Glide-symmetric magnetic topological crystalline insulators with inversion symmetry

Heejae Kim, Ken Shiozaki, and Shuichi Murakami

Phys. Rev. B 100, 165202 – Published 8 October 2019

Abstract

It is known that three-dimensional magnetic systems with glide symmetry can be characterized by a $Z_{2}$ topological invariant together with the Chern number associated with the normal vector of the glide plane, and they are expressed in terms of integrals of the Berry curvature. In the present paper, we study the fate of this topological invariant when inversion symmetry is added while time-reversal symmetry is not enforced. There are two ways to add inversion symmetry, leading to space group Nos. 13 and 14. In space group No. 13, we find that the glide- $Z_{2}$ invariant is expressed solely from the irreducible representations at high-symmetry points in $k$ space. It constitutes the $Z_{2} \times Z_{2}$ symmetry-based indicator for this space group, together with another $Z_{2}$ representing the Chern number modulo 2. In space group No. 14, we find that the symmetry-based indicator $Z_{2}$ is given by a combination of the glide- $Z_{2}$ invariant and the Chern number. Thus, in space group No. 14, from the irreducible representations at high-symmetry points we can only know possible combinations of the glide- $Z_{2}$ invariant and the Chern number, but in order to know each value of these topological numbers, we should calculate integrals of the Berry curvature. Finally, we show that in both cases, the symmetry-based indicator $Z_{4}$ for inversion symmetric systems leading to the higher-order topological insulators is directly related with the glide- $Z_{2}$ invariant and the Chern number. As an independent approach to these results, we also construct all invariants from the layer construction for these space groups, and we show complete agreement with the above results for the topological invariants constructed from $k$ -space topology.

Received 5 November 2018

DOI:https://doi.org/10.1103/PhysRevB.100.165202

Physics Subject Headings (PhySH)

Symmetry protected topological states Topological insulators Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Heejae Kim ¹, Ken Shiozaki², and Shuichi Murakami^1,3

¹Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
²Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
³TIES, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan

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Vol. 100, Iss. 16 — 15 October 2019

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Images

Figure 1
Upper half of the Brillouin zone in SG Nos. 7, 13, and 14. $Γ$ , A, E, Y, Z, B, C, and D denote the high-symmetry points in SG No. 13 and SG No. 14.
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Figure 2
Layer constructions (a) $(001; 0)$ , showing a glide- $Z_{2}$ topological phase ( $\tilde{ν} = 1, n_{Ch} = 0$ ) and (b) $(010; \frac{1}{2})$ , showing a Chern insulator ( $\tilde{ν} = 0, n_{Ch} = 1$ ) with SG No. 13. The parities at high-symmetry points for the models (a) and (b) are shown in (c) and (d), respectively.
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Figure 3
Layer constructions for (a) a glide- $Z_{2}$ topological phase and (b) a Chern insulator with SG No. 14. The parities at the high-symmetry points for the models (a) and (b) are shown in (c) and (d), respectively. Unlike the case of SG No. 13, the combinations of parities at high-symmetry points are equivalent between (c) and (d).
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Figure 4
Chern invariant $δ_{n_{Ch}, i = 1, 2, 3}$ and glide invariant $δ_{g}$ in layer constructions. The yellow planes are the planes in the layer construction, and the blue planes are glide planes. The red arrows represent the directions of the reciprocal lattice vector $G = m b_{1} + n b_{2} + l b_{3}$ , which specifies the orientation of the normal vector of the plane giving a positive Chern number $+ 1$ . (a) Chern invariant $δ_{n_{Ch}, 2}$ along the $a_{3} − a_{1}$ plane is determined as the number of intersections between the planes in the layer, and the $a_{2}$ axis within the lattice constant. The figure shows the case with $(021; 0)$ , which yields $δ_{n_{Ch}, 1} = 0, δ_{n_{Ch}, 2} = 2$ , and $δ_{n_{Ch}, 1} = 1$ . [(b)–(d)] Glide invariant $δ_{g}$ . (b) shows one of the nontrivial configurations of the layer. Two planes are perpendicular to the glide plane, and they are related by the glide operation, which means that they are separated by a glide vector $t_{∥}$ . Note that $2 t_{∥}$ is a lattice translation vector. (c) The configuration in (b) can be deformed into that in (d). (d) shows another nontrivial configuration, where a plane in the layer matches with the glide plane.
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Figure 5
Inversion invariant $δ_{i}, C_{2}$ rotation invariant $δ_{r}$ and $C_{2}$ screw invariant $δ_{s}$ in layer constructions. (a) Inversion invariant $δ_{i}$ . The plane includes the inversion center, which cannot be trivialized. [(b) and (c)] $C_{2}$ rotation invariant $δ_{r}$ . In (b), the plane is perpendicular to the $C_{2}$ rotation axis. This configuration is trivial, because as shown in (c) it can be trivialized while keeping the $C_{2}$ rotation symmetry. (d) $C_{2}$ screw invariant $δ_{s}$ . The two planes in the layer are perpendicular to the $C_{2}$ screw axis, and they are related by the screw operation. $t_{∥}$ is a screw vector, and is a half of a lattice translation vector. This configuration is nontrivial.
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