1/Nexpansion in correlated graphene

Abstract

We examine the $1/N$ expansion, where $N$ is the number of two-component Dirac fermions for Coulomb interactions in graphene with a gap of magnitude $\ensuremath{\Delta}=2m$. We find that for $N\ensuremath{\alpha}⪢1$, where $\ensuremath{\alpha}$ is graphene's ``fine-structure constant,'' there is a crossover as a function of distance $r$ from the usual three-dimensional Coulomb law, $V(r)\ensuremath{\sim}1/r$, to a two-dimensional Coulomb interaction, $V(r)\ensuremath{\sim}\text{ln}(N\ensuremath{\alpha}/mr)$, for ${m}^{\ensuremath{-}1}⪡r⪡{m}^{\ensuremath{-}1}N\ensuremath{\alpha}/6$. This effect reflects the weak ``confinement'' of the electric field in the graphene plane. The crossover also leads to unusual renormalization of the quasiparticle velocity and gap at low momenta. We also discuss the differences between the interaction potential in gapped graphene and usual quantum electrodynamics for different coupling regimes.