Chaos and Ergodicity in Extended Quantum Systems with Noisy Driving

TL;DR

This paper investigates how quantum chaos in local quantum circuits with random fields leads to the evolution operator behaving like a random matrix over time, linking spectral properties to local observable correlations and conservation laws.

Contribution

It analytically connects the spectral form factor with local observable correlations, elucidates the role of conservation laws, and explains different Thouless time scalings in chaotic quantum systems.

Findings

Spectral form factor matches RMT predictions at large times.

Thouless time relates to system's conservation laws.

Different size scalings of Thouless time observed depending on conservation laws.

Abstract

We study the time evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon we compute analytically the squared magnitude of the trace of the evolution operator -- the generalised spectral form factor -- and compare it with the prediction of Random Matrix Theory (RMT). We show that for the systems under consideration the generalised spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time $\tau_{\rm th}$ -- the time at which the generalised spectral form factor starts following the random matrix theory prediction -- and the conservation laws of the system. Moreover, we explain different scalings of $\tau_{\rm th}$ with the system size, observed for systems with and without the conservation laws.