Hyperuniformity of quasicrystals

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 $Z\left(k\right)$ to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of $Z\left(k\right)$ as $k$ tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, ${\sigma }^2\left(R\right)$, 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, $Z\left(k\right)\sim k^4$; for all others, $Z\left(k\right)\sim k^2$. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.

Figure 1

Projection of lattice points to create the 1D point set of interest. The red dots lie in the physical space $X$.

Figure 2

Scaling classes in reciprocal space for the Fibonacci projection tilings. Each large gray dot belongs to a distinct scaling class $pq$, with ${\kappa }_{pq}$ being the wave number of the element of that class lying between $2\pi /a$ and $2\pi \tau /a$ (dashed lines).

Figure 3

Scaling of $S\left(k\right)$ at small $k$ for Fibonacci projection tilings constructed from different window widths. The scaling sequences associated with the 15 smallest values of the invariant $I_{pq}$ are shown, each in a different color. Black and gray lines have slope 4 and 2, respectively. (a) The canonical case $\omega =1+1/\tau$. (b) $\omega =9-12/\tau$. (c) $\omega =\tau /2$, for which the window is not ideal.

Figure 4

Behavior of $Z\left(k\right)$ for a Fibonacci projection tiling computed by direct summation of the peak intensities in Fig. 3. Dashed lines indicate predicted upper and lower bounds with $c_{+}=0.6$ and $c_{-}=0.6{\tau }^{-4}$. The curve lies within these bounds for sufficiently small $k$. The scaling exponent $\gamma =4$ is in the strongly hyperuniform range.

Figure 5

Behavior of $Z\left(k\right)$ for a Fibonacci projection tiling computed by direct summation of the intensities of the 15 strongest scaling sequences for $\omega =\sqrt{2}$. Dashed lines indicate derived upper bound and apparent lower bound with exponent $\gamma =2$. Note that this exponent is smaller than the one indicated in Fig. 4 and hence does not correspond to strong hyperuniformity.

Figure 6

The analytically computed number variance for the canonical Fibonacci point set. The dotted (red) line shows the upper bound of exactly $1/4$.

Figure 7

A quasicrystal generated as a cut through a hyperlattice of curved atomic surfaces. The red points are the intersections of the curves and the physical line.

Figure 8

Overlap areas for calculation of variances. (a) A portion of the projection strip showing one position of the window of length $R$. (b) A view of one corner of the window. The dashed region indicates where window corner $A$ must lie in order for the marked gray point to be the leftmost point in the window. Exactly one of the doubly circled sites must be in the window for all positions of $A$ within the dashed region. (c) The region in which the window corner $B$ must lie in order for the marked gray point to be the rightmost point in the window. Exactly one of the doubly circled sites must be in the window for all positions of $A$ within the dashed region. (d) The overlapping regions that determine the variance in the number of points within a finite strip. The vector shown represents the relative displacement of $B$ with respect to $A$ modulo the lattice constant in both the horizontal and vertical directions. The numbers indicate the increasing number of points included in the strip for different locations of $B$.

Figure 9

Partition of the unit cell into regions with different overlap functions. (a) $\beta <2$. (b) $\beta >2$.

Figure 10

Left: Contour plot of the variance function on the unit cell for $\beta =(1+\sqrt{5})/2$ (the golden mean). Contour line values are not uniformly spaced. The color bar shows a linear scale. Right: The variance as a function of $R$, with $R$ measured in units of the 2D lattice constant.

Figure 11

Left: Contour plot of the variance function on the unit cell for $\beta =1+\sqrt{5}$ (twice the golden mean). Contour line values are not uniformly spaced. The color bar shows the same linear scale as in Fig. 10. Right: The variance as a function of $R$, with $R$ measured in units of the 2D lattice constant.

Figure 12

Schematic model of number variance calculations of a subset of points of a square lattice. The number variance expression has been derived for points in a rectangular window of width $w$ and length $2R$ as this window moves along the direction parallel to $R$ indicated by the dashed line.

Figure 13

The number variance for a nonideal Fibonacci quasicrystal as a function of $R$ and as obtained by Eq. (C6). The red dashed line represents the function $(1+lnR)/8$.

Figure 14

Two nonideal rational windows of equal width that yield crystals with different densities.