Edge modes, Hall conductivity and topological features of a band deformed Dice lattice

The evolution of the topological nature of a Chern insulator induced by the deformation of the bandstructure has been studied on a dice lattice. Such band deformation can be achieved by tuning the hopping amplitude (say, t₁) along one of the three neighbors (between A - B sublattices) compared to the other two (which we may denote as t). Further, the time-reversal symmetry of the system is broken by a complex second neighbour hopping. For the isotropic case, that is, for t₁ = t, the system shows gap at the two Dirac points, namely, the K and K^′ points along with the appearance of a zero energy flat band. As one increases t₁, the flat bands get distorted and the band extrema, either at K or at K^′ point are shifted. At a particular value of t₁, namely, t₁ ≈ 1.7t, the shifted band extrema points touch the distorted flat band, and hence the spectral gap vanishes. Beyond this value (that is, for t₁ >1.7t), the gap reopens. A closer inspection of the topological properties of such a manipulated bandstructure reveals that a phase transition occurs at the critical point (t₁ = 1.7t), beyond which the system continues to behave as a band insulator. The topological robustness is demonstrated via computing the edge modes, anomalous Hall conductivity and the Chern number. The latter aids in arriving at the phase diagram in the relevant parameter space, which conclusively shows the vanishing of the Chern insulating phase with Chern number, |C| = 2 to a trivial insulator with |C| = 0 at the critical point through a gap closing transition. Further, to decipher the properties of the edge modes, we study a semi-infinite nanoribbon that yields a pair of chiral edge modes at each edge for t₁ < 1.7t. The results receive robust support from the phase diagram obtained by us. Finally, the vanishing of the plateaus at 2e²/h in the Hall conductivity provides support to the topological phase transition occurring at t₁ = 1.7t.