Transport and integrability-breaking in non-Hermitian many-body quantum systems

The effective description of open quantum systems in terms of non-Hermitian Hamiltonians gives rise to non-unitary time evolution. Here, we study how non-unitary dynamics affects the emergent hydrodynamics in systems with a globally conserved quantity. To this end, we show how dynamical correlation functions can be generalized to the non-Hermititan case. Moreover, we demonstrate that the concept of dynamical quantum typicality provides a useful numerical approach to evaluate such correlation functions, albeit with non-unitary time evolution leading to certain subtleties compared to the usual Hermitian setting. As a convenient starting point for our numerical investigation, we consider the Hermitian spin-1/2 XXZ chain, whose high-temperature transport properties have been studied extensively in recent years. We add different non-Hermitian perturbations to the XXZ chain and explore their effect on the resulting hydrodynamics. In this context, we also discuss the role of integrability by studying the complex energy-level statistics of the non-Hermitian quantum models.