Eigenstate correlations, the eigenstate thermalization hypothesis, and quantum information dynamics in chaotic many-body quantum systems

TL;DR

This paper investigates eigenstate correlations in chaotic many-body quantum systems, focusing on entanglement and operator spreading, and proposes a maximum-entropy model that extends the eigenstate thermalization hypothesis to capture complex quantum information dynamics.

Contribution

It introduces a new correlation function and a maximum-entropy Ansatz that generalizes ETH to describe multi-eigenstate correlations and quantum information dynamics.

Findings

The correlation function captures universal features of eigenstate correlations.

The maximum-entropy Ansatz reproduces ETH for two eigenstates and models entanglement growth for four eigenstates.

Monte Carlo sampling of the Ansatz agrees with exact diagonalization results.

Abstract

We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterise entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalisation hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations amongst sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behaviour that are expected to be universal at long distances and low energies. We also propose a maximum-entropy Ansatz for the joint distribution of a small number $n$ of eigenstates. In the case $n = 2$ this Ansatz reproduces ETH. For $n = 4$ it captures both the growth with time of entanglement between subsystems, as characterised by the purity of the time-evolution operator, and also operator spreading, as characterised by the behaviour of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our Ansatz with exact diagonalisation studies of Floquet quantum circuits.