Characterizing the Hofstadter butterfly's outline with Chern numbers

TL;DR

This paper explores the Hofstadter butterfly's outline, revealing its connection to Chern numbers that quantify topological properties and charge transport, with universal features across different lattice geometries.

Contribution

It identifies a novel relationship between the Hofstadter butterfly's outline and specific Chern numbers, highlighting their universal topological significance across lattice types.

Findings

The Hofstadter butterfly's outline is associated with a sequence of Chern numbers.

Each stairway in the outline corresponds to a Chern number related to charge transport.

Universal properties are observed in both square and honeycomb lattice geometries.

Abstract

In this work, we report original properties inherent to independent particles subjected to a magnetic field by emphasizing the existence of regular structures in the energy spectrum's outline. We show that this fractal curve, the well-known Hofstadter butterfly's outline, is associated to a specific sequence of Chern numbers that correspond to the quantized transverse conductivity. Indeed the topological invariant that characterizes the fundamental energy band depicts successive stairways as the magnetic flux varies. Moreover each stairway is shown to be labeled by another Chern number which measures the charge transported under displacement of the periodic potential. We put forward the universal character of these properties by comparing the results obtained for the square and the honeycomb geometries.