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 $\ensuremath{\beta}(g(L))=\frac{d\mathrm{ln}g}{d\mathrm{ln}L}$ 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.