  We investigate the competition between charge-density-wave (CDW) states and a
Coulomb interaction-driven topological Mott insulator (TMI) in the honeycomb
extended Hubbard model. For the spinful model with on-site ($U$) and
next-nearest-neighbor ($V_2$) Coulomb interactions at half filling, we find two
peculiar six-sublattice charge-density-wave insulating states by using
variational Monte Carlo simulations as well as the Hartree-Fock approximation.
We observe that conventional ordered states always win with respect to the TMI.
The ground state is given in the large-$V_2$ region by a CDW characterized by a
220200 (001122) charge configuration for smaller (larger) $U$, where 0, 1, and
2 denote essentially empty, singly occupied, and doubly occupied sites. Within
the 001122-type CDW phase, we find a magnetic transition driven by an emergent
coupled-dimer antiferromagnet on an effective square lattice of singly occupied
sites. Possible realizations of the found states are discussed.
