  Using state of the art Hybrid-Monte-Carlo (HMC) simulations we carry out an
unbiased study of the competition between spin-density wave (SDW) and
charge-density wave (CDW) order in suspended graphene. We determine that the
realistic inter-electron potential of graphene must be scaled up by a factor of
roughly 1.6 to induce a semimetal-SDW phase transition and find no evidence for
CDW order. A study of critical properties suggests that the universality class
of the three-dimensional chiral Heisenberg Gross-Neveu model with two fermion
flavors, predicted by renormalization group studies and strong-coupling
expansion, is unlikely to apply to this transition. We propose that our results
instead favor an interpretation in terms of a conformal phase transition. In
addition, we describe a variant of the HMC algorithm which uses exact fermionic
forces during molecular dynamics trajectories and avoids the use of
pseudofermions. Compared to standard HMC this allows for a substantial increase
of the integrator stepsize while achieving comparable Metropolis acceptance
rates and leads to a sizable performance improvement.
