Characterizing eigenstate thermalization via measures in the Fock space of operators

TL;DR

This paper introduces measures to quantify the validity of the eigenstate thermalization hypothesis (ETH) in quantum systems by mapping them onto high-dimensional graphs and analyzing deviations, revealing insights into operator complexity and state distinguishability.

Contribution

It defines new measures, the n-weight and n-distinguishability, to characterize ETH deviations and operator effectiveness in non-integrable quantum systems.

Findings

Simple operators are better at distinguishing eigenstates than complex ones.

The n-weight measure exhibits a universal scaling form across non-integrable systems.

Complicated operators are less effective at eigenstate discrimination than naive expectations suggest.

Abstract

The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the non-integrable Ising model, we propose a quantity that we call the $n$-$weight$ to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, that we conjecture to hold true for all non-integrable systems. A closely related quantity, that we term the $n$-$distinguishability$, tells us how well two states can be distinguished if only $n$-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.