  We study the spinful fermionic Haldane-Hubbard model at half filling using a
combination of quantum cluster methods: cluster perturbation theory (CPT), the
variational cluster approximation (VCA), and cluster dynamical mean-field
theory (CDMFT). We explore possible zero-temperature phases of the model as a
function of on-site repulsive interaction strength and next-nearest-neighbor
hopping amplitude and phase. Our approach allows us to access the regime of
intermediate interaction strength, where charge fluctuations are significant
and effective spin model descriptions may not be justified. Our approach also
improves upon mean-field solutions of the Haldane-Hubbard model by retaining
local quantum fluctuations and treating them nonperturbatively. We find a
correlated topological Chern insulator for weak interactions and a
topologically trivial N\'eel antiferromagnetic insulator for strong
interactions. For intermediate interactions, we find that topologically
nontrivial N\'eel antiferromagnetic insulating phases and/or a topologically
nontrivial nonmagnetic insulating phase may be stabilized.
