Energy spectrum of a triangular lattice in a uniform magnetic field: Effect of next-nearest-neighbor hopping

The energy spectrum of a triangular lattice in a uniform magnetic field is studied within the tight-binding approximation. The effect of next-nearest-neighbor (NNN) hopping on the energy spectrum is investigated in detail. Numerical calculations show that, when NNN hopping is turned on, most of the subband widths for a rational magnetic Aux first decrease to certain critical values of NNN hopping, resulting in a very narrow band structure. Then, the widths increase with increasing NNN hopping strength, and the resultant energy spectrum shows a very complicated, fat band structure. These phenomena are explained by elucidating, in detail, the dependence of the subbands and the gaps on the NNN hopping strength. The symmetry of the energy spectrum in the presence of NNN hopping is also discussed.