Anderson Localization ind<~2Dimensions: A Self-Consistent Diagrammatic Theory

Abstract

A diagrammatic theory is presented for the density response function of a system of independent particles moving in a random potential in terms of a current relaxation kernel $M(\overrightarrow{\mathrm{q}},\omega )$ (essentially the inverse of the diffusion coefficient). In the presence of time-reversal invariance, $M(\overrightarrow{\mathrm{q}},\omega )$ is shown to have infrared divergencies in $d<~2$ dimensions. A self-consistent treatment of the divergent terms yields a finite static electric polarizability $\alpha$, a dynamical conductivity $\sigma \left(\omega \right)\propto {\omega }^2(\omega \to 0)$, and a finite localization length in $d<~2$ for arbitrarily weak disorder.