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

We use a nonperturbative functional renormalization group approach to calculate the renormalized quasiparticle velocity $v (k)$ and the static dielectric function $\epsilon(k)$ of suspended graphene as function of an external momentum $k$. We fit our numerical result for $v(k)$ to $v(k)/v_F = A + B \ln(\Lambda_0/k)$, 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 $\textit{stark}$ 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 $\textit{et al.}$ [Nat. Phys. $\textbf{7}$, 701 (2011)]. With in the same approximation, we also explore an alternative scheme in order to understand the true nature of the low energy (momentum) behavior in graphene.