Incommensurate inter-valley coherent states in ABC graphene: collective modes and superconductivity

Recent experiments in ABC trilayer graphene detected superconductivity on the border of a phase transition to a symmetry-broken phase. In this work, we use unrestricted Hartree-Fock to study the nature of this phase. We find a close competition between two incommensurate inter-valley coherent (IVC) phases: an IVC crystal where the ordering occurs at multiple wavevectors, and an IVC spiral with a single ordering wavevector. Focusing on one of the regimes where superconductivity is observed experimentally, we find a continuous (or very weakly first order) transition between a half metallic phase to an IVC crystal, followed by a first-order transition into an IVC spiral. Using time-dependent Hartree-Fock, we study the collective mode spectrum in the half-metalic phase. We find a soft inter-valley mode that can mediate superconductivity in a narrow sliver of density near the continuous transition, with a Tc that can reach a few hundreds of mK and a sign-changing s-wave order parameter. The spin stiffness in the half metal phase is found to be surprisingly low, of the order of a few degrees Kelvin.