Nontrivial topological states on a Mobius band

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 past decade. Here we extend these studies to a Mobius 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 Mobius 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 Mobius band. We also provide symmetry arguments to show the possible lattice structures and Hamiltonian terms for which topological states may exist in a Mobius band, and we locate our systems in the classification of topological states. Then, we propose a method to calculate Mobius dispersions from those of the cylinder, and we show the results for a honeycomb and a kagome Mobius 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 setup which could allow for the realization of the elusive quantum Hall effect in a Mobius band.