Hidden Quasicrystal in Hofstadter Butterfly

TL;DR

This paper uncovers a hidden quasicrystal structure within the Hofstadter butterfly, revealing universal features and topological invariants related to quantum Hall states, with implications for experimental observation.

Contribution

It introduces a topological quasicrystal framework for the Hofstadter butterfly, linking spectral gaps to a universal irrational number and demonstrating phase transition effects under periodic drive.

Findings

Identification of a quasicrystal structure in the Hofstadter butterfly

Relation of the structure to a universal irrational number $=2-\u221a{3}$

Amplification of fine spectral features through periodic driving

Abstract

Topological description of hierarchical sets of spectral gaps of Hofstadter butterfly is found to be encoded in a quasicrystal where magnetic flux plays the role of a phase factor that shifts the origin of the quasiperiodic order. Revealing an intrinsic frustration at smallest energy scale, described by $\zeta=2-\sqrt{3}$, this irrational number characterizes the universal butterfly and is related to two quantum numbers that includes the Chern number of quantum Hall states. 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 .