Dielectric constant near the Anderson transition

The authors study the effect of power-law localisation on the dielectric constant of a noninteracting disordered electron gas. If the envelope of the eigenstate goes as 1/r^s, then the authors find a divergent epsilon ₁ when s to d/2, where d is the dimensionality. The AC conductivity in this limit is sigma approximately omega ^2-ds/. If the states are not pure power law states and psi approximately r^-se^-r xi /, their results still apply for high frequencies where the characteristic length is smaller than xi and the 1/r^s component of psi dominates. It is argued that this power-law correction to psi may explain qualitatively the AC conductivity and its temperature dependence just below the Anderson transition in Si:P. For omega to 0, the authors find epsilon ₁(0) approximately xi ² independent of whether or not a minimum metallic conductivity exists.