Equivalence of the Fokker-Planck Approach and the Nonlinear<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>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 ( $L\ensuremath{\ll}\mathrm{Nl}$) to the insulating regime ( $L\ensuremath{\gg}\mathrm{Nl}$) and for arbitrary channel number. In the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ [with $L/(\mathrm{Nl})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{const}$] our expressions agree exactly with those of the nonlinear $\ensuremath{\sigma}$ model derived from microscopic Hamiltonians.