Critical behaviour of reduced QED${}_{4,3}$ and dynamical fermion gap generation in graphene

The dynamical generation of a fermion gap in graphene is studied at the infra-red Lorentz-invariant fixed point where the system is described by an effective relativistic-like field theory: reduced QED${}_{4,3}$ with $N$ four component fermions ($N=2$ for graphene), where photons are $(3+1)$-dimensional and mediate a fully retarded interaction among $(2+1)$-dimensional fermions. A correspondence between reduced QED${}_{4,3}$ and QED${}_3$ allows us to derive an exact gap equation for QED${}_{4,3}$ up to next-to-leading order. Our results show that a dynamical gap is generated for $\alpha >{\alpha }_c$ where $1.03 < \alpha_c < 1.08$inthecase$N=2$orfor$N < N_c$where$N_c$issuchthat$\alpha_c \to \infty$andtakesthevalues$3.24 < N_c < 3.36$. The striking feature of these results is that they are in good agreement with values found in models with instantaneous Coulomb interaction. At the fixed point: $\alpha = 1/137 \ll \alpha_c$, and the system is therefore in the semi-metallic regime in accordance with experiments.