Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries

In this work we consider whether nonsymmorphic symmetries such as a glide plane can protect the existence of topological crystalline insulators and superconductors in three dimensions. In analogy to time-reversal symmetric insulators, we show that the presence of a glide gives rise to a quantized magnetoelectric polarizability, which we compute explicitly through the Chern-Simons 3-form of the bulk wave functions for a glide-symmetric model. Our approach provides a measurable property for this insulator and naturally explains the connection with mirror-symmetry-protected insulators and the recently proposed ${\mathbb{Z}}_2$ index for this phase. More generally, we prove that the magnetoelectric polarizability becomes quantized with any orientation-reversing space group symmetry. We also construct analogous examples of glide-protected topological crystalline superconductors in classes $\mathrm{D}$ and $\mathrm{C}$ and discuss how bulk invariants are related to quantized surface thermal Hall and spin Hall responses.

Figure 1

(a) Bulk BZ of a glide- or mirror-symmetric crystal with the two invariant planes BZ in red (shaded) and invariant lines in blue (thick) if particle-hole symmetry is also present. (b) Surface BZ for a cut normal to $y$ with invariant lines in red (thick), labels for high-symmetry points, and a sketch of an occupied band pair along these lines. On the invariant planes ($k_z=0$ and $\pi$) the glide eigenvalue (color code) distinguishes the two bands. (c) Example of a crystal with glide symmetry $G$ that reflects the $z$ direction and translates by half of the unit cell in the $x$ direction, the glide operator exchanges the two sublattices ($△$ and $▽$). This pattern is repeated in parallel planes shifted perpendicular to the plane of the drawing.

Figure 2

(a) Band structure in slab geometry with 20 unit cells in the $y$ direction for glide-protected class $\mathrm{A}$ topological insulator. We use $H^{AG}$ with $t_{\mu }=1, m=2$, and $\varphi =0.4$. The surface Dirac cone is at a generic momentum on the high-symmetry line; the left- and right-moving branches are distinguished by their glide eigenvalues and cannot gap out. (b) Evolution of $\theta$ from numerics while tuning across the transition from the topological to the trivial phase without breaking the glide symmetry. The error bars indicate two standard deviations (95% confidence interval) of the Monte Carlo estimates.

Figure 3

Band structures in slab geometry for glide-protected class $\mathrm{D}$ topological superconductor. We use $H^{DG}$ with $t_{\mu }=-1$. (a) With $m=2.5$ the surface Majorana cone is at $\overline{M}$, while (b) with $m=0.5$ at $\overline{X}$. The left- and right-moving branches are distinguished by their glide eigenvalues on the invariant lines and cannot gap out. Inset: evolution of $\theta$ while tuning across the transition from the topological ($m<3$) to the trivial ($m>3$) phase from numerics. Note that at $m=3$ the gap closes and $\theta$ takes on an intermediate value not allowed in a gapped system. The error bars for the Monte Carlo results are smaller than the symbols.

Figure 4

Band structure in slab geometry for glide-protected class $\mathrm{C}$ topological superconductor. We use $H^{CG}$ with ${\mathrm{\Delta }}_{xy}={\mathrm{\Delta }}_{xz}={\mathrm{\Delta }}_0=1, \alpha =0.5$, and $m=1.5$. Note the symmetric pair of surface Majorana cones at a generic momentum, the left- and right-moving branches are protected from gapping out by different glide eigenvalues. This is the spectrum for one spin sector; for the full system there is an additional spin degeneracy. Inset: evolution of $\theta$ while tuning across the transition from the topological to the trivial phase from numerics. Note that at $m=2.5$ the gap closes and $\theta$ takes on an intermediate value not allowed in a gapped system. The error bars for the Monte Carlo results are smaller than the symbols.

Figure 5

(a) Evolution of $\theta$ during the gapped deformation $H_{\mathsf{k}}^{AG}$ from the trivial to the topological phase while breaking glide symmetry in class $\mathrm{A}$. We use parameters $t_{\mu }=1, \varphi =0.4,$ and $m=2$ (circles) or $m=2.5$ (squares) and we also show a deformation with $m=2$ but with the symmetry-allowed perturbation $\beta {\tau }_y$ with $\beta =0.5$ added to $H_{\mathbf{k}}^{AG}$ (triangles). (b) Same plot for class $\mathrm{C}$ using $H_{\mathsf{k}}^{CG}$ with ${\mathrm{\Delta }}_{xy}={\mathrm{\Delta }}_{xz}={\mathrm{\Delta }}_0=1, \alpha =0.5$, and $m=1.5$ (circles) or $m=2$ (squares) and $m=1.5$ with symmetry-allowed perturbation $\beta {\tau }_x$ with $\beta =0.5$ added to $H_{\mathbf{k}}^{CG}$. The error bars are smaller than the symbols.

Figure 6

Illustration of our definition of the glide Chern number. (a) Bulk BZ of a glide-symmetric crystal with a possible choice of the “bent” BZ in red (shaded). The glide Chern number is calculated by integrating the Berry flux following a single band around this torus. (b) Surface BZ for a cut normal to $y$ with projection of the “bent” BZ in red (thick) and a sketch of an occupied band pair along this loop. On the invariant planes ($k_z=0$ and $\pi$) the glide eigenvalue (color code) distinguishes the two bands. On the nonsymmetric part (other values of $k_z$), bands are generically nondegenerate. The glide Chern number is calculated by counting the winding number of $\varphi \left({\mathbf{k}}_s\right)$ following a single band (solid or dashed line) around the $\overline{X}-\overline{\mathrm{\Gamma }}-\overline{X}-\overline{M}-\overline{Z}-\overline{M}-\overline{X}$ loop. (c) Value of $\varphi \left({\mathbf{k}}_s\right)$ around the loop for the minimal model in the nontrivial phase. Color code indicates glide eigenvalue in the invariant planes. Note that following one band around, the phase winds $2\pi$, indicating that the glide Chern number is odd.