Symmetry-driven anisotropic coupling effect in antiferromagnetic topological insulator: Mechanism for a quantum anomalous Hall state with a high Chern number

Antiferromagnetic (AFM) topological insulators (TIs), which host magnetically gapped Dirac-cone surface states and exhibit many exotic physical phenomena, have attracted great attention. Here, we find that the coupled surface states can be intertwined to give birth to a set of $2n$ unique new Dirac cones, dubbed intertwined Dirac cones, through the anisotropic coupling enforced by crystalline $n$-fold ($n=2,3,4,6$) rotation symmetry ${C}_{nz}$ in the presence of a $PT$-symmetry breaking potential, for example, an electric field. Interestingly, we also find that the warping effect further drives the intertwined Dirac-cone state into a quantum anomalous Hall phase with a high Chern number ($C=n$). Then, based on first-principles calculations, we have explicitly demonstrated six intertwined Dirac cones and a Chern insulating phase with a high Chern number ($C=3$) in ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}/{({\mathrm{Bi}}_{2}{\mathrm{Te}}_{3})}_{\mathrm{m}}/{\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ heterostructures, as well as the $C=2$ and $C=4$ phases in HgS and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Ag}}_{2}\mathrm{Te}$ films, respectively. This work discovers the intertwined Dirac-cone state in AFM TI thin films, which reveals a mechanism for designing the quantum anomalous Hall state with a high Chern number and also paves a way for studying highly tunable high-Chen-number flat bands of twistronics.