Extended Hubbard model in undoped and doped monolayer and bilayer graphene: Selection rules and organizing principle among competing orders

Performing a leading-order renormalization group analysis, here we compute the effects of generic local or short-range electronic interactions in monolayer and the Bernal bilayer graphene. Respectively in these two systems, gapless chiral quasiparticles display linear and biquadratic band touching, leading to linearly vanishing and constant density of states. Consequently, the former system remains stable for weak enough local interactions and supports a variety of ordered phases only beyond a critical strength of interactions. By contrast, ordered phases can nucleate for sufficiently weak interactions in bilayer graphene. By tuning the strength of all symmetry allowed local interactions, we construct various cuts of the phase diagram at zero and finite temperature and chemical doping. Typically, at zero doping, insulating phases (such as charge density wave, antiferromagnet, quantum anomalous, and spin Hall insulators) prevail at the lowest temperature, while gapless nematic or smectic liquids stabilize at higher temperatures. On the other hand, at finite doping, the lowest temperature ordered phase is occupied by a superconductor. Besides anchoring such an organizing principle among the candidate ordered phases, we also establish a selection rule between them and the interaction channel responsible for the breakdown of linear or biquadratic chiral nodal Fermi liquid. In addition, we also demonstrate the role of the normal state band structure in selecting the pattern of symmetry breaking from a soup of preselected incipient competing orders. As a direct consequence of the selection rule, while an antiferromagnetic phase develops in undoped monolayer and bilayer graphene, the linear (biquadratic) band dispersion favors condensation of a spin-singlet nematic (translational symmetry breaking Kekule) superconductor in doped monolayer (bilayer) graphene, when the on-site Hubbard repulsion dominates in these systems. On the other hand, nearest-neighbor (next-nearest-neighbor) repulsion accommodates charge density ...