Weak eigenstate thermalization with large deviation bound

TL;DR

This paper examines the eigenstate thermalization hypothesis in quantum spin systems, showing that deviations from thermal behavior are exponentially rare, thus supporting the typicality of thermalization in large quantum systems.

Contribution

It demonstrates that the diagonal elements of local observables exhibit large deviation bounds, quantifying the rarity of non-thermal eigenstates in the weak ETH regime.

Findings

Diagonal elements follow large deviation bounds

Fraction of non-thermal eigenstates is exponentially small

Supports typicality of thermalization in large systems

Abstract

We investigate the eigenstate thermalization hypothesis (ETH) for a translationally invariant quantum spin system on the $d$-dimensional cubic lattice under the periodic boundary conditions. It is known that the ETH holds in this model for typical energy eigenstates in the sense that the standard deviation of the expectation values of a local observable in the energy eigenstates within the microcanonical energy shell vanishes in the thermodynamic limit, which is called the weak ETH. Here, it is remarked that the diagonal elements of a local observable in the energy representation shows the large deviation behavior. This result implies that the fraction of atypical eigenstates which do not represent thermal equilibrium is exponentially small.