Hyperuniformity of quasicrystals

The authors present a method for characterizing the hyperuniformity (the suppression of density fluctuations at long wavelengths) for quasicrystals (and other structures whose diffraction pattern includes a dense set of Bragg peaks) based on determining the behavior of the integrated spectral intensity Z(k) for small wavenumber k. Surprisingly, we find that quasicrystals with peaks at the same wavenumbers k can have different behavior for Z(k) and, hence, qualitatively different degrees of hyperuniformity. This effect has never been explored in the laboratory. The figure shows a one-dimensional example: two ``sidewalks'' with different widths (pitched at an incommensurate angle) built on a lawn with a square crystal pattern of spots. The construction rule is that, every time the sidewalk crosses a spot, the sidewalk slab is cut and a new one begins. This rule guarantees both sidewalks are quasicrystalline. But the upper one (a Fibonacci sidewalk) has a different degree of hyperuniformity than the lower one.