Instabilities, competing orders and emergent criticality of interacting Dirac fermions

I will discuss several non-universal and universal aspects of interacting Dirac fermions with a focus towards fermions on the honeycomb lattice. First, I will present our recent study of interaction-driven instabilities of electrons on the half-filled honeycomb lattice as a model for graphene employing realistic tight-binding and Coulomb interaction parameters. To this end, we employ an implementation of functional renormalization group equations allowing for a high-definition resolution of the interaction vertex' wave-vector dependence. In contrast to QMC simulations which have to avoid the occurrence of a sign problem, our approach is not limited in the choice of hopping and interaction parameters. Besides the antiferromagnetic ground state, we find other instabilities to become leading, e.g. an incommensurate charge density wave phase and a novel extended s-wave pairing state. By means of a low-energy effective model, I will further discuss effects of the competition of the antiferromagnetic and charge density wave order parameters. Finally, I will study the ordering transition of honeycomb fermions to a Kekule valence bond solid in terms of an effective model of Dirac fermions coupled to a Z3 order parameter field. The Landau-Ginzburg paradigm suggests a discontinuous phase transition due to the presence of cubic terms in the free energy. I will argue that scaling corrections due to fermionic quantum fluctuations can render the phase transition continuous.