Topological Insulators with Commensurate Antiferromagnetism

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.