Topological insulators with commensurate antiferromagnetism

We study the topological features of noninteracting 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 antiunitary 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 three-dimensional (3D) AFM insulators, but not for those in lower dimensions, and (iii) in 3D AFM insulators with a nontrivial $Z_2$ 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 topological insulator 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.

Figure 1

AFM configurations that are (a), (b) ${\Theta }_S$ invariant and (c) ${\Theta }_S$ variant. In (a), we plot the typical spin structure of a CuO${}_2$ layer in any parent compound of cuprate superconductors (Ref. 32). In (b), we show the spin structure of vacancy-doped iron-based superconductor Rb${}_2$Fe${}_4$Se${}_5$ (Ref. 33). In (c), we plot the ground state of a classical Heisenberg model on a triangular lattice. The basis vectors of the lattice ${\mathbf{a}}_{1,2}$ and those of the magnetic superlattice ${\mathbf{a}}_{1,2}^M$ are also shown. Note that in (a) and (b), a translation by $\mathbf{D}={\mathbf{a}}_1$ inverts all spins, while in (c) one can not find such a vector.

Figure 2

(a) The Feynman diagram expansion of the self-energy induced by the scattering of AFM field. (b) Represents the free-electron propagator in the momentum space without the AFM field. (c) The vertex representing the scattering by the AFM field of momentum $\mathbf{p}$.

Figure 3

(a) Magnetic structure of a sinusoidal antiferromagnet with wave vector $Q=\pi /5$ described in Eq. (40). (b) The energy splitting of the Kramers pair in a 1D TRS insulator in the presence of AFM shown in (a) at $k=0$ as a function of the spatial period $s$ of the AFM. Red dots correspond to odd and blue empty circles correspond to even $s$'s. (c) Magnetic structure given by Eq. (41) with $s=10$. (d) The energy splitting of the Kramers pair in a 1D TRS insulator in the presence of AFM shown in (c) at $k=0$ plotted against the AFM strength cubed $M^3$.

Figure 4

(a) Different terminations of a 2D AFM insulator with propagation vector $(\pi ,\pi )$, for which the coordinates of sites on the boundary satisfy $y=nx$. (b) The energy separation at the Dirac point of the system described by Eq. (59) with $k_z=0$, $m=1.5$, and $M=0.2$, as a function of the inverse slope of the cut for $n\in \text{odd}$. If $n\in \text{even}$, the energy separation is zero.

Figure 5

(a)–(c) Edge-state dispersion under different parameters without breaking the symmetries. It shows an adiabatic process, from (a) to (c), in which a full gap is open between the edge bands. (d)–(f) Edge and bulk dispersion of a 2D TRS TI described by Eq. (59) subject to AFM given by Eq. (63), in which the edge is along the $y$ axis. From (d) to (f), the bias potential is increased to separate the edge bands without either breaking the symmetry or closing the band gap. In (d), $M=0.2$ (perturbative) and in (e) and (f), $M=0.4$ (nonperturbative).

Figure 6

The dispersion of the central surface band in the MBZ on surface in a 3D AFM TI with magnetic structure given by Eq. (65) with $n=2,4,6,8$, respectively. The spectral weight for the first two layers of the surface is overplotted with the dispersion.

Figure 7

The $\mathbf{k}$-resolved spectral weight on the open surface of a 3D AFM TI with a magnetic structure given in Eq. (65). The spectral weight is plotted for $E=0.3,0.5,0.7$ in rows 1, 2, and 3, respectively. The first column corresponds to $M=0$, that is, a TRS TI; the second one corresponds to $M=0.2$ and $n=4$; the third column corresponds to $M=0.2$ and $n=8$. The color bar shows the distribution of a state on the first two layers of the surface where unity means completely bound to the surface.