Abstract
Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the and spacings from a given level are introduced. Analytical predictions are derived using a matrix model. The density is generalized to the spacing density, which allows for investigating long-range correlations. For larger the probability density of spacings is well described by a Gaussian. Using these spacings we propose the ratio of the to the as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.
Highlights
Random matrix theory (RMT) describes quite successfully the statistical properties of complex systems spectra of various origins, such as nuclear and condensed matter systems, and has applications in mathematics as well [1, 2]
One motivation for this comes from applications in the context of perturbation theory [9, 10], where it is of importance to know the statistics of the distance of a given level to the closer neighbour (CN) of its two nearest neigbors
The closest neighbour and farther neighbour spacing probability densities are introduced and predictions obtained for the random matrix ensembles GOE, GUE, and Gaussian symplectic (GSE) based on a 3 × 3 matrix modeling, and Poisson spectra
Summary
Random matrix theory (RMT) describes quite successfully the statistical properties of complex systems spectra of various origins, such as nuclear and condensed matter systems, and has applications in mathematics as well [1, 2]. The simplest are the densities of the spacings of the closest neighbour and that of the farther neighbour One motivation for this comes from applications in the context of perturbation theory [9, 10], where it is of importance to know the statistics of the distance of a given level to the closer neighbour (CN) of its two nearest neigbors. For the closest neighbour and farther neighbour spacing densities random matrix predictions are derived for the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) ensembles. Unfolding works best with an analytical expression for the smoothed density of eigenvalues, such as the Weyl formula for billiards [13], which is not always available To overcome this difficulty, the ratio of the closer neighbour spacing to the farther neighbour spacing (r = CN/FN) was. As the second closest neighbour could be the farther neighbour or the next-neighbour level in the direction of the closest, this ratio will be in general larger than r
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