Topology and Self-Similarity of the Hofstadter Butterfly

TL;DR

This paper explores the self-similar and topological properties of the Hofstadter butterfly spectrum, revealing a hidden quasicrystal structure, hierarchical topological quantum numbers, and their scaling, with implications for experimental observation.

Contribution

It identifies a dominant hierarchical structure linked to an irrational number and maps the butterfly's topology to an integral Apollonian gasket, uncovering hidden symmetries.

Findings

Dominant hierarchy associated with $$=2+√3

Mapping of topological hierarchy to an Apollonian gasket

Amplification of fine structure via periodic drive

Abstract

We revisit the problem of self-similar properties of the Hofstadter butterfly spectrum, focusing on spectral as well as topological characteristics. In our studies involving any value of magnetic flux and arbitrary flux interval, we single out the most dominant hierarchy in the spectrum, which is found to be associated with an irrational number $\zeta=2+\sqrt{3}$ where nested set of butterflies describe a kaleidoscope. Characterizing an intrinsic frustration at smallest energy scale, this hidden quasicrystal encodes hierarchical set of topological quantum numbers associated with Hall conductivity and their scaling properties. This topological hierarchy maps to an {\it integral Apollonian gasket} near-$D_3$ symmetry, revealing a hidden symmetry of the butterfly as the energy and the magnetic flux intervals shrink to zero. With a periodic drive that induces phase transitions in the system, the fine structure of the butterfly is shown to be amplified making states with large topological invariants accessible experimentally.