Correlated Phases in Spin-Orbit-Coupled Rhombohedral Trilayer Graphene

Recent experiments indicate that crystalline graphene multilayers exhibit much of the richness of their twisted counterparts, including cascades of symmetry-broken states and unconventional superconductivity. Interfacing Bernal bilayer graphene with a WSe$_2$ monolayer 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 field effects, long-range Coulomb repulsion, short-range (Hund's) interactions, and substrate-induced Ising spin-orbit coupling. In addition to generalized Stoner ferromagnets, we find various flavors of intervalley coherent ground states distinguished by their transformation properties under electronic time reversal, $\text{C}_3$ rotations, and an effective anti-unitary symmetry. We pay particular attention to broken-symmetry phases that yield Fermi surfaces compatible with zero-momentum Cooper pairing, identifying promising candidate orders that may support spin-orbit-enhanced superconductivity.