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:1609.09776

Weak eigenstate thermalization with large deviation bound

Takashi Mori

Published Sep 30, 2016• 24 citations
arXivHTMLPDF

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) for a translationally invariant quantum spin system on the $d$-dimensional cubic lattice under the periodic boundary conditions. It is known that the ETH holds in this model for typical energy eigenstates in the sense that the standard deviation of the expectation values of a local observable in the energy eigenstates within the microcanonical energy shell vanishes in the thermodynamic limit, which is called the weak ETH. Here, it is remarked that the diagonal elements of a local observable in the energy representation shows the large deviation behavior. This result implies that the fraction of atypical eigenstates which do not represent thermal equilibrium is exponentially small.

DOI
10.48550/arxiv.1609.09776

Related papers

Dynamical Typicality Approach to Eigenstate Thermalization

Peter Reimann

We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.

arXiv:1806.03193•Physical Review Letters•Jun 6, 2018•31 citations

Numerical Large Deviation Analysis of the Eigenstate Thermalization Hypothesis

Tôru Yoshizawa, Eiki Iyoda, Takahiro Sagawa

A plausible mechanism of thermalization in isolated quantum systems is based on the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates in the microcanonical energy shell have thermal properties. We numerically investigate the ETH by focusing on the large deviation property, which directly evaluates the ratio of athermal energy eigenstates in the energy shell. As a consequence, we have systematically confirmed that the strong ETH is indeed true even for near-integrable systems. Furthermore, we found that the finite-size scaling of the ratio of athermal eigenstates is a double exponential for nonintegrable systems. Our result illuminates the universal behavior of quantum chaos, and suggests that a large deviation analysis would serve as a powerful method to investigate thermalization in the presence of the large finite-size effect.

arXiv:1712.07289•Physical Review Letters•May 16, 2018•63 citations

Convergence of eigenstate expectation values with system size

Yichen Huang

Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.

arXiv:2009.05095•Advances in Theoretical and Mathematical Physics•Jan 1, 2022•2 citations

Eigenstate thermalization hypothesis and integrability in quantum spin chains

Vincenzo Alba

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg $(XXX)$ chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with $L$ the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the $XXX$ chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).

arXiv:1409.6096•Physical Review B•Apr 16, 2015•156 citations

Eigenstate thermalization for local versus translationally invariant observables

Rohit Patil, Marcos Rigol

Local observables and their translationally invariant counterparts are generally thought to provide the same predictions for experiments. While this equivalence holds for expectation values in clean systems (up to finite-size effects), it is often assumed to extend to correlation functions, where it need not hold. We examine this assumption from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the spectral functions of local and translationally invariant observables in the spin-1 tilted-field Ising chain with periodic and open boundary conditions. We identify the contexts in which these observables and boundary conditions differ and those in which they are interchangeable. We unveil an off-diagonal eigenstate thermalization in translationally invariant systems for matrix elements between energy eigenstates with different quasimomenta.

quant-pharXiv:2602.09087•Physical review. B./Physical review. B•Feb 11, 2026•0 citations

Atypicality of Most Few-Body Observables

Ryusuke Hamazaki, Masahito Ueda

The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a major candidate to explain thermalization in isolated quantum systems. According to the typicality argument, the maximum variations of such matrix elements should decrease exponentially with increasing the size of the system, which implies the ETH. We show, however, that the typicality argument does not apply to most few-body observables for few-body Hamiltonians when the width of the energy shell decreases at most polynomially with increasing the size of the system.

arXiv:1708.04772•Physical Review Letters•Feb 23, 2018•38 citations
We investigate the eigenstate thermalization hypothesis (ETH) for a translationally invariant quantum spin system on the ddd-dimensional cubic lattice under the periodic boundary conditions. It is known that the ETH holds in this model for typical energy eigenstates in the sense that the standard deviation of the expectation values of a local observable in the energy eigenstates within the microcanonical energy shell vanishes in the thermodynamic limit, which is called the weak ETH. Here, it is remarked that the diagonal elements of a local observable in the energy representation shows the large deviation behavior. This result implies that the fraction of atypical eigenstates which do not represent thermal equilibrium is exponentially small.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
A plausible mechanism of thermalization in isolated quantum systems is based on the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates in the microcanonical energy shell have thermal properties. We numerically investigate the ETH by focusing on the large deviation property, which directly evaluates the ratio of athermal energy eigenstates in the energy shell. As a consequence, we have systematically confirmed that the strong ETH is indeed true even for near-integrable systems. Furthermore, we found that the finite-size scaling of the ratio of athermal eigenstates is a double exponential for nonintegrable systems. Our result illuminates the universal behavior of quantum chaos, and suggests that a large deviation analysis would serve as a powerful method to investigate thermalization in the presence of the large finite-size effect.
Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by 1/O(N)1/O(N)1/O(N), where NNN is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.
We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg (XXX)(XXX)(XXX) chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with LLL the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the XXXXXXXXX chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).
Local observables and their translationally invariant counterparts are generally thought to provide the same predictions for experiments. While this equivalence holds for expectation values in clean systems (up to finite-size effects), it is often assumed to extend to correlation functions, where it need not hold. We examine this assumption from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the spectral functions of local and translationally invariant observables in the spin-1 tilted-field Ising chain with periodic and open boundary conditions. We identify the contexts in which these observables and boundary conditions differ and those in which they are interchangeable. We unveil an off-diagonal eigenstate thermalization in translationally invariant systems for matrix elements between energy eigenstates with different quasimomenta.
The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a major candidate to explain thermalization in isolated quantum systems. According to the typicality argument, the maximum variations of such matrix elements should decrease exponentially with increasing the size of the system, which implies the ETH. We show, however, that the typicality argument does not apply to most few-body observables for few-body Hamiltonians when the width of the energy shell decreases at most polynomially with increasing the size of the system.