Nests and Chains of Hofstadter Butterflies

TL;DR

This paper investigates the fractal structure of the Hofstadter butterfly spectrum, revealing how nested sub-images are scaled and characterized, and analyzing chains of these sub-images with respect to gap closures and accumulation points.

Contribution

It provides a detailed analysis of the self-similar nesting and scaling properties of the Hofstadter butterfly spectrum, highlighting unexpected simplicity in its fractal structure.

Findings

Determined sizes and scaling factors of nested sub-images.

Characterized semi-infinite chains of sub-images.

Identified gap closure and accumulation points in the spectrum.

Abstract

The \lq Hofstadter butterfly', a plot of the spectrum of an electron in a two-dimensional periodic potential with a uniform magnetic field, contains subsets which resemble small, distorted images of the entire plot. We show how the sizes of these sub-images are determined, and calculate scaling factors describing their self-similar nesting, revealing an un-expected simplicity in the fractal structure of the spectrum. We also characterise semi-infinite chains of sub-images, showing one end of the chain is a result of gap closure, and the other end is at an accumulation point.