Pairing in the Hubbard model on the honeycomb lattice with hopping up to the third-nearest-neighbor

The electronic properties of graphene can be approximately described by the standard Hubbard model on the honeycomb lattice. It was argued that the d+id-wave pairing is the dominant pairing symmetry in such a system. However, in the standard Hubbard model, only the nearest-neighbor hopping t1 is considered, while the second-nearest-neighbor hopping t2 and third-nearest-neighbor hopping t3 are ignored. In this paper, by doing the determinant quantum Monte Carlo simulations of the Hubbard model with hopping up to the third-nearest-neighbor, we examine the effect of t2 and t3 on the pairing symmetries. We find that the second- and third-nearest-neighbor hopping amplitudes have strong effect on the d+id-wave pairing symmetry.