  Using first-principle 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 the extended Hubbard model on the two
dimensional hexagonal lattice at half filling. We determine the phase diagram
in the space of on-site and nearest-neighbor couplings $U$ and $V$ in the
region $V<U/3$, which can be simulated without a fermion sign problem, and find
that a transition from semimetal to a SDW phase occurs at sufficiently large
$U$ for basically all $V$. Tracing the corresponding phase boundary from $V=0$
to the $V=U/3$ line, we find evidence for critical scaling in the Gross-Neveu
universality class for the entire boundary. With rather high confidence we rule
out the existence of the CDW ordered phase anywhere in the range of parameters
considered. We also discuss several improvements of the HMC algorithm which are
crucial to reach these conclusions, in particular the improved fermion action
with exact sublattice symmetry and the complexification of the
Hubbard-Stratonovich field to ensure the ergodicity of the algorithm.
