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.

arXiv comment21 pages, 7 figures

References in corpus (32)

• Anyons in an exactly solved model and beyond
• Topological Photonics
• Topological Photonics
• Classification of topological quantum matter with symmetries
• Topological Crystalline Insulators
• Artificial gauge potentials for neutral atoms
• Topological Acoustics
• Topolectrical circuit realization of topological corner modes
• Quantum Orders and Symmetric Spin Liquids
• Topological Bands for Ultracold Atoms
• Dirac Semimetals in Two Dimensions
• Photonic quadrupole topological phases
• Topology of crystalline insulators and superconductors
• Topological phonon modes and their role in dynamic instability of microtubules
• Topological quantum matter with cold atoms
• High-Temperature Majorana Corner States
• Observation of an acoustic octupole topological insulator
• Topological Classification and Stability of Fermi Surfaces
• Topology of nonsymmorphic crystalline insulators and superconductors
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• Nonsymmorphic symmetry-required band crossings in topological semimetals
• Projective Construction of Non-Abelian Quantum Hall Liquids
• 4D spinless topological insulator in a periodic electric circuit
• 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