NoncompactσModels and the Existence of a Mobility Edge in Disordered Electronic Systems near Two Dimensions

The properties of an electron in a disordered solid are discussed with use of a matrix nonlinear $\sigma$ model first introduced by Wegner and Schäfer. The model is defined on the noncompact space $\frac{\mathrm{O}\mathrm{}(M,M)\mathrm{}}{\left[\mathrm{O}\mathrm{}\right(M)\mathrm{}\times \mathrm{O}\mathrm{}(M\left)\right]}$ where $M$ is the number of replicas. This noncompact symmetry represents the assential physics of the problem. It is found that all states are localized in two dimensions; above two dimensions for weak disorder there are mobility edges, but these merge above a critical amount of disorder and all states become localized.