Scaling Equations from a Self-Consistent Theory of Anderson Localization

Abstract

The conductance of a disordered system of finite volume $L^d$ ($d$ spatial dimensions) is calculated by making use of a recently developed self-consistent theory. Scaling equations are derived for the dimensionless conductance $g$ and the scaling function $\beta \mathrm{}\left(g\right(L\left)\right)=\frac{d\ln g}{d\ln L}$ is explicitly calculated for arbitrary dimension $d$. It is shown that the theory obeys scaling in the sense of Wegner, Thouless, and Abrahams et al.