Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

Erdal C. Oğuz, Joshua E.S. Socolar, Paul J. Steinhardt, Salvatore Torquato

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law.

Original languageEnglish (US)
Pages (from-to)3-13
Number of pages11
JournalActa Crystallographica Section A: Foundations and Advances
Issue number1
StatePublished - Jan 2019

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Structural Biology
  • Biochemistry
  • General Materials Science
  • Inorganic Chemistry
  • Physical and Theoretical Chemistry


  • Diffraction
  • Hyperuniformity
  • Limit-periodic tilings
  • Non-Pisot tilings
  • Quasiperiodic tilings
  • Substitution tiling


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