Topological phases of a semi-Dirac Chern insulator in the presence of extended range hopping

We study the topological properties and the topological phase transitions therein of a semi-Dirac Haldane model on a honeycomb lattice in the presence of an extended range (third neighbor) hopping. While in the absence of third-neighbor hopping t(3) the system exhibits a gapless electronic spectrum, its presence creates an energy gap in the dispersion. However, the nature of the spectral gap, that is, whether it is trivial or topological, needs to be ascertained. We find that the answer depends on the value of t(3) and its interplay with the value of the on-site potential that breaks the sublattice symmetry, namely, the Semenoff mass Delta. To elucidate our findings on the topological phases, we demonstrate two kinds of phase diagrams using the available parameter space, one in which the phases are shown in the Delta-t(3) plane and one in a more familiar Delta-phi plane (phi is the Haldane flux). The phase diagrams depict the presence of Chern insulating lobes comprising Chern numbers +/- 2 and +/- 1 for a certain range of values for t(3), along with trivial insulating regions (zero Chern number). Thus, there are phase transitions from one topological regime to another which are characterized by abrupt changes in the values of the Chern number. To support the existence of the topological phases, we compute counterpropagating chiral edge modes in a ribbon geometry. Finally, the anomalous Hall conductivity obtained by us shows plateaus at either e(2)/h or 2e(2)/h corresponding to these topological phases.