The interplay between superconductivity and magnetism is thought to play a role in a variety of unconventional superconductors, including cuprates, heavy fermions, and moire graphene. Here, I will describe a new venue for examining this interplay by tuning the chemical potential through a van Hove singularity in simple allotropes of graphene, in particular rhombohedral trilayer and Bernal bilayer. In both systems, applying a perpendicular electric field gaps out a series of low-energy Dirac nodes, leading to large divergences in the density of states at low densities. Using both transport and compressibility measurements, we find that this regime is characterized by a cascade of phase transitions between states of differing fermi surface degeneracy. These include quarter- and half-metals with only one or two occupied (out of a possible four) combined spin- and valley flavors, as well as a variety of states showing partial polarization within the spin- and valley-isospin space. Most surprisingly, superconductivity arises near a number of phase boundaries. In the trilayer, we observe two superconducting states for hole doping; one arises from a normal state that preserves the spin and valley symmetry, and is suppressed by in-plane magnetic fields in accordance with the Clogston-Chandrasekhar limit, while the other arises from a full spin polarized half metallic state and is not affected by in plane magnetic fields. In bilayer graphene, superconductivity is not observed at B=0, but emerges only above a critical field in plane field, consistent with a magnetic field induced transition into a spin polarized ferromagnetic state with a superconducting ground state. I will lay out the many outstanding theoretical puzzles in these systems, as well as experimental opportunities enabled by the exceptionally high sample quality.
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