Universalities in the spectra of disordered and chaotic systems

The Wigner-Dyson distribution of level spacings plays a central role in the classification of quantum chaotic systems. However, this description provides no information about the dispersion of the energy levels ${\mathit{E}}_{\mathit{i}}$(X) in response to an external perturbation. For weakly disordered metallic grains we demonstrate that the response depends on the detailed properties of the system only through mean-level spacing, and a conductance which we define. We derive a nonperturbative expression for the autocorrelator of density-of-states fluctuations for systems taken from both the unitary and orthogonal ensembles, showing that the dependence on these two parameters can be removed by a rescaling. We argue that this description applies quite generally to arbitrary peturbations, and that after rescaling the statistical properties of the random functions, ${\mathit{E}}_{\mathit{i}}$(X) become universal, dependent only on the Dyson ensemble. We demonstrate that this classification can be generalized to a wider class of systems providing a new characterization of quantum chaos. The analytical results are confirmed by numerical simulation of disordered metallic rings and a chaotic billiard where Aharonov-Bohm flux or a background potential provides the external perturbation.