The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators

TL;DR

This paper explores the connection between the Eigenstate Thermalization Hypothesis and chaos indicators like out-of-time-order correlators, proposing a generalized ETH to include correlations relevant for chaotic dynamics.

Contribution

It introduces a generalized ETH framework that accounts for correlations between matrix elements, linking ETH to chaos measures such as Lyapunov exponents.

Findings

Correlations between matrix elements are essential for positive Lyapunov exponents.

A generalized ETH encompasses correlations relevant for slow dynamics and chaos.

The framework connects ETH with out-of-time-order correlators and dynamic heterogeneity.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out-of-time-order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.