Abstract
Non-Hermitian random matrices have been utilized in such diverse fields as dissipative and stochastic processes, mesoscopic physics, nuclear physics, and neural networks. However, the only known universal level-spacing statistics is that of the Ginibre ensemble characterized by complex-conjugation symmetry. Here we report our discovery of two other distinct universality classes characterized by transposition symmetry. We find that transposition symmetry alters repulsive interactions between two neighboring eigenvalues and deforms their spacing distribution. Such alteration 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 Wigner-Dyson's threefold universal statistics of Hermitian random matrices and serve as a basis for characterizing nonintegrability and chaos in open quantum systems with symmetry.
Highlights
Symmetry and universality are two important concepts of Hermitian random matrix theory
We show for the first time that the newly found universality classes manifest themselves in dissipative quantum many-body nonintegrable systems described by the Lindblad dynamics and non-Hermitian Hamiltonians
II, we briefly review 9 non-Hermitian symmetry classes characterized by a single symmetry among the 38 symmetry classes [35]
Summary
Symmetry and universality are two important concepts of Hermitian random matrix theory. While Dyson’s classification is about Hermitian matrices, non-Hermiticity plays a key role in such diverse systems as dissipative systems [11,12,13], mesoscopic systems [14], and neural networks [15] Many of these systems have been investigated in terms of non-Hermitian random-matrix ensembles introduced by Ginibre (Fig. 1), which are referred to as GinUE, GinOE, and GinSE as non-Hermitian extensions of GUE, GOE, and GSE [16]. We here investigate the nearest-neighbor spacing distributions of random matrices which belong to the simplest classes (i.e., classes with a single symmetry) in the non-Hermitian symmetry classification [33,34,35]. VI, we summarize the paper and discuss some future problems
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