Singular-Value Statistics of Non-Hermitian Random Matrices and Open Quantum Systems

Abstract

The spectral statistics of non-Hermitian random matrices are of importance as a diagnostic tool for chaotic behavior in open quantum systems. Here, we investigate the statistical properties of singular values in non-Hermitian random matrices as an effective measure of quantifying dissipative quantum chaos. By means of Hermitization, we reveal the unique characteristics of the singular-value statistics that distinguish them from the complex-eigenvalue statistics, and establish the comprehensive classification of the singular-value statistics for all the 38-fold symmetry classes of non-Hermitian random matrices. We also analytically derive the singular-value statistics of small random matrices, which well describe those of large random matrices in the similar spirit to the Wigner surmise. Furthermore, we demonstrate that singular values of open quantum many-body systems follow the random-matrix statistics, thereby identifying chaos and nonintegrability in open quantum systems. Our work elucidates that the singular-value statistics serve as a clear indicator of symmetry and lay a foundation for statistical physics of open quantum systems.

23 pages, 6 figures, 6 tables

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• Non-Hermitian Topology in Hermitian Topological Matter
• Krylov space approach to Singular Value Decomposition in non-Hermitian systems
• Field theory of non-Hermitian disordered systems
• Dissipative Quantum Chaos unveiled by Stochastic Quantum Trajectories
• Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field
• Probing quantum chaos through singular-value correlations in sparse non-Hermitian SYK model
• Multifractality of many-body non-Hermitian skin effect
• Singular Value Decomposition and Its Blind Spot for Quantum Chaos in Non-Hermitian Sachdev-Ye-Kitaev Models
• Decoherence rate in random Lindblad dynamics
• Diagnosing Quantum Many-body Chaos in Non-Hermitian Quantum Spin Chain via Krylov Complexity
• Universal hard-edge statistics of non-Hermitian random matrices
• Higher-order gap ratios of singular values in open quantum systems
• Integrability versus chaos in the steady state of many-body open quantum systems
• Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials
• Symmetry classification correspondence between quadratic Lindbladians and their steady states
• Limitations of SVD-Based Diagnostics for Non-Hermitian Many-Body Localization with Time-Reversal Symmetry
• Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI${}^{†}$ and AII${}^{†}$