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arXiv:2210.13183 · Physical review. E

Semiclassical study of diagonal and off-diagonal functions in the eigenstate thermalization hypothesis

Xiao 晓 Wang 王, Wen-ge Wang

Published Nov 12, 2025• 5 citations
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Abstract

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 off-diagonal 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 diagonal function, which includes higher-order contributions of ℏ. And, another semiclassical expression is derived for the scale of the height of a central platform of the latter off-diagonal function. These analytical results are checked numerically in the Lipkin-Meshkov-Glick model. Possible relevance of the latter result for future investigation is discussed.

DOI
10.1103/xbtc-hlxw

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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 off-diagonal 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 diagonal function, which includes higher-order contributions of ℏ. And, another semiclassical expression is derived for the scale of the height of a central platform of the latter off-diagonal function. These analytical results are checked numerically in the Lipkin-Meshkov-Glick model. Possible relevance of the latter result for future investigation is discussed.
In this letter, by an approach that employs Weyl symbols for operators, a semiclassical theory is developed for the off-diagonal function in the eigenstate thermalization hypothesis, which is for off-diagonal elements 〈E_{i}|O|E_{j}〉 of an observable O on the energy basis. It is shown analytically that the matrix of O has a banded structure, possessing a bandwidth w_{b} that scales linearly with ℏ, a phase-space gradient of the classical Hamiltonian 〈|∇H_{cl}|〉 and an O-dependent property. This predicts that the thermalization timescale of a quantum system may be inversely proportional to the phase-space gradient of the Hamiltonian, aligning with intuitions in classical thermalization. This approach also elucidates the origin of a ρ_{dos}^{-1/2} scaling of the off-diagonal function. The analytical predictions are checked numerically in the Lipkin-Meshkov-Glick model.
Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.
The eigenstate thermalization hypothesis (ETH) is foundational to modern discussions of thermalization in closed quantum systems. In this work, we expand on traditional explanations for the prevalence of ETH by emphasizing the role of operator locality. We introduce an operator-specific perturbation problem that can be thought of as a means of understanding the onset or breakdown of ETH for specific classes of operators in a given system. We derive explicit functional forms for the off-diagonal variances of operator matrix elements for typical local operators under various `scrambling ansatzes', expressed in terms of system parameters and parameters of the corresponding perturbation problem. We provide simple tests and illustrations of these ideas in chaotic spin chain systems.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.
We study the off-diagonal matrix elements of real-space observables in time-reversal-invariant single-particle chaotic systems. By analyzing the semiclassical expression for the off-diagonal variance derived from Berry's conjecture, we show that the banded structure of the observable matrix emerges naturally. For local observables, we identify a characteristic bandwidth associated with a late-time timescale inversely proportional to the particle velocity. We further show that, for systems with steep-wall confinement, the predicted magnitude follows the entropy scaling of the eigenstate thermalization hypothesis (ETH), multiplied by an additional kinetic-energy-dependent factor that is independent of spatial dimension and is not captured by conventional many-body ETH. We illustrate these results through a case study of quantum billiards and verify the semiclassical scaling numerically in a generalized quarter-Sinai billiard. Our results elucidate the dynamical implications of Berry's conjecture and provide a comparison between single-particle eigenstate thermalization and many-body ETH.