$Z_2$ topology in nonsymmorphic crystalline insulators: Möbius twist in surface states

It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize ${\mathit{Z}}_2$ topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such ${\mathit{Z}}_2$ topological matters. Here we report the ${\mathit{Z}}_2$ topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize ${\mathit{Z}}_2$ topological phases without assuming any antiunitary symmetry. The ${\mathit{Z}}_2$ topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the ${\mathit{Z}}_2$ phases have a surface consistent with the nonsymmorphic space group symmetry, and thus they support topological gapless surface states. Remarkably, the surface states have a unique energy dispersion with the Möbius twist, which identifies the ${\mathit{Z}}_2$ phases experimentally. We also provide the relevant structure in the $K$ theory.

Figure 1

(a) Two inequivalent sites $A$ and $B$ in the unit cell. (b)Topologically different trajectories $(x\left(k_x\right),y\left(k_x\right))$. (c) Insulating states I (top) and II (bottom).

Figure 2

Schematic illustration of edge states with Möbius twist. (a) The red (blue) line is an edge state in the eigensector of $U\left(k_x\right)$ with the eigenvalue $u=e^{-i{k}_x/2}$ $\left(u=-e^{-i{k}_x/2}\right)$. (b) An exchange process of the eigensectors, which is carried out by changing the sign of $y\left(k_x\right)$ in $H_{1\mathrm{D}}\left(k_x\right)$ of Eq. (7).

Figure 3

A surface state protected by glide reflection symmetry. The spectrum at $k_z=\Lambda$ has a Möbius twist in the $k_x$ direction: Along the $k_x$ direction, the red branch with the eigenvalue $e^{i{k}_x/2}$ of $G\left(k_x\right)$ turns into the blue one with the eigenvalue $-e^{i{k}_x/2}$.

Figure 4

Hamiltonian mapping. Two Hamiltonians $H_{\mathrm{R}}(\mathit{k},\theta )$ and $H_{\mathrm{R}}(\mathit{k},\theta )$ defined on $T^d\times [0,\pi ]$ are sewn at $\theta =0$ and $\pi$.

Figure 5

Upper half Brillouin zone.