Hyperuniformity of quasiperiodic tilings generated by continued fractions

Abstract

Hyperuniformity is a property of certain heteroneous media in which density fluctuations in the long wavelength range decay to zero. In reciprocal space this behavior translates into a decay of Fourier intensities in the range near small wavenumbers. In this paper quasiperiodic tilings constructed by word concatenation are under study. The lattice is generated from a parameter given by its continued fraction so that quasiperiodicity emerges for infinite when irrational generators are into consideration. Numerical simulations show a quite regular quadratic decay of Fourier intensities, regardless of the number considered for the generator parameter, which leads us to formulate the hypothesis that this type of media is strongly hyperuniform of order 3. Theoretical derivations show that the density fluctuations scale in the same proportion as the wavenumbers. Furthermore, it is rigorously proved that the structure factor decays around the origin according to the pattern $S(k) \sim k^4$. This result is validated with several numerical examples with different generating continued fractions.