The nonequilibrium steady states of Markovian open quantum many-body systems can undergo phase transitions due to the competition of unitary and dissipative dynamics. In the first part of the talk, we focus on fermionic and bosonic systems 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. Quadratic one-dimensional systems with translation-invariant finite-range interactions necessarily have exponentially decaying Green's functions. For the quasi-free case without quadratic Lindblad operators, we find 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. Technically, we employ the third quantization formalism to bring Liouvillians of quadratic models into a useful block-triangular form and to solve quasi-free models exactly. The second part of the talk discusses how block-triangular structures of Liouvillians also occur in non-quadratic models which, again, has important consequences for the spectrum and the occurrence of dissipative phase transitions.

[1] Y. Zhang and T. Barthel, "Criticality and phase classification for quadratic open quantum many-body systems", arXiv:2204.05346
[2] T. Barthel and Y. Zhang, "Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems", arXiv:2112.08344
[3] T. Barthel and Y. Zhang, "Super-operator structures and no-go theorems for dissipative quantum phase transitions", arXiv:2012.05505