Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: A renormalization group analysis

Motivated by recent experimental observations of correlated metallic phases and superconductivity in rhombohedral trilayer graphene (RTG), we perform an unbiased study electronic ordering instabilities hole-doped RTG. Specifically, focus on states energetically proximate to Van Hove singularities (VHSs), where a large density promotes different interaction-induced symmetry-breaking orders. To resolve the Fermi surface near VHSs, construct fermionic hot-spot model demonstrate that perpendicular electric field can tune nesting structures surface. Subsequently, apply renormalization group analysis describe low-energy phase diagrams our under both short-range repulsive interactions as well realistic (long-range) Coulomb interactions. Our shows towards either intervalley coherent or superconducting phases. The dominant pairing channel depends crucially nature -- interaction favors spin-singlet $d$-wave for relatively small displacement $i$-wave larger field. We argue diagram RTG be well-understood modeling sum density-density ferromagnetic spin-triplet coherence (IVC) Hund's coupling, while phonon-mediated have negligible effect this system, sharp contrast twisted multilayers.