Even though in its pristine state graphene behaves like a semi-metal, fermions can, in principle find themselves in a variety of insulating phases, depending on relative magnitude of different components of the finite ranged repulsive interaction. For example, a gapped insulating phase with staggered pattern either in electronic density or magnetization can be realized at sufficiently strong nearest-neighbor or onsite Hubbard interaction, respectively. Motivated by these possibilities we established a general low-energy theory of electrons, interacting via short-ranged interactions on graphene's honeycomb lattice at filling one-half.

We derived the Lagrangian with local four-fermion interactions consistent with the symmetries dictated by the lattice and the time reversal symmetry. Moreover, subjecting the interacting theory first to the rotational, then Lorentz and finally chiral symmetry, in conjunction with the Fierz identity, we managed to capture the metal-insulator transitions into the states either with broken time reversal or chiral symmetry. In this talk I will show that both the transitions belong to the Gross-Neveu universality class and the pseudo relativistic invariance is restored near the quantum criticality. I will mention various consequences of emergent symmetries on the interaction induced gap. Furthermore, the emergence of the Lorentz symmetry in the vicinity of all critical points will be established, even when the interacting theory enjoys the rotational symmetry only.

On the other hand, if a short-ranged interaction has an attractive component, a plethora of superconducting states becomes available for the Dirac fermions to condense. During this talk, I will assume that the dominant component of the finite-ranged attractive interaction is the nearest-neighbor one and the chemical potential is tuned right at the filling one-half. Under these assumptions, I will propose a spatially non-uniform superconducting state, that breaks the translational invariance of the lattice into Kekule pattern, as a electronic variation ground state, at least, at the mean-field level.