Spectral and steady-state properties of fermionic random quadratic Liouvillians

We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian superoperators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by $M$ random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, $m=M/(2N_F)$, we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs at $m=1/2$ and is concomitant with a qualitative change in both the steady-state and the spectral gap that describes the large-time dynamics. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators (above $m=1/2$). On the other hand, if the number of dissipation channels is decreased below $m=1/2$, the gap closes and the steady state decouples into an ergodic and a nonergodic sector, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.