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arXiv · 1911.09172

Renormalization and universality of the Hofstadter spectrum

Hans Koch·Sasha Kocic
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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.

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Hans Koch, Sasha Kocic. 2019-11-20. Renormalization and universality of the Hofstadter spectrum. https://doi.org/10.1088/1361-6544%2Fab8693

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