Distributions of consecutive level spacings of Gaussian unitary ensemble and their ratio: ab initio derivation

TL;DR

This paper derives exact analytical formulas for the distribution of ratios of consecutive energy level spacings in the Gaussian Unitary Ensemble, providing tools to quantify quantum chaos without spectral unfolding.

Contribution

It presents the first ab initio derivation of the joint distribution of consecutive level spacings and their ratios for GUE matrices at infinite size, using Tracy-Widom theory.

Findings

Analytic expressions for joint eigenvalue spacing distributions.

Distribution of spacing ratios derived from differential equations.

Comparison with zeros of the Riemann zeta function as a quantum-chaotic spectrum.

Abstract

In recent studies of many-body localization in nonintegrable quantum systems, the distribution of the ratio of two consecutive energy level spacings, $r_n=(E_{n+1}-E_n)/(E_{n}-E_{n-1})$ or $\tilde{r}_n=\min(r_n,r_n^{-1})$, has been used as a measure to quantify the chaoticity, alternative to the more conventional distribution of the level spacings, $s_n=\bar{\rho}(E_n)(E_{n+1}-E_n)$, as the former makes unnecessary the unfolding required for the latter. Based on our previous work on the Tracy-Widom approach to the Janossy densities, we present analytic expressions for the joint probability distribution of two consecutive eigenvalue spacings and the distribution of their ratio for the Gaussian unitary ensemble (GUE) of random Hermitian $N\times N$ matrices at $N\to \infty$, in terms of a system of differential equations. As a showcase of the efficacy of our results for characterizing an approach to quantum chaoticity, we contrast them to arguably the most ideal of all quantum-chaotic spectra: the zeroes of the Riemann $\zeta$ function on the critical line at increasing heights.