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In the thermodynamic limit, nonequilibrium steady states of open quantum many-body systems can undergo phase transitions due to the competition of unitary and driven-dissipative dynamics. Considering Markovian systems, I will describe recent results concerning preconditions for criticality, phase transitions, and irreducibly. (a) A large class of translation-invariant fermionic and bosonic systems can be characterized almost completely -- "quadratic" systems, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian [1]. In one dimension, such systems with finite-range couplings cannot be critical, i.e., steady-state correlations necessarily decay exponentially. For the quasi-free case without quadratic Lindblad operators, fermionic systems with finite-range couplings are non-critical for any number of spatial dimensions. Quasi-free bosonic systems in d>1 dimensions can be critical. Furthermore, for quadratic systems without symmetry constraints beyond particle-hole symmetries, all gapped Liouvillians belong to the same phase [2]. This also has implications for transitions in non-quadratic (interacting) systems above the upper critical dimension [3]. (b) A related scenario are open systems with dynamical constraints that make it possible to bring the Liouvillian into block-triangular form and to assess the spectrum through suitable operator basis transformations. I will discuss corresponding classes of systems where Weyl ordering relations establish the absence of dissipative phase transitions [4]. (c) Time permitting, we can discuss the important concept of (Davies) irreducibility in driven-dissipative systems, i.e., the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e., they have full rank. Extending seminal work by Davies, Frigerio and others from the 1970s, we found a powerful algebraic criterion for irreducibility [5]. [1] "Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems", arXiv:2112.08344, J. Stat. Mech. 113101 (2022) [2] "Criticality and phase classification for quadratic open quantum many-body systems", arXiv:2204.05346, PRL 129, 120401 (2022) [3] "Driven-dissipative Bose-Einstein condensation and the upper critical dimension", arXiv:2311.13561, PRA 109, L021301 (2024) [4] "Super-operator structures and no-go theorems for dissipative quantum phase transitions", arXiv:2012.05505, PRA 105, 052224 (2022) [5] "Criteria for Davies irreducibility of Markovian quantum dynamics", arXiv:2310.17641, J. Phys. A: Math. Theor. 57, 115301 (2024)

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Thomas Barthel, Duke University
Seminar Room 0.03, ETP
Contact: Sebastian Diehl