Diagrammatic, self-consistent treatment of the Anderson localization problem in $d\le 2$ dimensions

A standard diagrammatic theory is formulated for the density response function $\chi (\overrightarrow{\mathrm{q}}, \omega )$ of a system of independent particles moving in a random potential. In the limit of small $\overrightarrow{\mathrm{q}}$, $\omega$ the Bethe-Salpeter equation for the particle-hole vertex function may be solved for $\chi (\overrightarrow{\mathrm{q}}, \omega )$ in terms of a current relaxation kernel $M(\overrightarrow{\mathrm{q}}, \omega )$ [essentially the inverse of the diffusion coefficient $D(\overrightarrow{\mathrm{q}}, \omega )$]. $M(\overrightarrow{\mathrm{q}}, \omega )$ 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 $M(0, \omega )$ or $D(0, \omega )$ 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 $M(0, \omega )$ or $D(0, \omega )$ from a diffusion propagator. However, in the presence of time-reversal invariance, $M(\overrightarrow{\mathrm{q}}, \omega )$ is shown to have infrared divergencies in $d\le 2$, 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 $\alpha$, a dynamical conductivity $\mathrm{Re}\sigma \left(\omega \right)\propto {\omega }^2$ for $\omega \to 0$, and a finite localization length in $d\le 2$ for arbitrarily weak disorder. In $d=1\mathrm{}$ our results agree remarkably well with the exact solutions by Berezinsky and also Abrikosov and Ryshkin. Ryshkin.