| Summary: | We analyze the pairing instabilities for fermions on hexagonal lattices (both honeycomb and triangular ones) in a wide range of fermionic densities ranging from van Hove density at which a single large Fermi surface splits into two disconnected Fermi pockets, to a density at which disconnected pockets shrink to Fermi points (half-filling for a honeycomb lattice and full filling for a triangular lattice). We argue that for a generic doping in this range, superconductivity at weak coupling is of Kohn-Luttinger type, and, due to the presence of electronic interactions beyond on-site repulsion, is a threshold phenomenon, with superconductivity emerging only if the attraction generated by the Kohn-Luttinger mechanism exceeds the bare repulsion in some channel. For disconnected Fermi pockets, we predict that Kohn-Luttinger superconductivity, if it occurs, is likely to be f wave. While the Kohn-Luttinger analysis is adequate over most of the doping range, a more sophisticated analysis is needed near van Hove doping. We treat van Hove doping using a parquet renormalization group, the equations for which we derive and analyze. Near this doping level, superconductivity is a universal phenomenon, arising from any choice of bare repulsive interactions. The strongest pairing instability is into a chiral d-wave state (d + id). At a truly weak coupling, the strongest competitor is a spin-density-wave instability, however, d-wave superconductivity still wins. Moreover, the feedback of the spin density fluctuations into the Cooper channel significantly enhances the critical temperature over the estimates of the Kohn-Luttinger theory. We analyze renormalization group equations at stronger couplings and find that the main competitor to d-wave superconductivity away from weak coupling is actually ferromagnetism. We also discuss the effect of the edge fermions and show that they are unimportant in the asymptotic weak coupling limit, but may give rise to, e.g., a charge-density-wave order at moderate coupling strengths.
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