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

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.