Ordered level spacing probability densities

TL;DR

This paper introduces and analyzes the probability densities of ordered level spacings in quantum spectra, providing analytical models and numerical validation across different systems, revealing insights into spectral correlations and ratios.

Contribution

It develops analytical predictions for ordered neighbor spacings using a 3x3 matrix model and introduces a new ratio measure for spectral analysis, extending understanding of spectral correlations.

Findings

Gaussian approximation for large k-th spacings

Ratio of closest to second closest neighbor distinguishes spectral types

Good agreement between predictions and numerical data

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 closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3 \times 3$ matrix model. The closest neighbour density is generalized to the $k-$th closest neighbour spacing density, which allows for investigating long-range correlations. For larger $k$ the probability density of $k-$th closest neighbour spacings is well described by a Gaussian. Using these $k-$th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour 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.