Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

TL;DR

This paper analyzes the statistical properties of electron scattering matrices in disordered systems, comparing different Dyson ensembles, and explores phase shift distributions in various regimes including quasi-one-dimensional conductors and insulators.

Contribution

It provides explicit probability distributions for scattering matrices in Dyson ensembles and compares them with those of quasi-1D systems, including phase shift statistics and crossover behaviors.

Findings

Phase shift distributions match Dyson ensembles in quasi-1D metals.

Crossover from coupled to uncoupled ensemble behavior in localized regimes.

Deviations from isotropic Dyson distributions in higher dimensions.

Abstract

We consider elastic reflection and transmission of electrons by a disordered system characterized by a $2N\!\times\!2N$ scattering matrix $S$. Expressing $S$ in terms of the $N$ radial parameters and of the four $N\!\times\!N$ unitary matrices used for the standard transfer matrix parametrization, we calculate their probability distributions for the circular orthogonal (COE) and unitary (CUE) Dyson ensembles. In this parametrization, we explicitely compare the COE--CUE distributions with those suitable for quasi--$1d$ conductors and insulators. Then, returning to the usual eigenvalue--eigenvector parametrization of $S$, we study the distributions of the scattering phase shifts. For a quasi--$1d$ metallic system, microscopic simulations show that the phase sift density and correlation functions are close to those of the circular ensembles. When quasi--$1d$ longitudinal localization breaks $S$ into two uncorrelated reflection matrices, the phase shift form factor $b(k)$ exhibits a crossover from a behavior characteristic of two uncoupled COE--CUE (small $k$) to a single COE--CUE behavior (large $k$). Outside quasi--one dimension, we find that the phase shift density is no longer uniform and $S$ remains nonzero after disorder averaging. We use perturbation theory to calculate the deviations to the isotropic Dyson distributions. When the electron dynamics is no