Nontrivial topological states on a Möbius band

Abstract

In the field of topological insulators, the topological properties of quantum states in samples with simple geometries, such as a cylinder or a ribbon, have been classified and understood during the last decade. Here, we extend these studies to a Möbius band, and argue that its lack of orientability prevents a smooth global definition of parity-odd quantities such as pseudovectors. In particular, the Chern number, the topological invariant for the quantum Hall effect, lies in this class. The definition of spin on the Möbius band translates into the idea of the orientable double cover, an analogy used to explain the possibility of having the quantum spin Hall effect on the Möbius band. We also provide symmetry arguments to show the possible lattice structures and Hamiltonian terms for which topological states may exist in a Möbius band, and we locate our systems in the classification of topological states. Then, we propose a method to calculate Möbius dispersions from those of the cylinder, and we show the results for a honeycomb and a kagome Möbius band with different types of edge termination. Although the quantum spin Hall effect may occur in these systems when intrinsic spin-orbit coupling is present, the quantum Hall effect is more intricate and requires the presence of a domain wall in the sample. We propose an experimental set-up which could allow for the realization of the elusive quantum Hall effect in a Möbius band.

10 pages, 7 figures

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Cited by in corpus (12)

• Synthetic three-dimensional atomic structures assembled atom by atom
• Magnetochiral symmetry breaking in a Möbius ring
• Quantum simulation of non-trivial topology
• Quasicrystalline Chern Insulators
• Curvature-induced quantum spin-Hall effect on a Möbius strip
• Topological elasticity of non-orientable ribbons
• Floquet bound states around defects and adatoms in graphene
• Chern Insulators on Singular Geometries
• Edge Majoranas on locally flat surfaces - the cone and the Möbius band
• Topological Floquet states on a Möbius band irradiated by circularly polarised light
• Inseparable time-crystal geometries on the Möbius strip
• Dynamics of 2D topological quadrupole insulator and Chern insulator induced by real-space topological changes