Non-power-law universal scaling in incommensurate systems

Previous studies of incommensurate systems concluded that critical scaling in such systems 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 heat capacity of a Fermi gas. We argue that this scaling is generic of a broad class of incommensurate models.