Vertical absorption edge and universal onset conductance in semi-hydrogenated graphene

We show that for graphene with any finite difference in the on-site energy between the two sub-lattices ($\Delta$), The optical absorption edge is determined by the $\Delta$. The universal conductance will be broken and the conductance near the band edge varies with frequency as $1/\omega^2$. Moreover, we have identified another universal conductance for such systems without inversion symmetry, i.e., the onset conductance at the band edge is $% \sigma_c=2\sigma_0=\pi e^2/2h$, independent of the size of the band gap. The total integrated optical response is nearly conserved despite of the opening of the band gap.