Self-Consistent Theory of Anderson Localization: General Formalism and Applications

TL;DR

This paper reviews the self-consistent theory of Anderson localization, covering basic concepts, regimes, scaling, renormalization group approaches, and recent applications to classical waves, providing a comprehensive framework for understanding localization phenomena.

Contribution

It presents a unified self-consistent formalism for Anderson localization applicable to quantum particles and classical waves, including recent extensions and applications.

Findings

Quantitative description of static and dynamic transport properties.

Natural derivation of self-consistent equations for diffusion coefficients.

Application to classical waves and extensions beyond original theory.

Abstract

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.