Topological Insulators from Group Cohomology

We classify insulators by generalized symmetries that combine space-time transformations with quasimomentum translations. Our group-cohomological classification generalizes the nonsymmorphic space groups, which extend point groups by real-space translations; i.e., nonsymmorphic symmetries unavoidably translate the spatial origin by a fraction of the lattice period. Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology. In this sense, cohomology underlies band topology. Our claim is exemplified by the first theory of time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These insulators may be described as “piecewise topological,” in the sense that subtopologies describe the different high-symmetry submanifolds of the Brillouin zone, and the various subtopologies must be pieced together to form a globally consistent topology. The subtopologies that we discover include a glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow topology (exemplified by our recently proposed KHgSb material class), and quantized non-Abelian polarizations. Our cohomological classification results in an atypical bulk-boundary correspondence for our topological insulators.

Popular Summary

When spatial symmetries such as rotations or reflections are combined with real-space translations, the resultant group of symmetries describes crystals. A field of mathematics called group cohomology dictates that there are exactly 230 possible combinations of symmetries in three spatial dimensions. Here, we demonstrate that these combinations do not exhaust all possible symmetries in crystals.

The translational symmetry of crystals guarantees that each particle has a conserved quasimomentum. We, for the first time, combine rotations and reflections with quasimomentum translations, in addition to real-space translations. Our theory thus places real and quasimomentum space on equal footing. We then obtain a richer classification of symmetries through group cohomology.

Our expanded classification provides a unified framework to discuss insulators with topologically robust properties that are protected by space-time symmetries. We exemplify our theory with the large-gap insulator KHgSb, whose surface is uniquely characterized by unremovable fermions with an hourglass-shaped dispersion.

Figure 1

(a) Mirror (red, blue) and glide (green, brown) planes in the 3D Brillouin torus of $\mathrm{KHg}X$. These planes project to the high-symmetry line $\tilde{X}\tilde{U}\tilde{Z}\tilde{\mathrm{\Gamma }}\tilde{X}$ in the 2D Brillouin torus of the 010 surface. (b)–(d) Examples of possible piecewise topologies in the space group of $\mathrm{KHg}X$, as illustrated by their surface band structures along $\tilde{X}\tilde{U}\tilde{Z}\tilde{\mathrm{\Gamma }}\tilde{X}$. Panel (b) describes a “quantum glide Hall effect” (along $\tilde{Z}\tilde{\mathrm{\Gamma }}$) and an odd mirror-Chern number (along $\tilde{\mathrm{\Gamma }}\tilde{X}$); these two subtopologies must be pieced together at their intersection point $\tilde{\mathrm{\Gamma }}$. Panel (c) describes an hourglass-flow topology ($\tilde{X}\tilde{U}\tilde{Z}\tilde{\mathrm{\Gamma }}$) and an even mirror-Chern number ($\tilde{\mathrm{\Gamma }}\tilde{X}$). (d) A trivial topology is shown for comparison.

Figure 2

(a) 3D view of atomic structure. The Hg (red) and $X$ (blue) ions form a honeycomb layers with $AB$ stacking. The K ion (cyan) is located at an inversion center, which we also choose to be our spatial origin. (b) Top-down view of atomic structure that is truncated in the 010 direction; two of three Bravais-lattice vectors are indicated by ${\tilde{\mathit{a}}}_1$ and ${\tilde{\mathit{a}}}_2$. (c) Center: bulk Brillouin zone (BZ) of $\mathrm{KHg}X$, with two mirror planes of ${\overline{M}}_z$ colored red and blue. Top: 100-surface BZ. Right: 010-surface BZ.

Figure 3

(a) A constant-$k_z$ slice of quasimomentum space, with two of three reciprocal-lattice vectors indicated by ${\tilde{\mathit{b}}}_1$ and ${\tilde{\mathit{b}}}_2$. While each hexagon corresponds to a Wigner-Seitz primitive cell, it is convenient to pick the rectangular primitive cell that is shaded in cyan. A close-up of this cell is shown in (b). Here, we illustrate how the glide reflection (${\overline{M}}_x$) maps $(k_y,\pi /\sqrt{3}a,k_z)\to (k_y,-\pi /\sqrt{3}a,k_z)$ (red dot to brown), which connects to $(2\pi /a+k_y,\pi /\sqrt{3}a,k_z)$ (blue) through ${\tilde{\mathit{b}}}_2$. Panel (c) serves two interpretations. In the first, $T{\overline{M}}_z$ maps $(k_y,\pi /\sqrt{3}a,k_z)\to (-k_y,-\pi /\sqrt{3}a,k_z)$ (red dot to brown), which connects to $(2\pi /a-k_y,\pi /\sqrt{3}a,k_z)$ (blue) through ${\tilde{\mathit{b}}}_2$. If we interpret (c) as the $k_z=0$ cross section, the same vectors illustrate the effect of time reversal.

