A Appendix

Mutual information area laws for thermal free fermions

Abstract

We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel framework for computing determinants of Töplitz operators with smooth symbols, and for treating Töplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the one-dimensional lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put onto the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Töplitz determinants.

1 Introduction

How do the correlations of natural quantum states of many-body systems behave? If “natural” is taken to mean “generic” in the sense of a random pure state drawn from the Haar measure, then the answer to this question is: Subsystems will almost surely be very nearly maximally correlated with their complementary subsystems. However, this is not the situation that one is usually interested in in many-body and condensed-matter physics. Ground states of local Hamiltonians typically exhibit far less entanglement than that suggested by the previous argument. Indeed, there is a large body of evidence (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31); (32); (33), suggesting that ground states of gapped quantum many-body systems on lattices satisfy an area law (1) for the entanglement entropy. In other words, given the pure state of a lattice system, if one distinguishes a certain (connected) region of the lattice, then the von-Neumann (or Renyi) entropy of the reduced state associated with this region does not grow, as one might expect, like the number of degrees of freedom of this region (its ”volume”). Instead, it scales like the number of degrees of freedom on the boundary, hence as its “area”. Area laws have been shown for all gapped one-dimensional models with local interactions (24); (30); (31). Extensions to classes of higher-dimensional lattice models have also been obtained (12); (2); (29). These results and others therefore suggest that ground states of quantum many-body systems are much less entangled than they could be. These measures of entanglement – refined quantities revealing much more detailed information about the structure of correlations than more conventional correlation functions – in a way inherit the decay of correlations. This deep insight is also at the basis of classical efficient simulations of quantum many-body systems: The formulation and analysis of many-body systems in terms of matrix-product and other tensor network states have put this intuition on a solid theoretical footing (34); (35); (1). In numerical simulations of many-body systems, the importance of such approximations can hardly be overestimated. As a consequence, the last decade has seen an enormous amount of interest in the study of entanglement in quantum many-body systems in the condensed matter context and in issues of numerical simulation (34); (35); (1); (36), entanglement spectra (36); (37); (38); (39); (40); (41), and their relationship to quantifiers of topological order (38); (39); (40); (41).

Can this analysis be extended to thermal states? A moment of thought reveals that the entanglement entropy should in fact satisfy a volume law for thermal states. However, it turns out that the entanglement entropy is not the right quantity to grasp the correlations of a mixed quantum state. There exist many such measures, but the most natural and the most frequently adopted one is the mutual information. For a subset of sites of the lattice and its complement , it is defined as

 I(A:B)=S(ρA)+S(ρB)−S(ρ), (1)

where and are the reduced states of with respect to and , respectively. For pure states, the mutual information reduces to twice the entanglement entropy. One might then hope that this quantity fulfills an area law for thermal states of local hamiltonians on a lattice. This turns out to be true in fact for any fixed temperature (42); (43). Specifically, for the Gibbs states of a Hamiltonians with local interactions on a lattice, and with bounded operator norm , one finds that

 I(A:B)≤2β∥h∥|∂A|, (2)

where denotes the boundary area of the region labelled . This statement is in fact surprisingly simple to prove. It following rather directly from the extremality of Gibbs states with respect to the free energy.

The strength of this result - that it is completely general - also constitutes its weakness; Eq. (2) does not say anything about the correlation behaviour of thermal states of specific models. Given that the bound is linearly divergent in the inverse temperature, the law becomes less and less tight in the limit of small temperatures. In particular, at zero temperature, we know that there can be logarithmic corrections to the area law for critical systems. Hence, it would be desirable to have bounds which depend more explicitly on the properties of the system at hand. What is more, asymptotically exact results for important classes of “laboratory” models seem important.

In this work, we present asymptotically exact results for important classes of “laboratory” models. We introduce a framework capable of rigorously computing the mutual information in free fermionic lattice models. This framework is based on new approximation techniques as well as on proof tools for dealing with Töplitz matrices with smooth symbols. We separate the presentation in a discussion of the one-dimensional case and one of the higher-dimensional situation of a cubic lattice with the topology of a torus. We find rigorous area laws for the mutual information that are meaningful in the limit of small temperatures, and converge to the known results for the entanglement entropy for ground states (26); (10). This feature is even present in higher-dimensional fermionic lattice models (21); (22); (23); (25); (15).
For the critical model at low temperatures, we bound the mutual information by a logarithmic divergent term in the inverse temperature, in fact confirming predictions of conformal field theory. This scaling suggests that in order to detect criticality, extremely low temperatures are needed. This is an exponential improvement in the inverse temperature to the above bound. For the low temperature limit of the non-critical model we can bound the mutual information by a term that is exponentially decaying in the inverse temperature. The best decaying rate in our estimate can be bounded by the energy gap. Finally for the high temperature limit we prove an asymptotic decay of the mutual information proportional to the inverse square of the temperature.

