  The impact of local and nonlocal density-density interactions on the
electronic instabilities in the honeycomb lattice is widely investigated. Some
early studies proposed the emergence of interaction-induced topologically
nontrivial phases, but recently, it was denied in several works including
renormalization group calculations with refined momentum resolution. We use the
truncated unity functional renormalization group to study the many-body
instabilities of electrons on the half-filled honeycomb lattice, focusing on
the effect of the exchange interaction. We show that varying the
next-nearest-neighbor repulsion and nearest-neighbor exchange integral can lead
to diverse ordered phases, namely, the quantum spin Hall, the spin-Kekul\'e,
and some spin- and charge-density-wave phases. The quantum spin Hall phase can
be induced by a combination of the ferromagnetic exchange and pair hopping
interactions. Another exotic phase, the spin-Kekul\'e phase, develops in a very
small region of the parameter space considered. We encounter the
three-sublattice charge-density-wave phase in a large part of the parameter
space. It is replaced by the incommensurate charge density wave when increasing
the exchange integral. In order to reduce the computational effort, we derive
the explicit symmetry relations for the bosonic propagators of the effective
interaction and propose a linear-response-based approach for identifying the
form factor of order parameter. Their efficiencies are confirmed by numerical
calculations in our work.
