Hyperuniformity and non-hyperuniformity of quasicrystals

TL;DR

This paper introduces a comprehensive framework for analyzing hyperuniformity in quasicrystals, providing new examples of non-hyperuniform structures and establishing conditions for hyperuniformity based on mathematical properties.

Contribution

It offers a general framework for studying hyperuniformity in quasicrystals, presents new non-hyperuniform examples, and identifies conditions under which hyperuniformity occurs.

Findings

Non-hyperuniform quasicrystals can be non-limit-quasiperiodic.

Hyperuniformity is established for a broad class of quasicrystals.

Hyperuniformity depends on Diophantine properties of lattices and windows.

Abstract

We develop a general framework to study hyperuniformity of various mathematical models of quasicrystals. Using this framework we provide examples of non-hyperuniform quasicrystals which unlike previous examples are not limit-quasiperiodic. Some of these examples are even anti-hyperuniform or have a positive asymptotic number variance. On the other hand we establish hyperuniformity for a large class of mathematical quasicrystals in Euclidean spaces of arbitrary dimension. For certain models of quasicrystals we moreover establish that hyperuniformity holds for a generic choice of the underlying parameters. For quasicrystals arising from the cut-and-project method we conclude that their hyperuniformity depends on subtle diophantine properties of the underlying lattice and window and is by no means automatic.