Eigenstate Thermalization, Random Matrix Theory and Behemoths

TL;DR

This paper explores the eigenstate thermalization hypothesis (ETH) for nonlocal operators by constructing 'Behemoths' with unique distributions, revealing sub-ETH behavior and connecting to random matrix theory.

Contribution

It introduces highly nonlocal operators called Behemoths, analyzing their distributions and implications for ETH in nonlocal operators, supported by numerical simulations.

Findings

Behemoths have a singular distribution with width w~D^{-1}.

Local operators constructed from Behemoths follow Gaussian distribution with width w~D^{-1/2}.

Extrapolation suggests sub-ETH behavior for larger nonlocal operators with widths D^{-δ}.

Abstract

The eigenstate thermalization hypothesis (ETH) is one of the cornerstones in our understanding of quantum statistical mechanics. The extent to which ETH holds for nonlocal operators is an open question that we partially address in this paper. We report on the construction of highly nonlocal operators, Behemoths, that are building blocks for various kinds of local and non-local operators. The Behemoths have a singular distribution and width $w\sim \mathcal{D}^{-1}$ ($\mathcal{D}$ being the Hilbert space dimension). From them, one may construct local operators with the ordinary Gaussian distribution and $w\sim \mathcal{D}^{-1/2}$ in agreement with ETH. Extrapolation to even larger widths predicts sub-ETH behavior of typical nonlocal operators with $w\sim \mathcal{D}^{-\delta}$, $0<\delta<1/2$. This operator construction is based on a deep analogy with random matrix theory and shows striking agreement with numerical simulations of non-integrable many-body systems.