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We use the recently developed typical medium dynamical cluster (TMDCA)approach~[Ekuma \etal,~\textit{Phys. Rev. B \textbf{89}, 081107 (2014)}] toperform a detailed study of the Anderson localization transition in threedimensions for the Box, Gaussian, Lorentzian, and Binary disorderdistributions, and benchmark them with exact numerical results. Utilizing thenonlocal hybridization function and the momentum resolved typical spectra tocharacterize the localization transition in three dimensions, we demonstratethe importance of both spatial correlations and a typical environment for theproper characterization of the localization transition in all the disorderdistributions studied. As a function of increasing cluster size, the TMDCAsystematically recovers the re-entrance behavior of the mobility edge fordisorder distributions with finite variance, obtaining the correct criticaldisorder strengths, and shows that the order parameter critical exponent forthe Anderson localization transition is universal. The TMDCA is computationallyefficient, requiring only a small cluster to obtain qualitative andquantitative data in good agreement with numerical exact results at a fractionof the computational cost. Our results demonstrate that the TMDCA provides aconsistent and systematic description of the Anderson localization transition.

Finite Cluster Typical Medium Theory for Disordered Electronic Systems