Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2406.04684 · SciPost Physics

Out-of-equilibrium eigenstate thermalization hypothesis

Laura Foini, Anatoly Y. Dymarsky, Silvia Pappalardi

Published Apr 24, 2025• 2 citations
arXivHTMLPDF

Abstract

Understanding how out-of-equilibrium states thermalize under quantum unitary dynamics is an important problem in many-body physics. In this work, we propose a statistical Ansatz for the matrix elements of non-equilibrium initial states in the energy eigenbasis, in order to describe such evolution. The approach is inspired by the Eigenstate Thermalisation Hypothesis (ETH) but the proposed Ansatz exhibits different scaling. Importantly, we point out the exponentially small cross-correlations between the observable and the initial state matrix elements that determine relaxation dynamics toward equilibrium. We numerically verify scaling and cross-correlation, point out the emergent universality of the high-frequency behavior, and outline possible generalizations.

DOI
10.21468/scipostphys.18.4.136

Related papers

Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time

Jiaozi Wang, Mats H. Lamann, Jonas Richter +3 more

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

arXiv:2110.04085•Physical Review Letters•May 4, 2022•62 citations

Microscopic foundation of the Eigenstate Thermalization Hypothesis

Nils O. Abeling

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.

•Jan 1, 2023•1 citation

Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability

Elisa Vallini, Laura Foini, Silvia Pappalardi

The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.

cond-mat.stat-mechCross-listedarXiv:2511.23217•Dec 1, 2025•1 citation

Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability

Anonymous, Laura Foini, Silvia Pappalardi

The eigenstate thermalization hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention toward the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.

PRX Quantum•Aug 4, 2026•0 citations

Theory of Eigenstate Thermalisation

Tobias Helbig, Tobias Hofmann, Ronny Thomale +1 more

If we prepare an isolated, interacting quantum system in an eigenstate and perturb a local observable at an initial time, its expectation value will relax towards a thermal expectation value, even though the time evolution of the system is deterministic. The eigenstate thermalization hypothesis (ETH) of Deutsch and Srednicki suggests that this is possible because each eigenstate of the full quantum system acts as a thermal bath to its subsystems, such that the reduced density matrices of the subsystems resemble thermal density matrices. Here, we use the observation that the eigenvalue distribution of interacting quantum systems is a Gaussian under very general circumstances, and Dyson Brownian motion random matrix theory, to derive the ETH and thereby elevate it from hypothesis to theory. Our analysis provides a derivation of statistical mechanics which neither requires the concepts of ergodicity or typicality, nor that of entropy. Thermodynamic equilibrium follows solely from the applicability of quantum mechanics to large systems and the absence of integrability.

arXiv:2406.01448•Jun 3, 2024•0 citations

Eigenstate Correlations, the Eigenstate Thermalization Hypothesis, and Quantum Information Dynamics in Chaotic Many-Body Quantum Systems

Dominik Hahn, David J. Luitz, J. T. Chalker

We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024

arXiv:2309.12982•Physical Review X•Aug 16, 2024•18 citations
Understanding how out-of-equilibrium states thermalize under quantum unitary dynamics is an important problem in many-body physics. In this work, we propose a statistical Ansatz for the matrix elements of non-equilibrium initial states in the energy eigenbasis, in order to describe such evolution. The approach is inspired by the Eigenstate Thermalisation Hypothesis (ETH) but the proposed Ansatz exhibits different scaling. Importantly, we point out the exponentially small cross-correlations between the observable and the initial state matrix elements that determine relaxation dynamics toward equilibrium. We numerically verify scaling and cross-correlation, point out the emergent universality of the high-frequency behavior, and outline possible generalizations.
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
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.
The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
The eigenstate thermalization hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention toward the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
If we prepare an isolated, interacting quantum system in an eigenstate and perturb a local observable at an initial time, its expectation value will relax towards a thermal expectation value, even though the time evolution of the system is deterministic. The eigenstate thermalization hypothesis (ETH) of Deutsch and Srednicki suggests that this is possible because each eigenstate of the full quantum system acts as a thermal bath to its subsystems, such that the reduced density matrices of the subsystems resemble thermal density matrices. Here, we use the observation that the eigenvalue distribution of interacting quantum systems is a Gaussian under very general circumstances, and Dyson Brownian motion random matrix theory, to derive the ETH and thereby elevate it from hypothesis to theory. Our analysis provides a derivation of statistical mechanics which neither requires the concepts of ergodicity or typicality, nor that of entropy. Thermodynamic equilibrium follows solely from the applicability of quantum mechanics to large systems and the absence of integrability.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024