  We study the many-body instabilities of correlated electrons on the
half-filled honeycomb lattice with enhanced exchange coupling. The system is
described by an extended Hubbard model including the next-nearest-neighbor
Coulomb repulsion ($V_2$) and the nearest-neighbor exchange interaction ($J$).
We use the truncated unity functional renormalization group approach to
determine a schematic ground-state phase diagram with higher resolution in the
parameter space of $V_2$ and $J$. In the absence of the on-site repulsion and
presence of sizable next-nearest-neighbor repulsion and enhanced
nearest-neighbor exchange interaction, we encounter the quantum spin Hall
phase, the spin-Kekul\'{e} phase, and the three-sublattice and the
incommensurate charge-density-wave phases. We propose a scheme for combining
consistently the truncated unity functional renormalization group and the
mean-field approximation, which is distinct from the conventional one that
directly uses the renormalization-group results as an input for the mean-field
calculation. This scheme is used to study in detail the quantum spin Hall
phase, presenting some characteristics like the bulk gap, the Chern number and
the helical edge states.
