Classification of massive and gapless phases in bilayer graphene

I here classify of all the fully gapped massive and the gapless phases in bilayer graphene. The effective low-energy theory in bilayer graphene is constructed, and various discrete and continuous symmetries of the noninteracting system is analyzed. Spinless fermions, placed in a quantizing magnetic field, are considered. The quantum anomalous Hall insulator is properly defined. Constructing a particle-hole doubled 16 component Nambu-Dirac spinor, I recognize all the possible fully gapped and the gapless states, which, on the other hand, split the parabolic dispersion into two anisotropic Dirac-like conical ones. A thorough symmetry analysis of all the ordered states is performed. Altogether there are eight insulating and four superconducting phases in bilayer graphene, that can lead to a fully gapped spectrum. Among the gapped superconductors, three are spin singlet, which include uniform s-wave and two spatially inhomogeneous, translational symmetry-breaking Kekule superconductors. The triplet pairing exhibits an f-wave symmetry. Besides the gapped phases, there are eight semimetallic and eight gapless superconducting states in total, available for fermions to condense into. I also find interesting gapless superconducting states, which break the translational symmetry, dubbed "gapless Fulde-Farrell-Larkin-Ovchinikov" superconductors. I also discuss the role of the Coulomb interaction, and propose various experimental tools to determine the nature of the underlying ordered states.