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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.16509 |
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| _version_ | 1866917789071048704 |
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| author | Koga, Akihisa Sakai, Shiro Matsushita, Yushu Ishimasa, Tsutomu |
| author_facet | Koga, Akihisa Sakai, Shiro Matsushita, Yushu Ishimasa, Tsutomu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperuniform properties of the square-triangle tilings Koga, Akihisa Sakai, Shiro Matsushita, Yushu Ishimasa, Tsutomu Statistical Mechanics 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. |
| title | Hyperuniform properties of the square-triangle tilings |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2409.16509 |