Quantum transport in disordered wires: Equivalence of the one-dimensional sigma model and the Dorokhov-Mello-Pereyra-Kumar equation

The two known nonperturbative 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 [Phys. Rev. Lett. 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 that agree well with existing numerical simulations from the metallic into the localized regime.