FIG. 10.

Haldane model. (a) Haldane lattice geometry showing overall honeycomb lattice structure with the addition of next-nearest-neighbor hopping with Aharonov-Bohm tunneling phases ϕ_AB. The shaded (blue) box marks the two sites comprising a single unit cell. We also show the deformation of the underlying lattice to the brick-wall geometry, used to plot the dispersions with q_xa = q · a₁ and q_ya = q · a₂. (b) Haldane model phase diagram showing the two topological lobes immersed in a nontopological background. (c) Band structure computed with only nearest-neighbor tunneling [black solid lines in (a), showing the familiar pair of Dirac points from this “brick-wall” lattice. (d) Band structure computed in the topological phase at the marked red square in (b), with ϕ_AB = π/2 and “tilt” $\mathrm{\Delta }=0,J^{\prime }=J/10$. (e) Band structure at the topological transition, marked by the blue circle in (b), with ϕ_AB = π/2 and $\mathrm{\Delta }=-3\sqrt{3}{J}^{\prime },$ showing the formation of a single Dirac point.