Spectral and steady-state properties of fermionic random quadratic Liouvillians

TL;DR

This paper analyzes the spectral and steady-state behaviors of fermionic quadratic Liouvillians, revealing phase transitions and ergodic properties depending on dissipation strength and channel ratio, with implications for quantum system control.

Contribution

It introduces a detailed study of phase transitions in fermionic quadratic Liouvillians, highlighting the impact of dissipation channels on spectral and steady-state properties.

Findings

Two distinct spectral phases identified based on dissipation strength and channel ratio.

Transition at m=1/2 correlates with changes in spectral gap and steady-state purity.

Large dissipation channels lead to universal features similar to non-quadratic Liouvillians.

Abstract

We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators 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 for $m=1/2$ and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below $m=1/2$, the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. 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. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.