Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature
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Updated: Sep 25, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
A M García-García1, J J M Verbaarschot
1Department of Physics and Astronomy, SUNY, Stony Brook, New York, 11794, USA.
Abstract:
We investigate the spectral properties of a generalized 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 (CS) Hamiltonian. Taking advantage of the correspondence between CS particles and eigenvalues, and utilizing a recently conjectured expression by Kravtsov and Tsvelik for the finite temperature density-density correlations of the CS model, 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 or 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 Delta(3) statistic are accurately described by our analytical results.
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