2 × 2 random matrix ensembles with reduced symmetry: from Hermitian to -symmetric matrices

Abstract

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity–time ()-symmetric matrices. To illustrate the main idea, we first study 2 × 2 complex Hermitian matrix ensembles with O(2)-invariant constraints, yielding novel level-spacing statistics such as singular distributions, the half-Gaussian distribution, distributions interpolating between the GOE (Gaussian orthogonal ensemble) distribution and half-Gaussian distributions, as well as the gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2 × 2 -symmetric matrix ensembles with real eigenvalues. In particular, -symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of -symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian unitary ensemble) statistics or the ‘truncated-GUE’ statistics.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.