Topological states of matter in equilibrium, as well as out of equilibrium, have been thoroughly investigated during the last years in condensed-matter and cold-atom systems. However, the geometric topology of the studied samples is usually trivial,
such as a ribbon or a cylinder. In this paper, we consider a graphene Mobius band irradiated with circularly polarised light. Interestingly, due to the non-orientability of the Mobius band, a homogeneous quantum Hall effect cannot exist in this system, but the quantum spin Hall effect can. To avoid this restriction, the irradiation is applied in a longitudinal-domain-wall configuration. In this way, the periodic time-dependent driving term tends to generate the quantum anomalous Hall effect. On the other hand, due to the bent geometry of the Mobius band, we expect a strong spin-orbit coupling, which may lead to quantum spin Hall-like topological states. Here, we investigate the competition between these two phenomena upon varying the amplitude and the frequency of the light, for a fixed value of the spin-orbit coupling strength. The topological properties are analysed by identifying the edge states in the Floquet spectrum at intermediate frequencies, when there are resonances between the light frequency and the energy difference between the conduction and valence bands of the graphene system.
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 Mob
ius 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 set-up which could allow for the realization of the elusive quantum Hall effect in a Mobius band.
