Distribution of Level Spacing Ratios in Random Matrix Theory and Chaotic Quantum Systems: Variants and Applications

Abstract

Quantum systems with classically chaotic counterparts are studied in the realm of quantum chaos. A popular indicator of quantum chaos are the level spacing statistics, whose mathematical formulation is given by Random Matrix Theory (RMT). In this thesis, we study the distribution of ratios of spacings between eigenvalues of a random matrix or a Hamiltonian matrix corresponding to a quantum chaotic system. We also briefly consider other complex systems whose spectral fluctuations are described by random matrix theory. The main object of interest in this thesis, the spacing ratio, has recently been introduced, and has gained popularity in RMT as well as quantum chaos due to its ease of computation. We study variants of the spacing ratio, and show that its distribution takes different forms depending on the particular scenario considered. In order to study the effect of localized states on the spectral statistics of a quantum chaotic system, we propose a basic random matrix model for this interaction, and analytically derive a form for the distribution of spacing ratios for this model. We show that this model may be used to understand the strength of interaction between localized states and their generic neighbors, for various model systems. Next, we show numerically the form taken by the spacing ratio distribution over longer energy scales, which is an indicator of long-range correlations in the spectra of random matrices and complex systems that are modeled by them. Finally we how numerical evidence of scaling relationships in random matrices, for higher order ratio distributions, as well as for superpositions of random matrices. These results provide a straightforward but powerful application of the higher order ratios in determining the number of symmetries present in the Hamiltonian of a given quantum chaotic system.