Equivalence of Fokker-Planck approach and non-linear $\sigma$-model for disordered wires in the unitary symmetry class

TL;DR

This paper demonstrates the equivalence between the Fokker-Planck approach and the non-linear sigma model for disordered wires in the unitary class, providing exact correlation functions and conductance moments across regimes.

Contribution

It establishes the exact correspondence between two theoretical frameworks for disordered conductors, extending the understanding of their equivalence in the large channel limit.

Findings

Exact expressions for conductance moments valid across length scales

Agreement between Fokker-Planck and sigma-model results in large N limit

Method generalizes to arbitrary channel number and regimes

Abstract

The exact solution of the Dorokhov-Mello-Pereyra-Kumar-equation for quasi one-dimensional disordered conductors in the unitary symmetry class is employed to calculate all $m$-point correlation functions by a generalization of the method of orthogonal polynomials. We obtain closed expressions for the first two conductance moments which are valid for the whole range of length scales from the metallic regime ($L\ll Nl$) to the insulating regime ($L\gg Nl$) and for arbitrary channel number. In the limit $N\to\infty$ (with $L/(Nl)=const.$) our expressions agree exactly with those of the non-linear $\sigma$-model derived from microscopic Hamiltonians.