Participation ratio in the nonlinearσ-model representation of localization

Wegner’s analysis of the singularities in participation ratios near the mobility edge is extended to the case of spin-flip scattering and of random complex hopping. The critical exponents for the participation ratio and inverse participation ratio in 2+ε dimensions are respectively given by ${\mu }_2$=(2ε${)}^{\mathrm{-}1/2}$+O(ε) and ${\pi }_2$=${\epsilon }^{\mathrm{-}1}$-(2ε${)}^{\mathrm{-}1/2}$+O(1 ). The calculation is based on a unitary nonlinear σ-model representation for this localization problem.