Motivated by recent experiments [H. Zhou et al., Science 375, 774 (2022) and S. C. de la Barrera et al., arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by a Hubbard repulsion-driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion-driven quantum anomalous Hall (QAH) orders, lifts the fourfold degeneracy of electronic bands, yielding a quarter metal for small doping, as also observed in ABC trilayer graphene. With the disappearance of the QAH order, electronic bands recover twofold valley degeneracy, thereby forming a conventional or compensated (with majority and minority carriers) half metal at moderate doping, depending on the relative strength of LP and LAF. At even higher doping and for a weak $D$ field only LAF survives and the Fermi surface recovers fourfold degeneracy. We also show that a pure repulsive electronic interaction mediated triplet $f$-wave pairing emerges from a parent correlated nematic liquid or compensated half metal when an in-plane magnetic field is applied to the system.

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• Magnetism
• Nematic order
• Quantum anomalous Hall effect
• Superconductivity
• Graphene
• Honeycomb lattice
• Mean field theory
• Renormalization group