Recent experimental discovery of flavor symmetry breaking metallic phases in Bernal-stacked bilayer graphene points to the strongly interacting nature of electrons near the top (bottom) of its valence (conduction) band. Superconductivity was also observed in between these symmetry breaking phases when the graphene bilayer is placed under a small in-plane magnetic field or in close proximity to a monolayer ${\mathrm{WSe}}_2$ substrate. Here we address the correlated nature of the band edge electrons and obtain the quantum phase diagram of their many-body ground states incorporating the effect of proximity induced spin-orbit coupling. We find that in addition to the spin/valley flavor polarized half and quarter metallic states, two types of intervalley coherent phases emerge near the phase boundaries between the flavor polarized metals. Both spin-orbit coupling and in-plane magnetic field disfavor the spin-unpolarized valley coherent phase. Our findings suggest possible competition between intervalley coherence and superconducting orders, arising from the intriguing correlation effects in bilayer graphene in the presence of spin-orbit coupling.

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• Strong interaction
• Bilayer graphene
• Flavor symmetries