Multifractality and quantum diffusion from self-consistent theory of localization

Abstract

Multifractal properties of wave functions in a disordered system can be derived from self-consistent theory of localization by Vollhardt and Woelfle. A diagrammatic interpretation of results allows to obtain all scaling relations used in numerical experiments. The arguments are given that the one-loop Wegner result for a space dimension d=2+εmay appear to be exact, so the multifractal spectrum is strictly parabolical. The σ-models are shown to be deficient at the four-loop level and the possible reasons of that are discussed. The extremely slow convergence to the thermodynamic limit is demonstrated. The open question on the relation between multifractality and a spatial dispersion of the diffusion coefficient D(ω,q) is resolved in the compromise manner due to ambiguity of the D(ω,q) definition. Comparison is made with the extensive numerical material.

Latex, 19 pages, 7 figures included

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