Hyperuniform properties of the square-triangle tilings

Akihisa Koga; Shiro Sakai; Yushu Matsushita; Tsutomu Ishimasa

doi:10.1103/PhysRevB.110.094208

paper

Hyperuniform properties of the square-triangle tilings

arXiv:2409.16509 · doi:10.1103/PhysRevB.110.094208

Abstract

We study hyperuniform properties for the square-triangle tilings. The tiling is generated by a local growth rule, where squares or triangles are iteratively attached to its boundary. The introduction of the probability $p$ in the growth rule, which controls the expansion of square and triangle domains, enables us to obtain various square-triangle random tilings systematically. We analyze the degree of the regularity of the point configurations, which are defined as the vertices on the square-triangle tilings, in terms of hyperuniformity. It is clarified that for $p < p_{c} (p_{c} \sim 0.5)$ , the system can be regarded as a phase separation between square and triangular lattice domains and the variance of the point configurations obeys the scaling law $σ^{2} \sim O (R^{2 - α})$ with $α < 0$ . The configurations are antihyperuniform. On the other hand, for $p > p_{c}$ , the squares and triangles are spatially well mixed and the point configurations belong to the hyperuniform class III with the exponent $0 < α < 1$ . This means the existence of the hyperuniform-antihyperuniform transition at $p = p_{c}$ . We also examine the structure factor of the square-triangle tilings. It is clarified that the peak structures in the large-wave-number regime are mostly common to all square-triangle tilings, while those in the small-wave-number regime strongly depend on whether the point configurations are hyperuniform or antihyperuniform.

arXiv comment14 pages, 18 figures

Hyperuniform properties of the square-triangle tilings

Abstract

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