Quantum chaos in transition in sparsified random matrices

Abstract

Random matrix theory has become an indispensable tool in the study of chaos, thermalization and localization in quantum systems. Its apparent universality lies in its power to describe a class of systems by an random matrix ensemble belonging to the appropriate symmetry class. In connection with studies on holographic duality, it was proposed that sparsity of a matrix determines whether it belongs into a sparse or a dense universality class, which may introduce subtleties in comparing the physics of chaotic systems described by sparse or dense matrices. In this paper, we study the transition between the sparse and dense universality classes by directly sparsifying GUE matrices while preserving hermiticity. We find the defining signatures of sparse and dense ensembles in the near-edge behavior of the spectral density and find a large regime where the sparsified GUE matrices retain the spectral correlations of their dense counterparts and a small regime where the extremely sparse matrices remain chaotic but not maximally so that precedes the breakdown of spectral correlations.