Mathematically, we discuss new methods of approximation, allowing to use Töplitz matrix techniques to studying arbitrary subsystems of translationally invariant fermionic models. What is more, we introduce a novel second order Szegö theorem, simplifying Widom’s formula for smooth symbols. This approach allows to use the trace formula without encountering an infinite sum leading to arbitrarily highly oscillatory terms. Hence, apart from the application in the study of quantum many-body physics, we expect our results to be a highly useful tool in the abstract theory of Töplitz determinants. The paper is structured as follows:

• In Section 2, we introduce Renyi entropy mutual informations for free fermionic models, discuss fermionic covariance matrices and present new, easily computable and practical upper and lower bounds to mutual informations.

• In Section 3, we go on to formulate the problem, and we provide a synopsis of the mathematical argument. A general proof strategy for computing the mutual information is also provided.

• Section 4.1 contains the main theorem – the asymptotically exact expression for the mutual information – and the core of the proof. Since the argument is general enough to capture expressions deriving from all Renyi entropies, the complete knowledge of the spectrum of reduced states is also obtained in this fashion. This section reports also the main technical progress when it comes to dealing with Töplitz matrices with smooth symbols.

• Section 5 is dedicated to a thorough discussion of the temperature dependence of the area law, with an emphasis on the the very low temperatures behavior.

• We then turn to higher-dimensional free fermionic models on the torus in Section 6, which we can treat with similar methods.

• Finally, we present an outlook in Section 7, comparing the findings with predictions of conformal field theory, and discussing implications to the study of entanglement spectra (36); (37); (38); (39); (40); (41). We also consider the implications for noise-driven, open fermionic quantum many-body systems, as have recently been discussed also in the context of open Majorana wires and notions of noise-driven criticality (44); (45); (46); (47); (48).

2 Free fermionic models

In this work, we focus on isotropic translationally invariant fermionic models. For now, we assume the most general form for the couplings. Later, we will concentrate on the fermionic variant of the XX spin model (under the Jordan-Wigner transformation), which is a particularly important special case. In this section, we introduce the class of models discussed here.

2.1 Hamiltonians

We consider free fermionic models on cubic lattices for some even as becomes large. Bi-sected geometries on the torus can be related to the situation of having . For simplicity of notation, but without loss of generality, we will therefore present this one-dimensional case in the main part of this work, while discussing the higher-dimensional situation in Section 6. The Hamiltonian takes the general form

 H:=∑i,j∈\mathbbmZNf†jVj,kfk, (3)

with being a circulant matrix, referred to as the Hamiltonian matrix. We assume the coupling to be an specified, finite-ranged interaction of arbitrary interaction length . The couplings are defined by a sequence of numbers with the property that for . Then take

 dj:=v|j+1|,j=−N/2+1,…,N/2, (4)

having a natural reflection symmetry, , and consider the Hamiltonian matrix with entries

 Vi,j=d(i−j)modN (5)

for . This situation is depicted in Fig. 1. The most important special case is constituted by the fermionic variant of the XX model, which is originally a model for spin- systems. Here, we directly consider its fermionic instance, by virtue of the Jordan Wigner transformation. In this language,

 v1=a,v2=b,vk=0 (6)

for and , so that is a circulant matrix with on the main diagonal and on the first off-diagonal. All these models are integrable and exactly solvable. Thermal states are defined entirely by the collection of second moments of fermionic operators. The arbitrary finite range of interactions has in the main text been chosen for simplicity of notation only, and exponentially decaying interactions can easily be accommodated as well. Since is circulant, we find the spectrum of to satisfy

 1N∑k∈\mathbbmZNεke2πik/N=dk. (7)

For the well-studied fermionic variant of the XX model, see Eq. (6), one gets the familiar expression

 εk=a+2bcos(2πkN). (8)

2.2 Majorana fermions and covariance matrices

It is convenient to define fermionic covariance matrices in terms of Majorana fermions:

 ri:=fi+f†i√2,ri+N:=fi−f†i√2i,i=0,…,N−1, (9)

similar to the canonical coordinates for bosonic operators. These Majorana fermions are Hermitian, traceless, and form a Clifford algebra. We are interested in thermal states of the quadratic Hamiltonians, which are particular instances of (quasi)-free or Gaussian fermionic states. Such states are completely specified by their fermionic covariance matrix (49); (25); (44); (50) with entries

 Γi,j=itr(ρ[ri,rj]), (10)

, where the brackets denote the commutator. is always anti-symmetric,

 Γ=−ΓT, (11)

and satisfies .

2.3 Covariance matrices of Gibbs states and their reductions

Matrix functions of the covariance matrix can be computed exactly. This is the case because any covariance matrix is unitarily equivalent to a direct sum of covariance matrices, reflecting a situation of entirely uncoupled fermionic modes. This is the fermionic analogue of what is often called the Williamson normal form in the bosonic setting. As a consequence, one can identify an explicit expression for the covariance matrix of Gibbs states

 ρ=e−βHtr(e−βH) (12)

at inverse temperature . The covariance matrix of is given by

 Γ=(0η−η0), (13)

where

 η=f(V) (14)

where for all values of the smooth function is defined as

 f(x):=2e−βxe−βx+1−1=−tanh(βx2). (15)

The expression is to be interpreted as a matrix function, that is, is applied to the spectral values of Hermitian matrices (55). We suppress the temperature dependence here: Throughout this work, we will be concerned with Gibbs states with respect to some . We allow for arbitrary temperatures and will also later consider the asymptotic limits and .

Reduced states of Gaussian states are always Gaussian (as can be seen most easily by considering their Grassman representation; compare Ref. (49)). The covariance matrix of a reduced state is the appropriate principle sub-matrix of the full covariance matrix. Applied to Gibbs states, we find that the reduced state of a subset is a Gaussian state with covariance matrix

 Γ|A=(0f(V)|A−f(V)|A0), (16)

where denotes the principal sub-matrix associated with the degrees of freedom of subsystem .

2.4 Töplitz matrices

Töplitz matrices (51); (52); (53) may be viewed as principal sub-matrices of infinite circulant matrices. Such matrices will play a crucial role in this work. Throughout, we will encounter families of Töplitz matrices , whose entries are given by

 (Tn)i,j=ti−j (17)

for some sequence of reals . Colloquially speaking, such Töplitz matrices largely resemble circulant matrices, with the “upper right and lower left corners” deviating from a strict circulant matrix. In fact, most of the theory on Töplitz matrices (Töplitz determinants) is in one way or the other concerned with the error made when replacing a Töplitz matrix by a circulant matrix of the same dimension. Later on we will consider sub-matrices of large finite circulant matrices. It is clear that this sequence of Töplitz matrices is defined entirely in terms of the sequence of numbers . This sequence of numbers - and hence the sequence of Töplitz matrices - is most conveniently represented in therms of its symbol, defined as the inverse Fourier transform on the torus of ; i.e. the symbol 1 is defined as

 tk=12π∫2π0dθt(eiθ)e−ikθ. (18)

The decay of the Fourier coefficients is directly related to the regularity properties of the symbol. The summability of the coefficients (i.e., ) is sufficient to ensure . The relationship between the decaying behaviour of the Fourier coefficients and the regularity of the symbol will be used very frequently. In fact, given our assumptions on the interaction parameters, much better regularity results can be derived as we will show and use later on. Mathematically, is equivalent to the requirement that the associated Töplitz operator generates a continuous linear operator on . Furthermore, note that the spectra can be expressed entirely in terms of the symbol.

The starting point of our analysis is the following observation: covariance matrices of subsystems of thermal states of translationally invariant fermionic models are well approximated by Töplitz matrices. Since the Hamiltonian matrix in Eq. (3) is circulant, the matrix of a Gibbs state is, for any , circulant as well. That is to say,

 ηi,j=li−jmodN, (19)

for some real sequence , suppressing the temperature dependence. Indeed, sub-matrices for a region of the matrix (of the large but finite system) can be well approximated by the Töplitz matrix for large (see Section 4.2). The family of Töplitz matrices can be expressed in terms of the symbol

 λ=f∘ε (20)

where is defined as

 ε(x)=∑k∈\mathbbmZdkxk. (21)

Again, via the inverse Fourier transform on the torus, one recovers

 dk=12π∫2π0dθε(eiθ)e−ikθ, (22)

where is just the sequence of numbers that govern the coupling in the Hamiltonian matrix in Eq. (3).

2.5 Entropies of fermionic Gaussian states

Given that the states we consider are completely described by their covariance matrices, we are able to explicitly compute their Renyi, and specifically von-Neumann, entropies explicitly and efficiently. In particular, we consider covariance matrices of the form

 Γ=(0X−X0) (23)

where . For fermionic Gaussian states of modes with a covariance matrix of this form one finds

 Sα(ρ) = 11−αlog2tr(ρα)=tr(sα(X)), (24)

where the function is defined as

 sα(x):=11−αlog2((1+x2)α+(1−x2)α). (25)

Taking the limit as , one recovers the von-Neumann entropy,

 S(ρ)=tr(s(X)) (26)

with being

 s(x)=−1+x2log2(1+x2)−1−x2log2(1−x2). (27)

These expressions will be central for our analysis. With respect to an operational interpretation, the von-Neumann entropic version of the mutual information is by far the most natural quantity in this context, but we keep the generality at this point, partially also because the results found also hold in this setting. With similar techniques, other Renyi divergences can presumably also be treated.

