Universal Spectral Correlations in Diffusive Quantum Systems

TL;DR

This paper investigates spectral correlations in disordered quantum systems under flux, revealing universal behaviors and deviations from perturbative predictions during symmetry transitions.

Contribution

It provides numerical analysis of spectral properties, curvature distributions, and correlations during the GOE-GUE transition in disordered quantum systems.

Findings

Curvature distribution follows a modified Lorentz form with different exponents in GOE and GUE regimes.

Single level current correlations exhibit logarithmic behavior at low flux.

Universal relation between slope of levels and curvature distribution width is confirmed.

Abstract

We have studied numerically several statistical properties of the spectra of disordered electronic systems under the influence of an Aharonov Bohm flux $\varphi$, which acts as a time--reversal symmetry breaking parameter. The distribution of curvatures of the single electron energy levels has a modified Lorentz form with different exponents in the GOE and GUE regime. It has Gaussian tails in the crossover regime. The typical curvature is found to vary as $ -E_c\ln (E_c\varphi^2/\Delta)$ ($E_c$ is the Thouless energy and $\Delta$ the mean level spacing) and to diverge at zero flux. We show that the harmonics of the variation with $\varphi$ of single level quantities (current or curvature) are correlated, in contradiction with the perturbative result. The single level current correlation function is found to have a logarithmic behavior at low flux, in contrast to the pure symmetry cases. The distribution of single level currents is non--Gaussian in the GOE--GUE transition regime. We find a universal relation between $g_d$, the typical slope of the levels, and $g_c$, the width of the curvature distribution, as was proposed by Akkermans and Montambaux. We conjecture the validity of our results for any chaotic quantum system.