This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent}
β and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength (
WcU), required to localize all states, increases with increasing interactions (
U); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate
W close to
WcU is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential,
μ, is at or beyond the mobility edge energy (
ωϵ). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material's specific details as such provide an avenue for the possibility of studying electron localization in real materials.
The interplay of interactions and disorder is studied using the Anderson-Hubbard model within the typical medium dynamical cluster approximation. Treating the interacting, nonlocal cluster self-energy $[{\mathrm{\ensuremath{\Sigma}}}_{c}[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}](i,j\ensuremath{\ne}i)]$ up to second order in the perturbation expansion of interactions,
U2, with a systematic incorporation of nonlocal spatial correlations and diagonal disorder, we explore the initial effects of electron interactions (
U) in three dimensions. We find that the critical disorder strength (
WcU), required to localize all states, increases with increasing
U; implying that the metallic phase is stabilized by interactions. Using our results, we predict a soft pseudogap at the intermediate
W close to
WcU and demonstrate that the mobility edge (${\ensuremath{\omega}}_{\ensuremath{\epsilon}}$) is preserved as long as the chemical potential, $\ensuremath{\mu}$, is at or beyond the mobility edge energy.