Finite-size scaling of eigenstate thermalization

TL;DR

This paper investigates how eigenstate expectation value fluctuations in finite quantum systems scale with system size, revealing a universal power law behavior in non-integrable models supported by numerical evidence.

Contribution

It demonstrates a universal $D^{-1/2}$ power law scaling of fluctuations in non-integrable systems and analyzes the impact of integrability on this scaling.

Findings

Universal $D^{-1/2}$ scaling in non-integrable systems

Scaling affected by proximity to integrability

Numerical evidence from three model families

Abstract

According to the eigenstate thermalization hypothesis (ETH), even isolated quantum systems can thermalize because the eigenstate-to-eigenstate fluctuations of typical observables vanish in the limit of large systems. Of course, isolated systems are by nature finite, and the main way of computing such quantities is through numerical evaluation for finite-size systems. Therefore, the finite-size scaling of the fluctuations of eigenstate expectation values is a central aspect of the ETH. In this work, we present numerical evidence that for generic non-integrable systems these fluctuations scale with a universal power law $D^{-1/2}$ with the dimension $D$ of the Hilbert space. We provide heuristic arguments, in the same spirit as the ETH, to explain this universal result. Our results are based on the analysis of three families of models, and several observables for each model. Each family includes integrable members, and we show how the system size where the universal power law becomes visible is affected by the proximity to integrability.