  An antiferromagnetic Chern insulator (AFCI) can exist if the effect of the
time-reversal transformation on the electronic state cannot be compensated by a
space group operation. The AFCI state with collinear magnetic order is already
realized in noncentrosymmetric honeycomb structures through the
Kane-Mele-Hubbard model. In this paper, we demonstrate the existence of the
collinear AFCI in a square lattice model which preserves the inversion
symmetry. Our study relies on the time-reversal-invariant
Harper-Hofstadter-Hubbard model extended by a next-nearest-neighbor hopping
term including spin-orbit coupling and a checkerboard potential. We show that
an easy $z$-axis AFCI appears between the band insulator at weak and the easy
$xy$-plane AF Mott insulator at strong Hubbard repulsion provided the
checkerboard potential is large enough. The close similarity between our
results and the results obtained for the noncentrosymmetric Kane-Mele-Hubbard
model suggests the AFCI as a generic consequence of spin-orbit coupling and
strong electronic correlation which exists beyond a specific model or lattice
structure. An AFCI with the electronic and the magnetic properties originating
from the same strongly interacting electrons is promising candidate for a
strong magnetic blue shift of the charge gap below the N\'eel temperature and
for realizing the quantum anomalous Hall effect at higher temperatures so that
applications for data processing become possible.
