Hyperuniformity of quasicrystals

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, σ 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/τ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, Z(k) ∼ k 4 ; for all others, Z(k) ∼ k 2 . This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.

Scaling of diffraction intensities near the origin: some rigorous results

“…Starting from the idea to use the degree of '(hyper)uniformity' in density fluctuations in many-particle systems [39] to characterise their order, the scaling behaviour of the diffraction near the origin has emerged as a measure that captures the variance of the long-distance correlations. Recently, a number of conjectures on the scaling behaviour of the diffraction of aperiodically ordered structures were made [34,35], reformulating and extending earlier, partly heuristic, results by Luck [31] from this perspective; see also [3,27].…”

Landau level broadening, hyperuniformity, and discrete scale invariance

“…However, we emphasize that, contrary to the Fibonacci chain Ref. 17 (see also App. E), this power-law behavior is modulated by a bounded oscillating nonperiodic function as can be seen in Eqs.…”

“…As explained in Ref. 17, for a spectrum made of a dense set of Bragg peaks (discontinuous S), the integrated intensity function…”

“…establishing a relation between the broadening of the LLL and the so-called hyperuniformity exponent α that characterizes the potential. For α > 0, the potential is hyperuniform 17 whereas α < 0 refers to hypo-uniformity (or anti-hyperuniformity 18 ). The special case α = 0 corresponds to a potential with a constant S, such as the random potential considered previously [see Eq.…”

“…One way to investigate this issue is to follow the approach proposed in Ref. 17 for the Fibonacci chain (see App. E).…”