Scaling theory of localization: absence of quantum diffusion in two dimensions

Abstract

Arguments are presented that the T=0 conductance G of a disordered electronic system depends on its length scale L in a universal manner. Asymptotic forms are obtained for the scaling function β(G)=dlnG/dlnL, valid for both G≪G_c≃ e²/ℏ and G≫G_c. In three dimensions, G_c is an unstable fixed point. In two dimensions, there is no true metallic behavior; the conductance crosses over smoothly from logarithmic or slower to exponential decrease with L.