Optical Hall conductivity in 2DEG and graphene QHE systems

We have revealed from a numerical study that the Hall plateaus are retained in the optical Hall conductivity ${\sigma }_{xy}(\omega )$ in the ac ($\sim$ THz) regime in both of the ordinary two-dimensional electron gas and graphene in the quantum Hall regime, although the plateau height in ac deviates from integer multiples of $e^2\mathrm{/}h$. The effect remains unexpectedly robust against a significant strength of disorder, which we attribute to an effect of localization. We predict the ac Hall plateaus are observable through the Faraday rotation with the rotation angle characterized by the fine-structure constant $\alpha$. In this paper we clarify the relationship between plateau structures and the disorder strength by performing numerical calculation.