Non-power-law universality in one-dimensional quasicrystals

TL;DR

This paper reveals that in one-dimensional quasiperiodic systems, the scaling near localization transitions does not follow a power law but depends on the continued fraction expansion of the incommensurate parameter, indicating a non-power-law universality.

Contribution

It uncovers non-power-law universal scaling behavior in quasiperiodic models and explains it via a discrete renormalisation group approach, extending understanding of critical phenomena in these systems.

Findings

Scaling of characteristic energies is non-power-law and depends on the continued fraction of $eta$.

Universal behavior observed across models sharing the same $eta$ value.

Anomalous diffusion exponents relate to multifractal dimensions of the spectrum.

Abstract

We have investigated scaling properties of the Aubry-Andr\'e model and related one-dimensional quasiperiodic Hamiltonians near their localisation 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 behaviour depends strongly on the correlation length in a manner governed by the continued fraction expansion of the irrational number $\beta$ describing incommensurability in the system. This dependence is, however, found to be universal between a range of models sharing the same value of $\beta$. For the Aubry-Andr\'e model, we explain this behaviour in terms of a discrete renormalisation group protocol which predicts rich critical behaviour. 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.