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Effective Cluster Typical Medium Theory for 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, $W_c \sim (1/L_c)^{1/ν}$; whereas in two dimensions, the critical disorder strength decreases logarithmically with the linear cluster size. Our results are consistent with previous numerical work and in agreement with the one-parameter scaling theory.