  The Haldane-Hubbard model is a prime example of the combined effects of band
topology and electronic interaction. We revisit its spinful phase diagram at
half-filling as a consensus on the presence of SU($2$) symmetry is currently
lacking. To start, we utilize the Hartree-Fock mean-field method, which offers
a direct understanding of symmetry breaking through the effective mass term
that can acquire spin dependence. Our results, in agreement with previous
studies, provide an instructive insight into the regime where the Chern number
$C=1$, with only one spin species remaining topological. Besides that, we
numerically study the phase diagram of the Haldane-Hubbard model via a
large-scale infinite-density matrix renormalization group (iDMRG) method. The
phase boundaries are determined by the Chern number and the correlation lengths
obtained from the transfer-matrix spectrum. Unlike previous studies, the iDMRG
method investigates the Haldane-Hubbard model on a thin and infinitely long
cylinder and examines scenarios consistent with the two-dimensional
thermodynamic limit. Here, the phase diagram we obtained qualitatively goes
beyond the Hartree-Fock scope, particularly in the $C=1$ region, and serves as
a quantitative benchmark for further theoretical and experimental
investigations.
