  We investigate the quantum many-body instabilities of the extended Hubbard
model for spinless fermions on the honeycomb lattice with repulsive
nearest-neighbor and 2nd nearest-neighbor density-density interactions. Recent
exact diagonalization and infinite density matrix renormalization group results
suggest that a putative topological Mott insulator phase driven by the 2nd
nearest-neighbor repulsion is suppressed, while other numerically exact
approaches support the topological Mott insulator scenario. In the present
work, we employ the functional renormalization group (fRG) for correlated
fermionic systems. Our fRG results hint at a strong suppression of the
scattering processes stabilizing the topological Mott insulator. From analyzing
the effects of fermionic fluctuations, we obtain a phase diagram which is the
result of the competition of various charge ordering instabilities.
