Hall conductance, topological quantum phase transition, and the Diophantine equation on the honeycomb lattice

We consider a tight-binding model with the nearest-neighbor hopping integrals on the honeycomb lattice in a magnetic field. Assuming one of the three hopping integrals, which we denote by t_a , can take a different value from the two others, we study quantum phase structures controlled by the anisotropy of the honeycomb lattice. For weak and strong t_a regions, the Hall conductances are calculated algebraically by using the Diophantine equation. Except for a few specific gaps, we completely determine the Hall conductances in these two regions including those for subband gaps. In a weak magnetic field, it is found that the weak t_a region shows the unconventional quantization of the Hall conductance, σ_xy=-(e²/h)(2n+1) (n=0,±1,±2,…) , near the half filling, while the strong t_a region shows only the conventional one, σ_xy=-(e²/h)n (n=0,±1,±2,…) . From the topological nature of the Hall conductance, the existence of gap closing points and quantum phase transitions in the intermediate t_a region is concluded. We also study numerically the quantum phase structure in detail and find that even when t_a=1 , namely, in graphene case, the system is in the weak t_a phase except when the Fermi energy is located near the Van Hove singularity or the lower and upper edges of the spectrum.