Noncompact<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>Models and the Existence of a Mobility Edge in Disordered Electronic Systems near Two Dimensions

Abstract

The properties of an electron in a disordered solid are discussed with use of a matrix nonlinear $\ensuremath{\sigma}$ model first introduced by Wegner and Sch\"afer. The model is defined on the noncompact space $\frac{\mathrm{O}(M,M)}{[\mathrm{O}(M)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(M)]}$ 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.