Eigenvalue statistics for generalized symmetric and Hermitian matrices

Abstract

Random matrix theory predicts the level statistics of a Hamiltonian to exhibit either clustering or repulsion if the underlying dynamics is integrable or chaotic, respectively. In various physical systems it is also possible to observe intermediate spectral properties showing the transition between different symmetry classes. In this work, we study generalized random matrix ensembles by dropping the constraint of canonical invariance and considering different variances in the diagonal and off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit intermediate level statistics. The nearest neighbour spacing (NNS) distributions can be computed for generalized symmetric matrices exhibiting crossover from clustering to GOE-like repulsion. The analysis is extended to matrices where the distributions of NNS as well as ratio of nearest neighbour spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices () where crossovers also take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes to quantify the system size dependence of the crossover boundary.