Anderson localization and non-linear sigma model with graded symmetry

Abstract

A model of disordered single-particle systems is studied with regard to properties of the localized phase. Defined over a graded coset space, this model represents the correct non-perturbative extension of a non-linear sigma model introduced into localization theory by Schäfer and Wegner. An integral theorem is proven which allows us to change variables and execute the Grassmann integrations rather easily. In the localized phase, the invariant two-point functions are singular on the real axis. It is shown how to extract the singular contribution before evaluation of the functional integral. This is used to derive Efetov's solution of the Cayley tree model in a simple and transparent manner. Finally, a Monte Carlo algorithm is outlined which makes it possible to study Anderson localization in d > 2 dimensions.