Experimental observation of multifractality in Fibonacci chains
Mattis Reisner, Yanel Tahmi, Frédéric Piéchon, Ulrich Kuhl, Fabrice Mortessagne
Abstract
The tight-binding model for a chain, where the hopping constants follow a Fibonacci sequence, predicts multifractality in the spectrum and wave functions. Experimentally, we realize this model by chains of small dielectric resonators with a high refractive index (${\ensuremath{\epsilon}}_{r}\ensuremath{\approx}45$) 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 nonzero resonance width and size constraints. We measure the singularity spectrum and the fractal dimensions of the wave functions, and we find good agreement with theoretical predictions for the multifractality based on a perturbative description in the quasiperiodic limit.