Degenerate observables and the many Eigenstate Thermalization Hypotheses

TL;DR

This paper investigates how the eigenstate thermalization hypothesis (ETH) applies to highly degenerate observables, analyzing conditions on overlaps and phases between eigenbases to understand thermalization in quantum systems.

Contribution

It introduces a framework for understanding ETH in degenerate observables by examining overlap and phase conditions between observable and Hamiltonian eigenbases.

Findings

ETH can be extended to degenerate observables with specific overlap conditions

Phase relations between eigenbases influence thermalization behavior

The approach clarifies the emergence of ETH in realistic measurement scenarios

Abstract

Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian.