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

TL;DR

This paper investigates the critical behavior of dynamical fermion gap generation in graphene using reduced QED$_{4,3}$, deriving an exact gap equation and identifying conditions for gap formation consistent with experimental semi-metallic behavior.

Contribution

It derives an exact gap equation for reduced QED$_{4,3}$ and analyzes the critical coupling constants for fermion gap generation, providing insights into graphene's electronic properties.

Findings

Dynamical gap generated for coupling $\alpha > \alpha_c$ with $1.03 < \alpha_c < 1.08$

Critical fermion number $N_c$ between 3.24 and 3.36

Results align with models using instantaneous Coulomb interaction

Abstract

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. At the fixed point: $\alpha = 1/137 \ll \alpha_c$, and the system is therefore in the semi-metallic regime in accordance with experiments.