$\mathbb{Z}_2$-projective translational symmetry protected topological phases

TL;DR

This paper explores how $ ext{Z}_2$ projectively represented translational symmetries lead to novel topological phases with unique boundary states and degeneracies, expanding the understanding of symmetry-protected topological matter.

Contribution

It introduces the concept of $ ext{Z}_2$ projective translation symmetry and demonstrates its role in creating exotic topological phases with distinctive boundary phenomena.

Findings

Distinct commutation relations for $ ext{Z}_2$ projective translations

Emergence of boundary degeneracies and Dirac points

Topological insulator phase with Möbius twist edge bands

Abstract

Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel topological phases. We show that the $\mathbb{Z}_2$ projectively represented translational symmetry operators adopt a distinct commutation relation, and become momentum dependent analogous to twofold nonsymmorphic symmetries. Combined with other internal or external symmetries, they give rise to many exotic band topology, such as the degeneracy over the whole boundary of the Brillouin zone, the single fourfold Dirac point pinned at the Brillouin zone corner, and the Kramers degeneracy at every momentum point. Intriguingly, the Dirac point criticality can be lifted by breaking one primitive translation, resulting in a topological insulator phase, where the edge bands have a M\"{o}bius twist. Our work opens a new arena of research for exploring topological phases protected by projectively represented space groups.