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

TL;DR

This paper explores how lattice geometry and Brillouin zone modifications in twisted bilayer $ extit{ extbf{ extit{ extbf{ extalpha}}}}$-$T_3$ systems lead to complex topological band structures, revealing phase transitions and the impact of quantum interference on flat bands.

Contribution

It uncovers the emergence of topologically non-trivial sub-bands in twisted bilayer $ extit{ extbf{ extalpha}}$-$T_3$ lattices and analyzes their phase transitions and flatness dependence.

Findings

A trivial sub-band near charge neutrality

Emergence of a topologically non-degenerate sub-band away from neutrality

Multiple topological phase transitions as parameters vary

Abstract

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.