Random matrix universality in dynamical correlation functions at late times

TL;DR

This paper demonstrates that late-time two-time correlation functions in finite quantum systems exhibit universal ramp and plateau features linked to energy level correlations, with non-self-averaging behavior and dependence on symmetry class.

Contribution

It analytically and numerically shows the universal late-time behavior of correlation functions using Random Matrix Theory and the Eigenstate Thermalisation Hypothesis.

Findings

Correlation functions display a ramp and plateau similar to spectral form factors.

Plateau value depends on energy level correlations and symmetry class.

Correlation functions in the ramp are not self-averaging and differ from spectral form factors.

Abstract

We study the behavior of two-time correlation functions at late times for finite system sizes considering observables whose (one-point) average value does not depend on energy. In the long time limit, we show that such correlation functions display a ramp and a plateau determined by the correlations of energy levels, similar to what is already known for the spectral form factor. The plateau value is determined, in absence of degenerate energy levels, by the fluctuations of diagonal matrix elements, which highlights differences between different symmetry classes. We show this behavior analytically by employing results from Random Matrix Theory and the Eigenstate Thermalisation Hypothesis, and numerically by exact diagonalization in the toy example of a Hamiltonian drawn from a Random Matrix ensemble and in a more realistic example of disordered spin glasses at high temperature. Importantly, correlation functions in the ramp regime do not show self-averaging behaviour, and, at difference with the spectral form factor the time average does not coincide with the ensemble average.