3 Statement of the problem

3.1 Computing mutual informations

We will now turn to the main object of our study, the mutual information; a measure of the correlations between two non-overlapping subsystems. We will consider a one-dimensional lattice with lattice sites and let constitute one part and its complement. The quantum mutual information between and is defined as

 I(A:B)=S(ρA)+S(ρB)−S(ρAB), (28)

where again, is a state on the entire system, () is obtained by tracing out subsystem (). The mutual information naturally captures all correlations, quantum as well as classical, between and . It is a meaningful measure of correlation for mixed states, including finite temperature thermal states as a special case. For zero temperature, it reduces to (twice the) entanglement entropy. Similarly, expressions of the kind

 Iα(A:B)=Sα(ρA)+Sα(ρB)−Sα(ρ) (29)

for an different from are well defined mutual Renyi entropies. We will identify novel formulae for the asymptotic behavior of the mutual information for Gibbs states of isotopic translationally invariant free-fermionic models, and present bounds that allow to study the limit of large analytically. The formulae given will be exact in the asymptotic limit of large .

3.2 Structure of the argument for one-dimensional systems

A number of steps will be necessary in order to arrive at an asymptotically exact expression of the mutual information. To start with, we need to have a handle on how to make the intuition rigorous that we can compute (and ) as if it was the reduced state of an infinite system. The result has to fit then with the expression for for the entire system, in a way that only the boundary terms remain. This turns out to be a delicate affair, and requires very different tools than the pure state analysis, where the asymptotically exact entanglement entropy is obtained using the Fisher-Hartwig formalism (10). In the case of thermal states, the so-called “double scaling limit” makes sense, where is taken as a constant fraction of the total system size, and the total system is taken to infinity. Here, the reduced states of any part necessarily maintain a system size dependence, and one cannot simple compute spectra of reduced states of infinite systems. This is why the technical tools developed in Lemma 1 and Lemma 8 will be necessary. In fact, we show in these lemmas more than what is needed for the main result, in that bounds exponentially tight in the system size are being provided. Given the prominent status the computation of entanglement Renyi entropy has in the literature, and since it is important to have rigorous bounds also available for finite system sizes, we expect these bounds to be very valuable even outside the precise context of the present work.

We aim at computing an asymptotically exact approximation of the entropy of a subsystem

 Sα(ρA)=tr(sα(X|A)), (30)

where is given by

 Xk,l:=xk−l:=1NN−1∑j=0λ(e2πij/N)e−2πij(k−l)/N. (31)

The proof strategy is as follows:

• We perform a continuum limit on the full system

 X(N)k,l→X(∞)k,l:=x(∞)k−l:=12π∫2π0λ(eiϕ)e−i(k−l)ϕdϕ, (32)

and show that the entropy of a subsystem can be computed with an exponentially small error in .

• When computing entropies of sub-matrices , we can in this continuum limit invoke the theory of Töplitz matrices, even though the entries of the Töplitz matrices actually depend (very slightly) on the system size. We can therefore consider families of Töplitz matrices with symbol

 λ=f∘ε. (33)

The same approach is feasible for .

• We apply trace formulae of Töplitz matrices with smooth symbols. This will allow us to compute an asymptotically exact expression of the entropy of both subsystems.

• We find an expression for the entropy of the total system , asymptotically exact in the limit of large .

• The expression for the mutual information found in this way will contain an infinite sum, reflecting the infinite number of modes in momentum space in the asymptotic limit. As such, this formula can not be evaluated. This obstacle we will overcome by a new technique that we introduce, making use of the theory of highly oscillatory integrands. This technique is expected to be of significant interest also outside this context, when analysing properties of families of Töplitz matrices with smooth symbols.

The combination of these steps will allow us to provide an asymptotically exact rigorous and computable expression for the mutual information.

4 An exact asymptotic expression for the mutual information

In this section, we will state the main results for one-dimensional systems. We will first present the main theorem and then continue with the proof of the theorem. This will require a number of techniques that we will lay out in later subsection s. The specifically important case of the XX model, the temperature dependence as well as the situation of free fermions on the torus will be discussed in later sections.

In accordance with the analysis of the previous section, we will evaluate the mutual information by analyzing the determinants of the covariance matrices, as the system is taken to the thermodynamics limit. Covariance matrices of free fermionic systems in one dimension are Töplitz matrices, so we are free to use the tools available from the theory of Töplitz determinants. In particular, it is well known that Töplitz determinants behave in a very regular manner as the dimension of the matrices are taken to infinity. In the case when the symbol is continuous, Szegö’s strong limit theorem (57) gives the precise scaling of the determinant with the dimension of the Töplitz matrix. When the symbol has discontinuities, then it is necessary to use the Fisher Hartwig formula in order to obtain accurate asymptotics. As we will be dealing with thermal states, the symbols will always be continuous, even though a discontinuous symbol reflecting the Fermi surface can be arbitrarily well approximated for low temperatures. This creates a quite intriguing situation: We can “approximate” the situation covered by the Fisher-Hartwig theorem arbitrarily well with smooth symbols, and can hence “interpolate” between these situations. As mentioned before, the methods developed here are expected to be useful also outside the context of quantum many-body systems.

