Interpolation between Poisson and Circular Unitary Ensemblesfor the Number Variance in Level Statistics

Abstract

Using Gaudin's model (Nucl. Phys. 85 (1966) 545), we study the asymptotic behavior of the number variance of quantum levels whose statistics is subject to a one-parameter family of the predicted distributions which lie in the Poisson and the circular unitary ensembles. It is shown that the variance function ∑ ²(S) for S>> 1 is a sum of a linear term and a logarithmic term, originating from the Poisson and the GUE (Gaussian unitary ensemble) results, respectively. The linearity of the leading term is attributed to Gaudin's treatment of the system as a compressible gas, thus verifying the prediction in a recent paper by Moshe, Neuberger and Shapiro (Phys. Rev. Lett. 73 (1994) 1497). A possibility of alternative treatment for incompressible gases is discussed.