Criticality and Phase Classification for Quadratic Open Quantum Many-Body Systems

We study the steady states of translation-invariant open quantum 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. These systems are called quasi-free and quadratic, respectively. We find that steady states of one-dimensional systems with finite-range interactions necessarily have exponentially decaying Green's functions. For the quasi-free case without quadratic Lindblad operators, we show that fermionic systems with finite-range interactions are non-critical for any number of spatial dimensions and provide bounds on the correlation lengths. Quasi-free bosonic systems can be critical in $D>1$ dimensions. Lastly, we address the question of phase transitions in quadratic systems and find that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped Liouvillians belong to the same phase.