Spatial-temporal dispersion of the kinetic coefficients near the Anderson transition

Abstract

A generalization of the Vollhardt-Wolfle localization theory is proposed to make it possible to study the spatial-temporal dispersion of the kinetic coefficients of a d-dimensional disordered system in the low-frequency, long-wavelength range (ωℰF and q≪kF). It is shown that the critical behavior of the generalized diffusion coefficient D(q,ω) near the Anderson transition agrees with the general Berezinskii-Gor’kov localization criterion. More precisely, on the metallic side of the transition the static diffusion coefficient D(q,0) vanishes at a mobility threshold λc common for all q: D(q, 0)∝t=(λc−λ)/λc→0, where λ=1/(2πℰFτ) is a dimensionless coupling constant. On the insulator side, q≠0 D(q,ω)∝−iω as ω→0 for all finite q. Within these limits, the scale of the spatial dispersion of D(q,ω) decreases in proportion to t in the metallic phase and in proportion to ωξ2, where ξ is the localization length, in the insulator phase until it reaches its lower limit ∼λF. The suppression of the spatial dispersion of D(q,ω) near the Anderson transition up to the atomic scale confirms the asymptotic validity of the Vollhardt-Wolfle approximation: D(q,ω)≃D(ω) as |t|→0 and ω→0. By contrast, the scale of the spatial dispersion of the electrical conductivity in the insulator phase is of order of the localization length and diverges in proportion to |t|−v as |t|→0.