Topological defects in Haldane model and higher Chern numbers in monolayer graphene

We consider the Haldane model, a two-band model in monolayer graphene with non-trivial Chern numbers. Two types of topological defects, monopoles and merons, are derived from the model: (a) the monopole defects occur at the Dirac points, where the system experiences a topological transition and the Chern number C takes an indeterminate value. The sign-change of the mass term after this transition indicates different topological states labeled by different C numbers; (b) the meron defects occur as per a varying mass term Summing up the topological charges of the merons leads to the C evaluation for the energy bands of an insulating bulk, and the result we obtain is in full agreement to the past literature. Furthermore, in this paper we propose a high-C model through studying the limitation behavior of the Hamiltonian vector in the neighborhood of the topological defects. It is discovered that two conducting states may arise form the edges, where the lower band of the insulating bulk carries a higher Chern number, C = +/- 2.