Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond

Abstract

We perform a systematic symmetry classification of many-body Lindblad superoperators describing general (interacting) open quantum systems coupled to a Markovian environment. Our classification is based on the behavior of the many-body Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, many-body Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes. In both cases, it does not include classes with Kramer's degeneracy. Remarkably, our classification admits a straightforward generalization to the case of non-Markovian, and even non-trace-preserving, open quantum dynamics. While the abstract classification is completely general, we then apply it to general (long-range, interacting, spatially inhomogeneous) spin-$1/2$ chains. We explicitly build examples in all ten classes of Lindbladians in steady-state sectors, describing standard physical processes such as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance of our classification for practical physics applications. Finally, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations. We further find that statistics of levels constrained onto the real and imaginary axes or close to the origin are not universal due to spontaneous breaking of Lindbladian PT symmetry.

arXiv comment21 pages, 7 figures. v2: analytical proof of eigenvector overlap properties, minor class counting misprint corrected, and references added. v3: Sec. IIF rewritten, new Sec. IIH, new Fig. 1, references added, version as published in PRX

References in corpus (56)

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Non-Hermitian Physics
• Symmetry and Topology in Non-Hermitian Physics
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Spectral Density of the QCD Dirac Operator near Zero Virtuality
• Canonical form of master equations and characterization of non-Markovianity
• A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains
• Non-Hermitian Many-Body Localization
• Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors
• Tenfold Way for Quadratic Lindbladians
• Universality of random matrices in the microscopic limit and the Dirac operator spectrum
• Universal results from an alternate random matrix model for QCD with a baryon chemical potential
• Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics
• Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
• Eigenvector statistics in non-Hermitian random matrix ensembles
• Eigenvalue statistics of the real Ginibre ensemble
• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Universality classes of non-Hermitian random matrices
• Anderson transition in three-dimensional systems with non-Hermitian disorder
• Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov-de Gennes equation
• Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
• Universality classes of the Anderson Transitions Driven by non-Hermitian Disorder
• Topological classification of defects in non-Hermitian systems
• Symmetry classes of open fermionic quantum matter
• Hierarchy of relaxation timescales in local random Liouvillians
• PT-symmetric quantum Liouvillian dynamics
• Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
• Random matrix triality at nonzero chemical potential
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• Lindbladian dissipation of strongly-correlated quantum matter
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• Emergence of PT-symmetry breaking in open quantum systems
• The distribution of overlaps between eigenvectors of Ginibre matrices
• Random matrices beyond the Cartan classification
• Mean-field validity in a dissipative critical system: Liouvillian gap, $\mathbb{PT}$-symmetric antigap, and permutational symmetry in the $XYZ$ model
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Integrable nonunitary open quantum circuits
• Many-body quantum chaos and emergence of Ginibre ensemble
• Spectral transitions and universal steady states in random Kraus maps and circuits
• Symmetry-protected coherent relaxation of open quantum systems
• Spectral Properties of Disordered Interacting Non-Hermitian Systems
• Random generators of Markovian evolution: A quantum-classical transition by superdecoherence
• $\mathcal{P}\mathcal{T}$ phase transition in open quantum systems with Lindblad dynamics
• Gap Probabilities in Non-Hermitian Random Matrix Theory
• Topological phases protected by shifted sublattice symmetry in dissipative quantum systems
• Level statistics of real eigenvalues in non-Hermitian systems
• Relaxation to a Parity-Time Symmetric Generalized Gibbs Ensemble after a Quantum Quench in a Driven-Dissipative Kitaev Chain
• Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model
• Generic examples of PT-symmetric qubit (spin 1/2) Liouvillians
• New universality classes of the non-Hermitian Dirac operator in QCD-like theories
• A Wigner Surmise for Hermitian and Non-Hermitian Chiral Random Matrices
• Kramers' degeneracy for open systems in thermal equilibrium
• Non-Hermitian extensions of Wishart random matrix ensembles
• Approximation formula for complex spacing ratios in the Ginibre ensemble
• Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble

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• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
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• Integrability versus chaos in the steady state of many-body open quantum systems
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• Symmetry classes of classical stochastic processes
• Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems
• Correspondence principle, dissipation, and Ginibre ensemble
• Symmetry and Topology of Monitored Quantum Dynamics
• Multi-block exceptional points in open quantum systems
• Continuous time crystals as a PT symmetric state and the emergence of critical exceptional points
• Open-system dynamics in local Lindbladians with chaotic spectra
• Gauge freedoms in unravelled quantum dynamics: When do different continuous measurements yield identical quantum trajectories?
• Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI${}^{†}$ and AII${}^{†}$
• Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems
• Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting
• Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and $\mathcal{P}\mathcal{T}$ Symmetry Breaking