Self-Consistent Theory of Anderson Localization in Two Dimensions in View of Exact Transport Equation

Self-consistent theory of Anderson localization of two-dimensional non-interacting electrons is formulated in the context of the exact transport equation and conductivity expression derived by the present authors (YI). The irreducible scattering vertex by Vollhardt and Wolfle (VW) is used in this equation, determining the diffusion coefficient in the scattering vertex self-consistently, through Einstein relation. It predicts a similar localization length to that obtained by VW, but shows that the conductivity evaluated by the Kubo formula decays exponentially, as the system size approaches the localization length. The result is opposed to the prediction by VW, who showed different behaviour of the diffusion coefficient that is equivalent to our conductivity. Our calculation also implies that the localization may be described along with the Landau-Silin theory of Fermi liquid.