  We study the quantum many-body ground states of electrons on the half-filled
honeycomb lattice with short- and long-ranged density-density interactions as a
model for graphene. To this end, we employ the recently developed
truncated-unity functional renormalization group (TU-fRG) approach which allows
for a high resolution of the interaction vertex' wavevector dependence. We
connect to previous lattice quantum Monte Carlo (QMC) results which predict a
stabilization of the semimetallic phase for realistic \emph{ab initio}
interaction parameters and confirm that the application of a finite biaxial
strain can induce a quantum phase transition towards an ordered ground state.
In contrast to lattice QMC simulations, the TU-fRG is not limited in the choice
of tight-binding and interaction parameters to avoid the occurrence of a sign
problem. Therefore, we also investigate a range of parameters relevant to the
realistic graphene material which are not accessible by numerically exact
methods. Although a plethora of charge density waves arise under medium-range
interactions, we find the antiferromagnetic spin-density wave to be the
prevailing instability for long-range interactions. We further explore the
impact of an extended tight-binding Hamiltonian with second-nearest neighbor
hopping and a finite chemical potential for a more accurate description of the
band structure of graphene's $p_z$ electrons.
