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

Abstract

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, ${\ensuremath{\sigma}}_{xy}=\ensuremath{-}({e}^{2}/h)(2n+1)$ $(n=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\dots{})$, near the half filling, while the strong ${t}_{a}$ region shows only the conventional one, ${\ensuremath{\sigma}}_{xy}=\ensuremath{-}({e}^{2}/h)n$ $(n=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,\dots{})$. 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.