Level statistics of real eigenvalues in non-Hermitian systems

Abstract

Symmetries associated with complex conjugation and Hermitian conjugation, such as time-reversal symmetry and pseudo-Hermiticity, have great impact on eigenvalue spectra of non-Hermitian random matrices. Here, we show that time-reversal symmetry and pseudo-Hermiticity lead to universal level statistics of non-Hermitian random matrices on and around the real axis. From the extensive numerical calculations of large random matrices, we obtain the five universal level-spacing and level-spacing-ratio distributions of real eigenvalues, each of which is unique to the symmetry class. Furthermore, we analyse spacings of real eigenvalues in physical models, such as bosonic many-body systems and free fermionic systems with disorder and dissipation. We clarify that the level spacings in ergodic (metallic) phases are described by the universal distributions of non-Hermitian random matrices in the same symmetry classes, while the level spacings in many-body localized and Anderson localized phases show the Poisson statistics. We also find that the number of real eigenvalues shows distinct scalings in the ergodic and localized phases in these symmetry classes. These results serve as effective tools for detecting quantum chaos, many-body localization, and real-complex transitions in non-Hermitian systems with symmetries.

32 pages

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• Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos
• Non-Hermitian Topological Phases: Principles and Prospects
• Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond
• Singular-Value Statistics of Non-Hermitian Random Matrices and Open Quantum Systems
• Degeneracies and symmetry breaking in pseudo-Hermitian matrices
• Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis
• Field theory of non-Hermitian disordered systems
• Infernal and Exceptional Edge Modes: Non-Hermitian Topology Beyond the Skin Effect
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Eigenvector Correlations Across the Localisation Transition in non-Hermitian Power-Law Banded Random Matrices
• Universal hard-edge statistics of non-Hermitian random matrices
• Higher-order gap ratios of singular values in open quantum systems
• Boundary-driven many-body phase transitions in a non-Hermitian disordered fermionic chain
• Transfer learning from Hermitian to non-Hermitian quantum many-body physics
• Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI${}^{†}$ and AII${}^{†}$