Affiliation:
1. Graduate School of Natural Science and Technology, Shimane University , Matsue 690-8504 , Japan
Abstract
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 Jánossy 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 $N\rightarrow \infty$, 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.
Funder
Japan Society for the Promotion of Science
Publisher
Oxford University Press (OUP)