# Thomas Barthel: Criticality and phase transitions in quadratic open quantum many-body systems

The nonequilibrium steady states of open quantum many-body systems can undergo phase transitions due to the competition of unitary and dissipative dynamics. We consider translation-invariant 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.

Quadratic one-dimensional systems with 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 use that the Green’s function equations of motion for quadratic systems form closed hierarchies, that the Liouvillians can be brought into a useful block-triangular form, and that quasi-free models can be solved exactly using the formalism of third quantization as previously discussed by Prosen and Seligman.

References [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