  We theoretically study a spin Hamiltonian for spin-orbit-coupled ferromagnets
on the honeycomb lattice. We find that the effective Hamiltonian for magnons, a
quanta of spin-wave excitations from ordered states, is equivalent to the
Haldane model for electrons in the honeycomb lattice, which indicates the
nontrivial topology of the magnon bands and the existence of the associated
edge state. We also take the Schwinger boson (or bosonic spinon)
representations of spins that are applicable both to ordered and disordered
phases. We show that the spinon bands within the mean-field treatment are
topologically nontrivial, which causes a spin Nernst effect, by adopting the
properties of the Kane-Mele model. A feasible experimental realization of the
spin Hamiltonian is proposed.
