Non-Power-Law Universality in One-Dimensional Quasicrystals

In this talk I will present a study of the scaling properties of the Aubry–André model and related one-dimensional quasiperiodic models near their localisation transitions. We find numerically that the scaling of energies near the ground state, usually captured by a single dynamical exponent, does not obey a power law relation. Instead, the scaling behaviour 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. In particular, arbitrarily close values of β can result in qualitatively different behaviours very close to the localisation transition. We find that this behaviour is universal between a range of models and can, for the Aubry–André model, be understood in terms of a discrete renormalisation group protocol.