Universal hard-edge statistics of non-Hermitian random matrices

Random matrix theory is a powerful tool for understanding spectral correlations inherent in quantum chaotic systems. Despite diverse applications of non-Hermitian random matrix theory, the role of symmetry remains to be fully established. Here, we comprehensively investigate the impact of symmetry on the level statistics around the spectral origin – hard-edge statistics – and complete the classification of spectral statistics in all the 38 symmetry classes of non-Hermitian random matrices. Within this classification, we discern 28 symmetry classes characterized by distinct hard-edge statistics from the level statistics in the bulk of spectra, which are further categorized into two groups, namely the Altland-Zirnbauer_0 classification and beyond. We introduce and elucidate quantitative measures capturing the universal hard-edge statistics for all the symmetry classes. Furthermore, through extensive numerical calculations, we study various open quantum systems in different symmetry classes, including quadratic and many-body Lindbladians, as well as non-Hermitian Hamiltonians. We show that these systems manifest the same hard-edge statistics as random matrices and that their ensemble-average spectral distributions around the origin exhibit emergent symmetry conforming to the random-matrix behavior. Our results establish a comprehensive understanding of non-Hermitian random matrix theory and are useful in detecting quantum chaos or its absence in open quantum systems.