Between Poisson and GUE Statistics: Role of the Breit–Wigner Width
✍ Scribed by Klaus M. Frahm; Thomas Guhr; Axel Müller-Groeling
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 578 KB
- Volume
- 270
- Category
- Article
- ISSN
- 0003-4916
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✦ Synopsis
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 7 2 (r). The graded eigenvalue approach leads to an expression for X 2 (r) which is valid for all values of the parameter * 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 *> >1 the Breit Wigner width 1 1 measured in units of the mean level spacing D is much larger than unity. In this limit, closed analytical expressions for X 2 (r) and 7 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 1 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 -1 1 . This is rigorously shown and discussed in great detail. The Breit Wigner 1 1 width itself governs the approach to the Poisson limit for r Ä . Our analytical findings are confirmed by numerical simulations of an ensemble of 500_500 matrices, which demonstrate the universal validity of our results after proper unfolding.
1998 Academic Press
I. INTRODUCTION
One of the archetypical problems in quantum mechanics consists of calculating (certain properties of) the eigenvalue spectrum of a diagonal operator and a superimposed non diagonal one. Little can be said in general about this problem. In our paper, we focus on the particular case where the matrix representations H 0 and H 1