Non-perturbative renormalization group calculation of the quasi-particle velocity and the dielectric function of graphene

Using a non-perturbative functional renormalization group approach we calculate the renormalized quasi-particle velocity $v (k)$ and the static dielectric function $\epsilon ( k )$ of suspended graphene as functions of an external momentum $k$. Our numerical result for $v (k )$ can be fitted by $v ( k ) / v_F = A + B \ln ( \Lambda_0 / k )$, where $v_F$ is the bare Fermi velocity, $\Lambda_0$ is an ultraviolet cutoff, and $A = 1.37$, $B =0.51$ for the physically relevant value ($e^2/v_F =2.2$) of the coupling constant. In contrast to calculations based on the static random-phase approximation, we find that $\epsilon (k )$ approaches unity for $k \rightarrow 0$. Our result for $v (k )$ agrees very well with a recent measurement by Elias et al. [Nat. Phys. 7, 701 (2011)].