Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

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.