Theory of Anderson localization in weak magnetic fields

We derive self-consistency equations determining the transverse dynamical conductivity for the case of Wegner's local gauge invariant model in a weak magnetic field B. The solution in the critical regime connected with Anderson localization is given for dimensionalities d = 2, 3. In d = 2 the self-consistency equations generate a logarithmic singularity in second order in the coupling constant. This is shown to be in agreement with the loop expansion and yields localization for arbitrarily weak coupling. In d = 3 there is a metal-insulator transition. In its vicinity the self-consistency equations reduce to a two-parameter scaling law, which is consistent with the results of Khmelnitskii and Larkin.