Distant-neighbor hopping induced the Dirac points and the high Chern number topological phase on π-flux square lattice

For the two-band system, an important route to achieve the high Chern number (HCN) topological phase is to add the number of low-energy Dirac points. In this work, we try to study the possibility of realizing HCN phase in π-flux square lattice when the (n−1)-nearest-neighboring hoppings Nn are introduced. We investigate N4 and N6 intersublattice hopping which can generate new Dirac points, whose chiralities and mergings are analyzed. While for intrasublattice hopping which can gap the Dirac points, we studied N2 and N5 cases to obtain the HCN phase diagram. We further discuss the experimental detections of the HCN phase by the edge states excitations and the transverse Hall conductance response.