  We study the phase diagram of the fermionic Hubbard model on the hexagonal
lattice in the space of on-site and nearest neighbor couplings with
Hybrid-Monte-Carlo simulations. With pure on-site repulsion this allows to
determine the critical coupling strength for spin-density wave formation with
the standard approach of introducing a small mass term, explicitly breaking the
sublattice symmetry. The analogous mass term for charge-density wave formation
above a critical nearest-neighbor repulsion, on the other hand, would introduce
a fermion sign problem. The competition between the two and the phase diagram
in the space of the two coouplings can however be studied in simulations
without explicit sublattice symmetry breaking. Our results compare
qualitatively well with the Hartree-Fock phase diagram. We furthermore
demonstrate how spin-symmetry breaking by the Euclidean time discretization can
be avoided also, when using an improved fermion action based on an exponetial
transfer matrix with exact sublattice symmetry.
