Renormalization group approach to chiral symmetry breaking in graphene

We investigate the development of a gapped phase in the field theory of Dirac fermions in graphene with long-range Coulomb interaction. In the large-$N$ approximation, we show that the chiral symmetry is only broken below a critical number of two-component Dirac fermions $N_c=32/{\pi }^2$, that is exactly half the value found in quantum electrodynamics in $2+1$ dimensions. Adopting otherwise a ladder approximation, we give evidence of the existence of a critical coupling at which the anomalous dimension of the order parameter of the transition diverges. This result is consistent with the observation that chiral symmetry breaking may be driven by the long-range Coulomb interaction in the Dirac field theory, despite the divergent scaling of the Fermi velocity in the low-energy limit.

Figure 1

Quantum corrections to the vertex built from $\overline{\psi }\psi$ in the large-$N$ approximation, where the interaction between electrons is taken as the RPA dressed Coulomb potential (thick wavy line).

Figure 2

(Color online) Phase diagram obtained to leading order in the $1/N$ approximation, showing the regime with massless Dirac fermions $\left(m=0\right)$ and the phase with CSB $\left(m\ne 0\right)$.

Figure 3

(Color online) Self-consistent diagrammatic equation for the vertex $\Gamma \left(\mathbf{q};\mathbf{k}\right)$, equivalent to the sum of ladder diagrams built from the iteration of the bare Coulomb interaction (thin wavy line).

Figure 4

(Color online) Plot of the absolute value of the coefficients $c_1^{\left(n\right)}$ in the expansion of $c_1\left(\lambda \right)$ as a power series in the coupling $\lambda$.