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 $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ (essentially the inverse of the diffusion coefficient). In the presence of time-reversal invariance, $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ is shown to have infrared divergencies in $d<~2$ dimensions. A self-consistent treatment of the divergent terms yields a finite static electric polarizability $\ensuremath{\alpha}$, a dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$, and a finite localization length in $d<~2$ for arbitrarily weak disorder.