Mean field theory and ϵ-expansion for Anderson localization

We present a field theoretic formulation of Anderson localization of an electron in a random potential. In mean field theory we find a mobility edge at energy E_c separating a region with no states from one with conducting states. When the nearest neighbor hopping is a random variable with variance σ, E_c²=4zσ², where z is the coordination number. We study this mobility edge using the ?-expansion. We find an upper critical dimension of eight near which this mobility transition is in the same universality class as the statistics of dilute branched polymers (lattice animals).