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

A standard diagrammatic theory is formulated for the density response function χ(q⃗, ω) of a system of independent particles moving in a random potential. In the limit of small q⃗, ω the Bethe-Salpeter equation for the particle-hole vertex function may be solved for χ(q⃗, ω) in terms of a current relaxation kernel M(q⃗, ω) [essentially the inverse of the diffusion coefficient D(q⃗, ω)]. M(q⃗, ω) 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, ω) or D(0, ω) 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, ω) or D(0, ω) from a diffusion propagator. However, in the presence of time-reversal invariance, M(q⃗, ω) is shown to have infrared divergencies in d≤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 α, a dynamical conductivity Reσ(ω)∝ω² for ω→0, and a finite localization length in d≤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.