Z2topology in nonsymmorphic crystalline insulators: Möbius twist in surface states

Abstract

It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize ${\mathbit{Z}}_{2}$ topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such ${\mathbit{Z}}_{2}$ topological matters. Here we report the ${\mathbit{Z}}_{2}$ topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize ${\mathbit{Z}}_{2}$ topological phases without assuming any antiunitary symmetry. The ${\mathbit{Z}}_{2}$ topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the ${\mathbit{Z}}_{2}$ 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 M\"obius twist, which identifies the ${\mathbit{Z}}_{2}$ phases experimentally. We also provide the relevant structure in the $K$ theory.