Eigenstate Randomization Hypothesis: Why Does the Long-Time Average Equal the Microcanonical Average?

TL;DR

This paper introduces the eigenstate randomization hypothesis (ERH), explaining why long-time averages in quantum systems match microcanonical averages, and shows ERH's broad applicability including integrable systems.

Contribution

The paper proposes and supports the eigenstate randomization hypothesis, extending understanding of thermalization beyond ETH and applicable to integrable systems.

Findings

Derived an upper bound on observable differences between long-time and microcanonical averages.

Numerically verified and analytically supported ERH, including its relation to ETH.

ERH's validity range determines the applicability of the microcanonical ensemble.

Abstract

We derive an upper bound on the difference between the long-time average and the microcanonical ensemble average of observables in isolated quantum systems. We propose, numerically verify, and analytically support a new hypothesis, eigenstate randomization hypothesis (ERH), which implies that in the energy eigenbasis the diagonal elements of observables fluctuate randomly. We show that ERH includes eigenstate thermalization hypothesis (ETH) and makes the aforementioned bound vanishingly small. Moreover, ERH is applicable to integrable systems for which ETH breaks down. We argue that the range of the validity of ERH determines that of the microcanonical description.