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.Here c(r) is the wave velocity at position r in an inhomogeneous medium, assumed to be a randomly fluctuating quantity. The main difference between the Schrödinger

Anderson-like localization transition of random walks with resetting

“…Consequently, the solutions of (79) exhibit the same properties as exposed in Section 3.1, as well as the essential singularity in D = 2 [50]. In 3D, the SCT predicts ν = 1 for the correlation length exponent, see Eq.…”

Nonuniversal critical behavior in disordered pseudospin-1 systems

“…t l shown in Figs. 2 and 4, respectively, we find that the magnitude of  is about 2~3 times of t l , which is consistent with the known results reported in ordinary 1D disordered systems[5,6,62,63]. This result also implies the absence of a diffusive regime in pseudospin-1 systems, irrespective of the type of disorder.B.…”

Electron‐electron interactions and the metal‐insulator transition in heavily doped silicon

“…Theoretically, μ is usually inferred from the correlation-length critical exponent ν via Wegner scaling μ = ν(d − 2) where d is the spatial dimension of the MI systems [54]. In a self-consistent theory of Anderson localization (neglecting interactions), an exponent μ = 1 for d = 3 is suggested [2,3]. Field-theoretical approaches are discussed in [55].…”

“…σ(0) ∼ |δ − δ c | μ , has been a subject of considerable controversy, as will be discussed below. A self-consistant theory of the Anderson transition for noninteracting electrons was developed by Vollhardt and Wölfle [2,3]. However, using both carrier concentration N and uniaxial stress S to tune the MI transition, identifying the critical region of δ where critical behavior is expected to occur, and employing a dynamic finite-temperature scaling analysis, a consistent picture has emerged for Si:P which is the prototype material for a localization transition in three dimensions.…”

“…The dependence of the concentration of localized moments on the P concentration N has been mapped out systematically from specific-heat measurements for Si:P [26,27] and Si:(P,B) [28]. The concentration of local moments has been determined from the zero-field excess specific heat ΔC = C −γT −βT 3 , i.e., the conduction-electron and phonon contributions subtracted from the measured specific heat C, via the entropy S = (ΔC/T )dT = N S ln2.…”