Diagrammatic, self-consistent treatment of the Anderson localization problem ind≤2dimensions

Abstract

A standard diagrammatic theory is formulated for the density response function $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$ of a system of independent particles moving in a random potential. In the limit of small $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$, $\ensuremath{\omega}$ the Bethe-Salpeter equation for the particle-hole vertex function may be solved for $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$ in terms of a current relaxation kernel $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$ [essentially the inverse of the diffusion coefficient $D(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$]. $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$ is obtained as the sum of the imaginary part of the single-particle self-energy and the current matrix element of the irreducible kernel and is determined diagrammatically. A theorem is formulated, stating that any diagram for $M(0, \ensuremath{\omega})$ or $D(0, \ensuremath{\omega})$ containing a (bare) diffusion propagator belongs to a well-defined class of diagrams whose divergencies cancel each other, and an exact proof is presented. In particular, this implies that there are no divergent contributions to $M(0, \ensuremath{\omega})$ or $D(0, \ensuremath{\omega})$ from a diffusion propagator. However, in the presence of time-reversal invariance, $M(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}, \ensuremath{\omega})$ is shown to have infrared divergencies in $d\ensuremath{\le}2$, signalling a breakdown of the perturbation expansion in terms of the scattering potential which has first been discussed by Abrahams et al. A self-consistent treatment in the weak-coupling limit yields a finite static polarizability $\ensuremath{\alpha}$, a dynamical conductivity $\mathrm{Re}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$, and a finite localization length in $d\ensuremath{\le}2$ for arbitrarily weak disorder. In $d=1$ our results agree remarkably well with the exact solutions by Berezinsky and also Abrikosov and Ryshkin. Ryshkin.