  Interacting fermions on the half-filled honeycomb lattice with short-range
repulsions have been suggested to host a variety of interesting many-body
ground states, e.g., a topological Mott insulator. A number of recent studies
of the spinless case in terms of exact diagonalization, the infinite density
matrix renormalization group and the functional renormalization group, however,
indicate a suppression of the topological Mott insulating phase in the whole
range of interaction parameters. Here, we complement the previous studies by
investigating the quantum many-body instabilities of the physically relevant
case of spin-1/2 fermions with onsite, nearest-neighbor and
second-nearest-neighbor repulsion. To this end, we employ the multi-patch
functional renormalization group for correlated fermions with refined momentum
resolution observing the emergence of an antiferromagnetic spin-density wave
and a charge-density wave for dominating onsite and nearest-neighbor
repulsions, respectively. For dominating second-nearest neighbor interaction
our results favor an ordering tendency towards a charge-modulated ground state
over the topological Mott insulating state. The latter evades a stabilization
as the leading instability by the additional onsite interaction.
