Dannie Heineman Prize for Mathematical Physics (2020): The Hofstadter's butterfly: from playing with numbers to studying quantum materials

We discuss some results on the Harper's operator - the model behind the Hofstadter's butterfly: metal-insulator transitions of two kinds, the Ten Martini problem, the proof of Thouless' conjecture from the early 80s: that Hausdorff dimension of the spectrum is bounded by 1/2 for all irrational fluxes, as well as the discovery of self-similar (reflective-)hierarchical structure of eigenfunctions throughout the localization regime.
Partially based on joint works with S. Becker and R. Han, I. Krasovsky, and W. Liu .