Analytic trajectories for mobility edges in the Anderson model

Abstract

A basis of Bloch waves, distorted locally by the random potential, is introduced for electrons in the Anderson model. Matrix elements of the Hamiltonian between these distorted waves are averages over infinite numbers of independent site energies, and so take definite values rather than distributions of values. The transformed Hamiltonian is ordered, and may be interpreted as an itinerant electron interacting with a spin on each site. In this new basis, the distinction between extended and localized states is clear, and edges of the bands of extended states, the mobility edges, are calculated as a function of disorder. In two dimensions these edges have been found in both analytic and numerical applications of tridiagonalization, but they have not been found in analytical approaches based on perturbation theory, or the single-parameter scaling hypothesis; nor have they been detected in numerical approaches based on scaling or critical distributions of level spacing. In both two and three dimensions the mobility edges in this work are found to separate with increasing disorder for all disorders, in contrast with the results of calculations using numerical scaling for three dimensions. The analytic trajectories are compared with recent results of numerical tridiagonalization on samples of over ${10}^{9}$ sites. This representation of the Anderson model as an ordered interacting system implies that in addition to transitions at mobility edges, the Anderson model contains weaker transitions characterized by critical disorders where the band of extended states decouples from individual sites, and that singularities in the distribution of site energies, rather than its second moment, determine localization properties of the Anderson model.