Why ETH? On thermalization and locality

TL;DR

This paper investigates how operator locality influences the eigenstate thermalization hypothesis (ETH) in quantum systems, introducing a perturbation framework to understand ETH's onset or failure, with tests on chaotic spin chains.

Contribution

It develops an operator-specific perturbation approach to analyze ETH, emphasizing the role of locality and providing explicit formulas for operator matrix element variances.

Findings

Derived explicit formulas for off-diagonal variances of local operators.

Showed how operator locality affects ETH validity.

Validated ideas through tests on chaotic spin chains.

Abstract

The eigenstate thermalization hypothesis (ETH) is foundational to modern discussions of thermalization in closed quantum systems. In this work, we expand on traditional explanations for the prevalence of ETH by emphasizing the role of operator locality. We introduce an operator-specific perturbation problem that can be thought of as a means of understanding the onset or breakdown of ETH for specific classes of operators in a given system. We derive explicit functional forms for the off-diagonal variances of operator matrix elements for typical local operators under various `scrambling ansatzes', expressed in terms of system parameters and parameters of the corresponding perturbation problem. We provide simple tests and illustrations of these ideas in chaotic spin chain systems.