Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice

We investigate an interesting interplay of destructive interference due to lattice geometry and band folding due to enlargement of the Brillouin zone in generating and subsequently modifying the band topology in a twisted bilayer $\alpha$-$T_3$ system. The pronounced degeneracy of the emergent flat band in the dice limit of the $\alpha$-$T_3$ lattice is removed on alignment with h-BN layers, resulting in the formation of sub-bands with varying topological characteristics. Remarkably, while the sub-band near charge neutrality exhibits a trivial behavior, a topologically non-degenerate singular sub-band emerges away from charge neutrality. The topological band remains isolated from the rest of the bands for a substantial area of the $\alpha - \theta$ plane (where $\alpha$ and $\theta$ correspond to the hopping ratio and twist angle respectively) while exhibiting multiple phase transitions as a function of the aforementioned parameters via hybridization with its nearest bands. We study the evolution of the hybrid Wannier charge center and the Chern number to characterize the different emergent topological phases. Finally, the degree of flatness of the topological band is studied as a function of both $\alpha$ and $\theta$ to explicitly show the influence of quantum interference and band folding on the width of the topological band.