Eigenvalue statistics as indicator of integrability of non-equilibrium density operators

TL;DR

This paper introduces a method to determine the integrability of non-equilibrium density operators by analyzing their eigenvalue level spacing statistics, distinguishing integrable from non-integrable systems.

Contribution

It proposes using eigenvalue level spacing distribution as a novel indicator of integrability in non-equilibrium quantum systems, supported by extensive numerical evidence.

Findings

Poissonian level statistics indicate integrability.

Gaussian unitary ensemble statistics indicate non-integrability.

Eigenvalue analysis effectively identifies integrable non-equilibrium systems.

Abstract

We propose to quantify the complexity of non-equilibrium steady state density operators, as well as of long-lived Liouvillian decay modes, in terms of level spacing distribution of their spectra. Based on extensive numerical studies in a variety of models, some solvable and some unsolved, we conjecture that integrability of density operators (e.g., existence of an algebraic procedure for their construction in finitely many steps) is signaled by a Poissonian level statistics, whereas in the generic non-integrable cases one finds level statistics of a Gaussian unitary ensemble of random matrices. Eigenvalue statistics can therefore be used as an efficient tool to identify integrable quantum non-equilibrium systems.