Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2002-04-05
Phys.Rev. E67 (2003) 046104
Physics
Condensed Matter
Disordered Systems and Neural Networks
27 pages, 4 figures, modified conclusions in section four
Scientific paper
10.1103/PhysRevE.67.046104
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 Poincar\'e 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 is accurately described by our analytical results.
Garcia-Garcia Antonio M.
Verbaarschot Jacobus J. M.
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