Average Equilibration Time for Gaussian Unitary Ensemble Hamiltonians

TL;DR

This paper derives an analytical expression for the average equilibration time of Gaussian Unitary Ensemble Hamiltonians, showing it decreases with system size and highlighting limitations of RMT in modeling realistic chaotic systems.

Contribution

It provides the first analytical approximation for GUE equilibration times and demonstrates its independence from initial states and observables due to rotational invariance.

Findings

Analytical expression approximates average GUE equilibration time.

Equilibration time decreases with system size, vanishing in the thermodynamic limit.

Numerical simulations confirm the analytical approximation.

Abstract

Understanding equilibration times in closed quantum systems is essential for characterising their approach to equilibrium. Chaotic many-body systems are paradigmatic in this context: they are expected to thermalise according to the eigenstate thermalisation hypothesis and exhibit spectral properties well described by random matrix theory (RMT). While RMT successfully captures spectral correlations, its ability to provide quantitative predictions for equilibration timescales has remained largely unexplored. Here, we study equilibration within RMT using the framework of equilibration as dephasing, focusing on closed systems whose Hamiltonians are drawn from the Gaussian unitary ensemble (GUE). We derive an analytical expression that approximates the average equilibration time of the GUE and show that it is independent of both the initial state and the choice of observable, a consequence of the rotational invariance of the GUE. Numerical simulations confirm our analytical expression and demonstrate that our approximation is in close agreement with the true average equilibration time of the GUE. We find that the equilibration time decreases with system size and vanishes in the thermodynamic limit. This unphysical result indicates that the true equilibration timescale of realistic chaotic many-body systems must be dominated by physical features not captured by random matrix ensembles -- the GUE in particular.