Hyperuniformity of Quasicrystals

TL;DR

This paper rigorously analyzes the hyperuniformity of quasicrystals using spectral intensity measures, revealing distinct classes based on projection window widths and advancing understanding of their structural properties.

Contribution

It introduces a spectral intensity approach to characterize quasicrystal hyperuniformity and classifies quasicrystals into new categories based on projection parameters.

Findings

One-dimensional quasicrystals exhibit $Z(k) \\sim k^4$ or $Z(k) \\sim k^2$ scaling.

Distinct classes of quasicrystals are determined by the projection window width.

The spectral measure approach links hyperuniformity to structural classification.

Abstract

Hyperuniform systems, which include crystals, quasicrystals and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity, $Z(k)$, to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of $Z(k)$ as $k$ tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, $\sigma^2(R)$, in the number of points contained in an interval of length $2R$. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope $1/\tau$ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, $Z(k) \sim k^4$; for all others, $Z(k)\sim k^2$. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.