# A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes

## TL;DR

This paper introduces a geometric quasicrystal-based method to analyze the topological phase diagram of the Hofstadter butterfly, revealing symbolic sequences and inflation rules for band conductance at all magnetic fluxes.

## Contribution

It provides a unified geometric approach to the Hofstadter topological diagram, linking symbolic sequences, Sturmian coding, and self-similar rules to topological properties across all fluxes.

## Key findings

- Band conductance follows a two-letter symbolic sequence.
- Sequences obey inflation/deflation rules similar to quasicrystals.
- Method allows quick computation of Chern numbers at any flux.

## Abstract

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, the bands conductance follow a two letter symbolic sequence . As a result, bands conductance at different fluxes obey inflation/deflation rules as the ones observed to build quasicrystals. The bands conductance 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 any magnetic flux. This approach rationalize the previously observed topological equivalences between the Fibonacci and Harper potentials (also known as the almost Mathieu operator problem) or with other trigonometric potential, as well as the relationship with Farey sequences and trees.