Effective cluster typical medium theory for the diagonal Anderson disorder model in one- and two-dimensions.

We develop a cluster typical medium theory to study localization in disordered electronic systems. Our formalism is able to incorporate non-local correlations beyond the local typical medium theory in a systematic way. The cluster typical medium theory utilizes the momentum-resolved typical density of states and hybridization function to characterize the localization transition. We apply the formalism to the Anderson model of localization in one- and two-dimensions. In one-dimension, we find that the critical disorder strength scales inversely with the linear cluster size with a power law, Wc ∼ (1/Lc)(1/ν), whereas in two-dimensions, the critical disorder strength decreases logarithmically with the linear cluster size. Our results are consistent with previous numerical work and are in agreement with the one-parameter scaling theory.