This paper discusses a connection between two important classes of materials, namely quasicrystals and topological insulators as exemplified by the quantum Hall problem. It has been remarked that the quasicrystal “inherits” topological properties from the 2D quantum Hall model. The aim of this work is to show this explicitly by introducing the Fibonacci-Hall model as a link between a 1D quasicrystal and the magnetic problems. We show here how Chern numbers for bands in periodic approximants of quasicrystals can be computed, along with gap labels. The Chern numbers are thus seen as a consequence of a flux parameter ${\varphi }^S$ induced by the geometry of winding in 2D space of the quasicrystal. We show the existence of lines of Lifshitz transitions in the phase space of the model. These are marked by change of Chern number and in open systems by disappearance of edge states. Our extrapolation method can be generalized to higher dimensional 2D and 3D quasicrystals, where higher order Chern numbers could be computed, and importantly, related to experimentally measurable transport quantities.

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• Edge states
• Electronic structure
• Topological materials
• Topological phases of matter
• Lattice models in condensed matter
• Tight-binding model