Abstract: Efforts on quantum simulation and computation have lead to the realization of well-controlled quantum many-body systems. Due to practical constraints, they are inevitably open, i.e., coupled to the environment, which generally leads to decoherence but can also be used to stabilize interesting states. In the thermodynamic limit, the nonequilibrium steady states can undergo phase transitions due to the competition of unitary and driven-dissipative processes. After recalling general properties, we will discuss first simple examples. Considering spin chains, we will see how the competition can lead to divergent decoherence times, i.e., algebraic coherence decay [1]. When trying to replicate the groundstate phase transition of transverse Ising chains in a dissipative system, we will see how block-triangular structures in the Liouvillian, corresponding to certain dynamical constraints, hinder steady-state phase transitions in large classes of systems [2]. We will encounter further fundamental differences between closed and open systems for the quadratic case, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian [3]. In one dimension, such systems with finite-range couplings cannot be critical. For the quasi-free case without quadratic Lindblad operators, local fermionic systems are non-critical for any number of spatial dimensions. Local 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 [4]. Another important concept is irreducibility, which can be assessed algebraically [5]. A somewhat surprising conclusion is that phase transitions in irreducible open systems are always associated with an instability. As a nontrivial example, we will discuss driven-dissipative Bose-Einstein condensation using Keldysh field theory [6].
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[2] "Super-operator structures and no-go theorems for dissipative quantum phase transitions", PRA 105, 052224 (2022)
[3] "Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems", J. Stat. Mech. 113101 (2022)
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[5] "Criteria for Davies irreducibility of Markovian quantum dynamics", J. Phys. A: Math. Theor. 57, 115301 (2024)
[6] "Driven-dissipative Bose-Einstein condensation and the upper critical dimension", PRA 109, L021301 (2024)