Dissipation and criticality in the lowest Landau level of graphene

The lowest Landau level of graphene is studied numerically by considering a tight-binding Hamiltonian with disorder. The Hall conductance ${\sigma }_{\mathrm{x}\mathrm{y}}$ and the longitudinal conductance ${\sigma }_{\mathrm{x}\mathrm{x}}$ are computed. We demonstrate that bond disorder can produce a plateau-like feature centered at $\nu =0$, while the longitudinal conductance is nonzero in the same region, reflecting a band of extended states between $\pm E_c$, whose magnitude depends on the disorder strength. The critical exponent corresponding to the localization length at the edges of this band is found to be $2.47\pm 0.04$. When both bond disorder and a finite mass term exist the localization length exponent varies continuously between $\sim 1.0$ and $\sim 7\mathrm{/}3$.