Experimental Observation of Multifractality in Fibonacci Chains

TL;DR

This paper experimentally demonstrates multifractality in Fibonacci chains by measuring the local density of states in dielectric resonator arrays, confirming theoretical predictions of multifractal spectra and wavefunctions.

Contribution

It provides the first experimental realization and analysis of multifractality in Fibonacci chains using dielectric resonators, including recursive schemes for LDOS.

Findings

Measured fractal dimensions agree with theoretical models

Rearrangement by local surroundings improves understanding of fractality

Recursive construction schemes for LDOS are proposed

Abstract

The tight-binding model for a chain, where the hopping constants follow a Fibonacci sequence, predicts multifractality in the spectrum and wavefunctions. Experimentally, we realize this model by chains of small dielectric resonators with high refractive index ($\epsilon_r \approx 45$) of cylindrical form that exhibit evanescent coupling. We show that the fractality of the measured local density of state (LDOS) is best understood when the sites are rearranged according to the similarities in their local surrounding, i.e., their conumbers. This allows us to deduce simple recursive construction schemes for the LDOS for the two cases of dominant strong and weak coupling, despite our limited resolution due to non-zero resonance width and size constraints. We measure the singularity spectrum and the fractal dimensions of the wavefunctions and find good agreement with theoretical predictions for the multifractality based on a perturbative description in the quasiperiodic limit.