Tunable van Hove Singularities and Competing Orders in Bernal Bilayer Graphene

Recent experiments on hole-doped Bernal bilayer graphene under strong displacement field discovered that two seemingly separate knobs -- in-plane magnetic field or proximity spin-orbit coupling -- promote superconductivity. Without these knobs, a competing phase featuring high resistivity and non-linear charge transport reminiscent of the charge density wave depinning appears instead. While previous works pointed out various possible mechanisms for the superconductivity, the key question of how two separate knobs are promoting superconductivity as well as the nature of the competing phase have not been clear. Here, we study instabilities arising from repulsive interactions near van Hove singularities in Bernal bilayer graphene through parquet renormalization group. We note that both the in-plane field and the proximity spin-orbit coupling have the effect of lifting the spin degeneracy. This observation opens an angle to study the shared aspect of two knobs. When there is a spin degeneracy, we find that the intra-valley charge density wave wins over superconductivity. Upon lifting the spin degeneracy, superconductivity becomes the primary instability, followed by inter-valley charge density waves. Hence, we propose that the competing intra-valley charge density wave phase suppresses superconductivity in spin-degenerate Bernal bilayer graphene. Our analysis also suggests the spin-degenerate Bernal bilayer graphene as a possible platform for a PDW state and a charge-4e superconductivity.