General Theory of Momentum-Space Nonsymmorphic Symmetry

Abstract

As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups (k-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any k-NSGs in any dimensions, it can identify the real-space symmorphic space groups (r-SSGs) and construct the corresponding projective representation of the r-SSG that leads to the k-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all k-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance the classification crystalline topological phases.

21 pages, 7 figures

References in corpus (13)

• Topological Photonics
• Topological Crystalline Insulators
• Topological Acoustics
• Topology of crystalline insulators and superconductors
• Projectively enriched symmetry and topology in acoustic crystals
• Acoustic Möbius insulators from projective symmetry
• Switching spinless and spinful topological phases with projective $PT$ symmetry
• Brillouin Klein Bottle From Artificial Gauge Fields
• Realization of a Hopf insulator in circuit systems
• 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
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries

Cited by in corpus (13)

• Spin Space Groups: Full Classification and Applications
• Klein-bottle quadrupole insulators and Dirac semimetals
• Weyl points on non-orientable manifolds
• A New Moiré Platform Based on M-Point Twisting
• Higher-order topology protected by latent crystalline symmetries
• Projective symmetry determined topology in flux Su-Schrieffer-Heeger model
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• Quantized crystalline-electromagnetic responses in insulators
• Brillouin Platycosms and Topological Phases
• Exceptional Non-Hermitian Topology Associated with Non-Toroidal Brillouin Zones
• Local symmetry groups for arbitrary wavevectors
• Exceptional topology on nonorientable manifolds
• Asymmetric real topology of conduction and valence bands