * s ′ (q ′ ), (4.4) q| T M |q ′ = ∞ s,s ′ ≥1 ′ M r,r ′ ≥1
′ i ω s,s ′ ϕ s,r (q)ϕ * s ′ ,r ′ (q ′ ).
(4.5)
When the energy eigenvalues satisfy equation (4.1), these kernels are square integrable:
Ω Ω | q| T 1 |q ′ | 2 dσ(q) dσ(q ′ ) = ∞ s,s ′ ≥1 ′ 1 ω 2 s,s ′ < ∞, (4.6) Ω Ω | q| T M |q ′ | 2 dσ(q) dσ(q ′ ) = (M − 1)M ∞ s,s ′ ≥1 ′ 1 ω 2 s,s ′ < ∞.
(4.7)
We know that such operators are compact. The compactness of T M , coupled with its self-adjointness, means that T M has discrete spectrum and its eigenvectors span the entire H.
* n,r φ n,r . Under this representation, the Hamiltonian H M with domain D(H M ) is explicitly given by
formal operator T M can be assigned a dense subspace in H M , and it is canonically conjugate with the Hamiltonian in some dense subspace of H M , and it is essentially self-adjoint in its assigned domain. Likewise we call T M as the characteristic time operator for the M-degenerate Hamiltonian.(a) T M is Densely DefinableNow we show that if condition (3.5) is satisfied, then T M can be assigned the following dense subspace, ,r | s, r , |ϕ s,r | < ∞, N < ∞ , (3.6) and in this subspace it is essentially self-adjoint. The subspace D M is dense because for every | ψ in H M the sequence of elements in D M given by r| ψ | s, r , k = 1, 2, . . .
<
∞ is finite by assumption. Since | ϕ is an arbitrary element of D M , the bound (3.10) holds for every element of D M . Thus T M | ϕ is in H M for all | ϕ in D M . Since T M is densely defined in D M , we define T M as the densely defined operator T M : D M ⊆ H M → H M with domain D(T M ) = D M . (b) T M and H M are Canonically Conjugate Now we show that the pair of operators T M and H M form a canonical pair in a dense subspace D M c of D(H M T M ) ∩ D(T 2 H M ), i.e. (T M H M − H M T M ) | ϕ = i | ϕ for all | ϕ in D M c . Again we assume for the moment that D(H M T M ) ∩ D(T M H M ) is not empty. If | ϕ is in D(H M T M ) ∩ D(T M H M ), then
=r ϕ s ′ = −ϕ s,r . In which case equation (3.12) reduces to(T M H M − H M T M ) | ϕ = i | ϕ .(3.14)That is T M and H M satisfy the canonical commutation relation in the subspace of vectors satisfying equation (3.13).We have then to identify a dense subspace of D(T M ) satisfying (3.13) and at the same time belonging to D(H M T M ) ∩ D(T M H M ). Now we show that the vectors of the following proper subspace of D(T M ) satisfy these conditions,
set a i,j,k = 0 when either i or j is outside of its respective range, 1 ≤ j ≤ (L − 1), (j + 1) ≤ i ≤ L; or when both are outside of their ranges; or when i ≤ j. Similar calculation shows that the sum of the coefficients for each k vanishes. Thus the vectors in DM c satisfy condition (3.13). Thus to establish our claim, it is sufficient to show that D M c is dense and that D M c is a subspace of D(H M T M ) ∩ D(T M H M ). Now D M c is dense. Let us assume otherwise. Then there exists a vector | ψ in H M which is not the zero vector such that ξ| ψ = 0 for all
r | s, r which does not lie in the Hilbert space, a contradiction with the assumption that | ψ is in H M . Then if ξ| ψ = 0 for all | ξ in D M c we must have | ψ = 0. Thus D M c is dense. To complete the proof, now we show that D M c is a subspace of D(H M T M ) ∩ D(T M H M ). Since D M c is a subset of D(T M ) and D(T M ) is invariant under H M , T M H M is defined in the entire D M c . Now D(H M T M ) consists of those | φ in D(T M ) such that T M | φ is in D(H M ). Specifically the domain consists of those | φ such that
with equation (3.19), we find that every | ψ in D M c belongs to D(H M T M ), which implies that D M c is a subspace of D(H M T M ). Thus the operators H M and T M are canonically conjugate in the dense subspace D M c .
s,k − η s,l ) = ∓i (η σ,k − η σ,l ) .(3.29)Since | ψ belongs to the Hilbert space, we must have ∞ σ=1 |(η σ,k − η σ,l )| 2 < ∞. Now where is the contradiction? We note that H M is a direct sum of M Hilbert spaces spanned separately by the M degenerate eigenvectors of the HamiltonianH M , i.e. H M = H 1 ⊕ H 2 ⊕ · · · ⊕ H M , where H ρ for all 1 ≤ ρ ≤ M is spanned by | s, ρ , s = 1, 2, . . . . Let P ρbe the projection operator unto the subspace H ρ . Then each of the subspaces D ρ = P ρ D(H M ), ρ = 1, 2 . . . , M , is invariant under the Hamiltonian; and these subspaces consequently reduce the Hamiltonain. Then the restriction H Mρ = P ρ H M P ρ of the Hamiltonian H M on H ρ is a self-adjoint, nondegerate Hamiltonian on H ρ with the same eigenvalues E s as the eigenvalues of the original Hamiltonain H M . Since ∞ s=1 E −2 s < ∞, for some fixed ρ we can construct the following characteristic time operator for the non-degenerate Hamiltonian H Mρ on H ρ ,
essential self-adjointness of T M (for all 1 ≤ M < ∞) means that there exists a unique self-adjoint operator T M : D(T M ) ⊆ H → H whose reduction in D(T M ) is T M itself. Equivalently T M is the unique self-adjoint extention of T M . Similarly, for every bounded sequence α, there exists a unique self-adjoint operator T M,α : D(T M,α ) ⊆ H → H whose reduction in D(T M ) is T M,α . Because T M and T M,α are extensions of T M and T M,α , respectively, they remain canonically conjugate with the Hamiltonian H M in the same dense subspace D (M) c . The self-adjoint operators T M and T M,α are just the adjoints T M * and T M,α * , respectively. Thus, in this paper, we have explicitly proven the following
Theorem 4 . 1 .
41Given a self-adjoint Hamiltonian H possessing the following properties:To appear in the Proceedings of the Royal Society of London A
Then there exists a self-adjoint time operator T characteristic of the system which is canonically conjugate with the Hamiltonian in a dense subspace of D(TH) ∩ D(HT), i.e. (TH − HT) ⊂ i I, where I is the identity of the Hilbert space H. Moreover, T generates a class of uncountably many other self-adjoint time operators canonically conjugate with the same Hamiltonian in the same dense proper subspace of the Hilbert space.
1 .
1It has a pure point spectrum bounded from below which can be ordered according to size, i.e. −∞ < E 1 < E 2 < E 3 < · · · , 2. It has a constant finite degeneracy 1 ≤ M < ∞, 3. The sum of the reciprocal of the square of its eigenvalues is finite, i.e 4. Its eigenvectors span the entire Hilbert space.∞
s=1 E −2
s <
∞,
To appear in the Proceedings of the Royal Society of London A
| s, r (3.22)It must be that | ψ * is in the Hilbert space. Thus the domain of T * M consists of the following subspace of H M ,And since the adjoint is defined byInstead of showing that T * M is symmetric to prove the essential self-adjointness of T M , we demonstrate that if T M is not essentially self-adjoint then we can contradict our earlier conclusion on the non-degenerate case. This is to show that the nondegenerate and degenerate cases are intimately related. (It may be possible that D (T * M ) = D (T M ) already for some systems, in which case T M is immediately self-adjoint.)Let us assume the contrary that T M is not essentially self-adjoint. Then there exists a vector | η = ∞ (3.26)
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| [] |
[] | [
"Leonid Makar-Limanov ",
"Umut Turusbekova ",
"Ualbai Umirbaev [email protected] ",
"\nDepartment of Mathematics & Computer Science\nDepartment of Mathematics\nBar-Ilan University\n52900Ramat-GanIsrael\n",
"\nDepartment of Mathematics\nWayne State University\n48202DetroitMIUSA\n",
"\nEurasian National University Astana\n010008Kazakhstan\n"
] | [
"Department of Mathematics & Computer Science\nDepartment of Mathematics\nBar-Ilan University\n52900Ramat-GanIsrael",
"Department of Mathematics\nWayne State University\n48202DetroitMIUSA",
"Eurasian National University Astana\n010008Kazakhstan"
] | [
"Mathematics Subject Classification"
] | Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable. | 10.1016/j.jalgebra.2008.01.005 | [
"https://arxiv.org/pdf/0708.1148v1.pdf"
] | 14,640,020 | 0708.1148 | bdf9a095cc8931090c2714f48e4cf2d8e4aebb6f |
2000
Leonid Makar-Limanov
Umut Turusbekova
Ualbai Umirbaev [email protected]
Department of Mathematics & Computer Science
Department of Mathematics
Bar-Ilan University
52900Ramat-GanIsrael
Department of Mathematics
Wayne State University
48202DetroitMIUSA
Eurasian National University Astana
010008Kazakhstan
Mathematics Subject Classification
2000Automorphisms and derivations of free Poisson algebras in two variablesPoisson algebrasautomorphismsderivations
Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
Introduction
It is well known [6,9,10,11] that the automorphisms of polynomial algebras and free associative algebras in two variables are tame. It was recently proved [17,18] that polynomial algebras and free associative algebras in three variables in the case of characteristic zero have wild automorphisms. P. Cohn [4] proved that the automorphisms of a free Lie algebra with a finite set of generators are tame.
There are many other results, some of them quite deep, known about the structure of polynomial algebras, free associative algebras, and free Lie algebras. Though free Poisson algebras are very closely connected with these algebras, only few results are known about them up to now. Say, one of the fundamental results about free associative algebras is the Bergman Centralizer Theorem (see [3]) which says that the centralizer of any nonconstant element is a polynomial algebra on a single variable. An analogue of this theorem for free Poisson algebras in the case of characteristic zero was proved in [12].
The question on the tameness of automorphisms of free Poisson algebras in two variables was open and was formulated in [12,Problem 5]. Note that the Nagata automorphism [13,17] gives an example of a wild automorphism of a free Poisson algebra in three variables.
In [14] R. Rentschler proved that the locally nilpotent derivations of polynomial algebras in two variables over a field of characteristic 0 are triangulable. Using this result he gave a new proof of Jung's Theorem [9] on the tameness of automorphisms of these algebras.
In this paper we study automorphisms and locally nilpotent derivations of free Poisson algebras over a field of characteristic zero. In Section 2 we introduce several gradings of free Poisson algebras and describe some properties of homogeneous derivations of these algebras. In Section 3 we prove that the locally nilpotent derivations of two generated free Poisson algebras are triangulable and the automorphisms of these algebras are tame. These results are analogues of Rentschler's Theorem [14] and Jung's Theorem [9], respectively.
Homogeneous derivations
A vector space B over a field k endowed with two bilinear operations x · y (a multiplication) and {x, y} (a Poisson bracket) is called a Poisson algebra if B is a commutative associative algebra under x·y, B is a Lie algebra under {x, y}, and B satisfies the following identity (the Leibniz identity):
{x, y · z} = {x, y} · z + y · {x, z}.
Of course, the Leibniz identity just says that for every x ∈ B the map
ad x : B −→ B, (y → {x, y}),
is a derivation of B as an associative algebra.
The map ad x also satisfies another similar identity:
ad x {y, z} = {ad x (y), z} + {y, ad x (z)}.
It is just the Jacobi identity for B as a Lie algebra. Let us call a linear homomorphism D of B to B a derivation of B as a Poisson algebra if it satisfies both the Leibniz and Jacobi identities. In other words, D is simultaneously a derivation of B as an associative algebra and as a Lie algebra.
There are two important classes of Poisson algebras. 1) Symplectic algebras S n . For each n algebra S n is a polynomial algebra k[x 1 , y 1 , . . . , x n , y n ] endowed with the Poisson bracket defined by
{x i , y j } = δ ij , {x i , x j } = 0, {y i , y j } = 0,
where δ ij is the Kronecker symbol and 1 ≤ i, j ≤ n.
2) Algebras of Lie type. Let g be a Lie algebra with a linear basis e 1 , e 2 , . . . , e k , . . .. The symmetric algebra S(g) of g (i. e. the usual polynomial algebra k[e 1 , e 2 , . . . , e k , . . .]) endowed with the Poisson bracket defined by
{e i , e j } = [e i , e j ]
for all i, j, where [x, y] is the multiplication of the Lie algebra g is the Poisson algebra of type g.
From now on let g be a free Lie algebra with free (Lie) generators x 1 , x 2 , . . . , x n . It is well known (see, for example [15]) that in this case S(g) is a free Poisson algebra on the same set of generators. We denote this algebra by P = P x 1 , x 2 , . . . , x n .
By deg we denote the standard degree function of the homogeneous algebra P , i.e. deg(
x i ) = 1, where 1 ≤ i ≤ n. Note that deg {f, g} = deg f + deg g
if f and g are homogeneous and {f, g} = 0. By deg x i we denote the degree function on P with respect to x i . We have deg x i (x j ) = δ ij , where 1 ≤ i, j ≤ n. The homogeneous elements of P with respect to deg x i can be defined in the ordinary way.
If f is homogeneous with respect to each deg x i , where 1 ≤ i ≤ n, then f is called multihomogeneous. For every multihomogeneous element f ∈ P we put
mdeg(f ) = (m 1 , m 2 , . . . , m n ),
where deg x i f = m i for all i and 1 ≤ i ≤ n.
Let us choose a multihomogeneous linear basis
x 1 , x 2 , . . . , x n , [x 1 , x 2 ], . . . , [x 1 , x n ], . . . , [x n−1 , x n ], [[x 1 , x 2 ], x 3 ], . . .
Note that The algebra P = P x 1 , x 2 , . . . , x n coincides with the polynomial algebra on the elements (1). Consequently, the words
mdeg{e i , e j } = mdeg(e i ) + mdeg(e j ) if i = j. So if i < j then {e i , e j } isu = e i 1 e i 2 . . . e i k , i 1 ≤ i 2 ≤ . . . ≤ i k (2)
form a linear basis of P . The basis (2) is multihomogeneous since so is (1). Consider the Lie algebra Der(P ) of all derivations of the Poisson algebra P . For every system of elements f 1 , f 2 , . . . , f n of P denote by
D = n i=1 f i ∂ ∂x i (3) a unique derivation of P such that D(x i ) = f i where 1 ≤ i ≤ n. Then the derivations v = u ∂ ∂x i ,(4)
where 1 ≤ i ≤ n and u is an element of (2), constitute a linear basis of Der(P ). For every element v of the form (4) we put
mdeg(v) = mdeg(u) − ǫ i ,
where ǫ i ∈ Z n is the standard basis vector with 1 in the ith position and with zeroes everywhere else. Now one can define the multihomogeneous derivations of the algebra P and every element of Der(P ) can be uniquely represented as the sum of multihomogeneous derivations of different multidegrees.
To each nonzero vector w ∈ Z n we associate the so called w-degree (or weight degree) function wdeg on P and Der(P ). Put
wdeg(u) =< mdeg(u), w >, wdeg(v) =< mdeg(v), w >,
where u and v are elements of the form (2) and (4) respectively, and < , > is the standard inner product in R n . Let P m and Der m P be the subsets of all w-homogeneous elements of degree m of P and Der(P ), respectively. It is clear that the decompositions P = ⊕ m∈Z P m , Der(P ) = ⊕ m∈Z Der m P are gradings of the corresponding algebras. Moreover, for every element d ∈ Der m P we have
d(P k ) ⊆ P m+k .
There is another natural degree function on P , just the total degree on P as a polynomial ring, where the degree is one for all elements of the homogeneous basis (1). Denote it by pdeg and observe that
pdeg[a, b] = pdega + pdegb − 1 for any p-homogeneous a, b ∈ P if [a, b] = 0.
If v is an element of the form (4) then we put pdegv = pdegu − 1.
Let P * m and Der * m P be the subsets of all p-homogeneous elements of degree m of P and Der(P ), respectively. It is again clear that the decompositions
P = ⊕ m∈Z P * m , Der(P ) = ⊕ m∈Z Der * m P
are gradings of the corresponding algebras and that for every element d ∈ Der * m P we have
d(P * k ) ⊆ P * m+k .
Recall that a derivation D of an algebra R is called locally nilpotent if for every a ∈ R there exists a natural number m = m(a) such that D m (a) = 0. The statement of the next proposition is well known (see, for example [8, Proposition 5.1.15]).
Proposition 1 Let R = ⊕ m∈Z R m be a graded algebra and suppose D be a locally nilpotent derivation of R such that
D = D p + D p+1 + . . . + D q , D i (R m ) ⊆ R i+m , p ≤ i ≤ q, D q = 0.
Then D q is locally nilpotent.
Proof.
If
f = f r + f r+1 + . . . + f s ∈ R,
where f i ∈ R i , r ≤ i ≤ s, and f s = 0, then we put f = f s .
Let a ∈ R m and assume that D i q (a) = 0 for any i. It can be easily proved by induction on i that
D i (a) = D i q (a).
Consequently, D i (a) = 0 for any i and this gives a contradiction. 2 Let f be an arbitrary element of P and D be an arbitrary derivation of P of the form (3). We put
f D = n i=1 (f f i ) ∂ ∂x i .
Put also
S(f ) = {e i 1 , e i 2 , . . . , e i k } if f ∈ k[S(f )] and f / ∈ k[S(f i ) \ {e i j }], where 1 ≤ j ≤ k. For D we put S(D) = S(f 1 ) ∪ S(f 2 ) ∪ . . . ∪ S(f n ).
If x = e i then we denote by pdeg x the polynomial degree function with respect to x on P . Elements f ∈ P and D ∈ Der P can be uniquely written as
f = f 0 + xf 1 + . . . + x m f m , x / ∈ S(f i ), 0 ≤ i ≤ m, and D = D 0 + xD 1 + . . . + x m D m , x / ∈ S(D i ), 0 ≤ i ≤ m,
respectively Then
pdeg x D(f ) ≤ pdeg x D + pdeg x f.
This inequality becomes an equality iff l x (D)(l x (f )) = 0 and in this case
l x (D(f )) = l x (D)(l x (f )).
Proof. Without loss of generality we may assume that f is an element of the basis (2) Proof. If l x (D) is not locally nilpotent then there exists x i such that l x (D) k (x i ) = 0 for all k ≥ 0. Put a = l x (D)(x i ). Note that x / ∈ S(a) and l x (a) = a. Using this and Proposition 2, we get l x (D k (a)) = l x (D)(l x (D k−1 (a))) = . . . = l x (D) k−1 (l x (D)(a)) = l x (D) k (a) = 0.
Consequently, D is not locally nilpotent. 2 Proposition 3 Let D be a derivation of P of the form
D = D 0 + xD 1 + . . . + x m−1 D m−1 + x m ∂ ∂x 1 , x / ∈ S(D i ), 0 ≤ i ≤ m − 1,
where x is the minimal element of S(D). Let f be an element of P such that x 1 / ∈ S(f ). Then
pdeg x D(f ) ≤ m − 1 + pdeg x f.
This inequality becomes an equality iff
D ′ (l x (f )) = 0, where D ′ = D m−1 + mx ∂ ∂x 1 ,
and in this case
l x (D(f )) = D ′ (l x (f )).
Proof. The same considerations as in the proof of Proposition 2 show that
pdeg x (x m ∂ ∂x 1 (f )) ≤ m − 1 + pdeg x f and if ∂ ∂x 1 (l x (f )) = 0 then l x (x m ∂ ∂x 1 (f )) = ml x (x ∂ ∂x 1 (f )).
Note that D = D * + x m ∂ ∂x 1 and pdeg x (D * ) ≤ m − 1. So applying Proposition 2, we can complete the proof of Proposition 3. 2 Lemma 2 Let D be a locally nilpotent derivation of P of the form
D = D 0 + xD 1 + . . . + x m−1 D m−1 + x m ∂ ∂x 1 , x / ∈ S(D i ), 0 ≤ i ≤ m − 1,
where x is the minimal element of S(D). If x = x 1 then D m−1 + mx ∂ ∂x 1 is also locally nilpotent.
Proof. Assume that D ′ = D m−1 + mx ∂ ∂x 1 is not locally nilpotent. Then there exists x i such that D ′k (x i ) = 0 for all k ≥ 0. We put a = D ′2 (x i ). It is not difficult to show that Consequently, D is not locally nilpotent. 2 Lemma 3 Let D be a multihomogeneous derivation of P = P x 1 , x 2 and mdeg(D) = (m 1 , m 2 ). If m i ≥ 0 for i = 1, 2 then D is not locally nilpotent.
Proof. Let D be a counterexample to the lemma with the minimal deg(D). By Proposition 1, we can also assume that D is p-homogeneous. Let x be the minimal element of S(D). By Lemma 1, it follows that l x (D) is also locally nilpotent. Put mdeg(l x (D)) = (n 1 , n 2 ). We can assume that n 1 = −1 since deg(l x (D)) < deg(D). Then l x (D) = αx n 2 2 ∂ ∂x 1 . If x = x 1 then D contains a summand l x (D) = αx m 1 +1 1 x r 2 ∂ ∂x 1 . In this case D induces a nonzero locally nilpotent derivation of the polynomial algebra k[x 1 , x 2 ] with the same multidegree. It is impossible (see, for example [8], p. 91). So x = x 1 . If x = x 2 then m 1 = −1. So x > x 2 and D can be written as in Lemma 2. By Lemma 2, it follows that D ′ = D m−1 + mx ∂ ∂x 1 is a nonzero locally nilpotent derivation. Note that pdeg(D ′ ) = 0 and D ′ is p-homogeneous. Therefore D ′ is a derivation of the free Lie algebra g generated by x 1 , x 2 . Obviously, exp(D ′ ) gives a nonlinear automorphism of g. But all automorphisms of g are linear [4]. 2
The main results
Recall that a derivation of the free Poisson algebra P x 1 , x 2 , . . . , x n of the form (3) is called triangular if f i ∈ P x i+1 , x i+2 , . . . , x n for any i. It is clear that every triangular derivation is locally nilpotent. A derivation D of P x 1 , x 2 , . . . , x n is called triangulable if there exists an automorphism ϕ such that ϕ −1 Dϕ is triangular. R. Rentschler proved [14] that the locally nilpotent derivations of polynomial algebras in two variables over a field of characteristic 0 are triangulable. H. Bass gave [1] an example of a nontriangulable derivation of polynomial algebras in three variables.
Theorem 1 Let D be a locally nilpotent derivation of P = P x 1 , x 2 . Then there exist a tame automorphism ϕ of P and f (
x 2 ) ∈ k[x 2 ] such that ϕ −1 Dϕ = f (x 2 ) ∂ ∂x 1 .
Proof. Denote by I the ideal of P generated by {x 1 , x 2 }. Then P/I ∼ = k[x 1 , x 2 ] and D induces a locally nilpotent derivation D ′ of k[x 1 , x 2 ]. By Rentschler's theorem [14], there exists a tame automorphism ψ of k[x 1 ,
x 2 ] and f (x 2 ) ∈ k[x 2 ] such that ψ −1 D ′ ψ = f (x 2 ) ∂ ∂x 1 . Denote by ϕ the extension of ψ to P such that ϕ| k[x 1 ,x 2 ] = ψ. Replacing D by ϕ −1 Dϕ we can assume that D ′ = f (x 2 ) ∂ ∂x 1 .
Then
D = (f (x 2 ) + a) ∂ ∂x 1 + b ∂ ∂x 2 ,
where a, b ∈ I. We would like to show that a = b = 0. Assume it is not the case. Consider deg x 1 and the corresponding highest homogeneous derivation R which is locally nilpotent by Proposition 1. But R = c ∂ ∂x 1 + d ∂ ∂x 2 where c, d ∈ I and either c or d is not zero. So R cannot be locally nilpotent by Lemma 3. 2 Corollary 1 Let D be a locally nilpotent derivation of P = P x 1 , x 2 . Then D{x 1 , x 2 } = 0.
Proof. If D is triangular then D{x 1 , x 2 } = 0. Note that ϕ{x 1 , x 2 } = α{x 1 , x 2 } for every tame automorphism since it is true for every elementary automorphism. 2 Theorem 2 Automorphisms of free Poisson algebras in two variables over a field of characteristic zero are tame.
Proof. Let θ be an arbitrary automorphism of P = P x 1 , x 2 . Then θ induces an automorphism ψ of k[x 1 , x 2 ]. Denote by ϕ the extension of ψ to P such that ϕ| k[x 1 ,x 2 ] = ψ. By Jung's theorem [9], ψ and ϕ are tame. Changing θ to θϕ −1 we can assume that θ induces the identical automorphism of k[x 1 , x 2 ]. Then,
θ(x 1 ) = x 1 + a, θ(x 2 ) = x 2 + b, a, b ∈ I,
where I is the ideal of P generated by {x 1 , x 2 }.
For So every automorphism of P x 1 , x 2 preserves {x 1 , x 2 } up to the proportionality. An analogue of this result for free associative algebras is also true, i.e., every automorphism of the free associative algebra k < x 1 , x 2 > in the variables x 1 , x 2 preserves the commutator [x 1 , x 2 ] up to the proportionality. Moreover, the so called commutator test theorem [7] says that any endomorphism of k < x 1 , x 2 > which preserves [x 1 , x 2 ] is an automorphism. Problem 1 Is any endomorphism of the free Poisson algebra P x 1 , x 2 over a field of characteristic 0 which preserves {x 1 , x 2 } an automorphism?
every h ∈ k[x] denote by D h a derivation of P defined by D h (x 1 + a) = h(x 2 + b), D h (x 2 + b) = 0. This derivation is locally nilpotent. Now, D h = (h(x 2 ) + (h(x 2 + b) − h(x 2 )) − D h (a)) ∂ ∂x 1 − D h (b) ∂ ∂x 2 since D h (x 1 ) = h(x 2 ) + (h(x 2 + b) − h(x 2 )) − D
Note that the positive answer to Problem 1 implies the Jacobian Conjecture for k[x 1 , x 2 ] [8].
It is well known [6,11] that Aut k[x 1 , x 2 ] ∼ = Aut k < x 1 , x 2 >, where k < x 1 , x 2 > is the free associative algebra generated by x 1 , x 2 .
Corollary 3 Let k be a field of characteristic zero. Then,
Aut k[x 1 , x 2 ] ∼ = Aut k < x 1 , x 2 > ∼ = Aut P x 1 , x 2 .
This isomorphism is also interesting in the context of paper [2] since k < x 1 , x 2 > is a deformation quantization of P x 1 , x 2 and because it shows that the group Aut P x 1 , x 2 has a nice representation as a free amalgamated product of its subgroups (see, for example [5]).
of the free Lie algebra g and denote the elements of this basis by e 1 , e 2 , . . . , e m , . . . .
a linear combination of e m where all m > j.
. If f m = 0 then pdeg x (f ) = m and we put l x (f ) = f m . Put also pdeg x D = m and l x (D) = D m if D m = 0. Put e i < e j if i < j. Proposition 2 Let D be a derivation of P and x be the minimal element of S(D).
and D is an element of the basis(4).If f = uv then D(f ) = D(u)v + uD(v). So if the Proposition is true for u and v it is also true for f . Because of that we can assume that the polynomial degree of f is one. Let us prove that in this case pdegx D(f ) ≤ pdeg x D. If f = {u, v} then D(f ) = {D(u), v} + {u, D(v)}. Denote by L(x) the set of all elements e i such that e i > x. If, say D(u) = x d u 1 where S(u 1 ) ⊂ L(x) then {D(u), v} = {x d u 1 , v} = x d {u 1 , v} + dx d−1 u 1 {x, v}. As we remarked if i < j then {e i , e j } isa linear combination of e m where all m > j. So both S({u 1 , v}) and S({x, v}) are subsets of L(x) and we can conclude that pdeg x D(f ) ≤ pdeg x D if it is true for u and v. It remains to check that pdeg x D(f ) ≤ pdeg x D for f with deg(f ) = 1. Since we can assume that D = x d u ∂ ∂x i where u ∈ L(x) and 1 ≤ i ≤ n we have D(x j ) = 0 when j = i and D(x i ) = x d u. So we proved that pdeg x D(f ) ≤ pdeg x D+pdeg x f . To prove that l x (D(f )) = l x (D)(l x (f )) in the case of equality take f = x n f n and D = x m u ∂ ∂x i where pdeg x (f n ) = 0 and u ∈ L(x). Since D(f ) = x n D(f n ) + nx n−1 f n D(x) only x n D(f n ) can contain x n+m and we should show that l x (D(f n )) = x m D m (f n ) where D m = u ∂ ∂x i . It can be done exactly as above by reduction first to the case when pdeg(f n ) = 1 and then to the case when deg(f n ) = 1. 2 Lemma 1 Let D be a derivation of P and x be the minimal element of S(D). If D is locally nilpotent then so is l x (D).
x 1 , x / ∈ S(a).So l x (a) = a. Using this and Proposition 3, we get l x (D k (a)) = D ′ (l x (D k−1 (a))) = . . . = D ′k (a) = 0.
h (a) and D h (x 2 ) = −D h (b). The ideal I is invariant under every derivation. Hence h(x 2 +b)−h(x 2 ))−D h (a), D(b) ∈ I. Since D h is locally nilpotent it is possible only if D h (b) = h(x 2 +b)−h(x 2 ))−D h (a) = 0 (see the proof of Theorem 1). Therefore D h (x 1 ) = h(x 2 ), D h (x 2 ) = 0 and D h (a) = h(x 2 + b) − h(x 2 ). Put h = x. Then D x (a) = b. Note that deg D x (a) ≤ deg a since D x (x 1 ) = x 2 and D x (x 2 ) = 0. So deg b ≤ deg a.We can exchange x 1 and x 2 in the definition of D h , so deg a ≤ deg b and deg a = deg b. Of course, deg a = deg b ≥ 2 since a, b ∈ I. We now put h = x 2 . Then D h (a) = 2x 2 b + b 2 . Note that in this case deg D h (a) ≤ deg a+1 since D h (x 1 ) = x 2 2 and D h (x 2 ) = 0. Consequently, deg a+1 ≥ 2 deg b = 2 deg a, and deg a ≤ 1. This contradiction gives a = 0 and b = 0. 2 Corollary 2 Let ϕ be an arbitrary automorphism of P = P x 1 , x 2 . Then ϕ{x 1 , x 2 } = α{x 1 , x 2 }, where 0 = α ∈ k.
AcknowledgmentsThe authors wish to thank several institutions which supported them while they were working on this project: Max-Planck Institute für Mathematik (the first and the third authors), Department of Mathematics of Wayne State University in Detroit (the third author), and Instituto de Matemática e Estatística da Universidade de São Paulo (the first author).
A non-triangular action of G a on A 3. H Bass, J. of Pure and Appl. Algebra. 331H. Bass, A non-triangular action of G a on A 3 , J. of Pure and Appl. Algebra, 33(1984), no.1, 1-5.
Automorphisms of the Weyl Algebra. A Belov-Kanel, M Kontsevich, Letters in Mathematical Physics. 74A. Belov-Kanel, M. Kontsevich, Automorphisms of the Weyl Algebra, Letters in Mathematical Physics, 74 (2005), 181-199.
G M Bergman, Centralizers in free associative algebras. 137G. M. Bergman, Centralizers in free associative algebras, Trans. Amer. Math. Soc., 137 (1969), 327-344.
Subalgebras of free associative algebras. P M Cohn, Proc. London Math. Soc. 56P. M. Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc., 56 (1964), 618-632.
Free rings and their relations. P M Cohn, Academic PressLondon2nd EdP. M. Cohn, Free rings and their relations, 2nd Ed., Academic Press, London, 1985.
Automorphisms of a free associative algebra of rank 2, I, II. A G Czerniakiewicz, Trans. Amer. Math. Soc. 160A. G. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2, I, II, Trans. Amer. Math. Soc., 160 (1971), 393-401; 171 (1972), 309-315.
A commutator test for two elements to generate the free algebra of rank two. W Dicks, Bull. London Math. Soc. 14W. Dicks, A commutator test for two elements to generate the free algebra of rank two, Bull. London Math. Soc., 14 (1982), 48-51.
. A Van Den Essen, Progress in Mathematics. 190Birkhauser verlagPolynomial automorphisms and the Jacobian conjectureA. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhauser verlag, Basel, 2000.
Uber ganze birationale Transformationen der Ebene. H W E Jung, J. reine angew. Math. 184H. W. E. Jung, Uber ganze birationale Transformationen der Ebene, J. reine angew. Math., 184 (1942), 161-174.
On polynomial rings in two variables. W Van Der Kulk, Nieuw Archief voor Wiskunde. 3W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, (3)1 (1953), 33-41.
The automorphisms of the free algebra with two generators, Funksional. L Makar-Limanov, Functional Anal. Appl. 43Anal. i PrilozhenL. Makar-Limanov, The automorphisms of the free algebra with two generators, Funk- sional. Anal. i Prilozhen. 4(1970), no.3, 107-108; English translation: in Functional Anal. Appl. 4 (1970), 262-263.
Centralizers in free Poisson algebras. L Makar-Limanov, U U Umirbaev, Proc. Amer. Math. Soc. 1357L. Makar-Limanov, U. U. Umirbaev, Centralizers in free Poisson algebras, Proc. Amer. Math. Soc. 135 (2007), no. 7, 1969-1975.
On the automorphism group of k. M Nagata, Lect. in Math., Kyoto Univ. x, yM. Nagata, On the automorphism group of k[x, y], Lect. in Math., Kyoto Univ., Kinokuniya, Tokio, 1972.
Operations du groupe additif sur le plan. R Rentschler, C. R. Acad. Sci. 267ParisR. Rentschler, Operations du groupe additif sur le plan, C. R. Acad. Sci. Paris, 267 (1968), 384-387.
Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras, Algebra i logika. I P Shestakov, Algebra and Logic. 325English translationI. P. Shestakov, Quantization of Poisson superalgebras and speciality of Jordan Pois- son superalgebras, Algebra i logika, 32(1993), no. 5, 571-584; English translation: in Algebra and Logic, 32(1993), no. 5, 309-317.
Poisson brackets and two generated subalgebras of rings of polynomials. I P Shestakov, U U Umirbaev, Journal of the American Mathematical Society. 17I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two generated subalgebras of rings of polynomials, Journal of the American Mathematical Society, 17 (2004), 181-196.
Tame and wild automorphisms of rings of polynomials in three variables. I P Shestakov, U U Umirbaev, Journal of the American Mathematical Society. 17I. P. Shestakov and U. U. Umirbaev, Tame and wild automorphisms of rings of polynomials in three variables, Journal of the American Mathematical Society, 17 (2004), 197-227.
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| [] |
[
"Performance of Step Network Using Simulation Tool",
"Performance of Step Network Using Simulation Tool"
] | [
"Taskeen Zaidi ",
"Nitya Nand Dwivedi "
] | [] | [] | Nowadays distributed computing approach has become very popular due to several advantages over the centralized computing approach as it also offers high performance computing at a very low cost. Each router implements some queuing mechanism for resources allocation in a best possible optimize manner and governs with packet transmission and buffer mechanism. In this paper, different types of queuing disciplines have been implemented for packet transmission when the bandwidth is allocated as well as packet dropping occurs due to buffer overflow. This gives result in latency in packet transmission, as the packet has to wait in a queue which is to be transmitted again. Some common queuing mechanisms are first in first out, priority queue and weighted fair queuing, etc. This targets simulation in heterogeneous environment through simulator tool to improve the quality of services by evaluating the performance of said queuing disciplines. This is demonstrated by interconnecting heterogeneous devices through step topology. In this paper, authors compared data packet, voice and video traffic by analyzing the performance based on packet dropped rate, delay variation, end to end delay and queuing delay and how the different queuing discipline effects the applications and utilization of network resources at the routers. Before evaluating the performance of the connected devices, a Unified Modeling Language class diagram is designed to represent the static model for evaluating the performance of step topology. Results are described by taking the various case studies.The Unified Modeling Language user guide, the prime developers of the UML--Grady Booch, James Rumbaugh, and Ivar Jacobson have presented a tutorial to describe the essence of the language in a two-color format which is designed to facilitate the learning. Outset with a conceptual model of the UML, the book on UML is used to solve series of complex modeling problems across a variety of application domains by various researchers[11][12]. Object-oriented distributed architecture system through UML has been explained by Arora et al.[13].Various authors are using distribu ted computer system which has become very popular approach of computing as it delivers high-end performance at a low cost. In a distributed computing environment, autonomous computers are linked by means of a communication network, arranged in a geometrical way called network topology. A detailed study of network topologies is executed for the distributed computer systems. A most popular object-oriented modeling language adopted by OMG i.e. Unified Modeling Language (UML) is used for modeling the different network topologies. A comparative study for 2D Mesh, Torus, and Hypercube network topologies and their performance is also evaluated after designing the UML model for class, sequence, and activity diagrams[14]. It has the generation of networks, network layer architecture; kinds of topologies used in the networking, different types of communication styles, types of secure transmission and finally covers the aspects of wireless types of secure transmission. | null | [
"https://arxiv.org/pdf/1701.04792v1.pdf"
] | 14,655,851 | 1701.04792 | 857cebc6e62b4466bea2ffcb3c81f94e88ec1fc4 |
Performance of Step Network Using Simulation Tool
Taskeen Zaidi
Nitya Nand Dwivedi
Performance of Step Network Using Simulation Tool
Index Terms-PerformanceUMLSimulationQueuing Discipline Data communicationnetworksinternetprotocolssignals
Nowadays distributed computing approach has become very popular due to several advantages over the centralized computing approach as it also offers high performance computing at a very low cost. Each router implements some queuing mechanism for resources allocation in a best possible optimize manner and governs with packet transmission and buffer mechanism. In this paper, different types of queuing disciplines have been implemented for packet transmission when the bandwidth is allocated as well as packet dropping occurs due to buffer overflow. This gives result in latency in packet transmission, as the packet has to wait in a queue which is to be transmitted again. Some common queuing mechanisms are first in first out, priority queue and weighted fair queuing, etc. This targets simulation in heterogeneous environment through simulator tool to improve the quality of services by evaluating the performance of said queuing disciplines. This is demonstrated by interconnecting heterogeneous devices through step topology. In this paper, authors compared data packet, voice and video traffic by analyzing the performance based on packet dropped rate, delay variation, end to end delay and queuing delay and how the different queuing discipline effects the applications and utilization of network resources at the routers. Before evaluating the performance of the connected devices, a Unified Modeling Language class diagram is designed to represent the static model for evaluating the performance of step topology. Results are described by taking the various case studies.The Unified Modeling Language user guide, the prime developers of the UML--Grady Booch, James Rumbaugh, and Ivar Jacobson have presented a tutorial to describe the essence of the language in a two-color format which is designed to facilitate the learning. Outset with a conceptual model of the UML, the book on UML is used to solve series of complex modeling problems across a variety of application domains by various researchers[11][12]. Object-oriented distributed architecture system through UML has been explained by Arora et al.[13].Various authors are using distribu ted computer system which has become very popular approach of computing as it delivers high-end performance at a low cost. In a distributed computing environment, autonomous computers are linked by means of a communication network, arranged in a geometrical way called network topology. A detailed study of network topologies is executed for the distributed computer systems. A most popular object-oriented modeling language adopted by OMG i.e. Unified Modeling Language (UML) is used for modeling the different network topologies. A comparative study for 2D Mesh, Torus, and Hypercube network topologies and their performance is also evaluated after designing the UML model for class, sequence, and activity diagrams[14]. It has the generation of networks, network layer architecture; kinds of topologies used in the networking, different types of communication styles, types of secure transmission and finally covers the aspects of wireless types of secure transmission.
I. INTRODUCTION N the present scenario, distributed computing is widely adopted by many researchers for computation of parallel execution of tasks in optimum time period as it represents the autonomous collection of heterogeneous systems interconnected through heterogeneous network. The devices may be computer system, mobile system, laptop, tablet and other types of handheld devices. Various kinds of topologies are studied by distributed computing approach and according to Coulouris et al. [1], distributed system is an autonomous collection of heterogeneous devices communicating with the help of message passing technique. The characterizations of distributed system, system model, inter-process communication, web services, security issues in distributed system and designing of distributed systems are described by the authors. Hwang [2] has described architecture of various computer models, program behavior, architectural choices, scalability, programmability, performance issues related to parallel processing, designing high-performance computers, supporting software and applications for distributed and parallel computing. Network topologies are one major building block for data communication and Wahlisch [3] described how network entities are directly interconnected with each other, also explained how information is flowing from one device to another device. A structure build of node relations can be built on different layers resulting in a physical or logical topology, constructed while connecting devices by a physical medium. Data exchange on top of this structure can be arranged via the network and application layer creates a logical or overlay topology. The detail description about queue, scheduling technique, QoS in network, queue size, queuing delay and TCP window policy are well explained in [4]. Latha and Srivatsa [5] illustrated the goal of topological design of a computer communication network which is used to accomplish a specified performance at a minimal cost. Kamalesh and Srivatsa [6] observed the assignment of node number in a computer communication network using heuristic approach. Assessment of the performance of different topological structure based on Ant Colony Optimization Algorithm using simulator is available in [7]. Modeling of parallel and distributed applications was a captivation of numerous research groups in the past due to increasing the importance of applications on mixed shared memory parallelism with message passing method [8]. The various aspects of UML and different versions of UML are released by OMG group [9][10].
congestion control and techniques to improve QoS are well explained by the Frouzan [15]. A method using evolutionary structural optimization method has been designed [16] in which the quality of solution is improved by ignoring chain like sets of elements which are responsible for potential kinematic instabilities and local error estimators. It also refines the mesh, to obtain an accurate and stable solution. Zaidi and Saxena [17] have used the concept of graph theory to design a step topology for static interconnection of devices across distributed network. An object-oriented approach UML is used to design model for execution of task in critical section and represented via class and sequence view. Space complexity is also calculated and result is shown in the form of tables and graphs.
Distributed computing approach has become an essential part of many of the software companies. In [18], a well-known object-oriented Unified Modeling Language (UML) is used to create a model for the execution of the tasks across a distributed network for the faster execution of tasks on newly designed step topology. Static and dynamic observations of the execution of the tasks are characterized through UML class and state diagrams, respectively. To validate the proposed model the state diagram is converted into a Finite State Machine (FSM) and different test cases are generated across distributed network environment. Ricart and Agrawala inferred that if all sites must donate permission by sending replies, then the release messages are not required, since a reply involves an implicit release. It reduced message complexity [19]. Using a series of benchmark, author [20] studied comparison of algorithms for structural topology optimization based on (i) the artificial material model and (ii) a special microstructure with square voids done in [20]. Zaidi and Saxena [21] also used OPNET to simulate the results for video conferencing and voice packets connected across step networks and it is observed that delay is negligible; a UML class model was also designed to represent the optimized network.
III. BACKGROUND A.
Step Topology It is a new kind of topology called as step topology for static interconnection of handheld devices in step manner which may be desktop computer systems, laptops, mobile, etc. This topology is modification of bus topology by varying the steps which interconnect N numbers of computer systems. This topology works well if the link between two computers connected through bus topology fails. In this topology if individual node is busy then tasks can be executed on next node by using message passing technique and the devices can be connected in static as well as dynamically through adhoc network. A view of step topology through OPNET simulation tool is sketched in figure 1. The detailed narration about step topology can be found in [16].
B. OPNET Modeler
Optimized network engineering (OPNET) software depicts wide varieties of simulations in heterogeneous environment using different protocols. We have used OPNET modeler 16.0 for simulation purpose. OPNET tool simulates the network and performs the powerful functions. Firstly this software was developed for military but nowdays it is used as a network simulation tool as well as a research tool for designing and analysis of networks. The hierarchical structure is divided into three domains as network domain, process domain and node domain. It is also known as event based network simulation tool. The hierarchical structure is divided into three domains as network domain, process domain and node domain. It is also known as event based network simulation tool. Key features of OPNET are: (i) Modeling and simulation. (ii) Hierarchal Modeling (iii) Automatic Simulation Generation (iv) Implementing and developing new data communication networks. To simulate performance of various scheduling techniques, the architecture of step network is shown in Table 1. Table C. Queuing Disciplines Packets arrive from different route at intermediate device such as switch and router for processing. Real time traffic such as video conferencing and voice traffic need proper bandwidth with minimal latency, jitter or packet loss as more sensitive towards the network QoS. QoS build for the given networks as:
Table 1. List of Events for Transition
(1) Adequate bandwidth allocation for all types of traffic i.e. voice, video and data transmitted over a network. (2) Classification of packets on basis of priority as voice packets gets higher priority compared to video and FTP packet has lower priority. (3) Queuing occurs in routers and switches by creating different buffer or queue for different types of traffic.
FIFO
First in first out is a simplest method as in which packets wait in queue until the node is ready to process it. In this technique, first packet arrives is to be first transmitted. In this technique, all the packets have equal opportunity regardless of application or importance of packets as all the packets are treated equally.
PQ
In Priority queuing packets are processed according to priority The packets having highest priority are processed first and lowest priority packets are processed at last. The packets are sorted in buffer according to priority that reflects the importance and urgency for transmission of packets. PQ is made from different buffers and real time application as VoIP and video conferencing traffic given higher priority so that it can observe minimal delay. But the cons of this scheduling is higher priority packets gets continuous chance to be processed and lower priority packets never processed so starvation will be occur.
WFQ
A best scheduling technique is in which packets are assigned to different classes and allocated to different queues. But queues are weighted on priority, higher priority queue has higher weight and packets are processed in round robin manner by selecting packets from each queue corresponding to their weight. For example, if weights assigned to a queue are 5,3 and 1 then five packets are processed from first queue, three from second and one from third queue. If priority is not assigned on classes then all weights can be equal, so this technique is also called fair queuing with priority. In weighted Fair queuing packets are weighted based on priority of queue.
The system processes packets in each queue in round robin manner and the packets are selected from each queue based on weight.
D. Performance
QoS is overall measurement of network performance. The characteristics that measure network performance is bandwidth and can be measured in terms of bandwidth in hertz and bandwidth in bits per second. Throughput is also a measure that actually depicts how fast data is moving on a network. Latency or delay is also a factor that defines the time to send entire message at destination from the first bit sent out from the sender. The latency can be measured in terms of propagation time, transmission time, queuing time and processing delay of data. Jitter is another issue of delay related to real time application. It is delay that occurs packets related to time sensitive application as audio, video data.
E. UML
Model is designed to simulate what may or did happen in a real time situation. Modeling is basically done to optimize the real world problem. In presented model a UML class diagram in figure 2 represents structure of a system in general and shows classes, attributes, operations and relationships among classes is designed in which Internet service provider class provides distributed services to the users. The Synchronous Transfer Mode [STM] terminates the fiber optic network. UTP cable is connected to the layer_3 switch that is attached to the server and next layer switch i.e. L2_switch controls and connect the step topology across VLAN and on server final estimation of performance is evaluated. VoIP application Intercativevoice (6) PCM Quality Speech" for voice, "Interactive Voice.
Figure 2. UML Class Model for Performance Estimation
IV. EXPERIMENTAL STUDY
A. Packets Dropped
In case of FIFO, there is only one queue having size of 500 packets and as FIFO processed the packet on first come first serve basis so no priority is assigned to individual type of incoming traffic. When packets arrive more, then they get stored in a queue and if queue becomes full the incoming packets will be dropped. Another scenario occurs queue becomes full quickly with voice packets then other types of traffic like video and FTP client dropped. The voice packets and video packets are more dropped in FIFO as packets are treated on First come first served basis and no priority or weight is assigned to packets. Whereas in WFQ packets dropped occur due to congestion state. PQ priority is higher for voice packets than WFQ and FIFO as shown in figure 3.
In case of PQ, Voice packets are assigned highest priority. After voice, video packets get priority then FTP. Voice traffic has supreme priority so they are transmitted as soon as received, and video and FTP traffic has to wait when there is incoming voice packets. Again video traffic has priority over FTP in PQ. As WFQ handles shared buffer and when buffer gets full congestion arises, now interface enforced queue to be limited. When congestion occurs voice packet has to wait as video packets causing packet loss as queue becomes full which is opposed to PQ where voice traffic has higher priority and need not to be wait.
Figure 3. Packets Dropped
B. Delay Variation for Video conferencing Packets:
In case of PQ and WFQ scheduling delay variation is very small because end to end delay is very small for both; also there is no variation in end to end delay because video packets are given somewhat smaller priority in both PQ and WFQ then voice packets as shown in figures 4.. Delay variation is less for PQ and WFQ as compared to FIFO as in FIFO packets are not treated according to priority but allowed on First come First serve basis. PQ and WFQ have the negligible delay variation.
Figure 4. Delay Variation for Video Conferencing Packets
C. Traffic Received for Video Conferencing
Voice packets occupy in PQ, whole channel because whenever there is voice traffic other traffic is stopped hence traffic received for Video Conferencing is lower in PQ. Now for WFQ, at start traffic received is higher because system is not in congested state and WFQ is using shared buffer and video packets are having lesser weight then voice packets this means video packets are Multipart figures given lesser priority and resulting lesser traffic for video in WFQ as shown in figure 5 and 6. in FIFO Traffic received for voice as compared to the other two scenarios as FIFO implements first come first serve based mechanism, in which no priority is assigned to any type of traffic. Traffic received in case of WFQ decline a little bit due to congestion phenomenon in WFQ. Traffic received for video is less in case of FIFO as compared to WFQ and PQ, also in case of PQ the videoconferencing packets has lesser priority than voice packets and in WFQ at the start the packets are received at higher speed as the network is not in congested state but when congestion occur the traffic received rate will become slower and also video packets has assigned weight 40 while voice packets are given weight 60 this means video packets are given less priority which results in less traffic for video in WFQ.
Figure 6. Traffic Received for VoIP and Videoconferencing
E. End-to-End Delay for video conferencing and Voice Packets:
For PQ and WFQ, the end-to-end delays are almost minor, as maximum priority allocated voice packets as shown in figure 7. For WFQ discipline voice packets have allocated weight 60 and video packets have allocated weight 40 so voice packets has higher weight.
In PQ voice packets are given priority and all other traffic terminated when voice packets incoming and in case of WFQ voice packets have supremacy as higher weight is allocated. In case of FIFO video packets and voice packets has to wait for longer time as packets are treated on FCFS basis.
Figure 7. E-to-E delay for VoIP and Videoconferencing
V. CONCLUDING REMARKS
The performance of step topology is evaluated and the impact of different queuing scenarios on network is analyzed in terms of packet drops, end to end delay, delay variation, traffic received etc. There is an advantage of using the step topology in comparison of bus topology as link of bus topology fails then it will work in the step network. FIFO queuing is not best for voice and video packets as it handles the traffic on first come first serve basis so packet drop is more in this scenario where as in PQ voice packets are transferred at higher priority as the priority of voice packets are high compared to video packets. It was analyzed that PQ does not shares bandwidth equally to all other types of traffic so other packets have to wait or rest, WFQ is perform by combining the functionality FIFO and PQ therefore each queue gets turn in round robin fashion. Simulation results illustrate that PQ and WFQ has very small end to end delay and delay variation which is foremost for real time applications i.e VoIP and Videoconferencing. The simulation results also depicted that WFQ and PQ has better performance
Figure 5 .
5Traffic Received for VoIP D. Traffic Received for VoIP and Videoconferencing:
Taskeen Zaidi, Assistant Professor, Department of Computer Science, SRM University, Lucknow (U.P.) 225003, India,email:[email protected] 2 Nitya Nand Dwivedi, Research Scholar, Department of Computer Science, SRM University, Lucknow (U.P.) 225003 India, e-mail: [email protected]
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| [] |
[
"Offset errors in probabilistic inversion of small-loop frequency-domain electromagnetic data: a synthetic study on their influence on magnetic susceptibility estimation",
"Offset errors in probabilistic inversion of small-loop frequency-domain electromagnetic data: a synthetic study on their influence on magnetic susceptibility estimation"
] | [
"Christin Bobe \nDepartment of Environment\nGhent University\nGentBelgium\n",
"Ellen Van De Vijver \nDepartment of Environment\nGhent University\nGentBelgium\n"
] | [
"Department of Environment\nGhent University\nGentBelgium",
"Department of Environment\nGhent University\nGentBelgium"
] | [] | Small-loop frequency-domain electromagnetic (FDEM) devices measure a secondary magnetic field caused by the application of a stronger primary magnetic field. Both the in-phase and quadrature component of the secondary field commonly suffer from systematic measurement errors, which would result in a non-zero response in free space. The inphase response is typically strongly correlated to subsurface magnetic susceptibility. Considering common applications on weakly to moderately susceptible grounds, the in-phase component of the secondary field is usually weaker than the quadrature component, making it relatively more prone to systematic errors. Incorporating coil-specific offset parameters in a probabilistic inversion framework, we show how systematic errors in FDEM measurements can be estimated jointly with electrical conductivity and magnetic susceptibility. Including FDEM measurements from more than one height, the offset estimate becomes closer to the true offset, allowing an improved inversion result for the subsurface magnetic susceptibility. | 10.1190/gem2019-079.1 | [
"https://arxiv.org/pdf/1902.00371v1.pdf"
] | 119,233,670 | 1902.00371 | 7627dfa5c099cfe335911fed7f0c4a0e925e7f9d |
Offset errors in probabilistic inversion of small-loop frequency-domain electromagnetic data: a synthetic study on their influence on magnetic susceptibility estimation
1 Feb 2019
Christin Bobe
Department of Environment
Ghent University
GentBelgium
Ellen Van De Vijver
Department of Environment
Ghent University
GentBelgium
Offset errors in probabilistic inversion of small-loop frequency-domain electromagnetic data: a synthetic study on their influence on magnetic susceptibility estimation
1 Feb 2019Accepted for publication in International Workshop on Gravity, Electrical and Magnetic Methods and Their Applications (GEM) 2019 Xi'an. Copyright (2019) Society of Exploration Geophysicists (SEG) and Chinese Geophysical Society (CGS). Further reproduction or electronic distribution is not permitted. * Corresponding Author: [email protected], Department of Environment, Ghent Univer-sity, Gent, Belgium 1
Small-loop frequency-domain electromagnetic (FDEM) devices measure a secondary magnetic field caused by the application of a stronger primary magnetic field. Both the in-phase and quadrature component of the secondary field commonly suffer from systematic measurement errors, which would result in a non-zero response in free space. The inphase response is typically strongly correlated to subsurface magnetic susceptibility. Considering common applications on weakly to moderately susceptible grounds, the in-phase component of the secondary field is usually weaker than the quadrature component, making it relatively more prone to systematic errors. Incorporating coil-specific offset parameters in a probabilistic inversion framework, we show how systematic errors in FDEM measurements can be estimated jointly with electrical conductivity and magnetic susceptibility. Including FDEM measurements from more than one height, the offset estimate becomes closer to the true offset, allowing an improved inversion result for the subsurface magnetic susceptibility.
Introduction
Electrical conductivity (EC) and magnetic susceptibility (MS) determine frequency-domain electromagnetic measurements (FDEM). Using small-loop sensors, secondary field responses are usually small compared to the present primary field. Small measurement quantities are particularly sensitive to systematic calibration errors. These errors imply that a hypothetic free-space measurement would yield non-zero results (Sasaki et al., 2008). Therefore, one way of calibration is to quantify the offset error by performing a measurement far from any conductive matter. Unfortunately, this is often practically infeasible. Alternatively, the offset is often corrected by repeated measurements on multiple elevations above ground, including the offset as an additional parameter in an inversion of such data, under the assumption that the offset is constant. This procedure has been proven to result in more reliable estimates for subsurface EC (e.g., Sasaki et al. (2008) and Tan et al. (2018)).
However, the aforementioned studies focus on EC and do not include inversion for subsurface MS. The FDEM in-phase response usually shows a strong correlation to subsurface MS, but, is often smaller than its quadrature counterpart and therefore even more prone to offset errors. For that reason, the in-phase response is often not used in inversion for EC. We show, using a synthetic example, how multi-elevation measurements from multi-coil measurement setups can be used to derive reliable MS inversion results and an estimation for offset errors. The synthetic subsurface model consists of three horizontal layers, including variation in EC as well as in MS. FDEM responses for this subsurface are simulated using a one-dimensional solution to Maxwell's equations (Ward and Hohmann, 1988). Applying the Kalman ensemble generator (Nowak, 2009) for inversion of the simulated FDEM responses, we perform a probabilistic inversion, approximating probability distributions by an ensemble of subsurface realizations.
Similar to former studies (e.g., Sasaki et al. (2008) and Tan et al. (2018)), we add constant offset errors for each receiver coil as parameters in the inversion framework. The probabilistic framework allows to jointly estimate EC, MS and the offset errors. Unlike to the method proposed by Sasaki et al. (2008), we do not introduce further regularization to an iterative inversion process. Instead, the offset is estimated through a search of the prior model parameter space, resulting in an offset estimate including uncertainty. For our synthetic study, we will give offset priors where only in one case the true offset matches the prior mean.
The advantage of multiple elevations for a FDEM measurement inversion is demonstrated by comparing the result to an inversion in which only measurements from one height are considered. Finally, both results are compared to the inversion result for a data set without offset errors.
Forward simulation
The synthetic FDEM data are simulated using a one-dimensional forward model, allowing for vertical variation of EC and MS (Ward and Hohmann (1988), and Hanssens et al. (2019)). In small-loop systems, a primary field is computed by assuming an alternating current in a transmitter coil. A secondary field is simulated based on induction currents in conductive model layers. The simulated measurement quantity is the field strength as registered by the induced voltage in a receiver coil at a defined model position with respect to the transmitter coil. In the following, the secondary field is expressed in parts-per-million (ppm) of the primary field. The offset error is included in the forward model by shifting the responses by a constant value.
Inversion method
The previously described forward model, in the following called g, also serves as the kernel for the probabilistic Kalman ensemble generator inversion (KEG; Nowak (2009)). A full publication on this FDEM inversion procedure is currently under review with moderate revisions. In this probabilistic framework, we seek a posterior probability distribution ρ(m|d) for the inversion parameters m given FDEM measurements d. According to EC and MS for each discretized layer and a coil specific offset, the forward responses are computed:
d sim = g(m).
(1)
Inserting the true physical parameters m true in g, we derive the synthetic measurements d obs = g(m true ). Using Bayes theorem, one derives the posterior probability solving
ρ(m|d) = κ · ρ M (m)ρ D (d|m),(2)
with a normalization constant κ (Allmaras et al., 2013). The likelihood ρ D (d|m), for measurements d given a set of parameters m, is computed by comparing d sim to d obs (assuming a noise-free d obs for the synthetic data). The prior information on the Gaussian parameters is given by ρ M (m) (see below).
As mentioned above, for solving equation 2, we apply the KEG as introduced by Nowak (2009), a stationary implementation of the widely used Ensemble Kalman filter (EnKF; Evensen (2003)). The advantage of using a probabilistic approach to inversion lies in the simplicity of including the offset error as estimated parameter, and in the availability of an uncertainty estimate. The EnKF is an ensemble approximation to the original Kalman filter (Kalman, 1960). In EnKF, Gaussian probability density functions (PDFs) for model parameters are characterized by random samples drawn from these PDFs. These n ens random samples are single model realizations which are combined in an ensemble matrix A ∈ R (m×nens) , for which g is computed and stored in the response matrix G ∈ R (m×nens) . Incorporating random noise as standard deviation (STD) to the FDEM measurements for all n coils coils, also here n ens samples are drawn to derive the data matrix D ∈ R (n coils ×nens) . The inversion update is derived from the matrix equation (Evensen, 2003)
A U pdate = AA G T (G G T + D D T ) −1 (D − G),(3)
where the primed matrices mean that the mean value of each column is subtracted from this column.
Prior information
The number of inversion parameters is determined by (1) the number of discrete subsurface layers, to which we assign EC and MS values, and (2) the number of coils, to each of which we assign offsets for both the in-phase and the quadrature-phase measurements. To all parameters we attribute prior information through definition of their prior PDFs. To enforce positive values of EC and MS, we use log-normal distributions that are transformed to normal distributions before the update step.
To exclude spurious detail and limit the influence of the prior model, we restrict the prior to a model with uniform mean and STD for all discrete layers. The uniform prior is derived from Gauss sampling, i.e. mean and STD for EC and MS are derived considering 10 cm intervals of the synthetic true.
When no geological layer boundary is present, electric and magnetic properties of adjoining layers will be similar. A simple covariance matrix Σ is introduced in which only the diagonal and the first off-diagonals are filled with non-zero values. On the diagonal, we write the variance of the expectations µ, on the first offdiagonals we include a value smaller than variance as correlation between the layers (half of the variance in our case). Finally, the prior model is described as a multivariate Gaussian N ( µ, Σ).
The prior offsets shall be updated independently for each coil and each in-phase and quadrature component. Hence, no correlation is introduced in between offset parameters. Moreover, no logarithmic barrier is introduced here since the offset is assumed to take positive as well as negative values (often dependent on the specific coil geometry).
Inversion result
The Gaussian posterior is derived by computing the mean and the STD of the ensemble realizations for all parameters in A U pdate . We consider the mean as the best fit to the measurement data, while the standard deviation is considered as its corresponding uncertainty. The Gaussianity of the posterior is enforced by the KEG method, such that the derived uncertainty information can only be considered unbiased when the forward solutions around m prior do not deviate too strongly from linearity.
Synthetic example
The synthetic example consists of a subsurface including three horizontal layers, varying in EC as well as in MS (synthetic true shown in Figure 2). The simulated FDEM measurement setup consists of one transmitter coil and four receiver coils (Fig. 1). Two receiver coils are in a horizontal co-planar (HCP) configuration, the two other receivers in a perpendicular (PRP) configuration. Measurements are simulated at two height above ground, at 0.2 m and 1 m (Table 1).
Using the outlined KEG inversion, we compare three scenarios. Scenario 1 uses the inversion as described above, but without including offset parameters, i.e. using data simulated at one height (0.2 m) that are not contaminated with an offset shift. This is considered the benchmark for the other two scenarios. For scenarios 2 and 3, data was shifted with an offset as listed in Table 1. For scenario 2, we only consider measurement data from 0.2 m (Table 1) 3 uses additional simulated measurements at 1 m height. Thus, twice as many measurements are used in deriving the inverse model.
Another plausible scenario is not considered in this manuscript: using measurements contaminated by offset shifts yet not include the offset in the inversion parameters. Such a scenario leads to unphysical updates, since the measurement data are inherently inconsistent. The inconsistency is caused by the fact that the offset remains constant, while the measurement response gets weaker as the instrument is lifted. For EC, MS and the offset, identical prior models are chosen for all investigated scenarios -except for the offset prior in scenario 1, since there is no offset in this scenario. Inverse model layers have a thickness of 0.07 m. The STD for the FDEM data matrix are set to a negligibly small value of 0.05 ppm, expressing that the simulated measurements are considered to be noise free. We sample the prior PDFs creating an ensemble of 10,000 model realizations.
Except for the HCP 2m coil in-phase response, the chosen prior mean values are not identical to the true offset. For the quadrature component of the PRP 1.1 m coil, a large offset is given. It is also noted that the true offset is not within one STD of the prior model.
The results for the offset estimation are listed in Table 2. The corresponding inversion results for both EC and MS for all scenarios are shown in Figure 2. The best fit offset error estimation for scenario 3 is closer to the true offset than for scenario 2, except for the quadrature phase HCP 2 m coil where the results are similar. It is remarked that the offset uncertainty might be larger when less receiver coils are used or when the subsurface is more heterogeneous, also using measurements at two (or more) heights. In these situations, more elevations per inversion location should be considered until the desired precision for the posterior offset standard deviation is achieved.
As illustrated in Figure 2, the more precise offset estimation in scenario 3 also greatly improves the behavior of the best fit compared with the best fit for scenario 2. The scenario 2 best fit significantly underestimates and smooths the synthetic true model geometries. This is particularly evident for the MS inverse model, where there is almost no contrast resolved between the different MS layers. The scenario 3 best fit clearly resembles the best fit for the benchmark scenario 1. Likewise, derived posterior uncertainties look mostly alike.
A quantitative comparison of scenario 1, 2 and 3 is somewhat flawed since the benchmark scenario 1 and scenario 2 use only four independent FDEM measurements, while scenario 3 uses eight. Additionally, a less uncertain prior model for the offsets or for subsurface EC and MS, might as well improve the posterior inverse model of single-height FDEM measurement data.
Although, the in-phase HCP 2 m and PRP 2.1 m measurements got identical absolute prior offset shifts, their posterior models are different. This might be caused by a fortunate measurement sensitivity with depth for the PRP 2.1 m coil in relation to the layer boundaries. A detailed discussion of this phenomenon is beyond the scope of the presented work and might be discussed later.
To demonstrate the effect of a rather far-off prior guess, the quadrature-phase PRP 1.1 m coil offset is assigned a prior model with mean -180.00 ppm and a STD of 30.00 ppm while the true offset is -100.00 ppm. The offset best fit with one standard deviation is computed to -115.22 ±5.41 ppm for scenario 2 and to -100.04 ±0.07 ppm for scenario 3. Particularly in the latter, the posterior uncertainty is remarkably small. We therefore conclude that offsets estimations are expected to be relatively robust towards wrong prior guesses.
Conclusion
Offset shifts have a relatively stronger effect on posterior MS estimates than on EC estimates, when in-phase offset effects are in the same order of magnitude as the quadrature-phase offsets. Probabilistic inversion of FDEM measurements on two heights including coil-specific offset errors as extra inversion parameters allows for a reliable offset error estimation. Additionally, the posterior estimate for EC and MS is improved. In the probabilistic inversion framework, the uncertainty information for the best fit can be used as a relative measure for reliability of the offset estimates when further, independent FDEM measurements from a coil on different heights are added.
Acknowledgments
This project has received funding from the European Union's EU Framework Programme for Research and Innovation Horizon 2020 under Grant Agreement No 721185. Figure 2: Comparison of the best fit models (solid lines) for the three scenarios with the synthetic true model for EC ad MS. Two posterior standard deviations around the best fit are shown as transparent fills in corresponding colors. Note their asymmetry due to the assumed lognormal behaviour for EC and MS parameters.
Figure 1 :
1Simulated measurement setup: transmitter T with four receiver coils, two in horizontal co-planar (HCP) configuration, two in perpendicular (PRP) configuration.
Table 1 :
1Quadrature-and in-phase responses for FDEM measurements at 0.2 m and 1.0 m height. Measurements from both heights are shifted by the "True Offset". The mean and STD of the Gaussian prior models for the offset estimation are listed in the fourth and fifth column.Quadrature-phase [ppm]
Prior offset
0.2 m 1.0 m True Offset mean STD
HCP 1 m
183.61 85.60
13.00
18.00
3.00
PRP 1.1 m 129.18 27.96
-100.00
-180.00 30.00
HCP 2 m
832.36 550.23
24.00
22.00
5.00
PRP 2.1 m 724.37 272.41
-19.00
-22.00
5.00
In-phase [ppm]
HCP 1 m
2.99
1.93
19.00
12.00
3.00
PRP 1.1 m -10.13 -1.23
-17.00
-12.00
3.00
HCP 2 m
36.82
23.63
20.00
20.00
5.00
PRP 2.1 m -12.00 -5.03
-21.00
-20.00
5.00
Table 2 :
2Comparing the true offset for all coils to the modeled best fit offset and the corresponding STD (indicated by plusminus term) for the data simluated at one height (Scenario 2) and simulated at two heights (Scenario 3). .1 m -21.00 -29.87 ± 1.66 -20.79 ± 0.27Quadrature-phase [ppm]
True
Scenario 2
Scenario 3
HCP 1 m
13.00
10.21 ± 1.38
13.23 ± 0.72
PRP 1.1 m -100.00 -115.22 ± 5.41 -100.04 ± 0.07
HCP 2 m
24.00
26.05 ± 4.90
26.10 ± 5.70
PRP 2.1 m -19.00
19.91 ± 4.71
-19.32 ± 0.39
In-phase [ppm]
HCP 1 m
19.00
14.20 ± 1.89
19.21 ± 0.34
PRP 1.1 m -17.00 -18.32 ± 1.66 -16.95 ± 0.07
HCP 2 m
20.00
12.31 ± 3.71
21.47 ± 2.56
PRP 2
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| [] |
[
"Dynamic Mechanism Design with Interdependent Valuations *",
"Dynamic Mechanism Design with Interdependent Valuations *"
] | [
"Swaprava Nath \nIndian Statistical Institute\nNew Delhi\n",
"Onno Zoeter \nXerox Research Centre Europe\nMeylanFrance\n",
"Y Narahari \nIndian Institute of Science\nBangalore\n",
"Christopher R Dance \nXerox Research Centre Europe\nMeylanFrance\n"
] | [
"Indian Statistical Institute\nNew Delhi",
"Xerox Research Centre Europe\nMeylanFrance",
"Indian Institute of Science\nBangalore",
"Xerox Research Centre Europe\nMeylanFrance"
] | [] | We consider an infinite horizon dynamic mechanism design problem with interdependent valuations. In this setting the type of each agent is assumed to be evolving according to a first order Markov process and is independent of the types of other agents. However, the valuation of an agent can depend on the types of other agents, which makes the problem fall into an interdependent valuation setting. Designing truthful mechanisms in this setting is non-trivial in view of an impossibility result which says that for interdependent valuations, any efficient and ex-post incentive compatible mechanism must be a constant mechanism, even in a static setting.Mezzetti (2004)circumvents this problem by splitting the decisions of allocation and payment into two stages. However, Mezzetti's result is limited to a static setting and moreover in the second stage of that mechanism, agents are weakly indifferent about reporting their valuations truthfully. This paper provides a first attempt at designing a dynamic mechanism which is efficient, strict ex-post incentive compatible and ex-post individually rational in a setting with interdependent values and Markovian type evolution. * A preliminary version of this work was presented in the conference on Uncertainty in Artificial Intelligence, 2011. | 10.1007/s10058-015-0177-6 | [
"https://arxiv.org/pdf/1506.07631v1.pdf"
] | 17,309,415 | 1506.07631 | 38ec43e7742114312e824383d7dca66952e0a3d1 |
Dynamic Mechanism Design with Interdependent Valuations *
25 Jun 2015 June 2, 2015
Swaprava Nath
Indian Statistical Institute
New Delhi
Onno Zoeter
Xerox Research Centre Europe
MeylanFrance
Y Narahari
Indian Institute of Science
Bangalore
Christopher R Dance
Xerox Research Centre Europe
MeylanFrance
Dynamic Mechanism Design with Interdependent Valuations *
25 Jun 2015 June 2, 2015
We consider an infinite horizon dynamic mechanism design problem with interdependent valuations. In this setting the type of each agent is assumed to be evolving according to a first order Markov process and is independent of the types of other agents. However, the valuation of an agent can depend on the types of other agents, which makes the problem fall into an interdependent valuation setting. Designing truthful mechanisms in this setting is non-trivial in view of an impossibility result which says that for interdependent valuations, any efficient and ex-post incentive compatible mechanism must be a constant mechanism, even in a static setting.Mezzetti (2004)circumvents this problem by splitting the decisions of allocation and payment into two stages. However, Mezzetti's result is limited to a static setting and moreover in the second stage of that mechanism, agents are weakly indifferent about reporting their valuations truthfully. This paper provides a first attempt at designing a dynamic mechanism which is efficient, strict ex-post incentive compatible and ex-post individually rational in a setting with interdependent values and Markovian type evolution. * A preliminary version of this work was presented in the conference on Uncertainty in Artificial Intelligence, 2011.
Introduction
Organizations often face the problem of executing a task for which they do not have enough resources or expertise. It may also be difficult, both logistically and economically, to acquire those resources. For example, in the area of healthcare, it has been observed that there are very few occupational health professionals and doctors and nurses in all specialities at the hospitals in the UK (Nicholson, 2004). With the advances in computing and communication technologies, a natural solution to this problem is to outsource the tasks to experts outside the organization. Hiring experts beyond an organization was already in practice. However, with the advent of the Internet, this practice has extended even beyond the international boundaries, e.g., some U.S. hospitals are outsourcing the tasks of reading and analyzing scan reports to companies in Bangalore, India (Associated-Press, 2004). Gupta et al. (2008) give a detailed description of how the healthcare industry uses the outsourcing tool.
The organizations where the tasks are outsourced (let us call them vendors) have quite varied efficiency levels. For tasks like healthcare, it is extremely important to hire the right set of experts. If the efficiency levels of the vendors and the difficulties of the medical tasks are observable by a central management (controller), and if the efficiency levels vary over time according to a Markov process, the problem of selecting the right set of experts reduces to a Markov Decision Problem (MDP), which has been well studied in the literature (Bertsekas, 1995;Puterman, 2005). Let us call the efficiency levels and task difficulties together as types of the tasks and resources.
However, the types are usually observed privately by the vendors and hospitals (agents), who are rational and intelligent. The efficiencies of the vendors are private information of the vendors (depending on what sort of doctors they hire, or machines they use), and they might misreport this information in order to win the contract and to increase their net returns. At the same time the difficulty of the medical task is private to the hospital, and is unknown to the experts. A strategic hospital, therefore, can misreport the task difficulty to the hired experts as well. Hence, the asymmetry of information at different agents' end transforms the problem from a completely or partially observable MDP into a dynamic game among the agents.
Motivated by examples of this kind, in this paper, we analyze them using a formal mechanism design framework. We consider only cases where the solution of the problem involves monetary compensation in quasi-linear form. The reporting strategy of the agents and the decision problem of the controller is dynamic since we assume that the types of the tasks and resources are varying with time. In addition, the above problem has two characteristics, namely, interdependent values: in a selected team of agents, the valuation of an agent depends not only on her own skills but also on the skills of other selected agents, and exchange economy: a trade environment where both buyers (task owners) and sellers (resources) are present. In this paper, the theme of modeling and analysis would be centered around the settings of task outsourcing to strategic experts. We aim to have a socially efficient mechanism, and at the same time, that would demand truthfulness and voluntary participation of the agents.
Prior work
The above properties have been investigated separately in literature on dynamic mechanism design. Bergemann and Välimäki (2010) have proposed an efficient mechanism called the dynamic pivot mechanism, which is a generalization of the Vickrey-Clarke-Groves (VCG) mechanism (Vickrey, 1961;Clarke, 1971;Groves, 1973) in a dynamic setting, and serves to be truthful and efficient. Athey and Segal (2007) consider a similar setting with an aim to find an efficient mechanism that is budget balanced. Cavallo et al. (2006) develop a mechanism similar to the dynamic pivot mechanism in a setting with agents whose type evolution follows a Markov process. In a later work, Cavallo et al. (2009) consider periodically inaccessible agents and dynamic private information jointly. Even though these mechanisms work for an exchange economy, they have the underlying assumption of private values, i.e., the reward experienced by an agent is a function of the allocation and her own private types. Mezzetti (2004Mezzetti ( , 2007, on the other hand, explored the other facet, namely, interdependent values, but in a static setting, and proposed a truthful mechanism. The mechanism proposed in these two papers use a two-stage mechanism, since it is impossible to design a single-stage mechanism satisfying both truthfulness and efficiency even for a static setting (Jehiel and Moldovanu, 2001). However, the mechanism provides a weak truthfulness guarantee in the second stage of the game. A similar result in the setting of interdependent valuations with static types by Nath and Zoeter (2013) ensures that the truthfulness guarantee is strict. However, since both Nath and Zoeter (2013) and Mezzetti (2004) consider mechanisms that use two stages of information realization -in the first stage the types are realized and the allocation is decided, and in the second stage the valuations are realized by the agents and payments are decided -both of them require attention on how the information is revealed to the agents. In this paper, we follow an approach similar to Nath and Zoeter (2013) that guarantees strict truthfulness. However, the equilibrium concept used here is ex-post Nash because we assume agents play in an incomplete information setting, and contrast this with the mechanism of Mezzetti (2004). We also discuss how a complete information setting along with the equilibrium concept of subgame perfection plays an important role in these results. We explain this point in detail while presenting the main result of the paper.
Contributions
In this paper, we propose a dynamic mechanism named MDP-based Allocation and TRansfer in Interdependent-valued eXchange economies (abbreviated MATRIX), which is designed to address the class of interdependent values. It extends the results of Mezzetti (2004) to a dynamic setting, and with a certain allocation and valuation structure, serves as an efficient, truthful mechanism where agents receive non-negative payoffs by participating in it. The key feature that distinguishes our model and results from that of the existing dynamic mechanism literature is that we address the interdependent values and dynamically varying types (in an exchange economy) jointly and provide a strict ex-post incentive compatible mechanism. In Table 1, we have summarized the different paradigms of the mechanism design problem, and their corresponding solutions in the literature.
Valuations
STATIC DYNAMIC Independent VCG Mechanism Dynamic Pivot Mechanism (Vickrey, 1961;Clarke, 1971;Groves, 1973) (Bergemann and Välimäki, 2010;Cavallo et al., 2006) Interdependent Generalized VCG Mechanism MATRIX (Mezzetti, 2004) (this paper) Our main contributions in this paper can be summarized as follows.
• We propose a dynamic mechanism MATRIX, that is efficient, truthful (Theorem 1) and voluntary participatory (Theorem 2) for the agents in an interdependent-valued exchange economy.
◮ This extends the classic mechanism proposed by Mezzetti (2004) to a dynamic setting. ◮ It solves the issue of weak indifference by the agents in the second stage of the classic mechanism.
However, we will see that Theorem 1 is true with a restricted domain of subset allocation and peer-influenced valuations. These two properties were not needed to achieve a similar claim in the static setting (Nath and Zoeter, 2013). We do not know if these are the minimal requirements for efficiency and truthfulness, but it is important to note that these properties in the dynamic setting do not immediately follow from its static counterpart.
• We discuss why the dynamic pivot mechanism (Bergemann and Välimäki, 2010) does not satisfy all the properties that MATRIX satisfies (Section 3.2).
• We discuss that these results can be extended to a more general setting in Section 4.
We also discuss that MATRIX comes at a computational cost which is the same as that of its independent value counterpart (Section 3.4).
The rest of the paper is organized as follows. We introduce the formal model in Section 2, and present the main results in Section 3. In Section 4, we discuss about a generalization of the main results. We conclude the paper in Section 5 with some potential future works.
Background and Model
Let the set of agents be given by N = {1, . . . , n}, who interact with each other for a countably infinite time horizon indexed by time steps t = 0, 1, 2, . . .. The time-dependent type of each agent is denoted by θ i,t ∈ Θ i for i ∈ N. We will use the shorthands θ t ≡ (θ 1,t , . . . , θ n,t ) ≡ (θ i,t , θ −i,t ), where θ −i,t denotes the type vector of all agents excluding agent i. We will refer to θ t as the type profile at time t, θ t ∈ Θ ≡ × i∈N Θ i .
The allocation set is denoted by A. In each round t, the mechanism designer chooses an allocation a t from this set and decides a payment p i,t to agent i. The allocation leads to a valuation to agent i, v i : A × Θ → R. This is in contrast to the classical independent valuations (also called private values) case where valuations are assumed to depend only on i's own type; v i : A × Θ i → R. However, we assume for all i, |v i (a, θ)| ≤ M < ∞, for some M ∈ R and for all a and θ.
Stationary Markov Type Transitions, SMTT The combined type θ t follows a first order Markov process which is governed by the transition probability function F (θ t+1 |a t , θ t ), which is independent across agents, where a t is the allocation at period t.
Definition 1 (Stationary Markov Type Transitions, SMTT) We call the type transitions to follow stationary Markov type transitions if the joint distribution F of the types of the agents θ t ≡ (θ 1,t , · · · , θ n,t ), and the marginals F i 's exhibit the following for all t.
F (θ t+1 |a t , θ t , θ t−1 , · · · , θ 0 ) = F (θ t+1 |a t , θ t ), and
F (θ t+1 |a t , θ t ) = i∈N F i (θ i,t+1 |a t , θ i,t ).
(1)
We will assume the types to follow SMTT throughout this paper.
For an easier exposition of the more general properties that lead to the same conclusions as in this paper, we will restrict our attention to a restricted space of allocations and valuations. In Section 4, we comment on the generalization of our results by introducing certain assumptions that subsume the following two assumptions on the allocation and valuations.
Subset Allocation, SA Let us motivate this restriction with the medical task assignment example given in the previous section. The organizations outsource tasks to experts for a payment, where the expert may have different and often time-varying capabilities of executing the task. The task owners come with a specific task difficulty (type of the task owner), which is usually privately known to them, while the workers' capabilities (types of the workers) are their private information. A central planner's job in this setting is to efficiently assign the tasks to a group of workers. Clearly, in this setting, the set of possible allocations is the set of the subsets of agents, i.e., A = 2 N . Note that, for a finite set of players, the allocation set is always finite. So, we can formally define this setting as follows.
Definition 2 (Subset Allocation, SA) When the set of allocations is the set of all subsets of the agent set, i.e., A = 2 N , we call the domain a subset allocation domain. Similarly, A −i = 2 N \{i} denotes the set of allocations excluding agent i.
Peer Influenced Valuations, PIV Even though the valuation of agent i is affected by not only her private type but also by the types of others, it is often the case that the valuation is affected by the types (e.g. the efficiencies of the workers in a joint project) of only the selected agents. The valuation therefore is a function of the types of the allocated agents and not the whole type vector. We also assume that the value of a non-selected agent is zero. The set of valuations satisfying the above two conditions is called the set of peer influenced valuations (PIV).
Definition 3 (Peer Influenced Valuations, PIV) This is a special set of interdependent valuations in the SA domain, where the valuation of agent i is a function of the types of other selected agents, given by,
v i (a, θ) = v i (a, θ a ) if i ∈ a 0 otherwise,(2)
where θ a ∈ × i∈a Θ i , for an allocation a ∈ A = 2 N .
The properties SA and PIV together allow for a well-defined counterfactual social welfare in a world where a particular agent does not exist. See also Equation (8).
Efficient Allocation, EFF The mechanism designer aims to maximize the sum of the valuations of task owners and workers, summed over an infinite horizon, geometrically discounted with factor δ ∈ (0, 1). The discount factor accounts for the fact that a future payoff is less valued by an agent than a current stage payoff. We assume δ to be common knowledge. If the designer would have perfect information about the θ t 's, his objective would be to find a policy π t , which is a sequence of allocation functions from time t, that yields the following for all t and for all type profiles θ t ,
π t ∈ argmax γ E γ,θt ∞ s=t δ s−t i∈N v i (a s (θ s ), θ s ) ,(3)
where γ = (a t (·), a t+1 (·), . . .) is any arbitrary sequence of allocation functions. Here we use E γ,θt [·] = E[ · |θ t ; γ] for brevity of notation. We point to the fact that the allocation policy γ is not a random variable in this expectation computation. The policy is a functional that specifies what action to take in each time instant for a given type profile. Different policies will lead to different sequences of allocation functions over the infinite horizon, and the efficient allocation is the one that maximizes the expected discounted sum of the valuations of all the agents. In general, the allocation policy π t depends on the time instant t. However, for the special kind of stochastic behavior of the type vectors, namely SMTT, and due to the infinite horizon discounted utility, this policy becomes stationary, i.e., independent of t. We will denote such a stationary policy by π = (a(·), a(·), . . .). Thus, the efficient allocation under SMTT reduces to solving for the optimal action in the following stationary Markov Decision Problem (MDP).
W (θ t ) = max π E π,θt ∞ s=t δ s−t j∈N v j (a(θ s ), θ s ) = max a∈A E a,θt j∈N v j (a, θ t ) + δE θ t+1 |a,θt W (θ t+1 ) .(4)
Here, with a slight abuse of notation, we have used a to denote the actual action taken in t rather than the allocation function. The second equality comes from a standard recursive argument for stationary infinite horizon MDPs. We refer an interested reader to standard text (Puterman, 2005, e.g.) for this reduction and the general properties of MDPs. We have used the following shorthand, E θ t+1 |a,θt [·] = θ t+1 p(θ t+1 |θ t ; a t )[·]. We will refer to W as the social welfare. The efficient allocation under SMTT is defined as follows.
Definition 4 (Efficient Allocation, EFF) An allocation policy a(·) is efficient under SMTT if for all type profiles θ t , Figure 1: Graphical illustration of a candidate dynamic mechanism in an interdependent value setting.
a(θ t ) ∈ argmax a∈A E a,θt j∈N v j (a, θ t ) + δE θ t+1 |a,θt W (θ t+1 ), .(5)At time t v 2 (a(θ t ), θ t ) v n (a(θ t ), θ t ) . . . . . . θ 1,t θ 2,t θ n,t . . . a(θ t )v 1,t v 2,t . . .v n,t v 1 (a(θ t ), θ t ) θ 1,t θ 2,t . . . . . . θ n,t p(θ t ,v t )
Challenges in mechanism design with interdependent valuations The value interdependency among the agents poses a challenge for designing mechanisms. Even in a static setting, if the allocation and payment are decided simultaneously under the interdependent valuation setting, efficiency and Bayesian incentive compatibility (and therefore ex-post incentive compatibility) cannot be satisfied together (Jehiel and Moldovanu, 2001). In a later paper, Jehiel et al. (2006) show that the only deterministic social choice functions that are ex-post implementable in generic mechanism design frameworks with multi-dimensional signals, interdependent valuations, and transferable utilities are constant functions. In view of these impossibility results, we are compelled to split the decisions of allocation and payment in two separate stages. We would mimic the two-stage mechanism of Mezzetti (2004) for each time instant of the dynamic setting (see Figure 1). We consider a direct revelation mechanism. In the first stage of this two-stage mechanism, the agents observe their individual types θ i,t ∈ Θ i , i ∈ N. The strategies available to the agents are to report any typê θ i,t ∈ Θ i . The designer decides the allocation a(θ t ) depending on the reported typesθ t in first stage. The reported types of the agents are not revealed publicly in the first stage. This assumption plays a crucial role in the concept of incentive compatibility we use in this paper. We discuss this after the definition of incentive compatibility briefly and in detail in the next section. After the allocation, the agents observe their valuations v i (a(θ t ), θ t )'s, and report v i,t 's to the designer. The payment decision is made after this second stage of reporting. Our definition of incentive compatibility is accordingly modified for a two stage mechanism. Due to SMTT and the infinite horizon of the MDP, we will focus only on stationary mechanisms, that give a stationary allocation and payment to the agents in each round of the dynamic game. Let us denote a typical two-stage dynamic mechanism by M = a, p . The function a : Θ → A yields an allocation for a reported type profileθ t in round t. Depending on the reported types in the first stage, the mechanism designer decides the allocation a(θ t ), due to which agent i experiences a valuation of v i (a(θ t ), θ t ) in round t. Let us suppose that in the second stage, the reported value vector is given byv t . The payment function p is a vector where p i (θ t ,v t ) is the payment received by agent i at instant t. Combining the value and payment in each round we can write the expected discounted utility of agent i in the quasi-linear setting, denoted by u M i (θ t ,v t |θ t ), when the true type vector is θ t and the reported type and value vectors areθ t andv t respectively. This utility has two parts: (a) the current round utility, and (b) expectation over the future round utilities. The expectation over the future rounds is taken on the true types. Thus the effect of manipulation is limited only to the current round in this utility expression. This is enough to consider due to the single deviation principle of Blackwell (1965).
u M i (θ t ,v t |θ t ) = v i (a(θ t ), θ t ) + p i (θ t ,v t ) current round utility + E π,θt ∞ s=t+1 δ s−t (v i (a(θ s ), θ s ) + p i (θ s , v s )) expected discounted future utility(6)
Here π denotes the stationary policy of actions, (a(·), a(·), . . .). For the SMTT, the type evolution is dependent on only the current type profile and action. To avoid confusion, we will use π, a(θ t ), or a(θ s ), s ≥ t + 1, according to the context. Equipped with this notation, we can now define incentive compatibility.
Definition 5 (w.p. EPIC) A mechanism M = a, p is within period Ex-post Incentive Compatible (w.p. EPIC) if for all agents i ∈ N, for all possible true types θ t , for all reported typesθ i,t , for all reported valuesv i,t , and for all t,
u M i (θ t , (v i (a(θ t ), θ t ), v −i (a(θ t ), θ t ))|θ t ) ≥ u M i ((θ i,t , θ −i,t ), (v i,t , v −i (a(θ i,t , θ −i,t ), θ t ))|θ t )
That is, reporting the types and valuations in the two stages truthfully is an ex-post Nash equilibrium. We use 'ex-post' to denote that the agent chooses her action after observing her own type and valuation, and not the types of others, since that is not revealed to her according to the mechanism considered here. 1 The reported valuationv i,t is therefore a function of the types θ i,t andθ i,t and not of either θ −i,t andθ −i,t , according to the assumption above. An interesting question would be: what happens when the agents' type reports in the first stage are made public. The agents' valuation reports in the second stage can then depend on the type reports in the first stage. The appropriate equilibrium concept in that setting is the subgame perfect equilibrium. We present a detailed discussion on the implications of revealing the type reports in the first stage after presenting the proposed mechanism in the next section.
In this context, individual rationality is defined as follows.
1 Some readers may interpret the term 'ex-post' differently, since the term is conventionally used in the context of single stage mechanisms, i.e., where the decisions of allocation and transfer are decided simultaneously (see, e.g., Jehiel et al. (2006)) and it denotes that truthful reporting is optimal for every realization of the other agents' types even if the agent knew the other agents' types. In the context of two-stage mechanisms that we consider here, we feel that this equilibrium of full observability can be better called as 'subgame perfect' equilibrium. This is the equilibrium concept used in the static two stage mechanism by Mezzetti (2004), and we discuss in detail the difference of the two equilibria concepts in the next section.
Definition 6 (w.p. EPIR) A mechanism M = a, p is within period Ex-post Individually Rational (w.p. EPIR) if for all agents i ∈ N, for all possible true types θ t and for all t,
u M i (θ t , (v i (a(θ t ), θ t ), v −i (a(θ t ), θ t ))|θ t ) ≥ 0.
That is, reporting the types and valuations in the two stages truthfully yields non-negative expected utility.
The MATRIX Mechanism under SA and PIV
In the interdependent valuation setting, our goal is to design a mechanism which is efficient (Def. 4), w.p. EPIC (Def. 5), and w.p. EPIR (Def. 6). This is non-trivial because to achieve efficient allocation in a dynamic setting, one needs to consider the expected future evolution of the types of the agents, which would reflect in the allocation and payment decisions, and for this reason a fixed payment mechanism or a repeated VCG fails to satisfy efficiency (Def. 4). The value interdependency among the agents plays a crucial role here. Even in a static interdependent value setting, if the allocation and payment are decided simultaneously, one cannot guarantee efficiency and incentive compatibility together (Jehiel and Moldovanu, 2001). One way out is to split the decision of allocation and payment in two stages (Mezzetti, 2004). Following this observation, we propose MDP-based Allocation and TRansfer in Interdependent-valued eXchange economies (MATRIX), which we prove to satisfy EFF, w.p. EPIC and w.p. EPIR under the restricted setting of SA and PIV.
Given the dynamics of the game, illustrated in Figure 1, the agents report their types in the first stage, and then the allocation is decided. In the second stage, they report their experienced values and the payment is decided. The task of the mechanism designer, therefore, is to design the allocation and payment rules a, p in each time instant.
In the context of SA and PIV, the social welfare given by Eq. (4) is modified as follows.
W (θ t ) = max π E π,θt ∞ s=t δ s−t j∈N v j (a(θ s ), θ a(θs) ) = max a∈A E a,θt j∈N v j (a, θ a ) + δE θ t+1 |a,θt W (θ t+1 ) .(7)
We also define the maximum social welfare excluding agent i to be W −i (θ −i,t ), which is the same as Eq. (4) except now the sum of the valuations and the allocations are over all agents j = i. We also use the set of allocations excluding i to be A −i as defined by SA,
W −i (θ −i,t ) = max a −i ∈A −i E a −i ,θt j∈N \{i} v j (a −i , θ a −i ) + δE θ t+1 |a −i ,θt W −i (θ −i,t+1 ) .(8)
Note that, SA and PIV are crucial for defining this quantity. Also, when i is absent, the following two notations are equivalent:
E θ t+1 |a −i ,θt [·] = E θ −i,t+1 |a −i ,θ −i,t [·]
, since the type of i will be unchanged when she is not in the game. However, we adopt the former for consistency in notation. Using the definitions above and in the previous section, now we formally present MATRIX.
Mechanism 1 (MATRIX) Given the reported type profileθ t in stage 1, choose the agents a * (θ t ) as follows.
a * (θ t ) ∈ argmax a∈A E a,θt j∈N v j (a,θ a ) + δE θ t+1 |a,θt W (θ t+1 ),(9)
and transfer to agent i after agents reportv t in stage 2, a payment of,
p * i (θ t ,v t ) = j =iv j,t + δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) − W −i (θ −i,t ) − v i,t − v i (a * (θ t ),θ a * (θt) ) 2 .(10)
Algorithm 1 MATRIX for all time instants t do Stage 1: for agents i = 0, 1, . . . , n do agent i observes θ i,t ; agent i reportsθ i,t ; end for compute allocation a * (θ t ) according to Eq. 9; Stage 2: for agents i = 0, 1, . . . , n do agent i observes v i (a * (θ t ), θ a * (θt) ); agent i reportsv i,t ; end for compute payment to agent i, p * i (θ t ,v t ), Eq. 10; types evolve θ t → θ t+1 according to SMTT; end for
The last quadratic term in the above equation is agent i's penalty of not being consistent with the first stage report. The intuition of charging a penalty is to make sure that agent i be consistent with her reported typeθ i,t in the first stage and her value reportv i,t in the second stage, given that others are reporting their types and values truthfully. We will argue that when all agents other than agent i reports their types and values truthfully in the two stages of the mechanism, it is the best response for agent i to do so as well. This term distinguishes our mechanism from that given by Mezzetti (2004), where the agents are weakly indifferent between reporting true and false values in the second stage. We summarize the dynamics of MATRIX using an algorithmic flowchart in Algorithm 1.
We have used this quadratic term for the ease of exposition. However, it is easy to show that any non-negative function g(x, ℓ) having the property that g(x, ℓ) = 0 ⇔ x = ℓ would still satisfy the claims made in this paper. Nath and Zoeter (2013) use a similar term to ensure strict truthfulness in the second stage of a two stage static mechanism with interdependent valuations.
MATRIX and Subgame Perfection Since this paper is an extension of the results of Nath and Zoeter (2013) to a dynamic type setting, we can do similar comparisons of properties with the mechanism of Mezzetti (2004) (let us call this the classic mechanism). If we consider the case where the first stage type reports are made public by the mechanism, i.e., observable by all agents, then the agents have a chance of modifying their next stage report depending on that information. The concept of truthfulness should be modified to subgame perfect equilibrium in this context, which ensures that truth-telling is an equilibrium in every subgame of the two stage game. It can be shown that an agent i can misreport her type in the first stage from θ i toθ i when other agents are reporting their types truthfully and in this subgame, since the reported types are public, each agent's best response would be to report valuations consistent with the first stage's reported types v i (a * (θ t ),θ a * (θt) ) (and not the true valuations v i (a * (θ t ), θ a * (θt) )), which results in more utility to agent i than reporting types truthfully in the first stage (see Nath and Zoeter (2013), where Example 1 illustrates this and can be modified in the dynamic setting for a similar conclusion). Hence, if the first stage reports are made public, MATRIX does not ensure truthfulness in a subgame perfect equilibrium. The classic mechanism, on the other hand, continues to satisfy truth-telling in a subgame perfect equilibrium even in this complete information scenario, and this is because the utility of the agent is unaffected by her second stage valuation reports. So, to summarize, in the incomplete information setting, MATRIX provides a strict truthfulness guarantee in the second stage and the truthfulness is in an ex-post Nash equilibrium, but in a complete information setting, it does not ensure truthfulness in a subgame perfect equilibrium, while the classic mechanism is not strictly truthful in an ex-post Nash equilibrium for an incomplete information setting, but is weakly truthful in a subgame perfect equilibrium in the complete information setting. It is important to note that even though the classic mechanism is weakly truthful in the second stage, and every agent's utility is unaffected by their valuation report, the truthfulness in the type reports in the first stage requires that the agents be truthful in the second stage. Hence, one needs to assume in the mechanism by Mezzetti (2004) that the agents report their valuations truthfully even when their utilities are unaffected by their reports.
Efficiency and incentive compatibility
The following theorem shows that MATRIX satisfies two desirable properties in the unrestricted setting.
Theorem 1 Under SMTT, with SA and PIV, MATRIX is EFF and w.p. EPIC. In addition, the second stage of MATRIX is strictly EPIC.
MATRIX is a two stage mechanism, and we need to ensure that truth-telling is a best response in both these stages. Proof : Clearly, given true reported types, the allocation of MATRIX is efficient by Definition 4. Hence, we need to show only that MATRIX is w.p. EPIC.
To show that MATRIX is w.p. EPIC, let us assume that the true type profile at time t is θ t , and all agents j = i report their true types and values in each round s = t, t + 1, · · · etc. Only agent i reportsθ i,t andv i,t in the two stages. Therefore,θ t = (θ i,t , θ −i,t ) and v j,t = v j (a * (θ t ), θ a * (θt) ), for all j = i. Using the single deviation principle (Blackwell, 1965), we conclude that it is enough to consider only a single shot deviation from the true report of the type. Hence, without loss of generality, let us assume that agent i deviates only in round t of this game.
Let us write down the discounted utility to agent i at time t.
u MATRIX i ((θ i,t , θ −i,t ), (v i,t , v −i (a * (θ i,t , θ −i,t ), θ a * (θ i,t ,θ −i,t ) ))|θ t ) = v i (a * (θ t ), θ a * (θt) ) + p * i (θ t ,v t ) current round utility + E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s )) expected discounted future utility = v i (a * (θ t ), θ a * (θt) ) + j =iv j,t + δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) − W −i (θ −i,t ) − v i,t − v i (a * (θ t ),θ a * (θt) ) 2 + E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s ))
We use the shorthand π * to denote the allocation policy under MATRIX. This gives rise to the allocations a(·) in each round given the type profiles (either reported or true). The first equality is from Eq. (6). The second equality comes by substituting the expression of payment from Eq. (10). Now, from the previous discussion on thev j,t 's andθ j,t 's, j = i, we get,
u MATRIX i ((θ i,t , θ −i,t ), (v i,t , v −i (a * (θ i,t , θ −i,t ), θ a * (θ i,t ,θ −i,t ) ))|θ t ) = v i (a * (θ t ), θ a * (θt) ) + j =i v j (a * (θ t ), θ a * (θt) ) + δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) −W −i (θ −i,t ) − v i,t − v i (a * (θ t ),θ a * (θt) ) 2 + E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s )) ≤ v i (a * (θ t ), θ a * (θt) ) + j =i v j (a * (θ t ), θ a * (θt) ) + δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) −W −i (θ −i,t ) + E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s ))(11)
The equality comes because of the assumption that all agents j = i report their types and values truthfully. The inequality is because we are ignoring a non-positive term. Now, let us consider the last term of the above equation.
E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s )) = E π * ,θt ∞ s=t+1 δ s−t v i (a * (θ s ), θ a * (θs) ) + j =i v j (a * (θ s ), θ a * (θs) ) + δE θ s+1 |a * (θs),θs W −i (θ −i,s+1 ) − W −i (θ −i,s ) = E π * ,θt ∞ s=t+1 δ s−t j∈N v j (a * (θ s ), θ a * (θs) ) + δE θ s+1 |a * (θs),θs W −i (θ −i,s+1 ) − W −i (θ −i,s )
The first equality comes from Eq. (10). We can now rearrange the expectation for the first term above using the Markov property of θ t that gives, E π * ,θt [·] = E θ t+1 |a * (θt),θt [E π * ,θ t+1 [·]]. Therefore,
E π * ,θt ∞ s=t+1 δ s−t (v i (a * (θ s ), θ a * (θs) ) + p * i (θ s , v s )) = E θ t+1 |a * (θt),θt E π * ,θ t+1 ∞ s=t+1 δ s−t j∈N v j (a * (θ s ), θ a * (θs) ) + E π * ,θt ∞ s=t+1 δ s−t δE θ s+1 |a * (θs),θs W −i (θ −i,s+1 ) − W −i (θ −i,s ) = E θ t+1 |a * (θt),θt (δW (θ t+1 )) + E π * ,θt ∞ s=t+1 δ s−t δE θ s+1 |a * (θs),θs W −i (θ −i,s+1 ) − W −i (θ −i,s )(12)
The last equality comes from the definition of W (θ t+1 ). Let us now focus on the last term of the above equation.
E π * ,θt ∞ s=t+1 δ s−t δE θ s+1 |a * (θs),θs W −i (θ −i,s+1 ) − W −i (θ −i,s ) = ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ δ 2 E π * ,θt W −i (θ −i,t+2 ) − δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) + ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ δ 3 E π * ,θt W −i (θ −i,t+3 ) − ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ δ 2 E π * ,θt W −i (θ −i,t+2 ) + · · · − ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ δ 3 E π * ,θt W −i (θ −i,t+3 ) + · · · − · · · = −δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 )(13)
Combining Equations 11, 12, and 13, we get,
u MATRIX i ((θ i,t , θ −i,t ), (v i,t , v −i (a * (θ i,t , θ −i,t ), θ a * (θ i,t ,θ −i,t ) ))|θ t ) ≤ v i (a * (θ t ), θ a * (θt) ) + j =i v j (a * (θ t ), θ a * (θt) ) + δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) − W −i (θ −i,t ) + δE θ t+1 |a * (θt),θt [W (θ t+1 ) − W −i (θ −i,t+1 )](14)
We also note that,
E θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) = E θ t+1 |a * (θt),θt W −i (θ −i,t+1 )(15)
This is because when i is removed from the system in SA domain (while computing W −i (θ −i,t+1 )), the values of none of the other agents will depend on the type θ i,t+1 , due to PIV. And due to the independence of type transitions, i's reported typeθ i,t can only influence θ i,t+1 . Hence, the reported value of agent i at t, i.e.,θ i,t cannot affect W −i (θ −i,t+1 ).
Hence, Equation 14
can be rewritten to show the following inequality.
u MATRIX i ((θ i,t , θ −i,t ), (v i,t , v −i (a * (θ i,t , θ −i,t ), θ a * (θ i,t ,θ −i,t ) ))|θ t ) ≤ v i (a * (θ t ), θ a * (θt) ) + j =i v j (a * (θ t ), θ a * (θt) ) + ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ δE θ t+1 |a * (θt),θt W −i (θ −i,t+1 ) −W −i (θ −i,t ) + δE θ t+1 |a * (θt),θt [W (θ t+1 ) − ✭ ✭ ✭ ✭ ✭ ✭ ✭ W −i (θ −i,t+1 )] (from Eq. 15) = j∈N v j (a * (θ t ), θ a * (θt) ) + δE θ t+1 |a * (θt),θt W (θ t+1 ) − W −i (θ −i,t ) ≤ j∈N v j (a * (θ t ), θ a * (θt) ) + δE θ t+1 |a * (θt),θt W (θ t+1 ) − W −i (θ −i,t ) (by definition of a * (θ t ), Eq. 9) = u MATRIX i (θ t , (v i (a * (θ t ), θ a * (θt) ), v −i (a * (θ t ), θ a * (θt) ))|θ t ).(16)
This shows that utility of agent i is maximized whenθ i,t = θ i,t andv i,t = v i (a * (θ t ), θ a * (θt) ). This proves that MATRIX is within period ex-post incentive compatible. We now argue that the second stage is strictly EPIC for an agent i. This happens because of the quadratic penalty term v i,t − v i (a * (θ t ),θ a * (θt) ) 2 in the payment p * i (Eq. (10)). Notice that if all the agents except i report the types and values truthfully, and agent i also reports her type truthfully in the first stage, then the penalty term will always penalize her ifv i,t is different from v i (a * (θ t ), θ a * (θt) ), which is her true valuation. Hence, the best response of agent i would be to report the true values in the second stage, which makes MATRIX strictly EPIC in this stage.
Why a dynamic pivot mechanism would not work in this setting
It is interesting to note that, if we tried to use the dynamic pivot mechanism (DPM), (Bergemann and Välimäki, 2010), unmodified in this setting, the true type profile θ t in the first summation of Eq. (14) would have been replaced byθ t , since this comes from the payment term (Eq. (10)). The proof for the DPM relies on the private value assumption (see the beginning of Section 2 for a definition) such that, when reasoning about the valuations for the other agents j = i, we have v j (a * ((θ i,t , θ −i,t )), (θ i,t , θ −i,t )) = v j (a * (θ t ), θ j,t ), with which the EPIC claim of DPM can be shown. But in the interdependent value setting, we cannot do such a substitution, and hence the proof of EPIC in DPM does not work. We have to invoke the second stage of value reporting in order to satisfy the EPIC.
Ex-post individual rationality
With SA and PIV, we now show that MATRIX is individually rational.
Theorem 2 (Individual Rationality) Under SMTT, with SA and PIV, MATRIX is w.p. EPIR.
Proof : Due to SA, the set of allocations excluding agent i, denoted by A −i = 2 N \{i} , is already contained in the set of allocations including i, denoted by A = 2 N . Formally, this means a −i ∈ A −i ⊆ A ∋ a. Therefore, the policies π −i ∈ A ∞ −i ⊆ A ∞ ∋ π. Hence in the ex-post Nash equilibrium, the utility of agent i is given by,
u MATRIX i (θ t , (v i (a(θ t ), θ t ), v −i (a(θ t ), θ t ))|θ t ) = j∈N v j (a * (θ t ), θ t ) + δE θ t+1 |a * (θt),θt W (θ t+1 ) − W −i (θ −i,t ) = W (θ t ) − W −i (θ −i,t ) ≥ 0.
The first equality comes from the last equality in Equation 16 and the second equality is by definition of W (θ t ) and a * (θ t ). The last inequality is an immediate consequence of SA and PIV, as the allocation that maximizes the social welfare excluding agent i is already in the potential allocations when i is present. This proves that MATRIX is within period ex-post individually rational.
Complexity of computing the allocation and payment
The non-strategic version of the resource to task assignment problem was that of solving an MDP, whose complexity was polynomial in the size of state-space (Ye, 2005). Interestingly, for the proposed mechanism, the allocation and payment decisions are also solutions of MDPs (Equations 9, 10). Hence the proposed mechanism MATRIX has polynomial time complexity in the number of agents and size of the state-space, which is the same as that of the dynamic pivot mechanism (Bergemann and Välimäki, 2010).
Discussions on a General Result
We can generalize the assumptions of SA and PIV in the following way that would result in the same conclusions as in this paper. These definitions also serve to show the minimal requirements of the proofs. Consider a set of all possible allocations denoted by A. The valuations are called independent of irrelevant agents (IIA) with respect to a set of allocations A ⊆ A if for all i ∈ N,
∃ A −i ⊆ A s.t. ∀ a −i ∈ A −i , v j (a −i , θ) = v j (a −i , θ −i ) v i (a −i , θ) = 0
SA and PIV together constitute a special case of IIA valuations. However, there exist not-so-restrictive examples as well. Consider a set of agents N = {1, 2, . . . , n} having types θ 1 , θ 2 , . . . , θ n and a dummy agent D who does not have any type. Let A = 2 N ∪{D} and A = 2 N . We define A −i = 2 N ∪{D}\{i} , the power set of agents where the dummy replaces agent i. Since the dummy does not have any type, the valuations of other agents after replacing agent i with D depends only on θ −i . Note, in particular, that A −i A.
If now, in addition, A −i ⊆ A, then the allocations are called monotone. SA is a monotone set of allocations and PIV is IIA over that.
We can show that Theorem 1 extends with IIA valuations and Theorem 2 extends with IIA valuations with respect to monotone allocations. We omit the proofs since they follow identical arguments.
Conclusions and Future Work
This paper provides a first attempt of designing a dynamic mechanism that is strict ex-post incentive compatible and efficient in an interdependent value setting with Markovian type evolution. In a restricted domain, which appears often in real-world scenarios, we show that our mechanism is ex-post individually rational as well. This mechanism, MATRIX, extends the mechanism proposed by Mezzetti (2004) to a dynamic setting and connects it to the mechanism proposed by Bergemann and Välimäki (2010).
We have discussed the interesting and challenging domain of mechanism design with dynamically varying types and interdependent valuations. There has been very little work where dynamic types and interdependent values have been addressed together. Hence, there is very little known on the limits of achievable properties in this domain. We have provided one mechanism, namely MATRIX, that is w.p. EPIC, strict in the second stage, and under a restricted domain, even w.p. EPIR. However, we do not know what mechanism characterizes those properties in this domain. For example, a question that may arise is "Is this the only efficient dynamic mechanism that satisfies strict w.p. EPIC in an interdependent value setting?". For the static setting with independent values we have the Green-Laffont characterization result that answers this question for efficiency and DSIC. However, such a characterization result is absent for interdependent valuations for both static and dynamic mechanisms. Developing such a full characterization would be worthwhile.
Table 1 :
1The different paradigms of mechanism design problems with their solutions.
AcknowledgementsWe are grateful to Ruggiero Cavallo, David C. Parkes, two anonymous referees and the associate editor for useful comments on the paper. This work was done when the first author was a student at the Indian Institute of Science and was supported by Tata Consultancy Services (TCS) Doctoral Fellowship. This work is part of a collaborative project between Xerox Research and Indian Institute of Science on incentive compatible learning.
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| [] |
[
"Model Checking with Program Slicing Based on Variable Dependence Graphs",
"Model Checking with Program Slicing Based on Variable Dependence Graphs"
] | [
"Masahiro Matsubara \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n",
"Kohei Sakurai \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n",
"Fumio Narisawa \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n",
"Masushi Enshoiwa \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n",
"Yoshio Yamane \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n",
"Japan Hisamitsu Yamanaka \nHitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan\n"
] | [
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan",
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan",
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan",
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan",
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan",
"Hitachi Research Laboratory\nHitachi Advanced Digital, Inc\nHitachi Automotive Systems, Ltd\nHitachiLtdJapan"
] | [
"FTSCS 2012 EPTCS 105"
] | In embedded control systems, the potential risks of software defects have been increasing because of software complexity which leads to, for example, timing related problems. These defects are rarely found by tests or simulations. To detect such defects, we propose a modeling method which can generate software models for model checking with a program slicing technique based on a variable dependence graph. We have applied the proposed method to one case in automotive control software and demonstrated the effectiveness of the method. Furthermore, we developed a software tool to automate model generation and achieved a 35% decrease in total verification time on model checking. | 10.4204/eptcs.105.5 | [
"https://arxiv.org/pdf/1301.0041v1.pdf"
] | 10,465,937 | 1301.0041 | 430151c2e2867e273e621c43996a3b4d5344520c |
Model Checking with Program Slicing Based on Variable Dependence Graphs
2012
Masahiro Matsubara
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Kohei Sakurai
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Fumio Narisawa
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Masushi Enshoiwa
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Yoshio Yamane
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Japan Hisamitsu Yamanaka
Hitachi Research Laboratory
Hitachi Advanced Digital, Inc
Hitachi Automotive Systems, Ltd
HitachiLtdJapan
Model Checking with Program Slicing Based on Variable Dependence Graphs
FTSCS 2012 EPTCS 105
201210.4204/EPTCS.105.5
In embedded control systems, the potential risks of software defects have been increasing because of software complexity which leads to, for example, timing related problems. These defects are rarely found by tests or simulations. To detect such defects, we propose a modeling method which can generate software models for model checking with a program slicing technique based on a variable dependence graph. We have applied the proposed method to one case in automotive control software and demonstrated the effectiveness of the method. Furthermore, we developed a software tool to automate model generation and achieved a 35% decrease in total verification time on model checking.
Introduction
In embedded control systems, potential risks to system safety have been increasing because programs are getting larger due to electrification, enhancement of diagnosis, etc. Functional safety standards such as IEC 61508 or ISO 26262 have been established to ensure these systems do not fall into dangerous situations. However, the problem of test coverage still remains. Furthermore, there are corner cases which are difficult to be found with usual tests or simulations.
For example, hardware malfunctions could cause software faults, so it is not sufficient to test the software only. Inspection techniques to test combinations of hardware and software such as HILS (Hardware in-the-Loop Simulation) have been in practical use, but failures caused by hardware malfunctions or timing problems caused by software interruptions are difficult to be detected because there could be large number of test cases.
To solve these problems, model checking is applied. In model checking, all of the state transitions of the system are fully searched to detect corner cases including timing problems. Implementation bugs can also be found if models are made out of the source code. However, the problem in applying model checking is well known: a state explosion that verification does not complete. To avoid a state explosion, the scope of verification has to be limited. However, too limited a model misses some causes of malfunctions, so that the part of the source code to be verified has to be determined appropriately.
We are trying to adopt model checking to the development of automotive control software, using it to find software defects which occur rarely on real systems by means of modeling both the software from the source code and the hardware. To solve the state explosion problem mentioned above, we introduced a program slicing technique [10] based on a variable dependence graph which enables an flexible adjustment of the boundaries between the source code to be modeled as software model in detail and the external environment to the software model. Also we introduced external environment The contributions of this paper are as follows:
• A practical modeling method to verify a large scale embedded control program. The main point of the method is the slicing technique based on a variable dependence graph.
• The algorithm for analysis of variable dependence graph, which could be adapted to the analysis of system dependence graph.
• An experimental evaluation of the tool which automates the above method.
The rest of this paper is structured as follows: Section 2 presents the modeling method and the slicing technique. Section 3 presents an example case where the method is applied, to show the usefulness of the method. Section 4 presents the tool which automates the method. Section 5 presents the related work. Finally, Section 6 concludes this paper by summarizing its main points.
Modeling Method
System Modeling
The way of modeling an embedded control system is shown in Figure 1. To find defects in implementations, the software model of the target system is converted from the source code which is related to the selected variables by a verifier. Those selections are done in accordance with a verification point. The code to be converted is sliced out from the whole program to avoid a state explosion, using a slicing technique based on a variable dependence graph (see 2.2).
On the other hand, the hardware, controlled instruments (if needed for verification), and a part of the software are modeled as external environments. These external environment models can have unsteady behavior or faults if they are concerned with the content to be verified.
Software
Hardware, Plants The slicing technique based on the variable dependence graph makes it easy to adjust the boundaries between the software models and the external environment models. The reason that such adjustments are needed is that the appropriate size of the software model differs depending on the verification points. Too large a model leads to a state explosion, and too small a model can miss the cause of a malfunction.
An example of a boundary adjustment is shown in Figure 2. There is a communication driver between the software and the hardware, and this driver has complex processing. It is better to omit the driver to reduce the state number of the software model, if the driver has no relation to a malfunction. In this case, it is appropriate to treat the driver as a part of the external environment, and simplify its processing. This method could make the modeling highly automated but not fully automated. The use of design knowledge is expected for modeling, so that the scope of the software model has to be changed according to the verification point.
Slicing Technique
The data to be used for slicing is usually a program dependence graph (PDG) or a system dependence graph (SDG) [7]. The criteria for slicing include a start statement, an end statement, and variables in the end statement.
In this method, the variable dependence graph (VDG) in Figure 3 is used as data for slicing, which is a kind of data flow to connect dependence among the variables. VDG is equivalent to PDG (or SDG) except for the data unit (a statement in PDG), but it is more suitable for expressing dependencies in a tree format. Tree format is better to pursue related variables and code sequentially, and to adjust the boundaries between code converted to the software model and code treated as external environments. In PDG, data dependence and control dependence are used as an edge. On the other hand, in VDG, dependence brought by an assignment which is similar to data dependence is also used as an edge. Nodes in VDG are variables distinct with positions and execution paths in the source code.
There are two directions in VDG trees. In one, the root node is the goal of the dependencies in the tree (we call it the goal tree): in the other, the root node is the start (the start tree). These root nodes are one side of the slicing criteria, and leaf nodes including those which are on a boundary adjusted by a verifier are another side. Variable Dependence Graph (VDG) *extracted The steps of slicing are as follows:
x (L9@main) b (L6@main) a (L6@main) x (L4@intr) x (L4@intr) if use def
1. Select variables which are related to the verification point in the source code of the target system. These variables are those which could be affected by defects, and are written in properties or assertions.
2. Analyze and extract the VDG goal / start trees from the source code, where the root nodes are the variables selected in step 1.
3. Adjust the boundaries in the VDG trees. (Figure 4) 4. Execute slicing to extract code which has a relation to the variables in the VDG trees extracted in
Step 2 and adjusted in Step 3.
In step 3, a variable on a boundary becomes interface between the software model and a external environment model. It is possible to set a boundary between functions by setting the boundary between variables which belong to different functions.
In step 4, code to be extracted are not only statements which include a variable in VDG directly, but also declarations of the variables, functions and control statements which include or relates to the above statements in control flow.
Statements (except function calls) which do not include a variable cannot be extracted, so that these statements have to be picked up additionally on a control flow if needed.
The benefits of this slicing technique are as follows: • High accuracy of slicing, equal to PDG-based slicing.
× × × × External Environment 005: void main ( ) { 006: b = a; 007: c = b; c (L4@f) c (L9@main) c (L7@main) b (L7@main) b (L6@main) a (L6@main) use use def use def use adjustment × × × × Hardware Resister
• Easy to adjust boundaries between software models and external environments, especially in the tree expression of variable dependencies.
• Interface variables between software models and external environments are clear. Information on the interface is important to make or select external environment models.
• Variables whose range has to be changed are clear when input / output values have data mapping.
The sliced code is converted into software models in the language of the model checker, such as Promela of SPIN [5]. This conversion is done via AST (Abstract Syntax Tree) according to the predetermined rules.
External Environment Models
External environment models are necessary to detect malfunctions caused by interactions between the hardware and software. Specifically, they set every possible combination of input values to software models, or limit those combinations to reduce the state number according to the specification of the external environment. They also simulate the behavior outside of the software so that it becomes possible to judge whether a property for the external environment is held or not.
External environment models are written manually. When a verifier makes a new model or selects one from the existing models of the external environment, information about the interface variable given in VDG is useful. Usually, input values to a software model or determinants of them are set with randomness. Processes are added to external environment models.
Furthermore, a verifier can put unsteady behavior or faults into the external environment models which occur at random realized with non-determinism. If there is a malfunction when an external environment model has unsteady behavior, and vice versa, that behavior seems to be one of the factors which cause the malfunction, so that a verifier could analyze a counter example from that point of view.
Case Study
In this section, we apply the modeling method described in section 2 to show its usefulness.
Example Case
The target to be verified is automotive software for a diagnosis of a power IC in a controller, as shown in Figure 5. The purpose of the verification was to investigate the reason for a malfunction. The malfunction was an incorrect error detection in the diagnosis on a long-term test. The diagnosis reports an error, but no fault was found in the power IC. Standard test methodology was not enough to reproduce the symptom and to find the cause of the malfunction, so model checking was applied.
The verification was done in two steps. In the first step, we focused on the serial communication driver between the micro controller and the power IC, but no defects were found. In the second step, we modeled the software for the diagnosis and the power IC. The communication driver was simplified as an external environment model, because it is proved the driver has no defect in the first step.
The modeling method was applied in the second step. Figure 6 shows the model. For the slicing, the variable which shows the result of the diagnosis was selected. The boundaries were adjusted so that the communication driver and some other part of the software were omitted. One of the inputs given by external environment models is the power IC status to be diagnosed, and another is the controller mode which is decided in the omitted software depending on the power supply. In the transition of the controller mode, a path was added that transits from the power-on state to the power-off state suddenly and abnormally. This path is intended to express the effect of an instantaneous drop of the voltage in the power supply.
In model checking, it is important to reduce the number of concurrent processes in order to reduce the state number. Although the micro controller and power IC run concurrently, the software model and the external model was set in the same one process, and the external environment models change its states before the diagnosis which is a cyclic task. If no malfunction was detected in this model, the external environment models could be separated into a different process. Furthermore, the data mapping was applied to the power IC status to be diagnosed. The sizes of the source code and models are shown in Table 1; logical LOC refers to non-comment lines of code.
The result was that one defect was detected in the software which could cause a malfunction when the controller mode turns to the power-off state suddenly while the diagnosis software is in the specific state. It is confirmed that this malfunction could happen in the actual controller. The mechanism of the malfunction is as follows. Although the diagnosis is stopped by power off, the power IC continues to change its state while the power is off. After the power supply is restored, an inconsistency between the actual power IC status and the recognition for that status by the diagnosis software occurs. This symptom happens only in a specific situation, so this is a timing problem. It is difficult to detect this defect with usual tests or simulations like HILS because a verifier can hardly set conditions to make this malfunction occur.
This case shows that model checking is effective against timing problems, and the modeling method can treat large-scale control software. An obstacle to applying this modeling method to a development process is that it is time consuming. To improve the efficiency of the method, we developed a software tool to automate the method of generating verification models.
Tool
Functionalities
The tool we developed has two main functions as shown in Figure 7. One is slicing C language code based on a variable dependence graph; another is conversion from C language code to Promela.
When a user indicates one or more functions as execution start points of the program, the tool will analyze the variable dependencies. A variable dependence graph in a tree format will be extracted and displayed according to the variables that the user has selected. The tool can extract variable dependencies which extend to other execution paths, so that the effects of interruptions or other tasks can be considered.
The user can limit the variable dependencies to adjust the boundaries between the software models and external environment models. Such an operation is reflected as a color classification of the source code. In the tool, variable dependence trees, function call trees, and source code files are displayed in association with each other, as shown in Figure 8. When a user selects a variable node on the VDG trees, related function node and code will be highlited to help the user understand the structure of the program, and to help the user to decide the boundaries. In a tree format, the VDG could become larger than in a network format, so there are some refinements in the projection of the tree. Redundant tree paths (the same partial trees as other parts) are replaced with a node which indicates redundancy. Every node can be gathered to the function node on which the variable is written, so that a boundary adjustment between functions becomes easier. Moreover, the tool has search functions for variables on VDG trees or execution paths to help operations of the user.
At present, the limitations of the tool are as follows. Source code has to be pre-processed before the input. Pointers can be analyzed, but only one variable is pointed by a pointer, and pointed variables given by loop controls are ignored. Function calls by function pointers are also ignored. Slicing and model conversion are separated into different tools, and model conversion is done for each C code file. This tool is not open to the public.
Algorithm for Analysis of VDG
The algorithm to analyze VDG used in this tool is shown in Figure 9. Dependencies are connected one by one along the control flow graph. Every node, which is a variable distinguished with the position in code and the execution path, is assigned information about the stack trace and associated to the control flow graph (CFG). Whether the nodes connect or not is judged using that information of the dependency source node, the sink node, and the nodes that have been already connected to the sink node. Candidate nodes to connect obtained in A and C can be reduced in consideration of the attributes of the node to be connected, such as the substance (memory area) and the position in the CFG. Dependencies due to loop statements are analyzed in the similar way. Different from [7] [8], initialization vertexes or finalization vertexes are not used. The advantage of this algorithm we think is scalability. Actually, analysis was successful for code of which the size is over 350k logical LOC.
The process of the analysis for the sample code is shown in Figure 10, where nodes in a conditional statement belong to a "if" node for the simplicity of appearance. There are no function parameters and arguments in the sample code, but they can be treated in the same way.
After the analysis, a VDG for an execution path is obtained. This VDG becomes a network if there is a loop statement. Goal/start trees are extracted from VDG with a selection of the variables, considering shared variables between different execution paths. Figure 11 shows the verification process with model checking we employed. Besides the slicing tool and the model converter, the verification assist tool was used in the process. This tool actuates checking on more than one property simultaneously, and shows a variable value table of a counter example in time series to help the user understand the mechanism that the violation of the property happens. If a false positive / false negative error was detected in the verification, the step "Model Revision" is taken additionally to refine the model, and another verification will be done. Table 2 shows the condition of the experiment. The target is automotive control software, and it had never been verified with model checking. Verifications were done without the tools first, and with the Figure 9: Algorithm for Analysis of VDG tools next. Table 3 shows measured time, and those values are normalized, as the total time in "without tools" is 1. Highlighted rows are the steps to which the tools were applied. The same measured time as the first try is set for some steps in the second try which should have no effects of the tools, because the verifier could work faster the second time. The step "Model Revision" was taken because there was a false positive error at first. When tools are used, the whole verification time on model checking is reduced 35% compared to the verification without the tools. Although no tool is used in the step "Simulation Check" in which the verifier confirms the correctness of the model with simulation, the time is reduced because the model conversion by the tool is precise. There is no reduction of time in the step "Verification" because the number of properties is only two in this case. For the four steps in which the time reduction is obtained, it is expected that more reduction will be achieved when the tools are improved.
Evaluation
Except for the user operations, the analysis and the slicing were completed within 5 minutes. (PC Spec: Intel(R) Core(TM)2 Duo E6550 @2.33GHz) Figure 11: Verification Process for model checking. SLAM verifies programs with software stubs as external environments. Such a static analyzer can detect defects automatically, but verification items are fixed. It is important for a flexible verification to limit a target to be verified in the software, and the slicing technique in this paper enables it with less time than the ordinal slicing.
Conclusion
In embedded control systems, the potential risks of software defects have been increasing because of growing software complexity. To detect software defects which are difficult to be found with usual tests or simulations, we proposed a modeling method which can generate software models from source code for model checking, with a program slicing technique based on a variable dependence graph to avoid a state explosion. VDG provides the measure to adjust a boundary between the software model converted from the source code and the external environment model, so that a large scale program can be verified with flexibility. VDG also provides the information of interface variables between software models and external environment models, so that it becomes easier for a verifier to make or select external environment models to the software models.
We applied the proposed method to one case in automotive control software to find the cause of a non-reproducible malfunction, and showed the effectiveness of the method by clarifying that it was a timing problem caused by concurrent operations of the different hardware. Furthermore, we developed the algorithm to analyze VDG, and we also developed a software tool to automate the generation of the model. We achieved a 35% decrease in total verification time on model checking. It is expected that more time reduction will be achieved when the tool is improved.
Figure 1 :
1Modeling Method for Embedded Control System models which include unsteady behavior or faults in them to detect malfunctions caused by interactions of hardware and software.
Figure 2 :
2Abstraction of Boundary between Software and Hardware
Figure 3 :
3Variable Dependence Graph of Sample Code
Figure 4 :
4Adjustment Boundary on VDG
Figure 5 :
5System Structure of Sample Case
Figure 7 :
7Verification Model Generation Tool
A .. nodes already acquired C .. nodes already acquired, in conditional statements C .. nodes already acquired, in conditional statements(Highlight) (Highlight)
Source Code
Source Code
File Explorer
File Explorer
(Highlight) (Highlight)
File Explorer
File Explorer
Variable Search
Result
Variable Search
Result
VDG Trees
VDG Trees
Function Call Trees
Function Call Trees
Result Result
Figure 8: Screen Shot
construct CFG
for all start point p of execution paths on CFG
statement s ← statement of p
while s exists do
T ← stack trace of s
T ← stack trace of s
get variable nodes V corresponding to s, and assign T to V
connect use to def in V
for all node vi in V do
for all def d in A do
if d is able to connect to vi then
connect d to vi
end if
end if
end for
for all use u in C do
if u is able to connect to vi then
connect u to vi
end if
end for
end for
end for
add V to A (and C)
s ← next statement on CFG
end while
end for
Table 2 :
2Condition of Experiment
[ main, call f, enter f ] [ main, call f, enter f ] def useFigure 10: VDG Analysis Steps of Sample Codeb (L6@main)
a (L6@main)
def
use
L6@main
Acquired Nodes
.. In condition
005: void main ( ) {
006: b = a;
007: c = b;
008: q = p;
c (L7@main)
b (L7@main)
b (L6@main)
a (L6@main)
def
use
def
use
def
use
L7@main
008: q = p;
009: if ( x > c ) {
010:
f ( );
011: }
012: else {
013:
g ( );
014: }
003: void f ( ) {
004: y = c;
005: }
c (L7@main)
b (L7@main)
b (L6@main)
a (L6@main)
def
use
def
use
def
use
L8@main
(none)
013:
g ( );
014: }
015: z = y;
016: }
005: }
003: void g ( ) {
004: r = q;
005: }
q (L8@main)
p (L8@main)
def
use
def
use
L9@main
(same as previous)
x (L9@main)
[ main ]
[ main ]
Stack Trace
L9@main
(same as previous)
c (L9@main)
if
b (L6@main)
a (L6@main)
c (L7@main)
b (L7@main)
b (L6@main)
a (L6@main)
def
use
[ main ]
[ main ]
[ main ]
[ main ]
c (L7@main)
b (L7@main)
b (L6@main)
a (L6@main)
def
use
def
use
y (L4@f)
c (L4@f)
def
use
L10@main
L4@ f
c (L7@main)
b (L7@main)
def
use
y (L4@f)
c (L4@f)
L4
@f
def
use
q (L8@main)
p (L8@main)
def
use
def
use
x (L9@main)
c (L9@main)
if
@f
def
use
#
Step
(1) without tools (2) with tools Difference Decrease
Remarks
1
Planning
0.032
0.032
0.00
0.0% same as (1)
2
Slicing
0.193
0.098
0.09
49.2%
49.2% 49.2%
49.2%
3 Architecture Desigin
0.148
0.148
0.00
0.0% same as (1)
4 Model Conversion
0.131
0.030
0.10
77.4%
77.4% 77.4%
77.4%
5 Impl. External Model
0.102
0.102
0.00
0.0% same as (1)
6 Model Synthesis
0.006
0.006
0.00
0.0% same as (1)
7 Simulation Check
0.181
0.074
0.11
59.1%
59.1% 59.1%
59.1%
8 Setting Properties
0.008
0.008
0.00
0.0% same as (1)
9
Verification
0.002
0.002
0.00
0.0% 2 properties
10
Analysis of CE
0.119
0.074
0.05
38.0%
38.0% 38.0%
38.0% 2 counter examples
11
Model Revision
0.078
0.078
0.00
0.0% same as (1)
Sum
1.000
0.652
0.35
34.8%
Table 3 :
3Result of Evaluation (execution times are normalized)Bandera[4] slices Java code and converts them into languages for model checker such as SPIN. Modex / Feaver[6] slices C code using CFG, and converts it to Promela with a convert table and a test harness. However, these tools provide no functionality to adjust an extent of the software to be verified. The Wisconsin Program-Slicing Tool[8] [9] can analyze PDG / SDG, but it provides no functionality to adjust boundaries on PDG/SDG either. CBMC[3], BLAST[2] and SLAM[1] input source code directly5 Related Work
T Ball, V K Levin & S, Rajamani, 10.1145/1965724.1965743A Decade of Software Model Checking with SLAM. Communications of the ACM. 54T. Ball, V. Levin & S. K. Rajamani (2011): A Decade of Software Model Checking with SLAM. Communi- cations of the ACM 54(7), pp. 68-76, doi:10.1145/1965724.1965743.
The software model checker BLAST. D Beyer, T A Henzinger, R Jhala, & R Majumdar, 10.1007/s10009-007-0044-zInternational Journal on Software Tools for Technology Transfer (STTT). 95D. Beyer, T. A. Henzinger, R. Jhala & R. Majumdar (2007): The software model checker BLAST. International Journal on Software Tools for Technology Transfer (STTT) 9(5), pp. 505-525, doi:10.1007/s10009-007- 0044-z.
A tool for checking ANSI-C programs. E Clarke, D Kroening, & F Lerda, 10.1007/978-3-540-24730-2_15Proceedings of Tools and Algorithms for the Construction and Analysis of Systems. Tools and Algorithms for the Construction and Analysis of SystemsE. Clarke, D. Kroening & F. Lerda (2004): A tool for checking ANSI-C programs. In: Proceedings of Tools and Algorithms for the Construction and Analysis of Systems, pp. 168-176, doi:10.1007/978-3-540-24730- 2 15.
Bandera: Extracting Finite-state Models from Java Source Code. J C B Corbett & M, Dwyer, 10.1109/ICSE.2000.870434Proceedings of the 22nd International Conference on Software Engineering. the 22nd International Conference on Software EngineeringJ. C. Corbett & M. B. Dwyer et al. (2000): Bandera: Extracting Finite-state Models from Java Source Code. In: Proceedings of the 22nd International Conference on Software Engineering, pp. 439-448, doi:10.1109/ICSE.2000.870434.
The model checker SPIN. G J Holzmann, 10.1109/32.588521IEEE Transaction on Software Engineering. 235G. J. Holzmann (1997): The model checker SPIN. IEEE Transaction on Software Engineering 23(5), pp. 279-295, doi:10.1109/32.588521.
G J C Holzmann & T, Ruys, 10.1007/11537328_3Proceedings of the 12th International SPIN Workshop. the 12th International SPIN WorkshopEffective Bug Hunting with SPIN and ModexG. J. Holzmann & T. C. Ruys (2005): Effective Bug Hunting with SPIN and Modex. In: Proceedings of the 12th International SPIN Workshop, pp. 24-24, doi:10.1007/11537328 3.
Interprocedural slicing using dependence graphs. S Horwitz, T Reps, & D Binkley, 10.1145/53990.53994Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation. the ACM SIGPLAN 1988 conference on Programming Language design and ImplementationS. Horwitz, T. Reps & D. Binkley (1988): Interprocedural slicing using dependence graphs. In: Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation, pp. 35-46, doi:10.1145/53990.53994.
T Reps, S Horwitz, M Sagiv, & G Rosay, 10.1145/193173.195287Proceedings of the 2nd ACM SIGSOFT Symposium on the Foundations of Software Engineering. the 2nd ACM SIGSOFT Symposium on the Foundations of Software EngineeringSpeeding up SlicingT. Reps, S. Horwitz, M. Sagiv & G. Rosay (1994): Speeding up Slicing. In: Proceedings of the 2nd ACM SIGSOFT Symposium on the Foundations of Software Engineering, pp. 11-20, doi:10.1145/193173.195287.
T Reps, & G Rosay, 10.1145/222124.222138Proceedings of the 3rd ACM SIGSOFT Symposium on the Foundations of Software Engineering. the 3rd ACM SIGSOFT Symposium on the Foundations of Software EngineeringPrecise interprocedural choppingT. Reps & G. Rosay (1995): Precise interprocedural chopping. In: Proceedings of the 3rd ACM SIGSOFT Symposium on the Foundations of Software Engineering, pp. 41-52, doi:10.1145/222124.222138.
Mark Weiser, Proceedings of the 5th International Conference on Software Engineering. the 5th International Conference on Software EngineeringProgram SlicingMark Weiser (1981): Program Slicing. In: Proceedings of the 5th International Conference on Software Engineering, pp. 439-449.
| [] |
[
"Reconfigurable Topological Photonic Crystal",
"Reconfigurable Topological Photonic Crystal"
] | [
"Mikhail I Shalaev \nDepartment of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA\n",
"Sameerah Desnavi \nDepartment of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA\n",
"Wiktor Walasik \nDepartment of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA\n",
"Natalia M Litchinitser \nDepartment of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA\n"
] | [
"Department of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA",
"Department of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA",
"Department of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA",
"Department of Electrical Engineering\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloNew YorkUSA"
] | [] | Topological insulators are materials that conduct on the surface and insulate in their interior due to non-trivial topological order. The edge states on the interface between topological (non-trivial) and conventional (trivial) insulators are topologically protected from scattering due to structural defects and disorders. Recently, it was shown that photonic crystals can serve as a platform for realizing a scatter-free propagation of light waves. In conventional photonic crystals, imperfections, structural disorders, and surface roughness lead to significant losses. The breakthrough in overcoming these problems is likely to come from the synergy of the topological photonic crystals and silicon-based photonics technology that enables high integration density, lossless propagation, and immunity to fabrication imperfections. For many applications, reconfigurability and capability to control the propagation of these non-trivial photonic edge states is essential. One way to facilitate such dynamic control is to use liquid crystals, which allow to modify the refractive index with external electric field. Here, we demonstrate dynamic control of topological edge states by modifying the refractive index of a liquid crystal background medium. Background index is changed depending on the orientation of a liquid crystal, while preserving the topological order of the system. This results in a change of the spectral position of the photonic bandgap and the topologically protected edge states. The proposed concept might be implemented using conventional semiconductor technology, and can be used for robust energy transport in integrated photonic devices, all-optical circuity, and optical communication systems. | 10.1088/1367-2630/aaac04 | [
"https://arxiv.org/pdf/1706.05325v1.pdf"
] | 3,280,620 | 1706.05325 | 214502658505e32b6a7bab1a341bd968a5ae0cfd |
Reconfigurable Topological Photonic Crystal
Mikhail I Shalaev
Department of Electrical Engineering
University at Buffalo
The State University of New York
14260BuffaloNew YorkUSA
Sameerah Desnavi
Department of Electrical Engineering
University at Buffalo
The State University of New York
14260BuffaloNew YorkUSA
Wiktor Walasik
Department of Electrical Engineering
University at Buffalo
The State University of New York
14260BuffaloNew YorkUSA
Natalia M Litchinitser
Department of Electrical Engineering
University at Buffalo
The State University of New York
14260BuffaloNew YorkUSA
Reconfigurable Topological Photonic Crystal
(Dated: June 19, 2017)numbers: 0365Vf4270Qs4279Kr Keywords: Topological phases (quantum mechanics)Photonic bandgap materialsLiquid crystals in optical devices
Topological insulators are materials that conduct on the surface and insulate in their interior due to non-trivial topological order. The edge states on the interface between topological (non-trivial) and conventional (trivial) insulators are topologically protected from scattering due to structural defects and disorders. Recently, it was shown that photonic crystals can serve as a platform for realizing a scatter-free propagation of light waves. In conventional photonic crystals, imperfections, structural disorders, and surface roughness lead to significant losses. The breakthrough in overcoming these problems is likely to come from the synergy of the topological photonic crystals and silicon-based photonics technology that enables high integration density, lossless propagation, and immunity to fabrication imperfections. For many applications, reconfigurability and capability to control the propagation of these non-trivial photonic edge states is essential. One way to facilitate such dynamic control is to use liquid crystals, which allow to modify the refractive index with external electric field. Here, we demonstrate dynamic control of topological edge states by modifying the refractive index of a liquid crystal background medium. Background index is changed depending on the orientation of a liquid crystal, while preserving the topological order of the system. This results in a change of the spectral position of the photonic bandgap and the topologically protected edge states. The proposed concept might be implemented using conventional semiconductor technology, and can be used for robust energy transport in integrated photonic devices, all-optical circuity, and optical communication systems.
INTRODUCTION
Topological insulators (TIs) build a class of materials that act as insulators in their interior and conduct on the surface, while having a non-trivial topological order [1][2][3][4][5]. The insulating properties result from the absence of conducting bulk states in a certain energy range, known as the bandgap. The interface between materials with different topological order supports strongly confined topologically protected edge states. For these states, the energy transport is robust against structural disorders and imperfections that do not change the system's topology. Until now, many theoretical and experimental demonstrations of TIs have been reported for fermionic (electronic) systems, but most of them work at low temperatures and require strong external magnetic fields which impedes their practical applications [1,[5][6][7][8][9]]. Alternatively, system preserving the time-reversal symmetry that support spin-and valley-Hall effects have been implemented [10,11].
Recent studies have shown the existence of oneway protected topological edge states in bosonic (photonic) systems. With the use of time-reversal symmetry breaking, an analogue of the quantum-Hall effect * These authors made equal contributions. † Corresponding author: [email protected] was achieved [12][13][14][15][16]. Later, analogues for spin-Hall and valley-Hall effects that do not require breaking of the time-reversal symmetry were realized using photonic TIs (PTIs) [1,2,17,18]. A considerable interest has been shown in manipulating photons by the use of an artificial gauge field, which acts as an effective magnetic field for photons [21,22]. Several approaches to engineer synthetic gauge potential emulating an effective magnetic field have been realized [21][22][23][24] by dynamic modulation of the system parameters. Examples include temporal [21,[25][26][27] and spatial modulation using an array of helical waveguides that imitate a breaking of the time-reversal symmetry by breaking the mirror symmetry along the propagation direction [28]. Some of the other proposed realizations include use of metacrystals [29][30][31][32][33][34]. Although these approaches demonstrate the possibility to realize PTIs, most of the designs either operate in the microwave regime or are bulky. Implementation of TIs in photonic systems can pave the way for robust light propagation unhindered by the influence of back-scattering losses. The breakthrough is likely to come from the synergy of the PTI concept and silicon-based photonics technology that enables high integration density, reconfigurability, and immunity to fabrication imperfections. In particular, silicon-based photonic crystals (PCs) offer a promising solution to integration of the fields of silicon photonics and topological photonics [1,2,18]. Indeed, FIG. 1. Schematic view of the structure and working principle of the reconfigurable photonic-crystal-based topological insulator. (a) The structure consists of silicon pillars (magenta) surrounded by a liquid crystal molecules (green) enclosed between conducting electrodes (yellow). It can be switched by applying voltage to the electrodes, such that when the voltage is applied, LC molecules orient along the pillars, resulting in the background refractive index n bg = 1.69 (b) for transverse-magnetic polarization considered here (the corresponding electric field polarized along the pillars is indicated by ↕E). (c) The energy density distribution for the case when switch is ON: light is guided along rhombus-shaped path shown in green. (d) When there is no voltage applied, background refractive index, n bg = 1.51, corresponding to the OFF-state, (e) Power penetrates to the crystal interior (shown in red) resulting in low transmittance along the interface path.
PCs enable implementation of topological effects. Nowadays, the majority of proposed PTIs operate in a fixed wavelength range and their mode of operation cannot be dynamically reconfigured at a high speed. Here, we propose a reconfigurable PTI structure based on PC design to realize the photonic analogue of the spin-Hall effect.
The tunability of transmission properties for the system is facilitated by the liquid crystal (LC) environment surrounding the PC. This structure offers compatibility with CMOS integrated systems, allows for switching at MHz frequencies, and can be designed to operate at telecommunication wavelengths.
RESULTS
Photonic crystals offer an excellent platform to control the flow of light by virtue of the periodicity of the dielectric constants of its constitutive materials [35]. The backbone of the proposed structure is the PC built of silicon pillars immersed in LC environment and enclosed between conducting electrodes; as shown schematically in Fig. 1(a). The design of the PC providing topological protection is based on the work of Wu and Hu [1]. The structure consists of two regions: one with trivial order, and another with topological. At the interface between crystals with different topological properties (shown in cyan), edge states are supported. Each region is built of a triangular lattice containing six pillars per unit cell that build a meta-molecule. Depending on the spacing between the pillars, the structure features a topological or a trivial order. Topological edge states for this type of PC emerge due to the optical analogue of the spin-Hall effect, and were analyzed in Ref. [1] where the PC was surrounded by air.
The PC in our design is immersed in a nematic LC environment, which offers a possibility of refrac-tive index tuning with unprecedented amplitude, reaching 10% at MHz switching speed [13][14][15]. This tuning is enabled by an external electric field supplied by the electrodes bounding the PC from top and bottom [8,10]. Here, we restrict ourself to considering transverse-magnetic-polarized waves at telecommunications wavelength 1.55 µm and assume the use of the E7 nematic LC [9]. When voltage is applied, corresponding to ON-state, the LC molecules align along the silicon pillars, resulting in a background refractive index n bg = n e = 1.69 (extraordinary refractive index) [ Fig. 1(b)]. In this case, light is efficiently guided along a rhombus-shaped path in an edge state located in the bulk bandgap, as shown in Fig. 1(c). If there is no voltage applied, corresponding to the OFF-state, the LC molecules orient perpendicular to the pillars due to the anchoring forces at the electrodes, and light experiences the background refractive index n bg = n o = 1.51 (ordinary refractive index), as shown Fig. 1(d). The change of the background index does not modify the topological properties of the structure, but shifts the location of the bandgap, and bulk states are present at the wavelength of interest, enabling light scattering into the bulk of the PC. For this case, the structure does not support light propagation and energy penetrates into the PC interior resulting in low transmittance along the interface, as shown in Fig. 1(e).
Tunable topological edge states
Topologically protected edge states offer unprecedented possibilities for designing robust guided-wave photonic structures and components, due to their feasibility for supporting light propagation along arbitraryshaped interfaces between trivial and topological regions. In this paper, we consider a structure exhibiting an optical equivalent of the spin-Hall effect. For this case, there is always a pair of essentially decoupled states that propagate in opposite directions and have different spins. In sharp contrast to edge states in standard (trivial) PCs, the pair of topological states do not couple to each other even in presence of disorders and sharp turns along the propagation path.
Let us consider scattering of light in standard PCs by obstacles on the light's path, such as rapid turns, crystal imperfections, or defects. When light propagating in the forward direction impinges on an obstacle, the wavevectors matching backward-propagating state may be introduced. For standard PC, the field distributions of forward-and backward-propagating states possess inversion symmetry along the propagation path, resulting in significant scattering of light from the forward state to the backward one [1,2,4,17], degrading the PC performance and resulting in scattering losses. On the contrary, for topologically protected states, the field shows vortexlike distribution for forward-and backward-propagating states (opposite spins), breaking the inversion symmetry. In this case, the field distributions of opposite spins do not overlap, and therefore the states do not scatter one to another, resulting in the suppression of backward scattering for the photonic analog of spin-Hall effect.
Here, we define the conditions required for achieving such scatter-free propagation. Firstly, at the desired frequency, a non-trivial edge state should exist. To confirm the presence of an edge state, we considered a ribbonshaped PCs shown schematically in the insets in Fig. 2. The band structure of this system reveals the presence of the bulk and edge states. However, the existence of an edge state is not sufficient to have loss-free propagation. Indeed, if besides the non-trivial edge state there is at least one bulk state at the frequency of interest, any obstacle on the light propagation path will cause undesired scattering of light into the PC interior, resulting in losses of energy, as shown in Figs. 2 and 3. Hence, the second necessary condition is the absence of any bulk states at the frequency where topologically-protected propagation is desired.
Light can be confined along the z-direction using two alternative mechanisms. The use of metal electrodes allows for strong light confinement inside the PC slab, however, this approach introduces an additional source of loss in the system at optical frequencies. Alternatively, the electrodes can be located at a distance from the PC slab. The space between electrodes and the slab can be filled with the LC which has much lower index than the PC itself. In this case, the light confinement stems from the total internal reflection between the high-index silicon slab and the low-index cladding. In both cases, these 3-dimensional (3D) systems can be well approximated using 2D analysis. Figures 2(d) and (g) present the band diagrams for the interface between the trivial and topological regions for uniform background refractive indices n tr = n to = 1.51 and n tr = n to = 1.69, respectively. In both cases, two edge states-one corresponding to a pseudo-spin-up (denoted by a minus sign and simply referred to as spinup in the following text) and one corresponding to a pseudo-spin-down (denoted by a plus sign and referred to as spin-down)-are present in the bandgap separating the bulk bands, showing that the change of the background refractive index does not change the topological properties of the system. There is a small frequency range where neither the edge states nor the bulk states exist. This global gap is a result of the avoided crossing of the edge states caused by their interaction due to the broken C 6 symmetry at the conducting interface [1,2,4]. The position and size of the bandgap is affected by variations of the background refractive index. For the case of background index of 1.51, shown in Fig. 2(c), the bandgap spans the normalized frequency range ωa 0 (2πc) ∈ [0.441, 0.462], whereas for the background index of 1.69 shown in Fig. 2(g), the bandgap extends from 0.433 to 0.447. The shrinking and redshifting of the bandgap observed here is consistent with the results presented in Ref. [43]. The effective working wavelength in the medium stays the same, but as a result of an increase in the effective index, the corresponding frequency (free space wavelength) becomes lower (higher) [35].
We choose to analyze the behavior of the structure at a normalized frequency of 0.433 that is located outside of the bandgap for n bg = 1.51, and inside of the bandgap for n bg = 1.69. . Increase of the distance between the meta-molecule center and the pillars, R, leads to stronger coupling between neighboring meta-molecules. This results in the opening of the bandgap with a non-trivial topological order. In contrast, for smaller R, the bandgap opens while maintaining trivial topological order. (d)-(g) Dispersion relations for ribbonlike photonic crystals formed by two regions with trivial and non-trivial topology (single periods along the x-direction are shown in the insets). The mechanism of formation of a bulk bandgap for this case is illustrated in the outside panels by combining the band structure diagrams for bulk topological and trivial crystals with triangular lattice (c) and corresponding refractive indices. Green shading shows the frequency range where topologically protected propagation is supported: in this case an edge state (shown in orange) exists while no bulk states are present. The field distributions for spin-down states with normalized frequency ωa0 (2πc) = 0.433 propagating along a rhombus-like interface between trivial and topological regions are presented in the corresponding plots in Fig. 3 0.433 is located below the bandgap for both trivial (red curve) and topological (green curve) PC geometry. On the contrary, this frequency is located inside the bandgap of both trivial (red curve) and topological (green curve) structures for n bg = 1.69 [dashed curves in the right panel of Fig. 2(g)]. Therefore, as seen in Fig. 3(d), the light propagates along the rhombus-like shaped interface between trivial and topological material without scattering to the bulk. The electric field and the Poynting vector distributions in the vicinity of the waveguiding in- terface for the spin-down eigen-mode of the system with uniform background index n bg = 1.69 at the normalized frequency ωa 0 (2πc) = 0.436 is shown in Fig. 3(e), indicating a strong light localization at the interface between the trivial and topological PC configurations. Moreover, the Poynting vector shows the vortex-like character of energy propagation along the interface associated with the spin-Hall-effect nature of the TI.
We have also analyzed the behavior of mixed configurations where the background refractive indices of the trivial and topological regions have different values. This can be achieved by independent control of the voltages on the electrodes sandwiching the trivial and topological regions. To this end, the top electrodes should be sep- Fig. 2 show that the normalized frequency of interest, ωa 0 (2πc) = 0.433, is located below the bandgap both for the configuration with n to = 1.51, n tr = 1.69 and the inverse configuration with n to = 1.69, n tr = 1.51. Therefore, in both cases, the light scatters into the bulk, and it penetrates the region with the lower background refractive index, as can be seen in Figs. 3(b) and (c). This behavior again can be explained using the right and left panels in Figs. 2(e) and (f), respectively, by analyzing whether the studied frequency lays inside or outside of the bandgap for the bulk trivial and topological PCs. The examples described above show that the transmission of light in the edge states at the interface between the trivial and topological PCs can have drastically different character depending on the background refractive index. Therefore, the control scheme over the background index presented above enables us to modify the transmission properties of the system.
Transmission properties of the reconfigurable topological structure
Let us consider light propagation along an interface between the trivial and topological regions with a 60°r hombus-shaped path. Transmission calculation through the rhombus allowed us to identify spectral positions of the topologically protected guided modes for different configurations of background refractive indices in trivial (n tr ) and topological regions (n to ). The transmission spectrum is calculated according to the following expression:
T (ω) = ∫ out P x (ω)dxdy ∫ in P x (ω)dxdy ,(1)
where P x denotes the x-component of the Poynting vector and the surface integrations are performed over the input and output regions shown in Fig. 4(a). From the typical distribution of P x when the transmission is close to unity, as depicted in Fig. 4(a), we observe that the energy is well localized near the edge.
Here, we consider the dependence of the transmission spectrum on the background refractive index variation for three cases: (i) background index is simultaneously changed in the topological and trivial regions [ Fig. 4(b)];
(ii) background index is modified in the topological region and fixed in the trivial region, n tr = 1.51 [ Fig. 4(c)]; (iii) background index is kept constant in the topological region, n to = 1.51, and varied in the trivial region [ Fig. 4(d)]. Spectral position of topologically protected modes is defined by the top and bottom frequencies of the bulk bandgaps in both topological and trivial regions. These frequencies are dependent on the background refractive index, as shown in Fig. 2. Figures 4(b)-(d) show that when the topologically protected guiding conditions (existence of edge state and lack of bulk states at given frequency) are satisfied, the transmission is high and close to unity. All three of the plots show the red-shift of the guided region with an increase in refractive index, due to the higher average index values. The width of the guided region is decreased for higher background indices as a result of reduced refractive index contrast between the silicon pillars and the background.
Choosing an appropriate operation frequency allows for switching between high and low transmission modes. For instance, propagation at the normalized frequency ωa 0 (2πc) = 0.436 results in low transmission for n to = n tr = 1.51, and topologically protected quasi-unitary transmission for n to = n tr = 1.69. Choosing a 0 = 675 nm results in operational wavelength of 1550 nm. Alternatively, one could operate at ωa 0 (2πc) = 0.457 where the high and low transmission modes are reversed.
CONCLUSION
In this paper, we have proposed a dynamically tunable topological photonic crystal enabled by the photonic analog of the spin-Hall effect. The structure supports edge states at the interface between the trivial and topological parts of the crystal. These edge states are topologically protected and robust against structural disorders and imperfections. Their propagation is supported along arbitrarily shaped paths and around defects. The reconfigurability is facilitated by immersing the photonic crystal into a nematic liquid crystal background. With the help of an external field applied to the liquid crystal, its molecules can be reoriented, causing variation in background refractive index and shifting the spectral position of edge states. We have shown that with rise of background permittivity, edge states exhibit red-shift due to rise in average refractive index of the crystal. The transmission characteristics through the structure can be dynamically tuned by modifying the spectral position of the non-trivial bandgap. Moreover, the topologically protected bandwidth decreases with an increase of the background refractive index because of the reduction in the index contrast between the background and high-index material. We have defined the conditions that are necessary for supporting topologically protected propagation to be: the presence of non-trivial edge states, along with an absence of bulk state(s) at desired guided frequencies.
When these conditions are satisfied, the structure supports topologically protected modes with transmittance close to 100%. Shifting the bandgap position results in scattering of light into the crystal interior, and a decrease in the transmittance through the structure. The reconfigurable photonic topological insulator proposed here is silicon based, and supports operation at telecommunication frequencies, making it attractive for practical applications. An alternative mechanism for transmission control could be achieved by dynamical switching between trivial and topological states of the structure. This concept is outside of the scope of this paper and requires further investigation.
METHODS
Band structure calculations
The band diagrams for bulk PCs with different background refractive indices shown in Fig. 2(c) were obtained using a plane-wave-expansion method. The band diagrams for the edge states shown in Figs. 2(d)-(g) and the energy distributions shown in Fig. 3 were calculated using COMSOL Multiphysics. In order to compute the band diagrams for the edge states we have analyzed a ribbonshaped PC infinitely periodic along the x-direction with a finite size of 30 unit cells of both trivial and topological regions along the y-direction.
Transmission calculations
For transmission calculations, we used commercially available Lumerical FDTD Solutions software. The time domain calculations were carried in the simulation domain shown in Fig. 4(a), and the spectral response was obtained by Fourier transform method. The simulation domain is 50 × 20 unit cells large, and is surrounded by perfectly-matched layers (PMLs). The size of the rhombus-shape path modification is 3 × 3 unit cells.
The system was excited with a spin-down (right circularly polarized light rotating counter-clockwise) dipole point source H + = H x + iH y placed near the interface between trivial and topological parts of the crystal, matching well the profile of the mode propagating in the positive direction of x-axis. The dipole position is shown with the star in Fig. 4(a). Injection of a short [full-width half-maximum of 7 electromagnetic wave periods at frequency ωa 0 (2πc) = 0.417] broadband pulse covering the frequency range ωa 0 (2πc) ∈ [0.417, 0.4837] guaranteed excitation of all potentially guided states for any considered background refractive index combinations. For accurate calculation of transmission, we used the simulation time equal to 1250 electromagnetic wave periods ensuring that all the energy coupled into the crystal is absorbed by PML domains.
ACKNOWLEDGMENTS
This work was supported by Army Research Office (ARO) grants: W911NF-16-1-0270 and W911NF-11-1-0297.
AUTHOR CONTRIBUTIONS
All authors contributed significantly to the work presented in this paper.
SUPPLEMENTARY MATERIALS FOR RECONFIGURABLE TOPOLOGICAL PHOTONIC CRYSTAL
Topological properties of the structure Let us consider a 2D photonic crystal (PC) with a structure of a honeycomb lattice of infinitely long silicon pillars with diameter d (refractive index of silicon is taken to be 3.5) surrounded by a uniform material with the refractive index n bg schematically presented in Fig. 1(a) in the main text of the manuscript. The lattice constant of this honeycomb lattice is equal to a and the unit cell containing two pillars is schematically shown in Fig. S1(a) (dashed lines). In our 2D model, the PC structure is invariant along the z-axis, and we only analyze the behavior of the transverse-magnetic (TM) modes because, simple geometrical modifications of the crystal lattice allow for introduction of topological properties for this light polarization [1]. It is important to notice that the honeycomb lattice described above is equivalent to a larger triangular lattice containing six silicon pillars in a single unit cell. The dimensions of the unit cell of the triangular lattice are: the lattice constant a 0 = √ 3a, the spacing between the unit cell center and the center of the pillar R = a 0 3, and the pillar diameter used here is d = 2 9a 0 . Description of the system in terms of a larger triangular lattice is required to characterize the topologically trivial and non-trivial PC geometries.
Band diagrams for the TM modes of the structure discussed above were calculated using the plane-wave-expansion (PWE) method for background refractive indices of 1.51 and 1.69, and are presented in Figs. S1(d) and (g). The allowed eigen-frequencies ω of the structure are shown in function of the wavevectors lying along the high-symmetry directions in the first Brillouin zone (BZ) of the large triangular structure shown in Fig. S1(b). The Floquet periodicity used for PC analysis assumes the phase difference φ = ⃗ ka n on the opposite unit cell boundaries, where a n denotes one of the lattice vectors. For the periodicity along the x-direction, this phase can be expressed as φ = k x a 0 = (k
(1 st BZ) x + 2π
a0 m)a 0 . Here, k x denotes any of the allowed k-vectors, k Fig. S1(b)] [2]. The Dirac cones are located at points K H and K ′ H of the honeycomb lattice BZ. The doubly-degenerate Dirac cones are the consequence of the C 6 symmetry of the honeycomb lattice [1]. Now, we modify the geometry of the PC by varying the pillar spacing R. This transformation does not affect the C 6 system symmetry but can lead to changes in the topological properties of the PC band structure. For values of R ≠ a 0 3, the lattice can no longer be described as a honeycomb lattice, and therefore, introduction of the triangular lattice with a larger unit cell was necessary. Variations of R lead to a spectral gap opening at the Γ point, where the Dirac cones were located. Although the band-gap is opened regardless of whether we increase or decrease the pillar spacing, the topological character of the gap is different in both cases. Smaller values of R result in weaker coupling between the adjacent meta-molecules and the topology of the system remains trivial. On the contrary, increased values of R result in stronger inter-cell coupling and the gap has a non-trivial topological character (see next section of the Supplementary materials). The band diagrams for the topologically trivial system with R = 0.316a 0 , and for the non-trivial configuration with R = 0.359a 0 are shown in Figs. S1(c), (f) and S1(e), (h), respectively.
The conducting topological edge states exist at an interface between the materials with overlapping band-gaps but different topological orders, defined by the topological invariant known as Chern number [3]. The number of edge states, residing inside the bulk band-gap, is determined by the difference between Chern numbers of the two materials. The Chern number of the crystal lattice studied here can be evaluated using Hamiltonians derived from k ⋅ P theory [1,4] or tight-binding approximation [2]. In the following section, we present the tight-binding approach to the Chern number calculation.
Tight-binding model
The photonic crystal considered in the main text of the manuscript is composed of a unit cell made of six silicon pillars, referred later as a meta-molecule (MM). For the TM-modes of the system, the z-component of the electric field ψ in the unit cell can be represented as a combination of Wannier states:
ψ = X=A,B...F A X ψ X e i ⃗ k⋅⃗ r , (S.1)
where ψ X is the field distribution localized at the atom X, A X is the corresponding complex amplitude, ⃗ k = [k x , k y ] is the light wavevector, and ⃗ r = (x, y) denotes the position. In the matrix notation, the electric field distribution in our system can be represented using a six-element columnvector. The elements of the vector represent the field amplitudes at the corresponding atoms: In this notation, the Wannier basis consists of the following six vectors:
ψ → ψ⟩ = A A , A B , A C , A D , A E , A F T . (S.2)A⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , B⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , D⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 1 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , E⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , F ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (S.3)
and the wavefunction is written as:
ψ⟩ = X=A,B...F A X X⟩. (S.4)
The interaction Hamiltonian of the system can be described using nearest-neighbor hopping potentials. In this model, the interaction Hamiltonian H W (written in the basis of Wannier states) can be separated into the intra-MM coupling term, H 1 , and the inter-MM coupling term, H 2 . The coupling energy between the nearest-neighbor atoms in the same MM is denoted by t 1 , and its contribution to the Hamiltonian is:
H 1 = −t 1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (S.5)
The interaction energy between the neighboring atoms from different MMs is denoted by t 2 , and its contribution to the Hamiltonian is described by:
H 2 = −t 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 e i ⃗ k⋅⃗ a1 0 0 0 0 0 0 e i ⃗ k⋅⃗ a2 0 0 0 0 0 0 e i ⃗ k⋅(⃗ a2−⃗ a1) e −i ⃗ k⋅⃗ a1 0 0 0 0 0 0 e −i ⃗ k⋅⃗ a2 0 0 0 0 0 0 e −i ⃗ k⋅(⃗ a2−⃗ a1) 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (S.6)
where the e i ⃗ k⃗ r XY terms in the interaction between the X and Y atoms in neighboring MMs (the centers of the MMs are separated by the vector ⃗ r XY ) appear as a result of the Floquet periodic boundary conditions. The full interaction Hamiltonian H W = H 1 +H 2 is given by:
H W = − ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 t 1 0 t 2 e i ⃗ k⋅⃗ a1 0 t 1 t 1 0 t 1 0 t 2 e i ⃗ k⋅⃗ a2 0 0 t 1 0 t 1 0 t 2 e i ⃗ k⋅(⃗ a2−⃗ a1) t 2 e −i ⃗ k⋅⃗ a1 0 t 1 0 t 1 0 0 t 2 e −i ⃗ k⋅⃗ a2 0 t 1 0 t 1 t 1 0 t 2 e −i ⃗ k⋅(⃗ a2−⃗ a1) 0 t 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (S.7)
Let us first analyze the system at the Γ point ( ⃗ k = ⃗ 0). The eigen-energies are computed to be:
E s = −2t 1 − t 2 , E p = −t 1 + t 2 , E d = t 1 − t 2
, and E f = 2t 1 + t 2 . They correspond to the eigen-states of the system: Topological properties of any structure can be inferred from a topological invariant called Chern number. Chern number describes the global [computed in the entire first Brillouin zone (BZ)] topological characteristics of the field distributions in a given energy band. In order to compute the Chern number, first we calculate the dispersion of light propagating in our topological structure. Using the PWE method, we obtain the ω( ⃗ k) surfaces for the first 6 bands of the system together with the corresponding electric field distributions E z (⃗ r, ⃗ k) = ψ k . As mentioned above, we are interested only in states p ± ⟩ and d ± ⟩ showing the pseudo-spin character. For each of the energy bands, we compute the Berry connection ⃗ A ± = [A ± x ; A ± y ] = ∬ unit cell ψ ± * k ∇ k ψ ± k dxdy, where minus and plus signs correspond to the spin-up and spin-down states, respectively, and ∇ k denotes a two-dimensional gradient operator in the k-space. Based on the Berry connection, we can compute the Berry curvature as:
s⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 1 1 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , p + ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 e iπ 3 e 2iπ 3 e 3iπ 3 e 4iπ 3 e 5iπ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , d + ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 e 2iπ 3 e 4iπ 3 e 6iπ 3 e 8iπ 3 e 10iπ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , p − ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 e −iπ 3 e −2iπ 3 e −3iπ 3 e −4iπ 3 e −5iπ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , d − ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 e −2iπ 3 e −4iπ 3 e −6iπ 3 e −8iπ 3 e −10iπ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , f ⟩ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 −1 1 −1 1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (S.8)Ω ± = ∂A ± y ∂k x − ∂A ± x ∂k y , (S.11)
Plots of the Berry curvature in the vicinity of the Γ point for our topological structure are shown in Fig. S3. Finally, the Chern number is calculated as:
C ± = 1 2πi ∬ 1 st BZ Ω ± dk x dk y . (S.12)
In our numerical approach, the eigen-spectrum of the Hamiltonian H pd given by Eq. (S.10) is fitted to the ω( ⃗ k) curves obtained from the PWE method allowing us to determine the t 1 and t 2 parameters. Then, the eigen-states of H pd are used to compute the Berry curvature and the Chern number using the approach presented in Ref. [7].
Reconfigurability enabled by liquid crystals
Liquid crystals (LCs) are a state of matter that simultaneously exhibit properties of liquids (fluidity) and crystals (anisotropy resulting from orientation order of the molecules). LCs offer a possibility of refractive index tuning with unprecedented amplitude reaching 10% [8]. The elongated nematic LC molecules, schematically shown in Fig. S4(a), are preferentially aligned along a particular direction, called director, resulting in uniaxial anisotropy of the LC medium. The refractive index experienced by the extraordinarily polarized eigen-wave, n e (θ), changes depending on the angle θ between the directorn and the light propagation direction ⃗ S: n e (θ) = n e n o n 2 e cos 2 (θ) + n 2 o sin 2 (θ)
1 2 , (S.13)
where n o denotes the ordinary refractive index and n e is the index of the wave polarized along the optical axis of the molecule. In this paper, we consider only the two extreme orientations of the LC molecules: (i) perpendicular to the PC pillars, for which the TM-polarized wave experiences the background index n bg = n o ; and (ii) parallel to the PC pillars, for which θ = 90°and n bg = n e , as explained below. The PC structure studied in the main text of the manuscript can be bound from top and bottom by metal plates, as illustrated in Fig. 1(a) in the main text of the manuscript, that serve two goals. Firstly, for sufficiently small separation between the plates, the field distribution inside the PC structure remains uniform in the z-direction. Alternatively, the electrodes can be located at a distance from the PC slab. The space between electrodes and the slab can be filled with the LC which has much lower index than the PC itself. In this case, the light confinement stems from the total internal reflection between the high-index silicon slab and the low-index cladding. In both cases, these 3-dimensional (3D) systems can be well approximated using 2D z-invariant modeling. This approximated treatment does not take into account the losses resulting from light penetrating into the metal regions and additional optimization is required to minimize these losses. Our 2D PC supports transverse-electric and transverse-magnetic modes. Here, we are only interested in the TM modes because the studied structure exhibits topological properties only for the this light polarization [1]. The TM mode has non-zero magnetic field components H x and H y lying in the x-y plane perpendicular to the silicon pillars and an electric field component E z along the z-axis parallel to the pillars.
Secondly, the conductive plates serve as electrodes supplying an external electric field along the z-direction that allows for LC reorientation [8,10]. Due to the birefringent nature of LCs, the electric field exerts a torque on the LC molecules and, in case of positive birefringence (∆n = n e − n o > 0), reorients them along the field direction. In our structure, the surface of the electrodes is prepared in such a way that the anchoring energy holds the surface molecules aligned parallel to the electrodes. In the absence of the electric field, the bulk molecules align along the surface molecules to minimize the global LC free energy, as shown in Fig. 1(d) in the main text of the manuscript. In this case, the TM-polarized wave is the ordinary wave and it experiences the background refractive index n bg = n o , irrespectively of the propagation direction in the x-y plane and the in-plane director orientation. Application of an external electric field, with the amplitude above the threshold of the Fréedericksz transition, results in rotation of the LC molecules [11], as shown in Fig. 1(b) in the main text of the manuscript. Then, the director lies along the z-axis and is perpendicular to the light propagation direction (θ = 90°). In this configuration, the background refractive index experienced by the TM-polarized wave (extraordinary wave) is n bg = n e , irrespectively of the in-plane propagation direction.
Orientations of the LC molecules at intermediate values of the angle θ, in addition to being difficult to achieve experimentally, also break the reflection symmetry of the PC structure with respect to the x-y plane. As a result, the TM-polarized wave excites both the ordinary and the extraordinary waves, and experiences polarization rotation due to the birefringence of the LC background. Systems with intermediate angles θ are not considered here and require further investigation.
Rotation of the LC molecules takes a finite amount of time due to their inertia and viscosity [12]. Typically, this switching time is of the order of microseconds allowing for device operation at MHz frequencies. Recently, submicrosecond switching times were also reported [13][14][15]. In this paper, we use the E7 LC at the room temperature T = 25°C which permits background index changes in the range between n o = 1.51 and n e = 1.69 [9,10]. Additional tunability can be added exploiting the fact that the values of n o and n e can be modified by temperature tuning, as seen in Fig. S4(b). For low temperatures, the LC birefringence is high, and the reorientation of the molecules leads to large changes in the background refractive index. At higher temperatures, the tunability is possible only in a limited range of background refractive indices, and therefore only a part of the parameter space shown in Figs. 4(b)-(d) in
FIG. 2 .
2Figures 3(a) and (d) show the propagation of light along an interface with a rhombus-shaped defect at ωa 0 (2πc) = 0.433 for background refractive indices of 1.51 and 1.69, respectively. When the background index has the value of 1.51, the edge state does not exist and the light couples to the bulk modes of both the trivial region located on the top of the structure, as well as the topological region on the bottom. This behavior can be understood by looking at the band structures for infinitely periodic triangular PCs shown in the left and right panels in Figs. 2(d) and (g), respectively. For n bg = 1.51 (see solid curves), the normalized frequency Dispersion relations for the trivial and topological photonic crystals and the edge state diagrams at the interfaces with different topological orders. (c) Band diagrams for trivial and topological PCs with hexagonal unit cell (a) and different background refractive indices. The first Brillouin zone of the triangular lattice is shown in panel (b)
FIG. 3 .
3Light propagation along the interface between trivial and topological photonic crystals for different values of background refractive indices. (a)-(d) Energy density distributions of the spin-down states at normalized frequency ωa0 (2πc) = 0.433, indicated by the dash-dot line in Fig. 2, for the four different configurations of background index values described inFig. 2. The color maps show that topologically protected propagation is supported only for the case (d) where there is no bulk state allowed for both topological and trivial regions at the considered frequency, as shown in the right panel ofFig. 2(g). For the cases shown in (b) and (c), light penetrates inside the topological and trivial parts, respectively, due to the presence of bulk states for these regions, as shown in the right and left panels in Figs. 2(e) and (f), respectively. Bulk states are allowed in non-trivial and trivial regions for the case (a) resulting in light penetration to both regions [seeFig. 2(d)]. (e) Energy density (color map) and the Poynting vector (white arrows) for a spin-down mode in the structure with ntr = nto = 1.69 at the normalized frequency ωa0 (2πc) = 0.436, corresponding to the mode indicated by the green point inFig. 2(g).
FIG. 4 .
4Transmission characteristics of the reconfigurable topological insulator. (a) Normalized Poynting vector distribution along the x-axis, Px, at the normalized frequency ωa0 (2πc) = 0.4436 showing light propagation along a rhombusshaped interface between the trivial and topological photonic crystals. The light is excited with the magnetic dipole source H+ = Hx + iHy (indicated with a star). (b)-(d) Transmission spectrum dependent on background refractive index for three different cases: (b) background refractive index varied in both the topological and trivial regions simultaneously; (c) background index is varied in the topological part and is fixed in the trivial one ntr = 1.51; (d) background index is fixed in the topological region nto = 1.51 and varies in the trivial part. For all cases, with the increase of the background refractive index, the index contrast in the structure decreases, leading to the narrower transmission bandwidth. Furthermore, for higher background indices, the average refractive index increases, resulting in the red-shift of the spectral location of the guided region. arated by a thin insulating layer. The band diagrams for these systems are presented in Figs. 2(e), (f) and the corresponding light propagations through the rhombuslike defect is shown in Figs. 3(b), (c). The band diagrams presented in
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( 1
1st BZ) x is its equivalent from the first BZ, and m ∈ [0, 1, 2, . . . ] denotes the number of full field oscillations in the unit cell. We observe the presence of a doubly-degenerate Dirac cone at the Γ point, which is a result of folding of the bands from the larger BZ corresponding to the smaller unit cell of the . S1. (a) Schematic illustration of the unit cells of the triangular lattice (shaded in green) and the honeycomb lattice (dashed lines) with the geometrical parameters specified in the text. (b) The first Brillouin zones of the triangular lattice (solid lines) and the honeycomb lattice (dashed lines). The high symmetry points of the triangular lattice are Γ = = [4π (3a0), 0], and M = [π a0, π ( √ 3a0)]. High symmetry points of the honeycomb lattice are, K H = [0, −4π ( √ 3a0)], M H = [π a0, − √ 3π a0], and K ′ H = [2π a0, −2π ( √ 3a0)]. (f)-(h) Full photonic band structures, and zooms on the band-gap regions (c)-(e) of the honeycomb lattice (triangular lattice with R = a0 3-blue) exhibiting symmetry protected double Dirac cones (d), (g); triangular lattice with R = 0.316a0 (red) and opened trivial gap (c), (f); and topological triangular lattice with R = 0.359a0 (green) with non-trivial gap (e), (h). Solid lines and dashed lines correspond to the background refractive indices of 1.51 and 1.69, respectively.
FIG. S2 .
S2Meta-molecules (MMs) made of six silicon pillars forming the triangular lattice of our system. A unit cell containing a single MM is shown in green. The lattice vectors are given by: a1 = [ √ 3; 0]a and a2 = [ √ 3 2; 3 2]a. The intra-MM and inter-MM coupling coefficients are represented by t1 and t2, respectively.
FIG. S3 .
S3Berry curvatures Ω± and Chern numbers C± of corresponding bands in the vicinity of the Γ point for the system with R = 0.359a0 and n bg = 1.51.
. S4. (a) Schematic illustration of a liquid crystal molecule showing the refractive indices along the long (director) axis ne, along the short axis (ordinary index) no, and the extraordinary refractive index ne(θ). The angle between the director axisn and the light propagation direction ⃗ S is called θ. (b) Dependence of the refractive indices no and ne on temperature T for the E7 liquid crystal at 1550 nm obtained using model from Ref.[9].
COMPETING FINANCIAL INTERESTSThe authors declare no competing financial interests.The energy levels E p and E d are doubly degenerate. The eigen-states of the system at the Γ point with the lowest and highest energies resemble the atomic orbitals s and f , respectively. The eigen-states with intermediate energies, have a spin character and can be represented as a linear combination of atomic orbitals p x , p y , d xy , and d x 2 −y 2 :The field distributions for the eigen-states p±⟩ have the phase e ±iφ , where the angle φ is measured in the coordinate frame centered in the middle of the unit cell. These states have a strong p character. Similarly, field distributions for the eigen-states d±⟩ have the phase e ±2iφ , and these states have a strong d character. Both p and d states possess rotating (vortex-like) phase distributions analogous to the orbital angular momentum (pseudo-spin). The minus and plus sign corresponds to the clockwise (spin-up) and counterclockwise (spin-down) phase rotation direction, respectively.In order to investigate the interaction between the spin-up and spin-down states, we express the Hamiltonian H W in the basis of s⟩, p ± ⟩, d ± ⟩, and f ⟩ orbitals. The new Hamiltonian is obtained by a standard procedure of change of the basis and is given by:where M is a 6 × 6 matrix of the form:whose columns represents the new basis vectors (s, p ± , d ± , f ) in terms of the old basis vectors (Wannier functions). The Hamiltonian in the new basis has the form:where the terms e i ⃗ k⃗ r XY have been expressed using the Taylor expansion up to the terms linear (quadratic) in k x and k y in the off-diagonal (diagonal) elements.Our main interest is to investigate the interaction between the spin states p ± ⟩ and d ± ⟩ because their interaction leads to the topological protection in the system. Therefore, in the following, we will consider a reduced 4 × 4 Hamiltonian containing only contributions from p ± ⟩ and d ± ⟩ states [highlighted in Eq. (S.9)]. This Hamiltonian has a block diagonal form[5]:and σ x , σ y , and σ z are the Pauli matrices. The Hamiltonian H pd has the same form as the Bernevig-Hughes-Zhang Hamiltonian describing the quantum spin-Hall effect in a quantum well systems[6]. Thus, our structure is capable of supporting optical pseudo-spin states. The term t 1 − t 2 is proportional to the energy difference at the Γ point between the p ± ⟩, d ± ⟩ states. Depending on the sign of t 1 − t 2 , the character of the bands might be inverted leading to a nontrivial topological properties of the system. For t 1 > t 2 , the intra-MM coupling dominates and the energy of p ± ⟩ bands is lower than the energy of d ± ⟩ states, leading to the trivial state of the system. On the contrary, for t 2 > t 1 , the inter-MM coupling is dominant, leading to inversion to the p ± ⟩ and d ± ⟩ bands at the Γ point and resulting in non-trivial topological properties[1]. When t 2 = t 1 , the p and d bands are degenerate at the Γ point, resulting in the formation of the Dirac cone.
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| [] |
[
"Industrial DevOps",
"Industrial DevOps"
] | [
"Wilhelm Hasselbring ",
"Sören Henning ",
"Björn Latte ",
"Armin Möbius \nIBAK Helmut Hunger GmbH & Co. KG\nWehdenweg 12224148KielGermany\n",
"Thomas Richter \nKrause-Biagosch GmbH\nPaul-Schwarze-Straße 533649BielefeldGermany\n",
"Stefan Schalk ",
"Maik Wojcieszak ",
"\nDepartment of Computer Science\nSoftware Engineering Group\nKiel University\nWittland 2-424098, 24109Kiel † wobe-systems GmbH, KielGermany\n"
] | [
"IBAK Helmut Hunger GmbH & Co. KG\nWehdenweg 12224148KielGermany",
"Krause-Biagosch GmbH\nPaul-Schwarze-Straße 533649BielefeldGermany",
"Department of Computer Science\nSoftware Engineering Group\nKiel University\nWittland 2-424098, 24109Kiel † wobe-systems GmbH, KielGermany"
] | [] | The visions and ideas of Industry 4.0 require a profound interconnection of machines, plants, and IT systems in industrial production environments. This significantly increases the importance of software, which is coincidentally one of the main obstacles to the introduction of Industry 4.0. Lack of experience and knowledge, high investment and maintenance costs, as well as uncertainty about future developments cause many small and medium-sized enterprises hesitating to adopt Industry 4.0 solutions. We propose Industrial DevOps as an approach to introduce methods and culture of DevOps into industrial production environments. The fundamental concept of this approach is a continuous process of operation, observation, and development of the entire production environment. This way, all stakeholders, systems, and data can thus be integrated via incremental steps and adjustments can be made quickly. Furthermore, we present the Titan software platform accompanied by a role model for integrating production environments with Industrial DevOps. In two initial industrial application scenarios, we address the challenges of energy management and predictive maintenance with the methods, organizational structures, and tools of Industrial DevOps. | 10.1109/icsa-c.2019.00029 | [
"https://arxiv.org/pdf/1907.01875v1.pdf"
] | 155,107,306 | 1907.01875 | a5ed7c613806aa47328a6d92910f352d64e0700f |
Industrial DevOps
Wilhelm Hasselbring
Sören Henning
Björn Latte
Armin Möbius
IBAK Helmut Hunger GmbH & Co. KG
Wehdenweg 12224148KielGermany
Thomas Richter
Krause-Biagosch GmbH
Paul-Schwarze-Straße 533649BielefeldGermany
Stefan Schalk
Maik Wojcieszak
Department of Computer Science
Software Engineering Group
Kiel University
Wittland 2-424098, 24109Kiel † wobe-systems GmbH, KielGermany
Industrial DevOps
https://www.industrial-devops.org
The visions and ideas of Industry 4.0 require a profound interconnection of machines, plants, and IT systems in industrial production environments. This significantly increases the importance of software, which is coincidentally one of the main obstacles to the introduction of Industry 4.0. Lack of experience and knowledge, high investment and maintenance costs, as well as uncertainty about future developments cause many small and medium-sized enterprises hesitating to adopt Industry 4.0 solutions. We propose Industrial DevOps as an approach to introduce methods and culture of DevOps into industrial production environments. The fundamental concept of this approach is a continuous process of operation, observation, and development of the entire production environment. This way, all stakeholders, systems, and data can thus be integrated via incremental steps and adjustments can be made quickly. Furthermore, we present the Titan software platform accompanied by a role model for integrating production environments with Industrial DevOps. In two initial industrial application scenarios, we address the challenges of energy management and predictive maintenance with the methods, organizational structures, and tools of Industrial DevOps.
I. INTRODUCTION
The digital transformation of the conventional, manufacturing industry enables a new level of automation in production processes. More and more technical machines and production plants become increasingly intelligent and autonomous. Equipped with network capabilities, they are able to consume and supply data to others. This new trend, often referred to as Industrial Internet of Things or Industry 4.0 [1], confronts the production operator with the challenge of connecting and monitoring the individual machines and devices [2]. While software used to be only part of the production process, nowadays software increasingly defines the production process itself [3], so that the resulting software systems inevitably become more complex. In order to ensure a smooth production process these software systems have to be designed with a special focus on reliability, scalability, and adaptability [4].
As is usual with complex systems, however, this complicates the design, development, and maintenance of such a system. In particular for small and medium-sized enterprises (SMEs), this poses enormous challenges as these SMEs often do not This research is funded by the Federal Ministry of Education and Research (BMBF, Germany) in the Titan project (https://www.industrial-devops.org, contract no. 01IS17084B).
have suitable software development departments. Instead, software is often developed by domain experts (e.g., mechanical engineers) with basic programming skills, but no education in software engineering. Alternatives such as establishing a dedicated software development department or delegating software development to specialized external companies are also risky. Business or domain requirements and their technical implementations often diverge, so adaptations are cumbersome, costly, and the time until they are released is long.
Agile and iterative principles, methods, and techniques that are common in other fields of software engineering, such as e-commerce systems [5], [6], can provide solutions to this. Section II of this paper describes how these methods can be transferred to the domain of industrial production environments and highlights necessary changes in processes and culture. Section III presents a software platform with which these methods can be applied in the manufacturing industry. Section IV describes a role model for applying the required organizational and technical structures and Section V outlines our two initial application scenarios. Section VI concludes this paper and points out future work.
II. INDUSTRIAL DEVOPS
The traditional separation of software operation from its development leads to several issues due to a lack of communication, collaboration, and integration. DevOps [7] is a movement to bridge this gap. We propose Industrial DevOps as an approach for transferring DevOps values, principles, and methods to industrial systems integration. In this way, we expect that development and operation of such an integrating software system will be improved and the discrepancy between production operators and software developers will be reduced, while ensuring high software quality. The following principles are of crucial importance for Industrial DevOps.
1) Continuous Adaption and Improvement Process: The core element of Industrial DevOps is a coherent, cyclic, and continuous process as illustrated in Fig. 1. During its operation, the software system as well as the production itself is monitored and analyzed. Based on analysis results, new requirements are identified. The implementation of these requirements will either be done by adapting the software or by reconfiguration in the IT operation or production. Adjustments are tested automatically and after passing all tests, the new arXiv:1907.01875v1 [cs.SE] 3 Jul 2019 Applying this iterative process aims to enable continuous adaptions and improvements of the software. Thus, complex systems are going to be created by starting from simple ones that are extended in incremental steps. Another advantage can be found in the maintenance of the systems. While IT systems that run in production after a development phase are provided with updates for a certain period of time, followed by a certain period of service, the actual operating time is often considerably longer. As a result, operation becomes increasingly complex and risky, but at the same time migrating to another system becomes more and more difficult. The cyclic process of Industrial DevOps is intended to achieve that development, updates, and service become continuous processes that run permanently parallel to the operation of the system.
2) Lean Organizational Structure: When applying external software solutions, this often requires enterprises to adapt their production processes to these solutions. Industrial DevOps comes with an organizational structure to prevent this and instead support the alignment of the software with the production process.
A key factor of Industrial DevOps is that requirements are discussed between all stakeholders and people from different business units are brought together. This practice is referred to as BizDevOps as a more general term [8]. Industrial DevOps extends this by applying BizDevOps to integrating systems and data of industrial production environments. For the implementation of these requirements, domain experts and developers work closely together. Moreover, the development provides appropriate means, which allow domain experts to solve the domain problems themselves to a certain extent. This is accomplished by an extendable software platform, which is owned by the user. Ownership can be direct or indirect if the software is owned by an open source community.
Information from all levels is available for anyone in the value stream through monitoring. Even if the information that is required differs, the source of information is consistent and allows investigation about cause and effect (culture of causality). This should enable lean organizational learning [9].
3) Customer-Centric Value Generation: Organizations who apply customer-centric value generation align all departments to contribute to this goal [10]. All activities in the organization which do not directly or indirectly contribute to this goal are abandoned. Some activities which are required for organizational or compliance reasons are organized to stay out of the way of the overall goal. Instead of wasting huge amounts of money in market research, experiments are started in a way that new features, functions, or products are offered to customers and direct measurements are made by utilizing IT to find out the customers response. These measurements are not only done once, but continuously.
III. THE TITAN PLATFORM
Titan is a software platform for integrating and monitoring industrial production environments using Industrial DevOps. In order to integrate tools or infrastructure (e.g., machines, IoT devices, software, or databases), modular software components
Business and Production
IT and Platform
Process
Monitoring Fig. 2. The Titan role model are created. These components, in Titan called bricks, serve for integrating exactly one such system. This allows on the one hand to exchange the underlying tool or infrastructure easily, but on the other hand also to replace the integrating software implementation. To connect bricks (and thus the underlying systems of the production), Titan applies the principles of flow-based programming [11] and calls a network of connected bricks a flow. Titan provides a graphical modeling language, which enables the organization to graphically model the integration and, based on this, to configure the integrated system. Hence, there is no programmer required to perform these configurations or (pre-configured) changes. Instead, a person that has received training on the modeling language can work together with a domain expert to do this.
Thus, this platform allows to react efficiently and effectively to changing requirements in production. Furthermore, it enables conducting experiments with little effort and to adjust quickly in case of experiment failures. Titan integrates the required monitoring, which includes resource consumption, system utilization, and business data.
IV. THE TITAN ROLE MODEL
Industrial DevOps involves people from different domains, business units, and organizations. The Titan role model (see Fig. 2) defines roles to be assigned to people or groups that use, operate, or develop the Titan platform and describes their responsibilities as well as how they interact.
A Developer is a trained software engineer. This person has learned to develop software in many programming languages. Many different aspects of software development like code quality or version management are covered. In Industrial DevOps, many developers who are spread over many teams in different organizations work together on a single system. The role model introduces the following sub-roles: The Platform Developer works on new features and improvements of the Titan platform itself. The Tool Developer creates software tools or interfaces for hardware that is integrated into the system. The Brick Developer creates bricks and works closely together with the Tool developers.
The IT Operator is trained in installing, configuring, and running IT infrastructure. This person is also responsible for installing, updating, configuring, and running software applications.
The Flow Manager is a person that is closely related to the business, but with a special training to master all aspects of flow modeling. This person works as a connection between business and technology. The Flow Manager implements new flows and flow changes according to business requirements.
The Business Unit Manager can be located on all levels of the hierarchy. This person knows about the business require-ments and works together with the Flow Manager, the Domain Expert, and the Process Operator to refine requirements.
The Domain Expert owns deep knowledge about the domain at hand.
The Process Operator benefits from a new flow or a change. This person is involved in utilizing the flow in the actual production environment.
The IT Security Expert is responsible for the IT security domain of the whole organization. The IT Security Expert works together with development and operations through all levels of the system.
As an extension to regular DevOps roles, the Titan role model introduces business roles, domain experts, flow managers, and process operators, which do not need to be developers.
V. INITIAL INDUSTRIAL APPLICATION SCENARIOS
Enterprises, in particular in the manufacturing industry, face the challenge of reducing and optimizing their energy consumption for economic and ecological reasons. This requires in-depth monitoring and analysis of the individual energy consuming devices, machines, and plants. Doing this, also referred to as energy management, serves as an initial application scenario for Industrial DevOps and Titan. We designed a monitoring infrastructure, which monitors, analyses, and visualizes the electrical power consumption in industrial production environments [12]. Employing the Titan platform, it is able to integrate different kinds of sensors that use different data schemata, formats, and protocols. Referring to the Titan role model, this monitoring infrastructure enables Business Unit Managers, Domain Experts, and Process Operators to gain detailed insights into the energy usage and to identify saving potentials. Thus, it is possible to perform and evaluate optimization measures across departmental boundaries within the company.
Predictive maintenance is another initial application scenario we address with Industrial DevOps and Titan. Titan can serve as a platform, which detects faults in machines and production plants in advance on the basis of data from integrated sensors and systems. An essential aspect here is that not only data from a single machine and production plan is considered for analysis. Instead, measurements and events from of the entire production as well as environmental influences can be taken into account. The predictive maintenance function thus has a much broader amount of data available and can therefore also include non-obvious influences. With the insights gained from the data, maintenance intervals can be optimized. Additionally by returning analysis results to the machines, they can optimize themselves.
VI. CONCLUSIONS AND FUTURE WORK
We propose Industrial DevOps as an approach to transfer practices, methods, and culture of DevOps for integrating systems in industrial production environments. It suggests an organizational structure in which domain experts, business managers, developers, and operators work together on solutions. Production environments should be integrated, configured, and operated in a validated learning loop to manage changes and to control costs.
Future work lies in evaluating the proposed principles, methods, and organizational structures of Industrial DevOps in different industrial application scenarios. Therefore, we further develop the Titan platform. To overcome the disadvantages of proprietary integration systems, we will release it as free and open source software. We are already using Clean Code approaches in the development to prevent software quality from degrading over time [13]. Along with including quantitative quality characteristics such as performance in the DevOps cycle [14] as well as rigorous monitoring in operation, this is supposed to contribute in creating a long-living software platform [15].
Fig. 1 .
1The continuous adaption and improvement process of Industrial DevOps software replaces the old one in production, where it is monitored again.
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| [] |
[] | [
"Björn Ahrens \nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA\n\nInstitut für Physik\nCarl-von-Ossietzky Universität\n26111OldenburgGermany\n",
"Jianping Xiao \nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA\n",
"Alexander K Hartmann \nInstitut für Physik\nCarl-von-Ossietzky Universität\n26111OldenburgGermany\n",
"Helmut G Katzgraber \nDepartment of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA\n\nMaterials Science and Engineering Program\nTexas A&M University\n77843-3003College StationTXUSA\n\nTheoretische Physik\nETH Zurich\nCH-8093ZurichSwitzerland\n"
] | [
"Department of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA",
"Institut für Physik\nCarl-von-Ossietzky Universität\n26111OldenburgGermany",
"Department of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA",
"Institut für Physik\nCarl-von-Ossietzky Universität\n26111OldenburgGermany",
"Department of Physics and Astronomy\nTexas A&M University\n77843-4242College Station, TexasUSA",
"Materials Science and Engineering Program\nTexas A&M University\n77843-3003College StationTXUSA",
"Theoretische Physik\nETH Zurich\nCH-8093ZurichSwitzerland"
] | [] | We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models. | 10.1103/physrevb.88.174408 | [
"https://arxiv.org/pdf/1302.2480v2.pdf"
] | 119,202,118 | 1302.2480 | c21ef2472a25c30e74d15ab09572f391f0077aef |
13 Nov 2013 (Dated: November 14, 2013)
Björn Ahrens
Department of Physics and Astronomy
Texas A&M University
77843-4242College Station, TexasUSA
Institut für Physik
Carl-von-Ossietzky Universität
26111OldenburgGermany
Jianping Xiao
Department of Physics and Astronomy
Texas A&M University
77843-4242College Station, TexasUSA
Alexander K Hartmann
Institut für Physik
Carl-von-Ossietzky Universität
26111OldenburgGermany
Helmut G Katzgraber
Department of Physics and Astronomy
Texas A&M University
77843-4242College Station, TexasUSA
Materials Science and Engineering Program
Texas A&M University
77843-3003College StationTXUSA
Theoretische Physik
ETH Zurich
CH-8093ZurichSwitzerland
13 Nov 2013 (Dated: November 14, 2013)arXiv:1302.2480v2 [cond-mat.dis-nn] Diluted antiferromagnets in a field seem to be in a different universality class than the random-field Ising modelnumbers: 6460De7510Nr7540-s7550Lk
We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.
I. INTRODUCTION
The random-field Ising model 1 (RFIM) is of paramount importance in the field of disordered systems. [2][3][4][5] A plethora of problems across disciplines can be studied via the RFIM, ranging from the thermodynamics of disordered magnets, 6 hysteresis in magnetic systems and Barkhausen noise, 7-9 tunable domain-wall pinning, 10 the random pinning of polymers, 11 and even water seepage in porous media. As such, the RFIM is still under intense theoretical, as well as numerical and experimental scrutiny.
More recently, the RFIM has been realized in diluted dipolar magnets in a transverse field such as LiHo x Y 1−x F 4 . However, most experimental studies focus on diluted antiferromagnets in a field (DAFF), such as Fe x Zn 1−x F 2 . 3,12-15 Fishman and Aharony 16 were the first to note that a random antiferromagnet in a field can be described by the RFIM, and Cardy 17 predicted, using a mean-field argument, that the critical behavior of both models should be in the same universality class in the limit of small fields. The work of Fishman and Aharony, 16 as well as Cardy, 17 therefore opened the door for intense experimental investigation of the RFIM via DAFF materials. However, early experiments and simulations already hinted towards discrepancies between experimental and numerical estimates of the critical exponents. 3,18,19 On the other hand, exact groundstate calculations using moderate system sizes suggested an agreement between the critical exponents for both models when the random fields are Gaussian distributed, however not when the random fields are drawn from a bimodal distribution. 18,19 This result, however, has been revised recently, 20 i.e., the universality class of the RFIM is independent of the form of the implemented randomfield distribution.
In this paper we perform detailed Monte Carlo sim-ulations of both the RFIM and the DAFF. The latter is studied at 30% dilution, i.e., below the percolation threshold for vacancies. Using a finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation function, and the susceptibility, we show that even in the limit of small fields-where the Cardy mapping 17 is expected to work-both models seem to be in different universality classes. Therefore, care should be taken when making predictions for the critical behavior of the RFIM using experiments on DAFF materials. Finally, we present heuristic analytical expressions based on our numerical data for the phase boundaries of both models to help guide experimental studies. The manuscript is structured as follows. In Sec. II we introduce both the RFIM and the DAFF, followed by an explanation of the used algorithms in Sec. III, as well as the measured quantities in Sec. IV. In Sec. V we show our numerical results, followed by a detailed discussion of the phase boundaries and universality between both models in Sec. VI.
II. MODELS
The Hamiltonian of the diluted antiferromagnet in a field (DAFF) is given by
H DAFF = +J i,j ε i ε j S i S j − B i ε i S i ,(1)
and the Hamiltonian for the random-field Ising model (RFIM) is
H RFIM = −J i,j S i S j − h i δ i S i .(2)
In Eqs. (1) and (2) S i ∈ {±1} represent Ising spins, J = 1 is the coupling constant between two adjacent spins, and i, j denotes a sum over nearest neighbors. The linear term in S i couples to an external field: For the DAFF it is an externally-applied uniform field B, whereas for the RFIM the spins couple to a random field of strength hδ i , where the δ i are quenched random variables chosen from a Gaussian distribution with zero mean and standard deviation unity. This means that the typical field has strength h. In the DAFF ε i ∈ {0, 1} represents the site dilution, where each site is randomly and independently occupied by a spin (ε i = 1) with probability p.
Here, we fix the dilution to 1 − p = 0.3. Both models are studied in three space dimensions on a lattice with N = L 3 spins, L being the linear size of the lattice.
III. ALGORITHM
The simulations are done using the Machta-Newman-Chayes replica-exchange (MNC) algorithm 21 combined with single-spin Metropolis Monte Carlo. 22,23 The MNC algorithm is a mixture of the Swendsen-Wang exchange algorithm 24 and simulated tempering Monte Carlo. 25,26 Note that the latter is not efficient when simulating random-field systems. 27 The advantage of the MNC algorithm over standard parallel tempering lies in the fact that we can choose any path in the field-temperature plane. Although parallel tempering can also be implemented with a variable field, the method does not perform efficiently when systems have disorder. 28 In the MNC algorithm 21 a cluster of connected spins is grown between two replicas with the same disorder but at different points in the parameter space, i.e., (T, B) and (T ′ , B ′ ), where T represents the temperature and B the external field (here for the case of the DAFF). Starting from an arbitrary spin with different sign in both realizations, adjacent spins pointing in the same direction are successively added to the cluster with probability
p(β, β ′ ) = 1 − exp{−2(β + β ′ )} ,(3)
where β = 1/T is the inverse temperature. Once no more spins can be added to the cluster C, it flips with the Metropolis probability 29 min{1, exp(−Σ)}, where
Σ DAFF = 2sign(C) (β−β ′ )(n ++ −n −− )+(B−B ′ )|C| (4)
for the DAFF, and for the RFIM
Σ RFIM = 2sign(C) (β − β ′ )(n ++ − n −− ) − (h − h ′ ) i∈C δ i .(5)
Here |C| is the number of spins in the cluster, sign(C) the orientation of the spin in the replica having inverse temperature β, n ++ and n −− are the number of bonds connecting to nearest neighbors of the cluster with spin up and spin down in both replicas, respectively. After each cluster update, (L/2) 3 attempts to flip single spins are performed, where L is the linear size of the system. As stated before, the MNC algorithm enables us to perform simulations along any arbitrary path in parameter space. We denote such path a replica chain (RC). The phase boundaries for the RFIM and DAFF in the field-temperature plane are well described by ellipses (see below). To reduce corrections to finite-size scaling 30,31 we therefore choose paths in the field-temperature plane that cut the phase boundaries at as orthogonal an angle as possible. This means that, in general, T ∼ h for the RFIM and T ∼ B for the DAFF. To ensure efficient mixing and therefore fast convergence of the Monte Carlo method, we additionally connect the point with the highest field within the disordered phase to another RC that runs parallel to the approximated phase boundary to a temperature T > T c and B = 0 (h = 0 for the RFIM), where T c is the critical temperature of the model at zero field (see Fig. 5, light dashed lines). This end point of the second RC is simulated efficiently by the Wolff cluster algorithm. 32 Simulation parameters are listed in Tables I and II for the RFIM and DAFF, for the first RCs, respectively.
Finally, we also study the DAFF at zero temperature using the method introduced in Refs. 33 and 34. Here, the DAFF is mapped onto a graph 35 with N nodes (N is the number of spins) attached to a source and a sink node, all connected in a distinct manner via edges with positive edge weights. The edge weights are calculated depending on the local staggered field, i.e., ±B. The maximum flow/minimum cut is obtained using the algorithm introduced in Ref. 36. The minimum cut is a direct representation of the ground-state spin configuration from which derived quantities, such as a zerotemperature Binder ratio, can be calculated. Note that the method takes the ground-state degeneracy into account. The simulation parameters for the DAFF at zero temperature are shown in Table III.
IV. OBSERVABLES
Both the DAFF and RFIM undergo second-order phase transitions as a function of temperature and field. To pinpoint the transition temperature, we measure the Binder cumulant, 37 as well as the two-point finite-size correlation function. [38][39][40] To compute these observables, we measure the magnetization per spin
M = 1 N N i S i .(6)
For the DAFF we measure the staggered magnetization, i.e., each second spin is counted opposite to its orientation in a three-dimensional checker-board manner. For simplicity, we refer to the staggered magnetization also as M . An antiferromagnetically-ordered spin configuration has therefore M = 1. A Binder cumulant for M can
g(T, L) = 1 2 3 − [ M 4 ] av [ M 2 2 ] av ,(7)
where · · · represents a thermal average and [· · · ] av an average over disorder (field or dilution configurations) for a fixed value of h (RFIM) or B (DAFF). Close to criticality the Binder ratio scales as
g(T, L) =G[L 1/ν (T − T c )] ,(8)
whereG is a universal function. Note that for the DAFF, when
T = 0, g(B, L) =G ′ [L 1/ν (B − B c )].
To com-
χ(k) = 1 N j S j e ikxj 2 av .(9)
The two-point finite-size correlation function is then given by
ξ L = 1 2 sin(k min /2) χ(0) χ(k min ) − 1(10)
with k min = (2π/L, 0, 0). The two-point finite-size correlation function scales as
ξ L (T, L)/L =X[L 1/ν (T − T c )] .(11)
Using both the Binder ratio and the two-point finitesize correlation function allows us to perform a detailed finite-size scaling analysis to determine the critical exponent ν, as well as to test if both models share the same universality class using the method introduced in Ref. 41. To obtain an optimal data collapse we use a Levenberg-Marquardt minimization combined with a bootstrap analysis, see Ref. 41. This allows us to determine the optimal values of the critical parameters T c and ν with a statistical error bar by fitting the data to a thirdorder polynomial that approximates the scaling functions
G(x) andX(x) close to x = 0, where x = L 1/ν (T − T c ).
Finally, to determine the critical exponent η, we determine the peak position of the connected susceptibility given by
χ = 1 T M 2 av − [ M ] 2 av ,(12)
where the magnetization M is given by Eq. (6). Note that the connected susceptibility is related to Eq. (9) in the limit of zero wave vector. Furthermore, in the thermodynamic limit [ M ] av = 0 for T = T c so, in principle, Eq. (9) could also be used for the analysis. In general, the susceptibility scales as
χ ∼ L 2−η C[L 1/ν (T − T c )] .(13)
Therefore, when T = T c the function C is a constant independent of the system size and χ ∼ L 2−η from which the exponent η can be determined.
V. RESULTS
The critical parameters for both the RFIM and the DAFF have been computed via a finite-size scaling analysis of the two-point finite-size correlation function [Eq. (11)] along the different simulation paths. Finitesize corrections can be large for small system sizes and are strongly field dependent, which is why for some external fields in both models we do not include small systems in the finite-size scaling analysis used to determine the critical parameters. To illustrate the typical behavior, in Fig. 1, left panel, we show the two-point finite-size correlation function for the DAFF for B = 1.0 and different system sizes. The data cross at a point, therefore signaling the existence of a phase transition. Note that for this particular field corrections to scaling are manageable and the data scale well, as can be seen in Fig. 1, right panel. However, this is not always the case, especially when the external field is large. For the RFIM corrections to scaling are considerably stronger, even at small fields, see Fig. 2.
Using finite-size scaling we determine the location of the critical points, as well as the associated critical exponent ν for the different simulation paths. In addition, we also compute the critical exponent η by studying the finite-size behavior of the susceptibility peak. Data for the RFIM are summarized in Table IV, for the DAFF in Table V. To determine the critical field B c at zero temperature for the DAFF we compute ground states with the algorithm introduced in Ref. 34. The same finite-size scaling technique as used for the two-point finite-size correlation function (see above) can be used to analyze the ground-state Binder cumulant. The data collapse is shown in Fig. 3. The results for the critical point and the correlation-length exponent at zero temperature are stated in the last line of Table V. We also determine the peak position of the fluctuations of the staggered magnetization of the ground states:
F (B) = L 3 [M 2 ] av − [M ] 2 av .(14)
This approach has proven to be quite accurate in previous studies for the susceptibility. 47 Because the fluctuations peak at the putative transition, we fit a Gaus-sian to the peak and determine its precise location. Error bars are determined via a configurational bootstrap analysis. 48 Figure 4, left panel, shows the fluctuations at zero temperature and as a function of the applied field B. The peaks are well described by Gaussians. The right panel of Fig. 4 shows an extrapolation of the peak (2) [ω = 1.25 (9)], in agreement with the estimate using the Binder cumulant, see Table V.
Combining the data in Table V
+ T c T 0 c 1.80 = 1(16)
with h 0 c = 2.27 46 and T 0 c = 4.5115. 45 Note that the critical phase boundary points T 0 c and h 0 c have been determined to high precision in the literature; see Refs. 45 and 46, respectively. Furthermore, for the RFIM with bimodal disorder, a similar elliptical phase diagram has been proposed in Ref. 51. For the DAFF,T 0 c andB 0 c are approximated but agree with the numerical estimates we present. In Fig. 5 we show the phase boundaries for the DAFF (left panel) and the RFIM (right panel), together with the simulated critical points. The dashed lines represent the simulation paths taken.
VI. DISCUSSION
Cardy 17 predicted an equivalence between the DAFF and the RFIM for small applied fields using a mean-field argument. This equivalence is often quoted in experimental studies where materials which are diluted antiferromagnets in a field are then described using the RFIM (see, for example, Refs. 3,12-15).
Equation (15)
h(T ) = p(1 − p)(T pure c /T ) 2 (B/T ) 2 (1 − θ MF /T ) 2 .(17)
Here, p = 0.7, T pure c = 4.5115, and θ MF = 2dJ = 6 is the mean-field coupling strength. We can now use the obtained phase boundaries [Eqs. (15) and (16)] to compare both models. Figure 6 shows the phase boundary for the RFIM (solid line, the circles represent the obtained critical points along the different simulation paths) together with the phase boundary for the DAFF mapped onto the RFIM space using Eq. (17) (dashed line, the squares represent the obtained critical points along the different simulation paths for the DAFF). For random-field strengths of up to h ≈ 1.2-which means field strengths of up to B ≈ 1.6 for the DAFF-there is an approximate correspondence between both models. However, as the figure clearly illustrates, strictly speaking the correspondence only seems to work in the limit of h → 0 (h 0.3). Given the mean-field nature of the Cardy argument, the agreement of the phase boundaries is rather good. On the other hand, it is not surprising that for larger disorder, they do not agree exactly. It is of importance to take these limitations of the Cardy mapping 17 into account when studying diluted antiferromagnets in an external field experimentally while attempting to describe the data analytically using the RFIM. Furthermore, a basic finite-size scaling analysis leads to no systematic deviations of the correlation-length exponent ν. Including the estimates for rough simulations at high fields, our results support ν = 1.39 (17)
for the range of fields studied, in agreement with previous studies, such as ν RFIM = 1.37(9), 46 ν = 1.20(5) 14 from experiments on Fe 0.85 Zn 0.15 F 2 (p = 0.85), or ν = 1.40(6) from the disconnected part of the susceptibility of Fe 0.93 Zn 0.07 F 2 (p = 0.93). 52 Note that our results are also compatible with the value ν = 1.10(15) computed by Fernandez et al. 42 obtained for their largest system size using the quotient method. They do find other values of ν for smaller system sizes. Our results are summarized in Fig. 7. As can be clearly seen, the difference between the estimates for the critical exponent of the correlation length for both models is marginal and within error bars: The average estimate for the RFIM isν RFIM = 1.37(12) (red line in Fig. 7), whereas for the DAFFν DAFF = 1.41(15) (blue line in Fig. 7). This apparent agreement of the critical exponent is quite good, given that the proposed equivalence is based on a meanfield argument that typically leads to quite different exponents compared to the true non-mean-field values. However, the error bars are large and therefore a more detailed study needs to be performed. To truly discern if both models are in the same universality class, in addition to having one (apparently) agreeing critical exponent, one would have to compute a second critical exponent. We also analyzed the behavior of the magnetic susceptibility χ which has a peak at the phase transition. By studying the finite-size behavior of the peak height (not shown), we determine the critical exponent η using the finite-size scaling form of the susceptibility, Eq. (13). Our estimates of the critical exponent η along the phase boundary are shown in Fig. 8 and summarized in Tables IV and V for the RFIM and DAFF, respectively. Fluctuations are very large, especially for large fields, but suggest that both the RFIM and the DAFF might not share the same universality class. For the DAFF, a clear systematic trend is visible that shows that η might be strongly field dependent for B 1.6, i.e., in the curved portion of the phase boundary. However, note that the exponent η is very difficult to compute, as recently shown in Ref. 20. A different approach is the computation of the critical exponent α that describes the divergence of the specific heat. However, for both the RFIM and the DAFF α is close to zero. 34,46 Therefore, simulations of very large system sizes that are currently not accessible numerically are required. Fortunately, there is a simple yet more sensitive method to verify if two different systems share the same universality class without having to compute any critical exponents: 41,53 Both the Binder cumulant and the two-point finite-size correlation function divided by the system size are dimensionless quantities. By plotting one as a function of the other, nonuniversal quantities cancel out. 41 For a given system, once large enough system sizes are reached such that corrections to scaling are negligible, the data for all system sizes collapse onto a universal curve within error bars. If two systems share the same critical exponent ν, we expect that all data should collapse onto the same universal curve within error bars and, in particular, that the estimates of the Binder cumulant and the two-point finite-size correlation function agree at the putative critical point(s). We therefore would expect that data sets of g(ξ L /L) for both the DAFF and the RFIM should agree for all simulated temperatures and, in particular, for T = T c . Figure 9 shows the Binder cumulant as a function of the two-point finite-size correlation function divided by the system size for both the DAFF and the RFIM. The left set of points (reddish/light tones, circles) are for the RFIM. Data for the different simulation paths used collapse onto a master curve. The right set of points (greenish/dark tones, squares) are for the DAFF. Again, all data collapse onto a master curve for all simulation paths taken. This shows that for this type of analysis the finite-size corrections are small for both models and within the statistical fluctuations. However, the data sets for the RFIM and the DAFF do not agree, except in the trivial limit where g(T ) → 1. The large circles for the RFIM (squares for the DAFF) represent our estimates of g(ξ L /L) at T = T c . As can be seen, the data for both models do not agree (i.e., a large circle should sit on top of a large square), something which is even more clear when zooming into the boxed area (inset). Note that the large error bars are due to the uncertainty of the critical temperature. This discrepancy reveals the differences between the DAFF and the RFIM which could not be detected within the scope of a mean-field calculation.
VII. CONCLUSIONS
We have performed extensive Monte Carlo simulations of the diluted antiferromagnet in a field at 30% dilution (p = 0.7) and the random-field Ising model. Using these data we show that the phase boundaries for both models are well described by ellipses (see Fig. 5). In addition, using zero-temperature heuristic methods, we compute the zero-temperature critical point for the DAFF with 30% dilution (p = 0.7). We expect that the phase boundary for other dilutions will be similar, albeit with different nonuniversal parameters.
Furthermore, we numerically study the equivalence of the RFIM and the DAFF as predicted by Cardy. 17 Our results show that only in the limit of small fields do both phase boundaries map onto each other.
Finally, we perform a finite-size scaling analysis to determine the critical exponent ν of the correlation length. Our results from the two-point finite-size correlation function suggest that the exponent ν agrees within error bars for both the RFIM and the DAFF. However, error bars are large. To circumvent this problem, we study the Binder cumulant as a function of the two-point finite-size correlation function divided by the system size and show that both models apparently do not share the same universality class. A computation of the exponent η is extremely difficult and plagued by finite-size effects. Clearly, more detailed simulations need to be performed to fully discern the critical behavior of both models and fully determine their universality classes. It would be interesting to also measure the critical behavior of the specific heat (critical exponent α). However, because the exponent is close to zero for both models, large system sizes are needed; sizes that are currently not accessible via simulations. We conclude by cautioning researchers when using the equivalence of both models. : (Color online) Binder ratio g as a function of the two-point finite-size correlation function divided by the system size ξL/L for several system sizes and simulation paths. Note that also small system sizes are included, i.e., corrections to finite-size scaling are small. The data for the RFIM and DAFF collapse onto two distinct set of curves, suggesting that both models do not share the same universality class. The left set of points (reddish/light tones, circles) are for the RFIM. The right set of points (greenish/dark tones, squares) are for the DAFF. The large circles for the RFIM (large squares for the DAFF) represent our estimates of g(ξL/L) at T = Tc. The inset zooms into the important region (large box), where the Cardy mapping should apply. Clearly, both data sets are different, suggesting that the RFIM and the DAFF do not share the same universality class.
FIG. 2 :
2(Color online) Left: Two-point finite-size correlation function ξL/L vs temperature T for the RFIM with h = 0.225 and different linear system sizes L. Finite-size corrections are large. Right: Finite-size scaling collapse of the data in the left panel. Because of the large corrections to scaling, only data for L ≥ 28 are used in the scaling collapse. Data for L ≤ 24 (light shaded) are not included in the data collapse and shown to illustrate the corrections to scaling. The best collapse is obtained with Tc ≈ 4.481 and ν ≈ 1.39.
FIG. 3 :
3(Color online) Data collapse of the zero-temperature Binder cumulant of the DAFF as a function of the reduced scaling variable (B − Bc)L 1/ν for different system sizes. The best collapse is obtained for B 0 c ≈ 2.32 and ν ≈ 1.43. position to infinite system size assuming the functional form B c (L) = B c + aL −ω . The best fit is obtained for B c = 2.34
FIG. 4 :FIG. 5 :FIG. 6 :
456(Color online) Left: Fluctuations of the staggered magnetization of the DAFF as a function of applied field B for different system sizes. The peak positions signals the presence of a transition. The data are well described by a Gaussian close to the peak (solid lines). To determine the thermodynamic critical field Bc we extrapolate the data to infinite system size (right panel) using Bc(L) = Bc + aL −ω . The best fit is obtained for Bc = 2.34(2) and ω = 1.25(9). The red (filled) point represents the thermodynamic extrapolation, Bc = 2.34(2).(Color online) Left: Empirical phase boundary of the DAFF (p = 0.7). The red point is from Ref. 42, the coral point from Ref. 43, and the purple points from Ref. 44. Our data agree within error bars with these previous studies. The blue (solid) curve is given by Eq. (15). The dashed lines represent the parts of the simulation paths that cross the phase boundary. The light shaded line is an example of the second RC that runs parallel to the approximated phase boundary to a temperature T > Tc and B = 0 to speed up equilibration. Right: Empirical phase boundary of the RFIM. The zero-field critical temperature is T 0 c = 4.5115 45 and h 0 c = 2.270 46 (gray open circles). The red (solid) curve is given by Eq. (16). The dashed lines represent the parts of the simulation paths that cross the phase boundary. Again, the light shaded line shows an example of the second RC that runs parallel to the approximated phase boundary to a temperature T > Tc and h = 0 to speed up equilibration. (Color online) Phase boundary of the RFIM (solid line, from Fig. 5, right panel). The temperature axis has been normalized with Tc(h = 0) = 4.5115. The circles represent the different estimates of the critical points along the different simulation paths. The dashed line is the phase boundary computed by applying Eq. (17) to the data of the DAFF. Squares represent the different critical points simulated for the DAFF along the different simulation paths. An approximate correspondence between the phase boundaries only works for fields h 1.2 (B 1.6 for the DAFF).
FIG. 7 :
7(Color online) Critical exponent ν as a function of the field h (RFIM) and B (DAFF). The labels on the upper axis correspond to the random-field strength h (RFIM), those on the lower axis to the external field B (DAFF). The weighted mean is ν = 1.39(17) (gray line) and the weighted error is represented by the shaded (light blue) area. The difference betweenνDAFF = 1.41(15) (blue dashed line) andνRFIM = 1.37(12) (red dashed line) is marginal in comparison to the error-bars of the data points. The RFIM ground-state value is taken from Ref. 46.
FIG. 8 :
8(Color online) Critical exponent η as a function of the field h (RFIM) and B (DAFF). The labels on the upper axis correspond to the random-field strength h (RFIM), those on the lower axis to the external field B (DAFF). For comparison, we also add the estimates for the three-dimensional Ising ferromagnet (filled circle at h = 0, marked with 'Ising magnet'), 45 the RFIM at T = 0 (open circle at h = hc, marked with 'RFIM GS'), 50 and the DAFF at T = 0 and B = Bc computed from our ground-state data [η(T = 0) ≈ 0.68(1), filled square, marked with 'DAFF GS']. Note that we find very large fluctuations, i.e., a detailed determination of the different universality classes is difficult.
FIG. 9: (Color online) Binder ratio g as a function of the two-point finite-size correlation function divided by the system size ξL/L for several system sizes and simulation paths. Note that also small system sizes are included, i.e., corrections to finite-size scaling are small. The data for the RFIM and DAFF collapse onto two distinct set of curves, suggesting that both models do not share the same universality class. The left set of points (reddish/light tones, circles) are for the RFIM. The right set of points (greenish/dark tones, squares) are for the DAFF. The large circles for the RFIM (large squares for the DAFF) represent our estimates of g(ξL/L) at T = Tc. The inset zooms into the important region (large box), where the Cardy mapping should apply. Clearly, both data sets are different, suggesting that the RFIM and the DAFF do not share the same universality class.
TABLE I :
ISimulation parameters for the RFIM along dif-
ferent nontrivial paths of the type h = a + bT in the h-T
plane for different linear system sizes L (the first two path
types have b = 0). Nsa is the number of disorder realiza-
tions. NT corresponds to the number of temperatures (points)
along the simulation path. Tmin and Tmax are the lowest and
highest temperature simulated, respectively. The equilibra-
tion/measurement times are 2 x Monte Carlo sweeps.
simulation path
L
Nsa
NT
Tmin
Tmax
x
h = 0.225
8
1536
25
4.00
5.00
18
h = 0.225
10
827
25
4.00
5.00
18
h = 0.225
12
2048
17
4.30
4.80
18
h = 0.225
16
1024
19
4.35
4.70
18
h = 0.225
20
1024
19
4.35
4.70
18
h = 0.225
24
1024
26
4.40
4.69
18
h = 0.225
28
666
26
4.40
4.69
18
h = 0.225
32
406
26
4.40
4.69
18
h = 0.225
36
1017
26
4.40
4.69
18
h = 0.5
10
2503
17
4.20
4.60
18
h = 0.5
12
4035
17
4.20
4.60
18
h = 0.5
16
2048
17
4.20
4.60
18
h = 0.5
20
1024
14
4.30
4.50
18
h = 0.5
24
512
14
4.30
4.50
18
h = 1.22T − 3.43
10
4096
15
3.40
4.10
18
h = 1.22T − 3.43
12
3852
15
3.40
4.10
18
h = 1.22T − 3.43
16
1177
17
3.65
4.10
18
h = 1.22T − 3.43
18
862
17
3.65
4.10
18
h = 1.22T − 3.43
20
957
17
3.60
4.00
18
h = 1.22T − 3.43
24
976
17
3.60
4.00
18
h = 1.22T − 3.43
28
646
17
3.60
4.00
18
h = 1.22T − 3.43
32
379
17
3.60
4.00
18
h = 2.67T − 6.10
8
4071
25
2.80
3.06
18
h = 2.67T − 6.10
10
4045
25
2.80
3.06
18
h = 2.67T − 6.10
12
512
27
2.85
3.00
18
h = 2.67T − 6.10
14
512
27
2.85
3.00
18
h = 2.67T − 6.10
16
605
17
2.85
2.95
18
h = 2.67T − 6.10
18
1024
27
2.85
3.05
18
h = 2.67T − 6.10
20
512
31
2.86
2.93
18
h = 2.67T − 6.10
22
981
31
2.85
3.05
18
h = 2.67T − 6.10
24
1024
31
2.85
3.05
18
h = 4.94T − 6.80
16
1912
15
1.76
1.88
18
h = 4.94T − 6.80
18
2048
15
1.76
1.88
18
h = 4.94T − 6.80
20
1858
15
1.76
1.89
18
h = 4.94T − 6.80
24
906
15
1.76
1.89
18
h = 4.94T − 6.80
28
505
15
1.76
1.89
18
h = 4.94T − 6.80
32
627
15
1.76
1.89
18
then be defined via
TABLE II :
IISimulation parameters for the DAFF along nontrivial paths of the type B = a + bT in the B-T plane for different linear system sizes L (the first two path types have b = 0). Nsa is the number of disorder realizations. NT corresponds to the number of temperatures (points) along the simulation path. Tmin and Tmax are the smallest and the highest temperatures of the RC, respectively. The equilibration/measurement times are 2 x Monte Carlo sweeps.pute the two-point finite-size correlation function we first calculate the wave-vector-dependent susceptibility (alongsimulation path
L
Nsa
NT
Tmin
Tmax
x
B = 0.1
8
2166
26
2.50
3.50
18
B = 0.1
12
1208
26
2.50
3.50
18
B = 0.1
14
1042
18
2.70
3.30
18
B = 0.1
16
2048
19
2.80
3.30
18
B = 0.1
18
1104
19
2.80
3.30
18
B = 0.1
20
796
21
2.80
3.35
18
B = 0.1
24
444
21
2.80
3.35
18
B = 0.1
28
505
21
2.80
3.35
18
B = 0.1
32
322
21
2.80
3.35
18
B = 1.0
14
1271
21
2.70
3.20
18
B = 1.0
16
1718
21
2.70
3.20
18
B = 1.0
18
1215
21
2.70
3.20
18
B = 1.0
20
888
21
2.70
3.20
18
B = 1.0
24
491
21
2.70
3.20
18
B = 1.0
28
556
21
2.70
3.20
18
B = 1.0
32
352
21
2.70
3.20
18
B = 0.2T
8
1344
17
2.55
3.30
18
B = 0.2T
10
685
17
2.55
3.30
18
B = 0.2T
12
452
17
2.55
3.30
18
B = 0.2T
16
542
31
2.87
3.50
18
B = 0.2T
20
1564
31
2.87
3.50
18
B = 0.2T
22
825
31
2.87
3.50
18
B = 0.2T
24
189
31
2.87
3.50
18
B = 0.2T
26
128
31
2.87
3.50
18
B = 0.2T
28
115
31
2.87
3.50
18
B = 0.2T
30
558
31
2.87
3.50
18
B = 0.2T
32
383
31
2.87
3.50
18
B = 0.67T
10
1201
30
2.45
3.50
18
B = 0.67T
12
711
30
2.45
3.50
18
B = 0.67T
16
305
30
2.45
3.50
18
B = 0.67T
20
512
27
2.35
3.50
18
B = 0.67T
22
1024
27
2.35
3.50
18
B = 0.67T
24
2048
30
2.35
3.50
18
B = 0.67T
28
1024
27
2.35
3.50
18
B = 0.67T
32
741
30
2.37
3.50
18
B = 1.5T
10
1920
17
1.30
1.62
18
B = 1.5T
12
1984
17
1.30
1.62
18
B = 1.5T
16
2048
17
1.30
1.62
18
B = 1.5T
18
2048
26
1.30
3.50
18
B = 1.5T
20
1056
20
1.35
1.60
18
B = 1.5T
24
807
20
1.35
1.60
18
B = 1.5T
28
457
20
1.35
1.60
18
B = 1.5T
32
532
20
1.35
1.60
18
B = 1.5T
36
336
20
1.35
1.60
18
TABLE III :
IIISimulation parameters for the DAFF at zero temperature for different fields B and for different linear system sizes L. Nsa is the number of disorder realizations. Bmin and Bmax are the lowest and highest fields simulated, and NB corresponds to the number of fields simulated to perform a finite-size scaling analysis.L
Nsa
Bmin
Bmax
NB
24
10302
2.00
4.30
31
32
2091
2.40
2.70
16
48
2091
2.10
2.80
17
64
2091
2.30
2.70
21
72
2040
2.30
2.54
17
96
5100
2.30
2.54
17
128
3586
2.30
2.47
22
the x direction) via
TABLE IV :
IVCritical temperature Tc and critical field hc computed from a finite-size scaling analysis of the two-point finitesize correlation function for the RFIM. ν is the critical exponent of the correlation length. The exponent η is computed from the peak of the susceptibility.simulation path
Tc
hc
ν
η
h = 0.225
4.481(1)
0.225
1.39(4)
0.082(1)
h = 0.5
4.381(2)
0.5
1.30(5)
0.202(16)
h = 1.22T − 3.4 3.76(2)
1.16(3)
1.39(5)
0.92(40)
h = 2.70T − 6.1 2.89(5)
1.7(1)
1.3(1)
0.47(15)
h = 4.94T − 6.8 1.79(1)
2.01(5)
1.4(1)
0.85(4)
FIG. 1: (Color online) Left: Two-point finite-size correlation function ξL/L vs temperature T for the DAFF with B = 1.0 and different linear system sizes L. Finite-size corrections are small and the data cross at one point signaling a transition. Right: Finite-size scaling collapse of the data in the left panel. The best collapse is obtained with Tc ≈ 2.807 and ν ≈ 1.2.2.70
2.75
2.80
2.85
2.90
0.5
1.0
1.5
2.0
2.5
T
ξ L
L= 32
L= 24
L= 20
L= 18
L= 16
−4
−2
0
2
4
0.5
1.0
1.5
2.0
2.5
(T − T c )L 1 ν
ξ L
L= 32
L= 24
L= 20
L= 18
L= 16
4.45
4.47
4.49
4.51
0.5
1.0
1.5
2.0
T
ξ L
L= 36
L= 32
L= 28
L= 24
L= 20
L= 16
L= 12
L= 10
L= 8
−2
−1
0
1
2
0.6
0.8
1.0
1.2
(T − T c )L 1 ν
ξ L
L= 24
L= 20
L= 16
L= 12
L= 28
L= 32
L= 36
TABLE V :
VCritical temperature Tc and critical field Bc computed from a finite-size scaling analysis of the two-point finitesize correlation function for the DAFF. ν is the critical exponent of the correlation length. The exponent η is computed from the peak of the susceptibility. Note that estimating η was not possible for B = 1.5. The last line lists data from zero-temperature simulations (see text). The estimate of the critical field Bc is obtained from a finite-size scaling analysis of the zero-temperature Binder ratio.simulation path Tc
Bc
ν
η
B = 0.1
2.977(1) 0.1
1.34(5) 0.406(26)
B = 1.0
2.807(1) 1.0
1.2(2)
0.023(12)
B = 0.2T
2.908(4) 0.582(8) 1.36(7) 0.11(2)
B = 0.67T
2.42(1)
1.61(1)
1.5(3)
0.67(5)
B = 1.5
1.46(9)
2.2(1)
1.4(3)
-
T = 0
0
2.32(2)
1.43(2) 0.68(1)
with some values from the literature[42][43][44]49 we can approximate to good accuracy the phase boundary for the DAFF viaB c
B 0
c
1.81
+
T c
T 0
c
3.54
= 1
(15)
withT 0
c ≈ 2.980 andB 0
c ≈ 2.31. Similarly, using
the data from Table IV and known values from the
literature 45,46,50 we obtain for the RFIM
h c
h 0
c
1.95
in Ref. 17 maps the RFIM onto the DAFF:
universality, Phys. Rev. B 73, 224431 (2006).
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| [] |
[
"Control of Time-Varying Epidemic-Like Stochastic Processes and Their Mean-Field Limits",
"Control of Time-Varying Epidemic-Like Stochastic Processes and Their Mean-Field Limits"
] | [
"Yingdong Lu ",
"Mark S Squillante ",
"Wah Chai ",
"Wu "
] | [] | [] | The optimal control of epidemic-like stochastic processes is important both historically and for emerging applications today, where it can be especially important to include time-varying parameters that impact viral epidemic-like propagation. We connect the control of such stochastic processes with time-varying behavior to the stochastic shortest path problem and obtain solutions for various cost functions. Then, under a mean-field scaling, this general class of stochastic processes is shown to converge to a corresponding dynamical system. We analogously establish that the optimal control of this class of processes converges to the optimal control of the limiting dynamical system. Consequently, we study the optimal control of the dynamical system where the comparison of both controlled systems renders various important mathematical properties of interest.The authors are with the Mathematical Sciences | 10.23919/acc.2018.8430846 | [
"https://arxiv.org/pdf/1709.10345v1.pdf"
] | 52,014,970 | 1709.10345 | c973699dccb6ef3798a434dbeeaeef5becea08a2 |
Control of Time-Varying Epidemic-Like Stochastic Processes and Their Mean-Field Limits
Yingdong Lu
Mark S Squillante
Wah Chai
Wu
Control of Time-Varying Epidemic-Like Stochastic Processes and Their Mean-Field Limits
The optimal control of epidemic-like stochastic processes is important both historically and for emerging applications today, where it can be especially important to include time-varying parameters that impact viral epidemic-like propagation. We connect the control of such stochastic processes with time-varying behavior to the stochastic shortest path problem and obtain solutions for various cost functions. Then, under a mean-field scaling, this general class of stochastic processes is shown to converge to a corresponding dynamical system. We analogously establish that the optimal control of this class of processes converges to the optimal control of the limiting dynamical system. Consequently, we study the optimal control of the dynamical system where the comparison of both controlled systems renders various important mathematical properties of interest.The authors are with the Mathematical Sciences
I. Introduction
The mathematical analysis of epidemic-like behavior has a very rich and important history, with the seminal work of Bernoulli on epidemiological models as a starting point in the 1766 [1]. More recently, mathematical models of epidemic-like behavior have received considerable attention in the research literature, which include models of various aspects of large networks such as the complex structures and behaviors of communication networks, social media/networks, viral-propagation networks (e.g., epidemics, computer viruses and worms), and financial networks; refer to, e.g., [4] and the references therein. However, research on the control and optimization of such epidemic-like mathematical systems has been much more limited; see, e.g., [3]. Even more importantly, this entire body of work has focused solely on static (nontime-varying) model parameters that impact the complex structures and behaviors of the large epidemic-like systems of interest. In contrast, time-varying behaviors often arise in many emerging applications of epidemic-like systems, especially those where one observes behaviors that lead to forms of exacerbated complex dynamics and actions frequently found in communication, financial, social, and viral-propagation networks.
We consider in this paper variants of the classical mathematical model of epidemic-like behavior analyzed by Kurtz [5,Chapter 11], extending the analysis to first incorporate time-varying behavior for the infection and cure rates of the model and to then study aspects of the corresponding stochastic optimal control problem. Specifically, we start by formally presenting an epidemiclike continuous-time, discrete-state stochastic process in which each individual comprising the population can be either in a non-infected state or in an infected state, and where the rate at which the non-infected population is infected and the rate at which the infected population is cured are both functions of time. Then, we investigate the optimal control problem associated with this general class of epidemic-like stochastic processes. Making connections to the well-studied stochastic shortest path problem, we exploit structural properties to obtain the corresponding optimal control policy; in one special case of interest we derive the explicit control policy, whereas in other cases we compute the value functions through efficient (linear program based) methods instead of value or policy iteration. Taking the limit as the population size tends to infinity under a mean-field scaling, we establish that the time-varying epidemic-like stochastic processes converge to a continuous-state nonautonomous dynamical system. Moreover, the control of such stochastic processes is shown to be asymptotically equivalent to the optimal control of the limiting dynamical system. We then investigate instances of the optimal control of the limiting dynamical system and establish structural properties of their equilibria and trajectories. Lastly, computational experiments compare the optimal control of both the stochastic process as a function of population size and the dynamical system. The paper is organized as follows. §II presents our model and control of the general class of epidemic-like stochastic processes with time-varying parameters. §III presents results for the mean-field limit of the stochastic process and its controlled counterpart. §IV presents our model and analysis of the corresponding limiting dynamical system. §V presents some computational experiments for both systems. We refer the reader to [6] for additional results, all proofs, related work, and technical details.
II. Epidemic-Like Stochastic Processes
A. Mathematical Model
Consider a sequence of Markov processesẐ n = {(X n (t),Ŷ n (t)); t ≥ 0} indexed by the total population size n ∈ Z + := {1, 2, . . .} and defined over the probability space (Ω n , F n , P n ), composed of the state spaceΩ n := {(i, j) : 0 ≤ i, j ≤ n, i + j = n}, σ-algebra F n and probability measure P n , with initial probability distribution α n . Each processẐ n (t) represents the ordered pair (X n (t),Ŷ n (t)) of non-infected and infected population at time t, where we assume connections among the population form a complete graph.
Define Ω ⊂ R 2 and Ω n := Ω ∩ { /n : ∈Ω n }. The time-dependent infinitesimal generator Q n (t) = [q (n) (i,j)(k,l) (t)] (i,j),(k,l)∈Ωn for the Markov processẐ n has time-dependent transition intensities q
(n) (i,j)(i−1,j+1) (t) = λ(t)i j n = nλ(t) i n j n , q (n) (i,j)(i+1,j−1) (t) = µ(t)j = nµ(t) j n ,
where the latter equalities bear the general form q (n) k,k+ (t) = nβ ,t (k/n), for k, k + ∈Ω n , with nonnegative functions β ,t (x) defined on Ω for ∈Ω n and t ≥ 0, continuous in t, Lipschitz continuous in x = k/n (by definition), and (x + /n) ∈ Ω n when β ,t (x) > 0, for x ∈ Ω n . Throughout the functions λ(t) and µ(t) are assumed to be continuous in t.
Note that the above definition of the epidemic-like stochastic processẐ n is slightly different from the corresponding (non-time-varying) model of Kurtz [5], in that we allow an infected individual who is cured to become infected at a later time. In any case, our results hold for both types of epidemic-like models and related variations thereof with time-varying transition rates.
B. Optimal Control and Analysis
Now consider the optimal control problem associated with the stochastic processẐ n for any population size n, fixing n henceforth in this subsection and omitting this parameter when clear by context. Let P (X(t)) denote the profits (rewards minus costs) as a function of the state of the systemX(t) at time t ∈ R + . The decision variables are based on the controlled infection and cure rates λ(t) and µ(t) deployed by the system that represent changes from the original infection and cure rates, now denoted in this subsection byλ(t) andμ(t), where the system incurs additional costsĈ λ (λ(t),λ(t)) andĈ µ (µ(t),μ(t)) as functions of the pairs of infection and cure rates, respectively. Throughout this subsection the control variables λ(t) and µ(t) are assumed to be continuous in t, withλ(t) and µ(t) continuously varying for all t. Define λ := (λ(t)) and µ := (µ(t)). The objective function of our optimal control formulation is then given by
max λ, µ f T 0 P (X(t)) −Ĉ λ (λ(t),λ(t)) −Ĉ µ (µ(t),μ(t)) dt ,(1)
where T denotes the time horizon, which can be finite or infinite, and f (·) represents an operator of interest. An appropriate form of expectation is of primary interest in this paper. Throughout this section, we assume that P (x), x ∈ [0, c], has a single maximum at x * which, e.g., occurs when P (·) is linear (in which case x * = 0 or x * = c = 1) or when P (·) is concave (in which case x * ∈ [0, c = 1]). Define [n] := {0, . . . , n} and let i * = arg max i∈[n] P (i) denote the integer(s) at which the profit function P (·) has a maximum value w.r.t. the states ofẐ. We further assume there is a unique i * in order to elucidate the exposition that follows; otherwise, there would be i * 1 = x * and i * 2 = x * where P (i * 1 ) = P (i * 2 ) and our analysis would apply w.r.t. both states i * 1 and i * 2 . To determine the optimal control policy for the Markov processẐ, we first exploit uniformization and consider the uniform version discrete-time Markov chain Z with transition probability matrix (p ij ) i,j∈ [n] and uniformization constant ν > max i∈ [n] {−Q ii }. Our objective then is to determine the policy π that maximizes the total expected profit over the entire time horizon
J π (i) = E ∞ t=1 h(i(t), u(i(t))) x 0 = i , J * (i) = max π J π (i),
or equivalently J * (i) = min π J π (i) and
J π (i) = E ∞ t=1 g(i(t), u(i(t))) x 0 = i ,(2)
where
−h(i(t), u(i(t))) = g(i(t), u(i(t))) = −P (i(t)) + C λ (λ(i(t)),λ(t)) +Ĉ µ (µ(i(t)),μ(t)) denotes the cost per stage for state i(t) ofẐ at time t ∈ Z + , for all i ∈ [n].
Next, the above expressions and assumptions allow us to determine the optimal control policy by solving the optimal control problem through a corresponding stochastic shortest path problem [2] over the set of states S ∪ {i * } where S := [n] \ {i * } and state i * is the special cost-free absorbing state in which the system remains at no further cost once reached. For all other states i ∈ S, a cost of g(i, u) will be incurred if action u ∈ U (i) is taken when the Markov chainẐ (t) is in state i at time t ∈ Z + . The objective is to determine the policy π such that, for all i ∈ S, J * (i) = min π J π (i) together with (2). From known results for the stochastic shortest path problem, specifically Proposition 7.2.1 in [2], we know that the optimal costs J * (i), ∀i ∈ S, satisfy Bellman's equation
J * (i) = min u∈U (i) g(i, u) + j∈[n] p ij (u)J * (j) (3)
and that a stationary policy γ is optimal if and only if for every state i ∈ S, γ(i) attains the minimum in (3). While the value functions and optimal policies in general can be computed through methods such as value or policy iteration, the remainder of this section considers several cases in which exact solutions can be derived or exact calculations can be applied to efficiently obtain the value functions. Our analysis will exploit structural properties of the Markov chainẐ , such as p ij = 0 for all j / ∈ {i − 1, i, i + 1} when i ∈ {1, . . . , n − 1}; p 0j = 0 for all j / ∈ {0, 1}; and p nj = 0 for all j / ∈ {n − 1, n}.
1) No Action Costs:
We start with the case where there are no costs for adjusting the infection and cure rates, i.e.,Ĉ λ (a, b) = 0 =Ĉ µ (a, b) for all a, b. Hence, g(i, u) = −P (i) = c(i) for any (i, u), i ∈ S, u ∈ U (i), and thus c(i) is monotone decreasing when i < i * and monotone increasing when i > i * . In conjunction with the special structure of the Markov chain, Bellman's equation for this case can be written as
J(i) = c(i) + min λ∈[0,λ],µ∈[0,μ] µ(n − i) ν J(i − 1) + 1 − µ(n − i) ν − λi(n − i) nν J(i) + λi(n − i) nν J(i + 1) .(4)
From the properties of c(i) and an analysis of the minimization in Bellman's equation, we observe that the optimal policy should push the Markov chainẐ to the right as hard as possible when i < i * and push to the left as hard as possible when i > i * . Based on this observation, we can "guess" an optimal control policy and the corresponding set of J(i), ∀i ∈ S, and then verify that this system of equations satisfies (4).
In particular, we respectively have for i ≤ i * and i ≥ i *
J(i) = c(i) + µ(n − i) ν J(i − 1) + 1 − µ(n − i) ν J(i) , J(i) = c(i) + 1 − λi(n − i) nν J(i) + λi(n − i) nν J(i + 1) ,
where in both systems of equations we set one of the control rates to be zero and set one of the other control rates to be its maximum. We therefore obtain
J(i) = J(i + 1) + nνc(i) λi(n − i) , i < i * ,(5)J(i) = J(i − 1) + νc(i) µ(n − i) , i > i * .(6)
It then can be readily verified that (5) and (6) are indeed a solution to Bellman's equation (4), which from Proposition 7.2.1 in [2] is the unique solution of (4).
2) Linear Action Costs:
Let us now consider the case where the costs for adjusting the infection and cure rates are linear. Specifically, to simplify the presentation, we assume the action costs to be linear functions of λ and µ,
i.e.,Ĉ λ (λ(t),λ(t)) = c λ ·λ(t) andĈ µ (µ(t),μ(t)) = c µ ·µ(t)
where c λ and c µ are two constants for the corresponding cost rates. In this case, Bellman's equation becomes
J(i) = c(i) + min λ∈[0,λ],µ∈[0,μ] c λ λ + c µ µ + µ(n − i) ν J(i − 1) + 1 − µ(n − i) ν − λi(n − i) nν J(i) + λi(n − i) nν J(i + 1) .
We can rewrite this expression to obtain 0 = c(i) + min
λ∈[0,λ],µ∈[0,μ] c λ + i(n − i) nν ∆(i + 1) λ + c µ − (n − i) ν ∆(i) µ ,
where ∆(i) := J(i) − J(i − 1). This optimization problem is clearly a linear program, which implies that only the vertices of the feasible region need to be considered. The vertices consist of the λ and µ that take on values of either 0 or its maximumλ andμ, respectively. We therefore can further reduce Bellman's equation as follows
0 = c(i) + min λ∈{0,λ},µ∈{0,μ} c λ + i(n − i) nν ∆(i + 1) λ + c µ − (n − i) ν ∆(i) µ .
This represents Bellman's equation for a stochastic control problem with finite states and finite controls, for which it is well known [2] that the problem can be solved as a linear program. Hence, we efficiently compute the solution of our control problem in this case via
max i η i J(i) s.t. 0 ≤ c(i) + c λ + i(n − i) nν (J(i + 1) − J(i)) λ + c µ − (n − i) ν (J(i) − J(i − 1)) µ , ∀λ ∈ {0,λ}, µ ∈ {0,μ}.
where η i are positive real numbers and J(i) are the variables for the linear program.
3) General Action Costs:
Lastly, consider the case where the infection and cure rate adjustment costs are general functions with the number of actions restricted to a finite number of possibilities, in which case we can extend our above linear program approach. More specifically, Bellman's equation in this case can be written as
0 = c(i) + min (λ,µ)∈U K c λ + i(n − i) nν ∆(i + 1) λ + c µ − (n − i) ν ∆(j) µ ,(7)
where U K represents the finite set of K possible combinations of λ and µ. The value functions then can be obtained by solving the following linear program
max i η i J(i) s.t. 0 ≤ c(i) + c λ + i(n − i) nν ∆(i + 1) λ + c µ − (n − i) ν ∆(j) µ , ∀(λ, µ) ∈ U K .
In general, we can always discretize the action space to obtain a version of (7), the solution of which serves as an approximation to the value functions of the original control problem whose accuracy can increase with K.
III. Mean-Field Limits
A. Epidemic-Like Stochastic Processes
Suppose that the Markov ChainẐ n (t) is as defined in §II with time-dependent transition intensities of the general form given therein. From the martingale-problem method (see, e.g., [5,Chapters 4,6]), we devise thatẐ n (t) has the integral representation
Z n (t) =Ẑ n (0) + W n t 0 β ,s Ẑ n (s) n ds ,(8)
where the W are independent standard Poisson processes. Define F t (z) := β ,t (z), z ∈ Ω n . Further define Z n (t) :=Ẑ n (t)/n on the state space Ω n with transition intensities q (n) i,j (t) = nβ n(j−i),t (i), i, j ∈ Ω n . Our strategy for the proof is to first obtain the integral representation of Z n (t), which leads to the generator of Z n (t) again through the martingale-problem method and the law of large numbers for the Poisson process. From this and the above we derive the desired expression
Z n (t) = Z n (0) + nW n t 0 β ,s (Z n (s))ds + t 0 F s (Z n (s))ds,(9)
whereW denotes the centered Poisson process, i.e., W (x) = W (x) − x. One of our main results can now be presented, upon noting the following basic fact:
lim n→∞ sup u≤v W (nu) n = 0, a.s. for v ≥ 0. Theorem 3.1: Suppose that for each compact set K ⊂ Ω | | sup x∈K β ,t (x) < ∞, ∀t ≥ 0, and there exists M K > 0 s.t. |F t (x) − F t (y)| ≤ M K |x − y|, ∀x, y ∈ K, t ≥ 0. (10)
Further supposing Z n (t) satisfies (9) and lim n→∞ Z n (0) = z 0 , and denoting Z(t) as the solution to
Z(t) = z 0 + t 0 F s (Z(s))ds, t ≥ 0,(11)
From Theorem 3.1, we then have that the stochastic process Z n (t) converges to a deterministic process Z(t), taking values in [0, c], a.s. as n → ∞ and that Z(t) satisfies the integral form of the general nonautonomous dynamical system given in (11) where the specific details of the process and the corresponding set of ODEs depend upon F s (·) characterizing the averaging behavior of the original stochastic processẐ n (t). For epidemiclike models, the process Z n (t) converges to a deterministic process Z(t) = (X(t), Y (t)) a.s. as n → ∞ with Z(t) satisfying the pair of ordinary differential equations (ODEs):
dX(t) dt = −λ(t)X(t)Y (t) + µ(t)Y (t), dY (t) dt = λ(t)X(t)Y (t) − µ(t)Y (t)
, with proper initial conditions. This desired a.s. convergence result justifies the use of a continuous-state nonautonomous dynamical system to model a discrete-state real-world stochastic system.
B. Controlled Stochastic Processes
We next turn our attention to an optimal control problem associated with the class of epidemic-like stochastic processes, where our goal is to show that this control process is asymptotically equivalent to the optimal control of the corresponding set of ODEs as the population size tends to infinity under a mean-field scaling.
Consider a sequence of general controlled Markov processesẐ n (t), with the adaptive control process u n (t) that is realized w.r.t. the adaptive transition kernel nβ ,t (k/n), k, k + ∈Ω n , recalling β ,t (·) is continuous in t. For each system indexed by n, the optimal control u * n (t) is determined by solving the optimal control problem w.r.t. the cost functions c 1 (·) and c 2 (·):
J * n (z) = min un(t)Ĵ n (z) = min un(t) T 0 c 1 (Ẑ n (t), u n (t))dt + c 2 (Ẑ n (T )) ,s.t.Ẑ n (0) = z.
Here we assume the cost functions c 1 (z, u) and c 2 (z) are uniformly bounded, which is reasonable and justified by our interest in costs related only to the proportion of a population. Recall the integral representation ofẐ n (t) and Z n (t) in (8) and (9), respectively. For comparison towards our goal in this section, we also consider the corresponding optimal control problem associated with the limiting mean-field dynamical system of the above subsection. Namely, the optimal control u * (t) is determined by solving the corresponding optimal control problem w.r.t. the same cost functions c 1 (·) and c 2 (·), which can be formulated as
J * (z) = min un(t) J(z) = min un(t) T 0 c 1 (Z(t), u(t))dt + c 2 (Z(T )) , s.t. Z(0) = z,
where Z(t) follows the dynamics Z(t) = z + t 0 F s (Z(s))ds. Note that the function F s (·) encodes the control information.
We seek to show that the optimal control u * (t) in the limiting mean-field dynamical system provides an asymptotically equivalent optimal control u * n (t) for the original system indexed by n in the limit as n tends toward infinity. More specifically, we first establish the following main result.
Theorem 3.2: LetẐ n (t), Z n (t) and Z(t) be as above. We then have lim n→∞Ĵ * n (z) = J * (z).
Furthermore, let F * s (·) denote the function that encodes the optimal control u * (t) of the limiting mean-field dynamical system. Suppose the original stochastic procesŝ Z n (t) follows the deterministic state-dependent control policy determined by F * s (·). Then, asymptotically as n → ∞ under a mean-field scaling, both systems will realize the same objective function value in (13).
IV. Epidemic-Like Dynamical Systems A. Mathematical Model and Analysis
We next consider the continuous-time, continuousstate nonautonomous dynamical system z(t) = (x(t), y(t)) from the results of Section II. The starting state (x(0), y(0)) of the system at time t = 0 has initial probability distribution α. We assume throughout that λ(t) > 0 and µ(t) > 0. The state equations are then given by dx(t) dt = −λ(t)x(t)y(t) + µ(t)y(t) and
dy(t) dt = λ(t)x(t)y(t) − µ(t)y(t)
, where x(t) and y(t) respectively describe the fraction of non-infected and infected population at time t, with total population c = x(t) + y(t).
Since c = x(t) + y(t) and d(x(t)+y(t)) dt = 0, we have x(t) + y(t) = c = x(0) + y(0) for all t; i.e., the total population is constant. Upon substituting y(t) = c−x(t), we can equivalently rewrite the two-dimensional ODE as an one-dimensional ODE:
dx(t) dt = λ(t)x(t) 2 − (λ(t)c + µ(t))x(t)+µ(t)c.
We then have the following main result.
Theorem 4.1: For the dynamical system (x(t), y(t)) with 0 ≤ x(0), y(0) ≤ c and x(t) + y(t) = c, λ(t), µ(t) continuously varying for all t, the system has an asymptotic state at x * 1 (t) = µ(t) λ(t) and an equilibrium point at x * 2 = c, and stability properties given as follows. 1) 0 < µ(t) λ(t) < ξ < c, ∀t: The equilibrium point x * 2 is unstable. All trajectories of the dynamical system with initial state x(0) < c will converge towards being eventually near the asymptotic state x * 1 (t) w.r.t. a δ-neighborhood, i.e., x(t) − µ(t) λ(t) ≤ δ where δ is a nonnegative constant that depends on the rates of change of µ(t) and λ(t).
2) µ(t) λ(t) > c:
The equilibrium point x * 2 is stable, towards which all trajectories of the dynamical system will converge. 3) µ(t) λ(t) = c: There is one equilibrium point at x * 2 , which is neither stable nor unstable, towards which all trajectories of the dynamical system will converge. 4) µ(t) λ(t) = 0: There is one equilibrium point at x * 1 = 0, which is neither stable nor unstable, towards which all trajectories of the dynamical system will converge. To summarize, for the dynamical system of Theorem 4.1, all trajectories x(t) will approach a δneighborhood of µ(t) λ(t) when 0 < µ(t) λ(t) < c, will approach 0 when µ(t) λ(t) = 0, and will approach c when µ(t) λ(t) ≥ c.
B. Optimal Control and Analysis
Now we turn to consider the optimal control problem formulation (1) within the context of the dynamical system z(t), recalling our use of λ,λ, µ,μ from §II-B. Let λ * and µ * denote the optimal solution to (1) subject to the corresponding ODEs of the above subsection. This formulation represents the general case of the optimal control problem of interest for the dynamical system. Although there are no explicit solutions in general, this problem can be efficiently solved numerically using known methods from control theory.
To consider more tractable cases, and gain fundamental insights into the problem, we start by first considering a one-sided version of this general problem in equilibrium with a fixed constant infection rate λ =λ =λ(t) where the goal is to maximize the reward at the equilibrium point and only the parameter µ is under our control. The optimal control in this case is a stationary policy for the cure rate, i.e., a single control µ in equilibrium. Under a linear profit function with rate P and linear action-cost function of µ with rateĈ µ , we can rewrite the objective function (1) as
max µ P(x(∞)) −Ĉ µ (µ),
since the optimal control is a stationary policy for the cure rate. Upon substituting min{c, µ λ } for x(∞), we derive the optimal control policy to be
µ * = arg max µ≥0 P min c, µ λ −Ĉ µ (µ).(14)
Namely, the optimal stationary control policy employs for all time t the single control µ * that solves (14). An analogous formulation and result on λ * can be established for the opposite one-sided version of the problem in equilibrium with constant cure rate µ. Next, as another step toward the general formulation, consider the case where there are no costs for adjusting the infection and cure rates, i.e.,Ĉ λ (a, b) = 0 =Ĉ µ (a, b) for all a, b. Further assume, as in §II-B, that P (x) continues to have a single maximum at x * . We introduce the notion of an ideal trajectory denoted by (x I (t) = x * ) that maximizes the objective function (1) at all time in this problem instance. Hence, the optimal policy is to have µ(t) λ(t) = x * with λ(t) as large as possible, subject tô
µ(t)
λ(t) varying over time, since this governs the speed at which x(t) approaches and continually follows x * . More precisely, we now present a main result of interest for this instance of the general formulation showing that we can get arbitrarily close to the ideal trajectory, and thus the maximum objective.
Theorem 4.2: SupposeĈ λ (a, b) = 0 =Ĉ µ (a, b), for all a, b. For each > 0 with 0 ≤ x(0) < c − , there is â δ > 0 s.t. if λ(t), µ(t) >δ and µ(t)
λ(t) = x * for all t, then the optimal solution of (1) is realized within .
Let us now consider the above case where there are no costs for adjusting the infection and cure rates, but where there are constraints on the rates of change of the control variables λ(t) and µ(t), i.e., θ λ <λ < θ u λ and θ µ <μ < θ u µ . We again assume that P (x) has a single maximum at x * . Our above notion of an ideal trajectory remains the same, namely (x I (t) = x * ) maximizes the objective function (1) without constraints for all time t. We therefore have that the optimal policy consists of setting λ(t) and µ(t) so as to maximize the speed at which x(t) approaches and continually follows a maximum within an achievable neighborhood of x * , subject to the constraints onλ andμ and subject toμ (t)
λ(t)
varying over time.
More precisely, we establish a result showing that we can get arbitrarily close to a best state within a δ-neighborhood of the ideal trajectory, and thus the maximum objective, where δ is a nonnegative constant that depends on the rates of change ofλ(t) andμ(t), and on θ λ , θ u λ , θ µ , θ u µ . λ(t) =x * (t) := arg max x(t)∈D(t) P (x(t)) for all t, then the optimal solution of (1) under the constraints onλ andμ is realized within a δ-neighborhood of x * , i.e.,D, and in particular, the optimal reachable solutionx * (t) is realized within .
When the costs for adjusting the infection and cure rates are introduced to either of the above instances of the general formulation, the optimal policy will deviate from the ideal policies above where the deviation will depend on the initial state x(0), the cost functionsĈ λ (·, ·) andĈ µ (·, ·), the rates of change ofλ(t) andμ(t), and any constraints on the rates of change of λ(t) and µ(t). Even though the policy of following the ideal trajectory is not optimal in general, it can provide structural properties and insight into the complex dynamics of the system in a very simple and intuitive manner.
V. Computational Experiments
In this section we investigate various aspects of our theoretical results through computational experments. The behavior of the Markov decision process is clear from the results in §II-B; similarly for the behavior of the optimal control of the dynamical system from the results in §IV-B. We assume that the profit function P is continuous and has a single maximum at x * . Then, when sampled in n equal-spaced intervals, the set of P (i/n) for i = 1, · · · , n reaches its maximum at either one or two values of i. For simplicity, let us assume that for each n, P (i/n) is maximized at a single value of i * such that i * /n is closest to x * . Classical number theory shows that the difference between i * /n and x * is asymptotically no better than o(n −2 ) and this optimal rate is approached by the convergents of x * . Hence, the Markov decision process will converge towards this value of i * . Figures 1 and 2 illustrate how the value of i * behaves for x * = 1/4 and x * = φ − 1 = √ 5−1 2 , respectively.
We observe that the quantitative differences between the optimal control of the stochastic process and the dynamical system vanishes as n → ∞, and does so relatively quickly in accordance with classical results.
then we have, for every t ≥ 0, lim n→∞ sup s≤t |Z n (s) − Z(s)| = 0, a.s.
DefineD := {x(t) : x(t)−x * ≤ δ} and D(t) := {x(t) : x(t) ∈D and x(t) is reachable at time t} for all t. The main result of interest for this instance of the general formulation can then be expressed as follows. Theorem 4.3: SupposeĈ λ (a, b) = 0 =Ĉ µ (a, b), for all a, b, together with the constraints θ λ <λ < θ u λ and θ µ <μ < θ u µ . For each > 0 with 0 ≤ x(0) < c − , there is aδ > 0 s.t. if λ(t), µ(t) >δ and µ(t)
Fig. 1 .
1i * /n versus n, where i * is the state the MDP converges to. x * = 1 4 is indicated by the solid line.
Fig. 2 .
2i * /n versus n, where i * is the state the MDP converges to. x * = φ − 1 is indicated by the solid line.
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Dynamic Programming and Optimal Control, Vols. I and II. D Bertsikas, Athena ScientificD. Bertsikas Dynamic Programming and Optimal Control, Vols. I and II, Athena Scientific, 2005.
How to distribute antidote to control epidemics. Random Structures & Algorithms. C Borgs, J Chayes, A Ganesh, A Saberi, C. Borgs, J. Chayes, A. Ganesh, A. Saberi. How to distribute antidote to control epidemics. Random Structures & Algo- rithms, 2010.
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| [] |
[
"TRACE-derived temperature and emission measure profiles along long-lived coronal loops: the role of filamentation",
"TRACE-derived temperature and emission measure profiles along long-lived coronal loops: the role of filamentation"
] | [
"F Reale \nDip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo\nPiazza del Parlamento 1I-90134PalermoItaly\n",
"G Peres \nDip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo\nPiazza del Parlamento 1I-90134PalermoItaly\n"
] | [
"Dip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo\nPiazza del Parlamento 1I-90134PalermoItaly",
"Dip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo\nPiazza del Parlamento 1I-90134PalermoItaly"
] | [] | In a recent letter (ApJ 517, L155) Lenz et al. have shown the evidence of uniform temperature along steady long coronal loops observed by TRACE in two different passbands (171Å and 195Å filters). We propose that such an evidence can be explained by the sub-arcsecond structuring of the loops across the magnetic field lines. In this perspective, we present a model of a bundle of six thin parallel hydrostatic filaments with temperature stratification dictated by detailed energy balance and with temperatures at their apex ranging between 0.8 and 5 MK. If analyzed as a single loop, the bundle would appear isothermal along most of its length. | 10.1086/312414 | [
"https://arxiv.org/pdf/astro-ph/9911096v1.pdf"
] | 14,637,529 | astro-ph/9911096 | 6566ab7d7d26aca0b4f99e3c8f5698dbd48ef39f |
TRACE-derived temperature and emission measure profiles along long-lived coronal loops: the role of filamentation
5 Nov 1999
F Reale
Dip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo
Piazza del Parlamento 1I-90134PalermoItaly
G Peres
Dip. di Scienze Fisiche & Astronomiche -Sez. di Astronomia -Univ. di Palermo
Piazza del Parlamento 1I-90134PalermoItaly
TRACE-derived temperature and emission measure profiles along long-lived coronal loops: the role of filamentation
5 Nov 19991Subject headings: Sun: coronaSun: UV radiationSun: X-rays
In a recent letter (ApJ 517, L155) Lenz et al. have shown the evidence of uniform temperature along steady long coronal loops observed by TRACE in two different passbands (171Å and 195Å filters). We propose that such an evidence can be explained by the sub-arcsecond structuring of the loops across the magnetic field lines. In this perspective, we present a model of a bundle of six thin parallel hydrostatic filaments with temperature stratification dictated by detailed energy balance and with temperatures at their apex ranging between 0.8 and 5 MK. If analyzed as a single loop, the bundle would appear isothermal along most of its length.
Introduction
The solar X-ray emitting corona entirely consists of independent loop-like bright structures, in which the plasma is confined by the magnetic field (e.g. Vaiana et al. 1973). The coronal loops cover a wide range of sizes and brightness, and make the solar corona highly structured and contrasted in X-rays. Observations at high angular resolution (1") made with the Transition Region and Coronal Explorer (TRACE, e.g. Handy et al. 1999) show, once again, the high level of structuring of the solar corona, and, in particular, that coronal loops typically consist of several individual filaments, down to the telescope resolution limit. This raises even more questions on the structure, evolution, interaction, turning on, coherence and eventually the heating of the individual filaments (e.g. Proceedings of the Workshop on the Physics of the Solar Corona and Transition Region, 24 -27 August 1999Monterey, California, Solar Physics, 1999). In particular, one basic question is how the physical conditions of the several, energetically independent, loop filaments are related to the physical parameters derived from the analysis of TRACE data.
The temperature and density distribution along some loop structures observed by TRACE has been recently investigated by Lenz et al. (1999, hereafter L99). They have selected four relatively isolated loops extending above the solar limb and steady, at least for time intervals of 1-2 hr. The half length of three of the loops (loops a, b, and d) is L ≈ 10 10 cm, and that of the fourth (loop c) is L ≈ 5 × 10 9 cm. L99 have analyzed the brightness distribution along each loop by selecting four subimages of each loop (at 1/5, 1/3, 2/3 and 3/3 of the distance from the base to the loop top). Each subimage contains a few hundred to a few thousand pixels and includes the whole loop cross-section.
The temperature and the emission measure (EM) in the regions mentioned above are derived from the 171/195Å filter ratio and from the 171Å count rate, respectively, by assuming that all the plasma along the line of sight is at the same temperature and density. L99 find that the filter ratio varies very little along the four loops, therefore concluding that the temperatures profiles are constant along the loops, around 1.3 MK. This finding is at variance from a static, steady-state, non-isothermal loop model (e.g. Serio et al. 1981), uniformly heated and with a temperature of 1.3 MK at the apex: although the tem-perature profile would be rather flat in the corona according to this model, the observed profile is significantly (and incompatibly) flatter.
By assuming a uniform line-of-sight depth of 10 10 cm, L99 obtain emission measures increasing from the apex to the base, and ranging between 10 27 and 10 28 cm −5 . Such profiles are flatter, and at higher density, than predicted by single loop hydrostatic models at that temperature (Serio et al. 1981), and are better described by isothermal loops at T=1.3 MK.
Although aware that the static non-isothermal models used do not take into account possible additional effects, such as non-uniform heating, the presence of flows and mixing, and wave interaction with the background fluid, L99 conclude that "the lack of temperature variation in the EUV loops considered invites speculation that there is a class of such isothermal loops distinct from loops with a temperature maximum at the apex."
This letter revisits this interpretation in light of a more realistic scenario in which the loops detected by TRACE consist of bundles of filaments independent of each other, possibly in different physical conditions, and, in particular, at different temperatures. Each filament may then be described by a distinct nonisothermal loop model. We will show that TRACE diagnostic may actually yield an apparently almost isothermal profile along most of a loop consisting of such a bundle.
We conclude, therefore, that the evidence of isothermal loops shown by TRACE may just be a further signature of the filamentary structure of coronal loops and of the multi-temperature nature of each bundle of filaments across field lines.
In section 2 we model a loop observed by TRACE as a bundle of thin parallel hydrostatic filaments, synthesize its emission in the relevant TRACE bands and derive its effective temperature and emission measure with the procedures used to analyze real TRACE data. We discuss the results in section 3.
The modeling
We consider a loop of half length 5×10 9 cm (corresponding to loop c in L99). The choice of the shortest loop allows us to concentrate on the effects of loop filamentation, without the complication of a strong gravitational stratification expected for the longer loops.
We model the loop as a bundle of parallel, static and hydrostatic filaments. Each filament is assumed to be semicircular and symmetric with respect to the apex, and to lie on a plane perpendicular to the solar surface. Temperature and density stratification is taken into account, according to Serio et al. (1981), by solving numerically the equations for hydrostatic equilibrium and for energy balance among plasma thermal conduction, radiative losses and a heating source assumed uniform along each filament 3 . Table 1 shows the relevant parameters of the model loop filaments, all of the same length and differing for the heating rate. The latter determines the plasma pressure conditions inside each filament (according to the scaling laws of Rosner et al. 1978 andof Serio et al. 1981). We have considered six filaments, with heating rates such that the pressures at the footpoints are logarithmically equispaced and spanning between 0.03 and 10 dyn cm −2 . The corresponding filament maximum temperatures span between 0.8 and 5.2 MK. With this choice we sample the plasma conditions typical of non-flaring coronal loops, from quiet to active regions, and relevant for observations with TRACE 171Å and 195Å filters.
The computed density and temperature profiles along half of each filament (the other half is symmetric) have been used to synthesize the emission E per unit optical depth along the filament, in units of Data Number (DN) s −1 pix −1 cm −1 , in the two relevant TRACE passbands, according to:
E(s) = n(s) 2 G[T (s)](1)
where s is the coordinate along the filament (cm), n the plasma density (cm −3 ), T its temperature (K), G(T ) the response in each TRACE passband (DN s −1 pix −1 EM −1 ), computed with the MEKAL spectral code (Mewe et al. 1995), and shown in Fig. 1. The figure also shows the expected ratio of the emission detected in the 171Å and 195Å passbands vs temperature (assuming ionization equilibrium). The loop-shaped pictures in Fig. 2 show the predicted distributions of the emission in the two TRACE passbands for the six filaments in Table 1, put side-byside and ordered with pressure (and maximum temperature) increasing inwards. The grey scale (emission per unit optical depth) is saturated at 5 × 10 −10 DN s −1 pix −1 cm −1 .
Such pictures, which assume all filaments to have equal cross-section, provide a general impression of how a thermally structured bundle of filaments would look like in the TRACE passbands. The outermost (and coldest) filament is barely visible in the 171 A passband, even if most of it is at a temperature not far from the peak of the response, because of its low emission measure, and analogously invisible in the 195Å passband. In the 171Å passband the hotter filaments have comparable brightness in the chosen scale, except the hottest one which is clearly brighter, because of its higher emission measure. In the 195Å passband filament 3 is brighter than filaments 2 and 4, because the temperature of most of its plasma is close to the peak of the filter response. Again, the hottest filament is the brightest one because of its high emission measure. Fig. 3 shows the ratio of the filtered emission in the 171Å to the 195Å passband computed along (half of) each of the six filaments, from the footpoint to the apex, according to Eq. (1). These profiles can be easily understood in terms of the filter ratio curve shown in Fig. 1. Above the lowest 10 9 cm, the ratio is quite high for the coolest models 1 and 2 (≫ 10 for model 1), because the temperature along most of them is in the branch below log T = 6.2, and it is much lower than one along most of filament 3 whose temperature is mostly around the ratio minimum occurring at log T ≈ 6.3. The temperature of the other hotter models mostly falls in the branch of the ratio curve above log T ≈ 6.3, which is more weakly dependent on temperature and with a value ≈ 1. The ratio profile of model 4 has a minimum close to 10 9 cm (corresponding to the minimum at log T ≈ 6.3 in the ratio curve of Fig. 1), and then gradually increases to ∼ 1 upwards along the loop. For models 5 and 6, the hottest ones, the minimum is more localized and located well below the level of 10 9 cm: above, the ratio is constantly ∼ 1.
Our key point is now how such a bundle of filaments would be detected by TRACE if analyzed as a single loop, as in L99. We then sum all the emission profiles shown in Fig. 2, to obtain two single profiles, one for each filter passband. The profile of the ratio of the two resulting emission distributions is shown in Fig. 3 (thick solid line): the ratio is virtually constant (≈ 0.7) along the whole loop except at the very base (below 5 × 10 8 cm). Of course, such a ratio is determined mostly by the brightest filaments, and in particular model 6, the hottest one, which contributes 30% of the total emission above 10 9 cm in the 171 A band; the faintest models 1 to 5 contribute to lower the value of the ratio and to make it even more uniform.
The most straightforward interpretation of such a flat ratio profile would be a temperature uniformly distributed along the loop, and in the region around log T = 6.1, i.e. close to the temperature of maximum formation of the 171Å and 195Å lines, if one restricts the range of possible temperatures to be in the monotonic branch of the filter ratio curve around log T = 6.1. The temperature distribution, shown in Fig. 4, is flat for s ≥ 5 × 10 8 cm, and around 1.4 MK, i.e. very close to value obtained by L99 for the four loops they have analized.
From the ratio of the expected emission to the value of the response function at the given temperature, by assuming a line-of-sight depth of 10 10 cm, as done in L99, one obtains the distribution of the emission measure shown in Fig. 4. The region of interest, which can be compared to the results of L99, is the one above 10 9 cm: in such 4/5 of the loop the emission measure spans between ≈ 4 × 10 28 cm −5 and ≈ 4 × 10 27 cm −5 . For loop c L99 report quite a flat profile, with a value around 5 × 10 27 cm −5 . The EM profiles shown in Fig. 4 are not flat above 10 9 cm, but such profiles are obtained in the assumption of a constant line-of-sight depth. This may not be entirely realistic, as mentioned also in L99, and a smaller depth due to a thinning of the loop near and at the chromosphere cannot be excluded (a factor of two may be easily in agreement with observations). If the shrinking is not very fast, this should not affect significantly the model results, obtained in the assumption of a constant cross-section. Note that the choice of the depth cannot influence the key effect on the filter ratio discussed in this letter, since it depends on the combination of the several filaments emission, all six equally influenced by the depth value.
Discussion
Base on the indication of isothermal loops, observed by TRACE, L99 conjecture the possibility of a new class of isothermal loops, distinct from the nonisothermal hydrostatic loops.
The problem of the interpretation of the evidence of large isothermal loops, although at temperatures lower than those in L99, and probably in physical conditions very different from those in L99 (Brekke et al. 1997), have already been faced by Peres & Orlando (1996) and Peres (1997), who propose that such loops may be non-steady and strongly dynamic.
On the other hand, in the course of modeling a loop ignition observed by TRACE, Reale et al. (1999) have found that the temperature profile of a model loop of total length 10 10 cm and heated at one footpoint becomes relatively flat after a few thousand seconds. Since evidence for isothermality is invariably found in all loops analyzed in L99, one wonders whether the phenomena discussed by Reale et al. (1999) may be so common. Reale et al. (1999) themselves have found that the loop they study is certainly not activated by heat deposition at one footpoint, but more likely higher in corona. In the same work it is shown that relatively flat temperature profiles are obtained if the heating is high for a short time and then decaying slowly. However, such a possibility, as well as the non-steadiness proposed for large cool loops, seems unlikely for the TRACE loops analyzed by L99, since they are observed to be steady for times longer than the loop characteristic cooling times.
The modeling illustrated in the present work shows that a bundle of conventional (Serio et al. 1981) nonisothermal and uniformly-heated static filaments with apex temperatures ranging between 0.8 and 5 MK, observed by TRACE in the 171Å and 195Å passbands, if analyzed as a single loop, would appear as an isothermal loop with a filter ratio typical of ∼ 1.4 MK and with emission measures in rough agreement with the observed ones.
We have obtained this result by selecting a bundle of six loop filaments with base pressure logarithmically equispaced, simply summing their emission with equal weights. One may wonder how the results are influenced by this particular (although unbiased) choice of the parameters; indeed we obtain very similar results with a different choice of the pressures, provided that hot filaments are included. The fundamental result is therefore quite robust. Observations in all three TRACE filters may provide further constraints on the thermal structure of the loop bundle. However a more conclusive word may require future instruments with even higher spatial resolution, so as to better resolve the single filaments and to obtain enough signal-to-noise ratio from each of them.
The key result obtained here crucially depends on the major role played by the hottest components of the bundle: a) their temperature is mostly in the region in which the 171/195 filter ratio is weakly depen-dent on temperature, thus yielding a flat filter ratio profile along more than 4/5 of the length and showing a dip (corresponding to the dip in the filter ratio curve around log T = 6.3) only very close to the footpoints; b) their emission measure is high and contributes significantly to the total emission in the TRACE band even though their temperature is not close to the peak of the filter response. For such hot filaments, and, in general, for all loops with 6.3 < log T < 7, the temperature diagnostics offered by TRACE two-filter observations should be taken with care, due to the non-monotonic and weak dependence of the 171/195 filter ratio on the temperature in that range.
Our conclusion is that the isothermal profiles obtained by L99 could be a consequence of the high structuring of the observed loops across the magnetic field lines, i.e. "filamentation", and, likely, an indication of the presence of high temperature threads (∼ 5 MK). The TRACE images, indeed also those in L99, show a high level of loop filamentation.
This result of course needs further support by future observation and analysis, possibly in coordination with other instruments with spectral capabilities, such as those on board SOHO.
We thank S. Serio for useful suggestions. We acknowledge support from Ministero della Ricerca Scientifica e Tecnologica and from Agenzia Spaziale Italiana. Table 1, ordered with pressure (and maximum temperature) increasing inwards. The grey scale (emission per unit optical depth) is saturated at 5 × 10 −10 DN s −1 pix −1 cm −1 . Table 1 (thin lines). The ratio obtained by merging the six loops into a single loop, i.e. by summing the emission of the loops is also shown (thick solid line). and emission measure profiles along the bundle of six filaments (Table 1); we have summed the synthesized emission of all six filaments in the 171Å and 195Å passbands and used the monotonic branch of the filter ratio curve around log T = 6.1. The solid line shows the emission measure profile derived, as in L99, using a constant G(T) value of 2 × 10 −27 DN s −1 pix −1 EM −1 for the 171Å passband; the dashed line shows the analogous result using the G(T) proper.
Fig. 1 .Fig. 2 .
12-TRACE response of 171Å (solid) and 195Å (dashed) passbands (EM is in units of cm −5 ), and filter ratio (thick solid) of the 171Å and 195Å passbands vs temperature of the emitting plasma volume. -The loop-shaped structures show the distribution of the emission per unit volume in the TRACE 171 A and 195Å passbands, synthesized from the six model loops in
Fig. 3 .
3-Ratio of the of the 171Å to the 195Å passband expected emission computed along each of the six model loops in
Fig. 4 .
4-Temperature (dashed-dotted line)
Table 1 :
1Parameters of the model loop filaments Model Base pressure T maxHeating
Apex density
Serio et al. 1981 have shown that non-uniform heating distributions do not change dramatically the model results, and, in particular, the temperature stratification, due to the effectiveness of thermal conduction, unless the heating is very localized.
This 2-column preprint was prepared with the AAS L A T E X macros v4.0.
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. G S Vaiana, A S Krieger, A F Timothy, Sol. Phys. 3281Vaiana, G. S., Krieger, A. S., Timothy, A. F., 1973, Sol. Phys., 32, 81.
| [] |
[
"Classification AMS 2000: Primary 37F, Secondary 32U40",
"Classification AMS 2000: Primary 37F, Secondary 32U40"
] | [
"Tien-Cuong Dinh ",
"Viêt-Anh Nguyên ",
"Nessim Sibony "
] | [] | [] | We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C k , under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps, to regular polynomial automorphisms of C k and to their small pertubations. | 10.1016/j.aim.2008.07.006 | [
"https://arxiv.org/pdf/0710.4007v2.pdf"
] | 15,045,900 | 0710.4007 | 89c8795050840c049eae841f6ba062662b17efab |
Classification AMS 2000: Primary 37F, Secondary 32U40
6 Sep 2008 September 6, 2008
Tien-Cuong Dinh
Viêt-Anh Nguyên
Nessim Sibony
Classification AMS 2000: Primary 37F, Secondary 32U40
6 Sep 2008 September 6, 2008Dedicated to Professor Gennadi Henkin on the occasion of his 65th birthdayhorizontal-like mapGreen currentequilibrium measureentropymixingLyapounov exponentstructural disc of currents
We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C k , under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps, to regular polynomial automorphisms of C k and to their small pertubations.
Introduction
The abstract theory of non-uniformly hyperbolic systems is well-developed, see e.g. Katok and Hasselblatt [28], Pesin [32], L.-S. Young [38]. It is however difficult to show that a concrete example is a non-uniformly hyperbolic system. The main questions are to construct a measure of maximal entropy, to study the decay of correlations and to show that the Lyapounov exponents do not vanish. Such problems have been studied in dimension 2 for real Hénon maps by Benedicks-Carleson, L.-S. Young, Viana etc., see e.g. [3,4,39]. In this paper we consider these questions for holomorphic horizontal-like maps in C k using tools from complex analysis: positive closed currents, estimates for solutions of the dd cequation and appropriate spaces of test forms. The complex analytic methods permit to avoid the delicate arguments used in the real setting.
In [13] the first and the third authors studied the dynamics of polynomiallike maps in several complex variables using adapted spaces of test functions. This approach permits to study convergence problems, in particular, the decay of correlations for the measure of maximal entropy. Recall that a polynomial-like map is a proper holomorphic map f : U → V between convex open sets U ⋐ V (or more generally pseudoconvex open sets) in C k . Such a map is somehow "expanding", but it has in general a non-empty critical set; so, it is not uniformly hyperbolic in the dynamical sense, see [28]. It is shown in [13] that the measure of maximal entropy is hyperbolic if the topological degree is strictly larger than the other dynamical degrees. This condition is natural and is stable under small pertubations on the map. Holomorphic endomorphisms of P k can be lifted to polynomial-like maps in some open sets of C k+1 . So, their dynamical study is a special case of polynomial-like maps. Small transcendental pertubations of such maps provide large families of examples.
Here, we consider the quantitative aspects of the dynamics of horizontal-like maps f in any dimension, inside a product of convex open sets D = M × N in C p × C k−p . They are basically holomorphic maps which are somehow "expanding" in p directions (horizontal directions) and "contracting" in the other k −p directions (vertical directions), see Section 3 for the precise definition. They partially look like a horseshoe. But, the expansion and contraction are of global nature, and in general, these maps are not uniformly hyperbolic. Small pertubations of horizontal-like maps are horizontal-like provided that we shrink slightly the domain of definition. When p = k we obtain polynomial-like maps.
Hénon maps in C 2 were studied by Bedford-Lyubich-Smillie [2] with the equilibrium measure introduced by the third author of the present article, see also [22]. The case of horizontal-like maps in dimension 2, i.e. k = 2 and p = 1, has been studied by Dujardin with emphasis on biholomorphic maps (Hénon-like maps) [20] and was developed by Dujardin, the first and the third authors to deal with random iteration of meromorphic horizontal-like maps [12]. It turns out that horizontal-like maps are the building blocks for polynomial maps of "saddle type". In particular, they were used to study rates of escape to infinity for polynomial mappings in C 2 . The randomness comes from the indeterminacy points at infinity, see also [37].
In this paper, we continue our study in the higher dimensional case. In order to simplify the notation, we only consider invertible maps. However, a large part of our study can be extended to the general case. Some basic objects and the first properties for such maps (Green currents T ± , equilibrium measure µ, entropy, mixing, etc) were constructed and established in [17]. The Green current T + is positive closed of bidegree (p, p), invariant under f * and is vertical: its support does not intersect the vertical boundary ∂M × N of D. The Green current T − is positive closed of bidegree (k −p, k −p), invariant under f * and is horizontal. The equilibrium measure µ is an invariant probability measure which is equal to the wedge-product T + ∧ T − of the Green currents. The definition of wedge-product relies on an intersection theory for positive closed currents.
The main technical problem is the use of currents of bidegree (p, p), p ≥ 1. For that purpose, a geometry on the space of positive closed (p, p)-currents was introduced using as basic objects: structural discs of currents. Roughly speaking, in order to travel from a positive closed current R 1 of bidimension (k −p, k −p) to another one R 2 , we construct a family of currents parametrized by a holomorphic disc ∆ ⊂ C. These currents appear as the slices of a positive closed current R of bidimension (k − p + 1, k − p + 1) in ∆ × D; the currents R 1 and R 2 are seen as two points of the disc, i.e. two currents obtained by slicing R with {θ 1 } × D and {θ 2 } × D for some θ 1 , θ 2 in ∆. We use properties of subharmonic functions on those structural discs in order to define the wedge-product of currents of higher bidegree and in order to prove the convergence results in the construction of T ± and µ. More formally as in [19] we use super-functions, i.e. functions defined on horizontal currents which are p.s.h. on structural discs of currents.
In the present article, we study the quantitative properties of these basic dynamical objects. For a horizontal-like map f , one associates a main dynamical degree d ≥ 2 which is an integer. The topological entropy of f and the entropy of µ are equal to log d. We will define the other dynamical degrees d ± s in Section 3. One of our main results is the following. Theorem 1.1. Let f be an invertible horizontal-like map on a convex domain D = M × N in C p × C k−p . Assume that the main dynamical degree d of f is strictly larger than the other dynamical degrees. Then the Green currents T + and T − of f are the unique, up to a multiplicative constant, invariant vertical and horizontal positive closed currents of bidegrees (p, p) and (k − p, k − p) respectively. The equilibrium measure µ of f is exponentially mixing and is hyperbolic. More precisely, µ admits k − p strictly negative and p strictly positive Lyapounov exponents.
We study the speed of convergence towards the Green currents T ± and the equilibrium measure µ, and also the regularity of these objects. The regularity is studied by considering on which space of forms or functions the currents or measures act continuously. We show in particular that µ is PB, that is, plurisubharmonic functions (p.s.h. for short) are µ-integrable. The main tools here are estimates and localization of the support for good solutions of the dd c -equation. We obtain these estimates through integral formulas (a classical result by Andreotti-Grauert is crucial here). They permit to apply the dd c -method and the duality method as in [13,15,16,17]. The speed of convergence towards the Green currents is a basic ingredient in the proof of the decay of correlations for µ.
For Hénon like-maps (k = 2, p = 1), the hypothesis on the dynamical degrees is always satisfied. Theorem 1.1, except for the decay of correlations (exponential mixing), was proved in [20]. The decay of correlations for Hölder observables and for Hénon maps was investigated by the first author in [11]. The hyperbolicity of the equilibrium measure is considered in a very general context for meromorphic maps on compact Kähler manifolds by de Thélin [9]. We follow his method.
We end this introduction by giving another large family of examples. Consider a polynomial automorphism f of C k . We still denote by f its meromorphic extension to P k . When the indeterminacy sets I + and I − of f and f −1 in the hyperplane at infinity L ∞ are non-empty and have no intersection, we say that f is regular. Then there is an integer p such that dim I + = k − p − 1 and dim I − = p − 1. We refer to [35] for the basic dynamical objects and properties of such maps, see also Section 6 below. Let z = (z 1 , . . . , z k ) denote the coordinates in C k and denote [z 0 : · · · : z k ] the homogeneous coordinates of P k . The hyperplane at infinity L ∞ := P k \ C k is given by the equation z 0 = 0.
I + = {z 0 = z 1 = · · · = z p = 0} and I − = {z 0 = z p+1 = · · · = z k = 0}.
Let B R s denote the ball of center 0 and of radius R in C s . Then, if R is large enough, any holomorphic map on B R p ×B R k−p , close enough to f , is horizontal-like. Moreover, its equilibrium measure is exponentially mixing and hyperbolic.
Note that the above pertubation of f may be transcendental and that Corollary 1.2 produces large families of examples.
Here is a brief outline of the paper. In Section 2, the main tools, in particular, several classes of currents and the solution of the dd c -equation, are introduced. In Section 3, we recall the dynamical objects associated to a horizontal-like map. Theorem 1.1 is proved in Sections 4 and 5. Corollary 1.2 is deduced from Theorem 1.1 and from Proposition 6.1 in the last section. Also in the last section open questions are stated.
Notation and convention. Throughout the paper, D := M × N is a bounded convex domain in C p × C k−p . The estimates we obtain are valid in the interior of D and might be bad near the boundary, but this is harmless for the type of maps we consider. So, we sometimes reduce D slightly in order to have maps and currents defined in a neighbourhood of D; this simplifies the exposition. We will also choose strictly convex domains with smooth boundary M ′′ ⋐ M ′ ⋐ M and N ′′ ⋐ N ′ ⋐ N and consider the domains D ′ := M ′ × N ′ and D ′′ := M ′′ × N ′′ . When we consider vertical currents R or horizontal currents S, Φ, our choice is so that R is supported on M ′′ × N and S, Φ are supported on M × N ′′ . When we consider a horizontal-like map f on D, we assume that f −1 (D) ⊂ M ′′ × N and f (D) ⊂ M × N ′′ . So, f restricted to D ′ or D ′′ is horizontal-like. The convex domains M , M, N , N are chosen so that M ⋐ M ⋐ M and N ⋐ N ⋐ N . Note also that when we consider the convergence of a family of vertical or horizontal currents, we assume that they have support in the same vertical or horizontal set.
Currents and dd c -equation
In this section, we will introduce the tools used in this work. We will give some geometrical and analytical properties of several classes of currents. In particular, we will define structural discs of currents and solve the dd c -equation with estimates and with controlled support. Recall that d c := i 2π (∂ − ∂). • Vertical, horizontal currents and their intersection. We call vertical (resp. horizontal) boundary of D the sets
∂ v D := ∂M ×N (resp. ∂ h D := M ×∂N). A subset E of D is vertical (resp. horizontal) if E does not intersect ∂ v D (resp. ∂ h D)
. Let π 1 and π 2 denote the canonical projections of D onto M and N. Then E is vertical or horizontal if and only if π 1 (E) ⋐ M or π 2 (E) ⋐ N. A current on D is vertical or horizontal if its support is vertical or horizontal. Let C v (D) denote the cone of positive closed vertical currents of bidegree (p, p) on D. Consider a current R in C v (D). Since π 2 is proper on supp(R), (π 2 ) * (R) is a positive closed current of bidegree (0, 0) on N. Hence, (π 2 ) * (R) is given by a constant function on N that we denote by R v . Convergence in C v (D) is the weak convergence of currents with support in a fixed vertical set.
Recall from Theorem 2.1 in [17] that the slice measure R, π 2 , w is defined for every w ∈ N, and that its mass is equal to R v which is independent of w. We say that R v is the slice mass of R. For every smooth probability measure Ω with compact support in N, we have R v := R, (π 2 ) * (Ω) . When R v = 1 we say that R is normalized. Let C 1 v (D) denote the set of such currents. This convex set is relatively compact in the cone of positive closed currents on D. In particular, the mass of normalized currents R on a compact set of D is bounded uniformly on R. In order to avoid convergence problems on the boundary, we will also use the convex set C 1 v (M × N ) of positive closed currents which are vertical in M × N with slice mass 1 for some neighbourhood N of N .
The slice mass · h , the sets C h (D), C 1 h (D) and the convergence for horizontal currents of bidegree (k − p, k − p) are defined similarly. If R is a current in C v (D) and S is a current in C h (D) we can define the intersection R ∧ S. This is a positive measure of mass R v S h with support in supp(R) ∩ supp(S), see [17]. It depends linearly on R and on S and is continuous with respect to the plurifine topology in the following sense. Let (R θ ) and (S θ ′ ) be structural discs in
C 1 v (D) and C 1 h (D), see the definition below. Assume that supp(R θ ) ∩ supp(S θ ′ ) is contained in an open set Ω ⋐ D.
If ϕ is a p.s.h. function on a neighbourhood of Ω, then R θ ∧ S θ ′ , ϕ is either a p.s.h. function of (θ, θ ′ ) or equal identically to −∞, see Proposition 3.4 and Remark 3.8 in [17]. Basically, for a suitable choice, with R 1 = R, S 1 = S and R θ , S θ ′ smooth when θ = 1, θ ′ = 1, we obtain R ∧ S as the limit of R θ ∧ S θ ′ , R θ ∧ S, R ∧ S θ ′ for θ → 1 and θ ′ → 1. It is also shown in [17] that for a p.s.h. function ϕ on D R ∧ S, ϕ = lim sup R ′ ∧ S ′ , ϕ = lim sup R ′ ∧ S, ϕ = lim sup R ∧ S ′ , ϕ , with R ′ , S ′ smooth in C v (D), C h (D) converging respectively to R and S.
• Structural discs of currents. Let X be a complex manifold. Consider a positive closed (p, p)-current R in X × D. We assume that the support of R is contained in X × M ′ × N for some open set M ′ ⋐ M. Let π : X × D → X denote the canonical projection. It is shown in [17] that the slice R, π, x exists for every x ∈ X. They can be considered as the intersection of R with the current of integration on π −1 (x). This is a positive closed (p, p)-current on {x} × D that we identify with a current R x on D which is vertical. When R is a smooth form, the slice R x is simply the restriction of R to π −1 (x). The slice mass of R x does not depend on x. So, multiplying R with a constant, we can assume that this mass is 1. We obtain a map τ :
X → C 1 v (D) with τ (x) := R x .
In general, R x does not depend continuously on x with respect to the usual topology on X. The dependence is continuous with respect to the plurifine topology, i.e. the coarsest topology for which p.s.h. functions on X are continuous. We call structural variety of C 1 v (D) the map τ or the family (R x ). This notion can be easily extended to
C 1 v (M × N). Consider a vertical positive closed (p, p)-current R in C 1 v (M × N).
So, R is a vertical current of slice mass 1 on M ′ × N for some convex open sets M ′ ⋐ M and N ⋑ N. Let ∆ denote a small neighbourhood of the interval [0, 1] in C. We constructed in [17] a particular structural disc (R θ ) θ∈∆ in C 1 v (M × N) parametrized by ∆ such that R 1 = R and R 0 is independent of R. The current R θ is obtained as a regularization of R. More precisely, we consider some holomorphic family of linear endomorphisms h a,b,θ : C k → C k parametrized by (a, b, θ) ∈ C p × C k−p × ∆ with h a,b,1 = id. The current R θ is obtained using a smooth probability measure ν with compact support in C p × C k−p :
R θ := (h a,b,θ ) * (R)dν(a, b).
The convexity of M × N and the fact that R is defined on M ′ × N permit to define the smoothing and to obtain vertical currents R θ in C 1 v (M × N). The size of ∆ depends only on M, M ′ , N and N . The considered structural discs satisfy the following important properties. The currents R θ depend continuously on θ, linearly on R and are smooth for θ = 1. The continuity is with respect to the weak topology on R θ and the usual topology on θ. Moreover, R θ depend continuously on θ and on R with respect to the usual topology on θ ∈ ∆ \ {1}, the C ∞ topology on R θ and the weak topology on R. When R is smooth, the last property also holds for θ ∈ ∆.
• PSH currents and p.s.h. functions. A real (k − p, k − p)-horizontal current Φ on D is called PSH if dd c Φ ≥ 0 1 . Let PSH h (D) denote the set of horizontal PSH currents. It is endowed with the following topology. A sequence (Φ n ) converges to Φ in PSH h (D) if Φ n → Φ weakly and if Φ n and Φ have their supports in a fixed horizontal set of D.
Recall that an upper semi-continuous function φ : D → R ∪ {−∞} is p.s.h. if it is not identically −∞ and if its restriction to any holomorphic disc in D is subharmonic or equal to −∞. Let PSH(D) denote the cone of such functions. It is relatively compact in L p loc (D) for 1 ≤ p < +∞. Note that an L 1 loc function φ : D → R ∪ {−∞} is p.s.h. if it is strongly upper semi-continuous and if dd c φ is a positive closed current. The strong upper semi-continuity means φ(a) = lim sup z→a φ(z) for a ∈ D and z ∈ A where A is any measurable subset of full measure in D. Denote by PSH(D) the cone of p.s.h. functions defined in a neighbourhood of D.
• Extension of spaces of test forms and super-functions. Let R be a current in C 1 v (D). It acts on horizontal smooth forms of bidegree (k − p, k − p). We will extend this space of test forms. Let H h (D) denote the space of real horizontal currents Φ of bidegree (k − p, k − p) with dd c Φ = 0. We consider the following topology on H h (D): a sequence (Φ n ) converges to Φ in H h (D) if Φ n → Φ weakly and Φ n have support in a fixed horizontal set.
R, Φ) → R, Φ with Φ ∈ H h (D)
is bilinear and continuous in (R, Φ). In particular, R, Φ is bounded on compact subsets of C 1 v (D) × H h (D). Proof. Observe that if Φ is a current in H h (D) we can use a slight dilation and a convolution in order to regularize Φ. So, there are smooth forms Φ n converging to Φ in H h (D). This implies the uniqueness, the linearity and the positivity of the extension. Recall that the positivity means R, Φ ≥ 0 for Φ ≥ 0. We prove now the existence of the extension on H h (D) and the continuity.
Shrinking D allows to assume that R is defined on M ′ × N with M ′ ⋐ M and N ⋑ N. Consider the structural disc (R θ ) as above. Define h(θ) := R θ , Φ . As in [17,Thm. 2.1], h is a harmonic function on ∆ \ {1}. If Φ is smooth, the function is defined and is harmonic on ∆. Define h n (θ) := R θ , Φ n . The above description of properties of R θ implies that h n converge locally uniformly to h on ∆ \ {1}. Since h n are harmonic on ∆ and locally uniformly bounded, by maximum principle, the limit h can be extended to a harmonic function on ∆ and h n converge to h on ∆. Observe that the limit does not depend on the choice of Φ n .
We have R, Φ = h(1) when Φ is smooth. Define R, Φ := h(1) the extension of R to all Φ in H h (D). Recall that R θ , for θ = 1, depends continuously on R with respect to the C ∞ topology on R θ . Hence, h depends continuously on (R, Φ). The continuity of R, Φ follows.
We will extend R to a linear form on PSH h (D), but the extension can take the value −∞. Recall that R is a current on M ′ × N.
→ R θ , Φ is subharmonic on ∆ and we have R, Φ = lim sup R ′ , Φ with R ′ → R in C 1 v (M × N).
Proof. We can assume that Φ is supported on M × N ′ and that R is vertical in M ′′ × N . So, we can assume that the considered currents Φ ′ are horizontal on D.
Consider first the case where Φ is smooth. Let Φ n be a sequence of smooth forms converging to Φ in PSH h (D). Define h(θ) := R θ , Φ and h n (θ) := R θ , Φ n . These functions are subharmonic and continuous on ∆, see [17, Thm 2.1] (the subharmonicity is deduced from the positivity of dd c (R ∧ Φ n ) and of its pushforward to ∆). We also have h n → h on ∆ \ {1}. It follows from the classical Hartogs' lemma [27]
that lim sup h n (1) ≤ h(1). So, lim sup R, Φ ′ ≤ R, Φ .
On the other hand, since R θ is obtained from R by smoothing using an averaging on a group of linear transformations, a coordinate change implies that
R θ , Φ = R, Φ θ where Φ θ is obtained from Φ by a similar smoothing. The fact that Φ is defined on M × N ′ guarantees that Φ θ is horizontal in D. We also have Φ θ → Φ when θ → 1 for the C ∞ topology. Since h is continuous we deduce that R, Φ θ → R, Φ when θ → 1. So, R, Φ = lim sup R, Φ ′ when Φ is smooth.
In other words, R, Φ := lim sup R, Φ ′ defines an extension of R to all Φ in PSH h (D). It is clear that the extension does not depend on the choice of N.
For a general current Φ, there are smooth forms Φ n converging to Φ in PSH h (D). Define h n and h as above. The function h is defined on ∆ \ {1}. The functions h n are continuous subharmonic, bounded from above and converge to h on ∆ \ {1}. It follows that h can be extended to a subharmonic function on ∆. By Hartogs' lemma, we have
h(1) ≥ lim sup h n (1) = lim sup R, Φ n . It follows that h(1) ≥ R, Φ = lim sup R, Φ ′ .
On the other hand, since h is subharmonic, we have h(1) = lim sup h(θ) = lim sup R, Φ θ when θ → 1. We deduce as above that h(1) = R, Φ . Since h depends linearly on R and Φ, R, Φ depends linearly on R and Φ. We also obtain that θ → R θ , Φ is subharmonic on ∆.
It remains to prove that R,
Φ = lim sup R ′ , Φ with R ′ → R in C 1 v (M × N ). This property implies that R, Φ is independent of the choice of M ′ . Since R, Φ = lim sup R θ , Φ for θ → 1, we have R, Φ ≤ lim sup R ′ , Φ with R ′ → R. Now, if (R ′ θ ) is the structural disc associated to R ′ and if h ′ (θ) := R ′ θ , Φ , then h ′ (θ) → h(θ) for θ = 1.
We deduce from Hartogs' lemma that h(1) ≥ lim sup h ′ (1) which implies that R, Φ ≥ lim sup R ′ , Φ and completes the proof. Remark 2.3. We can consider R as a vertical current and Φ as a horizontal one in appropriate domains D ′ ⋐ D and define R, Φ on D ′ instead of D. We will obtain the same value. Indeed, in order to define (R θ ) we can find smoothings which are adapted for both D and D ′ , see [17] for details.
Remark 2.4. Let R be a current in C v (D), S in C h (D)
and ϕ a p.s.h. function on D. If ϕ is integrable with respect to the trace measure S ∧ ω p of S then ϕS defines a current in PSH h (D). We deduce from the above results that
R ∧ S, ϕ = lim sup θ→1 R θ ∧ S, ϕ = lim sup θ→1 R θ , ϕS = R, ϕS . Definition 2.5. Let Λ : C 1 v (M × N) → R ∪ {−∞} be an upper semi-continuous function which is not identically −∞. We say that Λ is a p.s.h. super-function if it is p.s.h. or identically equal to −∞ on each structural variety in C 1 v (M × N )
, and Λ is pluriharmonic if both Λ and −Λ are p.s.h., see also [19].
Proposition 2.6. Let Φ be a real horizontal (k − p, k − p)-current on D. If Φ is dd c -closed, then R → R, Φ defines a pluriharmonic super-function. If Φ is PSH, then R → R, Φ is a p.s.h. super-function.
Proof. We only have to prove the second assertion. Consider a structural variety (R x ) x∈X as above. Without loss of generality, we can assume that R
x are vertical in M ′ × N and Φ is horizontal in M × N ′ , see Remark 2.3. We want to prove that x → R x , Φ is identically equal to −∞ or p.s.h. If Φ is smooth,R x , Φ = lim sup θ→1 R x,θ , Φ = lim sup θ→1 R x , Φ θ ,
where (R x,θ ) is the particular structural disc constructed as above using the same smoothing for each R x . We deduce from the regularity of R x,θ that R x,θ , Φ is locally uniformly bounded on (x, θ) ∈ X × (∆ \ {1}). Since θ → R x , Φ θ is p.s.h., it follows from the maximum principle that R x,θ , Φ is locally uniformly bounded from above on X × ∆. Hence, the upper semi-continuous regularization
of x → R x , Φ is p.s.h. or identically −∞. It is enough to show that x → R x , Φ is upper semi-continuous.
For every a ∈ X, we have lim sup x→a R x , Φ θ = R a , Φ θ for θ = 1. Since the functions θ → R x , Φ θ are subharmonic, we deduce using Hartogs' lemma that lim sup x→a R x , Φ θ ≤ R a , Φ θ for every θ. This implies the result.
• PB, PC currents and measures. Let T be a vertical current of bidegree (p, p) in C v (D). We say that T is PB if T, Φ is bounded when Φ is in a relatively compact subset of PSH h (D). We say that T is PC if it can be extended to a continuous linear form on PSH h (D) with respect to the topology we have introduced. Observe that this extension coincides with the extension in Proposition 2.2. PC currents are PB. PB and PC horizontal currents of bidegree (k − p, k − p) are defined in the same way. In the case of bidegree (1, 1), PB and PC currents correspond to currents with bounded and continuous local potentials, see also [13,15,16].
A positive measure µ with compact support in D is said to be PB if µ, φ is bounded when φ are smooth functions in a relatively compact subset in PSH(D). Since p.s.h. functions on a neighbourhood of D can be approximated by decreasing sequences of smooth ones, µ is PB if and only if p.s.h. functions on a neighbourhood of D are µ-integrable. PB measures have no mass on pluripolar sets, i.e. sets which are contained in the pole set {φ = −∞} of a p.s.h. function φ. The measure µ is said to be PC if it can be extended to a linear continuous form on PSH(D). Denote by µ, φ the value of this extension on φ. Note that by continuity the extension is unique and µ, φ is equal to the usual integral µ, φ of φ. Any PC measure is PB.
• Solution of dd c -equation. We consider the dd c -equation on D. We will need negative solutions with horizontal or vertical support and with estimates on the mass. The behavior near the rest of the boundary is not important in our study. The following theorem is obtained using classical results. Recall that
d c := i 2π (∂ − ∂). Theorem 2.7. Let M ′ and M be convex domains in C p such that M ′ ⋐ M. Let N ′ and N ′′ be convex open sets in C k−p such that N ′′ ⋐ N ′ . Let Ω be a horizontal positive closed current of bidegree (k − p + 1, k − p + 1) on M × N ′′ . Then there is a horizontal negative L 1 form Φ of bidegree (k − p, k − p) on M ′ × N ′ such that dd c Φ = Ω on M ′ × N ′ and Φ M ′ ×N ′ ≤ c Ω M ×N ′′
with c > 0 independent of Ω. Moreover, Φ is defined by an integral formula, and depends linearly and continuously on Ω.
In what follows, the solutions of d, ∂ or dd c equations are given by classical integral formulas. Consequently, the linearity, the continuous dependence on data and the estimate on the mass of solutions are satisfied. Therefore, we will focus our attention only on the support of the solutions.
Lemma 2.8. Let D ′ and D be convex domains in C k with D ′ ⋐ D. Let Ω be a positive closed current of bidegree (k − p + 1, k − p + 1) on D. There is a negative L 1 form Ψ of bidegree (k − p, k − p) on D ′ , smooth out of the support of Ω, such that dd c Ψ = Ω on D ′ .
Proof. We can assume that D is contained in the ball of center 0 and of radius 1/2. Define for coordinates (z, ξ) on C k × C k the kernel
K(z, ξ) := log z − ξ (dd c log z − ξ ) k−1 .
Observe that K is negative when z < 1/2, ξ < 1/2, and dd c K is equal to the current of integration on the diagonal of C k × C k . Let χ be a cut-off function,
0 ≤ χ ≤ 1, with compact support in D, such that χ = 1 on a neighbourhood U of D ′ . Define Ψ ′ (z) := ξ χ(ξ)Ω(ξ) ∧ K(z, ξ).
Hence, Ψ ′ is a negative L 1 form depending continuously on Ω. If z is outside the support of Ω, then Ψ ′ (z) is given by an integration outside the singularities of K. So, Ψ ′ (z) is smooth there. Let π 1 and π 2 denote the canonical projections of C k × C k on its factors. If Ω is smooth we have
Ψ ′ = (π 1 ) * (π * 2 (χΩ) ∧ K). Since Ω is closed and dd c K = [z = ξ], we deduce that Ω ′ := dd c Ψ ′ − Ω is equal on U to Ω ′ = ξ dχ(ξ) ∧ Ω(ξ) ∧ d c K(z, ξ) − ξ d c χ(ξ) ∧ Ω(ξ) ∧ dK(z, ξ) + ξ dd c χ(ξ) ∧ Ω(ξ) ∧ K(z, ξ).
The last formula is valid for arbitrary Ω by regularization. So, Ω ′ is defined by
integration on {dχ(ξ) = 0} where K(z, ξ) is smooth if z ∈ U.
It follows that Ω ′ is smooth. We also have good estimates on C r norm of this form on compact subsets of U.
Since Ω ′ is closed and smooth, it is classical to obtain smooth solution of the equation dd c Ψ ′′ = Ω ′ with estimates (we first solve a d-equation and then a ∂-equation, the method will be described below with details in a situation where more estimates are needed). One checks that dd c Ψ = Ω for Ψ := Ψ ′ − Ψ ′′ − cω k−p where ω := dd c z 2 is the standard Kähler form on C k and c > 0 is large enough in order to guarantee that Ψ is negative on D ′ . Now, we need to control the support of the solution. We shrink slightly M and extend slightly N ′′ . This allows to assume that Ω is defined in M × F for some fixed compact set F in N ′′ . Using the previous lemma, we can find Ψ on M × N, smooth outside the support of Ω such that dd c Ψ = Ω. Let χ be a cut-off function equal to 1 on a neighbourhood of M × F and equal to 0 near M × ∂N ′′ and on M × (N \ N ′′ ). In particular, χ = 1 on the support of Ω and Ψ is smooth on {dχ = 0}. Define Φ 1 := χΨ and Ω ′ := dd c Φ 1 − Ω. This is a smooth horizontal closed form of bidegree (k − p + 1, k − p + 1) with support in M × N ′′ . Moreover, Ω ′ vanishes near M × F and has a controlled C r norm. We will find a smooth positive solution of the equation dd c Φ 2 = Ω ′ with horizontal support in M ′ × N ′ . The current Φ := Φ 1 − Φ 2 satisfies Theorem 2.7.
A construction using an integral formula as in the book [5, pp. 37-39 and 61-63] by Bott and Tu implies that there is a real smooth form Ψ which is horizontal in M × N ′′ such that dΨ = Ω ′ (shrink M and extend N ′′ if necessary). Of course, it satisfies the desired estimates in C r norms. Moreover, we can write
Ψ = Ψ ′ +Ψ ′′ with Ψ ′ of bidegree (k − p, k − p + 1) and Ψ ′′ of bidegree (k − p + 1, k − p) such that Ψ ′′ = Ψ ′ . Lemma 2.9. There is a smooth horizontal form Φ ′ on M ′ × N ′ , of bidegree (k − p, k − p), such that ∂Φ ′ = Ψ ′ .
Proof. Recall that we can, in each step of the proof, shrink or extend slightly the considered domains M, N ′ or N ′′ . This permits to avoid the problem near the boundary and to assume that they are strictly convex with smooth boundary.
Since dΨ is of bidegree (k − p, k − p), we have ∂Ψ ′ = 0. So, using a classical integral formula (see, for example [25,34]) we can find a smooth form
Φ * of bidegree (k − p, k − p) on M × N such that ∂Φ * = Ψ ′ .
Its support is not necessarily horizontal. So, we have ∂Φ * = 0 outside the support of Ψ ′ . We will apply a result of Andreotti-Grauert [26, p.109] in order to solve the equation
∂H = Φ * on M ′ × (N \ N ′′ ) with H smooth of bidegree (k − p, k − p − 1). Let χ be a cut-off function equal to 0 on M ′ × N ′′ and 1 in a neighbourhood of M ′ ×(N \N ′ ). The form χH is defined on M ′ ×N. It is clear that Φ ′ := Φ * −∂( χH) is horizontal in M ′ × N ′ and satisfies ∂Φ ′ = Ψ ′ , which completes the proof.
In order to apply the Andreotti-Grauert theorem, i.e. to solve the ∂-equation We need a much weaker result than Theorem 12.7 in [26]. Let ρ 1 be a smooth strictly convex function on N such that ρ 1 (z) → ∞ when z → ∂N and N ′′ = {ρ 1 < 1}. Since M ′ is strictly convex, we may find an unbounded exhaustion function ρ 0 for M ′ which is smooth and strictly convex. Define
for a ∂-closed form of bidegree (l, k − s), s ≥ p, in M ′ × (N \ N ′′ ),ρ(z) := ρ 0 (z ′ ) + cρ 1 (z ′′ ) + κ(ρ 1 (z ′′ )), z = (z ′ , z ′′ ) ∈ M ′ × (N \ N ′′ ).i∂∂(κ • ρ 1 ) = κ ′ · i∂∂ρ 1 + κ ′′ · i∂ρ 1 ∧ ∂ρ 1 ,
and κ ′′ (t) ≫ |κ ′ (t)| as t → 1 + , dd c ρ admits, at every point, at least one strictly positive eigenvalue with respect to the variable z ′′ . This completes the proof.
End of the proof of Theorem 2.7.
Define Φ ′′ := −iπ(Φ ′ − Φ ′ )
. This is a real smooth horizontal form in M ′ × N ′ . We have
dd c Φ ′′ = ∂∂(Φ ′ − Φ ′ ) = ∂Ψ ′ + ∂Ψ ′ = dΨ = Ω ′ .
The smooth form Φ ′′ is not necessarily positive. We can assume that it has support in M ′ × F for some compact subset F of N ′ . We now construct a horizontal closed form
U on M ′ × N ′ of bidegree (k − p, k − p) which is strictly positive on M ′ × F . Then, the form Φ 2 := Φ ′′ + cU, with c > 0 large enough, is positive and satisfies dd c Φ 2 = Ω ′ . For every point z ∈ M ′ × F there is a complex plane P of dimension p passing through z which does not intersect M ′ × ∂N ′ . This plane defines by integration a positive closed (k − p, k − p)-current [P ]
. Using a convolution, we obtain by averaging on small pertubations of [P ], a smooth positive closed form U z which is horizontal in M ′ × N ′ and is strictly positive at z. By continuity, such a form is strictly positive in a neighbourhood of z. It is enough to take a finite sum of such forms in order to obtain a form U which is strictly positive on M ′ × F . This completes the proof.
Remark 2.10. If Ω is a continuous form then Φ C 1 (M ′ ×N ′ ) ≤ c Ω C 0 (M ×N ′′ )
with a constant c > 0 independent of Ω. Indeed, we are using a solution given by a "good" kernel.
Horizontal-like maps
In this section we introduce the class of horizontal-like maps, the main dynamical objects of our study, and we give some basic properties. 2. pr 1|Γ is injective; pr 2|Γ has finite fibers. This property characterizes horizontal-like maps and we often use it in order to check that a map is horizontal-like. Since Γ is a submanifold of D × D, when z tends to ∂f −1 (D) ∩ D, f (z) tends to ∂ v D. When z tends to ∂f (D) ∩ D, f −1 (z) tends to ∂ h D. So, the vertical part of ∂f −1 (D) is sent into the vertical part of ∂f (D). If g is another horizontal-like map on D, f • g is also a horizontal-like map. When p = k, we obtain the polynomial-like maps studied in [13].
Γ does not intersect
∂ v D × D nor D × ∂ h D.
If pr 2|Γ is injective, we say that f is invertible. In this case, up to a coordinate change (an exchange of horizontal and vertical directions), f −1 : pr 2 (Γ) → pr 1 (Γ) is a horizontal-like map. When k = 2 and p = 1, we obtain the Hénon-like maps [20,12]. In order to simplify the paper, we consider only invertible horizontal-like maps.
Small pertubations of an invertible horizontal map are still horizontal and invertible if one shrinks slightly the domain D. Therefore, it is easy to construct large families of such maps.
Define f n := f • · · · • f (n times) the iterate of order n of f and f −n := f −1 • · · · • f −1 (n times) its inverse. Let K + (resp. K − ) denote the set of points z ∈ D such that f n (resp. f −n ) are defined at z for every n ≥ 0. In other words, we have K + := ∩ n≥0 f −n (D) and K − := ∩ n≥0 f n (D). It is easy to check that K ± are closed in D; K + is vertical and K − is horizontal. We call K + the filled Julia set of f and K − the filled Julia set of f −1 . Their boundaries are called Julia sets. Define also K := K + ∩ K − . This is a compact subset of D. We have [17].
f −1 (K + ) = K + , f (K − ) = K − and f ±1 (K ) = K , see
• Dynamical degrees, Green currents and equilibrium measure. The operator f * := (pr 2|Γ ) * • (pr 1|Γ ) * acts continuously on horizontal currents. If S is a horizontal current or form, so is f * (S). The operator f * := (pr 1|Γ ) * •(pr 2|Γ ) * acts continuously on vertical currents. If R is a vertical current or form, so is f * (R).
The continuity of f * , f * for non-invertible maps is treated in [18]. Recall from [17] the following proposition for positive closed currents of the right bidegree.
d ≥ 1 such that f * (S) h = d S h for every S ∈ C h (D v ). The operator f * : C v (D h ) → C v (D v ) is well-defined and continuous. If R belongs to C v (D h ), we have f * (R) v = d R v .
The integer d is called the main dynamical degree of f . In the sequel, it is often denoted by d(f ). Note that the previous proposition implies that d(f ) = d(f −1 ) and d(f n ) = d n . Consider a vertical subvariety L of dimension k − p in D. The projection π 2 : L → N defines a (ramified) covering. If m is the degree of this covering, the current [L] has slice mass m. We deduce from the previous proposition that f −1 (L) is a vertical subvariety of degree md. For m = 1, we obtain that d is an integer. There is an analogous picture when we push forward a horizontal subvariety. Note also that the projection of Γ onto the product of the first factor N with the second factor M defines a (ramified) covering of degree d. The following results were proved in [17].
(f n ) * (R) (resp. d −n (f n ) * (S)) converge to a current T + in C 1 v (D) (resp. T − in C 1 h (D)
) which does not depend on R (resp. S) and d −2n (f n ) * (R) ∧ (f n ) * (S) converge to the probability measure µ := T + ∧ T − . The current T + (resp. T − ) is supported on the Julia set ∂K + (resp. ∂K − ) and is invariant under
d −1 f * (resp. under d −1 f * ).
The measure µ is invariant under f * , f * and is supported on ∂K + ∩ ∂K − .
The current T + (resp. T − ) is the Green current associated to f (resp. f −1 ). The measure µ is called the equilibrium measure of f .
(f n ) * S M ′′ ×N ′′ 1/n ≤ lim sup n→∞ sup S (f n ) * S M ′ ×N ′ 1/n = lim sup n→∞ sup S (f n−1 ) * S ′ M ′ ×N ′ 1/n ≤ d + s .
This implies the first part of the lemma. Since f * preserves the mass of positive measures on f −1 (D), we obtain that d + 0 ≤ 1. If S is a probability measure on K then (f n ) * (S) is also a probability on K . So, d + 0 = 1. We obtain in the same way that d − 0 = 1. Assume that S is of bidegree (k − p, k − p). By definition of slices, we have S h S M ′ ×N and as we already discussed in Section 2,
S M ′ ×N S h . So, d n (f n ) * S M ′ ×N d n ,
which implies that d + p = d. We obtain in the same way that d − k−p = d.
• Action on super-functions. We reduce slightly D and assume that f is defined in a neighbourhood of D. Let Φ be a current in PSH h (D) and Λ the super-function associated to Φ defined on C 1 v (M × N), i.e. Λ(R) := R, Φ , see Proposition 2.6. The following lemma is useful in our calculus. for R ∈ C v (D) and Φ ∈ PSH h (D).
Proof. Let Λ ′ denote the function R → Λ d −1 f * (R) and Λ ′′ the super-function associated to d −1 f * (Φ). It is clear that Λ ′ (R) = Λ ′′ (R) for R smooth. We have to prove this equality for general R.
Let R be the current in ∆ × D associated to the structural disc (R θ ) constructed in Section 2. If F : ∆ × f −1 (D) → ∆ × f (D) is the map given by F (θ, z) := (θ, f (z)), one can check that the current d −1 F * (R) defines a struc-
tural disc (R ′ θ ) with R ′ θ = d −1 f * (R θ ). Since Λ is p.s.h., Λ ′ (R θ ) = Λ(R ′ θ ) is subharmonic on θ ∈ ∆.
The super-function Λ ′′ is also subharmonic on the disc (R θ ) and coincide with Λ ′ at R θ with θ = 1 because R θ is smooth for θ = 1. Hence, Λ ′ and Λ ′′ coincide also at R 1 = R, that is, Λ ′ (R) = Λ ′′ (R).
• Product maps. Let f i be horizontal-like maps on
D i = M i × N i . Define the product map F (x 1 , x 2 ) := (f 1 (x 1 ), f 2 (x 2 )) on D 1 × D 2 .
Up to a permutation of coordinates, we can identify D 1 ×D 2 to (M 1 ×M 2 ) ×(N 1 ×N 2 ). One checks easily that F is a horizontal-like map on this domain. If d i denote the main dynamical degree of f i , the main degree of F is d 1 d 2 . We can deduce from Theorem 3.3 the following properties. If T i,± are the Green currents associated to f ±1 i , the Green currents associated to F ±1 are T 1,+ ⊗T 2,+ and T 1,− ⊗T 2,− . If µ i are the equilibrium measures of f i , the equilibrium measure of F is µ 1 ⊗ µ 2 . In what follows, we will use the product • About the hypothesis on dynamical degrees. The hypothesis we need in this paper is that the main dynamical degree is larger than the other dynamical degrees. The following proposition shows that the family of the maps f satisfying this condition is open. Proof. It is clear that f ǫ is horizontal-like on D ′ . Since d can be interpreted as the degree of a covering, the main dynamical degree of f ǫ is also d. Let d + s and d + ǫ,s denote the dynamical degrees of order s of f and f ǫ . Fix a constant δ such that d + s < δ < d and a domain D ′′ = M ′′ × N ′′ in D ′ such that D ′ \ D ′′ is small enough. So, f and f ǫ restricted to D ′′ are horizontal-like. Consider a horizontal positive closed (k − s, k − s)-current S of mass 1 in D ′′ . By Lemma 3.5, there is an integer n 0 independent of S such that the mass of (f n 0 ) * S on D ′ is smaller than δ n 0 /2. If ω denotes the standard Kähler form on C k , we have since S is supported on D ′′ and f −n 0
(D ′′ ) ⊂ M ′′ × N (f n 0 ) * S D ′′ = D ′′ (f n 0 ) * S ∧ ω s = f −n 0 (D ′′ )∩D ′′ S ∧ (f n 0 ) * ω s . If f ǫ is close enough to f , (f n 0 ǫ ) * ω s − (f n 0 ) * ω s is a small form on f −n 0 (D ′ ) ∩ D ′ and f −n 0 ǫ (D ′′ ) ∩ D ′′ ⊂ f −n 0 (D ′ ) ∩ D ′ . Hence, (f n 0 ǫ ) * S D ′′ ≤ (f n 0 ) * S D ′ + f −n 0 ǫ (D ′′ )∩D ′′ S ∧ (f n 0 ǫ ) * ω s − (f n 0 ) * ω s .
It follows that (f n 0 ǫ ) * S D ′′ ≤ δ n 0 . The estimate is independent of S and implies by iteration that (f n ǫ ) * S D ′′ δ n for n ≥ 1 uniformly on S. Hence, d + ǫ,s ≤ δ < d. We get a similar results for f −1 ǫ and its dynamical degrees.
Convergence theorems
In this section we will give several quantitative versions of Theorem 3.3 under the hypothesis that the main dynamical degree d is strictly larger than the degrees δ + := d + p−1 and δ − := d − k−p−1 . We will see that this hypothesis is natural and is satisfied for large families of maps. A similar condition was considered in the context of polynomial-like maps, see [13].
• Convergence towards the Green currents. We will use the PSH horizontal currents as test "forms". The above solution of the dd c -equation allows to write such a test current as the sum of a PSH current with good estimates and a dd c -closed one. We obtain in particular the following result. We first consider the dd c -closed test currents. The following result shows that in this case, without any hypothesis on the dynamical degrees, the convergence is exponentially fast and uniform. N), Ψ ∈ H and n ≥ 0.
| d −n (f n ) * R − T + , Ψ | ≤ A 0 λ −n 0 for all R ∈ C 1 v (M ′ ×
Proof. Reducing D allows to assume that R is in
C 1 v (M ′ × N ) and H is compact in H h ( M × N ′ ). There is a constant A ′ > 0 such that | d −n (f n ) * R, Ψ | ≤ A ′ for all R ∈ C 1 v (D)
, Ψ ∈ H and n ≥ 0. This follows from Proposition 2.1 since
(R, Ψ) → R, Ψ is continuous. If Ψ ′ is in C 1 h ( M × N ′ )
, we have and T + , Ψ ′ = 1 and d −n (f n ) * R, Ψ ′ = 1 for every R ∈ C 1 v (D). By adding to Ψ a multiple of Ψ ′ , we can assume that T + , Ψ = 0 and we only need to prove the estimate under this assumption. Assume also for simplicity that A ′ = 1.
Denote by Λ Ψ the super-function Λ Ψ (R) := R, Ψ and L :
= d −1 f * the linear operator from C 1 v (D) into C 1 v (M ′ × N ). Since T + is invariant, we have Λ Ψ • L n (T + ) = 0. Let F denote the set of pluriharmonic super-functions Λ on C 1 v (M × N )
such that Λ(T + ) = 0 and Λ ∞ ≤ 1. Then, by Lemma 3.6 and the assumption that A ′ = 1, Λ Ψ • L n belongs to F for n ≥ 1 and we have
d −n (f n ) * R − T + , Ψ = Λ Ψ • L n (R).
So, by induction, it is enough to show that Λ • L ∞ ≤ 1/λ 0 for Λ in F and for some constant λ 0 > 1.
Assume that no constant λ 0 satisfies the above condition. Then there are Λ ∈ F and R ′ ∈ C 1 v (M ′ × N ) such that |Λ(R ′ )| is as close to 1 as we want. Recall that as in Section 2 we can construct a structural disc τ ′ (resp. τ ) such that τ ′ (1) = R ′ (resp. τ (1) = T + ). Moreover, τ ′ (0), τ (0) are equal to a fixed current R 0 . These discs are parametrized by a fixed neighbourhood ∆ of [0, 1]. By Harnack's inequality applied to the non-vanishing harmonic function 1−Λ•τ ′ on ∆, |Λ(R 0 )| is close to 1. Applying again the Harnack's inequality to 1 − Λ • τ , we deduce that |Λ(T + )| is close to 1. This contradicts the definition of F .
Proof of Theorem 4.1. Fix a constant δ such that δ + < δ < d. Consider a test current Φ in a fixed compact set of PSH h (D). Define Ω 0 := dd c Φ and Ω n := (f n ) * Ω 0 . The currents Ω n are positive of bidegree (k − p + 1, k − p + 1) and by definition of δ + , we have Ω n M ′ ×N ≤ Aδ n Ω 0 with A > 0 independent of Φ. By Theorem 2.7 applied to M ′′ and M ′ , there are negative horizontal L 1 forms Φ n such that dd c Φ n = Ω n with Φ n M ′′ ×N δ n . Then, δ −n Φ n belong to a fixed compact set of PSH h (M ′′ × N). Define Ψ 0 := Φ − Φ 0 and Ψ n := f * (Φ n−1 ) − Φ n for n ≥ 1. We have dd c Ψ n = 0 and since f * is continuous, δ −n Ψ n belong to some compact set in H h (M ′′ × N).
Fix a current R in C 1 v (D). We can assume that M ′′ is chosen so that R is supported on M ′′ × N. We have since Φ = Ψ 0 + Φ 0
d −n (f n ) * R, Φ = d −n (f n ) * R, Ψ 0 + d −n+1 (f n−1 ) * R, d −1 f * (Φ 0 ) = d −n (f n ) * R, Ψ 0 + d −n+1 (f n−1 ) * R, d −1 Ψ 1 + d −n+1 (f n−1 ) * R, d −1 Φ 1 .
By induction and using the identity f * (Φ n ) = Ψ n+1 + Φ n+1 , we obtain
d −n (f n ) * R, Φ = 0≤j≤n−1 d −n+j (f n−j ) * R, d −j Ψ j + R, d −n f * (Φ n−1 ) = 0≤j≤n d −n+j (f n−j ) * R, d −j Ψ j + R, d −n Φ n .(1)
Now assume that R is smooth and let n → ∞. The estimate on Φ n implies that the last term tends to 0. Recall that δ −n Ψ n belong to a compact set in H h (M ′′ ×N) and that δ < d. On the other hand, by Theorems 3.3, d −n+j (f n−j ) * R tends to T + when n − j → ∞. Proposition 4.2 and Lebesgue's convergence theorem, applied to the series in the identity (1), imply that for Φ smooth
T + , Φ = T + , j≥0 d −j Ψ j .(2)
Observe that the last sum is pluriharmonic and depends continuously on Φ in PSH h (D). It follows from Proposition 2.1 that the right hand side of the last identity depends continuously on Φ. So, T + is a PC current and the identity (2) holds for all Φ in PSH h (D).
The following propositions give the speed of convergence towards the Green current.
d −n (f n ) * R − T + , Φ ≤ Aλ −n for all R ∈ P v , Φ ∈ D h and n ≥ 0.
Proof. Observe that when Φ belongs to a compact family in PSH h (D), δ −n Φ n and δ −n Ψ n belong to compact families in PSH h (M ′ × N) and in H h (M ′ × N) for some M ′ ⋐ M. It follows from the identities (1) and (2)
that d −n (f n ) * R−T + , Φ is equal to 0≤j≤n d −j d −n+j (f n−j ) * R − T + , Ψ j − j≥n+1 d −j T + , Ψ j + d −n R, Φ n . Proposition 4.2 implies that | d −n+j (f n−j ) * R − T + , Ψ j | λ −n+j 0 δ j .
We also deduce from Proposition 2.1 applied to M ′ ×N instead of D, that | T + , Ψ j | δ j . Since Φ n is negative, the last term in the previous sum is negative. This implies the desired estimate for 1 < λ < min(λ 0 , d/δ). The following result gives a strong ergodic property for the action of f on vertical currents. Theorem 4.6. Let f be an invertible horizontal-like map as above with d > δ + .
| d −n (f n ) * R − T + , Φ | ≤ Aλ −n for all R ∈ P ′ v , Φ ∈ D h and n ≥ 0. Proof. As in Proposition 4.3, it is enough to estimate | R, Φ n |. We have | R, Φ n | δ n since RThen d −n (f n ) * R converge to T + uniformly on R ∈ C 1 v (D). In particular, T + is the unique current in C 1 v (D) which is invariant under d −1 f * . Proof.
Since smooth horizontal test forms are generated by the PSH ones, it is enough to test smooth PSH horizontal forms. Using identity (1) for Φ smooth, we only have to show that d −n R, Φ n tend to 0 uniformly on R. Recall that Φ n is negative, so d −n R, Φ n is negative. For simplicity, we reduce the size of D and we replace R by d −1 f * (R). So, we can assume that f is defined in a neighbourhood D = M × N of D = M × N and that R, Φ n are vertical or horizontal on M ′′ × N and M ×N ′′ respectively. Recall that the convex sets M, M , N and N are chosen so that M ⋐ M ⋐ M and N ⋐ N ⋐ N . We can also assume that the C 1 norm of f −1 on D is bounded by a constant A > 0.
Assume by contradiction that there is an increasing sequence (n i ) such that R i , Φ n i ≤ −2cd n i for some positive constant c > 0 and some sequence ( N). Let (R i,θ ) θ∈∆ denote the structural discs associated to R i as in Section 2. Define ϕ i (θ) := δ −n i R i,θ , Φ n i with δ + < δ < d. The properties of R i,θ and of Φ n imply that ϕ i belong to a compact family of subharmonic functions on ∆. It is then classical that for every compact subset K of ∆ there are constants C > 0 and α > 0 such that e −αϕ i L 1 (K) ≤ C, see e.g. [27].
R i ) in C 1 v (M ′ ×
The currents R i,θ are obtained by smoothing of R. Using a coordinate change, we obtain that
R i,θ , Φ n i = R i , Φ n i ,θ
where Φ n i ,θ is a smoothing of Φ n i . With the notation in Section 2, we have Φ n i ,θ := (h a,b,θ ) * (Φ n i )dν(a, b).
Since the family h a,b,θ is holomorphic and h a,b,1 = id, we obtain (see also [17,
Lemma 2.7]) Φ n i ,θ − Φ n i ∞,D |θ − 1| Φ n i C 1 ( e D)
for θ close to 1. On the other hand, the C 1 norm of f −n is bounded by A n , hence Theorem 2.7 and Remark 2.10 imply that
Φ n C 1 ( e D) (f n ) * (dd c Φ) C 1 ( b D) A 2kn .
Therefore, Φ n i ,θ − Φ n i ∞,D |θ − 1|A 2kn i and since the mass of R i is bounded
| R i , Φ n i ,θ − Φ n i | |θ − 1|A 2kn i .
Hence, for θ in a disc of center 1 and of radius ≃ A −2kn i , we have R i,θ , Φ n i ≤ −cd n i and then ϕ i (θ) ≤ −cd n i δ −n i . This contradicts the above uniform integrability of e −αϕ i .
• Convergence towards the equilibrium measure. The main result in this section is the following property of the equilibrium measure. . Indeed, if Θ is a smooth vertical (p, p)-form, then ϕ → Θ, ϕT − is continuous, since it is upper semi-continuous when Θ is positive and is continuous when Θ is positive closed. Using the PC property of T + and the identity µ, ϕ = T + , ϕT − , see Remark 2.4, we obtain that µ, ϕ depends continuously on ϕ. Therefore, µ is PC.
We can now prove estimates on the speed of convergence towards the equilibrium measure.
Proposition 4.8. Let f be as in Theorem 4.7 with d > δ + and d > δ − . Let P v (resp. P h ) be a compact family of currents in C 1 v (D) (resp. in C 1 h (D)). Then, there exist constants A > 0 and λ > 1 such that
d −2n (f n ) * R ∧ (f n ) * S − µ, ϕ ≤ Aλ −n
for all R ∈ P v , S ∈ P h , ϕ p.s.h. on D with |ϕ| ≤ 1 and n ≥ 0.
Proof. Since µ = T + ∧ T − , we can write d −2n (f n ) * R ∧ (f n ) * S − µ, ϕ as the sum of the following two integrals
d −2n (f n ) * R ∧ (f n ) * S − d −n (f n ) * R ∧ T − , ϕ = d −n (f n ) * S − T − , ϕd −n (f n ) * R and d −n (f n ) * R ∧ T − − T + ∧ T − , ϕ = d −n (f n ) * R − T + , ϕT − .
Since R is in a compact family in C 1 v (D), d −n (f n ) * R belong also to a compact family in C 1 v (D) independent of n ≥ 0. Indeed, their supports are controlled. Hence, for |ϕ| ≤ 1, ϕd −n (f n ) * R belong to a compact family in PSH v (D). By Proposition 4.3 applied to f −1 , the first integral is λ −n for some λ > 1. Since ϕT − belongs to a compact family in PSH h (D), the second integral is also λ −n for some λ > 1. The proposition follows. Proposition 4.9. Let f be as in Theorem 4.7 with d > δ + and d > δ − . Let P v (resp. P h ) be a bounded family of PB currents in C 1 v (D) (resp. in C 1 h (D)). Then, there exist constants A > 0 and λ > 1 such that
| d −2n (f n ) * R ∧ (f n ) * S − µ, ϕ | ≤ Aλ −n for all R ∈ P v , S ∈ P h , ϕ p.s.h. on D with |ϕ| ≤ 1 and n ≥ 0.
Proof. We proceed as in the proof of Proposition 4.8 using Proposition 4.4 instead of Proposition 4.3.
Properties of the equilibrium measure
In this section, we prove two important properties of the equilibrium measure for horizontal-like maps with large main dynamical degree.
• Decay of correlations. It was proved in [17] that the equilibrium measure is mixing for a general invertible horizontal-like map. Under our hypothesis on dynamical degrees, we have the following result.
Theorem 5.1. Let f be an invertible horizontal-like map as above with d > δ + and d > δ − . Then the equilibrium measure µ of f is exponentially mixing. More precisely, for all test functions φ of class C α and ψ of class C β on D with 0 < α, β ≤ 2, the following estimate holds
| µ, (φ • f n )ψ − µ, φ µ, ψ | ≤ A α,β λ −nαβ φ C α ψ C β
where A α,β > 0 is a constant independent of φ, ψ, n and λ > 1 is a constant independent of α, β, φ, ψ, n.
Recall that the measure µ is mixing means that the left hand side of the above inequality tends to 0 when n goes to infinity. It follows from the theory of interpolation between Banach spaces [36] that the previous inequality for general α, β is deduced from the case where α = β = 2, see [11] for details. In the case of Hénon-like maps, i.e. k = 2, we have δ + = δ − = 1. So, the hypothesis in the previous theorem is automatically satisfied and we obtain the following corollary.
I n (φ, ψ) := µ, (φ • f n )ψ − µ, φ µ, ψ .
Observe that since I n+1 (φ, ψ) = I n (φ • f, ψ), it is enough to consider the case where n is even. Note also that since µ is invariant, I n (φ, ψ) = 0 when φ or ψ is constant.
Near supp(µ) we can write φ and ψ as differences of functions which are strictly p.s.h. on a neighbourhood of D. So, we can assume that dd c φ ≥ dd c z 2 , dd c ψ ≥ dd c z 2 and that φ, ψ have C 2 norms bounded by a fixed constant. This allows to fix a constant A > 0 large enough such that (φ(z) + A)(ψ(z ′ ) + A) and
(−φ(z) + A)(ψ(z ′ ) − A) are p.s.h. on (z, z ′ ) in D 2 .
We have to bound from above I 2n (φ, ψ) = I 2n (φ + A, ψ + A) and
−I 2n (φ, ψ) = I 2n (−φ + A, ψ − A).
We will consider the first quantity, the proof for the second one is similar. For that purpose, we will apply Proposition 4.3 and Remark 4.5 to the product F of the horizontal-like maps f and f −1 defined in Section 3. Define ϕ(z, z ′ ) := (φ(z) + A)(ψ(z ′ ) + A). Let ∆ denote the diagonal of D × D and [∆] the current of integration on ∆. We have since µ is invariant
I 2n (φ + A, ψ + A) = µ, (φ • f n + A)(ψ • f −n + A) − µ, φ + A µ, ψ + A .
Lifting these integrals to D × D and using the identity
d −2 F * (T + ⊗ T − ) = d 2 F * (T + ⊗ T − ) = T + ⊗ T − ,
we obtain that I 2n (φ + A, ψ + A) is equal to
(T + ⊗ T − ) ∧ [∆], ϕ • F n − µ ⊗ µ, ϕ = (T + ⊗ T − ) ∧ d −2n (F n ) * [∆], ϕ − (T + ∧ T − ) ⊗ (T + ∧ T − ), ϕ = (T + ⊗ T − ) ∧ d −2n (F n ) * [∆], ϕ − (T + ⊗ T − ) ∧ (T − ⊗ T + ), ϕ = d −2n (F n ) * [∆] − T − ⊗ T + , ϕ(T + ⊗ T − ) .d −2n (F n ) * dd c ϕ ∧ (T + ⊗ T − ) = dd c (ϕ • F n ) ∧ (T + ⊗ T − )
decreases exponentially (we reduce the size of D if necessary). We have
dd c (ϕ•F n )∧(T + ⊗T − ) D 2 = D 2 dd c z 2 +dd c z ′ 2 k−1 ∧dd c (ϕ•F n )∧(T + ⊗T − ).
In the last wedge-product, T + depends only on z and T − depends only on z ′ . Then we expand
dd c (ϕ • F n ) = dd c (φ(f n (z)) + A)(ψ(f −n (z ′ )) + A) .
In this product, the terms containing mixed derivatives
dφ(f n (z)) ∧ d c ψ(f −n (z ′ )) and d c φ(f n (z)) ∧ dψ(f −n (z ′ ))
vanish when wedged with dd c z 2 + dd c z ′ 2 k−1 ∧ (T + ⊗ T − ) by bidegree consideration. This, combined with the fact that ϕ and ψ are bounded, implies that
dd c (ϕ • F n ) ∧ (T + ⊗ T − ) D 2 dd c (φ • f n ) ∧ T + D + dd c (ψ • f −n ) ∧ T − D .
We have
dd c (φ • f n ) ∧ T + = d −n (f n ) * (dd c φ ∧ T + ) and dd c (ψ • f −n ) ∧ T − = d −n (f n ) * (dd c ψ ∧ T − ).
Since d > δ + and d > δ − , the masses of these currents decrease exponentially. This completes the proof.
Remark 5.3. We can prove the converse of Theorem 4.1: if the current T + of f is PB then d > δ + . This will allow to prove that F satisfies also the hypothesis on dynamical degrees if d > δ + and d > δ − , hence we can apply directly Proposition 4.3. However, the proof requires a long development on the notion of superfunctions introduced in Section 2, and we prefer to avoid it here, see also [19].
• Lyapounov exponents. We will show that when the main dynamical degree of f is larger than the other ones, the measure µ is hyperbolic: it admits p strictly positive and k−p strictly negative Lyapounov exponents. We follow the approach by de Thélin [9].
Recall that the measure µ is mixing and is supported on the filled Julia set K := K + ∩ K − which is compact in D, see [17]. Using the theory of Oseledec-Pesin [32], we can decompose the tangent space of C k at µ-almost every point x into a direct sum of vector subspaces T x = ⊕ m i=1 E i,x with the following properties:
-The integer m and the dimension of each E i,x do not depend on x.
-The decomposition T x = ⊕ m i=1 E i,x is unique and depends in a measurable way on x.
-The vector bundle E i,x is invariant under f , that is, the differential Df of f defines an isomorphism between E i,x and E i,f (x) .
- -There are distinct real numbers λ i independent of x such that
The decomposition T x = ⊕ m i=1 E i,lim n→±∞ 1 |n| log Df n (v) v = ±λ i uniformly on v in E i,x \ {0}.
The constants λ i are the Lyapounov exponents of µ. The multiplicity of λ i is the dimension of E i,x . So, µ admits k Lyapounov exponents counted with multiplicities 2 . The Lyapounov exponents of f n are nλ i even for n negative. When there is no zero Lyapounov exponent, µ is said to be hyperbolic.
Theorem 5.4. Let f be an invertible horizontal-like map as above with dynamical degrees d, d + s and d − s . Define δ + := max s≤p−1 d + s and δ − := max s≤k−p−1 d − s . If δ + < d, then µ admits p strictly positive Lyapounov exponents larger than or equal to 1 2k log(d/ δ + ). If δ − < d, then µ admits k − p strictly negative ones which are smaller than or equal to − 1 2k log(d/ δ − ). We prove the first assertion. The second one is treated in the same way using f −1 instead of f . We will need the following lemmas where ω v denotes the restriction to M ′′ × N of the standard Kähler form ω on C k . Define d + q := max s≤q d + s for 0 ≤ q ≤ p − 1. Lemma 5.5. Let δ be a constant strictly larger than d + q . Then there exists a constant C > 0 such that for all positive closed current S of bidegree (k − q, k − q) supported on M × N ′ we have
S ∧ (f n 1 ) * ω v ∧ . . . ∧ (f nq ) * ω v ≤ Cδ n 1 S D
for all integers n 1 ≥ · · · ≥ n q ≥ 0.
2 if f is considered as a real map, the multiplicity of λ i is 2 dim E i,x and µ has 2k Lyapounov exponents; this is the reason for the coefficients 1 2 in Theorem 5.4.
Proof. We prove the lemma by induction on q. Clearly, the lemma is valid for q = 0. Suppose it holds for the rank q − 1. This, applied to f restricted to
D v := M ′ × N and to S ′ := (f nq ) * S ∧ ω, implies that Dv S ′ ∧ (f n 1 −nq ) * ω v ∧ . . . ∧ (f n q−1 −nq ) * ω v ≤ Cδ n 1 −nq S ′ Dv .
By definition of d + s , there is a constant c > 0 such that
S ′ Dv = (f nq ) * S Dv ≤ cδ nq S D .
Consequently,
Dv S ′ ∧ (f n 1 −nq ) * ω v ∧ . . . ∧ (f n q−1 −nq ) * ω v ≤ Cδ n 1 S D .
for some constant C > 0. The left hand side of the last inequality is equal to
f −nq (Dv) S ∧ (f n 1 ) * ω v ∧ . . . ∧ (f nq ) * ω v .
This implies the lemma for rank q. Note that the last integral does not change if
we replace f −nq (D v ) by D since (f nq ) * ω v is supported on f −nq (D v ).
Let Γ n denote the graph of (f, . . . , f n ), i.e. the set of points (x, f (x), . . . , f n (x)) in D n+1 . We will use the standard Kähler metric ω n in D n+1 ⊂ C k(n+1) . If Π j , with 0 ≤ j ≤ n, denote the projections from D n+1 onto its factors D, we have ω n = Π * j (ω). Let π j denote the restriction of Π j to Γ n and vol n (S) the mass of π * 0 (S) on ∩ 0≤j≤n π −1 j (M ′′ × N).
Lemma 5.6. Let δ be a constant strictly larger than d + q . Then there exists a constant C > 0 such that for all positive closed current S of bidegree (k − q, k − q) supported on M × N ′ we have vol n (S) ≤ Cδ n S D .
Proof. Observe that f j can be identified with π j • π −1 0 . This allows to write vol n (S) as the following sum of (n + 1) q integrals vol n (S) = π * 0 (S),
π * j (ω v ) q = 0≤n j ≤n S ∧ (f n 1 ) * ω v ∧ . . . ∧ (f nq ) * ω v .
Lemma 5.5 applied to a constant δ ′ > d + q implies that vol n (S) ≤ C ′ n q δ ′ n S for some constant C ′ > 0. We obtain the result by choosing a δ ′ smaller than δ.
A subset A of D is said to be (n, ǫ)-separated if f j is defined on A with f j (A) ⊂ D ′′ := M ′′ × N ′′ for 0 ≤ j ≤ n and for every distinct points a, b in A the distance between f j (a) and f j (b) is larger than ǫ for at least one j with 0 ≤ j ≤ n. Define for a subset X of D the topological entropy of f on X by
h X (f ) := sup ǫ>0 lim sup n→∞ 1 n log max # A ⊂ X, A (n, ǫ)-separated .
We have the following version of the Gromov's inequality, see also [24,14,9].
Proposition 5.7. Let δ be a constant strictly larger than d + q with q ≤ p − 1. Let X be a horizontal subvariety of dimension q of D. Then for every ǫ > 0 there is a constant C ǫ > 0 such that every (n, ǫ)-separated subset in X contains at most C ǫ δ n points. In particular, we have h X (f ) ≤ log d + q .
Proof. We can choose N ′ such that X is contained in M × N ′ . We can also assume that ǫ is small enough. So, X defines a horizontal positive closed current [X] of bidegree (k − q, k − q). Lemma 5.6 applied to M ′ instead of M ′′ , implies that the volume of π −1 0 (X) in ∩ 0≤j≤n π −1 j (M ′ × N) is smaller than Cδ n for some constant C > 0.
Consider an (n, ǫ)-separated subset A of X. For every a in A denote by B a the ball of center (a, f (a), . . . , f n (a)) and of diameter ǫ in D n+1 . Since A is (n, ǫ)separated, these balls are disjoint. Since ǫ is small and the center of B a is in ∩ 0≤j≤n π −1 j (D ′′ ), these balls are contained in ∩ 0≤j≤n π −1 j (M ′ × N). It follows that the total volume of B a ∩ π −1 0 (X) is bounded by Cδ n . On the other hand, an inequality of Lelong [30] says that the volume of B a ∩ π −1 0 (X) is bounded from below by a constant depending only on ǫ. Hence, the number of the balls B a is δ n . This implies that #A δ n and completes the proof.
Recall that it is proved in [17] that µ is of maximal entropy log d. This also holds for f −1 since the main dynamical degree of f −1 is also equal to d. Let B −n (x, ǫ) denote the Bowen (−n, ǫ)-ball with center x, i.e. the set of the points y such that f −j (y) is defined and f −j (y) − f −j (x) ≤ ǫ for 0 ≤ j ≤ n. The entropy h(µ) for f −1 can be obtained by the following Brin-Katok formula [6] h(µ) := sup
ǫ>0 lim inf n→∞ − 1 n log µ(B −n (x, ǫ))
for µ-almost every x. So, for every θ > 0, there are positive constants C, ǫ and a Borel set Σ 0 with µ(Σ 0 ) > 3/4 such that µ(B −n (x, 6ǫ)) ≤ Ce −n(log d−θ) for x ∈ Σ 0 and n ≥ 0.
Proof of Theorem 5.4. Assume in order to reach a contradiction that µ admits at least k − p + 1 Lyapounov exponents strictly smaller than 1 2k log(d/ δ + ). Let q ≤ p − 1 be an integer and λ < 1 2k log(d/ δ + ) a positive constant such that µ admits exactly k − q Lyapounov exponents strictly smaller than λ and the other ones are larger than or equal to 1 2k log(d/ δ + ). We are going to construct a complex subspace F of dimension q, contradicting the estimate in Proposition 5.7, i.e. with too many (n, ǫ)-separated points. Fix a positive constant θ such that θ ≪ λ and θ ≪ 1 2k log(d/ δ + ) − λ. By Oseledec-Pesin theory (replacing f by an iterate f n and θ, λ, d, δ + by nθ, nλ, d n , δ n + if necessary), we can assume that there is a decomposition T x = E x ⊕ F x for µ almost every x with the following properties:
-E x and F x are vector spaces of dimension k − q and q respectively.
-The vector bundles E x and F x are f -invariant.
-There is a Borel set Σ ⊂ K with µ(Σ) ≥ 1/2 and a constant η > 0 such that
Df −1 (v) ≥ e −λ v , Df −1 (u) ≤ e −λ−7θ u , ∡ E f −n (x) , F f −n (x) ≥ ηe −nθ for v ∈ E x , u ∈ F x ,
x ∈ Σ and n ≥ 0.
We now identify each T x with C k and consider x as the origin. Fix coordinate systems on E x and F x so that the associated distances coincide with the distances induced by the standard metric on C k . On T x = E x ⊕ F x we use the coordinate system induced by the fixed coordinates on E x and F x . We call it dynamical coordinate system. Note that the angle between E x and F x , with respect to the standard coordinates, might be small and in this case there is a big distorsion of the dynamical coordinates with respect to the standard ones.
Fix a positive constant c small enough, c ≪ η and c ≪ ǫ where ǫ is the constant associated to θ as above. Let B x −n denote the (small) ball of radius ce −n(λ+7θ) of center x −n := f −n (x) in E x −n . We are interested in graphs in T x −n = E x −n ⊕ F x −n of holomorphic maps over B x −n . Claim 1. For every x ∈ Σ there are holomorphic maps h n :
B x −n → F x −n with graph V x −n such that h n (0) = 0, Dh n ≤ e −4nθ and f sends V x −n−1 into V x −n .
The proof of this claim is by induction. For n = 0, it is enough to choose h 0 = 0. We will obtain V x −n as an open set in f −1 (V x −n+1 ). Consider the map f −1 on a small neighbourhood of x −n+1 with image in a neighbourhood of x −n . In dynamical coordinates for T x −n+1 and T x −n we can write f −1 (z) = l(z) + r(z) with l = (l ′ , l ′′ ) and r = (r ′ , r ′′ ) where l(z) is the linear part of f , i.e. the differential Df −1 at x −n+1 , and r(z) is the rest which is of order ≥ 2 with respect to z.
We have l ′ : E x −n+1 → E x −n and l ′′ : F x −n+1 → F x −n . We also have l ′ (z ′ ) ≥ e −λ z ′ for z ′ ∈ E x −n+1 and l ′′ (z ′′ ) ≤ e −(λ+7θ) z ′′ for z ′′ ∈ F x −n+1 . In the standard coordinates, the derivatives of f −1 are bounded. Taking into account the distortions of dynamical coordinates, we have Dr(z) ≤ Ae 6nθ z with A > 0 independent of c, n, θ. Now, consider two points z = (z ′ , z ′′ ) and w = (w ′ , w ′′ ) in E x −n+1 ⊕ F x −n+1 which are contained in V x −n+1 . So, z and w are smaller than 2ce −(n−1)(λ+7θ) . Write z := ( z ′ , z ′′ ) = f −1 (z) and w := ( w ′ , w ′′ ) = f −1 (w). We deduce from the estimates on l ′ , Dr and Dh n−1 that
z ′ − w ′ ≥ l ′ (z ′ ) − l ′ (w ′ ) − r ′ (z) − r ′ (w) ≥ e −λ z ′ − w ′ − 2Ae 6nθ ce −(n−1)(λ+7θ) z − w ≥ e −λ z ′ − w ′ − 4Ae 6nθ ce −(n−1)(λ+7θ) z ′ − w ′ .
Hence, z ′ − w ′ ≥ e −(λ+θ) z ′ − w ′ since c, θ are small and θ ≪ λ. It follows that f −1 (V x −n+1 ) is a graph of a holomorphic map h n over an open set B of E x −n . The last estimate for w ′ = 0 implies that B contains the ball B x −n .
On the other hand, we have
z ′′ − w ′′ ≤ l ′′ (z ′′ ) − l ′′ (w ′′ ) + r ′′ (z) − r ′′ (w) ≤ e −(λ+7θ) z ′′ − w ′′ + 2Ae 6nθ ce −(n−1)(λ+7θ) z − w ≤ e −(λ+7θ) e −4(n−1)θ z ′ − w ′ + 4Ae 6nθ ce −(n−1)(λ+7θ) z ′ − w ′ .
Therefore, z ′′ − w ′′ ≤ e −4nθ z ′ − w ′ since θ ≪ λ and c is small. It follows that Dh n ≤ e −4nθ and this finishes the proof of the claim.
Note that all the constructed graphs are small and contained in a small neighbourhood U of the filled Julia set K . We now come back to the standard metric on C k . Let F ′ x denote the orthogonal of E x . We use coordinate systems on F ′ x which induce the standard metric. Let B ′ x −n denote the ball of center 0 and of radius c ′ e −n(λ+10θ) in E x −n with c ′ > 0 small enough. We claim that V x −n contains some flat graph V ′ x −n . Claim 2. For every x ∈ Σ, V x −n contains the graph V ′
x −n of a holomorphic map h ′ n : B ′ x −n → F ′ x −n such that h ′ n (0) = 0 and Dh ′ n e −nθ .
With the considered coordinates on E x −n , F x −n and F ′ x −n , denote by τ :
E x −n ⊕ F x −n → E x −n ⊕F ′
x −n the linear map of coordinate change. Since the angle between E x −n and F x −n is larger than ηe −nθ , we can write τ = (τ ′ , τ ′′ ) with τ ′ (z) − z ′ e nθ z ′′ and τ ′′ (z) ≤ z ′′ for z = (z ′ , z ′′ ) in E x −n ⊕ F x −n . Claim 2 is proved using analogous estimates as in Claim 1 where we replace f −1 by τ . We will not give the details here.
We continue the proof of Theorem 5.4. Let A be a subset of Σ ∩ Σ 0 such that the balls B −n (x, 3ǫ) with centers x ∈ A are disjoint. We choose A maximal satisfying this property. So, the balls B −n (x, 6ǫ) with centers x ∈ A cover Σ ∩ Σ 0 . Since µ(Σ ∩ Σ 0 ) ≥ 1/4 and µ(B −n (x, 6ǫ)) ≤ Ce −n(log d−θ) , A contains at least (4C) −1 e n(log d−θ) points. Consider the graphs V x −n and V ′ x −n constructed above for x ∈ A. Since the balls B −n (x, 3ǫ) are disjoint, the set of x −n are (n, 3ǫ)separated. Claim 1 implies that the diameter of V x −n is smaller than ǫ. So, if we replace each x −n by a point x ′ −n in V x −n the resulting set is always (n, ǫ)-separated.
Let Π be an orthogonal projection of C k = C p × C k−p onto a subspace E of dimension k − q. If E is a product of a subspace of C p with C k−p , then the fibers of Π which are close enough to K (in particular the fibers which intersect U ) are horizontal in D. This property holds for the projection on any small perturbation of E. So, we can choose a finite number of projections Π 1 , . . ., Π N on E 1 , . . ., E N satisfying this property, and a constant θ 0 > 0 such that any subspace F of dimension q in C k has an angle ≥ θ 0 with at least one of E i . We deduce from Claim 2 that for each of the considered graphs V ′
x −n , the volume of Π i (V ′ x −n ) is ≥ c ′′ e −2n(k−q)(λ+10θ) for at least one projection Π i with a fixed constant c ′′ > 0. Choose an i such that this property holds for at least N −1 #A graphs V ′ x −n . Since #A ≥ (4C) −1 e n(log d−θ) , the sum of the volumes of Π i (V x −n ) is e n(log d−θ)−2n(k−q)(λ+10θ) . Hence, there is a fiber F of Π i which intersects e n(log d−θ)−2n(k−q)(λ+10θ) graphs V x −n . It follows that F contains an (n, ǫ)-separated subset of e n(log d−θ)−2n(k−q)(λ+10θ) ≥ e n(log e δ + +θ) points since θ ≪ 1 2k log(d/ δ + ) − λ. This contradicts Proposition 5.7 for X = F since δ + ≥ d + q , and finishes the proof of Theorem 5.4.
Remark 5.8. The above bound 1 2k log(d/ δ + ) can be replaced by the infimum of the numbers 1 2(k−q) log(d/ d + q ) for q ≤ p − 1. Remark 5.9. The fact that we are in the holomorphic setting is used only in Proposition 5.7 in order to get an estimate on the topological entropy on analytic manifolds of dimension q. The result still holds for real C 1+α horizontal-like maps (i.e. non-uniformly hyperbolic horshoes) with an ergodic invariant measure with compact support. We only need that the entropy of the measure is strictly larger than the entropy on vertical subspaces of dimension ≤ k − p − 1 and horizontal manifolds of dimension ≤ p − 1, see also Newhouse, Buzzi and de Thélin [31,9].
Examples and open problems
Consider a polynomial automorphism f of C k . We extend f to a birational map on the projective space P k . Let I + and I − denote the indeterminacy sets of f and f −1 . They are in the hyperplane at infinity L ∞ := P k \ C k and we assume that they are non-empty. When I + and I − have empty intersection, f is said to be regular. This class of automorphisms was introduced and studied in [35]. In dimension k = 2, they are the Hénon type maps and any polynomial automorphism of positive entropy is conjugated to a regular automorphism.
There is an integer p such that dim I + = k − p − 1 and dim I − = p − 1. If d + and d − denote the algebraic degrees of f and f −1 , we have d p + = d k−p − . At infinity we have f (L ∞ \ I + ) = I − and f −1 (L ∞ \ I − ) = I + . Define the filled Julia sets by K + := z ∈ C k , (f n (z)) n≥0 bounded in C k and Since R is large, it follows that f ′ (z) > R for z in the vertical boundary of Question 6.3. Let f be an invertible horizontal-like map as above. Is the sequence (d + s ) 0≤s≤p of dynamical degrees of f increasing? Question 6.4. Let f be an invertible horizontal-like map as above. Is the Green current T + laminar? More precisely, is it decomposable into currents of integration on complex manifolds, not necessarily closed, in D?
We refer to [10,8] for recent results on laminar currents in higher dimension. The following problems are also open for regular polynomial automorphisms. Question 6.5. Is the equilibrium measure µ the intersection in the geometrical sense of T + and T − ? More precisely, is it possible to decompose T + and T − into currents of integration on complex manifolds and to obtain µ as an average on the intersections of such manifolds? Question 6.6. Are saddle periodic points equidistributed with respect to µ? It is not difficult to show that there are d n periodic points of period n counted with multiplicities.
Question 6.7. Is the Hausdorff dimension of µ positive? Is there a relation between this dimension and the Lyapounov exponents of µ?
Corollary 1 . 2 .
12Let f be a regular polynomial automorphism of C k . Assume that the indeterminacy sets of f and f −1 are linear and defined by
Proposition 2 . 1 .
21The action of R can be extended in a unique way to a positive continuous linear form on H h (D). Moreover, (
Proposition 2 . 2 .
22The limit R, Φ := lim sup R, Φ ′ with Φ ′ smooth converging to Φ in PSH h (M ′ × N),defines an extension of R to PSH h (D). The extension depends linearly on R, Φ. It takes values in R ∪ {−∞} and does not depend on the choice of M ′ and N . The function θ
we only have to prove that M ′ × (N \ N ′′ ) satisfies the right convexity property. More precisely, one should construct a smooth exhaustion function ρ on M ′ × (N \ N ′′ ) such that dd c ρ has at every point p + 1 strictly positive eigenvalues. The domain is completely strictly p-convex in the terminology of [26, p.65].
with κ(t) := 1 t−1 and c > 0 large enough. The function ρ is an exhaustion function on M ′ × (N \ N ′′ ). The p eigenvalues of dd c ρ with respect the the variable z ′ are strictly positive. On the other hand, since
•
Horizontal-like maps and Julia sets. A horizontal-like map f on D is not necessarily defined on the whole domain D but only on a vertical subset f −1 (D) of D. It takes values in a horizontal subset f (D) of D. Horizontal-like maps are defined by their graphs Γ as follows [17]. Let pr 1 and pr 2 be the canonical projections of D × D on its factors.Definition 3.1. A horizontal-like map f on D is a holomorphic map with graph Γ such that 1. Γ is a submanifold of D × D.
The last property is equivalent to the fact that the projections of Γ on the first factor M and the last factor N in D × D are relatively compact. The map f = pr 2 • (pr 1|Γ ) −1 is defined on f −1 (D) := pr 1 (Γ) and its image is equal to f (D) := pr 2 (Γ). There exist open sets M ′ ⋐ M and N ′ ⋐ N such that f −1 (D) ⊂ D v := M ′ × N and f (D) ⊂ D h := M × N ′ . We have Γ ⊂ D v × D h .
Proposition 3. 2 .
2The operator f * : C h (D v ) → C h (D h )is well-defined and continuous. Moreover, there exists an integer
Theorem 3. 3 .
3Let f be an invertible horizontal-like map on D = M × N, d its main dynamical degree and K ± , K the filled Julia sets as above. Let R and S be smooth forms in C 1 v (D) and C 1 h (D) respectively. Then d −n
Theorem 3. 4 .
4With the notation of the previous theorem, the topological entropy of f on K is equal to log d and µ is a measure of maximal entropy log d.The notion of entropy will be recalled in Section 5. We now introduce the other dynamical degrees of f . Recall that the open setsM ′ ⋐ M and N ′ ⋐ N are chosen so that f −1 (D) ⊂ M ′ × N and f (D) ⊂ M × N ′ . So, the restriction of f to M ′ × N ′ is also horizontal-like. For every 0 ≤ s ≤ p, let d + s = d s (f ) := lim sup n→∞ sup S (f n ) * S M ′ ×N1/n , the supremum being taken over all positive closed horizontal currents S of bidegree (k − s, k − s) on D ′ = M ′ × N ′ such that S D ′ = 1. For every 0 ≤ s ≤ k − p, define d − s = d s (f −1 ) := lim sup n→∞ sup R (f n ) * R M ×N ′ 1/n , the supremum being taken over all positive closed vertical currents R of bidegree (k − s, k − s) on D ′ = M ′ × N ′ such that R D ′ = 1. In the sequel we will write for short δ + := d + p−1 and δ − := d − k−p−1 . These are the dynamical degrees which have to be compared to d.
Lemma 3. 5 .For
5The dynamical degrees do not depend on the choice of the particular convex domains M ′ and N ′ . Moreover, we haved + 0 = d − 0 = 1 and d + p = d − k−p = d.Proof. Let M ′′ and N ′′ be convex open sets such that M ′′ ⋐ M ′ ⋐ M, N ′′ ⋐ N ′ ⋐ N and f −1 (D) ⊂ M ′′ × N, f (D) ⊂ M × N ′′ . If in the previous definition, we replace M ′ by M ′′ and N ′ by N ′′ , we obtain δ + s and δ − s . It is enough to prove that δ + s = d + s and δ − s = d − s . We prove the first equality; the second one is obtained in the same way. Let S be a horizontal positive closed current of bidegree (k − s, k − s) on M ′′ × N ′′ . Since f is horizontal-like, f * ( S) is horizontal in M × N ′′ and there is a constant A > 0 independent of S such that f * ( S) M ′ ×N ′ ≤ A S M ′′ ×N ′′ . In particular, if S is horizontal in M ′ × N ′ then we have (f n ) * S M ′ ×N ′ ≤ A (f n−1 ) * S M ′′ ×N ′′ for n ≥ 2. If, moreover, S M ′ ×N ′ = 1, then S ′ := f * (S) is horizontal in M ′′ × N ′′ with bounded mass. S horizontal in M ′′ × N ′′ with S M ′′ ×N ′′ = 1, define also S ′ := f * (S). Then S ′ is horizontal in M ′ × N ′ with bounded mass and we have
Lemma 3 . 6 .
36The function R → Λ d −1 f * (R) is the super-function associated to d −1 f * (Φ).In other words, we havef * (R), Φ = R, f * (Φ)
F of the horizontal-like maps f 1 := f and f 2 := f −1 defined on D = M × N as above. In this case, we have M 1 = N 2 = M and M 2 = N 1 = N; the Green currents of F and F −1 are T + ⊗ T − and T − ⊗ T + . We can perturb F in order to obtain new families of examples.
Proposition 3 . 7 .
37Let f be a horizontal-like map on D = M × N with the main dynamical degree d as above and D ′ := M ′ × N ′ a domain such that D ′ ⋐ D and that D \ D ′ is small enough. Then every small pertubation f ǫ of f is a horizontallike map on D ′ of the same main dynamical degree d. If the dynamical degree of order s of f is strictly smaller than d, then the dynamical degree of order s of f ǫ satisfies the same property.
Theorem 4 . 1 .
41Let f be an invertible horizontal-like map on D = M × N and d, δ + its dynamical degrees as above. Assume that d > δ + . Then the Green current T + of f is PC.
Proposition 4 . 2 .
42Let H be a compact family of currents in H h (D). Then there are constants A 0 > 0 and λ 0 > 1 such that
Proposition 4 . 3 .
43Let f be as in Theorem 4.1 with d > δ + . Let P v be a compact family of currents in C 1 v (D) and D h a compact family of test currents in PSH h (D). Then, there exist constants A > 0 and λ > 1 such that
Proposition 4 . 4 .
44Let f be as in Theorem 4.1 with d > δ + . Let P ′ v be a bounded family of PB currents in C 1 v (D) and D h a compact family of test currents in PSH h (D). Then, there exist constants A > 0 and λ > 1 such that
belongs to a bounded family of PB currents in C 1 v (M ′ × N) for some M ′ ⋐ M. This implies the proposition.
Remark 4 . 5 .
45In Propositions 4.3 and 4.4, the condition d > δ + is superflous if the mass of d −n (f n ) * (dd c Φ) decreases to 0 exponentially and uniformly on Φ ∈ D h when n goes to infinity. We will use this observation in the proof of Theorem 5.1.
Theorem 4. 7 .
7Let f be an invertible horizontal-like map as above with d > δ + and d > δ − . Then the equilibrium measure µ of f is PC.Proof. By Theorem 4.1, T + , T − are PC on D and also on D ′ := M ′ × N ′ . If ϕ is a p.s.h. function on D, ϕ is locally integrable with respect to the trace measure T − ∧ ω p of T − . Hence, ϕT − defines a PSH horizontal current. Moreover, the fact that T − is PC implies that ϕ → ϕT − is continuous on ϕ ∈ PSH(D) with values in PSH h (D)
Corollary 5 . 2 .
52Let f be a Hénon-like map. Then the equilibrium measure of f is exponentially mixing.Proof of Theorem 5.1. We only have to consider the case where α = β = 2. Define
x has a tempered distortion. More precisely, if I and J are disjoint subsets of {1, . . . , m}, define E I,x := ⊕ i∈I E i,x and E J,x := ⊕ i∈J E i,x . Then, the angle ∡ E I,f n (x) , E J,f n (x) between E I,f n (x) and E J,f n (x) satisfies lim n→±∞ 1 n log sin ∡ E I,f n (x) , E J,f n (x) = 0.
this was proved in [17, Lemma 2.2]. For the general case, we have
The current [∆] is not horizontal but F * [∆] is horizontal. So, we can apply Proposition 4.3 and Remark 4.5 for F −1 .For Remark 4.5, we need to show that the mass of
In other situations, we often assume that Φ is of order 0 or negative. This is necessary in particular when one defines the pull-back by a non-invertible map[18]. Note that a p.s.h. function is defined everywhere but not a PSH current.
In the case of regular polynomial automorphisms, since µ = (dd c G + ) p ∧ (dd c G − ) k−p and G + , G − are Hölder continuous, µ gives no mass to sets of small Hausdorff dimension, see e.g.[35,Théorème 1.7.3].We refer to Dupont[21], Ledrappier-Young[29]and the references therein for analogous problems in other contexts.The dependence of Lyapounov exponents on the map can be studied following the works by Bassanelli-Berteloot[1]and Pham[33].K − := z ∈ C k , (f −n (z)) n≥0 bounded in C k .These sets are invariant under f −1 , f and satisfy K + = K + ∪I + , K − = K − ∪I − . One associates to f and f −1 the following functions, called Green functionsandwhere log + := max(log, 0). These functions are continuous p.s.h. on C k . It follows from[15,Proposition 2.4] that G + and G − are Hölder continuous. They satisfyIt is shown in[19]that the Green currentsare, up to a multiplicative constant, the unique positive closed currents of bidegrees (p, p) and (k − p, k − p) with support in K + and K − respectively. These currents are invariant:Note that to prove the uniqueness we do not assume invariance.The family of regular automorphisms is large but for simplicity we restrict to the case where the indeterminacy sets I + and I − are linear. In what follows, we assume thatandwhere [z 0 : · · · : z k ] denotes the homogeneous coordinates of P k , C k is identified to the chart {z 0 = 1} and the hyperplane at infinity L ∞ is given by the equation z 0 = 0. The following proposition allows to apply the results in the previous sections to the small (possibly transcendental) pertubations of f and proves Corollary 1.2.Proposition 6.1. Let f be a regular polynomial automorphism of C k as above. Let B R s denote the ball of center 0 and of radius R in C s . Then, if R is large enough, any holomorphic mapis a horizontal-like map of main dynamical degree d which is strictly larger than the other dynamical degrees. Write, using the coordinates (z 1 , . . . ,Since f (L ∞ \ I + ) = I − , the equation of I − implies that the components of f ′′ have degree ≤ d + − 1 and the components of f ′ have degree d + . Moreover, if f + j denotes the homogeneous part of degree d + of f j , the equation of I + implies that f + 1 = · · · = f + p = 0 only when z 1 = · · · = z p = 0. The restriction of f to I − defines an endomorphism of algebraic degreeThis proves that f restricted to B R p × B R k−p is horizontal-like. In order to avoid confusion, let us denote by f the horizontal-like map on D := B R p × B R k−p associated to f . Since K + = K + ∪ I + , the equation of I + implies that K + restricted to D is vertical. The restriction of T + to D is vertical and invariant under d −1 f * . So, the main dynamical degree of f is equal to d. It remains to check that the other dynamical degrees are strictly smaller than d.Fix an α > 0 small enough so thatConsider the family Q h of horizontal positive closed currents of bidegree (k − s, k − s) and of mass 1 in D ′′ with s ≤ p − 1. We will show that the mass of (f n ) * S on D ′′ for S ∈ Q h , is of order O(d s + ). This implies that the dynamical degree d + s of f is ≤ d s + and then is strictly smaller than d. The proof is analogous for the degrees d − s associated to f −1 .Observe that S ′ := f * (S) is horizontal in D ′ and has bounded mass. Let ω FS := dd c H, with H := log(1 + z 2 ) 1/2 , be the Fubini-Study form on P k . Since the standard Kähler form on C k and ω FS are comparable in compact sets of C k , it is enough to estimate the mass of ω s FS ∧ (f n ) * S on D ′′ . We haveIt was shown in[35]that d −n + log + f n (z) converge locally uniformly to G + . We deduce easily that d −n + H • f n converge also locally uniformly to G + . It follows from the theory of intersection of currents, see[7,23]that the family of currentsis relatively compact. Hence, the integrals in (3) are d sn + and the mass of (f n ) * S on D ′′ is d sn + . This completes the proof. Remark 6.2. The restriction of K + , K − , T + and T − to D = B R p × B R k−p coincide with the filled Julia sets and the Green currents constructed for f . Note that in the context of horizontal-like maps, T + is not the unique positive closed (p, p)current with support in K + . For the horseshoes, this current can be decomposed into currents of integration on vertical submanifolds of D.Many questions have to be considered in the context of horizontal-like maps even when we assume that the condition on the dynamical degrees is satisfied. We refer to the paper by Dujardin[20]for the case of dimension 2, see also[2,12].
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. T.-C Dinh, UMR 7586UPMC Univ Paris 06Institut de Mathématiques de JussieuT.-C. Dinh, UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu, F-
Mathematics Section, the Abdus Salam International Centre for Theoretical. France Paris, Strada costiera. 1134014PhysicsParis, France. [email protected], http://www.math.jussieu.fr/∼dinh V.-A. Nguyên, Mathematics Section, the Abdus Salam International Centre for Theo- retical Physics, Strada costiera, 11, 34014
Italy Trieste, [email protected] N. Sibony. Université Paris-Sud, 91405 Orsay, France. [email protected], Italy. [email protected] N. Sibony, Mathématique-Bâtiment 425, UMR 8628, Université Paris-Sud, 91405 Or- say, France. [email protected]
| [] |
[
"The VIMOS Ultra-Deep Survey (VUDS): fast increase of the fraction of strong Lyman alpha emitters from z=2 to z=6 ⋆",
"The VIMOS Ultra-Deep Survey (VUDS): fast increase of the fraction of strong Lyman alpha emitters from z=2 to z=6 ⋆"
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"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"Department of Astronomy\nUniversity of Geneva\nch. d'Écogia 16CH-1290VersoixSwitzerland",
"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"Max-Planck-Institut für Extraterrestrische Physik\n1312, D-85741Postfach, Garching bei MünchenGermany",
"Institut de Recherche en Astrophysique et Planétologie -IRAP\nCNRS\nUniversité de Toulouse\nUPS-OMP\n14, avenue E. BelinF31400ToulouseFrance",
"Geneva Observatory\nUniversity of Geneva\nch. des Maillettes 51CH-1290VersoixSwitzerland",
"INAF-IASF\nvia Bassini 15, I-20133MilanoItaly",
"INAF-Osservatorio Astronomico di Roma\nvia di Frascati 33, I-00040, Monte Porzio CatoneItaly",
"Department of Physics and Astronomy (DIFA)\nUniversity of Bologna\nV.le Berti Pichat, 6/2 -40127BolognaItaly",
"Department of Physics and Astronomy (DIFA)\nUniversity of Bologna\nV.le Berti Pichat, 6/2 -40127BolognaItaly",
"Research Center for Space and Cosmic Evolution\nEhime University\nBunkyo-cho 2-5790-8577MatsuyamaJapan",
"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"INAF-Osservatorio Astronomico di Bologna\nvia Ranzani,1, I-40127BolognaItaly",
"INAF-IASF Bologna\nvia Gobetti 101, I-40129BolognaItaly",
"LAM (Laboratoire d'Astrophysique de Marseille)\nUMR 7326\nAix Marseille Université\nCNRS\n13388MarseilleFrance",
"Institut d'Astrophysique de Paris\nCNRS\nUniversité Pierre et Marie Curie\n98 bis Boulevard AragoUMR7095, 75014ParisFrance",
"Institut de Recherche en Astrophysique et Planétologie -IRAP\nCNRS\nUniversité de Toulouse\nUPS-OMP\n14, avenue E. BelinF31400ToulouseFrance",
"Department of Astronomy\nUniversity of Geneva\nch. d'Écogia 16CH-1290VersoixSwitzerland",
"Centro de Estudios de Física del Cosmos de Aragón\nTeruelSpain",
"Institut d'Astrophysique de Paris\nCNRS\nUniversité Pierre et Marie Curie\n98 bis Boulevard AragoUMR7095, 75014ParisFrance",
"Department of Astronomy\nCalifornia Institute of Technology\n1200 E. California Blvd., MC 249-1791125PasadenaCAUSA"
] | [] | Aims. The aim of this work is to constrain the evolution of the fraction of strong Lyα emitters among UV selected star-forming galaxies at 2 < z < 6, and to measure the stellar escape fraction of Lyα photons over the same redshift range. Methods. We exploit the ultradeep spectroscopic observations with VIMOS on the VLT collected by the VIMOS Ultra-Deep Survey (VUDS) to build an unique, complete and unbiased sample of ∼ 4000 spectroscopically confirmed star-forming galaxies at 2 < z < 6. Our galaxy sample UV luminosities brighter than M * FUV at 2 < z < 6, and luminosities down to one magnitude fainter than M * FUV at 2 < z < 3.5. Results. We find that 80% of the star-forming galaxies in our sample have EW 0 (Lyα) < 10Å, and correspondingly f esc (Lyα)< 1%. By comparing these results with literature, we conclude that the bulk of the Lyα luminosity at 2 < z < 6 comes from galaxies that are fainter in the UV than those we sample in this work. The strong Lyα emitters constitute, at each redshift, the tail of the distribution of the galaxies with extreme EW 0 (Lyα) and f esc (Lyα). This tail of large EW 0 (Lyα) and f esc (Lyα) becomes more important as the redshift increases, and causes the fraction of strong Lyα with EW 0 (Lyα)> 25Å to increase from ∼5% at z ∼ 2 to ∼30% at z ∼ 6, with the increase being relatively stronger beyond z∼ 4. We observe no difference, for the narrow range of UV luminosities explored in this work, between the fraction of strong Lyα emitters among galaxies fainter or brighter than M * FUV , although the fraction for the faint galaxies evolves faster, at 2 < z < 3.5, than for the bright ones. We do observe an anticorrelation between E(B-V) and f esc (Lyα): generally galaxies with high f esc (Lyα) have also small amounts of dust (and viceversa). However, when the dust content is low (E(B-V)<0.05) we observe a very broad range of f esc (Lyα), ranging from 10 −3 to 1. This implies that the dust alone is not the only regulator of the amount of escaping Lyα photons. | 10.1051/0004-6361/201423824 | [
"https://arxiv.org/pdf/1403.3693v2.pdf"
] | 29,615,287 | 1403.3693 | 01bcb3c258b4cd8121c679e1e6833e7a30be21ab |
The VIMOS Ultra-Deep Survey (VUDS): fast increase of the fraction of strong Lyman alpha emitters from z=2 to z=6 ⋆
14 Mar 2014 March 18, 2014
P Cassata [email protected]
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
L A M Tasca
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
O Le Fèvre
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
B Lemaux
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
B Garilli
INAF-IASF
via Bassini 15, I-20133MilanoItaly
V Le Brun
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
D Maccagni
INAF-IASF
via Bassini 15, I-20133MilanoItaly
L Pentericci
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
R Thomas
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
E Vanzella
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
G Zamorani
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
E Zucca
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
R Amorin
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
S Bardelli
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
P Capak
Department of Astronomy
California Institute of Technology
1200 E. California Blvd., MC 249-1791125PasadenaCAUSA
L Cassarà
INAF-IASF
via Bassini 15, I-20133MilanoItaly
M Castellano
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
A Cimatti
Department of Physics and Astronomy (DIFA)
University of Bologna
V.le Berti Pichat, 6/2 -40127BolognaItaly
J G Cuby
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
O Cucciati
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
Department of Physics and Astronomy (DIFA)
University of Bologna
V.le Berti Pichat, 6/2 -40127BolognaItaly
S De La Torre
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
A Durkalec
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
A Fontana
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
M Giavalisco
Astronomy Department
University of Massachusetts
01003AmherstMAUSA
A Grazian
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
N P Hathi
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
O Ilbert
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
C Moreau
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
S Paltani
Department of Astronomy
University of Geneva
ch. d'Écogia 16CH-1290VersoixSwitzerland
B Ribeiro
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
M Salvato
Max-Planck-Institut für Extraterrestrische Physik
1312, D-85741Postfach, Garching bei MünchenGermany
D Schaerer
Institut de Recherche en Astrophysique et Planétologie -IRAP
CNRS
Université de Toulouse
UPS-OMP
14, avenue E. BelinF31400ToulouseFrance
Geneva Observatory
University of Geneva
ch. des Maillettes 51CH-1290VersoixSwitzerland
M Scodeggio
INAF-IASF
via Bassini 15, I-20133MilanoItaly
V Sommariva
INAF-Osservatorio Astronomico di Roma
via di Frascati 33, I-00040, Monte Porzio CatoneItaly
Department of Physics and Astronomy (DIFA)
University of Bologna
V.le Berti Pichat, 6/2 -40127BolognaItaly
M Talia
Department of Physics and Astronomy (DIFA)
University of Bologna
V.le Berti Pichat, 6/2 -40127BolognaItaly
Y Taniguchi
Research Center for Space and Cosmic Evolution
Ehime University
Bunkyo-cho 2-5790-8577MatsuyamaJapan
L Tresse
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
D Vergani
INAF-Osservatorio Astronomico di Bologna
via Ranzani,1, I-40127BolognaItaly
INAF-IASF Bologna
via Gobetti 101, I-40129BolognaItaly
P W Wang
LAM (Laboratoire d'Astrophysique de Marseille)
UMR 7326
Aix Marseille Université
CNRS
13388MarseilleFrance
S Charlot
Institut d'Astrophysique de Paris
CNRS
Université Pierre et Marie Curie
98 bis Boulevard AragoUMR7095, 75014ParisFrance
T Contini
Institut de Recherche en Astrophysique et Planétologie -IRAP
CNRS
Université de Toulouse
UPS-OMP
14, avenue E. BelinF31400ToulouseFrance
S Fotopoulou
Department of Astronomy
University of Geneva
ch. d'Écogia 16CH-1290VersoixSwitzerland
C López-Sanjuan
Centro de Estudios de Física del Cosmos de Aragón
TeruelSpain
Y Mellier
Institut d'Astrophysique de Paris
CNRS
Université Pierre et Marie Curie
98 bis Boulevard AragoUMR7095, 75014ParisFrance
N Scoville
Department of Astronomy
California Institute of Technology
1200 E. California Blvd., MC 249-1791125PasadenaCAUSA
The VIMOS Ultra-Deep Survey (VUDS): fast increase of the fraction of strong Lyman alpha emitters from z=2 to z=6 ⋆
14 Mar 2014 March 18, 2014Astronomy & Astrophysics manuscript no. frac˙lya c ESO 2014 Received .....; accepted .....Cosmology: observations -Galaxies: fundamental parameters -Galaxies: evolution -Galaxies: formation
Aims. The aim of this work is to constrain the evolution of the fraction of strong Lyα emitters among UV selected star-forming galaxies at 2 < z < 6, and to measure the stellar escape fraction of Lyα photons over the same redshift range. Methods. We exploit the ultradeep spectroscopic observations with VIMOS on the VLT collected by the VIMOS Ultra-Deep Survey (VUDS) to build an unique, complete and unbiased sample of ∼ 4000 spectroscopically confirmed star-forming galaxies at 2 < z < 6. Our galaxy sample UV luminosities brighter than M * FUV at 2 < z < 6, and luminosities down to one magnitude fainter than M * FUV at 2 < z < 3.5. Results. We find that 80% of the star-forming galaxies in our sample have EW 0 (Lyα) < 10Å, and correspondingly f esc (Lyα)< 1%. By comparing these results with literature, we conclude that the bulk of the Lyα luminosity at 2 < z < 6 comes from galaxies that are fainter in the UV than those we sample in this work. The strong Lyα emitters constitute, at each redshift, the tail of the distribution of the galaxies with extreme EW 0 (Lyα) and f esc (Lyα). This tail of large EW 0 (Lyα) and f esc (Lyα) becomes more important as the redshift increases, and causes the fraction of strong Lyα with EW 0 (Lyα)> 25Å to increase from ∼5% at z ∼ 2 to ∼30% at z ∼ 6, with the increase being relatively stronger beyond z∼ 4. We observe no difference, for the narrow range of UV luminosities explored in this work, between the fraction of strong Lyα emitters among galaxies fainter or brighter than M * FUV , although the fraction for the faint galaxies evolves faster, at 2 < z < 3.5, than for the bright ones. We do observe an anticorrelation between E(B-V) and f esc (Lyα): generally galaxies with high f esc (Lyα) have also small amounts of dust (and viceversa). However, when the dust content is low (E(B-V)<0.05) we observe a very broad range of f esc (Lyα), ranging from 10 −3 to 1. This implies that the dust alone is not the only regulator of the amount of escaping Lyα photons.
Introduction
Narrowband surveys targetting the strong Lyα emission from star-forming galaxies (LAEs; Partridge&Peebles 1967; Djorgovski et al. 1985;Cowie &Hu 1998;Hu et al. 2004;Kashikawa et al. 2006;Gronwall et al. 2007;Murayama et al. 2007;Ouchi et al. 2008;Nilsson et al. 2009) and broadband surveys targetting the deep Lyman break (LBG; Steidel et al. 1999;Bouwens & Illingworth 2006;Bouwens et al. 2010;McLure et al. 2011) have been very successful to explore the high-redshift Universe. However, the overlap between the populations selected by the two techniques is still debated: LAEs are claimed to be forming stars at rates of 1 ÷ 10M ⊙ yr −1 (Cowie & Hu 1998;Gawiser et al. 2006;Pirzkal et al. 2007), to have stellar masses of the order of 10 8 ÷ 10 9 M ⊙ and to have ages smaller than 50 Myr (Pirzkal et al. 2007;Gawiser et al. 2007;Nilsson et al. 2009), while LBGs have in general a broader range of properties (Reddy et al. 2006;Hathi et al. 2012;Schaerer, de Barros & Stark 2011; but see also Kornei et al. 2010). Steidel et al. (2000) and Shapley et al. (2003) showed that only ∼ 20% of z∼ 3 LBGs have a Lyα emission strong enough to be detected with the narrowband technique. Recently, many authors have investigated the evolution with the redshift of the fraction of strong Lyα emitters among LBG galaxies. Stark et al. (2010; showed that this fraction evolves with redshift, and overall fraction is smaller (and that the rate of evolution is slower) for UV bright galaxies (−21.75 < M UV < −20.25) than for UV faint (−20.25 < M UV < −18.75) galaxies; they find that the fraction of UV faint galaxies with strong (EW 0 (Lyα)> 25 Å) Lyα emission is around 20% at z ∼ 2 ÷ 3 and reaches ∼ 50 ÷ 60% at z ∼ 6. At higher redshift (z > 6 ÷ 8), many authors claim a sudden drop in the fraction of spectroscopically confirmed LBGs with strong Lyα emission (Fontana et al. 2010;Pentericci et al. 2011;Ono et al. 2012;Schenker et al. 2012;Caruana et al. 2013), interpreting this as the observational signature of the increasing fraction of netural hydrogen between z ∼ 6 and z ∼ 7 due to tail-end of the reionization, although Dijkstra et al. (2014) has argued that the effect can be due to a variation of the average escape fraction over the same redshift range.
However, the bulk of studies of the Lyα fraction at 3 < z < 8 (Stark et al. 2010;Pentericci et al. 2011) are based on a hybrid photometric-spectroscopic technique: the denominator of the fraction (i.e. the total number of star-forming galaxies at those redshifts) is constrained by photometry only, and thus its determination relies on the strong assumption that the contamination by low-z interlopers and incompleteness are fully understood and well controlled. The numerator of the fraction is the number of the LBGs that are observed with spectroscopy, and for which a strong Lyα rest-frame Equivalent Width (EW 0 >25 Å) is measured. In fact, the LBGs for which this experiment is done have a UV continuum that is generally too faint to be detected, even with the most powerful spetrographs on 10 meter class telescopes. Recently, Mallery et al. (2012) combined a sample of LAEs and LBGs to constrain the evolution of this fraction, confirming earlier results by Stark et al. (2010;. Given the nature of the selection of these samples, it is important to make a robust estimate of the evolution of the Lya fraction covering as wide a range in redshift as possible, and based on larger samples.
The Lyα is not only interesting because it allows for the exploration of the high-redshift universe. In fact, its observed properties can give a lot of information about the physical condition of star-forming galaxies. Lyα is thought to be mainly produced by star formation, as the contribution of AGN activity to the Lyα population at z < 4 is found to be less than 5% (Gawiser et al. 2006;Ouchi et al. 2008;Nilsson et al. 2009;Hayes et al. 2010). Due to its resonant nature, Lyα photons are easily scattered, shifted in frequency, and absorbed by the neutral hydrogen and/or by the dust. As a result, in general, 2 < z < 2.7 2.7 < z < 3.5 3.5 < z < 4.5 4. Lyα emission is more attenuated than other UV photons, with the Lyα escape fraction (i.e., the fraction of the Lyα photons that escape the galaxies) that depends strongly on the relative kinematics of the HII and HI regions, dust content and geometry (Giavalisco et al. 1996;Kunth et al. 1998;Mas-Hesse et al. 2003;Deharveng et al. 2008;Hayes et al. 2014).
Predicting the escape fraction of the Lyα photons as a function of the galaxy properties involves including all the complex effects of radiative transfer of such photons. Developing on the first models by Charlot & Fall (1993), Verhamme et al. (2006;2008; and Dijkstra et al. (2006; made huge progress in predicting the shape of the Lyα emission as a function of the properties of the ISM, the presence of inflows/outflows and dust. Verhamme et al. (2006;2008) predicted a correlation between f esc (Lyα) and E(B-V), with the escape fraction being higher in galaxies with low dust content. Verhamme et al. (2012) and Dijkstra et al. (2012) studied the escape fraction of Lyα photons through a 3D clumpy medium, constraining the dependence on the column density of neutral hydrogen and on the viewing angle.
A lot of effort has been recently put to constrain the correlation between the Lyα properties and the general properties of star-forming galaxies (e.g. dust attenuation, SFR, stellar mass) in the local Universe. Hayes et al. (2014) and Atek et al. (2014) have found that Lyα photons escape more easily from galaxies with low dust content. At high redshift, although on samples that are much smaller than the one we use in this paper, a similar trend has been found by Kornei et al. (2010) and Mallery et al. (2012), respectively at z ∼ 3 and at 4 < z < 6. In this paper, we look for this correlation using a sample that is respectively five and ten times larger than the ones used by Mallery and Kornei. The aim of this paper is to estimate the evolution of the fraction of strong Lyα emitters as a function of the redshift, exploiting data from the new VIMOS Ultra-Deep Survey (VUDS). The goal is twofold: first, to put on firmer grounds the trends that have been found with photometric LBG samples (Stark et al. 2010; and improve on the knowledge of the evolution of the Lyα fraction; second, to offer the theoreticians a reference sample of galaxies with robust spectroscopic redshifts, with a well measured EW 0 (Lyα) distribution. In fact, in this paper, we select a sample of galaxies, sliced in volume limited samples according to different recipes, for which we have a spectroscopic redshift in ∼90% of the cases. The continuum is detected for almost all objects in the sample, thus allowing a robust measurement of the redshift based on the UV absorption features even in absence of Lyα.
Our selection is not based on LBG or narroband techniques, that are prone to incompleteness and contamination, but it is rather based on the magnitude in the i ′ −band and on the photometric redshifts measured on the full Spectral Energy Distribution (SED) of galaxies. The most important point to emphasize is that our flux selection is completely independent of the presence of Lyα, at least up to z ∼ 5, because it enters the photometric i ′ −band only at z > 5: since the i ′ −band does not contain the Lyα line, objects with strong Lyα emission have not a boosted i ′ −band magnitude. Moreover, when the photo-z are computed, some variable Lyα flux (as for other lines like OII, OIII and Hα) is added to the SED: this ensures that even objects with large Lyα flux are reproduced by the template set that is used to compute the photo-z. This implies also that if our selection is incomplete at some redshift, the incompleteness is also independent on the presence (or absence) of Lyα.
For these reasons, this sample is ideal to study the Lyα properties of a well controlled sample of star-forming galaxies. The fraction of strong Lyα emitters among star-forming galaxies is completely constrained by spectroscopy, as is also the case for non-Lyα emitters.
Throughout the paper, we use a standard Cosmology with Ω M = 0.3, Ω Λ = 0.7 and h = 0.7. Magnitudes are in the AB system.
Data
The data used in this study are drawn from the VIMOS Ultra-Deep Survey (VUDS), an ESO large program with the aim of collecting spectra and redshifts for around 10,000 galaxies to study early phases of galaxy formation at 2 < z < 6. To minimize the effect of cosmic variance, the targets are selected in three independent extragalactic fields: COSMOS , the CFHTLS-D1 Field (Cuillandre et al. 2012) and the Extended-Chandra-Deep-Field (ECDFS; e.g. see Cardamone et al. 2010). The survey is fully presented in Le Fèvre et al. (2014).
Photometry
The three extragalactic fields targetted by the VUDS survey are three of the most studied regions of the sky, and they have been imaged by some of the most powerful telescopes on earth and in the space, including CFHT, Subaru, HST and Spitzer. For more details, we refer the reader to Le Fèvre et al. (2014), where more detailed information can be found.
The COSMOS field was observed with HST/ACS in the F814W filter (Koekemoer et al. 2007). Ground based imaging includes deep observations in g ′ , r ′ , i ′ and z ′ bands from the Subaru SuprimeCam (Taniguchi et al. 2007) and u * band observations from CFHT Megacam from the CFHT-Legacy Survey. Moreover, the UltraVista survey is acquiring very deep near-infrared imaging in the Y, J, H and K bands using the VIRCAM camera on the VISTA telescope , and deep Spitzer/IRAC observations are available (Sanders et al. 2007;Capak et al. in prep.). The CANDELS survey (Grogin et al. 2012) also provided WFC3 NIR photometry in the F125W and F160W bands, for the central part of the COSMOS field.
The ECDFS field is covered with deep UBVRI imaging down to R AB = 25.3 (5σ, Cardamone et al. 2010 and references therein). For the central part of the field, covering ∼ 160arcmin 2 , observations with HST/ACS in the F435W, F606W,
Fig. 1. Top panel:
Absolute magnitude in the far-UV band as a function of the redshift, for all VUDS galaxies at 2 < z < 6 (black diamonds) and for the VUDS galaxies with EW 0 > 25 Å (red circles). The green continuous line indicates the evolving M * as a function of the redshift as derived from the compilation by Hathi et al. 2010; the dashed green lines indicates M * +1. Bottom panel: Redshift distribution of the all the VUDS galaxies at 2 < z < 6 (black line) and of the VUDS galaxies with EW 0 > 25Å (blue histogram) and EW 0 > 55Å (red histogram).
F775W and F850LP are available (Giavalisco et al. 2004), together with the recent CANDELS observations in the J, H and K bands. The SERVS Spitzer-warm obtained 3.6µm and 4.5µm (Mauduit et al. 2012) that complement those obtained by the GOODS team at 3.6µm, 4.5µm, 5.6µm and 8.0µm.
The VVDS-02h field is observed in the BVRI at the CFHT (Le Fèvre et al. 2004), and later received deeper observations in the u * , g ′ , r ′ and i ′ bands as part of the CFHTLS survey (Cuillandre et al. 2012). Deep infrared imaging has been obtained with the WIRCAM at CFHT in YJHK bands down to K s =24.8 (Bielby et al. 2012). This field was observed in all Spitzer bands as part of the SWIRE survey (Lonsdale et al. 2003), and recently deeper data were obtained as part of the SERVS survey (Mauduit et al. 2012).
Target selection
The aim of the VUDS survey is to build a well controlled and complete spectroscopic sample of galaxies in the redshift range 2 z 6. To achieve this goal, with the aim of being as inclusive as possible, we combined different selection criteria such as photometric redshifts, color-color and narrow-band selections. All the details of the selection can be found in Le Fèvre et al. (2014).
For this paper, we limited the analysis to the objects selected by the primary selection, that is based on photometric redshift and magnitude in the i ′ −band. In particular, only galaxies with auto magnitude in the i ′ −band 22.5 < m i < 25 are included.
If an object has a photometric redshift z p > 2.4 − σ z p (where σ z p denotes the 1-σ error of the photometric redshift) or if the second peak of the photometric redshift Probability Distribution Function (zPDF) z p,2 > 2.4, this object is included in the target list.
Spectroscopy
The spectroscopic observations were carried out with the VIMOS instrument on the VLT. A total of 640h were allocated, including overheads, starting in periods P85 and ending in P93 (end of 2013) to observe a total of 16 VIMOS pointings. The spectroscopic MOS masks were designed using the vmmps tool (Bottini et al. 2005) to maximize the number of spectroscopic targets that could be placed in them. In the end, around 150 targets were placed in each of the 4 VIMOS quadrants, corresponding to about 600 targets per pointing and about 9000 targets in the whole survey. The same spectroscopic mask was observed once for 14h with the LRBLUE grism (R=180) and for 14h with the LRRED grism (R=210), resulting in a continuous spectral coverage between λ = 3650 Å and λ = 9350 Å. Le Fèvre et al. (2013) used the data from the VVDS survey to estimate the redshift accuracy of this configuration, constraining it to σ z spec = 0.0005(1 + z spec ), which corresponds to ∼ 150km/s. The spectroscopic observations are reduced using the VIPGI code (Scodeggio et al. 2005). First, the individual 2D spectrograms coming from the 13 observing batches (OBs), in which the observations are splitted, are extracted. Sky subtraction is performed with a low order spline fit along the slit at each wavelength sampled. The sky subtracted 2D spectrograms are combined with sigma clipping to produce a single stacked 2D spectrogram calibrated in wavelength and flux. Then, the objects are identified by collapsing the 2D spectrograms along the dispersion direction. The spectral trace of the target and other detected objects in a given slit are linked to the astrometric frame to identify the corresponding target in the parent photometric catalogue. At the end of this process 1D sky-corrected, stacked and calibrated spectra are extracted. For more detail, we refer the reader to Le Fèvre et al. (2014).
The redshift determination procedure follows the one that was optimized for the VVDS survey (Le Fèvre et al. 2005), later used also in the context of the zCOSMOS survey (Lilly et al. 2007) and VIPERS (Guzzo et al. 2014): each spectrum is analyzed by two different team members; the two independent measurements are then reconciled and a final redshift with a quality flag are assigned. The EZ tool (Garilli et al. 2010), a cross-correlation engine to compare spectra and a wide library of galaxy and star templates, is run on all objects to obtain a first guess of the redshift; after a visual inspection of the solutions, it is run in manual mode to refine them, if necessary.
A quality flag is assigned to each redshift, repeating the same scheme already used for the VVDS, COSMOS and VIPERS survey. The flag scheme was thoroughly tested in the context of the VVDS survey, on spectra of similar quality than the one we have for VUDS, and it is remarkably stable, since the individual differences are smoothed out by the process that involves many people (Le Fèvre et al. 2013). In particular, Le Fèvre et al. (2013) estimated the reliability of each class: -Flag 4: 100% probability to be correct -Flag 3: 95-100% probability to be correct -Flag 2: 75-85% probability to be correct -Flag 1: 50-75% probability to be correct -Flag 0: no redshift could be assigned -Flag 9: the spectrum has a single emission line.
The Equivalent Width (EW) of the Lyα line was measured manually using the splot tool in the noao.onedspec package in IRAF, similarly to Tresse et al. (1999). We first put each galaxy spectrum in its rest-frame according to the spectroscopic redshift. Then, two continuum points bracketing the Lyα are manually marked and the rest-frame equivalent width is measured. The line is not fitted with a Gaussian, but the flux in the line is obtained integrating the area encompassed by the line and the continuum. This method allows the measurement of lines with asymmetric shapes (i.e. with deviations from Gaussian profiles), which is expected to be the case for most Lyα lines. The interactive method also allows us to control by eye the level of the continuum, taking into account defects that may be present around the line measured. It does not have the objectivity of automatic measurements, but, given the sometimes complex blend between Lyα emission and Lyα absorption, it does produce reliable and accurate measurements. We stress here that the m i < 25 selection ensures that the continuum around Lyα is well detected for all galaxies in our sample, even for galaxies with spectroscopic flag 1 (the lowest quality) and 9 (objects with a single emission line).
Absolute magnitudes and masses
We fitted the spectral energy distributions of galaxies in the survey using the Le Phare tool (Ilbert et al. 2006), following the same procedure described in Ilbert et al. (2013). The redshift is fixed to the spectroscopic one for objects with flags 1, 2, 3, 4 and 9. It is fixed to the photometric one for objects with spectroscopic flag 0. In particular, we used the suite of templates by Bruzual & Charlot (2003) with 3 metallicities (Z = 0.004, 0.008, 0.02), assuming the Calzetti et al. (2000) extinction curve. We used exponentially declining star formation histories, with nine possible τ values ranging from 0.1 Gyr (almost istantaneous burst) to 30 Gyr (smooth and continuous star formation). We included also two delayed SFH models with peaks at 1 and 3 Gyr. The age ranges from 0.05 Gyr and the age of the Universe at the reshift of each galaxy. The absolute magnitudes are then derived by convolving the best template with the filter responses. The output of this fitting procedure includes also the stellar masses, star formation rates and extinction E(B-V).
Emission lines are added to the synthetic spectra, with their luminosity set by the intensity of the SFR derived for that SED. Once the Hα luminosity is obtained from the SFR applying the classical Kennicutt (1998) relations, the theoretical Lyα luminosity is obtained assuming case B recombination (Brocklehurst 1971). Then, the actual Lyα luminosity that is added to the SED is allowed to vary between half and double the theoretical value.
The dataset
For this paper we limit the analysis to the redshift range 2 < z < 6. The lower limit is the lowest redshift for which the Lyα line is redshifted into our spectral coverage. For the upper limit, in theory, we could detect Lyα in emission up to z ∼ 6.5, but the scarcity of objects at z > 6 in the VUDS survey forced us to limit the analysis to z ∼ 6. We limit the analysis to the galaxies with m i < 25: at these magnitudes the continuum is always detected with signal to noise ratio per resolution element S/N∼ 10, and the redshift determination is quite reliable, for both spectra with Lyα in emission and absorption, since many UV absorption features are easily identified. As we already said in the Introduction, the Lyα line enters the i ′ −band only at z > 5: this ensures that no detection bias affects our analysis at z < 5. We have in our sample only 12 galaxies at 5 < z < 6: we choose to keep them for our analysis throughout the paper, but we will be extremely cautious to draw strong conclusions for that redshift range.
We include in the analysis also secondary objects, that is the objects that serendipitously fall in the spectroscopic slit centered on a target, for which a spectrum is obtained in addition to that of the target. In fact, also for these objects, if they are brighter than m i < 25, a spectroscopic redshift can be easily assigned. However, only 2% of the final sample is made by secondary objects, that in any case only marginally affect the main result of this paper. The database contain 4420 objects with m i < 25 that have been targetted by spectroscopy. Of these, 3129 have a high reliability spectroscopic redshift in the range 2 < z < 6, with a spectroscopic flag 2, 3 or 4. Of the remaining objects, 1058 have a more uncertain spectroscopic redshift, with a quality flag 1: statistically, Le Fèvre et al. (2013) showed that they are right in 50-75% of the cases. For the purpose of this paper, we decided to trust their spectroscopic redshift if the difference between the photometric and spectroscopic redshift is smaller than 10%; otherwise, we fix the redshift to the photometric one. We stress that almost all of the 1058 objects do not show any strong emission line in their spectra that could be interpreted as Lyα, and that could help to assign a reliable spectroscopic redshift. Thus, they will not be part of the sample of strong Lyα emitters, but they will contribute to the total sample of galaxies without Lya emission, hence setting a lower limit to the lya fraction (see below). In the end, only 601 of these 1058 objects with spectroscopic flag 1 survive the check against the photometric redshift (∼ 60%, not far from the 50-75% determined by Le Fèvre et al. (2013); the other 459 have a photometric redshift that is below z = 2 and are excluded by the dataset.
In the end we include in our final database 3730 objects with m i < 25 for which we have measured a redshift and assigned a spectroscopic flag from 1 to 9. Of them, 3650 are primary targets, and in addition we have 80 secondary objects with m i < 25. Moreover, 231 objects with a photometric redshift in the range 2 < z < 6 and m i < 25 have been targetted by spectroscopy, but no spectroscopic redshift could be measured (they are identified by the spectroscopic flag=0). In the next sections, we will take into account their possible contribution to the evolution of the fraction of the Lyα emitters.
In order to allow a fair comparison with other works in the literature, we define as strong Lyα emitters all the galaxies with a rest-frame equivalent width of Lyα in excess of 25 Å. In the end, 430 of the 3961 galaxies (∼ 11%) meet this definition.
The details about the number of objects for each flag class, as a function of the presence of strong Lyα emission, can be found in Table 1. The large majority of the galaxies used in this study has a spectroscopic redshift with very high reliability: in fact, 1438 objects (36% of the total) have a spectroscopic flag 2, meaning that they are right in 75-85% of the cases (Le Fèvre et al. 2013); 1593 objects (42% of the total) have a spectroscopic flag 3 or 4, that are proven to be right in more than 95% of the cases, 601 (15% of the total) are the objects with spectroscopic quality 1, but for which the spectro-z differs less than 10% from the photometric one and 98 objects (∼2% of the total) have a spectroscopic flag 9, meaning that only one feature, in their case Lyα, has been identified in the spectrum, and for which about 80% are proven to be right (Le Fèvre et al. 2014). Finally, 231 objects (∼6% of the total) have spectroscopic flag 0, meaning that a spectroscopic redshift could not be assigned.
From Table 1, it is evident that the vast majority of objects with strong Lyα (EW 0 > 25 Å) have been assigned a quality flag of 3 or 4: this is not surprising, and it reflects a tendency by the redshift measurers to assign an higher flag when the spectrum has Lyα in emission. Note as well that not all the galaxies with flag 9 are strong Lyα emitters, although all of them, of course, have Lyα in emission (it is the only spectral feature identified in their spectrum): only in ∼ 40% of the cases the emission is strong enough to pass the equivalent width treshold of 25 Å.
We show in Figure 1 the absolute magnitude in the Far Ultra-Violet as a function of redshift for the 3730 galaxies in the selected sample. We compare the distribution of our galaxies with the evolution of M * FUV as derived by fitting the values for M * FUV compiled by Hathi et al. (2010). In more detail, Hathi et al. (2010) derive the F UV luminosity function of starforming galaxies at z ∼2-3, constraining its slope and character-
We report this best fit on Fig. 1, together with the curve corresponding to M * FUV + 1: we can see that the data sample quite well the FUV luminosities brighter than M * up to redshift z ∼ 5. Similarly, we probe the luminosity down to one magnitude fainter than M * FUV up to redshift z ∼ 3.5. We also note that at z > 5, where the Lyα line and the Lyα forest absorptions by the IGM enter the i ′ −band, we only detect the brightest UV galaxies, while we completely miss galaxies around M * FUV . In the remaining of the paper, we will be cautious to include galaxies at z > 5 in our analysis, and where we will do so, we will discuss the consequences.
For the analysis that we present in the following sections we build two volume limited samples: the bright one, that contains all galaxies brighter than M * FUV at redshift 2 < z < 6; and the faint one, that contains galaxies with M FUV < M * FUV < M FUV + 1, limited at z < 3.5. This approach is slightly different than the one used in similar studies in literature: Stark et al. (2010Stark et al. ( , 2011 and Mallery et al. (2012), for example, rather use fixed intervals of absolute magnitudes at all redshift. However, we prefer here to account for the evolution of the characteristic luminosity of star-forming galaxies, comparing at different redshifts galaxies that are in the same evolutionary state.
The distribution of the rest-frame EW of Lyα
We show in Fig 2 the distribution of the rest-frame EW of Lyα in four redshift bins, for the bright and faint samples separately. Positive EW indicate that Lyα is in emission, and negative EW indicate that the line is in absorption.
Although we measured the equivalent width of Lyα for all the 3730 objects with a measured spectroscopic redshift (all the galaxies with spectroscopic flag 2, 3, 4 and 9, and also the objects with flag 1 for which the spectroscopic redshift differs less than 10% from the photometric one), this figure includes only the 3204 objects in the bright and faint volume limited samples. These are the largest volume limited samples of UV selected galaxies with almost full spectroscopic information ever collected in the literature, and they allow to constrain the EW distribution of the Lyα line from star-forming objects with strong Lyα in absorption compared to those with strong Lyα in emission. It can be seen that the shape of the distribution is similar at all redshifts: it is lognormal and it extends from -50Å to 200 Å, with the peak at EW 0 =0 at all redshift and for all luminosities.
In the first two redshift bins, 2 < z < 2.7 and 2.7 < z < 3.5, we can compare the EW distributions of the bright and faint sample, and we can see that they are quite similar. However, the extension of the tail of objects with large EW 0 (Lyα) evolves fast with redshift: while at 2 < z < 2.7 11% (7%) of the bright (faint) galaxies have EW 0 (Lyα)> 25Å, that fraction increses to ∼ 15% (12%) at 2.7 < z < 4 and to 25% at z∼5. Similarly, we observe an evolution with redshift of the upper EW 0 (Lyα) threshold which contains 80% of the sources: at 2 < z < 3.5 the threshold is around 10-12Å (for galaxies in both the bright and faint samples), at 3.5 < z < 4.5 it evolves to ∼18Å and at 4.5 < z < 6 it moves to ∼30Å.
We remark as well that the only 13 galaxies in the whole sample have EW 0 > 150Å (the highest value is EW 0 = 278.2 at z = 2.5661). So extreme EW 0 (Lyα) can not be easily produced by star formation with a Salpeter IMF, but must have a top-heavy IMF, a very young age < 10 7 yr and/or a very low metallicity (Schaerer 2003).
The evolution of the fraction of strong
Lyα emitters among star-forming galaxies at 2 < z < 6
We present in Figure 3 the evolution with the redshift of the fraction of star-forming galaxies that have an equivalent width of EW 0 (Lyα)> 25 Å (left panel) and EW 0 (Lyα)> 55 Å (right panel), for the bright sample (M FUV < M * ) and for the faint one (M * < M FUV < M * + 1) separately. In both panels we show the same fraction for galaxies with −21.75 < m FUV < −20.25, for consistency with previous studies (Stark et al. 2010;Stark et al. 2011;Mallery et al. 2012). Our fiducial case is obtained when we include all objects with spectroscopic flag 2, 3, 4 and 9, and we add also the "good" flag 1 (those objects for which the spectroscopic and photometric redshifts differ by less than 10%) to the distribution. However, it is possible that this combination slightly overestimates the true fraction, as we know that 231 objects with photometric redshift 2 < z < 6 have been observed in spectroscopy, but for them a spectroscopic redshift could not be assigned. So, it is possible that a fraction of them are actually at 2 < z < 6, and since no Lyα is present in the whole observed spectral range, they will decrease the fraction of strong Lyα emitters by a given amount. We discuss in Fig 4 the effect on the fraction of emitters of the choice of including objects with spectroscopic flag 0 and 1, that is quite minimal.
As we showed in Figure 1, while the bright sample (M FUV < M * ) is well represented up to z ∼ 6, the faint one is represented only up to z ∼ 3.5: in fact, the cut in observed magnitude at m i < 25, that we apply to be sure that the continuum is detected in spectroscopy with a S/N high enough to detect possible UV absorption features, basically prevents us by construction from having faint galaxies in our sample beyond z ∼ 3.5. In the same figure, for the three ranges of UV luminosities, we show the fractions obtained on a finer redshift grid (∆ z ∼ 0.3) and on a coarser grid, that highlights the general trend and smooth out variations due to cosmic variance. The fine grid extends on the whole 2 < z < 6 range for the bright sample: however, only the highest redshift bin contains galaxies at 5 < z < 6 and might be affected by the detection bias due to the Lyα line entering the i ′ −band at that redshift. In the case of the coarser grid we limited the analysis to the galaxies at z < 5, so to be sure that the results are not dependent on that effect.
For the bright sample, the evolution of the fraction of emitters with EW 0 (Lyα)> 25Å, shown in the left panel of Fig. 3, is characterized by a very modest increase of the fraction of Lyα emitters between redshift z ∼ 2 and z ∼ 4 (from ∼ 10 % at z ∼ 2 to ∼ 15 % at z ∼ 4), and then by a faster increase above z ∼ 4 (the fraction reaches ∼ 25 % at z ∼ 5 and ∼ 30 % at z ∼ 5.5). A very similar trend is observed when galaxies with −21.75 < m FUV < −20.25 are considered. If we then analyse the faint sample, and we compare it with the bright one, we find that the overall fraction of objects with EW 0 (Lyα)>25Å (or EW 0 (Lyα)>55Å) is similar to that of the bright sample between z ∼ 2 and z ∼ 3.5, but the evolution between z ∼ 2.3 and z ∼ 3 is much faster for the faint sample. This is in apparent disagreement with the results by Stark et al. (2010;, who found both a higher fraction of Lyα emitters and a steeper evolution of this fraction among faint UV galaxies (−20.25 < M FUV < −18.75) than among bright UV galaxies (−21.75 < M FUV < −20.25). However, we remark that the range of UV luminosities probed by our study is narrower than the one probed by Stark et al. (2010;.
The right panel of Figure 3 shows the effect on the fraction of Lyα emitters of changing the EW threshold from 25 to 55 Å. It can be seen that, as expected, the fraction drastically decreases at all redshifts. However, the general trend observed in the left panel of Fig. 3 is preserved: we observe that the fraction remains around 3-4% with a slight increase in 2 < z < 4, then it increases faster between z ∼ 4 and z ∼ 5 rising from 5% to 12%. The fiducial values, shown by the continuous thick lines, include all the galaxies with spectroscopic flag 2, 3, 4 and 9, and also all the galaxies with a spectroscopic flag 1 and a spectroscopic redshift that differs less than 10% from the photometric one. The dashed lighter lines show a finer binning in redshift. Right panel: same as left panel, but for galaxies with EW 0 (Lyα)> 55 Å An important point that needs to be stressed again here is that our selection criteria are completely independent on the presence and strength of the Lyα emission up to z ∼ 5. This selection ensures that there are no biases in the determination of this fraction over the range 2 < z < 5: if, for some reason, our selection is less complete in a given redshift range, it will be homogeneously incomplete for galaxies with and without Lyα, and thus the result shown in this Section will remain robust.
We report again the evolution of the fraction of Lyα emitters with EW 0 > 25Å and EW 0 > 55Å in Fig 4, where we simply highlight the results on the finer redshift grid and we show the effect of including objects with spectroscopic flag 0 and 1 in the analysis. In this Figure, as in Fig. 3, we consider our fiducial case the one including all the "good" flag 1 (objects with a spectroscopic flag 1, for which the spectroscopic redshift and the photometric one differ by less than 10%), together with flags 2, 3, 4 and 9. If objects with spectroscopic flag 1 are excluded, and only flags 2, 3, 4 and 9 are considered, the fraction of emitters increases by ∼2%, with respect to the fiducial value, at all redshifts and for all UV luminosities. This is quite obvious: since basically all the strong Lyα emitters have a spectroscopic flag 2, 3, 4 and 9 (see Table 1), this set of flags maximizes the fraction. On the other hand, if also the objects with flag 0 are considered, together with good flag 1 and all the flags 2, 3, 4 and 9, the fraction decreases by ∼2% with respect to the fiducial case. This effect is also easy to understand: objects with flag 0 are all non-emitters, because if an emission line had been identified they would have been assigned a redshift and a flag, and thus their net effect is to decrease the fraction. Although we can not know for sure how many of these objects with no spectroscopic redshift are indeed at the photometric redshift, their effect is rather negligible: for both Figure 3 and 4, the effect of considering flags 2, 3, 4 and 9 or of including good flag 1 and flag 0 is always below a few percent.
We remark also that our values are in good agreement with those published by Stark et al. (2010; and that are based on a completely different method that uses LBG technique to photometrically identify high-redshift galaxies (at z ∼4, 5 and 6) that are then observed in spectroscopy to look for strong Lyα emission. Our values are slightly higher than those by Mallery et al. (2012) though still compatible within the error bars.
Lyα escape fraction: driver of the Lyα fraction evolution?
The escape fraction of Lyα photons f esc (Lyα) is defined as the fraction of the Lyα photons that are produced within a given galaxy and that actually escape from the galaxy itself. Given the intrinsic resonant nature of the Lyα photons, it is thought to be dependent on the dust content, geometry of the inter-stellar gas (ISM) and relative kinematics of the ISM and stars. Atek et al. (2014) and Hayes et al. (2014), studying local samples of Lyα emitters, they found a correlation between f esc (Lyα) and E(B-V), with the escape fraction being larger on average in galaxies with low dust content. Kornei et al. (2010) and Mallery et al. (2012), although with smaller samples than Fig. 4. Same as Figure 3, but with a finer binning in redshift, and showing the effect of including galaxies with flags 0 and 1. The left panel shows the case when the galaxies with EW 0 (Lyα)> 25 Å are considered as emitters, and the right panel when the threshold is fixed at EW 0 (Lyα)> 55 Å. The fiducial values, shown by the continuous thick lines, include all the galaxies with spectroscopic flag 2, 3, 4 and 9, and also all the galaxies with a spectroscopic flag 1 and a spectroscopic redshift that differs less than 10% from the photometric one. The dotted line indicate the case when only flags 2, 3, 4 and 9 are considered; the dashed line is the same as the fiducial case, but also the galaxies with no spectroscopic redshift (flag=0) are included, with the redshift fixed to the photometric one. The red curves are for the bright volume limited sample; the blue ones are for the faint one; the green ones are for galaxies with −21.75 < m FUV < −20.25. For clarity, the error bars are shown only for the continuous curves. The cyan points are from Stark et al. (2010;, the yellow ones from Mallery et al. (2012). The red circles, blue triangles and green lozenges show the coarser binning in redshift adopted in Figure 3.
the one we use here, found a similar correlation at z ∼ 3 and at 3.5 < z < 6, respectively. The Lyα escape fraction is usually determined by comparing the Lyα luminosity with the dust-corrected Hα luminosity, once a recombination regime has been chosen. The Hα line in fact is not resonant and it is attenuated only by dust. However, for most of the redshift range of this study Hα is redshifted even beyond the reach of near-infrared spectrographs. An alternative method exploits the expected correlations between Lyα luminosity, Hα luminosity and the SFR of the galaxy: by comparing the SFR derived from the Lyα luminosity to an independent estimate of the total SFR of the galaxy we can estimate the Lyα escape fraction f esc (Lyα). In particular, we assume the case B recombination regime (Brocklehurst 1971), assuming that the SFR predicted by the SED fitting gives at least a crude estimate of the total SFR of the galaxy, and using the SFR-Hα conversion by Kennicutt (1998)
We stress here that the SFR inferred from fitting Bruzual & Charlot (2003) models to the SED of galaxies give only a crude estimate of the star formation rate and of the dust content of galaxies. This is expecially true in the redshift regime probed by VUDS, that is so far poorly explored, and for which independent estimates of the SFR from different methods are scarce. However, the SFR inferred from SED fitting are believed to be on average correct within a factor of 3 (Mostek et al. 2012;Utomo et al. 2014), and thus we choose to use these to obtain at least a crude estimation of the Lyα escape fraction.
We plot in Figure 5 the escape fraction f esc (Lyα) as a function of the redshift and of dust reddening E(B-V) for the galaxies in the bright and faint volume limited samples together. We tried to separate the two samples, to check for differences among the two them, but we did not find any, so we decided to show them together. For the galaxies with Lyα in absorption (i.e. EW 0 (Lyα) < 0, 1628 galaxies) we artificially set f esc (Lyα) to 10 −3 . For galaxies with EW 0 (Lyα)> 0 (1576 galaxies), the Lyα escape fraction ranges from 10 −4 to 1. We calculate as well the median escape fraction in bins of redshift, using the same coarse grid used for Fig. 3, and limiting the highest redshift bin to z = 5, to avoid possible detection biases affecting our selection at higher z.
It is clear from this figure that at each redshift and for each E(B-V) the strong Lyα emitters (with EW 0 (Lyα) > 25Å or EW 0 (Lyα) > 55Å) are the (rare) galaxies with the highest Lyα escape fraction. In more details, 80% of the galaxies with Lyα escape fraction as a function of redshift for the bright (grey diamonds) and faint (grey triangles) volume limited samples. Strong Lyα emitters with EW 0 > 25Å and EW 0 > 55Å are shown with cyan and magenta empty circles, respectively. Objects with formally negative equivalent width of Lyα, corresponding to negative Lyα luminosity, are set here to log[ f esc (Lyα)] = −3. The big red and blue circles indicate the median escape fraction for the galaxies with EW 0 [Lyα] > 55Å and EW 0 [Lyα] > 25 Å, respectively. The black (grey) arrows indicate the f esc (Lyα) below which 80% of the bright (faint) objects lie. Right panel: Lyα escape fraction as a function of the E(B-V). The symbols are the same than in the left panel. The green dashed line shows the prediction by Verhamme et al. (2006). escape fraction f esc (Lyα)> 10% have EW 0 (Lyα)> 55Å, and 70% of the galaxies with f esc (Lyα)> 3% have EW 0 (Lyα)> 25Å. The median escape fraction for galaxies EW 0 (Lyα) > 25Å is around 8% overall, evolving from 3% at z ∼ 2.3 to 8% at z ∼ 3 to 12% at z ∼ 4. The median escape fraction for galaxies EW 0 (Lyα) > 55Å is of course higher, evolving from 5% at z ∼ 2.3 to 12% at z ∼ 3 to 20% at z ∼ 4. For both threshold we observe a decrease of the median escape fraction between z ∼ 4 and z ∼ 5, which is probably due to low number statistics.
If we then consider the whole population in our sample, and we put together the bright and faint volume limited samples, we find that formally the median escape fraction is zero at all redshifts. In fact, the objects with Lyα in absorption (that have f esc (Lyα) fixed to 10 −3 ) are the majority, at all z, forcing the median f esc (Lyα) to zero. For this reason, we find more useful to show the evolution of the f esc (Lyα) below which 80% of the galaxies, at each redshift, lie (arrows in Fig. 5). Indeed, this threshold evolves from 1% at 2 < z < 2.7 to 1.5% at 2.7 < z < 3.5 to 2% at 3.5 < z < 5, with not much difference between the bright and the faint samples.
The comparison of the f esc (Lyα) with the E(B-V) is also interesting. From the right panel of Fig. 5 we can see that the E(B-V) anti-correlates with f esc (Lyα): for objects with high E(B-V) the median vaue of f esc (Lyα) is low (and viceversa). This is in qualitative agreement with the results by Hayes et al. (2014) and Atek et al. (2014) in the local Universe, and with Kornei et al. (2010) and Mallery et al. (2012) at high-z. Moreover, the median values for the galaxies with EW 0 (Lyα)> 25Å and EW 0 (Lyα)> 55Å correlates with the E(B-V) similarly to the prediction by Verhamme et al. (2006), although our data are better fitted by a flatter slope (∼-5 in comparison with -7.71 predicted by Verhamme et al. (2006). However, while galaxies with high E(B-V) never show large f esc (Lyα), the contrary is not true: when E(B-V) is low we observe a broad range of Lyα escape fractions, ranging from 10 −3 to 1. This implies that the dust content alone can not be the only factor to regulate f esc (Lyα), at least for galaxies with the UV luminosities similar to the ones probed in this paper.
Summary, discussion and conclusions
In this paper we used the unique VUDS dataset to build an unbiased and controlled sample of star-forming galaxies at 2 < z < 6, selected according to the photometric redshifts determined using the overall SED of the galaxies. This selection is complementary to the classical LBG technique, resulting in more complete and less contaminated samples of galaxies at high-z. For the purpose of this paper, even more imporant is the fact that the combination of the selection we use are independent on the presence of Lyα in emission, at least up to z ∼ 5: whatever incompleteness could affect our sample, it would affect galaxies with and without Lyα in the same way.
The sample is limited at m i < 25, ensuring that the continuum is detected with S/N∼ 10 per resolution element: this allows an accurate determination of the spectroscopic redshift through
Fig. 2 .
2Rest-frame equivalent width EW 0 of the Lyα line in four redshift bins. The dashed and dotted lines, respectively at EW 0 =25 and 55 Å, represent the two thresholds that we apply in the analysis. The red and blue histogram indicate the bright sample (M FUV < M * ) and the faint (M * < M FUV < M * + 1) one, respectively. 80% of the galaxies in each panel have an EW 0 (Lyα) below the value indicated by the arrow.
istic magnitude, and compare their values with other in literature between z ∼ 0 and z ∼ 8. With the aim of deriving an evolving M * FUV as a function of the redshift, we took the values published by Arnouts et al. (2005) at 0 < z < 3, Hathi et al. (2010) at 2 < z < 3, Reddy & Steidel (2009) at z ∼ 3, Ly et al. (2009) at z ∼ 2, Bouwens et al. (2007) at z ∼ 4, 5, 6 Sawicki et al. (2006) at z ∼ 4 and Mc Lure et al. (2009) at z ∼ 5, 6 and we fitted a parabola to them. In particular, we get this best-fit: M * (z) = −18.56 − 1.37 × z + 0.18 × z 2
Fig. 3 .
3Left panel: Our best estimate of the fraction of galaxies with EW 0 (Lyα)> 25 Å, as a function of the redshift, for three intervals of far-UV absolute magnitudes: faint objects (M * < M FUV < M * + 1) are shown in blue; bright objects (M FUV < M * ) are shown in red; objects with −21.75 < M FUV < −20.25 are shown in green.
we get: S FR(Lyα) = L Lyα /(1.1 × 10 42 ) (2) and thus: f esc (Lyα) = S FR(Lyα)/S FR(S ED) = L Lyα /(1.1 × 10 42 ) S FR S ED
Fig. 5 .
5Left panel:
Acknowledgements. We thank ESO staff for their continuous support for the VUDS survey, particularly the Paranal staff conducting the observations and Marina Rejkuba and the ESO user support group in Garching. This work is sup-P.Cassata et al.:Ly-α fraction vs redshift the identification of UV absorption features even for galaxies without Lyα in emission.We split this sample in two volume limited samples, using a far-UV luminosity cut that is evolving with redshift, following the observed evolution of M * FUV(Hathi et al. 2010): the bright sample include objects that at each redshift are brighter than M * FUV ; the faint one include objects with M * < M FUV < M * + 1. We use these two samples to constrain the distribution of the EW of Lyα of star-forming galaxies, that spans from objects with Lyα in absorption to objects with Lyα in emission. We find that ∼ 80% of the star-forming galaxies in our sample have a Lyα equivalent width EW 0 (Lyα) < 15Å.We use our sample to constrain the evolution of the fraction of strong Lyα emitters among star-forming galaxies at 2 < z < 6. We showed in Section 4 that the fraction of strong Lyα emitters with EW 0 (Lyα) > 25Å and EW 0 (Lyα) > 55Å monothonically increases with redshift, approximately at the same rate for the two EW thresholds. The evolution is characterized by a slower phase between z ∼ 2 and z ∼ 4, and by a faster evolution between z ∼ 4 and z ∼ 5.5. We see no difference, at 2 < z < 3.5 where both samples are well represented, between the fraction of strong emitters in the bright and faint volume limited samples. This is partly in contraddiction with results byStark et al. (2010;, who found that the fraction is higher, and the rate of evolution with redshift faster, for UV faint galaxies at 4 < z < 6. However, this might be due to the narrower range of UV luminosity probed by our work compared to the one probed byStark et al. (2010;.Moreover, slicing our sample with the same UV luminosity limits used by Stark (−21.75 < M FUV < −20.25) we see that the evolution of the fraction of strong Lyα emitters (for both EW 0 (Lyα) > 25Å and EW 0 (Lyα) > 55Å) is in very good agreement with the values byStark et al. (2010;, despite the different sample selection methods and available spectroscopy. This is a very important result, placing on firmer grounds the measures of the fraction of star-forming galaxies with Lyα in emission. In fact, their sample is LBG based and only the objects with strong Lyα emission are spectroscopically confirmed. In our case, on the other hand, we stress that all the galaxies, with and without Lyα, have a spectroscopic redshift.Finally, in Section 5, we have explored the possibility that the evolution of the fraction of strong Lyα emitters is primarly due to a change of the escape fraction of Lyα photons. We have found that, as expected, the strong Lyα emitters are the objects for which f esc (Lyα) is the largest. We find as well that the median f esc (Lyα) for the Lyα emitters (with not much difference between objects with EW 0 (Lyα)> 25Å and with EW 0 (Lyα)> 55Å) evolves from ∼5% at z ∼ 2.5 to ∼20% at z ∼ 5. If we try to estimate the median escape fraction for the whole population, we find that it is formally zero at all redshifts, since the majority of the galaxies in our sample have Lyα in absorption, and 80% of our galaxies have f esc (Lyα)< 1%. If we estimate at each redshift the f esc (Lyα) value below which 80% of the galaxies lie, we find that this value evolves from 1 to 2% between z ∼ 2 and z ∼ 5. It is interesting to compare these findings withHayes et al. (2011), who integrated the Lyα and UV luminosity functions from z ∼ 0 to z ∼ 8 and then compared the two to estimate the average f esc (Lyα) of the Universe at those redshifts. According toHayes et al. (2011)the average escape fraction is around 5% at z ∼ 2 and 20% at z ∼ 5, values that are much higher than those we obtain for our sample. This implies that for the galaxies with UV luminosities that we sample in this paper (M FUV < M * at 2 < z < 6 and M * < M FUV < M * + 1 at 2 < z < 3.5) the average escape fraction of Lyα photons is much smaller than the average escape fraction of the Universe. In other words, the bulk of the Lyα luminosity, at least in the redshift range 2 < z < 6 that is probed in this paper, is not coming from galaxies with the UV luminosities that are probed in this work, but from galaxies that are much fainter in the UV. In fact,Stark et al. (2011)showed that the fraction of strong (EW 0 (Lyα)>25Å) emitters is higher in galaxies with −20.25 < M FUV < −18.75 than in those with −21.75 < M FUV < −20.25, implying a larger escape fraction for faint UV galaxies. This is in line also with the results byAndo et al. (2006), who found a deficiency of strong Lyα emitters among UV bright galaxies and bySchaerer, de Barros & Stark (2011), who also found that the fraction of Lyα emitters rapidly increases among galaxies with fainter UV luminosities, indicating that the bulk of the Lyα luminosity in the universe comes from galaxies with M FUV > −20.Similarly toKornei et al. (2010)andMallery et al. (2012), we find also that there is an anti-correlation between f esc (Lyα) and the dust content E(B-V): galaxies with low f esc (Lyα) have preferentially a higher E(B-V), and viceversa. This implies that the dust is a key ingredient in setting the escape fraction of galaxies. However, we remark that galaxies with low extinction (E(B − V) < 0.05) have a very wide range of Lyα escape fractions, ranging from 10 −3 to 1: this means that the dust content, although important, is not the only ingredient to regulate the fraction of Lyα photons that escape the galaxy. In a forthcoming paper, we will further investigate on the dependence of f esc (Lyα) from other quantities as stellar mass, star formation rate and dust content, and on the evolution with redshift of these correlations.
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| [] |
[
"Weyl-type bounds for Steklov eigenvalues",
"Weyl-type bounds for Steklov eigenvalues"
] | [
"Luigi Provenzano ",
"Joachim Stubbe "
] | [] | [] | We present upper and lower bounds for Steklov eigenvalues for domains in R N +1 with C 2 boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues. | 10.4171/jst/250 | [
"https://arxiv.org/pdf/1611.00929v1.pdf"
] | 18,660,662 | 1611.00929 | 1e06939b6046c3908000d874ee26a89bd3a26200 |
Weyl-type bounds for Steklov eigenvalues
November 4, 2016
Luigi Provenzano
Joachim Stubbe
Weyl-type bounds for Steklov eigenvalues
November 4, 2016Steklov eigenvalue problemLaplace-Beltrami operatorEigenvalue boundsWeyl eigenvalue asymptoticsRiesz-meansmin-max principledistance to the boundarytubular neighborhood 2000 Mathematics Subject Classification: Primary 35P15; Secondary 35J2535P2058C40
We present upper and lower bounds for Steklov eigenvalues for domains in R N +1 with C 2 boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.
Introduction.
Let Ω ⊂ R N +1 be a bounded domain with boundary ∂Ω. We consider the Steklov eigenvalue problem on Ω: ∆u = 0, in Ω, ∂u ∂ν = σu, on ∂Ω,
(1.1)
where ∂u ∂ν = ∇u · ν denotes the derivative of u in the direction of the outward unit normal ν to ∂Ω. A classical reference for problem (1.1) is [38] where it was introduced to describe the stationary heat distribution in a body whose flux through the boundary is proportional to the temperature on the boundary. When N = 1 problem (1.1) can be intepreted as the equation of a free membrane the mass of which is concentrated at the boundary (see [33]). The eigenvalues of problem (1.1) can be also seen as the eigenvalues of the Dirichlet-to-Neumann map (see e.g., the survey paper [24]). We also mention that recently the analogue of the Steklov problem has been introduced for the biharmonic operator as well in [10] (see also [9]).
It is well known that under mild regularity conditions on the boundary ∂Ω (see e.g., [24] for a detailed discussion), in particular if ∂Ω is piecewise C 1 , problem (1.1) admits an increasing sequence of non-negative eigenvalues of the form 0 = σ 0 < σ 1 ≤ σ 2 ≤ · · · +∞, where the eigenvalues are repeated according to their multiplicity and satisfy the Weyl asymptotic formula (see [2])
lim j→∞ σ j j −1/N = 2π B −1/N N |∂Ω| −1/N ,(1.2)
with |∂Ω| denoting the N -dimensional measure of ∂Ω and B N = π N/2 Γ(1 + N/2) being the volume of the N -dimensional unit ball. It is an open problem to find bounds on σ j compatible with the Weyl-limit (1.2) except when N = 1 and ∂Ω is smooth (see [27]; see also [19] and the survey article [24]). The situation is different when we consider the eigenvalue problem for the Laplace-Beltrami operator on ∂Ω, that is − ∆ ∂Ω ϕ = λϕ on ∂Ω, and Weyl-type bounds of the form (see e.g., [14], [17])
λ j ≤ a ∂Ω + b N j 2/N |∂Ω| −2/N (1.5)
for some positive constants a ∂Ω , b N depending only on the geometry and the dimension of the manifold ∂Ω. We refer to [15] for an introduction to eigenvalue problems for the Laplace-Beltrami operator on Riemannian manifolds and to [17,18,19,26] and to the references therein for a more detailed discussion on upper bounds for the eigenvalues of the Laplacian on manifolds. The above asymptotic formulas suggest that at least for large j the Steklov eigenvalues σ j are related to the Laplacian eigenvalues λ j approximately via
σ j ≈ λ j .
(1.6)
The main result of our paper is a comparison between Steklov and Laplacian eigenvalues for all j compatible with the asymptotic relation (1.6).
Theorem 1.7. Let Ω ⊂ R N +1 be a bounded domain with boundary ∂Ω of class C 2 such that ∂Ω has only one connected component. Then there exists a constant c Ω such that for all j ∈ N λ j ≤ σ 2 j + 2c Ω σ j , σ j ≤ c Ω + c 2 Ω + λ j .
(1.8)
In particular,
σ j − λ j ≤ 2c Ω . (1.9)
The constant c Ω has the dimension of an inverse length and depends explicitely on the dimension N , the maximum of the mean of the absolute values of the principal curvatures κ i (x), i = 1, . . . , N , on ∂Ω and the maximal possible sizeh of a suitable tubular neighborhood about ∂Ω. For convex domains Ω we shall improve the estimates (1.8) such that they become sharp for all j when Ω is a ball of radius R and give the exact relation
λ j = σ 2 j + N − 1 R σ j
between Steklov and Laplacian eigenvalues on the N -dimensional ball and Ndimensional sphere of radius R respectively. Clearly Theorem 1.7 implies Weyl-type estimates for Steklov eigenvalues from the bounds (1.5) for Laplacian eigenvalues (see Corollary 4.8). Combining the sharp Weyl-type estimates for Laplacian eigenvalues on hypersurfaces obtained in [25] with the estimates of Theorem 1.7 we prove the following sharp bound for Riesz means of Steklov eigenvalues:
Theorem 1.10. Let Ω ⊂ R Nz ≥ 0 ∞ j=0 (z − σ j ) 2 + ≤ 2 (N + 1)(N + 2) (2π) −N B N |∂Ω| z + c Ω N +2 (1.11)
where c Ω is the constant from Theorem 1.7
The estimate (1.11) is asymptotically sharp since
lim z→∞ z −N −2 ∞ j=0 (z − σ j ) 2 + = 2 (N + 1)(N + 2) (2π) −N B N |∂Ω|
according to (1.2). Theorem 1.10 implies sharp upper bounds on the trace of the associated heat kernel (see Corollary 6.4) as well as lower bounds on the eigenvalues (see Corollary 6.6). The present paper is organized as follows: in Section 2 we recall some properties of the squared distance function from the boundary in a suitable tubular neighborhood of a C 2 domain. We exploit these properties in Section 3 in order to obtain estimates of boundary integrals of harmonic functions. In particular, we establish a comparison between the L 2 (∂Ω) norms of the normal derivative and of the tangential gradient of harmonic functions which is used in Section 4 together with the min-max principle to prove our main Theorem 1.7 and, as a consequence, Weyl-type upper bounds for Steklov eigenvalues. In Section 5 we consider the case of convex C 2 domains for which we refine the estimates (1.8), which become sharp in the case of the ball. Finally, in Section 6 we prove Theorem 1.10 as well as upper bounds on the trace of the Steklov heat kernel and lower bounds on Steklov eigenvalues which turn out to be asymptotically sharp.
The squared distance function from the boundary
In this section we collect a number of properties of the distance and squared distance functions from the boundary ∂Ω of a C 2 domain of R N +1 which will be used in the proof of the main result.
We set
d 0 (x) := dist(x, ∂Ω).
Let x ∈ ∂Ω and let ν(x) denote the outward unit normal to ∂Ω at x. We have the following characterization of ν(x) in terms of d 0 (x):
Lemma 2.1. Let Ω be a bounded domain in R N +1 of class C 2 . Then for x ∈ ∂Ω ν(x) = −∇d 0 (x).
We refer to [21,Ch.7,Theorem 8.5] for the proof of Lemma 2.1. Let h > 0. The h-tubular neighborhood ω h of ∂Ω is defined as
ω h := {x ∈ Ω : d 0 (x) < h} (2.2)
We have the following:
Theorem 2.3.
Let Ω be a bounded domain in R N +1 of class C 2 . Then there exists h > 0 such that every point in ω h has a unique nearest point on ∂Ω.
We refer to [30] for the proof of (2.5)
Throughout the rest of this section, we will denote by h a positive number such that h ∈]0,h[. In a tubular neighborhood ω h the distance function (and hence its square) is of class C 2 . This is stated in the following:
Theorem 2.6.
Let Ω be a bounded domain in R N +1 of class C 2 . Let ω h be as in (2.2). Then d 0 is of class C 2 in ω h . Moreover, for any x ∈ ∂Ω, the matrix D 2 (d 0 (x) 2 /2) represents the orthogonal projection on the normal space to ∂Ω at x and d 0 (x − pν(x)) = |p|,
∇d 0 (x − pν(x)) = −ν(x),
for any p ∈ R with |p| ≤ h.
We refer to [4, Figure 1.
Remark 2.7. From Theorem 2.6 it follows that the set Γ h is diffeomorphic to ∂Ω.
Let x ∈ ∂Ω and let κ 1 (x), ..., κ N (x) denote the principal curvatures of ∂Ω at x with respect to the outward unit normal. We refer e.g., to [23,Sec. 14.6] for the definition and basic properties of the principal curvatures of ∂Ω. We have the following: Let Ω be a bounded domain in R N +1 of class C 2 . Let x ∈ ω h and let y ∈ ∂Ω be the nearest point to x on ∂Ω. Then
Ω ω ϵ ∂Ω Γ ν(x)=ν(x') x x'=x-hν(x)1 − d 0 (x)κ i (y) > 0 (2.9)
for all i = 1, ..., N .
We refer to [34, Lemma 2.2] for a proof of Lemma 2.8. We note that the numberh in (2.4) provides an upper bound for the positive principal curvatures of ∂Ω. In fact we have
K + := max 1≤i≤N, x∈∂Ω max {0, κ i (x)} < 1 h .
(2.10)
We also define K − by
K − := min 1≤i≤N, x∈∂Ω min {0, κ i (x)} ≤ 0. (2.11)
and K ∞ by Clearly d(x) = h − d 0 (x) for all x ∈ ω h , hence d and η are of class C 2 in ω h . Let x ∈ ∂Ω and let x = x − hν(x) ∈ Γ h . Let now κ 1 (x ), ..., κ N (x ) denote the principal curvatures of Γ h at x with respect to the outward unit normal. The principal curvatures κ i (x ) and κ i (x) are related, as stated in the following: Lemma 2.13. Let Ω be a bounded domain in R N +1 of class C 2 . Let ω h and Γ h be defined by (2.2) and (2.5), respectively. Let x ∈ ∂Ω and let x = x − hν(x) ∈ Γ h . Then we have
K ∞ := max {K + , −K − } = maxκ i (x ) = κ i (x) 1 − hκ i (x)
(2.14)
for all i = 1, ..., N . Moreover, ν(x) = ν(x ).
The proof of Lemma 2.13 follows from [3, Theorem 3] and from the fact that d(x) = h − d 0 (x) (see also [37]). Now we are ready to state the following theorem concerning the eigenvalues of D 2 η.
Theorem 2.15. Let Ω be a bounded domain in R N +1 of class C 2 . Let ω h and Γ h be defined by (2.2) and (2.5), respectively. Let x ∈ ω h and let y = x+d(x)∇d(x) ∈ Γ h be the nearest point to x on Γ h . Then, denoting by ρ 1 (x), ..., ρ N (x) the eigen- values of D 2 η(x) it holds ρ i (x) = d(x)κ i (y ) 1+d(x)κ i (y ) , if 1 ≤ i ≤ N, 1, if i = N + 1.
The proof of Theorem 2.15 can be carried out in a similar way as in [6, Lemma 1] (see also [23,Lemma 14.17]). We also refer to [3,Theorem 4] and [4, Theorem 3.2] for an alternative approach.
From now on we will agree to order the eigenvalues
ρ i (x) of D 2 η(x) increas- ingly, so that ρ 1 (x) ≤ ρ 2 (x) ≤ · · · ≤ ρ N +1 (x) = 1.
We conclude this section by presenting some bounds for the eigenvalues ρ i (x) when x ∈ ω h . We have the following:
Lemma 2.16.
Let Ω, ω h and Γ h be as in Theorem 2.15. Let x ∈ ω h and let ρ i (x) denote the eigenvalues of D 2 η(x) for i = 1, ..., N . Then
hK − ≤ ρ i (x) ≤ hK + < 1. (2.17)
Proof. Let x ∈ ω h and let y be the unique nearest point to x on ∂Ω. From (2.14) and from the fact that d(
x) = h − d 0 (x) it follows that ρ i (x) = 1 − 1 − hκ i (y) 1 − d 0 (x)κ i (y)
.
(2.18)
We observe that the function κ → 1 − 1−hκ 1−dκ is increasing and convex for all 0 ≤ d ≤ h, provided κ < 1/h (which is always the case, see (2.10) and (2.11)). Moreover the function d → 1 − 1−hκ 1−dκ is decreasing and concave if κ ≥ 0 and increasing and concave if κ ≤ 0. Then
ρ i (x) ≤ 1 − 1 − hK + 1 − d 0 (x)K + ≤ hK + and ρ i (x) ≥ 1 − 1 − hK − 1 − d 0 (x)K − ≥ hK − , since K − ≤ 0 ≤ K + .
This concludes the proof.
Remark 2.19.
If Ω is a convex domain of class C 2 we have that κ i (x) ≥ 0 for all i = 1, ..., N and for all x ∈ ∂Ω, hence 0 ≤ ρ i (x) ≤ 1, for all i = 1, ..., N + 1 and for all x ∈ ω h . Moreover Theorem 2.15 holds for all h ∈]0, 1/K ∞ [ (see Section 5). This is not true for general non-convex domains, since it is not possible to estimate the size of the maximum tubular neighborhood ω h only in terms of the principal curvatures. In fact h can be much smaller than 1/K ∞ (see Figure 2). Ω 2h Figure 2: If the domain is not convex we can have arbitrary smallh while K ∞ is uniformly bounded.
Boundary integrals of harmonic functions
The aim of this section is to prove that for a function v ∈ H 2 (Ω) harmonic in Ω, the norms ∇ ∂Ω v L 2 (∂Ω) and ∂v ∂ν L 2 (∂Ω) are equivalent. Here ∇ ∂Ω v denotes the tangential gradient of a function v ∈ H 1 (∂Ω). This is the usual intrinsic gradient of v on the Riemannian C 2 -manifold ∂Ω with the induced Riemannian metric of R N +1 . We will denote by H m (Ω) (respectively H m (∂Ω)) the Sobolev spaces of real-valued functions in L 2 (Ω) (respectively L 2 (∂Ω)) with weak derivatives up to order m in L 2 (Ω) (respectively L 2 (∂Ω)). We will also denote by dσ the N -dimensional measure element of ∂Ω.
We start with the following generalized Pohozaev identity for harmonic functions:
Lemma 3.1. Let F : Ω → R N +1 be a Lipschitz vector field. Let v ∈ H 2 (Ω) with ∆v = 0 in Ω. Then ∂Ω ∂v ∂ν F · ∇vdσ − 1 2 ∂Ω |∇v| 2 F · νdσ + 1 2 Ω |∇v| 2 divF dx − Ω (DF · ∇v) · ∇vdx = 0, (3.2)
where DF denotes the Jacobian matrix of F .
Proof.
Since v is harmonic in Ω, we have ∆vF · ∇v = 0 in Ω. We integrate such identity over Ω. Throughout the rest of the proof we shall write ∂ i v for ∂v ∂x i and
∂ 2 ik v for ∂ 2 v ∂x i ∂x k . We have 0 = Ω ∆vF · ∇vdx = ∂Ω ∂v ∂ν F · ∇vdσ − Ω ∇v · ∇(F · ∇v)dx = ∂Ω ∂v ∂ν F · ∇vdσ − Ω (DF · ∇v) · ∇vdx − Ω (D 2 v · F ) · ∇vdx, (3.3)
where D 2 v denotes the Hessian matrix of v. Now let us consider the third summand in (3.3). We have
Ω (D 2 v · F ) · ∇vdx = Ω N +1 i,k=1 ∂ i v∂ 2 ik vF k dx = ∂Ω N +1 i,k=1 ∂ i v∂ i vF k ν k dσ − Ω N +1 i,k=1 ∂ i v∂ k (∂ i vF k )dx = ∂Ω |∇u| 2 F · νdσ − Ω |∇v| 2 divF dx − Ω (D 2 v · F ) · ∇vdx, thus Ω (D 2 v · F ) · ∇vdx = 1 2 ∂Ω |∇v| 2 F · νdσ − 1 2 Ω |∇v| 2 divF dx. (3.4)
We plug (3.4) in (3.3) and finally obtain (3.2). This concludes the proof of the lemma.
Remark 3.5. When F = x, formula (3.2) is usually referred as Pohozaev iden- tity. It reads ∂Ω ∂v ∂ν x · ∇vdσ − 1 2 ∂Ω |∇v| 2 x · νdσ + N − 1 2 Ω |∇v| 2 dx = 0, (3.6) for all v ∈ H 2 (Ω) with ∆v = 0. Formula (3.6)
when Ω is a ball in R N +1 allows to write the exact relations between the Steklov eigenvalues of Ω and the Laplace-Beltrami eigenvalues on ∂Ω without knowing explicitly the eigenvalues (see Subsection 5.2). For a general domain Ω of class C 2 it is natural to use F as in (3.7) here below.
Let Ω be a bounded domain of class C 2 in R N +1 . Let h ∈]0,h[, whereh is given by (2.4), and ω h be as in (2.2). Let F : Ω → R N +1 be defined by
F (x) := 0, if x ∈ Ω \ ω h , ∇η, if x ∈ ω h . (3.7)
By construction F is a Lipschitz vector field. We consider formula (3.2) with F given by (3.7). We use the fact that for v ∈ H 1 (Ω) (and hence for v ∈ H 2 (Ω)), (3.8), in order to compare the integrals of |∇ ∂Ω v| 2 and ∂v ∂ν 2 over ∂Ω, we have to estimate
|∇v| 2 | ∂Ω = |∇ ∂Ω v| 2 + ∂v ∂ν 2 . Moreover, we use the fact that F (x) = hν(x) when x ∈ ∂Ω. We have 0 = ∂Ω ∂v ∂ν 2 dσ − ∂Ω |∇ ∂Ω v| 2 dσ + 1 h ω h |∇v| 2 ∆η − 2(D 2 η · ∇v) · ∇vdx . (3.8) Let x ∈ ω h . From|∇v(x)| 2 ∆η(x) − 2(D 2 η(x) · ∇v(x)) · ∇v(x). (3.9)
We have the following lemma. Let Ω be a bounded domain in R N +1 of class C 2 . Let ω h be as in
(2.2). For any v ∈ H 1 (Ω) it holds ω h |∇v| 2 ∆η − 2(D 2 η · ∇v) · ∇vdx ≤ 1 + NH ∞ h Ω |∇v| 2 dx, (3.11) whereH ∞ := max x∈∂Ω 1 N N i=1 |κ i (x)| . Proof. Let x ∈ ω h . Let ξ i (x), i = 1, ..., N + 1 be the eigenvectors of D 2 η(x) associated with the eigenvalues ρ i (x) and normalized such that ξ i (x) · ξ j (x) = δ ij .
We can write then
∇v(x) = N +1 i=1 α i (x)ξ i (x), for some α i (x) ∈ R. We note that |∇v(x)| 2 = N +1 i=1 α i (x) 2 .
With this notation (3.9) can be re-written as follows:
Q(∇v(x)) := |∇v(x)| 2 ∆η(x) − 2(D 2 η(x) · ∇v(x)) · ∇v(x) = N +1 i=1 α i (x) 2 N i=1 ρ i (x) − 2 N +1 i=1 ρ i (x)α i (x) 2 . (3.12)
Suppose that ∇v = 0, otherwise inequality (3.11) is trivially true. We have that
Q(∇v(x)) = N +1 i=1 ρ i (x)(1 − 2α i (x) 2 )|∇v(x)| 2 , (3.13) whereα i (x) := α i (x) N +1 i=1 α i (x) 2 = α i (x) |∇v(x)| .
It is straightforward to see that
N i=1 ρ i (x) − 1 |∇v(x)| 2 ≤ N +1 i=1 ρ i (x)(1 − 2α i (x) 2 )|∇v(x)| 2 ≤ 1 + N i=2 ρ i (x) − ρ 1 (x) |∇v(x)| 2 . (3.14)
Now from (2.9),(2.17) and (2.18) it follows that
|ρ i (x)| − h|κ i (x)| = − d 0 (x)|κ i (y)|(1 − κ i (y)) 1 − d 0 (x)κ i (y) ≤ 0
for all x ∈ ω h and i = 1, ..., N , where y ∈ ∂Ω is the unique nearest point to x on ∂Ω. Hence for all
x ∈ ω h N i=2 ρ i (x) − ρ 1 (x) ≤ N i=1 |ρ i (x)| ≤ NH ∞ h. (3.15)
On the other hand, again from (2.9),(2.17) and (2.18) we have that
ρ i (x) − (h − d 0 (x))κ i (y) = (h − d 0 (x))d 0 (x)κ i (y) 2 1 − d 0 (x)κ i (y) ≥ 0,
for all x ∈ ω h and i = 1, ..., N , where y ∈ ∂Ω is the unique nearest point to x on ∂Ω. Hence for all
x ∈ ω h N i=1 ρ i (x) ≥ −N H ∞ h ≥ −NH ∞ h,(3.16)
where
H ∞ := max x∈∂Ω 1 N N i=1 κ i (x) (3.17)
denotes the maximal mean curvature of ∂Ω. Clearly (3.12), (3.13), (3.14), (3.15) and (3.16) imply
|Q(∇v(x))| ≤ (1 + NH ∞ h)|∇v(x)| 2
and therefore the validity of (3.11). This concludes the proof.
From now on we will write
c Ω := 1 2h + NH ∞ 2 ,
whereh is given by (2.4). We are ready to prove the following theorem:
dσ = ∂Ω |∇ ∂Ω v| 2 dσ − 1 h ω h |∇v| 2 ∆η − 2(D 2 η · ∇v) · ∇vdx ≤ ∂Ω |∇ ∂Ω v| 2 dσ + 1 h + NH ∞ ∂Ω ∂v ∂ν 2 dσ 1 2 , (3.22) which is equivalent to ∂Ω ∂v ∂ν 2 dσ − 1 h + NH ∞ ∂Ω ∂v ∂ν 2 dσ 1 2 − ∂Ω |∇ ∂Ω v| 2 dσ ≤ 0.
This is an inequality of degree two in the unknown ∂Ω Remark 3.24. We note that thanks to (3.14) and (3.16) we can use the maximal mean curvature H ∞ instead ofH ∞ in (3.22) and therefore in the inequality (3.23). Moreover, this and inequality (3.23) imply
Ω |∇v| 2 dx ≤ 1 2h + N H ∞ 2 + 1 2h + N H ∞ 2 2 + ∂Ω |∇ ∂Ω v| 2 dσ. (3.25)
for all v ∈ H 2 (Ω) with ∆v = 0 and ∂Ω v 2 dσ = 1.
Proof of Theorem 1.7
In this section we prove Theorem 1.7. Namely, we prove that the absolute value of the difference between the j-th eigenvalues of problems (1.1) and (1.3) is bounded by 2c Ω . Throughout the rest of the paper we shall assume that Ω is a bounded domain of class C 2 in R N +1 such that its boundary ∂Ω has only one connected component. This says that ∂Ω is a compact C 2 -submanifold of dimension N in R N +1 without boundary. In particular, ∂Ω is a Riemannian C 2 -manifold of dimension N with the induced Riemannian metric. The proof of Theorem 1.7 is carried out by exploiting Theorem 3.18 and the following variational characterizations of the eigenvalues of problems (1.1) and (1.3), namely
σ j = inf V ≤H 1 (Ω), dimV =j sup 0 =v∈V, ∂Ω v 2 dσ=1 Ω |∇v| 2 dx,(4.1)
for all j ∈ N, j ≥ 1, wherẽ
H 1 (Ω) := v ∈ H 1 (Ω) : ∂Ω vdσ = 0 , and λ j = inf V ≤H 1 (∂Ω), dimV =j sup 0 =v∈V, ∂Ω v 2 dσ=1 ∂Ω |∇ ∂Ω v| 2 dσ,(4.2)
for all j ∈ N, j ≥ 1, wherẽ
H 1 (∂Ω) := v ∈ H 1 (∂Ω) : ∂Ω vdσ = 0 .
It is useful to recall the following results on the completeness of the sets of eigenfunctions of problems (1.1) and (1.3) in L 2 (∂Ω).
Theorem 4.3.
Let Ω be a bounded domain in R N +1 of class C 2 . Let {σ j } ∞ j=0 be the sequence of eigenvalues of problem (1.1) and let {u j } ∞ j=0 ⊂ H 1 (Ω) denote the sequence of eigenfunctions associated with the eigenvalues σ j , normalized such that ∂Ω u i u k dσ = δ ik for all i, k ∈ N. Then {u j | ∂Ω } ∞ j=0 is an orthonormal basis of L 2 (∂Ω). Moreover, Ω ∇u i · ∇u k dx = σ i δ ik for all i, k ∈ N.
We refer e.g., to [5] for a proof of Theorem 4.3 (see also [7,20]).
Theorem 4.4.
Let Ω be a bounded domain in R N +1 of class C 2 . Let {λ j } ∞ j=0 be the sequence of eigenvalues of problem (1.3) and let {ϕ j } ∞ j=0 ⊂ H 1 (∂Ω) denote the sequence of eigenfunctions associated with the eigenvalues λ j , normalized such that ∂Ω ϕ i ϕ k dσ = δ ik for all i, k ∈ N. Then {ϕ j } ∞ j=0 is an orthonormal basis of
L 2 (∂Ω). Moreover, ∂Ω ∇ ∂Ω ϕ i · ∇ ∂Ω ϕ k dσ = λ i δ ik for all i, k ∈ N.
The proof of Theorem 4.4 follows from standard spectral theory for linear operators (see [7,20]) and from the compactness of the embedding H 1 (∂Ω) ⊂ L 2 (∂Ω).
We are now ready to prove Theorem 1.7.
Proof of Theorem 1.7. We start by proving i). Let u 1 , ..., u j be the Steklov eigenfunctions associated with σ 1 , ..., σ j normalized such that ∂Ω u i u k dσ = δ ik , so that Ω ∇u i · ∇u k dx = σ i δ ik for all i, k = 1, ..., j. Moreover ∂Ω u i dσ = 0 for all i = 1, ..., j. From the regularity assumptions on Ω, we have that u i are classical solutions, i.e., u i ∈ C 2 (Ω) ∩ C 1 (Ω) (see [1]). In particular, u i| ∂Ω ∈H 1 (∂Ω) and ∂u i ∂ν = σ i u on ∂Ω, for all i = 1, ..., j. Let V ⊂H 1 (∂Ω) be the space generated by u 1| ∂Ω , ..., u j | ∂Ω . Any function u ∈ V with ∂Ω u 2 dσ = 1 can be written as u = j i=1 c i u i| ∂Ω , where c = (c 1 , ..., c j ) ∈ R j is such that |c| = 1, i.e., c ∈ ∂B j and B j is the unit ball in R j . Moreover ∆u = 0 for all u ∈ V . From (4.2) and (3.19) we have
λ j ≤ max 0 =u∈V ∂Ω u 2 dσ=1 ∂Ω |∇ ∂Ω u| 2 dσ = max c∈B j c=(c 1 ,...,c j ) ∂Ω ∇ ∂Ω j i=1 c i u i 2 dσ ≤ max c∈B j c=(c 1 ,...,c j ) ∂Ω ∂ j i=1 c i u i ∂ν 2 dσ + 2c Ω ∂Ω ∂ j i=1 c i u i ∂ν 2 dσ 1 2 = max c∈B j c=(c 1 ,...,c j ) ∂Ω j i=1 c i σ i u i 2 dσ + 2c Ω ∂Ω j i=1 c i σ i u i 2 dσ 1 2 = max c∈B j c=(c 1 ,...,c j ) j i=1 c 2 i σ 2 i + 2c Ω j i=1 c 2 i σ 2 i 1 2 = σ 2 j + 2c Ω σ j .
This proves i). In an analogous way we prove ii). Let ϕ 1 , ..., ϕ j ∈ H 1 (∂Ω) be the eigenfunctions associated with the eigenvalues λ 1 , ..., λ j of problem (1.3), normalized such that ∂Ω ϕ i ϕ k dσ = δ ik for all i, k = 1, ..., j. Then ∂Ω ∇ ∂Ω u i · ∇ ∂Ω u k dσ = λ i δ ik for all i, k = 1, ..., j. Moreover ∂Ω ϕ i dσ = 0 for all i = 1, ..., j, thus ϕ i ∈H 1 (∂Ω). Now let φ i , i = 1, ..., j be the solutions to
∆φ i = 0, in Ω, φ i = ϕ i , on ∂Ω. (4.5)
It is standard to prove that for all i = 1, ..., j, problem (4.5) admits a unique solution φ i which is harmonic inside Ω and which coincides with ϕ i on ∂Ω (see e.g., [23,Theroem 2.14]. From the fact that Ω is of class C 2 and from standard elliptic regularity (see [1]) it follows that φ i ∈ C 2 (Ω) ∩ C 0 (Ω). Moreover ∂Ω φ i | ∂Ω dσ = ∂Ω ϕ i dσ = 0 for all i = 1, ..., j, thus φ i ∈H 1 (Ω) for all i = 1, ..., j. Let W ⊂ H 1 (Ω) be the space generated by φ 1 , ...φ j . Any function φ ∈ W with ∂Ω φ 2 dσ = 1 can be written as φ = j i=1 c i φ i with c = (c 1 , ..., c j ) ∈ B j . Moreover ∆φ = 0 for all φ ∈ V . Thanks to (3.20) and (4.1) we have
σ j ≤ max 0 =φ∈W ∂Ω φ 2 dσ=1 Ω |∇φ| 2 dx = max c∈B j c=(c 1 ,...,c j ) Ω ∇ j i=1 c i φ i 2 dx ≤ max c∈B j c=(c 1 ,...,c j ) ∂Ω ∂ j i=1 c i φ i ∂ν 2 dσ 1 2 ≤ c Ω + c 2 Ω + max c∈B j c=(c 1 ,...,c j ) ∂Ω ∇ ∂Ω j i=1 c i φ i 2 1 2 = c Ω + c 2 Ω + max c∈B j c=(c 1 ,...,c j ) ∂Ω ∇ ∂Ω j i=1 c i ϕ i 2 1 2 ≤ c Ω + c 2 Ω + max c∈B j c=(c 1 ,...,c j ) j i=1 c 2 i λ i 1 2 = c Ω + c 2 Ω + λ j .
This concludes the proof of ii) and of the theorem. Theorem 1.7 not only confirms the Weyl asymptotic behavior lim j→∞ λ j /σ j = 1, but says that the difference between the eigenvalues is given at most by a constant independent of j.
By combining (1.8) with (1.5) we can now bound the Steklov eigenvalues from above. To this purpose, it is convenient to specify the constants a ∂Ω and b N in (1.5) by recalling the following theorem from [14]. We will denote by Ric g (M ) the Ricci curvature tensor of a Riemannian manifold (M, g). Accordingly, Ric g (∂Ω) will denote the Ricci curvature tensor of the submanifold ∂Ω equipped with the induced Riemannian metric g.
λ j ≤ (N − 1)κ 2 4 + c N j V ol(M ) 2 N , (4.7)
where c N > 0 depends only on N .
From Theorems 1.7 and 4.6 it immediately follows Corollary 4.8.
Let Ω be a bounded domain of class C 2 in R N +1 such that ∂Ω has only one connected component. Then for all j ∈ N it holds
σ j ≤ a Ω + c 1 2 N j |∂Ω| 1 N , (4.9)
where a Ω > 0 depends on the dimension N , on the maximal mean curvature of ∂Ω, on a lower bound of the Ricci curvature of ∂Ω and on the maximal size of a tubular neighborhood about ∂Ω, and c N > 0 is as in Theorem 4.6 and depends only on the dimension N .
Proof. It suffices just to combine (4.7) with the second inequality in (1.8). We have
σ j ≤ c Ω + c 2 Ω + (N − 1)κ 2 4 + c N j V ol(M ) 2 N −1 ≤ 2c Ω + (N − 2)κ 2 + c 1 2 N j |∂Ω| 1 N −1 , (4.10)
where κ > 0 is such that Ric g (∂Ω) ≥ −(N − 2)κ 2 . Since ∂Ω is a compact submanifold in R N +1 of class C 2 and therefore Ric g (∂Ω) is continuous on ∂Ω, such a κ exists finite. From (3.25) and from the proof of Theorem 1.7, we note that c Ω in (4.10) can be replaced by 1 h + N H∞ 2 . This concludes the proof.
We conclude this section with some remarks.
Remark 4.11. We remark that in (4.9) we have separated the geometry from the asymptotic behavior of the Steklov eigenvalues. We also note that the constant c N in (4.7) (which depends only on the dimension) is not optimal, in the sense that it is strictly greater than the constant appearing in the Weyl's law of λ j , as highlighted in [14], thus the constant c 1 2 N in (4.9) is not optimal in this sense as well.
Remark 4.12. We remark that the constant c Ω in (4.9) may become very big when Ω presents very thin parts (like in the case of dumbell domains), and this can happen also if the curvature remains uniformly bounded (see Figure 2). In the case of convex sets, anyway, it is possible to improve the constant in (1.8)-(1.9) and therefore the bounds (4.9) (see Section 5).
Remark 4.13. We remark that Theorems 1.4 and 1.5 are usually stated for the eigenvalues of the Laplace-Beltrami operator on smooth Riemannian manifolds. Actually, it is sufficient that ∂Ω is a manifold of class C 2 for (1.4) and (1.5) to hold. In fact we can approximate ∂Ω with a sequence ∂Ω ε of C ∞ submanifolds such that ∂Ω = ψ ε (∂Ω ε ), where ψ ε is a diffeomorphism of class C 2 and
Id − ψ ε C 2 (∂Ωε) , Id − ψ (−1) ε C 2 (∂Ω) ≤ ε.
This follows from standard approximation of C k functions by C ∞ (or analytic) functions (see [39]). We also refer to [36,Sec. 4.4] for a more detailed construction of the approximating boundaries ∂Ω ε . It is then standard to prove that the eigenvalues of the Laplace-Beltrami operator on ∂Ω ε pointwise converge the eigenvalues of the Laplace-Beltrami operator on ∂Ω. This immediately follows from the min-max characterization of the eigenvalues (4.2) (we also refer to [32,35] for stability and continuity results for the eigenvalues of elliptic operators upon perturbations of some parameters entering the equation and to [11,12,13] and to the references therein for spectral stability results for eigenvalues upon perturbation of the domain). We also refer to [16,31] and to the references therein for more detailed information on the convergence of Riemannian manifolds and the convergence of the corresponding spectra of the Laplacian.
Moreover, from the fact that Id−ψ ε C 2 (∂Ωε) , Id−ψ
(−1) ε C 2 (∂Ω) ≤ ε, it follows that |∂Ω ε | → |∂Ω| and if κ > 0 is such that Ric g (∂Ω) ≥ −(N − 1)κ 2 , then there exists a sequence κ ε with κ ε → κ as ε → 0 such that Ric gε (∂Ω ε ) ≥ −(N − 1)κ 2 ε . Hence (1.4) and (1.5) hold if Ω is of class C 2 .
Examples: convex domains and balls
In this section we improve the constant in (1.8)-(1.9) and the bounds (4.9) in the case when Ω is a convex and bounded domain of class C 2 and show that the corresponding estimates become sharp when Ω is a ball.
Convex domains
Let Ω be a convex domain of class C 2 in R N +1 . It is well-known that in this case κ i (x) ≥ 0 for all i = 1, ..., N and for all x ∈ ∂Ω. Moreover Theorem 2.6 holds for any h ∈]0, 1/K ∞ [ (see also (2.12) for the definition of K ∞ ). This follows from Blaschke's Rolling Theorem for C 2 convex domains (see [8,22,28,29]) and from [23,Lemma 14.16].
From (3.14) and from the fact that 0 ≤ ρ i (x) ≤ 1 for all x ∈ ω h and i = 1, ..., N + 1 (see also Remark 2.19), it follows that
− Ω |∇v| 2 dx ≤ ω h |∇v| 2 ∆η − 2(D 2 η · ∇v) · ∇vdx ≤ N Ω |∇v| 2 dx. (5.1)
Then, by following the same lines of the proof of Theorems 1.7 and 3.18 and choosingh = 1/K ∞ , it is straightforward to prove the following:
Theorem 5.2.
Let Ω be a bounded and convex domain of class C 2 in R N +1 . Let σ j and λ j , j ∈ N, denote the eigenvalues of problems (1.1) and (1.3) respectively. Let K ∞ be defined by (2.12). Then
i) λ j ≤ σ 2 j + N K ∞ σ j ; (5.3) ii) σ j ≤ K ∞ 2 + K 2 ∞ 4 + λ j .
