  A two-dimensional topological insulator may arise in a centrosymmetric
commensurate N\'{e}el antiferromagnet (AF), where staggered magnetization
breaks both the elementary translation and time reversal, but retains their
product as a symmetry. Fang et al.[Phys. Rev. B 88, 085406 (2013)] proposed an
expression for a $Z_2$ topological invariant to characterize such systems.
Here, we show that this expression does not allow to detect all the existing
phases if a certain lattice symmetry is lacking. We implement numerical
techniques to diagnose topological phases of a toy Hamiltonian, and verify our
results by computing the Chern numbers of degenerate bands, and also by
explicitly constructing the edge states, thus illustrating the efficiency of
the method.
