We derive a necessary and sufficient condition for the thermalization of a local observable in a closed quantum system which offers an alternative explanation, independent of the eigenstate thermalization hypothesis, for the thermalization process. We also show that this approach is useful to investigate thermalization based on a finite-size scaling of numerical data. The condition follows from an exact representation of the observable as the sum of a projection onto the local conserved charges of the system and a projection onto the nonlocal ones. We show that thermalization requires that the time average of the latter part vanishes in the thermodynamic limit while time and statistical averages for the first part are identical. As an example, we use this thermalization condition to analyze exact diagonalization data for a one-dimensional spin model. We find that local correlators do thermalize in the thermodynamic limit, although we find no indications that the eigenstate thermalization hypothesis applies.
We derive a necessary and sufficient condition for the thermalization of a local observable in a closed quantum system which offers an alternative explanation, independent of the eigenstate thermalization hypothesis, for the thermalization process. We also show that this approach is useful to investigate thermalization based on a finite-size scaling of numerical data. The condition follows from an exact representation of the observable as the sum of a projection onto the local conserved charges of the system and a projection onto the nonlocal ones. We show that thermalization requires that the time average of the latter part vanishes in the thermodynamic limit while time and statistical averages for the first part are identical. As an example, we use this thermalization condition to analyze exact diagonalization data for a one-dimensional spin model. We find that local correlators do thermalize in the thermodynamic limit, although we find no indications that the eigenstate thermalization hypothesis applies.
Despite the unitary evolution of closed quantum systems, long-time expectation of local observables are well described by thermal ensembles, providing the foundation of quantum statistical mechanics. A promising route to understanding this quantum thermalization is the eigenstate thermalization hypothesis (ETH), which posits that individual energy eigenstates already appear locally thermal. Subsequent studies have extended this concept to the full ETH, which captures higher-order correlations among matrix elements through nontrivial relations. In this work, we perform a detailed exact-diagonalization study of finite-size corrections to these relations in the canonical ensemble. We distinguish two distinct sources of corrections: those arising from energy fluctuations, which decay polynomially with system size, and those originating from fluctuations within each energy window, which decay exponentially with system size. In particular, our analysis resolves the puzzle that, for certain observables, finite-size corrections exhibit anomalous growth with increasing system size even in chaotic systems. Our results provide a systematic and practical methodology for validating the full ETH in quantum many-body systems.
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.
We study the validity of the eigenstate thermalization hypothesis (ETH) and its role for the occurrence of initial-state independent (ISI) equilibration in closed quantum many-body systems. Using the concept of dynamical typicality, we present an extensive numerical analysis of energy exchange in integrable and nonintegrable spin-1/2 systems of large size outside the range of exact diagonalization. In the case of nonintegrable systems, our finite-size scaling shows that the ETH becomes valid in the thermodynamic limit and can serve as the underlying mechanism for ISI equilibration. In the case of integrable systems, however, indication of ISI equilibration has been observed despite the violation of the ETH. We establish a connection between this observation and the need of choosing a proper parameter within the ETH.
The thermalization phenomenon and many-body quantum statistical properties are studied on the example of several observables in isolated spin-chain systems, both integrable and generic nonintegrable. While diagonal matrix elements for nonintegrable models comply with the eigenstate thermalization hypothesis, the integrable systems show evident deviations and similarity to properties of noninteracting many-fermion models. The finite-size scaling reveals that the crossover between the two regimes is given by a scale closely related to the scattering length. Low-frequency off-diagonal matrix elements related to dc transport quantities also follow in a generic system a behavior analogous to the eigenstate thermalization hypothesis, however, unrelated to one of the diagonal matrix elements.
The eigenstate thermalization hypothesis provides a framework for understanding thermalization in isolated quantum many-body systems by characterizing statistical properties of local observables in energy eigenstates. Here we demonstrate that distributions of matrix elements in macroscopic systems may depend not only on the macrostate parameters, such as the densities of local conserved charges, but generally also on the properties of ensembles used in sampling eigenstates. To this end, we depart from the conventional analysis of microcanonical windows and consider statistical ensembles with an adjustable scale parameter prescribing the magnitude of charge fluctuations. We specifically consider an integrable field theory that permits efficient numerical sampling of matrix elements and reliable extrapolation to the thermodynamic limit. Moreover, in this system, we identify a class of states that enables explicit closed-form computation of the suppression rate of matrix elements. Our findings reveal an underlying multiscale structure of matrix elements captured by a non-analytic fluctuation-scale dependence of algebraic exponents governing their statistical properties.
The eigenstate thermalization hypothesis (ETH) has been highly influential in explaining thermodynamic behavior of closed quantum systems. As of yet, it is unclear whether and how the ETH applies to non-Hermitian systems. Here, we introduce a framework that extends the ETH to non-Hermitian systems, within which expectation values of local operators reproduce statistical and scaling predictions known from Hermitian ETH. We illustrate the validity of the framework on non-Hermitian random-matrix and Sachdev-Ye-Kitaev models. Further, we show numerically how the static ETH predictions become imprinted onto the dynamics of local observables. Finally, we present a prescription for observing both ETH-obeying and ETH-violating regimes in an optical-lattice experiment that implements a disordered interacting Hatano-Nelson model. Our results generalize the celebrated ETH to the non-Hermitian setting, and they show how it affects the system dynamics, and how the salient signatures can be observed in present-day cold-atom experiments.