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

Understand

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 $1.03 < \alpha_c < 1.08$ in the case $N=2$ or for $N < N_c$ where $N_c$ is such that $\alpha_c \to \infty$ and takes the values $3.24 < N_c < 3.36$.
• The striking feature of these results is that they are in good agreement with values found in models with instantaneous Coulomb interaction.