Theory of interacting electrons on the honeycomb lattice

The general 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 four-component field dictated by the lattice is identified as ${D}_{2}\ifmmode\times\else\texttimes\fi{}{U}_{c}(1)\ifmmode\times\else\texttimes\fi{}\text{time}$ reversal, where ${D}_{2}$ is the dihedral group, and ${U}_{c}(1)$ is a subgroup of the ${\text{SU}}_{c}(2)$ ``chiral'' group of the noninteracting Lagrangian that represents translations in Dirac language. The Lagrangian describing spinless particles respecting this symmetry is parametrized 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. The latter is proposed to govern the continuous transition in both the Thirring and Nambu-Jona-Lasinio models in $2+1$ dimensions and with a single Dirac field. In the site-localized ``atomic'' limit of the interacting Hamiltonian, under the assumption of emergent Lorentz invariance, 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 simple 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. In contrast to these two critical quantities, we argue that the Fermi velocity is noncritical as a consequence of the dynamical exponent being fixed to unity by the emergent Lorentz invariance near criticality. Possible effects of the long-range Coulomb interaction and the critical behavior of the specific heat and conductivity are discussed.

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