Figure 4

Possible Wilson spectra along $\tilde{\mathrm{\Gamma }}\tilde{Z}$.

Figure 5

Comparison of the spectrum of $P_{\perp }\widehat{y}{P}_{\perp }$ for a system without any symmetry (a) and one with time-reversal, spatial-inversion, and glide symmetries (b). Only the spectrum for two spatial unit cells is shown.

Figure 6

(a),(b) Possible Wilson spectra along $\tilde{X}\tilde{U}$. (c) A hypothetical spectrum that is ruled out by a continuity argument.

Figure 7

Illustrating the single-energy criterion to determine the parity of the mirror-Chern number (${\mathcal{C}}_e$): ${\mathcal{C}}_e=-1$ in (a) and $-2$ in (b). Red solid (dashed) lines correspond to a Wilson band with mirror eigenvalue ${\lambda }_z=+i$ ($-i$).

Figure 8

Possible Wilson spectra along $\tilde{\mathrm{\Gamma }}\tilde{X}$. (a)–(c) With nontrivial glide polarization (${\mathcal{P}}_{\tilde{\mathrm{\Gamma }}}^{\eta }$), the mirror-Chern numbers (${\mathcal{C}}_e$) are, respectively, $-1$, $+1$, and $-3$. (d)–(f) With trivial ${\mathcal{P}}_{\tilde{\mathrm{\Gamma }}}^{\eta }$, ${\mathcal{C}}_e$ equals 0, $-2$, and $-4$, respectively.

Figure 9

Possible Wilson spectra along $\tilde{Z}\tilde{U}$.

Figure 10

Origin of $W$-glide and $W$-time-reversal symmetries. (a)–(d) Constant-$k_z$ slices of the bulk Brillouin zone. Panel (a) illustrates how the glide (${\overline{M}}_x$) maps momenta from the glide plane $k_x=\pi$. Under ${\overline{M}}_x$, the Wilson loop is mapped from the red vertical arrow in (a) to the red vertical arrow in (b). Panels (c) and (d) describe the $k_z=0$ plane and illustrate a similar story for the time reversal $T$.

Figure 11

(a) Simple example of a 2D nonsymmorphic crystal. The two sublattices are colored, respectively, dark blue and cyan. Panels (b)–(d) illustrate the effect of a glide reflection.

Figure 12

To clarify our notations for Wilson loops and lines, we draw several examples, all of which occur at fixed ${\mathit{k}}_{\parallel }$ and variable $k_y$ (vertical axis). The arrow indicates the orientation for parallel transport. (a) The Wilson loop at base point $-\pi$ (encircled) is labeled by ${\mathcal{W}}_{-\pi }\equiv {\mathcal{W}}_{\pi \leftarrow -\pi }$. (b) ${\mathcal{W}}_{-2\pi }\equiv {\mathcal{W}}_{0\leftarrow -2\pi }$. (c) ${\mathcal{W}}_{\pi \leftarrow -\pi /2}$. (d) ${\mathcal{W}}_{\pi \leftarrow 0}$. (e) ${\mathcal{W}}_{\pi \leftarrow 2\pi }$. ${\mathcal{W}}_{r,{\overline{k}}_y}$ denotes a Wilson loop with base point ${\overline{k}}_y$ and oriented in the direction of decreasing $k_y$, e.g., ${\mathcal{W}}_{r,2\pi }$ in (f) and ${\mathcal{W}}_{r,\pi }$ in (g).

Figure 13

(a)–(c) Pictorial representation of Eq. (b42), with solid line indicating a product of projections that is path ordered according to the arrow on the same line, and $V_{+}$ indicating an insertion of the spatial-embedding matrix $V(2\pi \overrightarrow{y})$. Panels (d)–(f) represent Eq. (b52), with $V_{+}$ indicating an insertion of $V(-2\pi \overrightarrow{y})$.

Figure 14

Bulk band structures with glide and time-reversal symmetries, for systems with spin (a),(b) and without (c). $\mathrm{\Gamma }\equiv (0,0,k_z=0)$ and $Z\equiv (0,0,k_z=\pi )$ are high-symmetry points that are selected because the fractional translation (in the glide) is parallel to $\overrightarrow{z}$. Panel (a) either has no spin-orbit coupling or has spin-orbit coupling with an additional spatial-inversion symmetry. Panel (b) has spin-orbit coupling but breaks spatial-inversion symmetry. The crossings between orthogonal mirror branches (indicated by arrows) are movable along $\mathrm{\Gamma }Z$ but unremovable so long as glide and time-reversal symmetries are preserved. The glide eigenvalues are indicated at $\mathrm{\Gamma }$ and $Z$ for one of the two hourglasses. Panel (c) could apply to an intrinsically spinless system (e.g., bosonic cold atoms and photonic crystals), and also to an effectively spinless system (e.g., a single-spin species in an electronic system without spin-orbit coupling).