Frequency dependence of the polarization catastrophe at a metal-insulator transition and related problems

A scaling assumption is made in order to analyze the critical behavior of the complex dielectric constant ε(ω) at low frequencies near a metal-insulator or a superconductor-normal-conductor transition in a disordered system. Percolative transitions as well as microscopic or quantum transitions are discussed. On either side of the metal-insulator transition, Reε(ω) is found to have a peak at ω=0 whose width tends to zero and whose height diverges at the transition. Similar behavior is found for the normal component of conductivity on both sides of the superconducting transition, and the effective penetration depth is found to diverge as (p-pc)-t2 near the percolation threshold in a composite superconductor.