Super-operator structures and no-go theorems for dissipative quantum phase transitions

Abstract

In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville super-operator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The super-operator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems, and are also demonstrated for various spin models.

8 pages; added discussion of block-triangular representations for quadratic fermionic and bosonic systems (see arXiv:2112.08344 and arXiv:2204.05346), further minor improvements; published version

References in corpus (21)

• Many-Body Physics with Ultracold Gases
• The density-matrix renormalization group in the age of matrix product states
• Quantum fluids of light
• Simulated Quantum Computation of Molecular Energies
• Quantum States and Phases in Driven Open Quantum Systems with Cold Atoms
• Multi-party entanglement in graph states
• Strongly Interacting Polaritons in Coupled Arrays of Cavities
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Third quantization: a general method to solve master equations for quadratic open Fermi systems
• Dissipative Phase Transition in Central Spin Systems
• Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems
• Observation of a dissipative phase transition in a one-dimensional circuit QED lattice
• Variational principle for steady states of dissipative quantum many-body systems
• Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions
• Universal non-equilibrium properties of dissipative Rydberg gases
• Analysis of quantum semigroups with GKS--Lindblad generators II. General
• Reservoir engineering and dynamical phase transitions in optomechanical arrays
• Superoperator nonequilibrium Green's function theory of many-body systems; Applications to charge transfer and transport in open junctions
• Mean field theory and fluctuation spectrum of a pumped, decaying Bose-Fermi system across the quantum condensation transition
• Antiferromagnetic long-range order in dissipative Rydberg lattices
• Stationary discrete solitons in circuit QED

Cited by in corpus (9)

• Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems
• Rapid thermalization of spin chain commuting Hamiltonians
• Criticality and Phase Classification for Quadratic Open Quantum Many-Body Systems
• Effect of quantum jumps on non-Hermitian system
• Criteria for Davies Irreducibility of Markovian Quantum Dynamics
• Mixing Time of Open Quantum Systems via Hypocoercivity
• Driven-Dissipative Bose-Einstein Condensation and the Upper Critical Dimension
• Closed dynamical recursion equations for correlation functions and the application on the construction of Liouvillian spectrum in Lindbladian systems
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter