Quantum Transport in Disordered Wires: Equivalence of One-Dimensional Sigma Model and Dorokhov-Mello-Pereyra-Kumar Equation

TL;DR

This paper demonstrates the equivalence of two major non-perturbative theories of localization in disordered wires, resolving previous discrepancies and confirming results with numerical simulations across different symmetry classes.

Contribution

It proves the equivalence of the sigma model and DMPK equation approaches for all symmetry classes, correcting previous calculation errors and aligning theory with numerical data.

Findings

Confirmed the equivalence of the two theories across symmetry classes.

Corrected the implementation of Kramers degeneracy in calculations.

Achieved good agreement with numerical simulations for conductance moments.

Abstract

The two known non-perturbative theories of localization in disordered wires, the Fokker-Planck approach due to Dorokhov, Mello, Pereyra, and Kumar, and the field-theoretic approach due to Efetov and Larkin, are shown to be equivalent for all symmetry classes. The equivalence had been questioned as a result of field-theoretic calculations of the average conductance by Zirnbauer [PRL 69, 1584 (1992)], which disagreed with the Fokker-Planck approach in the symplectic symmetry class. We resolve this controversy by pointing to an incorrect implementation of Kramers degeneracy in these calculations, and we derive modified expressions for the first two conductance moments which agree well with existing numerical simulations from the metallic into the localized regime. ***Submitted to Physical Review B.***