Where can I get an introduction to the mathematics behind Hofstadter's Butterfly?

Are there any good books that give good mathematical/physical background to the workings of the Hofstadter's Butterfly? I'd appreciate some references. Books or Public access papers will work.

Preferably, I'd like something that gives some insight to things like the alteration of the Hamiltonian for magnetic fields among other things not covered in introductory QM course. Also, I'm interested in the numerical procedure used to actually plot the Butterfly. I'm willing to learn though, so if you post a prerequisite list, that'd work too.

• I'm curious why this post was made "community wiki". Is it because it's a resource recommendation question? I don't see anything about that in the linked policy. – LarsH Mar 23 '16 at 14:34
• No reason really, questions just seemed to be better received this way. – Zach466920 Mar 23 '16 at 16:41

3 Answers

The book Physics by Computer has a unit on Hofstadter's Butterfly. The "presentation of the theoretical foundations can only be seen as a first introduction," according to the book's introduction, but it does have source code for computing the figure.

This web page about the book has many broken links, but you can find the Java source code for the book's Butterfly unit on the Wayback Machine.

I have ported the java code to HTML5 canvas/javascript here, where you can watch the figure fill in as q increases:

I also recommend this pithy half-page summary of the math by Oliver Knill, that was linked to by the page that @Jitter posted a link to. It describes a function $f(x,y)$ for the Butterfly, using a Lyapunov exponent. Prerequisite: knowledge of matrices and determinants.

I created a Jupyter Notebook http://nbviewer.jupyter.org/github/empet/Mathematical-Physics/blob/master/Hofstadter-buterfly.ipynb, that presents how the Hofstadter butterfly is defined, as well as an interactive plot that shows on hover the corresponding rational magnetic flux and the energy.

The book called << Butterfly in the quantum world >> has very detailed explanation about many (almost all) expects about Hoftstadter's butterfly and beyond. The references in it are also very useful during the study.