Brillouin Klein Bottle From Artificial Gauge Fields

Abstract

A Brillouin zone is the unit for the momentum space of a crystal. It is topologically a torus, and distinguishing whether a set of wave functions over the Brillouin torus can be smoothly deformed to another leads to the classification of various topological states of matter. Here, we show that under ${\mathbb{Z}}_2$ gauge fields, i.e., hopping amplitudes with phases $\pm 1$, the fundamental domain of momentum space can assume the topology of a Klein bottle. This drastic change of the Brillouin zone theory is due to the projective symmetry algebra enforced by the gauge field. Remarkably, the non-orientability of the Brillouin Klein bottle corresponds to the topological classification by a ${\mathbb{Z}}_2$ invariant, in contrast to the Chern number valued in $\mathbb{Z}$ for the usual Brillouin torus. The result is a novel Klein-bottle insulator featuring topological modes at two edges related by a nonlocal twist, radically distinct from all previous topological insulators. Our prediction can be readily achieved in various artificial crystals, and the discovery opens a new direction to explore topological physics by gauge-field-modified fundamental structures of physics.

References in corpus (22)

• Topological Insulators
• Topological insulators and superconductors
• Topological Photonics
• Topological Photonics
• Classification of topological quantum matter with symmetries
• 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
• Topological phonon modes and their role in dynamic instability of microtubules
• Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals
• Observation of an acoustic octupole topological insulator
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Spin-Orbit Semimetals in the Layer Groups
• 4D spinless topological insulator in a periodic electric circuit
• Projectively enriched symmetry and topology in acoustic crystals
• Acoustic Möbius insulators from projective symmetry
• Filling-enforced Magnetic Dirac Semimetals in Two Dimensions
• ${\mathbb{Z}}_2$-projective translational symmetry protected topological phases
• Switching spinless and spinful topological phases with projective $PT$ symmetry
• The gauge-field extended $k\cdot p$ method and novel topological phases

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• Spinful topological phases in acoustic crystals with projective PT symmetry
• Classification of time-reversal-invariant crystals with gauge structures
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• General Theory of Momentum-Space Nonsymmorphic Symmetry
• 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
• Weyl points on non-orientable manifolds
• Revealing the spatial nature of sublattice symmetry
• Projective symmetry determined topology in flux Su-Schrieffer-Heeger model
• Projective Spacetime Symmetry of Spacetime Crystals
• Brillouin Platycosms and Topological Phases
• Quantized crystalline-electromagnetic responses in insulators
• Exceptional Non-Hermitian Topology Associated with Non-Toroidal Brillouin Zones
• Critical Behavior and Duality in Dimensionally Reduced Planar Chern-Simons Superconductors
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Non-Abelian interference of topological edge states
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field
• Non-orientable Exceptional Points in Twisted Boundary Systems
• Single $\pi$-flux hosting topological defect modes in bilayer acoustic metamaterials
• Exceptional topology on nonorientable manifolds
• Asymmetric real topology of conduction and valence bands