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

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.

Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions

“…Fitting the data for the two largest clusters starting from W ≈ 1.0 with a power law TDOS (ω = 0) = a 0 |W − C| β , we obtain β > 1.40, which is greater than a single site TMT value of β T MT = 1.0; but it is still smaller than the most recently reported β ≈ 1.67 [20,21,28]. We note that other mean-field methods reported β 1.0 [29]. In our method, we note that it is unlikely that we can calculate the critical disorder strength and exponents as precisely as diagonalization and transfer matrix methods [4,17,18,[20][21][22]27].…”

Scaling properties of dynamical localization in monochromatically perturbed quantum maps: Standard map and Anderson map

“…Next, we give a theoretical explanation for the scaling properties obtained numerically in the last two sections is given based on the self-consistent mean-field theory (SCT) of the Anderson localization in 2DDS [22].…”

Self-Consistent Theory of Anderson Localization in Two Dimensions in View of Exact Transport Equation

“…2)-7) Among others, Vollhardt and Wölfle (VW) have developed a self-consistent theory that covers from weakly to strongly localized regimes, taking into account the hydrodynamic singularity which is determined by the diffusion coefficient in question. 5)- 7) The main conclusions of this theory are now widely accepted. A point which may be further questioned is that for d = 2, where d is the dimensionality, it does not predict the exponential decay of the diffusion coefficient, as a function of the system size in the vicinity of the localization length.…”