Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time

TL;DR

This paper investigates the limits of Random Matrix Theory in describing observable matrix elements in quantum systems, revealing deviations from RMT near thermalization times through a novel numerical approach.

Contribution

It introduces a new numerical method to analyze correlations in matrix elements beyond exact diagonalization, challenging the assumption that RMT fully describes thermalization processes.

Findings

RMT does not fully describe observable matrix elements even near thermalization time.

Residual correlations influence the dynamics of out-of-time-ordered correlation functions.

The new approach extends analysis to larger Hilbert spaces than traditional methods.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) explains emergence of the thermodynamic equilibrium by assuming a particular structure of 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 physical operators can be described by RMT, more precisely at which energy scale strict RMT description applies, 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 for 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 genuine RMT behavior is absent even for narrow energy windows corresponding to time scales of the order of thermalization time $\tau_\text{th}$ of the respective observables. We also demonstrate that residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.