Mathematical Butterflies

Since Michael Aichinger mentioned the famous Hofstadter butterfly in his recent post, I cannot resist to jump on the bandwagon - what is this famous butterfly?

In the 1970s,  Douglas Hofstadter worked on a model to describe electrons moving in a periodic lattice under the influence of a perpendicular magnetic field, as part of his PhD thesis supervised by G. Wannier. While conceptually simple, this system gives rise to stunningly complex physical behaviour - its energy spectrum as a function of the magnetic field is a fractal object. Hofstadter published the spectrum under the name "Gplot" in a 1976 paper. The term "fractal", which was coined by Benoit Mandelbrot, only entered english language texts a few years later. The figure became known to a wider audience through Hofstadter's 1979 book "Gödel, Escher, Bach" and nowadays goes under the name of "Hofstadter's butterfly". Some people have even gone as far as calling the fractal "a picture of god".

I don't want to put you on the rack any longer: here's how the "butterfly" looks like:

At the time of its inception, people were of course wondering whether the self-similar, recursive structure of the spectrum was just an artefact of the model, or whether it could really be realised in nature. When the x-axis of the plot is scaled in "natural", dimensionless units ("flux quanta per unit cell" in the figure above), one can notice a peculiar grouping of the black areas in the graph: at 1/2 flux quanta per unit cell, the spectrum splits in 2 "bands" (dashed red line in the figure). At, say, 2/5 flux quanta per unit cell, it splits in 5 bands, which are grouped in 2 bands on top and bottom, plus one in the middle (blue dashed line). At 4/9 flux quanta, it is 9 bands, grouped in 4 bands at top/bottom, plus one in the middle. For irrational numbers, things are quite a bit more complicated - but does nature really "know" about the difference between rational and irrational numbers, and does it like to do fraction arithmetic?

First glimpses of the recursive spectrum were found only in 2001, and since a very recent (2013) Nature article things seem to be definite - Hofstadter's elusive butterfly is not just a beautiful mathematical figure, but is something that can really be measured in a laboratory.

But how does this peculiar fraction arithmetic come about? Stay tuned...

P.S.: Hofstadter's model was deliberately kept very simple, due to the computer power available at that time. Together with Eduardo Hernandez, Michael Aichinger and I recently tried to crack the "full problem" of solving Schrödinger's equation for that system, using a highly efficient method for certain types of eigenvalue problems.