Low dimensions electron localization in the beyond real space super cell approximation

Metal to insulator phase transition due to electron localization in disordered alloys (Anderson transition) and interacting electrons (Mott transition) systems is one of major problem in these fields. Multi site electron scattering is responsible for localization which can’t be seen by single site approximations such as coherent potential approximation (CPA) and dynamical mean field theory (DMFT). Here we develop a multi site technique to calculate multi site electron scattering for observation of phenomenons such as electron localization especially in low dimension systems. Our self-energy in first Brillouin zone (FBZ) is casual, in contrast to previous approximation fully crystal electron wave vector, q, dependent and continuous with respect to q. It recovers coherent potential approximation in the single site approximation and is exact when the number of sites in the super cell approaches to the total number of lattice sites. We illustrate that this approximation undertakes electrons localization for one and two dimensional alloy systems which isn’t observed by previous multi site approximations such as dynamical cluster approximation (DCA).