We note that when Ω is a bounded and convex domain of class C 2 , Ric g (∂Ω) ≥ 0. Accordingly, as a consequence of Theorem 4.6, we have the following:
Corollary 5.4.
Let Ω be a bounded and convex domain of class C 2 in R N +1 . Let σ j and λ j , j ∈ N, denote the eigenvalues of problem (1.1) and (1.3) respectively. Let K ∞ be defined by (2.12). Then
σ j ≤ K ∞ + c 1 2 N j |∂Ω| 1 N .
We note that the geometry of the set enters in the estimate only by means of the maximum of the principal curvatures.
Balls
Let Ω be a ball of radius R in R N +1 . We can suppose without loss of generality that it is centered at the origin. We are allowed to take h = R − δ for all δ ∈]0, R[ through Sections 2,3 and 4. By letting δ → 0, the expression for the vector field given by F in (3.7) simplifies to F (x) = x for all x ∈ Ω. We use F (x) = x in (3.2) and we obtain that for all v ∈ H 2 (Ω) with ∆v = 0 in Ω it holds:
i) ∂Ω |∇ ∂Ω v| 2 dσ = ∂Ω ∂v ∂ν 2 dσ + N − 1 R Ω |∇v| 2 dσ; (5.6) ii) ∂Ω ∂v ∂ν 2 dσ = ∂Ω |∇ ∂Ω v| 2 dσ − N − 1 R Ω |∇v| 2 dσ. (5.7)
We find then that i)
λ j ≤ σ 2 j + (N − 1) R σ j ; (5.8) ii) σ j ≤ (N − 1) 2 4R 2 + λ j − N − 1 2R . (5.9)
Inequality 5.8 follows immediately from (5.6) by the same arguments as in the proof of Theorems 3.18 and 1.7. For (5.9), we note that if ϕ j ∈ H 1 (∂Ω) is an eigenfunction associated with the eigenvalue λ j of (1.3) and if we denote by φ j the unique solution to (4.5), then from (5.7) we have
0 = λ j − ∂Ω ∂φ j ∂ν 2 dσ − N − 1 R Ω |∇φ j | 2 dσ ≤ λ j − Ω |∇φ j | 2 dx 2 − N − 1 R Ω |∇φ j | 2 dx.
This in particular implies
Ω |∇φ j | 2 dx ≤ (N − 1) 2 4R 2 + λ j − N − 1 2
and therefore, by the min-max principle (4.1), the validity of (5.9). Combining (5.8) with (5.9) we immediately obtain the exact relation among the eigenvalues of problems (1.1) and (1.3) on Ω and ∂Ω respectively, without knowing explicitly the eigenvalues. Namely we have the following:
λ j = σ 2 j + (N − 1) R σ j . (5.10)
For the reader convenience, we briefly recall the explicit formulas for the Laplacian eigenvalues on ∂Ω and the Steklov eigenvalues on Ω. An eigenvalue λ of the Laplace-Beltrami operator on ∂Ω is of the form λ = l(l+N −1) .
Throughout the paper we have considered bounded domains of class C 2 . This is a sufficient condition to ensure the validity of Theorems 2.3 and 2.6. Actually, Theorems 2.3 and 2.6 may hold also under lower regularity assumptions on Ω. It is known that the existence of a tubular neighborhood ω h of ∂Ω as in Theorem 2.3 implies that the distance function from ∂Ω is a function of class C 1,1 on ω h . We refer to [21,Ch.7] for a more detailed discussion on sets of positive reach.
We construct now a convex subset Ω of R 3 of class C 1,1 such that the set of points in Ω where the distance function is not differentiable has zero Lebesgue measure (in particular, it is a segment) and such that
3 i=1 ρ i (x) − 1 ≥ 0. Let x = (x 1 , x 2 , x 3 )
denotes an element of R 3 . Let L, R > 0 be fixed real numbers. Let x + 0 := (0, 0, L) and x − 0 := (0, 0, −L). Let Ω ⊂ R 3 be defined by
Ω := Ω 1 ∪ Ω 2 ∪ Ω 3 ,
where
Ω 1 := x ∈ R 3 : |x − x + 0 | < R ∩ x ∈ R 3 : x 3 ≥ L , Ω 2 := x ∈ R 3 : x 2 1 + x 2 2 < R 2 ∩ x ∈ R 3 : −L ≤ x 3 ≤ L and Ω 3 := x ∈ R 3 : |x − x − 0 | < R ∩ x ∈ R 3 : x 3 ≤ −L .
By construction Ω is of class C 1,1 but it is not of class C 2 . Moreover it is convex. We note that we can take h = R − δ for all δ ∈]0, R[. Hence, as in the case of the ball, we can take in (3.2) the vector field defined by
F (x) = x − x + 0 , if x ∈ Ω 1 , (x 1 , x 2 , 0), if x ∈ Ω 2 , x − x + 0 , if x ∈ Ω 3 .
By construction, F is a Lipschitz vector field. Standard computations show that
ρ i (x) = 1,
for all x ∈ Ω 1 ∪ Ω 3 and for i = 1, 2, 3 and λ 1 (x) = 0, λ 2 (x) = λ 3 (x) = 1,
for all x ∈ Ω 2 . Hence 2 i=1 ρ i (x) − 1 ≥ 0 for all x ∈ Ω. Then for the Steklov eigenvalues σ j on Ω we have σ j ≤ c 6 Proof of Theorem 1.10
In this section we prove Theorem 1.10, namely we prove asymptotically sharp upper bounds for Riesz means of Steklov eigenvalues. As a consequence we provide asymptotically sharp upper bounds for the trace of the Steklov heat kernel and lower bounds for Steklov eigenvalues.
Proof of Theorem 1.10. For the Laplacian eigenvalues λ i on ∂Ω the following asymptotically sharp inequality has been shown in [25]: where z 0 := N 2 4 H 2 ∞ and H ∞ is given by (3.17). We note that z 0 ≤ c 2 Ω . It follows from the first inequality of (1.8) of Theorem 1.7 that
∞ j=0 (z − λ j ) + ≥ ∞ j=0 (z − σ 2 j − 2c Ω σ j ) + . (6.2)
Defining a new variable ζ by ζ := z + c 2 Ω − c Ω it is easily shown that (6.2) is equivalent to
∞ j=0 (ζ 2 + 2c Ω ζ − λ j ) + ≥ 2(ζ + c Ω ) ∞ j=0 (ζ − σ j ) + − ∞ j=0 (ζ − σ j ) 2 +
and therefore it is equivalent to the differential inequality
d dζ ∞ j=0 (ζ − σ j ) 2 + ζ + c Ω ≤ ∞ j=0 (ζ 2 + 2c Ω ζ − λ j ) + (ζ + c Ω ) 2 . (6.3)
Integrating the differential inequality (6.3) between 0 and ζ and performing an integration by parts on the right-hand side of the resulting inequality, we obtain ∞ j=0 (ζ − σ j ) 2 + ζ + c Ω ≤ ∞ j=0 (ζ 2 + 2c Ω ζ − λ j ) + 4(ζ + c Ω ) 3 + 3 4 ζ 0 ∞ j=0 (s 2 + 2c Ω s − λ j ) 2 + (s + c Ω ) 4 ds.
We apply estimate (6.1), replace z 0 by c 2 Ω and compute the resulting integral. We get the inequality The estimate is sharp as t tends to zero since (6.5) implies the exact bound lim sup
t→0 + t N ∞ j=0
e −σ j t ≤ (2π) −N B N Γ(N + 1)|∂Ω|.
From (6.5) we immediately obtain Weyl-type lower bounds on Steklov eigenvalues.
Since (j + 1)e −σ j t ≤ ∞ k=0 e −σ k t for all j ∈ N and Γ(N + 3, c Ω t) ≤ Γ(N + 3) we get from (6.5) after optimizing with respect to t the following: Corollary 6.6.
Let Ω be a bounded domain of class C 2 in R N +1 such that ∂Ω has only one connected component. Then for all j ∈ N:
σ j ≥ r N 2πB −1/N N j + 1 |∂Ω| 1 N − c Ω with r N = N eΓ(N + 1) 1/N ≤ 1.
a connected and sufficiently regular ∂Ω (see Remark 4.13) admits an increasing sequence of non-negative eigenvalues of the form 0 = λ 0 < λ 1 ≤ λ 2
Theorem 2.3 (see also [21, Ch.6, Theorem 6.3] and [23, Lemma 14.16]). Throughout the rest of the paper we shall denote byh the maximal possible tubular radius of Ω, namelȳ h := sup {h > 0 : every point in ω h has a unique nearest point on ∂Ω} . (2.4) From Theorem 2.3 it follows that if Ω is of class C 2 suchh exists and is positive. For any h ∈]0,h[ we denote by Γ h the set Γ h := ∂ω h \ ∂Ω.
Figure 1 :
1Tubular neighborhood of a C 2 planar domain.
introduce the functions d and η from ω h to R defined by d(x) := dist(x, Γ h ) and η(x) := d(x) 2 2 .
( 3 .
323) Since(3.23) holds true for all h ∈]0,h[, it is true with h =h. This concludes the proof of ii) and of the theorem. Theorem 3.18 states that for harmonic functions v in Ω the L 2 (∂Ω) norms of ∂v ∂ν and of ∇ ∂Ω v are equivalent. This will be used in the next section to compare the Steklov eigenvalues on Ω with the Laplace-Beltrami eigenvalues on ∂Ω.
Theorem 4 . 6 .
46Let (M, g) be a compact Riemannian manifold without boundary of dimension N such that Ric g (M ) ≥ −(N − 1)κ 2 , κ > 0. Then
Remark 5. 5 .
5Suppose that Ω is a convex and bounded domain of class C 2 such that N i=1 ρ i (x) − 1 ≥ 0 for all x ∈ ω h . Then by (5.1) and by the same arguments in the proof of Theorems 1.
R 2 ,
2with l ∈ N. Let us denote by H l a spherical harmonic of degree l in R N +1 . An eigenfunction associated with the eigenvalue l(l+N −1)R 2 is of the form H l (x/R), x ∈ ∂Ω.Hence the multiplicity of the eigenvalue λ = l(l+N −1) R 2 equals the dimension d l of the space of the spherical harmonics of degree l in R N +1 , namely d l = (2l + N − 1) (l+N −2)! l!(N −1)! . On the other hand, a Steklov eigenvalue σ on Ω is of the form σ = l R with l ∈ N. The corresponding eigenfunctions are the restriction to Ω of the harmonic polynomials on R N +1 of degree l. Clearly the eigenvalues l(l+N −1) R 2and l R have the same multiplicity d l . It is now immediate to see that formula (5.10) holds true.
) −N B N |∂Ω|(ζ + c Ω ) inequality (1.11) of Theorem 1.10 yields the following upper bound on the trace of the heat kernel for the Steklov operator: Corollary 6.4. Let Ω be a bounded domain of class C 2 in R N +1 such that ∂Ω has only one connected component. Then ∞ j=0 e −σ j t ≤ 1 (N + 1)(N + 2) (2π) −N B N |∂Ω|t −N e c Ω t Γ(N + 3, c Ω t) (6.5) for all t > 0, where Γ(a, b) = ∞ b t a−1 e −t dt denotes the incomplete Gamma function.
+1 be a bounded domain with boundary ∂Ω of class C 2 such that ∂Ω has only one connected component. Then for all
Theorem 3.18. Let v ∈ H 2 (Ω) be such that ∆v = 0 in Ω and normalized such that ∂Ω v 2 dσ = 1. Then it holdsi)
∂Ω
|∇ ∂Ω v| 2 dσ ≤
∂Ω
∂v
∂ν
2
dσ + 2c Ω
∂Ω
∂v
∂ν
2
dσ
1
2
;
(3.19)
ii)
∂Ω
∂v
∂ν
2
dσ
1
2
≤ c Ω + c 2
Ω +
∂Ω
|∇ ∂Ω v| 2 dσ.
(3.20)
Proof. Let h ∈]0,h[. We start by proving i). From Lemmas 3.1 and 3.10 we have
∂Ω
|∇ ∂Ω v| 2 dσ =
∂Ω
∂v
∂ν
2
dσ +
1
h
ω h
|∇v| 2 ∆η − 2(D 2 η · ∇v) · ∇vdx
≤
∂Ω
∂v
∂ν
2
dσ +
1
h
+ NH ∞
Ω
|∇v| 2 dx
=
∂Ω
∂v
∂ν
2
dσ +
1
h
+ NH ∞
∂Ω
v
∂v
∂ν
dx
≤
∂Ω
∂v
∂ν
2
dσ +
1
h
+ NH ∞
∂Ω
v 2 dσ
1
2
∂Ω
∂v
∂ν
2
dσ
1
2
=
∂Ω
∂v
∂ν
2
dσ +
1
h
+ NH ∞
∂Ω
∂v
∂ν
2
dσ
1
2
, (3.21)
where we have used the following Green's identity
Ω
∆vvdx =
∂Ω
∂v
∂ν
vdσ −
Ω
|∇v| 2 dx,
the fact that ∆v = 0 in Ω, Hölder's inequality and the fact that ∂Ω v 2 dσ = 1.
Since (3.21) holds true for all h ∈]0,h[, it is true with h =h. This proves i). We
repeat a similar argument for ii). We have
∂Ω
∂v
∂ν
2
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| [] |
[
"Many-body topological invariants from randomized measurements in synthetic quantum matter",
"Many-body topological invariants from randomized measurements in synthetic quantum matter"
] | [
"Andreas Elben ",
"Jinlong Yu ",
"Guanyu Zhu ",
"† ",
"Mohammad Hafezi ",
"Frank Pollmann ",
"Peter Zoller ",
"Benoît Vermersch "
] | [] | [] | Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by siteresolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays. Many-body topological invariants from randomized measurements in synthetic quantum matter. Sci. Adv. 6, eaaz3666 (2020). | 10.1126/sciadv.aaz3666 | null | 186,206,927 | 1906.05011 | f604d7210d5d628ffd6cc63dc2a024e543b5c2cb |
Many-body topological invariants from randomized measurements in synthetic quantum matter
Andreas Elben
Jinlong Yu
Guanyu Zhu
†
Mohammad Hafezi
Frank Pollmann
Peter Zoller
Benoît Vermersch
Many-body topological invariants from randomized measurements in synthetic quantum matter
Elben et al., Sci. Adv. 2020; 6 : eaaz3666 10 April 2020 S C I E N C E A D V A N C E S | R E S E A R C H A R T I C L E 1 of 8 P H Y S I C S
Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by siteresolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays. Many-body topological invariants from randomized measurements in synthetic quantum matter. Sci. Adv. 6, eaaz3666 (2020).
INTRODUCTION
There is an increasing interest in realizing topological quantum phases in synthetic quantum systems (1)(2)(3)(4)(5), including ultracold atoms in optical lattices (2), Rydberg atoms (3), trapped ions (4), and superconducting qubits (5). These experimental platforms offer unique possibilities for preparing, controlling, and probing quantum states, with prospects of studying these exotic states of matter, e.g., the fractional Hall state (6,7), and in light of possible applications such as topological quantum computing (8). The characterization and identification of topological phases in an experimental setting represents, however, a substantial challenge: In contrast to symmetry-breaking phases of Landau's theory with local order parameters (9), topological phases are characterized by global properties, which cannot be revealed with local probes. Thus, measurement protocols need to be developed to access these global properties. For noninteracting systems, the measurement of topological invariants (such as the Chern number) has been achieved in seminal experiments in cold atom setups (10)(11)(12)(13), microwave networks (14), and photonic systems (15). Below, we address the generic interacting case, and we present measurement protocols that allow us to access many-body topological invariants (MBTIs) of interacting topological states with global symmetries (16,17).
MBTIs are highly nonlocal quantized correlators of the manybody wave function that have been originally derived in the context of the description of symmetry-protected topological (SPT) order (18), and in particular from the classification of unidimensional bosonic SPT phases (16,17,19). An MBTI identifies from a manybody wave function the projective representation of a global symmetry (16,17). For any realization of a topological state with a given symmetry, for instance, the spatial reflection symmetry or the time-reversal symmetry, the corresponding MBTI takes a non-zero quantized value. MBTIs can be considered as generalizations of string order parameters that were introduced (20) and measured (21,22) to detect SPT phases protected by internal symmetries. MBTIs can particularly identify all one-dimensional bosonic SPT phases, even in the absence of internal symmetries, and therefore of string order (17,23). MBTIs are not restricted to the description of SPT phases: They have been now understood in the general mathematical framework of topological quantum field theory (17,24,25), suggesting that they can identify many types of topological phases beyond SPT orders. In particular, recent theoretical works have shown that MBTIs can identify fermionic SPT phases (24,26) and topological quantum phase transitions (27) and can also distinguish various symmetry-enriched topological (SET) phases with intrinsic topological orders (28,29). Whereas MBTIs have become key quantities to characterize topology in synthetic quantum systems, the question of their measurement has remained so far elusive.
Our approach to measure MBTIs consists in using the information contained in statistical correlations between randomized measurements. These measurements are realized by applying to a quantum state a sequence of random unitary operations before performing projective measurements. Recently, randomized measurement protocols have been developed to measure entanglement (30,31), including an experimental demonstration in a trapped-ion quantum simulator (32), and out-of-time order correlators (33). Here, our approach is based on local random unitaries that can be implemented in experiments with high fidelities (32), and the key idea is to use distributions of such random unitaries with different symmetries. From the statistics of such "symmetric" randomized measurements, one can extract the MBTI associated with a particular symmetry. Our analysis of the protocol, including the study of statistical errors, shows that MBTIs can be measured via our protocols with current technology in various spin systems, such as Rydberg atoms, trapped ions, and superconducting qubits, and can be used to experimentally classify interacting many-body topological quantum phases.
Our article is organized as follows. Having in mind current experimental possibilities, e.g., with Rydberg atom quantum simulators (22), we study a model Hamiltonian with SPT phases as ground states. We consider MBTIs associated with reflection and timereversal symmetries to identify the trivial/nontrivial topology of the SPT phases as well as the symmetry-broken phase, and present the corresponding measurement protocols as experimental recipes; the other MBTIs and corresponding protocols are presented in the Supplementary Materials. We then discuss the role of statistical errors and imperfections in our scheme. We also illustrate our protocols via two physical examples that can be realized in experiments: We show how to monitor the dynamical building-up of topology by these MBTIs during adiabatic state preparation, and we also discuss how the breaking and protection mechanisms of symmetries can be probed experimentally. Last, we discuss how our protocols can be applied beyond the case of SPT phases.
RESULTS
Model
For concreteness, we present our approach in the context of the extended bosonic Su-Schrieffer-Heeger (SSH) model ( Fig. 1A) (22,(34)(35)(36)(37)
H eSSH = J ─ 2 ∑ i=1 N/2 ( σ 2i−1 x σ 2i x + σ 2i−1 y σ 2i y + δσ 2i−1 z σ 2i z ) + J′ ─ 2 ∑ i=1 N/2−1 ( σ 2i x σ 2i+1 x + σ 2i y σ 2i+1 y + δσ 2i z σ 2i+1 z )(1)
Here, i ( = x, y, z) are the Pauli matrices for the spin state at site i. J and J′ are alternating nearest-neighbor spin-exchange coefficients, and denotes the exchange anisotropy. The case of = 1 corresponds to the bond-alternating Heisenberg model (35,36), whereas the case of = 0 corresponds to the bosonic version of the (non-interacting) SSH model (34), as realized recently with Rydberg atoms (22). Note that, expect for = 0 and = 1, the model is generally not integrable and thus has no single-particle correspondence. The alternating spinexchange coefficients can be engineered, e.g., by loading bosonic atoms into optical superlattices (37) or dimerized optical tweezer arrays (22), or by coupling bosonic atoms to dynamical gauge fields (38,39). As a final remark, we emphasize that all protocols presented below can also be generalized to other spin Hamiltonians, e.g., spin-1 Haldane chain (40), straightforwardly.
As shown below, the model Hamiltonian in Eq. 1 hosts three different phases: a trivial phase, a topological Haldane phase (40), and a symmetry-broken antiferromagnetic phase. The trivial and topological phases are SPT phases protected by any one of the following three symmetries (23,41): reflection (inversion) symmetry at the center bond, time-reversal symmetry, and dihedral group D 2 of -rotations of spins around the x, y, and z axes.
Partial reflection invariant
We now show how to measure MBTIs via randomized measurements. First, SPT phases protected by reflection symmetry can be classified using the partial reflection MBTI ˜ Z ℛ = Z ℛ / √ __________________ [ Tr ( ρ I 1 2 ) + Tr ( ρ I 2 2 ) ] / 2 (17), with Z ℛ = Tr ( ρ I ℛ I ) (2) Here, I = Tr S − I (|⟩⟨|) is the reduced density matrix of a manybody quantum state |⟩, and the interval I = I 1 ∪ I 2 consists of two partitions I 1 and I 2 , each with n sites; S denotes all the sites of the system. The nonlocal operator ℛ I "spatially swaps" I 1 and I 2 with respect to the reflection center. On every basis state |s I ⟩ = |s 1 ,s 2 , …, s 2n ⟩ (s i = ↑, ↓ for i ∈ I), it acts as ℛ I |s I ⟩ = |s 2n , s 2n − 1 , …, s 1 ⟩ ≡ |ℛ I (s I )〉. This operation is graphically shown in Fig. 1B, where the state of each site of I, represented as a blue line, is "contracted" with the state of the mirror symmetric site.
The MBTI ˜ Z ℛ probes the action of the reflection symmetry on the many-body state |⟩. Using tensor-network theory, one can show analytically that, for the ground state of a gapped many-body Hamiltonian (e.g., H eSSH ), ˜ Z ℛ approaches a quantized value in the thermodynamic limit n, N → ∞ (17). The typical value of n required to achieve convergence is determined by the correlation length in the system and is discussed in detail below. For our model Eq. 1, the phase diagram evaluated by the MBTI ˜ Z ℛ , calculated numerically using the density matrix renormalization group (DMRG) method (see Materials and Methods for details), is shown in Fig. 1C. Three phases can be identified therein: (i) a phase with antiferromagnetic order where reflection symmetry is spontaneously broken with ˜ Z ℛ = 0 , (ii) the trivial SPT phase with ˜ Z ℛ = + 1 , and (iii) the nontrivial SPT phase with ˜ Z ℛ = − 1 . The MBTI ˜ Z ℛ , which is a highly nonlocal and nonlinear functional of the reduced density matrix I , can be measured with randomized measurements, with the following recipe (as illustrated in Fig. 1D): (i) One first prepares the ground state |⟩ via, e.g., adiabatic state preparation (see a later section for details). (ii.a) One applies to |⟩ a unitary operation U ℛ of the form
U ℛ = ⊗ i=1 2n U i , with U i = U 2n−i+1 .
The unitaries U i (i = 1,2, …, n) are drawn randomly from the circular unitary ensemble (CUE) defined on the local Hilbert spaces of individual spins. This type of random unitaries with spatial reflection symmetry (i.e., with a configuration U 1 U 2 …U n | U n …U 2 U 1 as shown schematically in Fig. 1D) will be essential to be able to extract ˜ Z ℛ from randomized measurements. Each local unitary U i can be decomposed in products of spin rotations along two axes (x,z) and can thus be generated with high fidelity in quantum simulators with single-site control, as also shown in recent experiments (32). Note that the impact of potential imperfections, such as miscalibration and decoherence, has been studied in detail in (42), showing the robustness and the applicability of protocols relying on randomized measurements in state-of-the-art quantum devices based on Rydberg atoms, trapped ions, or superconducting qubits. (ii.b) One measures the occupation probabilities P U ℛ ( s I ) = 〈 s I | U ℛ I U ℛ † | s I 〉 of the basis states s I , by performing projective measurements in the basis s I . (iii) One repeats (i) to (ii) for many independently sampled random unitaries U ℛ . Given the set of outcome probabilities P U ℛ (s I ), one obtains first Z ℛ from
Z ℛ = 2 n ∑ s I (− 2) − 1 _ 2 D[ s I , ℛ I ( s I )] ¯ P U ℛ ( s I )(3)Tr ( I 1 2 ) = 2 n ∑ s I 1 ,s ′ I 1 (− 2) −D[ s I 1 ,s ′ I 1 ] ¯¯ P U ℛ ( s I 1 ) P U ℛ (s ′ I 1 ) (4) 3 of 8 with the reduced probabilities P U ℛ ( s I ) = Tr ( | s I 1 〉〈 s I 1 | U ℛ I U ℛ † ) . Thus,
we obtain the normalized MBTI from the second-order correlations of randomized measurements, implemented with local random operations with a distribution that is tailored to identify a certain symmetry (here, the reflection symmetry) of the many-body quantum state. This is the key idea in our approach, and we show below how to apply it to measure any MBTI. For illustration, we show in Fig. 1E the value of ˜ Z ℛ (i) calculated from the DMRG method (line) and (ii) estimated from simulated randomized measurements (dots). They coincide with each other within the statistical errors that originate from the finite number of unitaries N U and the finite number of projective measurements per unitary N M . A detailed discussion about the statistical errors and imperfections for the MBTIs ( ˜ Z ℛ here and ˜ Z T below) estimated from randomized measurements can be found in a later section and the Supplementary Materials.
Partial time-reversal invariant
We now present the protocol to measure the MBTI associated with the time-reversal symmetry ˜
Z T = Z T / ( [ Tr ( I 1 2 ) +Tr ( I 2 2 ) ] / 2 ) 3/2 (17, 25), with Z T = Tr ( ρ I u T ρ I T 1 u T † )(5)
Here, T 1 denotes the partial transpose operation on the partition I 1 , and u T = ∏ i∈ I 1 i y is the unitary part of the time-reversal operator. The contraction operation resulting in Z T is illustrated graphically in Fig. 2A.
The MBTI ˜ Z T is a nonlinear functional of two copies of the (partially transposed) density matrix I , which can be measured via the following recipe (Fig. 2B). After (i) the state preparation, we perform two experiments: (ii.a.1) In the first experiment, we apply
from cross-correlations of the two experiments. In addition, the purity to normalize Z T to ˜ Z T is obtained from the same experimental data using the relation Eq. 4. Equation 6, which is also proven in Materials and Methods, shows that the partial time-reversal MBTI can be accessed from correlations between measurements using random unitary operations, which are complex conjugated. In Fig. 2C, we compare values of ˜ Z T obtained with the DMRG method with the ones estimated from finite number of randomized measurements. We see similar behavior of ˜ Z T in Fig. 2C compared with the one of ˜ Z ℛ as in Fig. 1E but with larger error bars and deviation. This is because of the fact that the statistical errors scale differently as functions of N U , N M , and n (see the Supplementary Materials). The deviation and error bars can be reduced by increasing both N U and N M . Moreover, the solid lines in Figs. 1E and 2C are similar for the current case, because both the reflection and time-reversal symmetries are present in the Hamiltonian. The MBTIs can behave completely different for the case that one of the protecting symmetries is broken but the many-body ground state is still topological (see below and the section on 'Probing the breaking and the protection of symmetries').
In Fig. 2D, we also show that by extracting ˜ Z T (or similarly, ˜ Z ℛ , which is not shown here for conciseness) for different n, one can measure the correlation length of SPT phases, i.e., the characteristic length above which MBTIs become quantized. In particular, one can identify quantum critical points separating different SPT phases from the divergence of .
The two examples given above illustrate how to access MBTIs from the statistics of measurements performed after correlated local random unitary operations. In the Supplementary Materials, we show how to access MBTIs for internal symmetries and combination of symmetries. We also show how to identify the breaking/protection of different symmetries in a later section. Combined together, they provide a complete set of protocols to experimentally probe the classification of one-dimensional bosonic SPT phases.
Statistical errors and imperfections
Having described our main results relating randomized measurements to the MBTIs ˜ Z ℛ and ˜ Z T , we now comment on various potential sources of errors in implementing our protocol. First, statistical errors are due to the finite number of repetitions of the experiment used to estimate the statistical correlations between randomized measurements. As detailed in the Supplementary Materials, we find that the typical required number of measurements to access MBTIs within a given accuracy (scaling as 2 1.5n to access Z ℛ for instance) are very similar to the requirements to measure state purities (31,32) and thus compatible with state-of-the-art experimental platforms of Rydberg atoms, trapped ions, and superconducting qubits with high repetition rates. Randomized measurements also feature a natural robustness with respect to decoherence, readout errors, and errors in the implementations of random unitaries (33,42), because they are based on extracting relevant quantities from ensemble averages (and not from individual measurements). We thus expect our protocols to allow faithful measurements of MBTIs in various experimental platforms. In the following, we illustrate our protocols by means of two important applications: the dynamical building-up of nontrivial topology during the adiabatic preparation of an SPT phase and the identification of the protecting symmetry group.
Monitoring the dynamical building-up of topology
The MBTIs ˜ Z ℛ and ˜ Z T can be defined for an arbitrary many-body quantum state |⟩ besides the ground state as described above. Thus, we can also use the presented measurement protocols to monitor the preparation of an SPT state |(t)⟩ as a function of time, which facilitates the visualization of the dynamical building-up of topology experimentally. For concreteness, we consider the adiabatic state preparation with a time-dependent Hamiltonian
H(t ) = H eSSH + f (t ) H N e el(7)
where H N e el = ∑ i (− 1) i i z is a staggered magnetic field term with a strength ≫ J′, J. We always set the function f(t) to satisfy f(t = 0) = 1 and f(t = t F ) = 0. At time t = 0, the system is initialized in the Néel state |(t = 0)⟩ = |↓↑↓ …⟩. As an example, we adopt the function f(t) = (t/t F − 1) 4 to adiabatically drive the system to the ground state of H eSSH at the final time t = t F . Our protocols give access to the time-dependent values of MBTI ˜ Z T (t ), ˜ Z ℛ (t) , obtained using the experimental recipe described above with random unitaries applied on the time-dependent many-body quantum state |(t)⟩. We illustrate the emergence of quantized values of the MBTI ˜ Z ℛ (t) [the results for ˜ Z T (t) are similar and are not shown for conciseness], associated with the preparation of the SPT phases, in Fig. 3A. Note that the preparation time Jt F = 20 is compatible with the coherence time achieved in the Rydberg experiment realizing the Haldane phase of the bosonic SSH model (22). As shown in Fig. 3B, the values of ˜ Z ℛ ( t F ) at the end of the preparation t = t F can be used to detect the quality of the preparation of an SPT phase: For Jt F ≫ 1, the preparation is perfectly adiabatic, and the values of the MBTI correspond to the ones of the ground state wave function (as presented in Figs. 1 and 2).
For Jt F ∼ 1, the correlations in the wave function do not extend over the full system, as in the true SPT ground state, but only extend to certain characteristic length scale n c . Consequently, for n ≫ n c , the many-body invariant tends to zero. We expect a similar behavior for a scenario where |(t = t F )⟩ is replaced by a thermal state, and n c by a "thermal length" describing the range of correlations. Our protocols can also be used to probe topology in non-equilibrium systems (43).
Probing the breaking and the protection of symmetries
The MBTIs ˜ Z ℛ and ˜ Z T behave similarly (cf. Figs. 1E and 2C) for the model Hamiltonian H eSSH in Eq. 1, because both reflection and time-reversal symmetries are respected. In addition to identifying the topology, measuring MBTIs also provides us with the ability to experimentally study the protection mechanism of SPT phases. In particular, SPT order can still exist in the absence of certain internal symmetries (thus, string order being absent), provided at least one protecting symmetry is present (41). To illustrate this effect with MBTIs, we add here the term
H B = B ∑ j=1 N−1 ( j x j+1 z − j z j+1 x )(8)
to the original Hamiltonian H eSSH . In the Hamiltonian H = H eSSH + H B , the reflection and D 2 symmetries are explicitly broken, but the time-reversal symmetry is respected (43). Thus, the ground state of H = H eSSH + H B can still exhibit nontrivial SPT order, protected solely by the time-reversal symmetry. This is encoded in the values of the MBTIs and can thus be revealed experimentally via our protocols. As shown in Fig. 4, the partial time-reversal MBTI ˜ Z T converges to ±1 for n → ∞, whereas the partial reflection MBTI ˜ Z ℛ approaches 0 as n → ∞.
DISCUSSION
To conclude, the use of randomized measurements to probe topological properties of the wave function is a new paradigm that enables the experimental classification of many-body topological quantum phases. While we have focused our study on unidimensional SPT phases, our protocols also open the possibilities for probing two-dimensional SPT phases (44), as well as identifying different SET phases (28,29,45,46). The accompanied symmetries, e.g., reflection and time-reversal symmetries, for SET phases (29) can be distinguished via the same MBTIs as for SPT phases (defined in compactified one-dimensional geometries) and thus can also be probed via randomized measurements. We also note that our protocolpresented in this work for spin systems-can also be realized in fermionic systems (24) via global random unitaries implemented for example with random quenches in Hubbard systems (31).
As a future direction, our work also suggests that anyonic statistics describing the essence of topologically ordered states can be accessed via randomized measurements, extending, in particular, approaches based on impurities (47) or linear response (48) to measure the manybody Chern number of fractional quantum Hall states. Moreover, modular matrices revealing anyonic statistics (49) can be expressed as spatial reflection operators in a form analog to ˜ Z ℛ on torus geometries (50) and could thus be measured via randomized measurements.
MATERIALS AND METHODS
In this part, we present the proofs of Eqs. 3 to 6, relating MBTIs to statistical correlations of randomized measurements, together with the details on our DMRG and TEBD (time-evolving block decimation) simulations. Here, we also focus on the case of spin-1/2 systems. Our formulas can, however, be extended straightforwardly to the cases with higher internal dimensions (spins 1, 3/2, etc.).
Random unitary calculus
We begin by summarizing elementary properties of random unitaries from the CUE. We discuss the minimal case of two spins, each with Hilbert space ℋ. These can be either (i) two spins located at different lattice sites in a single many-body system (partial inversion invariant) or (ii) two spins located at the same site but realized in two different, sequentially performed, experiments (time-reversal invariant). Given a two-spin operator O acting on both spins with total Hilbert space H ⊗2 , we define the unitary twirling channel
(O ) ≡ ¯¯¯ U † ⊗ U † OU ⊗ U(9)
where … ¯ denotes the average over random unitaries U taken from the CUE (i.e., the average with respect to the Haar measure on the group of unitary matrices on ℋ). Using the two-design identities of the CUE, we find (51).
(O ) = 1 ─ 3 ( Tr [ O ] − 1 ─ 2 Tr [ O ] ) 1 2 + 1 ─ 3 ( Tr [ O ] − 1 ─ 2 Tr [ O ] )
where = ∑ s,s′ | s, s′〉〈s′, s| denotes the swap operator. We also define the closely related isotropic twirling channel (52)
(O ) ≡ ¯¯¯ U † ⊗ ( U * ) † OU ⊗ U * = [( O T 2 )] T 2(10)
Here, ( · ) T 2 denotes the partial transpose with respect to the second spin. For the following proofs, we will use an operator ˜ O ≡ 2 ∑ s, s ′ (− 2) −D[s, s ′ ] | s, s ′ ⟩⟨s, s ′ | , which is diagonal in the computational basis, and fullfills (51)
( ˜ O ) = ,(11)Ψ( ˜ O ) = T 2 = ∑ s,s′ | s, s〉〈 s ′ , s ′ | ≡(12)
In the following, we show how to use the identities (Eqs. 11 and 12) to prove Eqs. 3 to 6 relating randomized measurements and MBTIs.
Partial reflection invariant from randomized measurements
The MBTI Z ℛ is inferred from statistical correlations of randomized measurements, performed on a quantum state I , which are implemented by applying spatially correlated local random unitaries of the form
U ℛ = ⊗ i=1 2n U i , with U i = U 2n−i+1 for i = 1, …, n.
To prove Eq. 3, we first note that its right-hand side can be rewritten as an expectation value of an operator
O R E ℛ ≡ 2 n ∑ s I (− 2) − 1 _ 2 D[ s I , ℛ I ( s I )] ¯ P U ℛ ( s I ) = Tr [ ¯¯ U ℛ † O ℛ U ℛ I ] = Tr [ ⊗ i=1 n ¯¯¯¯ ( U i † ⊗ U i † ) O ℛ,i ( U i ⊗ U i ) I ] with O ℛ = ⊗ i=1 n O ℛ,i ,¯¯¯¯ ( U i † ⊗ U i † ) O ℛ,i ( U i ⊗ U i ) = ℛ I[i]
and therefore obtain
E ℛ = Tr [ ⊗ i=1 n ℛ I [ i ] ρ I ] = Z ℛ
Partial time-reversal invariant from randomized measurements
The MBTI Z T is inferred from the statistical correlations of correlated randomized measurements on two (sequential) experiments, both preparing a quantum state I . These are implemented by applying to the sites in an interval I = I 1 ∪ I 2 local random unitaries U T (1) = U I 1 u T ⊗ U I 2 (experiment 1) and U T (2) = U I 1 * ⊗ U I 2 (experiment 2)
with U I 1,2 = ⊗ i∈ I 1,2 U i and u T = ⊗ i∈ I 1 i y , respectively. To prove Eq. 6, we rewrite its right-hand side as
E T ≡ 2 2n ∑ s I , s I ′ ( − 2 ) −D [ s I , s I ′ ] ¯¯¯ P U T ( 1 ) ( s I ) P U T ( 2 ) ( s I ′ ) = Tr [ ¯¯¯ ( U T ( 1 ) ) † ⊗ ( U T ( 2 ) ) † O T U T ( 1 ) ⊗ U T ( 2 ) ( ρ I ⊗ ρ I ) ] = Tr [ ⊗ i∈ I 1 ¯¯¯¯ U i † ⊗ ( U i * ) † O T,i U i ⊗ U i * ⊗ i∈ I 2 ¯¯¯ U i † ⊗ U i † O T,i U i ⊗ U i ( ˜ ρ I ⊗ ρ I ) ](14)
Here, we have defined ~ ρ I ≡ ( u T ⊗ 1 I 2 ) ρ I ( u T † ⊗ 1 I 2 ) and used the
Details on the DMRG and TEBD simulations DMRG and TEBD simulations for the ground states and timedependent states, respectively, were realized using the ITensor Library (http://itensor.org) in the framework of matrix product states. To get a ground state, the model was numerically solved with open-boundary conditions, with an additional small pinning field acting on the first site p 1 z , with p = 0.05J, to select one of the two degenerate ground states present in the topological phase for open boundary conditions (53). Note that in experiment with large system size N, the system would always choose one of the degenerate ground states because a cat state (i.e., the superposition of the two degenerate ground states) is always fragile to perturbations (as simulated by the small pinning field). We used a maximum bond dimension of D = 512.
The quasi-exact MBTIs were extracted from direct contractions of the matrix product states representing the ground states (as shown by the solid lines in Figs. 1, 2, and 4). The estimations for randomized measurements were obtained using a sampling algorithm of the occupation probabilities P U (s) for matrix product states (54). The simulations for the time-dependent state for the adiabatic state preparation (as in Fig. 3) were realized via the TEBD algorithm with a time step Jdt = 0.001 and a maximum bond dimension D = 512.
Fig. 1 .
1Measuring the MBTI Z ℛ for the extended bosonic SSH model. (A) Schematic illustration of the model Eq. 1, where the nearest-neighbor spin-exchange coefficients alternate between the bonds. (B) The partial reflection invariant Z ℛ (Eq. 2) is defined as the expectation value of a partial reflection operator ℛ I (visualized by the blue lines) for the many-body state |〉. The dashed line between the intervals I 1 and I 2 indicates the reflection center. (C) In terms of the normalized invariant Z ℛ , the full-phase diagram of the extended bosonic SSH model is revealed here for a system size of N = 48 spins and n = 6 reflected pairs of spins. We find three phases with different quantized values of Z ℛ . (D) Protocol to measure Z ℛ via statistical correlations between randomized measurements, implemented with local random unitaries applied symmetrically around the central bond. (E) The results of simulated experiments allow us to identify topological phase transitions. The solid lines are results from DMRG, whereas the dots with error bars represent estimations from simulated randomized measurements with N U = 512 unitaries and N M = 256 measurements per unitary. U T (1) = U I 1 u T ⊗ U I 2 , with U I 1 = ⊗ i=1 n U i and U I 2 = ⊗ i=n+1 2n U i , each U i being taken independently from the CUE. (ii.b.1) We measure the probabilities P U T (1) ( s I ) (see the left panel of Fig. 2B). (ii.a.2) In a second experiment, we use the unitaries U T (2) = U I 1 * ⊗ U I 2 . (ii.b.2) We measure P U T (2) ( s I ) (see the right panel of Fig. 2B). (iii) We repeat the two experiments (i and ii) with different unitaries U i and estimate Z T = 2 2n ∑ s I ,s ′ I (− 2) −D[ s I ,s ′ I ] ¯¯¯ P U T (1) ( s I ) P U T (2) (s ′ I )
Fig. 2 .
2Probing the MBTI Z T with randomized measurements. (A) Graphical representation of the definition of the time-reversal invariant Z T (Eq. 5) involving partial transpose (red lines) and partial swap (blue lines) operations. (B) Experimental protocol to measure Z T with two experiments, which are correlated using randomized measurements. To account for the anti-unitarity of the time-reversal symmetry, the local random unitaries applied in I 1 (red) in the two experiments are complex conjugate to each other. (C) Simulated measurements of Z T (dots with statistical error bars, with N U = 768, N M = 512), revealing the topological phase transitions in the extended bosonic SSH model as a function of J′ = J for two values of . Solid lines are calculated with the DMRG method, in a system with N = 48 sites, and n = 6 per interval I 1 and I 2 . (D) Z T converges as a function of the partition size n to the quantized values ±1 for the case of = 0.25. Different colors represent different values of J′ = J. Inset: The divergence of the corresponding correlation length , extracted from an exponential fit on the first three values of n, can be used to detect the quantum critical point between the topological trivial (with Z T = 1) and nontrivial (with Z T = −1) phases.
Fig. 3 .
3Monitoring the adiabatic preparation of an SPT state. (A) Starting from a trivial Néel state without reflection symmetry Z ℛ (t) , the ground state of H eSSH is adiabatically prepared. This is monitored by the evolution of Z ℛ (t) , which evolves to quantized values ±1 at late times. The dynamical buildup of long-range SPT orderfor intermediate times up to a certain length scale-is indicated at intermediate times by the increasing magnitude of Z ℛ (t) for decreasing number n of reflected pairs of spins. Here, we set Jt F = 20. (B) The convergence of Z ℛ ( t F ) to ±1 as a function of the total preparation time t F indicates that, for sufficiently long preparation times, the ground states in trivial and topological states are prepared with high fidelity. For the simulations, we use the time-evolving block decimation (TEBD) algorithm (as detailed in Materials and Methods) and set the parameters as = 0.25, = 40J, and N = 48.
Fig. 4 .
4Detecting the protecting symmetries for the SPT states. In the presence of the symmetry-breaking perturbation H B (Eq. 8), the topological phase in the modified Hamiltonian H = H eSSH + H B is (only) protected by the time-reversal symmetry. (A) This is detected by the partial time-reversal MBTI Z T -converging to the quantized values ±1 for increasing n-which still identifies the topological phase transition. (B) On the contrary, the partial reflection MBTI Z ℛ -approaching 0 with increasing n-shows that the reflection symmetry is explicitly broken for a nonzero B in Eq. 8. We choose B = 0.1J, = 0.3, and N = 48.
(i
spatial) tensor product structure of the operatorO T = ⊗ i ∈ I O T , i with O T,i = 2 ∑ s i , s i ′ ( − 2 ) −D [ s i , s i′ ] | s i 〉〈 s i | ⊗ | s i ′ 〉〈 s i ′ |(15)Using Eqs. 11 and 12 with the identification i → , i → , and O T,i → ˜ O , we thus directly obtainE T = Tr [ ⊗ ( ˜ ρ I ⊗ ρ I ) ] = Tr [ ( ˜ ρ I ) T I 1 ρ I ] = Z T
I|s i ≠ s 2n−i + 1 } is the Hamming distance between |s I ⟩ and |ℛ I (s I )〉. Equation 3 can be proven using the twodesign identities of the CUE (see Materials and Methods) and shows that the MBTI Z ℛ can be directly extracted from the statistics of randomized measurements. Second, the purity Tr ( I1 2 ) (and similarlyHere, ⋯
¯ denotes the ensemble average over the random unitaries
and D[s I , ℛ(s I )] ≡ #{i ∈ Tr [ I 2
2 ] ) is estimated using the relation (31, 32)
which is a tensor product of operatorsO ℛ,i = 2 ∑ D[ s I[i] , ℛ I ( s I[i] )] | s I[i] 〉〈 s I[i] |(13)acting on pairs of spins I[i] = (i, 2n − i + 1). We also used the independence of the unitaries U i and U i′ (for i ≠ i′ with i, i′ = 1, …, n)s I[i]
(− 2) − 1 _
2 A
B
′
′
applied to different pairs of spins I[i] and I[i′], respectively. Using Eq. 11 with the identification ℛ I[i] → , and O ℛ,i → ˜ O , we find
SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/6/15/eaaz3666/DC1
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| [] |
[
"A NEW SUB-STELLAR COMPANION AROUND THE YOUNG STAR HD 284149",
"A NEW SUB-STELLAR COMPANION AROUND THE YOUNG STAR HD 284149"
] | [
"Mariangela Bonavita \nDraft version\n\n",
"Sebastian Daemgen \nDraft version\n\n",
"Silvano Desidera \nDraft version\n\n",
"Ray Jayawardhana \nDraft version\n\n",
"Markus Janson \nDraft version\n\n",
"David Lafrenière \nDraft version\n\n"
] | [
"Draft version\n",
"Draft version\n",
"Draft version\n",
"Draft version\n",
"Draft version\n",
"Draft version\n"
] | [] | Even though only a handful of sub-stellar companions have been found via direct imaging, each of these discoveries has had a tremendous impact on our understanding of the star formation process and the physics of cool atmospheres. Young stars are prime targets for direct imaging searches for planets and brown dwarfs, due to the favorable brightness contrast expected at such ages and also because it is often possible to derive relatively good age estimates for these primaries. Here we present the direct imaging discovery of HD 284149 b, a 18 − 50 M Jup companion at a projected separation of 400 AU from a young (25 +25 10 Myr) F8 star, with which it shares common proper motion | 10.1088/2041-8205/791/2/l40 | [
"https://arxiv.org/pdf/1406.7298v1.pdf"
] | 119,166,822 | 1406.7298 | c54e98d286347c818295d72cb08734045d4cd720 |
A NEW SUB-STELLAR COMPANION AROUND THE YOUNG STAR HD 284149
27 Jun 2014 Draft version July 1, 2014 July 1, 2014
Mariangela Bonavita
Draft version
Sebastian Daemgen
Draft version
Silvano Desidera
Draft version
Ray Jayawardhana
Draft version
Markus Janson
Draft version
David Lafrenière
Draft version
A NEW SUB-STELLAR COMPANION AROUND THE YOUNG STAR HD 284149
27 Jun 2014 Draft version July 1, 2014 July 1, 2014Preprint typeset using L A T E X style emulateapj v. 04/17/13Subject headings: stars: individual (HD 284149)(stars:) planetary systems(stars:) brown dwarfsstars: pre-main sequencemethods: observationalinstrumentation: adaptive op- tics
Even though only a handful of sub-stellar companions have been found via direct imaging, each of these discoveries has had a tremendous impact on our understanding of the star formation process and the physics of cool atmospheres. Young stars are prime targets for direct imaging searches for planets and brown dwarfs, due to the favorable brightness contrast expected at such ages and also because it is often possible to derive relatively good age estimates for these primaries. Here we present the direct imaging discovery of HD 284149 b, a 18 − 50 M Jup companion at a projected separation of 400 AU from a young (25 +25 10 Myr) F8 star, with which it shares common proper motion
INTRODUCTION
In recent years, the rapid improvement of highcontrast imaging instrumentation and techniques have led to the discovery of a number of wide sub-stellar companions to nearby young stars, down to planetary mass (e.g. Chauvin et al. 2005;Luhman et al. 2006;Marois et al. 2008Marois et al. , 2010Lagrange et al. 2009;Currie et al. 2014).
Several of these discoveries, such as AB Pic b (Chauvin et al. 2005), HN Peg B (Luhman et al. 2007), 1RXS J1609 b (Lafrenière et al. 2010(Lafrenière et al. , 2008, HIP 78530 b (Lafrenière et al. 2011) and the recently discovered HD 106906 b (Bailey et al. 2014) and ROXS 42B b (Currie et al. 2014), have mass ratios with respect to their parent stars of only ∼ 1% and seriously challenge the current planet formation paradigm. In particular, their large separations are hard to explain and suggest they might be extreme outcomes of their underlying formation mechanism, regardless of whether it is based on core accretion or disk instability.
Our previous survey of 91 stars in the USco region (Lafrenière et al. 2014) implies a frequency of wide companion for such regions of 4-5%, in agreement with other studies (Ireland et al. 2011). This suggests a frequency of wide companions in star forming regions comparable to the values for young moving groups or the field, reported for example by Lafrenière et al. (2007); Metchev & Hillenbrand (2009) ;Chauvin et al. (2010).
Most recently we also confirmed three new companions with masses of ∼ 40 − 100 M Jup and separations of ∼ 40 − 230 AU in the Scorpius-Centaurus (Sco-Cen) region (Janson et al. 2012b). These companions represent an interesting intermediate between stellar companions and the ∼ 10 − 20 M Jup ones described above in the Upper Scorpius (USco) region. The existence of such a seemingly continuous population might imply that binary formation extends all the way down to planetary masses for wide separations, or at least that mass alone is not a clear-cut diagnostic for distinguishing between formation mechanisms. In order to further address these issues, we conducted a survey of 74 stars in the Taurus star forming region with ALTAIR/NIRI (Herriot et al. 2000;Hodapp et al. 2003). The results of the full survey will be presented in a dedicated paper (Daemgen et al. 2014, ApJ Submitted). Here we present the discovery of a 18 − 50 M Jup companion at a projected separation of ∼ 400 AU from the F8 star HD 284149. A dedicated analysis of the host properties is also presented in Sec. 3, addressing the question of its questionable Taurus membership.
OBSERVATIONS AND DATA REDUCTION
HD 284149 was observed during six epochs between October 2011 and March 2014 on Gemini North with the adaptive-optics assisted NIRI instrument (Hodapp et al. 2003) in J, H, and Ks-band. The f/32 camera provided a sampling of 21.9 mas/pixel and a field of view (FoV) of 22 ′′ ×22 ′′ . Total integration times varied between 9 sec (J) and ∼7 sec (Ks) and were taken as a series of coadds in a 5-point dither pattern to increase dynamic range and allow sky subtraction. The details of the observing times for each epoch, together with the mean airmass and seeing at each observing date are reported in Tab. 2. After subtraction of a striping pattern frequently observed in NIRI images, all images were flat fielded, bad pixel corrected, and sky subtracted. The field distortion was corrected as described in Lafrenière et al. (2014) who determine a residual astrometric uncertainty of 15 mas, 25 mas, and 50 mas at radii 4 ′′ , 8 ′′ , and 12 ′′ from the center, respectively.
The left panel of Fig. 1 shows one of the fully reduced images of HD 284149 and its companion obtained with NIRI in 2012B. The achieved full width at half maximum of the point spread function is 0. ′′ 08, and the companion, at a separation of 3.7 ′′ is detected at 14σ. As part of our survey for faint companions in Taurus (Daemgen et al. 2014, subm.), we also obtained deep exposures of HD 284149 in H and J band, which confirm the presence of the companion with high S/N > 200. These observations, however, saturate the central star and render the relative astrometry and photometry less precise than in the lower-S/N images analyzed here, and are not further used.
HOST STAR PROPERTIES
HD 284149 was included among the members of Taurus-Auriga association by Wichmann et al. (1996), but it is not considered in the compilations by Kenyon et al. (2008) and Esplin et al. (2014). Therefore, a re-assessment of the stellar properties is needed. HD 284149 was classified as F8 by Nguyen et al. (2012) and G1 by Wichmann et al. (2000).
A young age of the star is supported by the large lithium EW (Wichmann et al. 2000), the large X-ray luminosity as revealed by ROSAT, the photometric variability and fast rotation (a period of 1.079 days is reported by Grankin et al. 2007). The short-term RV monitoring by Nguyen et al. (2012) was able to exclude the possibility of a tidally-locked binary. However with a 3 Km/s difference between mean RV from Wichmann et al. (2000) and Nguyen et al. (2012) being about 3 km/s, a binarity with periods of months or years cannot be excluded 6 From the G1-F8 spectral classification, an effective temperature of 5970-6100 K is derived following Pecaut & Mamajek (2013). Photometric colors are broadly consistent with such temperatures, with a detailed comparison hampered by obervational scatter (e.g. peak-to valley differences larger than 0.2 mag in V band), possibly linked to the photometric variability of the star. Adopting the V magnitude from ASAS (9.653 ± 0.060), the V − K s color is 1.55 mag. Comparison with the premain sequence (pre-MS) intrinsic colors of young stars by Pecaut & Mamajek (2013) suggests a reddening E(B-V) of about 0.05-0.08 mag for a G1 and F8 star, respectively. Slightly smaller amounts of reddening are indicated by the B-V and V-I colors. Such amount of reddening is not unexpected at the distance of the star. The trigonometric parallax from van Leeuwen (2007) is 9.24 ± 1.58 mas. A comparison with members of young moving groups (MGs) (see left panel of Fig. 2) indicates that the lithium equivalent width of HD 284149 (208 mÅ, Wichmann et al. 2000) is comparable with that of members of β Pic, Tuc-Hor, Columba and Carina moving groups of similar temperatures, and clearly above that of Pleiades open cluster and AB Dor moving group. A similar result is obtained for a comparison of the X-ray luminosity (log L X /L bol = −3.3 for HD 284149).
The position of HD 284149 on an HR diagram (see right panel of Fig. 2) is close to the 25 Myr isochrone using the theoretical models by Bressan et al. (2012), with ages between 15 to 100 Myr also compatible with the data. This isochronal age is on average older than that obtained by other authors (14-16 Myr, Wichmann et al. 2000;Palla & Stahler 2002a) because of the revision in the trigonometric parallax, with respect to the value reported in the Hipparcos catalogue (Perryman & ESA 1997).
The resulting kinematic parameters, adopting the van Leeuwen (2007) parallax and proper motion and the mean of the RVs obtained by Wichmann et al. (2000) and Nguyen et al. (2012), are U = −12.3 km/s, W = −6.4 km/s and W = −8.8 km/s. This is similar to that of the Octans association discussed in Torres et al. (2008), whose proposed members are however all very far from HD 284149 on the sky. Recently Zuckerman et al. (2013) identified an additional group of young stars with similar kinematics to the Octans association but with a smaller distance from the Sun and a different sky distribution, labeled as Octans-Near. While the link between Octans and Octans-Near groups and the existence of the latter as a true moving group deserves further investigation, we note that one of the Octans-Near proposed members, HIP 19496, is separated on the sky by about 5 deg from HD 284189, it has a comparable distance (98 vs 108 pc), and the space velocities of the two stars differ by just 2.7 km/s. Zuckerman et al. (2013) estimate an age of 30 Myr for HIP 19496, similar to our determination for HD 284189.
In summary, HD 284149 is significantly older than the bulk of the Taurus-Auriga association. Membership is still possible in case of earlier start of star formation in the outer regions of the association (Palla & Stahler 2002b). Independently of the Tau-Aur membership, the Li EW well above the Pleiades locus coupled with the position on HR diagram above ZAMS and other age diagnostics indicate an age of about 25 Myr, with minimum and maximum values of about 15 and 50 Myr respectively. A summary of the stellar parameters is given in Table 1.
COMPANION PROPERTIES
The relative position of HD 284149 and its companion were determined with PSF photometry using daophot in IRAF. The bright star HD 284149 was used as PSF reference to obtain relative photometry and astrometry of the companion. Statistical uncertainties were inferred from the rms noise between the individual dither exposures for each epoch and filter. Systematic flux uncertainties are estimated from the residuals after PSF subtraction to be 5%, and systematic astrometric uncertainties are dominated by the uncertainty of the distortion correction at the position of HD 284149b of 15 mas. The resulting astrometry and photometry are listed in Tab. 2, while a summary of the derived properties is given in Tab. 1.
The right panel of Fig. 1 shows the relative change of separation and position angle of the companion between our previous observations (filled right-facing triangle) with respect to the most recent one (filled circle). We conclude that the point source we imaged is consistent with a co-moving companion at ∼ 400 AU from HD284149 with > 99% confidence according to a χ 2 test.
As discussed in Sec. 3, the age of this system is controversial, as it appears to be older than other members of the Taurus association. With an adopted age of The black star represents our target, orange circles represent members of beta Pic moving gropus and TW Hya association (age 8-16 Myr) red circles members of Tucana, Columba and Carina MG (age 30 Myr); light blue circles: members AB Dor MG (age 100 Myr); green circles: Pleiades OC (age 125 Myr); purple circles: Hyades OC (age 625 Myr). The filled squares represent the median values of EW Li for the corresponding color bin. Li EW data are from Torres et al. (2006) for young assocition, Soderblom et al. (1993) for the Pleiades, and Thorburn et al. (1993) for the Hyades. Right: Position of HD284149 (filled dots) on HR diagram for the temperatures corresponding to F8 and G1 spectral classification. Overplotted are the 5, 12, 20, 30, and 70 Myr isochrones from Bressan et al. (2012).
+25
−10 M yr, the Ks brightness of the companion suggests a mass of 32 +18 −14 M Jup , according to the DUSTY models by Chabrier et al. (2000). Using the DUSTY models by Chabrier et al. (2000) we derived an effective temperature of 2337 +95 −182 K. Together with the color measurements (J-H ∼ 0.8, H-K ∼ 0.4) this suggests a spectral type between M8 and L1 (see e.g. Pecaut & Mamajek 2013, and www.pas.rochester.edu/∼emamajek/ for the extended table), but further measurements are required to better constrain this.
DISCUSSION AND CONCLUSIONS
We presented here the detection of a substellar ( 32 M Jup assuming an age of 25 Myr, see Fig. 3 a and Tab. 1) companion orbiting the young star HD 284149 at a separation of ∼ 400 AU. Fig. 3 shows a comparison of HD 284149 mass-ratio and separation with the values of the planetary companions found by radial velocities (RV) and transit methods, as well as other directly imaged planetary and brown dwarf companions. The group of objects with separation < 100 AU and mass ratio < 0.01 seems to be well separated from the one including companions with larger separation and mass ratio, suggesting that different formation mechanisms could be at play. The small mass ratio of the first group might suggest planet-like formation, but objects with similar mass-ratios at larger separations are difficult to explain.
HD 284149b shows very similar properties to objects like ROXs 42Bb or AB Pic b which, as suggested by Currie et al. (2014), places it between the bona-fide planets and the lowest mass brown dwarfs imaged so far. Together with these and other companions of similar mass and separation, such as HN Peg B and HD 106906, HD 284149b represents a challenge for our understanding of the formation of low-mass companions at very wide separations. The high mass ratio of these systems might suggest a planet-like origin, but at the same time, Baraffe et al. (2003) and DUSTY (dashed lines) models by Chabrier et al. (2000). The position of HD 284149 B is marked by a filled star. Right: Mass ratio vs separation of HD 284149 (filled star), compared to those of other known low-mass companions discovered so far. The companions discovered using the radial velocity (RV) technique are marked with blue crosses, the ones transiting their parent stars with green triangles. Finally, directly imaged companions are represented by red circles if the stellar age is less than 500 Myr (young companions), and with red circles otherwise.
Parameter
Host Grankin et al. (2007) their estimated mass is well above the deuterium burning limit, suggesting a stellar-like formation.
The existence of such companions suggests that mass ratio alone is not sufficient to distinguish between planetlike and star-like formation, at least for wide companions (see also Janson et al. 2012a).
Finally, the findings of dedicated RV campains around young stars seem to suggest a paucity of close-in planetary companions around these targets. Only few close in companions have been detected around young early-G and F-type stars, such as HD 70573 (Setiawan et al. 2007) and HD 113337 (Borgniet et al. 2014). Their small number seem to imply a lower frequency of such companions if compared to the more massive, more distant ones as HD 284149B. If confirmed, this could suggest that multiple planet formation mechanisms are at play around these objects.
The object brightness and separation makes HD 284149 b a very well suited target for detailed characterization of both the host star and the companion. Efforts toward this direction are already under way and will be presented in further publications.
Fig. 1 .
1-Left: Candidate companion detected in Ks with GEMINI/NIRI in one of the images obtained in 2012B. Right: Evaluation of common proper motion for the detected companion to HD 284149. The continuous line shows the motion of a background star and the filled symbols its positions of at the various epochs. The open symbols represent the corresponding measurements of the position of HD 284149 B at the same epochs. If the star and companion are co-moving, the open symbols with error bars should be close to the filled circle in the origin, which represents our most recent epoch used as reference. If the companion candidate does not move with respect to the background, then the open symbol is consistent with the location of the identical filled symbol.
Fig. 2 .
2-Left: Lithium equivalent width vs B-V of HD284149 compared to that of members of nearby groups and clusters of young age.
Fig. 3 .
3-Left: Absolute magnitude in K S band vs age of a companion of 20, 30 and 50 M J up (blue, cyan and purple curve, respectively) according to the COND (solid lines) models by
ACKNOWLEDGMENTS
This work was supported by grants from the NSERC of Canada and the University of Toronto McLean Award to R.J. SD is supported by a McLean Postdoctoral Fellowship. M.B. is founded through the Progetti Premiali funding scheme of the Italian Ministry of Education, University, and Research, and through the PRIN-INAF 2010 Planetary systems at young ages and the interactions with their active host stars. Based on observations obtained at the Gemini Observatory, operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France
TABLE 1
1Wichmann et al. (2000); [8] Nguyen et al. (2012); [9]Summary of properties of both HD 284149 and its
companion.
References: [1] This paper; [2] ASAS (Pojmanski 2002); [3]
SIMBAD; [4] TASS (Richmond et al. 2000); [5] 2MASS
Cutri et al. (2003); [6] van Leeuwen (2007); [7]
TABLE 2
2Detailes of the observations setup and conditions, and relative astrometry and photometry of HD 284149 and its companion
In order to reduce the impact of possible binarity on the space velocity, we decided to use the average velocity. Efforts are underway to better understand the multiplicity status of HD 284149 and will be presented in further publications. There are no indication from non-detection on line profile alteration inNguyen et al. 2012 and lack of direct detection in our own images that the companion contributes significantly to the integrated flux. The expected impact on the age indicators is then minor.
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| [] |
[
"Two Power Series Models of Self-Similarity in Social Networks",
"Two Power Series Models of Self-Similarity in Social Networks"
] | [
"Subhash Kak "
] | [] | [] | Two power series models are proposed to represent self-similarity and they are compared to the Zipf and Benford distributions. Since evolution of a social network is associated with replicating self-similarity at many levels, the nature of interconnections can serve as a measure of the optimality of its organization. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjoining terms, our distribution considers the middle term to be the geometric mean. In one of the power series models, the scaling factor at one level is shown to be the golden ratio. A model for evolution of networks by oscillations between two different selfsimilarity measures is described. | 10.1016/j.ins.2016.10.010 | [
"https://arxiv.org/pdf/1506.07497v1.pdf"
] | 6,169,626 | 1506.07497 | b460eaadd5bd839fed47e78ea7801bf81c87b646 |
Two Power Series Models of Self-Similarity in Social Networks
Subhash Kak
Two Power Series Models of Self-Similarity in Social Networks
1Social networksself-similarity80-20 phenomenonconnectivitygolden ratio
Two power series models are proposed to represent self-similarity and they are compared to the Zipf and Benford distributions. Since evolution of a social network is associated with replicating self-similarity at many levels, the nature of interconnections can serve as a measure of the optimality of its organization. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjoining terms, our distribution considers the middle term to be the geometric mean. In one of the power series models, the scaling factor at one level is shown to be the golden ratio. A model for evolution of networks by oscillations between two different selfsimilarity measures is described.
Introduction
A social network consists of N nodes, labeled 1, 2, … n, that may be people, firms, or other entities. The network is represented by a graph where the connection between the nodes i and j is shown by a link between the two. A network may be valued for the function it performs to nodes outside of it, or it may be valued for the function it serves to the nodes within (Aral and Walker, 2012). Unlike engineered networks whose function (to nodes outside the network) is welldefined, goodness of social networks cannot be easily quantified because the entity that is scarce is attention which requires a different kind of economics (Simon, 1971;Goldharber, 1997;Essig and Arnond, 2001;Yu and Kak, 2014) and also because of the elusiveness of the inner experience of well-being. Nevertheless, for the entire network one speaks of some general function such as value, utility, well-being, or welfare of the group whose definition is driven by extraneous considerations of theory or ideology (Kirman, 1997).
In economic networks (Bienenstock and Bonacich, 1993;Ellison, 1993;Slikker and van den Nouweland, 2000) where value must be associated with the nodes within, one speaks of Pareto efficiency, or Pareto optimality, which is a state of allocation of resources in which it is impossible to make any one individual better off (in terms of a suitable measure of well -being) without making at least one individual worse off. A state is Pareto efficient or optimal when no further improvements can be made. The allocation, normally in terms of resources, could also be in terms of some property of the connectivity. We are not concerned with the cost of the connections and interested primarily in natural connectivity as determined by fundamental considerations.
If one were considering connectivity as value, the total value of the network is proportional to n (n -1), that is, roughly, since in a network with n nodes each can make (n -1) connections with other nodes. This ignores the fact that in an evolutionary network the connection capacity of the nodes must vary as new nodes can only gradually get connected with others and some other n 2 2 nodes might leave the network. A careful analysis indicates that the potential value of a network of size n grows in proportion to n log n which is connected to the need for self-similarity in natural systems (Briscoe et al., 2006).
Any theoretical definition of utility for a social network is at best arbitrary; therefore the actual connectivity can provide insight into value. Such connectivity is determined by cognitive, sociological, cultural and economic considerations, but one defining feature of it appears to be self-similarity across different layers. It is significant that self-similarity shows up in a variety of complex systems such as the World Wide Web, social, biological, cellular and protein-protein interaction networks and these are well characterized by power-law distributions. The logic behind the emergence of self-similarity in such networks is that of local and hierarchical interactions. The value of the network may be examined from how close it is to self-similarity across many different layers. This may be contrasted with theories that argue that value can be freely transferred across individuals in the society and, therefore, a sign of an efficient network is the total of the values of the individuals. Since value is relative and enhanced by scarcity, aggregate wealth cannot be a good measure of the success of a society.
This paper presents an approach to investigating structure in social networks from the perspective of self-similarity, which one may assume leads to social efficiency. Two power series models have been used for this idea and they have been compared to the Zipf and Benford distributions. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjacent terms, our distribution considers the middle term to be the geometric mean. A model for the evolution of networks by means of oscillation between two modes is advanced. This may be contrasted with the preferential attachment model of generating scale-free networks. An internal mechanism involving two modes is desirable because it represents an inner tension that drives the dynamics.
Scale-free and small-world properties
Complex physical and biological networks are characterized by the small-world and the scalefree properties (Albert et al., 1999;Dutta et al. 1998;Watts, 2001;Watts and Strogatz, 1998). In a small-world network one can reach a given node from another one in a small number of steps that are characteristic of social networks. For a complex network, the probability distribution of the number of links per node, p(k), is generally given by a power-law (scale-free) with a degree exponent λ usually in the range 2 ≤ λ ≤ 3:
K m m k ck k p ,..., 1 , , ) ( 1 (1)
where c is for the normalization of the expression (1) and m and K are the lower and the upper cutoffs of the distribution. The value k=K represents the nodes with the largest number of connections which have the least probability. As k becomes large,
) 1 ( ) 1 ( ) ( 1 1 1 k p k p k p
, and this points to the geometric nature of this distribution for such values. Power-law distributions have long tails.
In the related Zipf's law for natural languages (Zipf, 1949), the frequency of a word is inversely proportional to its rank in the frequency table:
1 ) ( k k p z
(k is the rank; for English the most 3 commonly occurring word is "the" which as a probability of about 7% in the Brown Corpus).
The intermediate term now is the harmonic mean
) 1 ( ) 1 ( ) 1 ( ) 1 ( 2 ) ( k p k p k p k p k p z z z z z of the values of ) 1 ( k p z and ) 1 ( k p z .
Zipf's distribution is also called the discrete Pareto distribution which is used in many social and scientific phenomena.
In one study where the population of cities was considered, a variant of the Zipf's law provided the best fit with the distribution of
07 . 1 ) ( k k p .
Zipf saw the principle of least effort at the basis of his eponymous law.
Let δ be the characteristic path length, which is the average shortest path length of all node pairs in the network. Mathematically, whereas for a random network δ has a logarithmic relationship with the total number of nodes N:
N ln
, for a scale-free network with 2 < λ < 3 it appears to be N ln ln (Cohen and Havlin, 2003).
One approach to understanding long-tailed behavior is through the underlying counting phenomenon. A counting process may be considered to be uniformly distributed over the range {1, …, S}. For a very large number of such processes with random values of S, the set of numbers will satisfy Benford's law and the leading digit d (d ∈ {1, ..., b}) occurs with probability (Benford, 1938) )
1 1 ( log ) ( d d p b b (2)
When b = 2, the probability of the first digit being 1 is trivially equal to 1. Table 1 compares the first digit frequencies with the corresponding Zipf's law frequencies where we have assumed that the highest frequency for the second case is also 0.301 . Figure 1 shows how both these distributions are characterized by long tails. It would be correct to assume that if empirical data follows Benford's Law, it is generated by a mixture of independent uniform processes (Janvresse and de la Rue, 2004).
Self-similarity versus combinatorial randomness
The mathematical approach to randomness is one of combinatorics, whereas biological and natural systems are characterized by self-similarity. The basis of self-similarity appears to be the evolutionary process of recursion across levels and it operates both for internal and external cognitions (Dixon et al., 2014). This self-similarity provides a certain order in randomness that is of value in biological structure, social relationships, as well as evolving physical systems and it can also be useful in checking engineered systems (Nigrini, 2012).
It is of course obvious that an evolving system will not be always perfectly self-similar. Thus the definition of self-similarity must come with a permissible range of values that is appropriate for the system to have the capacity to replicate. Evolution is transformation from one state to another and this requires a potential that drives the transformation.
The dynamic at the basis of evolutionary change in biological networks may be seen in the oscillation between two different modes which defines a kind of a ping-pong model. Such systems appear to have informational and cognitive drivers as in the theory that social size in primates is determined by cognitive capacity. Dunbar, who advanced this theory, argues that the size of the brain is correlated with the complexity of function and he develops an equation, which works for most primates, that relates the neocortex ratio of a particular species -the size of the neocortex relative to the size of the brainto the largest size of the social group (Dunbar, 1992). For humans, the maximum group size appears to ne 147.8, or about 150, and this represents Dunbar's estimate of the maximum number of people who can be part of a close social relationship. Support for this idea come from the community of Hutterites, followers of the sixteenth century Jakob Hutter of Austria, who are pacifists and believe in community property and live in a shared community called colony. Several thousand Hutterites relocated to North America in the late 19th century and their colonies are mostly rural. A colony consists of about 10 to 20 families, with a population of around 60 to 150. When the colony's population approaches the upper figure, a daughter colony is established. But for those living in a technology society where the individual's cognitive capacities have been enhanced by a variety of tools one would expect that Dunbar's number would not strictly apply.
Information is also seen as a driver in the evolution of quantum systems (Kak, 2007;Licata and Fiscaletti, 2014;Kak, 2015). Indeed, according to the orthodox interpretation of quantum theory, the very process of observation has an influence on the state of the system.
Single-layered self-similar system: p 2 distribution
Consider a single-layered self-similar system where all members of the network are at the same hierarchical level. In such a system we define self-similarity in terms of the relationship amongst members in adjacent ranks. If the aggregate connectivity is S and the node with the largest connectivity has A links, then the connectivity of the next one is A/a, where a > 1, with the following one with A/a 2 , and so on. For each number, the consecutive ratios satisfy the geometric mean
1 1 n n n x x x .
In other words, the probability distribution with respect to the rank order of connectivity is:
K m m k ca k p k ,..., 1 , , ) ( 2 (3)
This may be contrasted with the power-law distribution (1) where the probabilities are in an inverse rank order. As k becomes large, both this geometric series and the power law distributions become long-tailed, although (3) falls faster. The un-normalized distributions (1) and (3) Since the number of nodes is N, we can write for total connectivity generated by p 2 :
) 1 ... 1 1 1 ( 1 2 N a a a A S(4)
This is solved to give:
1 1 ) ( 1 a Aa a a a A S N (5)
The value of S will be larger than the right hand side of (5) depending on the length of tail since the minimum number of connections to any node in a connected graph must be 1 and also the constraint that N > A since the first ranked node has A connections.
Since 2 ) 1 /( 1 a da dS
, the rate of change of the aggregate number of connections depends on its variance from unity.
A small example of such a network for A= 10 and a ≈ 1.4 is given in Figure 2 below for which the actual number of connections is 40 whereas equation (4) corresponds to the value 35.
In Figure 2, node 1 has 10 connections (for it is connected to all other nodes), node 2 has 7 connections, node 3 has 5 connections, and so on. These are nominal numbers of connections. In a real-world distribution the actual connections will be random with these representing means for a large set of networks. Figure 2. An example of geometric series connectivity with a ≈ 1.4, A=10
A comparison of this geometric series power law (3) with the Zipf's law is given in Figure 3.
m m x x x x x p , ) ( 1 (6)
Given that the aggregate connectivity for the geometric distribution is S, we need to find the number of terms i, so that for (4) Elements in rank order. Geometric: red: a=1.5, blue: a=1.4. Zipf: green Probability Figure 4 plots the value of a for different number of aggregate nodes ranging from 10 to 55 for 20% of the nodes to contribute 80% of the connectivity. As we see the value of a goes down as the number of aggregate nodes increases.
N N a a a a(8)
can be put into larger groups such as:
... ) 1 1 1 ( 1 ) 1 1 1 ( 1 ) 1 1 1 ( 2 6 2 3 2 a a a a a a a a(9)
Whereas the individual elements have a relationship given by 1/a, the groups of three have a relationship given by 1/a 3 .
Since for a collection where N is ∞, S= 1/(1-a), the relationship between the two is like the one in the mathematical structure of a decimal sequence (Kak and Chatterjee, 1981;Kak, 1985) which is associated with randomness.
Hierarchical self-similar system: p 3 distribution
Now we consider another distribution where the probability of an element (or a group of elements) in a ranked list equals the probability of the next two elements (or the next group of double size) in the sequence shown below:
... ) ( ) 3 ( ) 2 ( ) 1 ( 7 4 3 3 3 3 i p p p p i(11)
If the groups are not defined as above (that is associated with powers of 2), then we will have the result (where g represents a group of elements):
) 2 ( ) 1 ( ) ( 3 3 3 g p g p g p(12)
For a small list, the relation (12) will be approximately correct for many groups. The ranked elements may be seen as constituting hierarchically arranged layers. The relationship between the first ranked element and those of the elements in the second layer will be replicated across the elements of the second and the third layers.
If we wished for the first equality of (10) or (11) to have the same self-similarity as of (3), we have:
2 1 1 1 a a (13)
This implies 1 2 a a , whose solution is the golden ratio φ:
... 6180339 . 1 2 5 1 (14)
For the next group, the relationship will be: can be reduced to 1 2 4 a a by factoring out a+1 on both sides. This indicates how a renormalization of the scaling factor will have to be performed as we go down the hierarchy.
For a system where there is a ternary division (we will associate it with b)
... ) ( ) 4 ( ) 3 ( ) 2 ( ) 1 ( 13 5 3 3 3 3 3 i p p p p p i (16) 9
The equation to solve at the next stage for self-similarity of (3) will be:
3 2 1 1 1 1 b b b (17)
Its solution is b=1.8383..
Oscillation between two self-similarity measures
Let us consider two states of the system that are associated with different measures of selfreplication (a and b) in the proposed geometric series model.
) 1 ... 1 1 1 ( 1 2 N a a a a A S(18)) 1 ... 1 1 1 ( 1 2 M b b b b B S(19)
We propose a dynamics bases on the oscillation between these two modes. This constitutes a method that is different from that of preferential attachment which leads to scale-free networks. For simplicity, we consider the distribution (10) rather than (3).
Example. Let a=2 and b=3. Similar to how for a=2 the second layer has two elements, for b=3, the second layer will have three elements and each of them will have a probability of 1/3 of the first. Let the first stage be governed by a. This will create the initial network with connections as below:
Stage The theoretical connections in stage 2 are by probabilities alone and they fail the constraint that the total number of connections should be even. Stage 2, Actual, provides a realizable network that is quite close to the theoretical values. Notice, in the transition node 1 has gained 5 new connections, nodes 2 and 3 have gained two each, node 4 has gained 2, nodes 5 through 7 have remained as before, and new nodes 8 through 13 are each connected with one link.
