Localization and critical diffusion of quantum dipoles in two dimensions

We discuss quantum propagation of dipole excitations in two dimensions. This problem differs from the conventional Anderson localization due to existence of long range hops. We found that the critical wavefunctions of the dipoles always exist which manifest themselves by a scale independent diffusion constant. If the system is T-invariant the states are critical for all values of the parameters. Otherwise, there can be a metal-insulator transition between this ordinary diffusion and the Levy-flights (the diffusion constant logarithmically increasing with the scale). These results follow from the two-loop analysis of the modified non-linear supermatrix $sigma$-model.