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Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

A. M. Garcia-Garcia·J. J. M. Verbaarschot
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Abstract

We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the $Δ_3$-statistic is accurately described by our analytical results.

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A. M. Garcia-Garcia, J. J. M. Verbaarschot. 2002-10-07. Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature. https://doi.org/10.1103/physreve.67.046104

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