4.1 Explicit result for one-dimensional systems

We consider the mutual information in the large limit, where and in asymptotic Landau notation, meaning that both subsystems grow essentially linearly in . Specifically we assume

 A={0,1,…,⌈qN⌉−1} (34)

and

 B={⌈qN⌉,⌈qN⌉+1,…,N−1}. (35)

The main result for one-dimensional bi-sected systems can be stated as follows.

Theorem 4.1 (Asymptotically exact expression for the mutual information)

For any inverse temperature , and for any , the mutual information is given by the, in asymptotically exact, expression

 Iα(A:B)=14π2∫π−πdϕ∫π−πdθsα(λ(eiθ))−sα(λ(eiϕ))λ(eiθ)−λ(eiϕ)λ′(ϕ)−λ′(θ)tan((ϕ−θ)/2)+o(1). (36)
Proof

The proof of Theorem 4.1 follows from Proposition 1, which still contains an infinite sum in , and Theorem 4.4, which take care of the infinite sum.

Here, the temperature dependence is only implicit in (since depends on ), while denotes the derivative of , which has to be understood in the following way,

 λ′(θ):=ddθλ(eiθ). (37)

This is an asymptotically exact and simply evaluated expression. The most important instance is the one for the von-Neumann mutual information.

Proposition 1 (Mutual information as an infinite sum)

For any inverse temperature , and for any , the mutual information is given by the, in asymptotically exact, expression

 Iα(A:B) = 12π2∞∑k=1∫π−πdϕ∫π−πdθsα(λ(eiθ))−sα(λ(eiϕ))λ(eiθ)−λ(eiϕ) (38) × sin(k(θ−ϕ))(λ′(ϕ)−λ′(θ))+o(1).

The proof of this statement will require some preparation. For clarity of the main argument, we present some of the technical steps in the appendix.

4.2 Approximation statements

In this subsection, we collect approximation statements that are being used in the proof of Theorem 4.1. We have a more detailed look at the expression

 X(N)k,l=x(N)k−l=1NN−1∑j=0λ(e2πij/N)e−2πij(k−l)/N (39)

as well as at

 Xk,l=xk−l=12π∫2π0λ(eiϕ)e−i(k−l)ϕdϕ, (40)

where the entries of the finite Töplitz matrices are always assumed mod . We know that

 Sα(ρA)=tr(sα(X(N)|A)),Sα(ρB)=tr(sα(X(N)|B)). (41)

Expression Eq. (32) is the continuum limit of Eq. (31). Therefore, from a physical perspective, it seems justified to replace by its infinite-system counterpart in the calculation of the entropy if the system is large enough. In the next lemma we will show that for any fixed temperature, the error made by this replacement decays exponentially in .

Lemma 1 (Approximation of entropies of subsystems)

For given by Eq. (34) and Eq. (35) the errors

 eA,β(N):=|tr(sα(X(N)|A))−tr(sα(X|A))| (42)

and

 eB,β(N):=|tr(sα(X(N)|B))−tr(sα(X|B))| (43)

are exponentially small in for fixed . More precisely, there exist some -independent constants , (depending on , and the explicit form of the symbol ), such that

 limN→∞exp(NαX,β)ek,β(N)=0 % for X∈{A,B}. (44)

This Lemma, along with other auxiliary Lemmas, will be proven in Appendix A.3.

4.3 Relating the mutual information to trace functions of Töplitz matrices

Given that we are interested not in the entropy of a subsystem, but in the mutual information, we will need to consider a second order asymptotic formula for the entropy functionals. This is due to the fact that the bulk contribution of the entropy is cancelled out in the expression for the mutual information, and we are left only with the contribution originating from the boundary. We will therefore require a Szegö’s strong limit theorem which contains explicit expressions for the second order contributions. In order to formulate the theorem, we will have to introduce some definitions and notation. It will be necessary to consider sequences of truncations of Töplitz matrices – in our case principal sub-matrices of covariance matrices. We will define the following classes of symbols. The Wiener algebra

 Missing or unrecognized delimiter for \left (45)

the Besov space

 B122:=⎧⎨⎩μ∈L2(T):∑n∈\mathbbmZ(|n|+1)∣∣∣12π∫2π0μ(eiθ)e−inθdθ∣∣∣2<∞⎫⎬⎭, (46)

the Krein algebra

 W∩B122, (47)

and the space of piecewise continuous symbols, denoted by play an important role; see Ref. (53) for more details. Consider a continuous symbol and a point . There exists a (unique up to some constant offset of the form with ) continuous argument function

 arg(λ−a):[−π,π]→\mathbbmR;ϕ↦arg[λ(eiϕ)−a]. (48)

Independent of the choice of offset, the winding number

 wind(λ,a):=12πarg(λ−a)(π)−arg(λ−a)(−π) (49)

is a well-defined integer. Before formulating Szegö’s limit theorem, we state an important lemma about the spectrum of Töplitz operators (explaining where the spectrum of the truncated matrices eventually concentrates).

