Criticality and phase classification for quadratic open quantum many-body systems

We study the steady states of translation-invariant open many-body systems governed by Lindblad master equations, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian. We find that one-dimensional systems with short-range interactions cannot be critical, i.e., steady-state correlations necessarily decay exponentially. For the quasi-free case without quadratic Lindblad operators, we show that fermionic systems with short-range interactions are non-critical for any number of spatial dimensions and provide upper bounds on the correlation lengths. Lastly, we address the question of phase transitions in quadratic fermionic systems and find that, without symmetry constraints beyond particle-hole symmetries, all gapped Liouvillians belong to the same phase.