The subsequent growth of the network, swinging between the two different measures of selfsimilarity, is shown in the tables below, where it is assumed that sometimes some connections are lost to conform to the theoretical numbers: We hypothesize that such a process does characterize real social networks. The change in the mode is determined by technology and policy regimes that evolve with time. Therefore, in the real world, a and b will be time-dependent and the average scaling factor at the second level will appear to be the golden ratio (if the dominant mode is a).
Conclusions
This paper considers the evolution of a social network through the lens of replicating selfsimilarity at many levels. Closeness to self-similarity in the interconnections is proposed as a measure of the optimality of the organization. Two power series models are proposed to represent self-similarity and they are compared to the Zipf and Benford distributions. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjoining terms, our distribution considers the middle term to be the geometric mean. A model for evolution of networks by oscillations between two different self-similarity measures is also discussed and it is shown that the scaling at the second level is according to the golden ratio.
Figure 1 .
1Benford's (20 digits, in red) andZipf's (green)
Figure 3 .
3Geometric and Zipf distribution frequencies Long-tailed distributions are characterized by what has been called the 80-20 rule or the Pareto principle associated with the Pareto distribution:
Figure 4 .
4The value of a for the 80-20 phenomenon in the power series model5. Hierarchical self-similarityNow consider self-similarity in a hierarchical sense for the geometric series distribution.
Table 1 .
1First digit frequencies for b=10 and Zipf's law frequencies Zipf's freq 0.301 0.155 0.103 0.075 0.060 0.051 0.043 0.038 0.0341
2
3
4
5
6
7
8
9
1st digit
0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
1.Nodes
1
2,3
4,5,6,7
Total connections
Connections 4
2 each
1 each
12
This network is realized by the following graph: (1,2), (1,3), (1,4), (1,5), (2,6), (2,7). Stage 2 will
be governed by b and it leads to stage 2 below:
Stage 2. Theoretical
Nodes
1
2,3,4
5 to 13
Total connections
Connections 9
3 each
1 each
27
Stage 2. Actual
Nodes
1
2,3,4
5 to 13
Total connections
Connections 9
4,4,3
1 each
30
Note that in Stage 4, nodes 5 through 15 have lost one link each. Connections 32 16,16 8,8,8,8 4 each 2 each 1 each 192 and so on.Stage 3. Theoretical
Nodes
1
2,3
4,5,6,7
8-15
16-31
Total connections
Connections 16 8,8
4,4,4,4
2 each 1 each
80
Stage 4. Theoretical
Nodes
1
2,3,4 5-13
14-40
Total connections
Connections 27 9,9,9 3 each
1 each
108
Stage 5. Theoretical
Nodes
1
2,3
4,5,6,7 8-15
16-31 32-63
Total connections
The diameter of the World Wide Web. R Albert, H Jeong, A-L Barabasi, Nature. 401130Albert, R. Jeong, H, Barabasi, A-L. (1999) The diameter of the World Wide Web. Nature 401,130.
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| [] |
[
"Is Texture Predictive for Age and Sex in Brain MRI?",
"Is Texture Predictive for Age and Sex in Brain MRI?"
] | [
"Nick Pawlowski [email protected] \nBiomedical Image Analysis Group\nImperial College London\nUK\n",
"Ben Glocker [email protected] \nBiomedical Image Analysis Group\nImperial College London\nUK\n"
] | [
"Biomedical Image Analysis Group\nImperial College London\nUK",
"Biomedical Image Analysis Group\nImperial College London\nUK"
] | [] | Deep learning builds the foundation for many medical image analysis tasks where neural networks are often designed to have a large receptive field to incorporate long spatial dependencies. Recent work has shown that large receptive fields are not always necessary for computer vision tasks on natural images. We explore whether this translates to certain medical imaging tasks such as age and sex prediction from a T1-weighted brain MRI scans. | null | [
"https://arxiv.org/pdf/1907.10961v1.pdf"
] | 140,120,425 | 1907.10961 | a3b4bb00558a5466b58cd25094ab40167b190d21 |
Is Texture Predictive for Age and Sex in Brain MRI?
Nick Pawlowski [email protected]
Biomedical Image Analysis Group
Imperial College London
UK
Ben Glocker [email protected]
Biomedical Image Analysis Group
Imperial College London
UK
Is Texture Predictive for Age and Sex in Brain MRI?
Proceedings of Machine Learning Research -Under Review:1-4, 2019 Extended Abstract -MIDL 2019 submission Editors: Under Review for MIDL 2019Age and Sex PredictionBrain MRINeural Networks
Deep learning builds the foundation for many medical image analysis tasks where neural networks are often designed to have a large receptive field to incorporate long spatial dependencies. Recent work has shown that large receptive fields are not always necessary for computer vision tasks on natural images. We explore whether this translates to certain medical imaging tasks such as age and sex prediction from a T1-weighted brain MRI scans.
Introduction
Deep learning has become the de-facto standard method in many computer vision and medical image analysis applications (Litjens et al., 2017). Recently introduced BagNets (Brendel and Bethge, 2019) have shown that on natural images, neural networks can perform complex classification tasks by only interpreting texture information rather than global structure. BagNets interpret a neural network as a bag-of-features classifier that is composed of a localised feature extractor and a classifier that acts on the average bag-encoding.
We explore whether texture information is sufficient for certain tasks in medical image analysis. For this, we generalise BagNets to arbitrary regression tasks and 3D images and examine the performance of different receptive fields. We apply BagNets to age regression and sex classification tasks on T1-weigthed brain MRI to examine the dependency of modern deep learning architectures on local texture in these medical imaging tasks. We find that the bag-of-local-features approach yields comparable results to larger receptive fields.
Method
BagNets (Brendel and Bethge, 2019) are adaptations of the ResNet-50 architecture (He et al., 2016), that restrict the receptive field by replacing 3 × 3 convolutional kernels with 1 × 1 kernels. A regular ResNet-50 has a receptive field of 177 pixels, whereas BagNets explore receptive fields of 9, 17 and 33 pixels. The use of small receptive fields enforces locality in the extracted features. After extracting local features a global spatial average builds the bag-of-local-features and enforces the invariance to spatial relations. The bag of features is then processed by a linear layer to provide the final prediction. Because of the linearity of the average operation and the final linear layer, it is possible to exchange the order of those operations, which enables the extraction of localised prediction maps.
We test the BagNets on the public Cambridge Centre for Ageing and Neuroscience (Cam-CAN) dataset (Taylor et al., 2017). The dataset contains T1-and T2-weighted brain MRI of 652 healthy subjects within an age range of 18 to 87. We only use the T1-weighted scans for our experiments and randomly split the scans into training, validation and test sets with 456, 65 and 131 subjects each. All scans have an isotropic resolution of 1 mm. We use skull-stripped, bias-fiel corrected images and extract random crops of shape [128×160×160] during training. We whiten the images with statistics extracted from within the brain mask.
We use the architecture from (Brendel and Bethge, 2019) but replace 2D with 3D convolutions and half the number of feature maps. We train the network with batch size 1 and accumulate gradients over 16 batches. To alleviate the effects of small batches we use instance normalization (Ulyanov et al., 2016) instead of batch normalization (Ioffe and Szegedy, 2015). We use a cross-entropy loss for the sex classification and MSE loss for the age regression. We use the Adam optimizer (Kingma and Ba, 2014) with a learning rate of η = 0.001, = 10 −5 and employ an L 2 -regularization of λ = 0.0001. We train the network for 500 epochs and decay the learning rate by a factor of 10 every 100 epochs. We use the checkpoint with the best validation performance for evaluation on the test data. Table 1 shows the mean absolute error (MAE) and accuracy for the age and sex prediction for different receptive fields. We achieve a MAE between 3.86 − 5.53 years for age and an accuracy between 80.9 − 84.0% for sex. Age regression has a stronger dependency on the receptive field than the sex classification. However, we find that the larger receptive field exhibits better training performance and might be prone to overfitting. Table 1: Results for the age regression and sex classification task for different receptive fields. We report the mean absolute error for age and classification accuracy for sex. We find that small receptive fields yield comparable results on those tasks.
Receptive Field Age Sex (9mm) 3 5.53 83.2% (17mm) 3 5.32 84.0% (33mm) 3 4.98 84.0% (177mm) 3 3.86 80.9%
We examine the local predictions on two examples from the test set in Figure 1 for age regression and Figure 2 for sex classification. The sex classification predicts 0 for male and 1 for female. The first row shows a 20 year old male subject, the second row shows an 80 year old female. The columns respectively show the middle slice of the T1-weighted MRI, the local predictions with receptive fields 9, 17, 33 and 177. Similarly to (Brendel and Bethge, 2019), we find that small receptive fields lead to more localised predictions, whereas larger receptive fields show more spread out predictions. Interestingly, the age regression exhibits very high variance predictions, where only few very high values contribute to the mean prediction of the volume. Generally, we find that the local predictions we get from our model do not seem as interpretable as in (Brendel and Bethge, 2019).
Discussion & Conclusion
We have generalised the concept of BagNets (Brendel and Bethge, 2019) to the setting of 3D images and general regression tasks. We have shown that a BagNet with a receptive field of (9mm) 3 yields surprisingly accurate predictions of age and sex from T1-weight MRI scans. However, we find that localised predictions of age and sex do not yield easily interpretable insights into the workings of the neural network which will be subject of future work. Further, we believe that more accurate localised predictions could lead to advanced clinical insights similar to (Becker et al., 2018;Cole et al., 2018).
Figure 1 :
1Localised age prediction on a 20 year old male subject (first row) and 80 year old female subject (second row). The columns show the middle slice of the T1weighted MRI and the localised predictions for receptive fields 9, 17, 33 and 177.
Figure 2 :
2Localised sex classification on a 20 year old male subject (first row) and 80 year old female subject (second row). The different columns show the middle slice of the T1-weighted MRI and the localised predictions for receptive fields 9, 17, 33 and 177. The network predicts male as 0 and female as 1.
c 2019 N. Pawlowski & B. Glocker.
AcknowledgmentsNP is supported by Microsoft Research PhD Scholarship and the EPSRC Centre for Doctoral Training in High Performance Embedded and Distributed Systems (HiPEDS, Grant Reference EP/L016796/1). BG received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 757173, project MIRA, ERC-2017-STG). We gratefully acknowledge the support of NVIDIA with the donation of one Titan X GPU.
Alzheimers Disease Neuroimaging Initiative, et al. Gaussian process uncertainty in age estimation as a measure of brain abnormality. Benjamin Gutierrez, Tassilo Becker, Christian Klein, Wachinger, NeuroImage. 175Benjamin Gutierrez Becker, Tassilo Klein, Christian Wachinger, Alzheimers Disease Neu- roimaging Initiative, et al. Gaussian process uncertainty in age estimation as a measure of brain abnormality. NeuroImage, 175:246-258, 2018.
Approximating cnns with bag-of-local-features models works surprisingly well on imagenet. Wieland Brendel, Matthias Bethge, arXiv:1904.00760arXiv preprintWieland Brendel and Matthias Bethge. Approximating cnns with bag-of-local-features models works surprisingly well on imagenet. arXiv preprint arXiv:1904.00760, 2019.
Brain age and other bodily ages: implications for neuropsychiatry. H James, Riccardo E Cole, Sarah E Marioni, Ian J Harris, Deary, Molecular psychiatry. 1James H Cole, Riccardo E Marioni, Sarah E Harris, and Ian J Deary. Brain age and other bodily ages: implications for neuropsychiatry. Molecular psychiatry, page 1, 2018.
Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016.
Batch normalization: Accelerating deep network training by reducing internal covariate shift. Sergey Ioffe, Christian Szegedy, arXiv:1502.03167arXiv preprintSergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network train- ing by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
Jeroen Awm Van Der Laak, Bram Van Ginneken, and Clara I Sánchez. A survey on deep learning in medical image analysis. Geert Litjens, Thijs Kooi, Babak Ehteshami Bejnordi, Arnaud Arindra Adiyoso Setio, Francesco Ciompi, Mohsen Ghafoorian, Medical image analysis. 42Geert Litjens, Thijs Kooi, Babak Ehteshami Bejnordi, Arnaud Arindra Adiyoso Setio, Francesco Ciompi, Mohsen Ghafoorian, Jeroen Awm Van Der Laak, Bram Van Ginneken, and Clara I Sánchez. A survey on deep learning in medical image analysis. Medical image analysis, 42:60-88, 2017.
The cambridge centre for ageing and neuroscience (cam-can) data repository: structural and functional mri, meg, and cognitive data from a cross-sectional adult lifespan sample. Nitin Jason R Taylor, Rhodri Williams, Tibor Cusack, Auer, A Meredith, Marie Shafto, Lorraine K Dixon, Richard N Tyler, Henson, Neuroimage. 144Jason R Taylor, Nitin Williams, Rhodri Cusack, Tibor Auer, Meredith A Shafto, Marie Dixon, Lorraine K Tyler, Richard N Henson, et al. The cambridge centre for ageing and neuroscience (cam-can) data repository: structural and functional mri, meg, and cognitive data from a cross-sectional adult lifespan sample. Neuroimage, 144:262-269, 2017.
Instance normalization: The missing ingredient for fast stylization. Dmitry Ulyanov, Andrea Vedaldi, Victor Lempitsky, arXiv:1607.08022arXiv preprintDmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016.
| [] |
[
"Tree-Structured Policy based Progressive Reinforcement Learning for Temporally Language Grounding in Video",
"Tree-Structured Policy based Progressive Reinforcement Learning for Temporally Language Grounding in Video"
] | [
"Jie Wu \nSun Yat-sen University\n\n",
"Guanbin Li [email protected] \nSun Yat-sen University\n\n",
"Si Liu [email protected] \nBeihang University\n\n",
"Liang Lin [email protected] \nSun Yat-sen University\n\n\nDarkMatter AI Research\n\n"
] | [
"Sun Yat-sen University\n",
"Sun Yat-sen University\n",
"Beihang University\n",
"Sun Yat-sen University\n",
"DarkMatter AI Research\n"
] | [] | Temporally language grounding in untrimmed videos is a newly-raised task in video understanding. Most of the existing methods suffer from inferior efficiency, lacking interpretability, and deviating from the human perception mechanism. Inspired by human's coarse-to-fine decision-making paradigm, we formulate a novel Tree-Structured Policy based Progressive Reinforcement Learning (TSP-PRL) framework to sequentially regulate the temporal boundary by an iterative refinement process. The semantic concepts are explicitly represented as the branches in the policy, which contributes to efficiently decomposing complex policies into an interpretable primitive action. Progressive reinforcement learning provides correct credit assignment via two task-oriented rewards that encourage mutual promotion within the treestructured policy. We extensively evaluate TSP-PRL on the Charades-STA and ActivityNet datasets, and experimental results show that TSP-PRL achieves competitive performance over existing state-of-the-art methods. | 10.1609/aaai.v34i07.6924 | [
"https://arxiv.org/pdf/2001.06680v1.pdf"
] | 210,839,610 | 2001.06680 | b8e744b1f33b1f3f53fcffb1dafd592e992694ff |
Tree-Structured Policy based Progressive Reinforcement Learning for Temporally Language Grounding in Video
Jie Wu
Sun Yat-sen University
Guanbin Li [email protected]
Sun Yat-sen University
Si Liu [email protected]
Beihang University
Liang Lin [email protected]
Sun Yat-sen University
DarkMatter AI Research
Tree-Structured Policy based Progressive Reinforcement Learning for Temporally Language Grounding in Video
Temporally language grounding in untrimmed videos is a newly-raised task in video understanding. Most of the existing methods suffer from inferior efficiency, lacking interpretability, and deviating from the human perception mechanism. Inspired by human's coarse-to-fine decision-making paradigm, we formulate a novel Tree-Structured Policy based Progressive Reinforcement Learning (TSP-PRL) framework to sequentially regulate the temporal boundary by an iterative refinement process. The semantic concepts are explicitly represented as the branches in the policy, which contributes to efficiently decomposing complex policies into an interpretable primitive action. Progressive reinforcement learning provides correct credit assignment via two task-oriented rewards that encourage mutual promotion within the treestructured policy. We extensively evaluate TSP-PRL on the Charades-STA and ActivityNet datasets, and experimental results show that TSP-PRL achieves competitive performance over existing state-of-the-art methods.
Introduction
We focus on the task of temporally language grounding in a video, whose goal is to determine the temporal boundary of the segments in the untrimmed video that corresponds to the given sentence statement. Most of the existing competitive approaches (Anne Hendricks et al. 2017;Gao et al. 2017;Liu et al. 2018;Ge et al. 2019;Xu et al. 2019) are based on extensive temporal sliding windows to slide over the entire video or rank all possible clip-sentence pairs to obtain the grounding results. However, these sliding window based methods suffer from inferior efficiency and deviate from the human perception mechanism. When humans locate an interval window associated with a sentence description in a video, they tend to assume an initial temporal interval first, and achieve precise time boundary localization through cross-modal semantic matching analysis and sequential boundary adjustment (e.g., scaling or shifting).
Looking deep into human's thinking paradigm (Mancas et al. 2016), people usually deduce a coarse-to-fine deliberation process to render a more reasonable and interpretable decision in daily life. Namely, people will first roughly determine the selection range before making a decision, then choose the best one among the coarse alternatives. This topdown coarse-to-fine deliberation has been explored in the task of machine translation, text summarization and so on (Xia et al. 2017). Intuitively, embedding this mode of thinking into our task can efficiently decompose complex action policies, reduce the number of search steps while increasing the search space, and obtain more impressive results in a more reasonable way. To this end, we formulate a Tree-Structured Policy based Progressive Reinforcement Learning framework (TSP-PRL) to imitate human's coarse-to-fine decision-making scheme. The tree-structured policy in TSP-PRL consists of root policy and leaf policy, which respectively correspond to the process of coarse and fine decisionmaking stage. And a more reasonable primitive action is proposed via these two-stages selection. The primitive actions are divided into five classes related to semantic concepts according to the moving distance and directions: scale variation, markedly left shift, markedly right shift, marginally left adjustment and marginally right adjustment. The above semantic concepts are explicitly represented as the branches into the tree-structured policy, which contributes to efficiently decomposing complex policies into an interpretable primitive action. In the reasoning stage, the root policy first roughly estimates the high-level semantic branch that can reduce the semantic gap to the most extent. Then the leaf policy reasons a refined primitive action based on the selected branch to optimize the boundary. We depict an example of how TSP-PRL addresses the task in Figure 1. As can be seen in the figure, the agent first markedly right shift the boundary to eliminate the semantic gap. Then it resorts to scale contraction and marginally adjustment to obtain an accurate boundary.
The tree-structured policy is optimized via progressive re-Figure 1: An example showing how TSP-PRL addresses the task in an iterative refinement manner. A more interpretable primitive action is proposed by the tree-structured policy, which consists of root policy and leaf policy to imitate human's coarse-to-fine decision-making scheme. inforcement learning, which determines the selected single policy (root policy or leaf policy) in the current iteration while stabilizing the training process. The task-oriented reward settings in PRL manages to provide correct credit assignment and optimize the root policy and leaf policy mutually and progressively. Concretely, the external environment provides rewards for each leaf strategy and the root strategy does not interact directly with the environment. PRL measures the reward for the root policy from two items: 1) the intrinsic reward for the selection of high-level semantic branch; 2) the extrinsic reward that reflects how the subsequent action executed by the selected semantic branch influences the environment.
Extensive experiments on Charades-STA (Sigurdsson et al. 2016;Gao et al. 2017) and ActivityNet (Krishna et al. 2017) datasets prove that TSP-PRL achieves competitive performance over existing leading and baseline methods on both datasets. The experimental results also demonstrate that the proposed approach can (i) efficiently improve the ability to discover complex policies which can hardly be learned by flat policy; (ii) provide more comprehensive assessment and appropriate credit assignment to optimize the tree-structured policy progressively; and (iii) determine a more accurate stop signal at an iterative process. The source code as well as the trained models have been released at https://github.com/WuJie1010/TSP-PRL.
Related work
Temporally Language Grounding in Video. Temporally language grounding in the video is a challenging task which requires both language and video understanding and needs to model the fine-grained interactions between the verbal and visual modalities. Gao et al. (Gao et al. 2017) proposed a cross-modal temporal regression localizer (CTRL) to jointly model language query and video clips, which adopts sliding windows over the entire video to obtain the grounding results. Hendricks et al.(Anne Hendricks et al. 2017) designed a moment context network (MCN) to measure the distance between visual features and sentence embedding in a shared space, ranking all possible clip-sentence pairs to locate the best segments. However, the above approaches are either inefficient or inflexible since they carry out overlapping sliding window matching or exhaustive search. Chen et al. (Chen et al. 2018a) designed a dynamic single-stream deep architecture to incorporate the evolving fine-grained frame-by-word interactions across video-sentence modalities. This model performs efficiently, which only needs to process the video sequence in one single pass. Zhang et al. (Zhang et al. 2019) exploited graph-structured moment relations to model temporal structures and improve moment representation explicitly. He et al. ) first introduced the reinforcement learning paradigm into this task and treated it as a sequential decision-making task. Inspired by human's coarse-to-fine decision-making paradigm, we construct a tree-structured policy to reason a series of interpretable actions and regulate the boundary in an iterative refinement manner. Reinforcement Learning. Recently, reinforcement learning (RL) technique (Williams 1992) has been successfully popularized to learn task-specific policies in various image/videobased AI tasks. These tasks can be generally formulated as a sequential process that executes a series of actions to finish the corresponding objective. In the task of multi-label image recognition, Chen et al. (Chen et al. 2018b) proposed a recurrent attentional reinforcement learning method to iteratively discover a sequence of attentional and informative regions. Shi et al. (Shi et al. 2019) implemented deep reinforcement learning and developed a novel attention-aware face hallucination framework to generate a high-resolution Figure 2: The overall pipeline of the proposed TSP-PRL framework. The agent receives the state from the environment (video clips) and estimates a primitive action via tree-structured policy. The action selection is depicted by a switch over the interface ⊥ in the tree-structured policy. The alignment network will predict a confidence score to determine when to stop. face image from a low-resolution input. Wu et al. (Wu et al. 2019a) designed a new content sensitive and global discriminative reward function to encourage generating more concrete and discriminative image descriptions. In the video domain, RL has been widely used in temporal action localization (Yeung et al. 2016) and video recognition (Wu et al. 2019b). In this paper, we design a progressive RL approach to train the tree-structured policy, and the task-oriented reward settings contribute to optimizing the root policy and leaf policy mutually and stably.
The Proposed Approach Markov Decision Process Formulation
In this work, we cast the temporally language grounding task as a Markov Decision Process (MDP), which is represented by states s ∈ S, action tuple a r , a l , and transition function T : (s, a r , a l ) → s . a r and a l denote the action proposed by root policy and leaf policy, respectively. The overall pipeline of the proposed Tree-Structured Policy based Progressive Reinforcement Learning (TSP-PRL) framework is depicted in Figure 2. State. A video is firstly decomposed into consecutive video units (Gao et al. 2017) and each video unit is used to extract unit-level feature through the feature extractor ϕ v (Tran et al. 2015;). Then the model resorts to uniformly sampling strategy to extract ten unit-level features from the entire video, which are concatenated as the global video representation V g . For the sentence query L, the skipthought encoder ϕ s (Kiros et al. 2015) is utilized to generate the language representation E = ϕ s (L). When the agent interacts with the environment, the above features are retained. At each time step, the action executed by the agent will change the boundary and obtain a refined video clip. The model samples ten unit-level features inside the boundary and concatenate these features as the current video feature V c t−1 , t = 1, 2, ..., T max . We explicitly involve the normal-
A EG t = σ(E) V g , A EC t = σ(E) V c t−1 , A EL t = σ(E) L t−1 ,(1)
where σ denotes the sigmoid activation function and is the Hadamard product. The above gated attention features are concatenated and fed into a fully-connected layer φ to obtain the state representation s t :
s t = φ(A EG t , A EC t , A EL t )(2)
An additional GRU (Cho et al. 2014) layer is adopted to process the state features before feeding them into the treestructured policy, which manages to develop high-level temporal abstractions and lead to a more generalizable model. Hierarchical Action Space. In our work, the boundary movement is based on the clip-level and each boundary consists of a series of video clips. All primitive actions can be divided into five classes related to semantic concepts according to the moving distance and directions, which results in a hierarchical action space on the whole. These semantic concepts are explicitly represented as the branches into the tree-structured policy, resulting in five high-level semantic branches to contain all primitive actions in this task: scale variation, markedly left shift, markedly right shift, marginally left adjustment and marginally right adjustment. i) The scale variation branch contains four primitive actions: extending/shortening ξ times w.r.t center point. ξ is set to 1.2 or 1.5; 2) Three actions are included in the markedly left shift branch: shifting start point/end point/start & end point backward ν. ν is fixed to N/10, where N denotes the number of the clip of the entire video; 3) The actions in the markedly right shift branch is symmetry with the markedly left shift: shifting start point/end point/start & end point forward ν; 4) Except for the moving scale, the actions in the marginally left adjustment branch is similar to the markedly left shift branch: shifting start point/end point/start & end point backward Z frame; The size of Z is constrained by the video lengths; 5) The marginally right adjustment branch also involves three primitive actions: shifting start point/end point/start & end point forward Z frame. Tree-Structured Policy. One of our key ideas is that the agent needs to understand the environmental state well and reason a more interpretable primitive action. Inspired by human's coarse-to-fine decision-making paradigm, we design a tree-structured policy to decompose complex action policies and propose a more reasonable primitive action via twostages selection, instead of using a flat policy that maps the state feature to action directly (He et al. 2019). As shown in the right half of Figure 2, the tree-structured policy consists of a root policy and a leaf policy at each time step. The root policy π r (a r t |s t ) decides which semantic branch will be primarily relied on. The leaf policy π l (a l t |s t , a r t ) consists of five sub-policies, which corresponds to five high-level semantic branches. The selected semantic branch will reason a refined primitive action via the corresponding sub-policy. The root policy aims to learn to invoke the correct sub-policy from the leaf policy in the following different situations: (1) The scaling policy should be selected when the scale of predicted boundary is quite mismatched with the ground-truth boundary;
(2) When the predicted boundary is far from the ground-truth boundary, the agent should execute the left or right shift policy; (3) The primitive action should be sampled from the left or right adjust policy when most of the two boundaries intersect but with some deviation. At each time step, the tree-structured policy first samples a r t from root policy π r to decide the semantic branch: a r t ∼ π r (a r t |s t ).
(3) And a primitive action is sampled from the leaf policy π l related to the selected semantic branch: a l t ∼ π l (a l t |s t , a r t ).
(4)
Tree-Structured Policy based Progressive Reinforcement Learning
Rewards. Temporal IoU is adopted to measure the alignment degree between the predicted boundary [l s , l e ] and ground-truth boundary [g s , g e ]:
U t = min (g e , l e t ) − max (g s , l s t ) max (g e , l e t ) − min (g s , l s t )
.
The reward setting for this task should provide correct credit assignment, encouraging the agent to take fewer steps to obtain accurate grounding results. We define two taskoriented reward functions to select an accurate high-level semantic branch and the corresponding primitive action, respectively. The first reward r l t is the leaf reward, which reveals the influence of the primitive actions a l t to the current environment. It can be directly obtained in the environment through temporal IoU. We explicitly provide higher leaf reward when the primitive action attempts to obtain better grounding results and the temporal IoU is higher than 0.5:
r l t = ζ + U t U t > U t−1 ; U t > 0.5 ζ U t > U t−1 ; U t ≤ 0.5 −ζ/10 U t−1 ≥ U t ≥ 0 −ζ otherwise ,(6)
where ζ is a factor that determines the degree of reward. The second reward is the root reward r r t , which should be determined deliberately since the action executed by root policy does not interact with the environment directly. To provide comprehensive assessment and correct credit assignment, r r t is defined to include two items: 1) the intrinsic reward term that represents the direct impact of a r t for semantic branch selection and 2) the extrinsic reward term reflects the indirect influence of the subsequent primitive action executed by the selected branch for the environment. In order to estimate how well the root policy chooses the high-level semantic branch, the model traverses through all possible branches and reasons the corresponding primitive actions to the environment, which results in five different IoU. The max IoU among these five IoU is defined as U max t .
Then the root reward r r t is designed as follow:
r r t = ζ intrinsic reward item + U t − U t−1 extrinsic reward item U t = U max t U t − U max t intrinsic reward item + U t − U t−1 extrinsic reward item otherwise ,(7)
where U 0 denotes the temporal IoU between initial boundary and the ground-truth boundary. The diagram of how the root reward and leaf reward are obtained in the framework is depicted in Figure 3. Progressive Reinforcement Learning. Progressive Reinforcement Learning (PRL) is designed on the basis of the advantage actor-critic (A2C) (Sutton and Barto 2018) algorithm to optimize the overall framework. Policy function π r (a r
Lroot(θπr ) = − 1 M M m=1 Tmax t=1 [logπ r (a r t |st)(R r t − V r (st)) + αH(π r (a r t |st))],(8)L leaf (θ π l ) = − 1 M M m=1 Tmax t=1
[logπ l (a l Figure 3: An illustration of how the tree-structured policy works iteratively. The solid blue line represents how the root reward and leaf reward are obtained from the proposed framework.
V l (s t , a r t ) denote the advantage functions in the A2C setting. H() is the entropy of policy networks and the hyperparameters α controls the strength of entropy regularization term, which is introduced to increase the diversity of actions. θ π r and θ π l are the parameters of the policy networks. Here, the model only back-propagates the gradient for the selected sub-policy in leaf policy. The reward of the following-up steps should be traced back to the current step since it is a sequential decision-making problem. The accumulated root reward function R r t is computed as follows:
R r t = r r t + γV r (s t ) t = T max r r t + γR r t+1 t = 1, 2, ..., T max − 1 ,(10)
where γ is a constant discount factor and the accumulated leaf reward R l t is obtained in a similar way. In order to optimize the value network to provide an estimation of the expected sum of rewards, we minimize the squared difference between the accumulated reward and the estimated value, and minimize the value loss:
L root (θ V r ) = 1 M M m=1 Tmax t=1 (R r t − V r (s t )) 2 , L leaf (θ V l ) = 1 M M m=1 Tmax t=1 (R l t − V l (s t , a r t )) 2(11)
where θ V r and θ V l are the parameters of the value networks.
Optimizing the root and leaf policies will simultaneously lead to the unstable training procedure. To avoid this, we design a progressive reinforcement learning (PRL) optimization procedure: for each set of K iterations, PRL keeps one policy fixed and only trains the other policy. When reaching K iterations, it switches the policy that is trained. The treestructured policy based progressive reinforcement learning can be summarized as:
ψ = i K mod 2,(12)L tree =ψ × [L root (θ π r ) + L root (θ V r )] +(1 − ψ) × [L leaf (θ π l ) + L leaf (θ V l )],(13)
where ψ is a binary variable indicating the selection of the training policy. i denotes the number of iterations in the entire training process. is the lower bound integer of the division operation and mod is the modulo function. These two policies promote each other mutually, as leaf policy provides accurate intrinsic rewards for root policy while the root policy selects the appropriate high-level semantic branch for further refinement of the leaf policy. The better leaf policy is, the more accurate intrinsic rewards will be provided. The more accurate the upper branch policy is selected, the better the leaf policy can be optimized. This progressive optimization ensures the agent to obtain a stable and outstanding performance in the RL setting. During testing, the treestructured policy takes the best actions tuple a r , a l at each time step iteratively to obtain the final boundary. Alignment Network for Stop Signal. Traditional reinforcement learning approaches often include stop signal as an additional action into the action space. Nevertheless, we design an alignment network to predict a confidence score C t for enabling the agent to have the idea of when to stop. The optimization of the alignment network can be treated as an auxiliary supervision task since the temporal IoU can explicitly provide ground-truth information for confidence score. This network is optimized by minimizing the binary crossentropy loss between U t−1 and C t :
L align = 1 M M m=1 Tmax t=1 [Ut−1 log σ(Ct)+(1−Ut−1) log(1−σ(Ct))].(14)
During testing, the agent will interact with the environment by T max steps and obtain a series of C t . Then the agent gets the maximum of C t , which indicates that the alignment network considers U t−1 has a maximal temporal IoU. So t − 1 is the termination step. The alignment network is optimized in the whole training procedure. The overall loss function in the proposed framework is summarized as:
L = L tree + λL align .(15)
where λ is a weighting parameter to achieve a tradeoff between two types of loss.
Experiments Datasets and Evaluation Metrics
Datasets. The models are evaluated on two widely used datasets: Charades-STA (Gao et al. 2017) and ActivityNet , we adopt two metrics to evaluate the model for this task. "IoU@ " means the percentage of the sentence queries which have temporal IoU larger than . "MIoU" denotes the average IoU for all the sentence queries.
Implementation Details
The initial boundary is set to L 0 = [N/4; 3N/4], where N denotes the clips numbers of the video. N/4 and 3N/4 denote the start and end clip indices of the boundary respectively. The parameters Z is set to 16 and 80 respectively for Charades-STA and ActivityNet Datasets. We utilize two mainstream structures of action classifiers (i.e., C3D (Tran et al. 2015) and Two-Stream )) for video feature extraction on Charades-STA dataset. For ActivityNet, we merely employ C3D model to verify the general applicability of the proposed approach. The size of the hidden state in GRU is set to 1024. In the training stage of TSP-PRL, the batch size is set to 16 and the learning rate is 0.001 with Adam optimizer. The factor ζ is fixed to 1 in the reward settings. The hyper-parameters α, γ and λ is fixed to 0.1, 0.4 and 1 receptively. For all experiments in this paper, we use K = 200 in TSP-PRL. T max is set to 20 to achieve the best trade off between accuracy and efficiency in the procedure of training and testing.
Experimental Results
Comparison with the state-of-the-art algorithms. In this subsection, we compare TSP-PRL with 12 existing state-of-the-art methods on the Charades-STA and Activi-tyNet datasets in Table 1. We re-implement ACRN (Liu et al. 2018 Our approach focuses more on localization optimization, and it is complementary to the above-mentioned feature modeling methods actually. On the one hand, TSP-PRL consistently outperforms these state-of-the-art methods, w.r.t all metrics with C3D feature. For example, our method improves [email protected] by 1.89% compared with the previous best (Xu et al. 2019) on the Charades-STA. For ActivityNet, the MIoU of TSP-PRL achieves the comparative enhancement over ABLR by 6.0%. MAN (Zhang et al. 2019) employs stronger I3D (Carreira and Zisserman 2017) to extract video features and obtain outstanding performance. Our method with the Two-Stream feature manages to improve [email protected] from 22.72% to 24,73% on the Charades-STA. On the other hand, TSP-PRL manages to obtain more flexible boundary, avoiding exhaustive sliding window searching compared with the supervised learningbased (SL) methods. SL methods are easy to suffer from overfitting and address this task like a black-box that lack of interpretability. While TSP-PRL contributes to achieving more efficient, impressive and heuristic grounding results.
(a) "Ours w/o TSP-10" with T grows (b) "Ours-10" with T grows (c) "Ours w/o TSP-10" with IoU increases (d) "Ours-10" with IoU increases Figure 4: The proportion curve of the selected semantic branch as time step (T ) Table 2: Comparison of the metrics (in %) of the proposed approach and four variants of our approach. "-j" denotes that we set the max episode lengths to j during testing.
Ablative Study
As shown in Table 2, we perform extensive ablation studies and demonstrate the effects of several essential components in our framework. The Charades-STA dataset adopts the Two-stream based feature and the ActivityNet dataset uses the C3D based feature. Analysis of Tree-Structured Policy. To validate the significance of the tree-structured policy, we design the flat policy (denote as "Ours w/o TSP") that removes the tree-structured policy in our approach and directly maps state feature into a primitive action. As shown in Table 2, flat policy declines [email protected] to 17.13%, 20.67%, and 22.40% at each level of T max , with a decrease of 5.72%, 4.06%, and 2.35% when compared with our approach. Furthermore, it's performance suffers from a significant drop as T max decreases, which reveals that the flat policy relies heavily on the episode lengths to obtain better results. However, our approach manages to achieve outstanding performance with fewer steps. In order to further explore whether the tree-structured policy can better perceive environment state and decompose complex policies, we summarize the proportion of the selected high-level semantic branch at each time step and IoU interval (0.05 for each interval). The percentage curves of two models ("Ours w/o TSP-10" and "Ours-10") are depicted in Figure 4. We can observe that the flat policy tends to choose the adjust based branches all the time and is not sensitive to the time step and IoU. However, our approach manages to select the shift based branches at first few steps to reduce the semantic gap faster. When the IoU increases or time step grows, the adjust based branches gradually dominant to regulate the boundary finely. Figure 4 clearly shows that tree-structured policy contributes to efficiently improving the ability to discover complex policies which can not be learned by flat policies. To sum up, it is more intuitive and heuristic to employ the tree-structured policy, which can significantly reduce the search space and provide efficient and impressive grounding results. Analysis of Root Reward. To delve deep into the significance of each term in the root reward, we design two variants that simply remove the intrinsic reward item (denotes as "Ours w/o IR") and extrinsic reward item (denotes as "Ours w/o ER") in the definition of the root reward. As shown in Table 2, removing the intrinsic reward term leads to an noticeable drop in performance. It indicates that the extrinsic reward item can not well reflect the quality of the root policy since this term is more relevant to the selected leaf policy. "Ours w/o ER" obtains 44.41% and 37.20% on [email protected] on two datasets respectively, but it is still inferior to our approach. Taking into account the direct impact (intrinsic reward) and indirect impact (extrinsic reward) simultaneously, our approach contributes to providing accurate credit assignment and obtaining a more impressive result. Analysis of Stop Signal. To demonstrate the effectiveness of the alignment network for stop signal, we design a variant (denote as "Ours w/o AN") that removes the alignment network and directly includes the stop signal as an additional action into the root policy. The baseline assigns the agent a small negative reward in proportion with the step numbers. As shown in Table 2, "Ours w/o AN" gets a less prominent performance, which may be due to the fact that it is difficult to define an appropriate reward function for the stop signal in this task. However, our approach manages to learn the stop information with stronger supervision information via the alignment network, and it significantly increases the performance of all metrics by a large margin.
Conclusions
We formulate a novel Tree-Structured Policy based Progressive Reinforcement Learning (TSP-PRL) approach to address the task of temporally language grounding in untrimmed videos. The tree-structured policy is invoked at each time step to reason a series of more robust primitive actions, which can sequentially regulate the temporal boundary via an iterative refinement process. The tree-structured policy is optimized by a progressive reinforcement learning paradigm, which contributes to providing the task-oriented reward setting for correct credit assignment and optimizing the overall policy mutually and progressively. Extensive experiments show that our approach achieves competitive performance over state-of-the-art methods on the widely used Charades-STA and ActivityNet datasets.
ized boundary L t−1 = [l s t−1 , l e t−1 ] into the state feature (He et al. 2019), where l s t−1 and l e t−1 denote the start point and end point respectively. Then the gated-attention (Chaplot et al. 2018) mechanism is applied to gain multi-modal fusion representation of verbal and visual modalities:
1 :
1The comparison performance (in %) with state-of-the-art methods. The approaches in the first group are supervised learning (SL) based approaches and methods of the second group are reinforcement learning (RL) based approaches. "-" indicates that the corresponding values are not available.(Krishna et al. 2017). Gao et al. (Gao et al. 2017) extended the original Charades dataset(Sigurdsson et al. 2016) to generate sentence-clip annotations and created the Charades-STA dataset, which comprises 12,408 sentence-clip pairs for training, and 3,720 for testing. The average duration of the videos is 29.8 seconds and the described temporally annotated clips are 8 seconds long on average. ActivityNet(Krishna et al. 2017) contains 37,421 and 17,505 video-sentence pairs for training and testing. The videos in ActivityNet are 2 minutes long on average and the described temporally annotated clips are 36 seconds long on average. ActivityNet dataset is introduced to validate the robustness of the proposed algorithm toward longer and more diverse videos. Evaluation Metrics. Following previous works(Gao et al. 2017;Yuan, Mei, and Zhu 2019)
), MAC (Ge et al. 2019) and RWM (He et al. 2019) and show their performance results in our experiments. The results of other approaches are taken from their paper. The well-performing methods, such as QSPN (Xu et al. 2019), ABLR (Yuan, Mei, and Zhu 2019) and MAN (Zhang et al. 2019) all delve deep into the multi-modal features representation and fusion between the verbal and visual modalities.
Table
grows and IoU increases. Correspondence between line color and semantic branch: 1) red : scale branch; 2) orange: left shift branch; 3) yellow: right shift branch; 4) dark blue: left adjust branch; 5) light blue: right adjust branch. Best viewed in color.Datasets
Charades-STA
ActivityNet
Metrics
[email protected]
[email protected]
[email protected]
[email protected]
Ours w/o TSP-10
17.13
38.06
32.09
49.35
Ours w/o TSP-20
20.67
41.31
34.39
51.96
Ours w/o TSP-30
22.40
43.38
35.32
52.77
Ours w/o IR
20.35
40.64
35.03
52.64
Ours w/o ER
23.18
44.41
37.20
55.78
Ours w/o AN
19.03
39.78
33.89
51.03
Ours-10
22.85
44.24
37.53
55.17
Ours-20
24.73
45.30
38.76
56.08
Ours-30
24.75
45.45
38.82
56.02
t |s t ) and π l (a l t |s t , a r t ) estimate the probability distribution over possible actions in the corresponding action space. These two policies are separate and each is equipped with a value approximator V r (s t ) and V l (s t , a r t ), which is designed to compute a scalar estimate of reward for the corresponding policy.Starting from the initial boundary, the agent invokes the tree-structured policy iteratively in the interaction process. We depict how the tree-structured policy works iteratively inFigure 3. From the figure, we can observe that the agent samples actions from root policy and leaf policy consecutively at each time step. The action will trigger a new state, which is fed into the tree-structured policy to execute the next actions. Given a trajectory in an episode Γ = { s t , π r (·|s t ), a r t , r r t , π l (·|s t , a r t ), a l t , r l t , t = {1, · · · , T max }}, PRL algorithm maximizes the objective of root policy L root (θ π r ) and leaf policy L leaf (θ π l ):
t |st, a r t )(R l t − V l (st, a r t )) + αH(π l (a l t |st, a r t ))],(9)where M denotes the size of a mini-batch and T max is the max time step in an episode. R r t − V r (s t ) and R l t −
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"Ida S Opstad \nDepartment of Physics and Technology\nUiT The Arctic University of Norway\nNO-9037TromsøNorway\n",
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] | [] | Labelfree nanoscopy encompasses optical imaging with a resolution in the 100-nm range using visible wavelengths. Here, we present a labelfree nanoscopy method that combines coherent imaging techniques with waveguide microscopy to realize a super-condenser featuring maximally inclined coherent darkfield illumination with artificially stretched wave vectors due to large refractive indices of the employed Si 3 N 4 waveguide material. We produce the required coherent plane wave illumination for Fourier ptychography over imaging areas 400 µm 2 in size via adiabatically tapered single-mode waveguides and tackle the overlap constraints of the Fourier ptychography phase retrieval algorithm two-fold: first, the directionality of the illumination wave vector is changed sequentially via a multiplexed input structure of the waveguide chip layout, and second, the wave vector modulus is shortend via step-wise increases of the illumination light wavelength over the visible spectrum. We test the method in simulations and in experiments and provide details on the underlying image formation theory as well as the reconstruction algorithm. While the generated Fourier ptychography reconstructions are found to be prone to image artefacts, an alternative coherent imaging method, rotating coherent scattering microscopy (ROCS), is found to be more robust against artefacts but with less achievable resolution. | 10.1364/oe.27.025280 | null | 202,141,785 | 1905.02401 | 21b344b0f5b27c0bb5981bf9e0d24bb9d330913c |
Super-condenser enables labelfree nanoscopy
Florian Ströhl *[email protected]
Department of Physics and Technology
UiT The Arctic University of Norway
NO-9037TromsøNorway
Department of Chemical Engineering and Biotechnology
University of Cambridge
CB3 0ASCambridgeUnited Kingdom
Ida S Opstad
Department of Physics and Technology
UiT The Arctic University of Norway
NO-9037TromsøNorway
Jean-Claude Tinguely
Department of Physics and Technology
UiT The Arctic University of Norway
NO-9037TromsøNorway
Firehun T Dullo
Department of Physics and Technology
UiT The Arctic University of Norway
NO-9037TromsøNorway
Ioanna Mela
Department of Chemical Engineering and Biotechnology
University of Cambridge
CB3 0ASCambridgeUnited Kingdom
Johannes W M Osterrieth
Department of Chemical Engineering and Biotechnology
University of Cambridge
CB3 0ASCambridgeUnited Kingdom
Balpreet S Ahluwalia
Department of Physics and Technology
UiT The Arctic University of Norway
NO-9037TromsøNorway
Clemens F Kaminski
Department of Chemical Engineering and Biotechnology
University of Cambridge
CB3 0ASCambridgeUnited Kingdom
Super-condenser enables labelfree nanoscopy
10.1364/OE.27.025280Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Labelfree nanoscopy encompasses optical imaging with a resolution in the 100-nm range using visible wavelengths. Here, we present a labelfree nanoscopy method that combines coherent imaging techniques with waveguide microscopy to realize a super-condenser featuring maximally inclined coherent darkfield illumination with artificially stretched wave vectors due to large refractive indices of the employed Si 3 N 4 waveguide material. We produce the required coherent plane wave illumination for Fourier ptychography over imaging areas 400 µm 2 in size via adiabatically tapered single-mode waveguides and tackle the overlap constraints of the Fourier ptychography phase retrieval algorithm two-fold: first, the directionality of the illumination wave vector is changed sequentially via a multiplexed input structure of the waveguide chip layout, and second, the wave vector modulus is shortend via step-wise increases of the illumination light wavelength over the visible spectrum. We test the method in simulations and in experiments and provide details on the underlying image formation theory as well as the reconstruction algorithm. While the generated Fourier ptychography reconstructions are found to be prone to image artefacts, an alternative coherent imaging method, rotating coherent scattering microscopy (ROCS), is found to be more robust against artefacts but with less achievable resolution.
Introduction
Conventional nanoscopy, optical microscopy with resolution below 100 nm, is based on fluorescence [1]. Often listed advantages of nanoscopy, especially in comparison to electron microscopy, are the simple sample preparation, live-cell compatibility, and molecular specificity. Though live-cell compatible, the introduction of fluorescent labels onto the molecular structures of interest are in living cells likely to cause both functional and structural aberrations, potentially leading to false conclusions, and is also associated with problems like photobleaching and phototoxicity, variable label specificity, imaging-and image reconstruction-related artifacts, and lengthy optimization protocols [2,3]. The advantage of label specificity also has its downside of excluding (ultra)-structural context of the specifically labeled structure, although this can be alleviated to some degree via multi-channel labeling. Synergistic approaches combining the advantages of label specificity from conventional nanoscopy together with ultra-structural context obtained via labelfree nanoscopy, could bring many new insights about cellular functions, especially as (contrary to correlative light and electron microscopy [4]), labelfree (optical) nanoscopy has the potential of also being applied to living cells and cellular systems. Excluding methods that have not gone beyond the proof-of-principle stage like hyperlensing [5] or superoscillation microscopy [6], suitable labelfree methods that have the potential to provide nanoscopic resolution can be sorted broadly into four groups: orophore properties that are required for ultra-high resolution nanoscopy like photoswitching [7] are normally not present in intrinsically fluorescent samples, structured illumination microscopy has been shown to resolve features in the 150 nm regime in unlabeled retinal tissue [8]. Note that intrinsic fluorescence is a property present only in some but not all samples.
2.
Nearfield scanning optical microscopy [9], which rasters a sample with an effective resolution below 100 nm. Akin to electron microscopy, this scanning optical approach has a low through-put and is challenging to combine with fluorescence-based nanoscopy.
3. Deep ultra-violet microscopy -a theoretically simple approach as resolution scales linearly with employed imaging wavelength. However, the limited availability and performance of optical components in this spectral range as well as the high phototoxicity associated with ultraviolet radiation offset the benefits gained by wavelengths below 400 nm illumination.
4.
Fourier ptychography, FP [10], a technique specifically developed for improving digital pathology [11].
In FP the sample is illuminated and imaged sequentially with plane waves from a multitude of directions that densely sample the illumination condenser numerical aperture (N A c ). The generated set of images is then synthesized into a super-resolved amplitude and phase image of resolution ∆x given by
∆x = λ N A c + N A o(1)
via a dedicated phase retrieval algorithm [10]. It is well known that in conventional microscopy the condenser N A c used for illumination should be matched to (or even slightly below) the objective N A o [12], resulting in an effective maximal resolution of ∆x = λ 2N A o , the Abbe resolution limit [12]. Crucially in FP, a numerical aperture N A c of the condenser can be larger than the objective's N A o in order to increase resolution with respect to the detection objective.
Theory
As the illumination in FP is coherent, the complex field (amplitude and phase) of the sample is probed rather than its intensity as in incoherent imaging. Even though the effective aperture of a coherent microscope is half the size of an incoherent one's, plane wave illumination at oblique angles re-positions the sample's field in the aperture, thus giving access to finer details as visualized in Fig. 1a-c. Further, because the down-modulation of sample spatial frequency information with the illumination's spatial frequency occurs before being low-pass filtered by the objective aperture, access to information beyond 2N A o is possible given high lateral spatial frequency of the illumination at greater angles than conventionally associated with N A o . To extract those finer details, multiple images are acquired sequentially using illumination angles spanning the entire condenser N A c , and combined into a super-resolved image computationally (see Fig. 1d). Despite its potential, an extended condenser N A c has so far almost exclusively been used to increase the space-bandwidth product [10] rather than performing nanoscopy. This is because, assuming the highest N A available for both illumination and detection, the resolution caps at the incoherent resolution limit and is hence in the orderhttps://www.overleaf.com/project/ of 200 nm. The largest illumination N A so far was demonstrated using an oil immersion condenser featuring an N A of 1.2 [13]. To resolve nanoscopic structures using FP, a super-condenser allowing illumination with spatial frequencies exceeding those offered by the best immersion objectives is necessary. In the following, we show how such a super-condenser can be implemented in the form of a photonic waveguide chip in conjunction with multi-spectral illumination. We show simulation results of the proposed method and perform proof-of-concept imaging of sub-diffraction-limit sized metal-organic framework (MOF) clusters. Figure 1e outlines the fundamental mechanism of the proposed super-condenser, which aims to optimize the magnitude of the lateral illumination wave vector components and to provide as dense coverage of the virtual condenser pupil as possible. To achieve largest lateral wave vector components, the illumination administered to the sample via photonic waveguides is intrinsically orthogonal to the detection objective and thus allows to maximize the wave vector magnitude geometrically. Furthermore, akin to microscopy with immersion media, the illumination wave vector is stretched by a factor determined by the refractive index of the waveguide material. Thus, by apt choice of material, a further tremendous increase in wave vector magnitude and image resolution can be achieved. To illustrate, a conventional fluorescence microscope imaging GFP (λ ex/em = 488/512 nm) with a high-performance 0.95 NA air objective offers a maximal resolution of λ em /2N A o ≈ 270 nm. The same sample imaged with the proposed super-condenser featuring Si 3 N 4 waveguides with refractive index n ≈ 2.08 yields a theoretical resolution of λ ex /(n+ N A o ) ≈ 160 nm. Furthermore, it should be considered that incoherent microscopy techniques (like fluorescence or brightfield) have a strongly damped optical transfer efficiency for higher spatial frequencies, whereas the effective transfer function produced by Fourier ptychography has close to unity transmission strength, thus obtaining greatly enhanced contrast for finer structural details (as visualised in Fig. 1) [14]. This feature is mainly used by an alternative coherent imaging technique, rotating coherent scattering microscopy (ROCS) [14], which averages over multiple azimuthal orientations to mainly increase contrast. The benefit of ROCS is then its robustness against reconstruction artefacts, which render the technique an interesting alternative to FP as is shown below.
Methods
Waveguide design
Photonic integrated systems out of high-index contrast materials have been used for various applications over the last decade and have recently been developed further for sensing tasks in the visible wavelength range [16][17][18][19][20][21][22][23][24][25]. Building on this previous work, the super-condenser waveguides were designed to provide a high refractive index (>2) and sufficiently high-intensity single-mode illumination (in the mW range) of multiple visible wavelengths over a large area (>100 µm 2 ), which can be switched between several distinct directions spanning a full circle. A sketch providing an overview and details on the waveguide geometry are presented in Figure 2a. Si 3 N 4 was used as guiding material in rib waveguide geometry [18,19]. A total slab thickness of 150 nm was chosen, which realizes a beneficial trade-off between reach of the evanescent field (a) Chip design: 8 inputs deliver visible light at various illumination angles to the imaging region, while simultaneously ensuring single-mode characteristics through bending (bend radii ≥ 2 mm) and adiabatic tapering. (b) Waveguide production steps: the surface of a silicon waver is thermally oxidized and subsequently covered with a layer of silicon nitride via low-pressure chemical vapor deposition (LPCVD). The waveguides structure is then created via photolithography and reactive ion etching (RIE) to produce the required 4 nm-sized rib. A protective wall between the waveguides is created via plasma-enhanced CVD of silicon oxide followed by LPCVD of polycrystalline silicon. RIE followed by chemical etching using hydrofluoric acid (HF) uncovers the waveguides again [15]. and coupling efficiency (coupling efficiency increases with waveguide thickness, whereas the evanescent field penetration depth decreases). To ensure a homogeneous field distribution over the waveguide surface while simultaneously keeping the guided light as mono-directional as possible, the waveguide geometry was designed to fulfil single-mode conditions [15,26]. The necessary waveguide design was carried out using the software package FIMMWAVE (Photon Design, Oxford, UK). In addition to optimising the slab thickness, simulations at wavelengths spanning the visible region (488 nm, 561 nm, 660 nm) showed that a 4 nm etched rib is necessary to enable single-mode condition at 488 nm, with larger wavelengths being feasible at taller rib heights [15]. Hence, an etched rib height of 4 nm was chosen. In width, the waveguide guide structures were limited to be no smaller than 1 µm, allowing the use of conventional photolithography with homogeneous results over a full 4" wafer, while still providing single-mode characteristics after coupling. Adjacent to the coupling region on the chip edge, the waveguides are designed to broaden out to enable larger fields-of-view. To maintain the initial single-mode conditions, broadening to a waveguide width of 25 µm was performed via adiabatic tapering with a linear taper of 1.8 mm length, which was found to achieve a high guiding efficiency (>90%) for all used wavelengths [26]. To enable illumination from multiple directions, eight inputs with a spacing of 127 µm were realized that bend towards a common imaging area, with no bend radius smaller than 2 mm. Simulations showed that less than 1 dB loss at 90 • for a 2 mm bend radius occurs for 488 nm and 561 nm, while 660 nm light is attenuated by~3.5 dB. The spacing of 127 µm was set between the arms to match conventional fiber-array adaptors that are standard in the telecommunication industry and could allow for fastest switching between the waveguide inputs in future set-ups. Since the guided light is not tightly confined to the rib due to its small dimensions, slab propagation is prone to occur at the chip input and at curvatures. The light guided in the slab can create cross-talk between neighboring rib structures, reducing the contrast between the illumination angles. This is avoided by a layer between the rib structures consisting of 200 nm of SiO 2 with 100 nm of polycrystalline silicon (p-Si) on top. The high-refractive index of p-Si effectively diverts the light coupled at the slab to itself, preventing it from leaking into the neighbored ribs. This absorbing layer follows along the rib structures, with a constant gap of 5 µm to the rib wall.
Production steps
Waveguide chips were produced at the Institute of Microelectronics Barcelona (IMB-CNM, Spain). The essential production steps are summarized in Figure 2b and details of the fabrication optimization and process can be found elsewhere [27]. In short, a silica layer with a thickness of 2 µm was first grown thermally on a silicon chip, followed by the deposition of a Si 3 N 4 layer using low-pressure chemical vapor deposition (LPCVD) at 800 • C. Then, standard photolithography was employed to define the waveguide geometry using photoresist, followed by reactive ion etching (RIE) to fabricate the delicate 4 nm height of the waveguide rib necessary for single-mode guiding. This can be structured into four sub-steps: (1) resist spinning on the continuous Si 3 N 4 layer; (2) light exposure through a mask and chemical development to remove resist on non-desired areas around the waveguide structures; (3) RIE to desired etch depth; and (4) removal of remaining resist through plasma ashing and solvent washing. After removing the remaining photoresist the absorbing wall between the rib structures was created. For this 200 nm of SiO 2 were deposited by plasma-enhanced chemical vapour depostion (PECVD) followed by 100 nm of polycrystalline silicon deposited by LPCVD. RIE removed the the p-Si and ca. 190 nm of the SiO 2 above the waveguide structures, with the remaining SiO 2 being etched away using hydrofluoric acid to prevent damage to the rib structures.
Optical microscope
For experiments, a custom-built upright microscope was used, which was described in detail elsewhere [28]. A 3D model of the system is shown in Fig. 2c and basic features are be reviewed here briefly. The microscope was based on a modular commercial system (CERNA, Thorlabs) and offers up to four LEDs with wavelengths centred at 385 nm, 490 nm, 565 nm, and 625 nm for episcopic illumination of the sample. The illumination light was produced in a four channel LED combiner (LED4D245, Thorlabs) and delivered to the main frame via a liquid light guide of 3 mm diameter (LLG0338-4, Thorlabs). To counter the broad emission of the 565 nm LED, a bandpass filter (#86-986, Edmund Optics) was installed in the LED combiner. The light guide output was focused using two lenses (AL2018-A and LBF254-040-A, Thorlabs) onto the front focal plane of a collimator lens (LBF254-040-A, Thorlabs), where an iris (SM1D12D, Thorlabs) was used to create Köhler illumination. Using a tube lens (LBF254-100-A, Thorlabs) in a 4f-system with the collimator lens, the LED illumination was then focused via a semi-transparent mirror (which was removed when using evanescent chip-illumination) onto the back focal plane of the objective lens (UPLSAPO40X2, Olympus).
Evanescent illumination of the sample was achieved by focusing laser light into the waveguide chip inputs. Three laser lines at 488 nm with 150 mW (OBIS 488LS, Coherent), 561 nm with 150 mW (OBIS 561LS, Coherent), and 647 nm with 120 mW (OBIS 647LX, Coherent) were used and custom combined using dichroic mirrors (ZT514rdc and ZT594rdc, Chroma) and coupled into a single-mode fibre (S405-XP-custom, Thorlabs) using a commercial fiber coupler (PAF2-A4A, Thorlabs). The fiber output was collimated using a reflective collimator (RC04FC-P01, Thorlabs) and focused with a 50× 0.5NA objective (LMPLFLN-BD 50X, Olympus) onto the inputs of the waveguide chip. To aid coupling into the single-mode waveguides, the reflective collimator and the focusing objective were installed on a differential micrometer stage with additional piezo fine controls (MDT630B/M with MAX302/M, Thorlabs).
The chip itself was resting on a vacuum stage (HWV001) and held in place with low vacuum. A three-axis long travel block (RB13M/M, Thorlabs) allowed coarse alignment of the chip with respect to the differential micrometer stage and lateral translation of the whole assembly was realized via a motorized two-axis translation stage (PLS-XY, Thorlabs). Focusing onto the sample was achieved by translating the imaging objective via a 1 inch travel module (ZFM2020, Thorlabs) with micrometer precision. Finally, the light captured by the imaging objective was focused by a tube lens (TTL180-A, Thorlabs) onto a CMOS camera (UI-3080CP-M-GL Rev.2, IDS), which featured a 3.45 µm pixel size. In all imaging experiments, the laser power and exposure time of the camera were set such that the full bit depth (12bit) of the camera was used.
Reconstruction algorithm
As our technique is based on Fourier ptychography it uses a phase retrieval algorithm [10], which was slightly modified and is depicted in the box of Fig. 3. The algorithm aims to invert the imaging pipeline and thus requires a detailed description of the image formation. Let the raw images be denoted as i k 0 ,k c (x), with k 0 being the illumination wave vectors' lateral component, and k c the coherent cut-off frequency of the used wavelength, i.e. k c = N A o λ . Using a coherent imaging model on the sample's complex field s(x), which is illuminated with plane waves featuring wave vector k 0 and imaged by an objective characterized by the coherent point spread function h c (x), the coherent image formation equation reads
i k 0 ,k c (x) = |[s(x) × exp (−ik 0 × x)] ⊗ h c (x)| 2 .(2)
In this equation, i is the imaginary unit, and ⊗ is the convolution operator. The coherent PSF h c (x) can be defined easily via its Fourier transform H c (k), which is described by a circle centred on the spatial frequency coordinate origin and with value one inside and zero outside a radius k c , the coherent cut-off frequency. The goal of the phase retrieval algorithm is to find the amplitude a(x) and phase φ(x) component of the complex sample s(x) = a(x) × exp(iφ(x)). Three pre-processing steps are performed: (1) the raw data is background corrected, (2) then low-pass filtered, and (3) finally an initial guess of the amplitudes a k 0 ,k c (x) is made. In analogy to the approach of Zheng [10] this is done by subtraction of a background estimate value b, and multiplication of a low-pass filter defined by the support of the incoherent optical transfer function to the image spectra I k 0 ,k c (k) to remove noise from outside the pass-band of the objective, i.e. beyond the incoherent cut-off spatial frequency. As the incoherent cut-off frequency is twice the coherent cut-off frequency, a scaled version of the coherent transfer function can be used, i.e.
H c k 2 . Note that Fourier analogues of real space functions, obtained via Fourier transform F will be denoted via capitalization, so e.g. H c (k) = F{h c (x)}, with k being the spatial frequency coordinate. The inverse Fourier transform is written as F − . After low-pass filtering, the real part of the square root is taken to approximate the field distribution that formed the recorded intensities:
a k 0 ,k c (x) = F − F i k 0 ,k c (x) − b × H c k 2 .(3)
The phase retrieval part of the algorithm is then initialized using the estimated amplitude of a brightfield image as starting guess f 0 (x) [29] for the high-resolution Fourier ptychography image. We note that in conventional FP any starting guess can be used [10]. In each iteration up to a total of n rounds, f j (x) is sequentially updated for all available coherent illumination wave vectors k 0 . The sequence is chosen such that the respective sub-sampled parts of Fourier space (which are centred around k 0 and with radius k c ), are spiralling out from lower to higher spatial frequencies. Formally in the algorithm, the individual updates are performed in three steps. First, a temporary low-resolution image t j (x) is calculated from the Fourier ptychography estimate f j (x) for the current respective illumination featuring wave vector k 0 , cut-off frequency k c and amplitude transfer function H c as
t j k 0 ,k c (x) = F − F j (k − k 0 ) × H c .(4)
The phase Φ(t(x)) of the temporary low-resolution image t j (x) is taken as an estimate of the phase distribution φ(x) of the sample s(x). Hence, only the amplitude of t j (x) is updated, i.e. replaced by the estimated amplitude a k 0 ,k c (x) of the respective pre-processed raw image
t j+1 k 0 ,k c (x) = a k 0 ,k c (x) × exp iΦ(t j (x)) .(5)
The updated temporary image's spectrum T j+1 k 0 ,k c (k) is successively used to replace the respective region in Fourier space of the Fourier ptychography image's spectrum F j+1 (k). This region is centred on k 0 within a support area defined by the coherent transfer function H c (k) of that respective wavelength:
F j+1 (k) = F j (k) × (1 − H c (k − k 0 ))) + H c (k − k 0 )) × T j+1 k 0 ,k c (k − k 0 ).(6)
After each loop the lower spatial frequencies can be updated using an incoherent brightfield image in analogy to the updating step with evanescent illumination. Note that this step is different to conventional FP [10] and was implemented to gain some form of access to oblique illumination information, which is necessary to avoid reconstruction artefacts [30,31]. After n full loops, the final Fourier ptychography image f (x) is produced via apodization and a successive inverse Fourier transform with enlarged Fourier support (potentially made to fit via additional zero-padding) to yield a smoother transform result
f (x) = F − {apo(pad(F(k)))} .(7)
For comparison to brightfield data, an intensity image can be created via squaring of the amplitude part of the Fourier ptychography reconstruction. The presented algorithm was tested on simulated data, as displayed in the top of Fig. 3 for a ground truth (GT) input and a successful FP intensity reconstruction.
Experimental results
To test waveguide-based FP experimentally, we imaged clusters of metal-organic frameworks (MOFs). The imaged MOFs belong to the group of Zirconium-MOFs with gold nano-rod core and have a small size distribution centered around 200 nm [32]. To ensure adherence of the MOFs to the waveguides, the waveguide chip was plasma-treated for 40 s at 40 W using a 0.35 mbar oxygen atmosphere. Then, a highly diluted aqueous solution of MOFs was drop-casted onto the waveguide chip imaging area and left to dry under a slight angle to provide a more even distribution of the particles. Note that the sample can be removed and the super-condenser cleaned for reuse via suitable sample-dependent solvents (e.g. acetone). Repeated plasma-treatment (required to aid sample adherence) will, however, destroy the single-mode characteristics of the waveguides. After imaging with the super-condenser, a ground truth image of the sample was generated via atomic force microscopy (AFM) using a commercial system (Bioscope RESOLVE, Bruker). The AFM was operated in tapping mode and RTESPA probes (Bruker) with a nominal spring constant of 6 N/m and resonant frequency 150 kHz. A line scanning resolution of 256 lines with 256 samples/line for 50×50 µm 2 was used to generate an overview and 1168 lines with 1168 samples/line for 20×20 µm 2 were used for greater detail of selected areas. To counter drift between frames, the individual raw frames were further aligned to each other using semi-automated alignment via the image processing software line ROI image alignment in Fiji [33]. As shown in Fig. 4, we retrieved images displaying both enhanced contrast and features beyond the incoherent Abbe diffraction limit. In panel (a), an overview of the waveguide chip geometry is shown using a brightfield reflectance image. As shown in (c), the sample is only barely visible in brightfield incoherent illumination image (for the same objective), which displays an unresolved cluster of 370 nm size in terms of full width at half maximum of a Gaussian fit (data not shown). Displayed in panel (d), the underlying distribution of individual MOFs in this cluster is shown using atomic force microscopy (AFM) and finest scanning reveals dense clustering of particles. The same distribution of clusters that is visible in the AFM image is also present in the Fourier ptychography image in panel (c), which is, however, plagued by image artefacts that do not resolve individual gold nanorod cores in the clusters. An alternative method for reconstruction of evanescent darkfield scattering data is rotating coherent scattering (ROCS) microscopy [34], which is shown in Fig. 5. Here, an image is simply generated by summation of the raw images over all azimuthal angles. Note that in ROCS no wavelength 'sweep' as in waveguide-based FP is necessary but the achievable resolution gains are small (according to the Rayleigh criterion) [14]. It is found that the ROCS imaging procedure results in dramatic enhancement of image contrast (compare Fig. 5c and d), and individual clusters of MOFs can easily be discerned. Concurrently, it offers resolution in accordance with the Abbe criterion.
Discussion and conclusion
Care must be taken in interpreting the results of FP as the discernible spots produced by the phase retrieval algorithm cannot be associated to individual particles in the AFM recordings. For instance, although the clusters visible in FP can be mapped onto the AFM ground truth, the individual particles are slightly misplaced or even missing completely. Furthermore, the full width at half maximum (FWHM) of reconstructed 'individual' particles is smaller then the theoretically achievable FWHM (∼ 50 nm versus ∼ 165 nm) and thus likely represents a reconstruction artefact. This can have multiple reasons. Firstly, the displacement might be traced back to drift of the sample during raw data acquisition. A shift of sample information even in the nanometer range might thus produce artefacts causing erroneous particle localization. Although drift was compensated for computationally, it cannot be avoided completely. Secondly, the gold core of the MOFs is asymmetric and causes scattering preferentially in certain directions. Paired with variable signal strength for different illuminations caused by higher losses of the waveguides in longer arms, this could lead to some particles outshining others in the reconstruction. Also note that the amplitude of the scattered light is wavelength dependent. Despite trying to adjust for this via variation of the laser illumination intensity and the exposure time of the camera, a limited maximum number of photons in certain raw frames restricted the achievable signal-to-noise ratio (SNR). Residual wavelength-dependent waveguide autofluorescence increases the challenge of limited SNR additionally. In the future, more advanced algorithms originally developed for conventional Fourier ptychography could be adopted to computationally alleviate some of these concerns [35]. Furthermore, axial chromatic offset could not be accounted for, which might stem from residual imperfections of the employed apochromatic objective and achromatic tube lens. This might be tackled via fine z-stepping of the objective (not possible in the presented set-up which offers no finer than 0.5 µm steps) and post-acquisition alignment or pre-acquisition calibration (potentially using multiple cameras to increase acquisition speeds). An additional complication is posed by the necessary image reconstruction, which had to be based on evanescent illuminations as well as a single bightfield image. In simulations, this restriction in terms of raw data was found to be only sufficient for artefact-free intensity reconstructions but had limited potential to extract the sample's phase [31]. The algorithm is further especially challenged when noise-corrupted data is used or when only little overlap of spectral information between raw images is present. Still, the produced data is promising for the young field of labelfree chip-nanoscopy. In contrast, the processing procedure of ROCS allowed visibly artefact-free imaging, albeit with less resolution. Looking ahead, a multitude of further developments are thinkable, both conceptually and practically. An example would be a change of the chip material. The catch here is the unavailability of established production methods for single-mode waveguides of more exotic character. Nevertheless, suitable candidates for alternative waveguide materials that also keep propagation losses low at short wavelengths are Ta 2 O 5 [36], TiO 2 [37], or graphene [38])see Fig. 6. TiO 2 is an especially interesting material, as it transmits even in parts of the near ultra-violet range, which would allow sub-100nm resolution:
∆x T iO = 405 nm (1.49 + 2.66) = 97.6 nm.(8)
A further current conceptual bottleneck, the limited field of view, could be alleviated through use of a slab region at the imaging area illuminated by un-tapered single-mode waveguides. Although initially propagating as circular waves (the two-dimensional analog of spherical waves produced by free-space point sources), the waves emanating from the waveguide outlets would be sufficiently close to plane waves already after a travelled distance of few wavelengths and thus suitable for quasi-coherent imaging. For a further field of view enhancement, illumination from a limited number of directions (potentially even only single-sided) could be feasible for amplitude-only samples [39] and would simplify both waveguide geometry and deliverable powers to the imaging area due to reduced bending losses. More speculatively, on-chip lasers [40] are an option to simplify the microscopy set-up and circumvent coupling losses. An intriguing alternative could further be the use of a broadly emitting fluorescent film to generate the illumination light with successive narrow-band emission filtering as proposed by Pang et al [30]. Although the illumination in this set-up is not coherent, FP algorithms have been developed that can be applied [29]. To tackle drift of the sample during image acquisition, a fully automated coupling procedure might speed up slow manual coupling of various wavelengths and to different inputs. A possible future improvement in this respect is input-multiplexing which is feasible via conventional fiber-array adaptors. These are standard in the telecommunication industry but are to date mostly available for infrared wavelengths.
In conclusion, we have developed a new microscopy illumination scheme for labelfree nanoscopy that combines coherent imaging with waveguide microscopy to realize a supercondenser. The waveguide geometry allows the use of maximally inclined coherent darkfield illumination and additionally makes use of the large refractive index of Si 3 N 4 as waveguide material to further double the illumination wave vector amplitudes as compared to air. We validated our method in silico and tested it in experimental imaging of metal organic frameworks that contained gold nano-rod cores. As shown by atomic force microscopy, we were able to image MOF clusters successfully with ROCS and found that FP produces images with visible image artefacts. Taken together, Fourier ptychography in combination with enlarged illumination wave vectors can be a promising avenue to enable nanoscopic imaging without the requirement of extrinsic labels as long as the reconstruction procedures can be improved. The more robust ROCS processing presents meanwhile a suitable alternative and could potentially be used in conjunction with fluorescence on-chip superresolution microscopy [41,42]. As such a combination would consist purely of widefield imaging techniques, a considerable increase in throughput is achievable as compared to scanning approaches like nearfield scanning optical microscopy.
Fig. 1 .
1Amplitude/modulation transfer functions using (a) coherent, (b) incoherent, and (c) oblique illumination. Amplitude transfer function sampling in (d) conventional FP and (e) waveguide-based Fourier ptychographic microscopy. E: electric field, I: intensity, O: objective, S: sample, and C: condenser. The arrow highlights the cut-off frequency for different imaging modalities.
Fig. 2 .
2Fig. 2. (a) Chip design: 8 inputs deliver visible light at various illumination angles to the imaging region, while simultaneously ensuring single-mode characteristics through bending (bend radii ≥ 2 mm) and adiabatic tapering. (b) Waveguide production steps: the surface of a silicon waver is thermally oxidized and subsequently covered with a layer of silicon nitride via low-pressure chemical vapor deposition (LPCVD). The waveguides structure is then created via photolithography and reactive ion etching (RIE) to produce the required 4 nm-sized rib. A protective wall between the waveguides is created via plasma-enhanced CVD of silicon oxide followed by LPCVD of polycrystalline silicon. RIE followed by chemical etching using hydrofluoric acid (HF) uncovers the waveguides again [15]. (c) The optical microscope as outlined in the main text: LED illuminator (LED), liquid light guide (L), fibre input for lasers (F), reflective collimator (R), vacuum stage (V), piezo stage (P), micrometer stage (M), sample stage (S), objectives (O 1/2 ), tube lens (T), (dichroic) mirrors (D 1/2/3 ), cameras (C 1/2/3 ).
(c) The optical microscope as outlined in the main text: LED illuminator (LED), liquid light guide (L), fibre input for lasers (F), reflective collimator (R), vacuum stage (V), piezo stage (P), micrometer stage (M), sample stage (S), objectives (O 1/2 ), tube lens (T), (dichroic) mirrors (D 1/2/3 ), cameras (C 1/2/3 ).
Fig. 3 .
3Phase-retrieval algorithm on simulated data. Details are provided in the text.
Fig. 4 .
4Imaging of metal-organic frameworks (MOFs) with FP. (a) Overview of the imaged region. (b) Raw evanescent scattering images under waveguide illumination with 488nm, 561nm, and 647nm laser light. (c) Brightfield image using sum of multiple LED wavelengths. (d) Intensity image created by Fourier ptychography. The red arrow in the inlay might be mistaken for individual particles but is most likely an image reconstruction artefact as its full width at half maximum (FWHM) is smaller then the theoretically achievable FWHM (∼ 50 nm versus ∼ 165 nm). (e) Atomic force microscopy image (line levelling artefacts prohibit a clear view of individual particles). Inlays show a zoomed region of a cluster of MOFs. The overview image (a) measures 100×100 µm 2 and the scalebars in (c-e) are 1 µm and 100 nm in the inlays respectively.
Fig. 5 .
5Imaging of metal-organic frameworks (MOFs) with ROCS. (a) Overview of the imaged region. (b) Raw evanescent scattering images under waveguide illumination with a 488 nm laser. (c) Brightfield image using sum of multiple LED wavelengths. The red circle highlights a cluster that is only visible under darkfield illumination. (d) Intensity image created by ROCS with (e) zoom onto MOF clusters. Although it is not possible to discern individual particles, the elongated shape of the clusters is visualised by ROCS in good agreement with (f) the atomic force microscopy image of the same region. The overview image (a) measures 100 × 100 µm 2 and the scalebars in (b-f) are 1 µm.
Fig. 6 .
6Theoretically achievable resolution given in nm via different waveguide materials and substrate/immersion objective combinations (assuming shortest illumination wavelength of 445 nm).
Funding
European Molecular Biology Organisation (7411); Marie Skłodowska-Curie actions (836355); European Research Council (336716); Physical Sciences Research Council (EP/H018301/1); Med-
Vol. 27, No. 18 | 2 Sep 2019 | OPTICS EXPRESS 25285
Author contributions statementFS and BSA conceived the idea of waveguide-chip based labelfree nanoscopy using Fourier ptychography. FS further expanded the idea towards the use of ROCS and to use different wavelengths to fill Fourier space, performed simulations, analyzed the data. JCT and FTD designed, produced, and characterized the waveguides. ISO prepared samples. FS and ISO built the chip-microscope and performed waveguide imaging. IM performed AFM measurements. JWMO provided metal-organic frameworks. FS wrote the first version of the manuscript and all authors contributed towards the writing of the manuscript. CFK and BSA provided research tools and supported the project.DisclosuresBSA has applied for patent GB1705660.7 on an Optical component for generating a periodic light pattern. The other authors declare no competing interests.
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