Drawing butterflies from the almost Mathieu operator

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.