Abstract. Recently much effort has been made towards the introduction of nonHermitian random matrix models respecting P T -symmetry. Here we show that there is a one-to-one correspondence between complex P T -symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. We conjecture that these ensembles represent universality classes for P Tsymmetric matrices. For the case of 2 × 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.In recent years there has been a surge of research interest in P T -symmetric quantum theories, accompanied by a multitude of experimental applications and realisations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For finite-dimensional systems represented by matrices, P Tsymmetry is equivalent to the reality of the characteristic polynomial [18]. That is, P T -symmetric matrices have either real or complex conjugate eigenvalues. Their eigenvectors are orthogonal with respect to a suitably defined CP T inner product [19]. It has recently been conjectured that P T -symmetry is closely related to split-quaternionic extensions of quantum theory [20,21]. Here we show that splitquaternionic extensions of Hermitian matrices are indeed a natural representation of P T -symmetric matrices. This equivalence allows us to introduce new P T -symmetric random matrix ensembles.In conventional quantum systems, random matrices play an important role due to their ability to describe spectral fluctuations in sufficiently complicated systems [22]. In particular, the famous Bohigas-Giannoni-Schmit conjecture states that the spectral fluctuations of quantum systems with chaotic classical counterparts are similar to those of certain random matrices [23,24]. There are three important universality classes for Hermitian quantum systems, depending on the time-reversal properties of the system, corresponding to the Gaussian orthogonal, unitary, and symplectic ensembles [25]. Non-Hermitian random matrix models, on the other hand, whose eigenvalues are in general complex, are widely studied, and have applications ranging from dissipative quantum systems and scattering theory to quantum chromodynamics (see, e.g., [26] and references therein). Several attempts towards defining P Tsymmetric random matrices and identifying universality classes for P T -symmetric systems have been made [27][28][29][30][31][32]. Most of them are restricted to 2 × 2 matrices, due to the lack of a natural parameterisation of larger P T -symmetric matrices. Here we introduce the split-complex and split-quaternionic versions of the Gaussian unitary arXiv:1505.07810v2 [math-ph]

A non-Hermitian P T -symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of P T -symmetry can lead to "stable" mixed regular chaotic behaviour without sinks or sources. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if P T -symmetry is broken. The amplitude dynamics of the classical map is investigated, and found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies, as well as in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow GOE statistics when the classical dynamics is regular, and show a cubic level repulsion when the classical dynamics is chaotic as previously identified as universal behaviour for other types of open quantum systems. The P T -symmetry of the system does not seem to influence this statistical feature. Similar features are observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

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