Metals, fractional metals, and superconductivity in rhombohedral trilayer graphene

Combining mean-field and renormalization group analyses, here we unveil the nature of recently observed superconductivity and parent metallic states in chemically doped rhombohedral trilayer graphene, subject to external electric displacement fields (D) [H. Zhou et al., Nature (London) 598, 434 (2021)]. We argue that close to the charge neutrality, on site Hubbard repulsion favors layer antiferromagnet, which when combined with the D-field induced layer polarization, produces a spin-polarized, but valley-unpolarized half metal, conducive to the nucleation of spin-triplet f-wave pairing (SC2). At larger doping valence bond order emerges as a prominent candidate for isospin coherent paramagent, boosting condensation of spin-singlet Cooper pairs in the s-wave channel (SC1), manifesting a "selection rule" among competing orders. Responses of these paired states to displacement and in-plane magnetic fields show qualitative similarities with experimental observation. With the onset of the quantum anomalous Hall order, the valley degeneracy of half metal gets lifted, forming a quarter metal at lower doping [H. Zhou et al., Nature (London) 598, 429 (2021)].