Renormalization and universality of the Hofstadter spectrum

TL;DR

This paper investigates a renormalization approach to the Hofstadter spectrum, revealing universal constants and scaling behaviors linked to the spectrum's fractal structure, especially over golden mean rotations.

Contribution

It introduces a renormalization transformation for Hofstadter-related maps, identifies periodic orbits, and conjectures infinite such orbits, linking scaling factors to the rotation number.

Findings

Periodic orbits of the renormalization map are found and conjectured to be infinite.

Scaling factors are algebraically related to the circle-rotation number.

The results suggest these factors describe local scaling properties of the Hofstadter spectrum.

Abstract

We consider a renormalization transformation $R$ for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of $R$ 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 $R$, 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.