Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling

Abstract

The Anderson model for independent electrons in a disordered potential is transformed analytically and exactly to a basis of random extended states leading to a variant of augmented space. In addition to the widely-accepted phase diagrams in all physical dimensions, a plethora of additional, weaker Anderson transitions are found, characterized by the long-distance behavior of states. Critical disorders are found for Anderson transitions at which the asymptotically dominant sector of augmented space changes for all states at the same disorder. At fixed disorder, critical energies are also found at which the localization properties of states are singular. Under the approximation of single-parameter scaling, this phase diagram reduces to the widely-accepted one in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson transition at infinitesimal disorder, there is a transition between two localized states, characterized by a change in the nature of wave function decay.