Graded symmetry and Anderson localization on the Bethe lattice for time-reversal invariant systems

We study the phenomenon of Anderson localization on a Bethe lattice via a supersymmetric nonlinear σ-model. We consider the time-reversal invariant case and compare our results to the technically simpler time-reversal noninvariant case. We found that the properties of the two cases are very similar. In particular, the density-density correlation function follows from the solution of a one-dimensional integral equation with a kernel having the same gross properties as in the time reversal noninvariant case. The inverse participation ratio is discontinuous at the transition point. The distance dependence of the density-density correlation function is derived with the help of the theory of random walks. Results have been supported and extended by numerical calculations. Due to elaborate grassmannian integrals no results have been obtained for the extended phase.