Parametric statistics of the scattering matrix: From metallic to insulating quasi-unidimensional disordered systems

TL;DR

This paper studies the statistical behavior of the scattering matrix in quasi-one dimensional disordered systems, revealing universal correlations in metallic regimes and their breakdown in localized regimes, with models explaining these phenomena.

Contribution

It introduces a Brownian-motion model for the scattering matrix and analyzes parametric correlations across metallic and insulating regimes, connecting them to delay times and conductance.

Findings

Universal parametric correlations in metallic regime

Breakdown of universality in localized regime

Correlation between Wigner time and conductance

Abstract

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 delay time 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.