Drawing butterflies from the almost Mathieu operator

TL;DR

This paper explores the fractal butterfly spectra of almost Mathieu operators, revealing systematic discontinuities in gap positions and proposing a formula to locate these discontinuities, supported by numerical evidence.

Contribution

It introduces a gap-labelling scheme based on K-theory to accurately depict butterfly spectra and presents a conjecture for locating spectral discontinuities.

Findings

Discontinuous gap positions are systematically identified.

A conjecture for locating discontinuities is proposed.

New visualizations of butterfly spectra are provided.

Abstract

Plotting spectra of a range of almost Mathieu operators reveals a beautiful fractal-like image that contains multiple copies of a butterfly image. We demonstrate that plotting the butterflies using a gap-labelling scheme based on K-theory or Chern numbers reveals systematic discontinuities in the gap positioning. A proper image is produced only when we take into account these discontinuities, and close the butterfly wingtips at the points of discontinuity. A conjecture is presented showing a simple formula for locating the discontinuities, and numerical evidence is given to support the conjecture. We also present new renderings of this butterfly.