  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
MnBi$_2$Te$_4$$/$(Bi$_2$Te$_3$)$_{\mathrm{m}}/$MnBi$_2$Te$_4$ heterostructures,
as well as the $C=2$ and $C=4$ phases in HgS and $\alpha$-Ag$_2$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.
