Towards the Realization of Self-Consistent Effective Medium Theory for Anderson Disorder Model*

A mean-field theory that properly characterizes the Anderson localization transition in three dimensions has remain elusive. Here, we present a systematic typical medium dynamical cluster approximation that provides a proper description of this phenomenon. Our method accurately provides a proper way to treat the different energy scales (close to the criticality) such that the characteristic re-entrant behavior of the mobility edge is obtained. This allows us to study the localization in different momenta cells, which renders the discovery that the Anderson localization transition occurs in a \textit{momentum cell-selective fashion}. As a function of cluster size, our method systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength with great improvement on the critical exponent of the order parameter ($\beta > 1.4$).