Lindbladian Dynamics of Disordered Fermions

To understand the effects of disorder in many-body systems driven out of equilibrium, we study ensembles of random Lindbladian fermions. We discuss both spectral properties and stationary distributions for pure dissipation and combinations of dissipative and unitary dynamics. The spectrum is obtained from non-Hermitian random matrix replica methods and displays multiple transitions by varying the number of dissipative channels and the relative strength of gain versus loss of particles. Different dynamical regimes are categorized by the number of connected components of the support of the spectrum. The stationary state is a Gaussian state determined by solving a random quantum kinetic equation. The eigenvalue distribution of the stationary density matrix can be either gapped or continuous and is generically multi-modal.