Full, three-quarter, half, and quarter Wigner crystals in Bernal bilayer graphene

Application of a displacement field opens a gap and enhances the Van-Hove singularities in the band structure of Bernal-stacked bilayer graphene. By adjusting the carrier density so that the Fermi energy lies in the vicinity of these singularities, recent experiments observe a plethora of highly correlated electronic phases including isospin polarized phases and high-resistance states with non-linear electric transport indicative of a possible Wigner crystal. We perform Hartree-Fock calculations incorporating long-range Coulomb interactions and allowing for translational and rotational symmetry breaking. We obtain the displacement field vs. carrier density phase diagram which shows isospin polarized metallic phases tracking the Van-Hove singularity in the valence band. Between these metallic phases we observe regions where the ground state is a Wigner crystal. The isospin polarization of the Wigner crystals tracks the isospin polarization of the nearby metallic phases. Depending on whether we have four, three, two or one isospin flavours, we obtain a full, three-quarter, half or quarter Wigner crystal.