Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions

Arguments are presented that the $T=0\mathrm{}$ 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 $\beta \mathrm{}\left(G\right)=\frac{d\ln G}{d\ln L}$, valid for both $G\ll G_c\simeq \frac{e^2}{\hslash }$ and $G\gg G_c$. 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 $L$.