Renormalization and universality of the Hofstadter spectrum
Mathematical Physics
2020-08-26 v1 math.MP
Spectral Theory
Quantum Physics
Abstract
We consider a renormalization transformation for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over a circle-rotations by the golden mean, we find several periodic orbits for , and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.
Keywords
Cite
@article{arxiv.1911.09172,
title = {Renormalization and universality of the Hofstadter spectrum},
author = {Hans Koch and Sasha Kocic},
journal= {arXiv preprint arXiv:1911.09172},
year = {2020}
}