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 nu
merical 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)].
We study the role of long-range electron-electron interactions in a system of two-dimensional anisotropic Dirac fermions, which naturally appear in uniaxially strained graphene, graphene in external potentials, some strongly anisotropic topological i
nsulators, and engineered anisotropic graphene structures. We find that while for small interactions and anisotropy the system restores the conventional isotropic Dirac liquid behavior, strong enough anisotropy can lead to the formation of a quasi-one dimensional electronic phase with dominant charge order (anisotropic excitonic insulator).