Lemma 2 (Spectrum of Töplitz operators and truncated Töplitz matrices (53))

Suppose and let be the infinite Töplitz matrix associated to the symbol . Assume is an open set and assume that the spectrum is a subset of . Then there exists an index such that implies . If is continuous the spectrum of follows entirely from geometric properties of the symbol,

 σ(T(λ))=range(λ)∪{x∈\mathbbmC|wind(λ,x)≠0}. (50)

Szegö’s Theorem holds for symbols in the Krein algebra (53):

Theorem 4.2 (A second order Szegö theorem (51); (53))

Let be a symbol and let be the family of associated Töplitz matrices. Let be an open subset that contains the spectrum of . For an analytic function the following trace formula holds,

 tr(g(Tn))=nGg(μ)+Eg(μ)+o(1) (51)

with

 Gg(μ) = 12π∫2π0g(μ(eiθ))dθ, (52) Eg(μ) = 12πi∫∂Ωg(λ)ddλlogE(μ−λ)dλ, (53) E(μ) = exp∞∑k=1k(logμ)k(logμ)−k, (54)

where

 (logμ)k:=12π∫2π0e−ikθlogμ(eiθ)dθ (55)

is the Fourier transform of .

Theorem 4.3 (Alternative expression for the second order term (52))

Let be absolutely continuous and let be the family of associated Töplitz matrices. Let be an open subset that contains the spectrum of and let be analytic on . Then the second order term from Theorem 4.2 can be written as

 Eg(μ) = 14π2∞∑k=1∫π−πdϕ∫π−πdθg(μ(eiθ))−g(μ(eiϕ))μ(eiθ)−μ(eiϕ) × sin(k(θ−ϕ))(μ′(ϕ)−μ′(θ)).

Again, the derivative is to be read as in Eq. (37).

Proof of Proposition 1. We are now in the position to prove Proposition 1. We can now collect the results from the previous sections. By the results from Subsection 4.2, specifically Lemma 1, we know that for any subset of sites whose cardinality is much larger than , we can work with infinite truncated Töplitz matrices and their symbols. The asymptotic expression for the entropy of the full system can be obtained directly from the continuum approximation. Remember that and subsystem is of size , where . Theorem 4.2 states that the block entropies are asymptotically equal to

 Sα(ρA)=tr(sα(X|A))=qNGsα(λ)+Esα(λ)+o(1) (56)

and

 Sα(ρB)=tr(sα(X|B))=(1−q)NGsα(λ)+Esα(λ)+o(1). (57)

We now turn to the computation of the entropy of the entire system. This is subtle, and one cannot employ the same formula for the larger system of : Lemma 1 would no longer be valid, and the boundary conditions would not be respected. But we can still find an asymptotically exact expression. The spectrum of is given by

 {(f∘ε)(e2πik/N):k=1,…,N}. (58)

Hence,

 Extra open brace or missing close brace (59)

Using Lemma 8 again, we find that this expression can be replaced by

 S(ρ)=NGs(λ)+o(1). (60)

Combining the terms, the mutual information is asymptotically equal to

 Iα(A:B)=2Esα(λ)+o(1). (61)

Since the symbol is analytic (and therefore absolutely continuous) Theorem 4.3 can be applied resulting in the desired expression

 Iα(A:B) = 12π2∞∑k=1∫02πdϕ∫02πdθsα(λ(eiθ))−sα(λ(eiϕ))λ(eiθ)−λ(eiϕ) × sin(k(θ−ϕ))(λ′(eiϕ)−λ′(eiθ))+o(1).

which completes the proof.∎

4.4 Simplifying Widom’s second order expression

We have almost proven the main theorem, but we still need to eliminate the infinite sum in in Eq. (38). The sum in is rather awkward; since the integrand is more and more oscillatory for larger and larger , it is a priori far from clear where one may truncate the sum in order to arrive at a reliable result. In this way, the expression cannot be easily computed, not even numerically, and practically only for low temperatures. We will hence go a technical step further and will prove the validity of Theorem 4.1, the main result for one-dimensional systems. A first step in this direction is the following observation.

