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.

24 pages, 4 figures

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