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 σ model first introduced by Wegner and Sch\"afer. The model is defined on the noncompact space O(M,M)[O(M)×O(M)] 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.