Lemma 3 (Function kernel)

For any and any ,

 ∑1≤k≤nsin(kϕ)=12cos(ϕ/2)−cos((n+1/2)ϕ)sin(ϕ/2)=:Kn(ϕ). (62)
Proof

We first note that the sum over sine functions encountered in Eq. (4.3) can be brought into a closed form reminiscent of the Dirichlet kernel. Indeed, if we truncate the sum at , then it can be expressed in closed form as

 ∑1≤k≤nsin(kϕ) = 12i∑1≤k≤n(eikϕ−e−ikϕ) (63) = 12i∑1≤k≤n((eiϕ)k−(e−iϕ)k) = 12i(1−e(n+1)iϕ1−eiϕ−1−e−(n+1)iϕ1−e−iϕ) = 12cos(ϕ/2)−cos((n+1/2)ϕ)sin(ϕ/2),

where the last line follows from elementary trigonometric identities. ∎

That is to say, the mutual information can be re-expressed in terms of the kernel as

 Iα(A:B) = 12π2limn→∞∫π−πdϕ∫π−πdθsα(λ(eiθ))−sα(λ(eiϕ))λ(eiθ)−λ(eiϕ) (64) × (λ′(eiϕ)−λ′(eiθ))Kn((θ−ϕ))+o(1).

We can further simplify the expression by invoking results from the theory of highly oscillatory integrals. Recall the subsequent fundamental lemma.

Lemma 4 (Riemann-Lebesgue)

Let , let be periodic (with period ) and assume (the probably most important special instances of such functions are the trigonometric functions and ) then

 limn→∞∫baf(t)g(nt)dt=0. (65)
Proof

If (hence is a compactly supported smooth function from to ), then partial integration gives

 ∫baf(t)g(nt)dt=−1n∫baf′(t)∫ntag(s)dsdt. (66)

Note that is bounded by assumption and that for every . Hence,

 limn→∞∫baf(t)g(nt)dt=0. (67)

For general , note that the smooth, compactly supported functions are dense in . Hence, for an arbitrary there exists with

 ∥f2−f∥1≤ε∥g∥∞(b−a). (68)

Therefore,

 limn→∞∣∣∣∫baf(t)g(nt)dt∣∣∣≤limn→∞(∣∣∣∫ba(f(t)−f2(t))g(nt)dt∣∣∣+∣∣∣∫baf2(t)g(nt)dt∣∣∣)≤ε (69)

by validity of the statement for compactly supported, smooth functions. Since was arbitrary, the result also holds true for general .∎

This well-known lemma simplifies the understanding of the limiting behaviour of these kernels action on sufficiently regular test functions.

Theorem 4.4 (Distributional convergence of the kernels Kn)

Let be locally Lipschitz continuous at then

 limn→∞∫π−πg(eiϕ)Kn(ϕ−θ)dϕ=12∫π−πg(eiϕ)−g(eiθ)tan((ϕ−θ)/2)dϕ. (70)
Proof

Without any loss of generality let (otherwise take ). Assume first. By Lipschitz continuity of , there exists some and some constant such that , whenever . By continuity, there exists some constant such that

 ∣∣∣ϕtan(ϕ/2)∣∣∣

for . Therefore

 ∣∣ ∣∣g(eiϕ)tan(ϕ/2)∣∣ ∣∣≤χ(−ε,ε)(ϕ)M⋅L+χ[−π,π]∖(−ε,ε)(ϕ)∣∣g(eiϕ)∣∣tan(ε/2), (72)

where is the characteristic function of the (measurable) set . Define . By the previous estimate and the Riemann-Lebesgue Lemma can be applied. Applying the Riemann-Lebesgue Lemma twice and using elementary angle-sum identities for trigonometric functions yields

 limn→∞∫π−πg(eiϕ)12cos(ϕ/2)−cos((n+1/2)ϕ)sin(ϕ/2)dϕ = limn→∞12∫π−πg(eiϕ)1−cos(nϕ)tan(ϕ/2)dϕ (73) = limn→∞12∫π−π~g(eiϕ)(1−cos(nϕ))dϕ = 12∫π−πg(eiϕ)1tan(ϕ/2)dϕ.

If write , apply the former result to the first summand and note that

 ∫π−πcos(ϕ/2)−cos((n+1/2)ϕ)sin(ϕ/2)dϕ=0, (74)

for every by anti-symmetry of the integrand. ∎

We need an extension to double integrals, a proof of which follows the same line of reasoning.

Theorem 4.5 (Distributional convergence of the kernels Kn for double integrals)

Let be Lipschitz continuous in a neighborhood of the diagonal

 DT2:={(x,y)∈T2|x=y} (75)

then

 limn→∞∫π−π∫π−πg(eiϕ,eiθ)Kn(ϕ−θ)dϕdθ=12∫π−π∫π−πg(eiϕ,eiθ)−g(eiθ,eiθ)tan(ϕ−θ2)dϕdθ. (76)

As a consequence, Eq. (64) can be rewritten in the following way:

 Missing or unrecognized delimiter for \right (77)

Let us conclude this subsection by commenting on the second order term in the trace formula of Theorem 4.3. In unpublished work that we learnt of upon completion of our work, another related approach aiming at a simplification of Widom’s formula has been discussed: The result of a masters’ thesis (56) is a derivation of a second order trace formula from asymptotic inverses and appropriate factorisations of the symbol. The presented final expression of the second order term does not contain -sum as well and finally only involves the calculation of an appropriate double integral – a result similar to ours. The real-valued integration approach presented here is elementary and readily gives rise to a computable formula in context at hand.

