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

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 Möbius twist. Our work opens a new arena of research for exploring topological phases protected by projectively represented space groups.

arXiv comment8 pages, 4 figures

References in corpus (41)

• Topological Insulators
• Topological insulators and superconductors
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Anyons in an exactly solved model and beyond
• Topological Photonics
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Quantum Spin Liquids
• Periodic table for topological insulators and superconductors
• Dirac semimetal in three dimensions
• Topological quantum chemistry
• Topological Acoustics
• The (High Quality) Topological Materials In The World
• Topolectrical circuit realization of topological corner modes
• Catalogue of Topological Electronic Materials
• Zoo of quantum-topological phases of matter
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Dirac Semimetals in Two Dimensions
• The space group classification of topological band insulators
• Nodal-chain metals
• Photonic quadrupole topological phases
• Hourglass Fermions
• The Flux-Phase of the Half-Filled Band
• Topological phonon modes and their role in dynamic instability of microtubules
• Nodal surface semimetals: Theory and material realization
• Superconductors are topologically ordered
• Möbius Insulator and Higher-Order Topology in MnBi${}_{2n}$Te${}_{3n+1}$
• Efficient Topological Materials Discovery Using Symmetry Indicators
• Topological Classification and Stability of Fermi Surfaces
• Observation of an acoustic octupole topological insulator
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Topology of nonsymmorphic crystalline insulators and superconductors
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• Wallpaper Fermions and the Nonsymmorphic Dirac Insulator
• Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups
• Nonsymmorphic symmetry-required band crossings in topological semimetals
• 4D spinless topological insulator in a periodic electric circuit
• Quantum symmetries in discrete gauge theories
• Möbius Kondo Insulators
• Filling-enforced Magnetic Dirac Semimetals in Two Dimensions
• Circumventing the no-go theorem: A single Weyl point surrounded by nodal walls

Cited by in corpus (24)

• Topological acoustics
• Projectively enriched symmetry and topology in acoustic crystals
• Acoustic Möbius insulators from projective symmetry
• Brillouin Klein Bottle From Artificial Gauge Fields
• The gauge-field extended $k\cdot p$ method and novel topological phases
• Spinful topological phases in acoustic crystals with projective PT symmetry
• Classification of time-reversal-invariant crystals with gauge structures
• Klein-bottle quadrupole insulators and Dirac semimetals
• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
• Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries
• Non-Abelian Braiding of Topological Edge Bands
• Multiplicative topological phases
• Mirror real Chern insulator in two and three dimensions
• Periodic Clifford symmetry algebras on flux lattices
• Revealing the spatial nature of sublattice symmetry
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Projective symmetry determined topology in flux Su-Schrieffer-Heeger model
• Encyclopedia of emergent particles in three-dimensional crystals
• Spinless topological chirality from Umklapp scattering in twisted 3D structures
• Floquet Möbius topological insulators
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field
• Topological phases protected by projective space-time inversion symmetry in alkaline-earth-metal-like atoms
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals