Symmetry classification correspondence between quadratic Lindbladians and their steady states

Abstract

Symmetry classification is crucial in understanding universal properties of quantum matter. Recently, the scope of symmetry classification has been extended to open quantum systems governed by the Lindblad master equation. However, the classification of Lindbladians and steady states remains largely separate. Because the former requires the non-Hermitian classification framework, while the latter relies on the classification scheme for Hermitian matrices. In this paper we build connections between symmetry classes of quadratic Lindbladian and its steady state, despite their different classification frameworks. We classify the full matrix representation of generic quadratic Lindbladians with particle conservation, showing they fall into 27 non-Hermitian symmetry classes. Among these, 22 classes lead to an infinite-temperature steady state. The remaining five classes have one-to-one correspondence with five steady-state Hermitian symmetry classes. Numerical simulations of random Lindbladian dynamics confirm the convergence to the correct steady-state symmetry classes at long time.

8 pages, 2 figures

References in corpus (43)

• Topological Insulators
• Topological insulators and superconductors
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Random Matrix Theories in Quantum Physics: Common Concepts
• Periodic table for topological insulators and superconductors
• Non-Hermitian Physics
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Open Quantum Systems. An Introduction
• Non-Hermitian skin effect and chiral damping in open quantum systems
• Topology by Dissipation in Atomic Quantum Wires
• Third quantization: a general method to solve master equations for quadratic open Fermi systems
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
• Topology by dissipation
• Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
• Spectral theorem for the Lindblad equation for quadratic open fermionic systems
• Universality classes of non-Hermitian random matrices
• Dissipative preparation of Chern insulators
• Density Matrix Topological Insulators
• Probing the topology of density matrices
• Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
• Topological classification of defects in non-Hermitian systems
• Symmetry classes of open fermionic quantum matter
• Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems
• Information Scrambling and Chaos in Open Quantum Systems
• Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics
• Lindbladian dissipation of strongly-correlated quantum matter
• Spectral and Steady-State Properties of Random Liouvillians
• Lindbladian dynamics of the Sachdev-Ye-Kitaev model
• Dynamical quantum phase transitions in SYK Lindbladians
• Topological phases protected by shifted sublattice symmetry in dissipative quantum systems
• Singular-Value Statistics of Non-Hermitian Random Matrices and Open Quantum Systems
• Topological phase transitions at finite temperature
• Symmetry-Preserving Quadratic Lindbladian and Dissipation Driven Topological Transitions in Gaussian States
• Spectral and steady-state properties of fermionic random quadratic Liouvillians
• Interaction-induced topological phase transition at finite temperature
• ${\mathbb{Z}}_2$ topological invariants for mixed states of fermions in time-reversal invariant band structures
• Probing the Topology of Fermionic Gaussian Mixed States with {U(1)} symmetry by Full Counting Statistics

Cited by in corpus (1)

• Dissipation-driven topological phase transitions in open quantum systems independent of system Hamiltonian