Localization transition in three dimensions: Monte Carlo simulation of a nonlinear σ model

We present a combination of analytical and numerical calculations for the critical behavior of a supersymmetric nonlinear {σ} model. This toy model is expected to describe at least qualitatively the localization transition of a disordered one-electron system. As a result, we obtain a localization length exponent and a set of inverse participation numbers in three dimensions. We find a continuous phase transition with the features of one-parameter scaling and multifractality at the critical point. {} 1996 The American Physical Society.