Asymmetric real topology of conduction and valence bands

Abstract

Previously, it was believed that conduction and valence bands exhibit a symmetry: They possess opposite topological invariants (e.g., the Chern numbers of conduction and valence bands for the Chern insulator are $\pm C$). However, we present a counterexample: The second Stiefel-Whitney numbers for conduction and valence bands over the Klein bottle may be asymmetric, with one being nontrivial while the other trivial. Here, the Stiefel-Whitney classes are the characteristic classes for real Bloch functions under $PT$ symmetry with $(PT{)}^2=1$, and the Klein bottle is the momentum-space unit under the projective anticommutation relation of the mirror reflection reversing $x$ and the translation along the $y$ direction. The asymmetry originates from the algebraic difference of real cohomology classes over the Klein bottle and torus. This discovery is rooted in the foundation of topological band theory, and has the potential to fundamentally refresh our current understanding of topological phases.

References in corpus (37)

• Topological Insulators
• Topological insulators and superconductors
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Photonics
• Topological Photonics
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Periodic table for topological insulators and superconductors
• Topological Acoustics
• Topolectrical circuit realization of topological corner modes
• Observation of a phononic quadrupole topological insulator
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Two-dimensional second-order topological insulator in graphdiyne
• Topological quantum matter with cold atoms
• Higher-Order Topological Insulator in Twisted Bilayer Graphene
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Non-Abelian band topology in noninteracting metals
• Two-dimensional higher-order topology in monolayer graphdiyne
• Demonstration of a quantized acoustic octupole topological insulator
• Experimental observation of non-Abelian topological charges and bulk-edge correspondence
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• Spin Space Groups: Full Classification and Applications
• 4D spinless topological insulator in a periodic electric circuit
• Observation of non-Abelian nodal links in photonics
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Second-Order Real Nodal-Line Semimetal in Three-Dimensional Graphdiyne
• Brillouin Klein Bottle From Artificial Gauge Fields
• Four-band non-Abelian topological insulator and its experimental realization
• Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal
• 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
• Weyl points on non-orientable manifolds