Trade-off between diagonal and off-diagonal elements in the eigenstate thermalization hypothesis

TL;DR

This paper introduces an observable-independent measure to analyze the eigenstate thermalization hypothesis, revealing a universal trade-off between diagonal and off-diagonal elements that governs thermalization in quantum systems.

Contribution

It establishes a universal trade-off relation connecting diagonal and off-diagonal elements of a new measure, unifying various ETH concepts and providing insights into thermalization mechanisms.

Findings

Off-diagonal elements are suppressed with system size in non-integrable systems.

Diagonal suppression fails in integrable systems due to lack of chaos.

The framework unifies subsystem ETH, weak ETH, and macroscopic observables.

Abstract

To bypass the reliance on local observables in verifying the eigenstate thermalization hypothesis (ETH), we introduce an observable-independent measure of distinguishability based on the variance of a rescaled local operator. We establish a universal trade-off relation between the diagonal and off-diagonal elements of this measure, rigorously connecting it to eigenstate typicality and spatially averaged observables. This trade-off reveals that exponential growth in the number of off-diagonal terms enforces their suppression, indirectly constraining diagonal deviations. Numerical simulations on a one-dimensional Ising spin chain with tunable transverse and longitudinal fields demonstrate stark contrasts between integrable and non-integrable regimes: While off-diagonal elements are universally suppressed with system size, diagonal suppression fails in integrable systems due to the absence of chaotic dynamics. Our results unify subsystem ETH, weak ETH, and macroscopic observables under a single framework, offering new insights into thermalization mechanisms.