Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings.

Area of Science:

• Mathematical Physics
• Materials Science
• Condensed Matter Physics

Background:

• Hyperuniformity describes materials with suppressed large-scale density fluctuations.
• Substitution tilings are a class of quasiperiodic and aperiodic structures with fractal properties.
• Understanding scaling properties is crucial for characterizing the physical behavior of these materials.

Purpose of the Study:

• To investigate the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in 1D substitution tilings.
• To predict the scaling exponent (α) of Fourier intensities at small wavenumbers.
• To numerically verify the theoretical predictions across different types of tilings.

Main Methods:

• Development of a simple theoretical argument to predict the scaling exponent α.
• Numerical computations to calculate Fourier intensities for various substitution tilings.
• Analysis of scaling behavior for quasiperiodic, singular continuous spectra, and limit-periodic tilings.

Main Results:

• A predictive argument for the exponent α governing Fourier intensity scaling was successfully presented.
• Numerical results confirmed the accuracy of the predictions for quasiperiodic, singular continuous, and limit-periodic tilings.
• The study demonstrates that α can be tuned within specific ranges (-1 to 3 for quasiperiodic/singular continuous, -1 to 1 for limit-periodic).

Conclusions:

• The scaling properties of hyperuniformity in 1D substitution tilings are well-characterized by the exponent α.
• The findings provide a theoretical framework and numerical validation for understanding density fluctuations in these complex structures.
• This work offers a method to construct tilings with specific hyperuniformity characteristics relevant to materials science.