We have studied numerically several statistical properties of the spectra of disordered electronic systems under the influence of an Aharonov-Bohm flux cphi, 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 Gaussian orthogonal ensemble (GOE) and the Gaussian unitary ensemble (GUE) regime. It has Gaussian tails in the crossover regime. The typical curvature is found to vary as ${\mathit{E}}_{\mathit{c}}$/Δ${\ln }^{1/2}$[Δ/(${\mathit{E}}_{\mathit{c}}$${\mathrm{\phi }}^2$)] (${\mathit{E}}_{\mathit{c}}$ is the Thouless energy and Δ the mean level spacing) and to diverge at zero flux. We show that the harmonics of the variation with cphi 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. The distribution of single-level currents is non-Gaussian in the GOE-GUE transition regime. We find a universal relation between ${\mathit{g}}_{\mathit{d}}$, the typical slope of the levels, and ${\mathit{g}}_{\mathit{c}}$ 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.

• Received 26 April 1994

DOI:https://doi.org/10.1103/PhysRevB.50.7776