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Experimental Observation of Multifractality in Fibonacci Chains
Disordered Systems and Neural Networks · Submitted 27 Jul 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The tight-binding model for a chain, where the hopping constants follow a Fibonacci sequence, predicts multifractality in the spectrum and wavefunctions. | background 1.00 | 0.08 | 0.24 | asserted 0.97 |
| s1 | 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. | method 0.98 | 0.93 | 0.04 | asserted 1.00 |
| s2 | 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. | interpretation 0.59 | 0.93 | 0.28 | asserted 0.99 |
| s3 | 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. | result 0.53 | 0.93 | 0.12 | asserted 0.95 |
| s4 | 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. | result 1.00 | 0.97 | 0.54 | asserted 1.00 |
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Original 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.
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3 arXiv papers cite this in 3 sentences
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Cites it as background 1 paper
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Localized (ergodic) states are characterized by $D_q(ψ)=0$ ($D_q(ψ)=1$), whereas intermediate $q$-dependent $0<D_q(ψ)<1$ implies an extended but non-ergodic character, and a multifractal geometry [ref] .
Particle dynamics and ergodicity-breaking in twisted-bilayer optical lattices · Eigenstates · 0.86
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