AC conductivity and the anomalous dielectric constant of a disordered two-dimensional system

The authors calculate the AC conductivity for a disordered two-dimensional system and find a transition between diffusive-like behaviour and hopping-like behaviour. In the diffusive region, (r²) approximately t^w with w<1 and sigma _diff approximately omega ^1-w. This behaviour is found to be consistent with power-law localisation with w=1/(1+s) where s is the exponent of the power-law wavefunction, mod psi mod approximately r^-s. In the hopping region, the authors find sigma _hop approximately omega ^2-3s/. This leads to an anomalous dielectric constant epsilon ₁ which diverges at two values of energy, at the mobility edge and at an energy for which s=3. For s>3, epsilon ₁ is finite and positive down to the mobility edge E_c. Below E_c, the dielectric constant diverges as epsilon ₁ approximately (E_c-E)^{-(4 nu -1)} where nu is the exponent of the localisation length xi approximately (E_c-=E)^{- nu}.