Domain wall competition in the Chern insulating regime of twisted bilayer graphene

We consider magic-angle twisted bilayer graphene (TBG) at filling $\nu =+3$, where experiments have observed a robust quantized anomalous Hall effect. This has been attributed to the formation of a valley- and spin-polarized Chern insulating ground state that spontaneously breaks time-reversal symmetry, and is stabilized by a hexagonal boron nitride (hBN) substrate. We identify three different types of domain wall, and study their properties and energetic selection mechanisms via theoretical arguments and Hartree-Fock calculations adapted to deal with inhomogeneous moir'e systems. We comment on the implications of these results for transport and scanning probe experiments.