Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

TL;DR

This paper studies how complex eigenvalue statistics in a quantum spin chain transition between different regimes, revealing separate breaking of Hermiticity and integrability at various disorder levels, using random matrix theory and Coulomb gas models.

Contribution

It uncovers distinct disorder-induced transitions in eigenvalue statistics related to Hermiticity and integrability breaking in a non-Hermitian quantum spin chain.

Findings

Transition from 1d Poisson to D-dimensional Poisson at small disorder

Matching of spectral statistics with non-Hermitian random matrix classes

Recovery of 2d Poisson statistics at large disorder

Abstract

Open quantum systems have complex energy eigenvalues which are expected to follow non-Hermitian random matrix statistics when chaotic, or 2-dimensional (2d) Poisson statistics when integrable. We investigate the spectral properties of a many-body quantum spin chain, the Hermitian XXZ Heisenberg model with imaginary disorder. Its rich complex eigenvalue statistics is found to separately break both Hermiticity and integrability at different scales of the disorder strength. With no disorder, the system is integrable and Hermitian, with spectral statistics corresponding to the 1d Poisson point process. At very small disorder, we find a transition from 1d Poisson statistics to an effective $D$-dimensional Poisson point process, showing Hermiticity breaking. At intermediate disorder we find integrability breaking, as inferred from the statistics matching that of non-Hermitian complex symmetric random matrices in class AI$^\dag$. For large disorder, as the spins align, we recover the expected integrability (now in the non-Hermitian setup), indicated by 2d Poisson statistics. These conclusions are based on fitting the spin chain data of numerically generated nearest and next-to-nearest neighbour spacing distributions to an effective 2d Coulomb gas description at inverse temperature $\beta$. We confirm such an effective description of random matrices also applies in class AI$^\dag$ and AII$^\dag$ up to next-to-nearest neighbour spacings.