Higher-dimensional quasicrystalline approach to the Hofstadter butterfly topological-phase band conductances: Symbolic sequences and self-similar rules at all magnetic fluxes

The topological properties of the quantum Hall effect in a crystalline lattice, described by Chern numbers of the Hofstadter butterfly quantum phase diagram, are deduced by using a geometrical method to generate the structure of quasicrystals: the cut and projection method. Based on this, we provide a geometric unified approach to the Hofstadter topological-phase diagram at all fluxes. Then we show that for any flux, bands conductances follow a two-letter symbolic sequence. As a result, conductances at different fluxes obey inflation/deflation rules as the ones observed to build quasicrystals. The symbolic sequences are given by the Sturmian coding of the flux and can be found by considering a circle map, a billiard, or trajectories on a torus. Simple and fast techniques are thus provided to obtain Chern numbers at all magnetic fluxes. This approach allows one to understand the global fractality of the butterfly in terms of Farey sequences and trees.