Instabilities of correlated electrons on the honeycomb lattice

Motivated by the possible relevance to graphene we analyze instabilities of electrons on a honeycomb lattice, interacting by Hubbard and longer-ranged interactions. Using a functional renormalization group scheme which takes into account the wavevector-dependence of the interactions throughout the Brillouin zone, we detect the leading ordering tendencies at low temperatures. Near half band filling and for dominant onsite repulsion, a critical minimal interaction strength is required for an instability toward antiferromagnetic order, in support of a previous large-$N$ work of Herbut [Phys. Rev. Lett. 97, 146401 (2006)] which focused on the Dirac points. We also present results for longer-ranged interactions and away from half band filling.