Moiré fractional Chern insulators. III. Hartree-Fock phase diagram, magic angle regime for Chern insulator states, role of moiré potential, and Goldstone gaps in rhombohedral graphene superlattices

We investigate in detail the ν=+1 displacement-field-tuned interacting phase diagram of L=3,4,5,6,7 layer rhombohedral graphene aligned to hBN (RLG/hBN). Our calculations account for the 3D nature of the Coulomb interaction, the inequivalent stacking orientations ζ=0,1, the effects of the filled valence bands, and the choice of "interaction scheme"for specifying the many-body Hamiltonian. We show that the latter has a dramatic impact on the Hartree-Fock phase boundaries and the properties of the phases, including for pentalayers (R5G/hBN) with large displacement field D, where recent experiments observed a Chern insulator at ν=+1 and fractional Chern insulators for ν<1. In this large D regime, the low-energy conduction bands are polarized away from the aligned hBN layer, making them well described by the folded bands of moiréless rhombohedral graphene at the noninteracting level. Despite this, the filled valence bands develop moiré-periodic charge density variations, which can generate an effective moiré potential, thereby explicitly breaking the approximate continuous translation symmetry in the conduction bands and leading to contrasting electronic topology in the ground state for the two stacking arrangements. Within time-dependent Hartree-Fock theory, we further characterize the strength of the moiré pinning potential in the Chern insulator phase by computing the low-energy q=0 collective mode spectrum, where we identify competing gapped pseudophonon and valley magnon excitations. Our results emphasize the importance of careful examination of both the single-particle and interaction model for a proper understanding of the correlated phases in RLG/hBN.