Semiclassical study of diagonal and offdiagonal functions in the eigenstate thermalization hypothesis Xiao Wang1 2and Wen-ge Wang1 2

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Semiclassical study of diagonal and oﬀdiagonal functions in the eigenstate

thermalization hypothesis

Xiao Wang1, 2 and Wen-ge Wang1, 2, ∗

1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

2CAS Key Laboratory of Microscale Magnetic Resonance,

University of Science and Technology of China, Hefei 230026, China

(Dated: October 17, 2024)

The so-called eigenstate thermalization hypothesis (ETH), which has been tested in various many-

body models by numerical simulations, supplies a way of understanding eventual thermalization

and is believed to be important for understanding processes of thermalization. Two functions

play important roles in the application of ETH, one for averaged diagonal elements and the other

for the variance of oﬀdiagonal elements of an observable addressed by ETH on the energy basis.

For the former function, a semiclassical expression is known of the zeroth order of ℏ, while, little

is known analytically for the latter. In this paper, a semiclassical expression is derived for the

former function, which includes higher-order contributions of ℏ. And, a semiclassical approximation

is derived for the latter function, under the assumption of negligible correlations among energy

eigenfuntions on an action basis. Relevance of the analytical predictions are tested numerically in

the Lipkin–Meshkov–Glick model.

I. INTRODUCTION

The idea of presuming eigenstate thermalization may

be traced back to early works of several authors (see, e.g.,

Refs.[1, 2]). While, the ﬁrst demonstration of this phe-

nomenon in a concrete model was given by Srednicki, in

a quantum chaotic model — rareﬁed hard-sphere gas [3],

based on the so-called Berry’s conjecture given by a semi-

classical analysis [4]. For a generic many-body quantum

chaotic system, eigenstate thermalization was expressed

as a hypothesis [5, 6], namely, eigenstate thermalization

hypothesis (ETH).

Technically, ETH is expressed as an ansatz for certain

type of observable Oof a many-body system, stating that

the matrix of Oon the energy basis of {|Ei⟩} exhibits

a special structure and its elements are written in the

following form,

Oij =O(Ei)δij +f(Ei, Ej)rij ,(1)

where O(e) and f(e, e′) are smooth functions, and rij =

r∗

ji are assumed to be independent random variables with

normal distribution (zero mean and unit variance). Orig-

inally, the function f(e, e′) was conjectured as being pro-

portional to e−S/2at the energy of e0= (e+e′)/2 for

enot far from e′[5], where Sis the thermodynamic en-

tropy which is proportional to the particle number Nof

the many-body system. From the perspective of quan-

tum mechanics, since the density of states ρdos increases

exponentially with N,f(e, e′) is sometimes written as

proportional to 1/√ρdos.

In recent years, the ETH ansatz has attracted lots

of attention (see, e.g., Refs.[7–10] and references cited

there). It is expected to supply a useful direction for the

∗wgwang@ustc.edu.cn

study of thermalization of isolated many-body quantum

systems (both for ﬁnal results and for processes), a long-

standing problem in physics which has attracted broad

attentions recently [7, 11–16]. The renewed interest was

caused by progresses achieved in theoretical and exper-

imental aspects, as well as improvements in the compu-

tation ability [7, 8, 17–28].

Although lacking a rigorous analytical proof, vast nu-

merical evidences have been found supporting validity of

the ETH ansatz [Eq.(1)] in many models, even in mod-

els that are not completely chaotic according to spectral

statistics (see, e.g., Refs.[26, 29–33] and references cited

in [7, 8]). The following behaviors of the function f(e, e′)

have been found numerically with the increase of |e−e′|.

That is, it usually shows a platform (scaling as 1/√ρdos)

at esuﬃciently close e′, followed by some power-law de-

cay and then an exponential-type decay at esuﬃciently

far from e′. Local observables Owere considered in the

original semiclassical arguments for ETH [5, 6], while,

later numerical simulations show that this requirement

of locality seems too restrictive. However, to make clear

the scope of the observable Ofor the ETH ansatz is a

task too hard for numerical simulations, which is still an

open problem.

To get a satisfactory understanding of the ETH ansatz,

at least, the following ﬁve topics should be studied.

(i) Analytical expression of the function O(e) for the

averaged diagonal elements.

(ii) The following properties of the function f(e, e′) for

the oﬀdiagonal elements. That is, (a) an estimate to the

height of the platform at small |e−e′|, including its ρ−1/2

dos

scaling behavior; (b) an estimate to the width of the

platform region, which may be related to the relaxation

time according to certain physical picture [34, 35]; (c)

scope of the power-law decay and the decay exponent;

and (d) exponent of the far exponential-type decay.

(iii) Meaning of the stated randomness of the quanti-

arXiv:2210.13183v2 [cond-mat.stat-mech] 16 Oct 2024

ties rij . In fact, these quantities should not be completely

random, because energy eigenfunctions (EFs) of chaotic

systems possess nonnegligible autocorrelation functions

according to the semiclassical theory [4]. Indeed, nonneg-

ligible correlations among rij were suggested by analyti-

cal arguments [36–38] and were found in recent numerical

simulations [34, 36, 38].

(iv) Relationship between f(e, e′) of e≃e′for the

oﬀdiagonal elements and f(e, e′) of e=e′for the diagonal

elements. In fact, according to heuristic arguments [2]

and numerical simulations [24, 25, 28, 33], the former

may be about half of the latter.

And, (v) scope of validity of the ETH ansatz, partic-

ularly regarding the type of the system and the type of

the operator O.

In this paper, we are to study the ﬁrst two topics dis-

cussed above. Regarding the function of O(e), a semiclas-

sical expression was proposed previously for it in the limit

of ℏ→0 [2], based on physical arguments about “reason-

able” operators, which have well-behaved classical limits.

More rigorously, a similar result was obtained by sophis-

ticated mathematical treatments (see references cited in

Ref.[5]). We are to show that a semiclassical expression

of O(e) is derivable mathematically by a much simpler

method, which utilizes the concept of Weyl-ordered op-

erator. And, even more, this method allows a direct com-

putation of quantum corrections to any order of ℏfrom

the aspect of the observable O.

Concerning the second topic discussed above, as is

known, derivation of a useful semiclassical expression of

the function f(e, e′) by making use of the original ver-

sion of Berry’s conjecture in the conﬁguration space is

not an easy task [5, 6]. In this paper, we use an alterna-

tive version of Berry’s conjecture, which is expressed on

an action basis [39], and study the possibility of deriving

some useful expression by this approach.

Moreover, one useful observation is that the above-

discussed semiclassical treatments to the two functions of

O(e) and f(e, e′) do not rely on the many-body feature of

the systems considered. As a consequence, the analytical

predictions may be tested in models with a few degrees

of freedom. Speciﬁcally, we are to employ the Lipkin-

Meshkov-Glick (LMG) model [40] for this purpose.

The paper is organized as follows. In Sec.II, semiclas-

sical understandings of EFs of quantum chaotic systems

are recalled, which is basically given by Berry’s conjec-

ture. In Sec.III, a semiclassical expression of O(e) is

derived by making use of Weyl-ordered operators. In

Sec.IV, a semiclassical study of f(e, e′) is given by mak-

ing use of Berry’s conjecture on the action basis. Numer-

ical simulations in the LMG model are given in Sec.V.

Finally, conclusions and discussions are given in Sec.VI.

II. SEMICLASSICAL APPROACH TO EFS IN

QUANTUM CHAOTIC SYSTEMS

In this section, we recall semiclassical descriptions of

EFs, basically known as Berry’s conjecture. Speciﬁcally,

some generic discussions on EFs are given in Sec.II A,

EFs in the conﬁguration space are discussed in Sec.II B,

and those on the action basis in Sec.II C.

A. Some generic discussions

In the past half century or so, a huge amount of knowl-

edge has been accumulated in the ﬁeld of quantum chaos

(see, e.g., [41, 42]). The most-often-used criterion for

quantum chaos was proposed in the early 80s of the

last century [43, 44], which conjectures that the spec-

tral statistics should be in agreement with predictions

of the random matrix theory. It then took more than

twenty years to justify the above conjecture by making

use of the semiclassical theory [45–49]. Compared with

spectral statistics, much less is known about statistical

properties of energy eigenstates of chaotic systems, more

exactly, of energy eigenfunctions (EFs). One major dif-

ﬁculty comes from the fact that EFs may behave quite

diﬀerently on diﬀerent bases.

In most of the studies that have been carried out

in EFs, three types of basis were employed: (i) Po-

sition/momentum basis (see, e.g., Refs.[3, 4, 50]); (ii)

eigenbasis of some integrable (or regular) system, as an

unperturbed basis (see, e.g., Refs.[15, 39, 51–56]); and

(iii) eigenbasis of a nearby system (see, e.g., Refs.[57, 58]).

Below, we are to discuss EFs on bases of the former two

types.

For EFs in the conﬁguration space (position basis), one

may distinguish between regions that are classically and

energetically allowed and those forbidden. For EF com-

ponents in the former regions, Berry’s conjecture states

that they may be regarded as Gaussian random variables

that possess certain correlations [4], the details of which

are to be discussed in the next section (II B). For EF

components in the latter regions, usually exponential be-

haviors are expected, as well known in WKB treatment;

meanwhile, certain perturbation approach may be useful

in a discretized conﬁguration space [59, 60].

Regarding bases of integrable systems, the simplest one

is given by eigenstates of the action. Based on Berry’s

conjecture in the conﬁguration space, one may discuss

EFs of quantum chaotic systems on an action basis. After

being appropriately rescaled by the average shape of EFs,

the rescaled components of EFs usually obeys a Gaussian

distribution [39], as to be discussed in Sec.II C. While, a

practically useful expression for autocorrelation functions

on this basis is still lacking.

