Topological properties of nearly flat bands in bilayer \(\alpha-\mathcal{T}3\) lattice

We study the effect of Haldane flux in the bilayer \(\alpha\)-\(\mathcal{T}_3\) lattice system, considering possible non-equivalent, commensurate stacking configurations with a tight-binding formalism. The bilayer \(\alpha\)-\(\mathcal{T}_3\) lattice comprises six sublattices in a unit cell, and its spectrum consists of six bands. In the absence of Haldane flux, threefold band crossings occur at the two Dirac points for both valence and conduction bands. The introduction of Haldane flux in a cyclically stacked bilayer \(\alpha\)-\(\mathcal{T}_3\) lattice system separates all six bands, including two low-energy, corrugated nearly flat bands, and assigns non-zero Chern numbers to each band, rendering the system topological. We demonstrate that the topological evolution can be induced by modifying the hopping strength between sublattices with the scaling parameter \(\alpha\) in each layer. In the dice lattice limit (\(\alpha = 1\)) of the Chern-insulating phase, the Chern numbers of the three pairs of bands, from low energy to higher energies, are \(\pm 2\), \(\pm 3\), and \(\pm 1\). Interestingly, a continuous change in the parameter \(\alpha\) triggers a topological phase transition through band crossings between the two lower energy bands. These crossings occur at different values for the conduction and valence bands and depend further on the next nearest neighbor (NNN) hopping strength. At the transition point, the Chern numbers of the two lower conduction and valence bands change discontinuously from \(\pm 2\) to \(\pm 5\) and \(\pm 3\) to \(0\), respectively, while leaving the Chern number of the third band intact.