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 $\ensuremath{\beta}(G)=\frac{d\mathrm{ln}G}{d\mathrm{ln}L}$, valid for both $G\ensuremath{\ll}{G}_{c}\ensuremath{\simeq}\frac{{e}^{2}}{\ensuremath{\hbar}}$ and $G\ensuremath{\gg}{G}_{c}$. 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 $L$.