Short-range plasma model for intermediate spectral statistics
Abstract
We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting the logarithmic pair interaction to a finite number k of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we show that the number variance has the asymptotic form \Sigma^2(L)\sim\chi L for large L and the nearest-neighbor distribution decreases exponentially when s\to \infty, P(s)\sim\exp (-\Lambda s) with \Lambda=1/\chi=k\beta+1, where \beta is the inverse temperature of the gas (\beta=1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of k=\beta=1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. P(s)=4s\exp(-2s). Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.