Staggered Potential and Elliptical Light Driven Topological Phase Transitions in $\alpha$-$\mathcal{T}_{3}$ Lattice

Abstract

We theoretically investigate the influence of hexagonal boron nitride (h-BN) on the electronic properties of an $\alpha$-$\text{T}_3$ lattice driven by an off-resonant elliptically polarized light field. The staggered potential $M$ breaks the sublattice inversion symmetry, transforming the initial semimetal into a trivial insulator with Chern number $C = 0$. We identify a fundamental geometric singularity at $\alpha = 1/\sqrt{2}$, independent of $M$, where the valley-resolved lower-gap threshold diverges, bounding a finite topological window where conduction--flat band inversion yields a Chern insulator with $C = 1$ carried by the flat band. Increasing the drive further closes the lower gap at the $K'$ valley, transferring the index to the valence band so that the flat band becomes trivial while the system remains $C = 1$. For $\alpha > 1/\sqrt{2}$ the lower gap closes at finite intensity, allowing a transition to $C = 2$ as the dice limit ($\alpha = 1$) is approached. The topological phases are characterized by quantized anomalous Hall plateaus at $\sigma_{xy} = e^2/h$ ($C = 1$) and $\sigma_{xy} = 2e^2/h$ ($C = 2$). The $C = 1$ plateau sits in a narrow gap and is the most fragile, while the $C = 2$ plateau is protected by a wider gap and remains robust to room temperature. A highly asymmetric thermoelectric Seebeck response further serves as an experimental fingerprint of each phase, providing a realistic framework for realizing stable high-Chern-number phases in substrate-supported $\alpha$-$\text{T}_3$ materials.