Non-coplanar magnetism, topological density wave order and emergent symmetry at half-integer filling of moir\'{e} Chern bands

Twisted double- and mono-bilayer graphene are graphene-based moir\'e materials hosting strongly correlated fermions in a gate-tunable conduction band with a topologically non-trivial character. Using unbiased exact diagonalization complemented by unrestricted Hartree-Fock calculations, we find that the strong electron-electron interactions lead to a non-coplanar magnetic state, which has the same symmetries as the tetrahedral antiferromagnet on the triangular lattice and can be thought of as a skyrmion lattice commensurate with the moir\'e scale, competing with a set of ferromagnetic, topological charge density waves featuring an approximate emergent O(3) symmetry, "rotating" the different charge density wave states into each other. Direct comparison with exact diagonalization reveals that the ordered phases are accurately described within the unrestricted Hartree-Fock approximation. Exhibiting a finite charge gap and Chern number $|C|=1$, the formation of charge density wave order which is intimately connected to a skyrmion lattice phase is consistent with recent experiments on these systems.