Universal Nonpower Law Scaling from a Chaotic Renormalization Group in the Harper-Hofstadter Model

Previous studies of incommensurate systems concluded that their critical scaling is sensitively dependent on the irrational, α, which determines the incommensuration. Contrary to this belief, in the canonical Harper-Hofstadter model, we show there is universal α-independent scaling for almost all α. This critical scaling is characterized by non-power-law time-length scaling t∼r^{ζloglogr}. We demonstrate this in the superfluid fraction of a Bose gas, and the specific heat of a Fermi gas. This scaling is generic of a broad class of generalized Harper-Hofstadter models. The α-independent scaling emerges as the number theoretic properties of almost all irrational numbers are statistically identical (à la Gauss-Kuzmin statistics). Consequently, we conjecture similar α-independent scaling applies in incommensurate models more generally.