Universal non-power-law scaling from a chaotic renormalisation group in the Harper-Hofstadter model

Abstract

Previous studies of incommensurate systems concluded that their critical scaling is sensitively dependent on the irrational, $\alpha$, which determines the incommensuration. Contrary to this belief, in the canonical Harper-Hofstadter model, we show there is universal $\alpha$-independent scaling for almost all $\alpha$. This critical scaling is characterized by non-power-law time-length scaling $t \sim r^{\zeta \log \log r}$. 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 $\alpha$-independent scaling emerges as the number theoretic properties of almost all irrational numbers are statistically identical (\emph{\`{a} la} Gauss-Kuzmin statistics). Consequently, we conjecture similar $\alpha$-independent scaling applies in incommensurate models more generally.