B. Chaotic EFs in the conﬁguration space

In this section, we recall basic contents of Berry’s con-

jecture. Consider a quantum system, which possesses

an eﬀective Planck constant ℏ, such that a classical-

counterpart system in a f-dimensional conﬁguration

space may be obtained in the limit of ℏ→0. Coordi-

nates in the classical phase space are indicated as (p,q),

where p= (p1, p2,··· , pf) and q= (q1, q2,··· , qf). The

classical counterpart system is assumed chaotic with a

positive maximum Lyapunov exponent.

The Hamiltonian of the system is denoted by Has a

function of ˆ

pand ˆ

q, namely, H(ˆ

p,ˆ

q). 1Its eigenenergies

are written as Ei, in the increasing energy order, and

eigenstates are indicated as |Ei⟩,

H|Ei⟩=Ei|Ei⟩.(2)

The EF of |Ei⟩in the conﬁguration space is indicated as

ψi(q), ψi(q) = ⟨q|Ei⟩.

The Wigner distribution function corresponding to

ψi(q), denoted by Wi(p,q), is written as

Wi(p,q) = 1

(2πℏ)fZ∞

−∞

ψ∗

i(q+r

2)ψi(q−r

2)eip·r/ℏdr.

(3)

The average shape of EFs within a narrow energy shell

centered at Eiis denoted by Πi(q),

Πi(q) = |ψi(q)|2.(4)

Here and hereafter, an overline indicates average taken

over an energy shell of the (perturbed) system H.2More

exactly, for a narrow energy window ϵ, one may consider

a coarse-grained δ-function, indicated as δϵ(e, Ei),

δϵ(e, Ei) = (1

ϵe∈[Ei−ϵ

2, Ei+ϵ

2],

0 otherwise,(5)

and take average within this window. The energy win-

dow ϵshould be small in the classical case, such that

change of the energy surface is negligible within the win-

dow; meanwhile, it be suﬃciently large in the quantum

case, such that many energy levels are included within

the window.

The average shape of EF may be computed from the

averaged Wigner function,

Πi(q) = ZdpWi(p,q).(6)

1For the sake of clearness in discussion, hats are written for the

operators of ˆ

pand ˆ

q(also for ˆ

Iof action).

2In the original work of Berry, the average was taken over a small

region in the conﬁguration space [4]. Later, it was argued that

a better method is to take average over a narrow energy shell

[3, 61].

Based on semiclassical arguments, it was conjectured

that the averaged Wigner function is approximately given

by the corresponding classical energy surface in phase

space [4, 39, 62–64], i.e.,

Wi(p,q)≃δ(Hcl(p,q)−Ei)

S(Ei),(7)

where S(E) represents the area of the energy surface with

Hcl(p,q) = E,

S(E) = Zdpdqδ(E−Hcl(p,q)).(8)

Here and hereafter, for an arbitrary operator function

O(ˆ

p,ˆ

q), Ocl(p,q) with a subscript “cl” refers to the clas-

sical quantity, which is obtained by directly replacing

(ˆ

p,ˆ

q) in O(ˆ

p,ˆ

q) by the classical coordinates (p,q).

By deﬁnition, the two-point autocorrelation function

is written as

C(q,q′)≡C(r,q0) = 1

Πi(q0)ψ∗

i(q0+r

2)ψi(q0−r

2),

(9)

where

q0=q+q′

2,r=q′−q.(10)

It has the following relation to the Wigner function,

C(r,q0) = 1

Πi(q0)ZdpWi(p,q0)e−ip·r/ℏ.(11)

Then, one sees that

C(r,q0)≃Rdpδ[Hcl(p,q0)−Ei]e−ip·r/ℏ

Rdpδ[Hcl(p,q0)−Ei].(12)

According to Berry’s conjecture [4], the EF ψis re-

garded as a Gaussian random function of q, whose sta-

tistical properties (probability distributions of ψand of

its derivatives, correlations between ψat two or more

points, etc.) are all determined by Π(q) and C(r,q0),

which are computed from the averaged Wigner distribu-

tion function given above.

C. EFs on the action basis

In this section, we recall a version of Berry’s conjecture,

which is expressed on an action basis [39]. The perturbed

Hamiltonian is written as

H=H0+λV, (13)

where H0indicates the Hamiltonian of an integrable sys-

tem and Vis a generic perturbation, with a parameter

λfor adjusting the perturbation strength. In terms of

operators for the action, H0is written as

H0=d·ˆ

I+c0,(14)

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SemiclassicalstudyofdiagonalandoffdiagonalfunctionsintheeigenstatethermalizationhypothesisXiaoWang1,2andWen-geWang1,2,∗1DepartmentofModernPhysics,UniversityofScienceandTechnologyofChina,Hefei230026,China2CASKeyLaboratoryofMicroscaleMagneticResonance,UniversityofScienceandTechnologyofChina,Hefei23002...

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Semiclassical study of diagonal and offdiagonal functions in the eigenstate thermalization hypothesis Xiao Wang1 2and Wen-ge Wang1 2

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