The Threefold Way in Non-Hermitian Random Matrices*

Non-Hermitian random matrices have been utilized in diverse scientific fields such as dissipative and stochastic processes, nuclear physics, and neural networks. However, the only known universal level-spacing statistics are those of the Ginibre ensemble characterized by complex-conjugation symmetry. In this talk, we report our discovery of two distinct universality classes characterized by transposition symmetry [1]. We find that transposition symmetry alters repulsive interactions between two neighboring eigenvalues and deforms their spacing distribution, which is not possible with other symmetries including Ginibre's complex-conjugation symmetry which can affect only nonlocal correlations. Our results complete the non-Hermitian counterpart of Dyson's threefold classification of Hermitian random matrices and serve as a basis for characterizing nonintegrability and delocalization in open quantum systems with symmetry [1,2].
[1] R. Hamazaki, K. Kawabata, N. Kura, and M. Ueda, arXiv preprint arXiv:1904.13082 (2019).
[2] R. Hamazaki, K. Kawabata, and M. Ueda, Phys. Rev. Lett. 123, 090603 (2019).