Hartree-Fock study of spin-orbit-coupled rhombohedral trilayer graphene

Recent experiments indicate that Bernal bilayer graphene and rhombohedral trilayer graphene exhibit much of the richness of their twisted counterparts, including a cascade of broken-symmetry states and superconductivity. Interestingly, interfacing Bernal bilayers with WSe2 was shown to dramatically enhance superconductivity—suggesting that proximity-induced spin-orbit coupling plays a key role in promoting Cooper pairing. Motivated by this observation, we study the phase diagram of spin-orbit-coupled rhombohedral trilayer graphene via self-consistent Hartree-Fock simulations, elucidating the interplay between displacement fields, long-range Coulomb repulsion, short-range Hund's coupling, and substrate-induced Ising spin-orbit interaction. We pay particular attention to broken-symmetry phases that yield band structures compatible with zero-momentum Cooper pairing, with the goal of identifying regimes that potentially support spin-orbit enhanced superconductivity as found in bilayers.