Distribution of the transfer matrix in disordered wires

TL;DR

This paper derives a comprehensive probability distribution for the transfer matrix in disordered one-dimensional wires, generalizing known results and applying the findings to persistent current calculations.

Contribution

It provides a closed-form distribution of the transfer matrix for weak, delta-correlated disorder, extending existing models and solving a diffusion equation on the SU(1,1) group.

Findings

Distribution generalizes known results for transmission and density of states

Solution of diffusion equation on SU(1,1) group

Application to persistent current in flux-threaded rings

Abstract

A closed expression is derived for the probability distribution of the transfer matrix of a particle moving in a one-dimensional system with delta-correlated, weak disorder. The change in the distribution as a function of wire length is described by a diffusion equation on the $SU(1,1)$ group, which is solved through the decomposition of the regular representation into irreducible components. The expression generalizes a number of well-known results, including the distributions of the transmission coefficient and local density of states. As an application, the average single energy-level contribution to the persistent current in a flux-threaded ring is derived.