Parametric Statistics of the Scattering Matrix: From Metallic to Insulating Quasi-Unidimensional Disordered Systems

We investigate the statistical properties of the scattering matrix $S$ describing the electron transport through quasi-one-dimensional disordered systems. For weak disorder (metallic regime), the energy dependence of the phase shifts of $S$ is found to yield the same universal parametric correlations as those characterizing chaotic Hamiltonian eigenvalues driven by an external parameter. This is analyzed within a Brownian motion model for $S$, which is directly related to the distribution of the Wigner-Smith time delay matrix. For large disorder (localized regime), transport is dominated by resonant tunneling and the universal behavior disappears. A model based on a simplified description of the localized wave functions qualitatively explains our numerical results. In the insulator, the parametric correlation of the phase shift velocities follows the energy-dependent autocorrelator of the Wigner time. The Wigner time and the conductance are correlated in the metal and in the insulator