Fermi velocity renormalization and dynamical gap generation in graphene

We study the renormalization of the Fermi velocity by the long-range Coulomb interactions between the charge carriers in the Dirac-cone approximation for the effective low-energy description of the electronic excitations in graphene at half-filling. Solving the coupled system of Dyson-Schwinger equations for the dressing functions in the corresponding fermion propagator with various approximations for the particle-hole polarization, we observe that Fermi velocity renormalization effects generally lead to a considerable increase of the critical coupling for dynamical gap generation and charge-density-wave formation at the semimetal-insulator transition.

Figure 1

Left: fermion mass function at zero momentum over the inverse coupling $1/\alpha$ together with two fits for values of the coupling close to the critical one (see text for details). Right: momentum dependence of the Fermi velocity dressing function, exemplified for the couplings $\alpha =6.0,7.0,7.8$ and $\alpha =9.0$, below and above the corresponding critical coupling ${\alpha }_c\approx 7.65\left(5\right)$, respectively. Note the logarithmic infrared divergence in the results for $\alpha =6.0,7.0$, whereas for $\alpha =7.8,9.0$ this infrared divergence is screened by the dynamically generated mass. All results are for $N_f=2$ here (see text for further details).

Figure 2

Left: mass dependence of the order parameter for $N_f=2$. Right: large-$N_f$ dependence of the critical coupling.