Multifractality and quantum diffusion from self-consistent theory of localization

TL;DR

This paper explores the multifractal nature of wave functions in disordered systems using self-consistent localization theory, revealing insights into scaling relations, the accuracy of one-loop results, and the limitations of sigma-models.

Contribution

It provides a diagrammatic interpretation of localization theory results, discusses the strict parabolicity of the multifractal spectrum, and analyzes the deficiencies of sigma-models at higher loop levels.

Findings

One-loop Wegner result may be exact for d=2+ε.

Multifractal spectrum is strictly parabolic.

Sigma-models are deficient at four-loop level.

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+\epsilon may appear to be exact, so the multifractal spectrum is strictly parabolical. The \sigma-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(\omega,q) is resolved in the compromise manner due to ambiguity of the D(\omega,q) definition. Comparison is made with the extensive numerical material.