Distribution of the Ratio of Consecutive Level Spacings for Different Symmetries and Degrees of Chaos

TL;DR

This paper introduces a family of distributions for the ratio of consecutive level spacings in quantum systems, capturing different symmetries and degrees of chaos, and highlights the model-dependent nature of these distributions.

Contribution

It proposes a one-parameter family of distributions $P(r;eta)$ for level spacings, extending the understanding of eigenlevel statistics across various quantum symmetries and chaos degrees.

Findings

Distribution family $P(r;eta)$ effectively models level spacing ratios.

Model-specific properties prevent a universal distribution formulation.

Information entropy guides the choice of ansatzs with minimal error.

Abstract

Theoretical expressions for the distribution of the ratio of consecutive level spacings for quantum systems with transiting dynamics remain unknown. We propose a family of one-parameter distributions $P(r)\equiv P(r;\beta)$, where $\beta\in[0,+\infty)$ is a generalized Dyson index, that describes the eigenlevel statistics of a quantum system characterized by different symmetries and degrees of chaos. We show that this crossover strongly depends on the specific properties of each model, and thus the reduction of such a family to a universal formula, albeit desirable, is not possible. We use the information entropy as a criterion to suggest particular ansatzs for different transitions, with a negligible associated error in the limits corresponding to standard random ensembles.