Between Poisson and GUE statistics: Role of the Breit-Wigner width

Abstract

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation function $X_2(r)$ and the number variance $\Sigma^2(r)$. The graded eigenvalue approach leads to an expression for $X_2(r)$ which is valid for all values of the parameter $\lambda$ governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. For $\lambda \gg 1$ the Breit-Wigner width $\Gamma_1$ measured in units of the mean level spacing $D$ is much larger than unity. In this limit, closed analytical expression for $X_2(r)$ and $\Sigma^2(r)$ can be derived by (i) evaluating the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations of $\Gamma_1$ manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by $\sqrt{\Gamma_1}$. This is rigorously shown and discussed in great detail. The Breit-Wigner $\Gamma_1$ width itself governs the approach to the Poisson limit for $r\to\infty$. Our analytical findings are confirmed by numerical simulations of an ensemble of $500\times 500$ matrices, which demonstrate the universal validity of our results after proper unfolding.