Indeed, the idea to approach the trace formula starting with the inverse function rather then the usual logarithm (the term from Szegö’s theorem is just the second order term of the logarithm) has many didactical and computational advantages in our opinion. First of all, this computationally inappropriate and practically incomputable sum in is missing right from the beginning – note that for larger and larger , the integrand becomes more and more oscillatory in a fashion that is hard to grasp. Furthermore, there exist well-known asymptotic expansions for asymptotic inverses even for higher order terms (which are very useful for approximate solutions of systems of linear equation with coefficient matrix being a Töplitz matrix for example). Last but not least the authors of Ref. (56) were able to derive second order formulas for block Töplitz matrices (which require – unlike the final result in the one-dimensional setting – an explicit factorisation of where is the matrix-valued symbol and is an arbitrary complex number on the integration contour).

5 Temperature dependence

5.1 General remarks

Our main result, Theorem 4.1, gives an asymptotically exact expression for the mutual information of two neighbouring blocks of fermions in one dimension. It has been shown that for a given inverse temperature ,

 Iα(A:B)=Iα,∞(A:B)+o(1) (78)

in the system size . The result given constitutes an easily computable expression that paves the way for studying a number of physically meaningful regimes. At this point, we would like to pause for a moment, however, and would like to come back to one of the questions posed in the introduction, namely of the possible asymptotic behaviour of for large and small inverse temperatures. As we have seen above, the general bound of Ref. (42) following from the extremality of the free energy suggests that the mutual information should scale like for large inverse temperatures,

 Iα,∞(A:B)=O(β). (79)

One might wonder whether this bound actually gives the proper asymptotic scaling on the temperature. Conformal field theory actually suggests a behaviour which is logarithmic rather than linearly in the inverse temperature (27); (2). In this section, we corroborate the prediction from conformal field theory by showing that the low temperature asymptotics are given by

 Iα,∞(A:B)=O(log(β)). (80)

This has a quite remarkable consequence: In order to see features of the criticality of the ground state, one has to go to extremely low temperatures. This dependence is also convincingly depicted in Fig. 2, where a logarithmic scale has been chosen – otherwise, signatures of ground state features could hardly be detected. Only at extraordinarily low temperatures, the familiar logarithmic divergence in the system size of the sub-block chosen becomes visible for a reasonably sized subsystem.

5.2 Analysis of the XX model

Needless to say, the low temperature asymptotics depends on the choice of the model parameters and . Specifically, the scaling of the mutual information reflects signatures of the zero temperature quantum phase transition of this model taking place at . Whenever the model is critical in its ground state, i.e., there is no energy gap. It is known that the entanglement entropy then exhibits a logarithmic divergence – signatures of that are also seen in the mutual information at small but non-zero temperature. If the model is gapped, i.e., whenever , the ground state is the vacuum or the fully occupied state, depending on the sign of . As expected, the mutual information converges exponentially quickly to zero in the non-critical phase at a rate proportional to the energy gap. The main insights for this model are summarised in the following theorem:

Theorem 5.1 (Temperature dependence)

The () mutual information of the thermal state of the fermionic XX model with parameters and satisfies

 limβ→0Iα,∞(A:B)β2=αb22log(2). (81)

Whenever , then

 limsupβ→∞Iα,∞(A:B)logβ (82) ≤ 2απ2log(2)(min{1,α})2min{272(min{1,α}),4|1−α|}.

Whenever , then for any

 limβ→∞exp(κβ)Iα,∞(A:B)=0. (83)

The proof of this statement is rather involved and requires a number of techniques developed in lemmas: Hence, for better readability of the main text, it will be presented in the appendix in Subsection A.5.

6 Free fermonic models on the torus

The results established above readily apply to the situation of a bi-sected, higher-dimensional fermionic lattice system on the torus. A quite similar strategy has already been exploited, e.g., in Ref. (25). Using appropriate discrete Fourier transforms, one can disentangle the constituents from each other with respect to all but one dimensions. The exception is the dimension in which the two regions labeled and are singled out, see Fig. 3. In this way, one arrives at a collection of suitably modulated and altered one-dimensional problems, to which the above statements apply. In this section, we highlight the results obtained in this manner. An important application of this is the computation of the mutual information in higher-dimensional tight binding models.

6.1 Geometry of the problem

Let us for simplicity consider in dimensions the geometry of slabs