Symmetries, interactions and phase transitions on graphene honeycomb lattice

Graphene, a monolayer of graphite, opened a new frontier in physics with reduced dimen- sionality. Due to the Dirac nature of the quasi particles it exhibits interesting experimental phenomena. It is believed that electron-electron interac- tions also play important role in graphene. We derive here a generalized theory of short ranged interactions consistent with the various discrete symmetries present on the lattice. Restrictions on the theory imposed by the atomic limit are also discussed. Within the framework of this model we calculated the beta functions governing the renormalization flow of the couplings to sub-leading order in 1/N. Our calculations show that charge den- sity wave and anti-ferromagnetic quantum critical points are in the Gross-Neveu universality class even beyond mean-field level. Thereafter we use the extended Hubbard model to extract the phase diagram. It shows that the semimetallic ground state is stabilized once we include corrections to sub-leading order in 1/N.