Scaling for the frequency-dependent conductivity in disordered electronic systems

The scaling theory of Abrahams et al. is extended to the dynamical conductivity $\sigma \left(\omega \right)$ and Hall conductivity ${\sigma }_H\left(\omega \right)$. It is shown, by means of a self-consistent scaling argument, that at the mobility edge $\sigma \left(\omega \right)\sim {\omega }^{\frac{\mathrm{}(d-2\mathrm{})\mathrm{}}{d}}$ and ${\sigma }_H\left(\omega \right)\sim {\omega }^{\frac{2(d-2\mathrm{})\mathrm{}}{d}}$ (the dimensionality $d>\mathrm{}2\mathrm{}$). Thus, at $d=3\mathrm{}$, the frequency-dependent part of the conductivity, $\Delta \sigma \left(\omega \right)\equiv \sigma \left(\omega \right)-{\sigma }_{\mathrm{dc}}$, exhibits a crossover near the mobility edge from ${\omega }^{\frac{1}{2}}$ to ${\omega }^{\frac{1}{3}}$ behavior. Similarly, the frequency-dependent part of the Hall conductivity crosses over from ${\omega }^{\frac{1}{2}}$ to ${\omega }^{\frac{2}{3}}$ behavior.