Non-power-law universality in one-dimensional quasicrystals

We have investigated scaling properties of the Aubry--Andr\'e model and related one-dimensional quasiperiodic Hamiltonians near their localization transitions. We find numerically that the scaling of characteristic energies near the ground state, usually captured by a single dynamical exponent, does not obey a power law relation. Instead, the scaling behavior depends strongly on the correlation length in a manner governed by the continued fraction expansion of the irrational number β describing incommensurability in the system. This dependence is, however, found to be universal between a range of models sharing the same value of β. For the Aubry--Andr\'e model, we explain this behavior in terms of a discrete renormalization group protocol which predicts rich critical behavior. This result is complemented by studies of the expansion dynamics of a wave packet under the Aubry--Andr\'e model at the critical point. Anomalous diffusion exponents are derived in terms of multifractal (R\'enyi) dimensions of the critical spectrum; non-power-law universality similar to that found in ground state dynamics is observed between a range of critical tight-binding Hamiltonians.