Random Matrix approach to the crossover from Wigner to Poisson statistics of energy levels

Abstract

We analyze a class of parametrized Random Matrix models, introduced by Rosenzweig and Porter, which is expected to describe the energy level statistics of quantum systems whose classical dynamics varies from regular to chaotic as a function of a parameter. We compute the generating function for the correlations of energy levels, in the limit of infinite matrix size. Our computations show that for a certain range of values of the parameter, the energy-level statistics is given by that of the Wigner-Dyson ensemble. For another range of parameter values, one obtains the Poisson statistics of uncorrelated energy levels. However, between these two ranges, new statistics emerge, which is neither Poissonnian nor Wigner. The crossover is measured by a renormalized coupling constant. The model is exactly solved in the sense that, in the limit of infinite matrix size, the energy-level correlation functions and their generating function are given in terms of a finite set of integrals.