arXiv2009
The low-energy theory of electrons interacting via repulsive short-range interactions on graphene's honeycomb lattice at half filling is presented. The exact symmetry of the Lagrangian with local quartic terms for the Dirac field dictated by the lattice is D_2 x U_c(1) x (time reversal), where D_2 is the dihedral group, and U_c(1) is a subgroup of the SU_c(2) "chiral" group of the non-interacting Lagrangian, that represents translations in Dirac language. The Lagrangian describing spinless particles respecting this symmetry is parameterized by six independent coupling constants. We show how first imposing the rotational, then Lorentz, and finally chiral symmetry to the quartic terms, in conjunction with the Fierz transformations, eventually reduces the set of couplings to just two, in the "maximally symmetric" local interacting theory. We identify the two critical points in such a Lorentz and chirally symmetric theory as describing metal-insulator transitions into the states with either time-reversal or chiral symmetry being broken. In the site-localized limit of the interacting Hamiltonian the low-energy theory describes the continuous transitions into the insulator with either a finite Haldane's (circulating currents) or Semenoff's (staggered density) masses, both in the universality class of the Gross-Neveu model. The picture of the metal-insulator transition on a honeycomb lattice emerges at which the residue of the quasiparticle pole at the metallic and the mass-gap in the insulating phase both vanish continuously as the critical point is approached. We argue that the Fermi velocity is non-critical as a consequence of the dynamical exponent being fixed to unity by the emergent Lorentz invariance. Effects of long-range interaction and the critical behavior of specific heat and conductivity are discussed.