  We study the topological features of non-interacting insulators subject to an
antiferromangetic (AFM) Zeeman field, or AFM insulators, the period of which is
commensurate with the lattice period. These insulators can be classified by the
presence/absence of an emergent anti-unitary symmetry: the combined operation
of time-reversal and a lattice translation by vector $\mathbf{D}$. For AFM
insulators that preserve this combined symmetry, regardless of any details in
lattice structure or magnetic structure, we show that (i) there is a new type
of Kramers' degeneracy protected by the combined symmetry; (ii) a new $Z_2$
index may be defined for 3D AFM insulators, but not for those in lower
dimensions and (iii) in 3D AFM insulators with a non-trivial $Z_2$ index, there
are odd number of gapless surface modes if and only if the surface termination
also preserves the combined symmetry, but the dispersion of surface states
becomes highly anisotropic if the AFM propagation vector becomes small compared
with the reciprocal lattice vectors. We numerically demonstrate the theory by
calculating the spectral weight of the surface states of a 3D TI in the
presence of AFM fields with different propagation vectors, which may be
observed by ARPES in Bi$_2$Se$_3$ or Bi$_2$Te$_3$ with induced
antiferromagnetism.
