Dynamical scaling analysis of the optical Hall conductivity in the quantum Hall regime

Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$ as a function of Fermi energy ${\epsilon }_F$ and frequency $\omega$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that ${\sigma }_{xy}({\epsilon }_F,\omega )$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are fitted and plugged in. A crossover from the dc-like bahavior to the ac regime is identified. The dynamical scaling analysis has enabled us to quantify the plateau in the ac Hall conductivity previously obtained, and to predict that the plateaux structure in ac is robust enough to be observed in the THz regime.