Higher order spacing ratios in random matrix theory and complex quantum systems

TL;DR

This paper explores higher order spacing ratios in random matrix theory and complex quantum systems, revealing their distributional properties and simplifying spectral fluctuation analysis without unfolding procedures.

Contribution

It demonstrates that higher order spacing ratio distributions in Wigner-Dyson ensembles are equivalent to standard distributions with a modified parameter, applicable across various physical systems.

Findings

Higher order ratios follow known distributions with adjusted parameters.

Results apply to Gaussian, circular ensembles, and physical systems.

Simplifies spectral analysis by avoiding unfolding procedures.

Abstract

The distribution of the ratios of nearest neighbor level spacings has become a popular indicator of spectral fluctuations in complex quantum systems like interacting many-body localized and thermalization phases, quantum chaotic systems, and also in atomic and nuclear physics. In contrast to the level spacing distribution, which requires the cumbersome and at times ambiguous unfolding procedure, the ratios of spacings do not require unfolding and are easier to compute. In this work, for the class of Wigner-Dyson random matrices with nearest neighbor spacing ratios $r$ distributed as $P_{\beta}(r)$ for the three ensembles indexed by $\beta=1,2, 4$, their $k-$th order spacing ratio distributions are shown to be identical to $P_{\beta'}(r)$, where $\beta'$, an integer, is a function of $\beta$ and $k$. This result is shown for Gaussian and circular ensembles of random matrix theory and for several physical systems such as spin chains, chaotic billiards, Floquet systems and measured nuclear resonances.