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  • Universality classes of non-Hermitian random matrices

    Ryusuke Hamazaki1, Kohei Kawabata1, Naoto Kura1, and Masahito Ueda1,2 1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    2RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan (Dated: February 25, 2020)

    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 charac- terized 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.

    I. INTRODUCTION

    Symmetry and universality are two important con- cepts of Hermitian random matrix theory. Dyson’s three- fold symmetry classes in terms of time-reversal symme- try (TRS) [1] led to three distinct universality classes (Fig. 1), where certain spectral statistics become inde- pendent of the detailed structures of matrices [2]. The three distinct universal statistics have found applications in diverse research fields including nuclear physics [3] and mesoscopic physics [4, 5], quantum chaos [6, 7], and infor- mation theory [8]. The most direct manifestation of the universality lies in the level-spacing distribution, which measures local correlations of eigenvalues. Level-spacing distributions of nonintegrable systems are described by those of Gaussian random matrices known as Wigner- Dyson’s universal statistics [7, 9, 10]. They belong to the Gaussian unitary ensemble (GUE) for the class with- out TRS (class A), the Gaussian orthogonal ensemble (GOE) for the class with TRS whose square is +1 (class AI), and the Gaussian symplectic ensemble (GSE) for the class with TRS whose square is −1 (class AII).

    TABLE I. Symmetry classes, constraints, and nearest- neighbor spacing distributions. The sign of each symmetry class indicates whether the square of the symmetry operator is +1 or −1. Only classes AI† and AII† show the distributions different from that of Ginibre’s unitary ensemble (GinUE).

    Class Symmetry Constraint pGinUE(s)? A None - Yes [11]

    AI (D†) TRS, + (PHS†, +) H = H∗ Yes [12] AII (C†) TRS, − (PHS†, −) H = ΣyH∗Σy Yes [11]

    AI† TRS†, + H = HT No AII† TRS†, − H = ΣyHTΣy No

    D PHS, + H = −HT Yes C PHS, − H = −ΣyHTΣy Yes

    AIII CS (pH) H = −ΣzH†Σz Yes AIII† SLS (CS†) H = −ΣzHΣz Yes

    While Dyson’s classification is about Hermitian ma- trices, non-Hermiticity plays a key role in such diverse systems as dissipative systems [12–14], mesoscopic sys- tems [15], and neural networks [16]. Many of these sys- tems 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 [11]. These three Gaussian ensembles are defined in terms of complex conjugation (TRS) and have thoroughly been investigated [17–22].

    Interestingly, three different symmetry classes (GinUE, GinOE, and GinSE for the Gaussian case) share the same universal level-spacing statistics of GinUE unlike the Hermitian case. In fact, TRS only creates nonlocal correlations between complex-conjugate pairs of eigenval- ues, but does not alter the repulsive interactions between neighboring eigenvalues away from the real axis. Thus, the Ginibre distribution is the only hitherto known uni- versal nearest-neighbor spacing distribution common to all three different symmetry classes with TRS, in contrast to the Hermitian case where TRS leads to three distinct universality classes. Here, symmetry classification is de- fined solely by algebraic structures of matrix ensembles, whereas the universality classification is defined by spec- tral statistics that can be the same for different matrix ensembles [2].

    Motivated by experiments of non-Hermitian sys- tems [23–32], symmetry classification of non-Hermitian matrix ensembles has witnessed remarkable develop- ments [33–35]. Quite recently, two of us have shown that there are 38 non-Hermitian symmetry classes [35] as an extension of the Hermitian symmetry classes by Altland and Zirnbauer [36]. The classification includes symme- try classes called AI† and AII† that arise from transposi- tion symmetry [33–35, 37–43], which is Hermitian conju- gate of TRS, i.e, TRS†. A rich variety of experimentally realizable systems belongs to these classes such as op- tical systems with gain and/or loss [44–46]. However, it has remained largely unexplored how these symme- try classes are related to the universality classification

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    Dyson’s symmetry classes → three universal statistics

    Ginibre’s symmetry classes

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