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

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

2008-08-26

arxiv.org/abs/0808.3440v2

Phys. Rev. B78, 235322 (2008)

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

35 pages, 15 figures; added references

Scientific paper

10.1103/PhysRevB.78.235322

We consider a tight-binding model with the nearest neighbour hopping integrals on the honeycomb lattice in a magnetic field. Assuming one of the three hopping integrals, which we denote 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, respectively, 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, \sigma_{xy}=-(e^2/h)(2n+1), (n=0,\pm 1,\pm 2,...), near the half-filling, while the strong t_a region shows only the conventional one, \sigma_{xy}=-(e^2/h)n,(n=0,1,2,...). From topological nature of the Hall conductance, the existence of gap closing points and quantum phase transitions in the intermediate t_a region are 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.

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