Hyperuniform properties of the square-triangle tilings

TL;DR

This paper investigates the hyperuniform properties of square-triangle tilings generated by a local growth rule, revealing a transition from antihyperuniform to hyperuniform states as the probability parameter varies.

Contribution

It introduces a systematic method to generate and analyze square-triangle tilings with varying hyperuniformity, identifying a phase transition at a critical probability value.

Findings

For p < p_c, configurations are antihyperuniform with variance scaling as R^{2-eta}

For p > p_c, configurations are hyperuniform class III with 0<α<1

Structural features in the structure factor depend on hyperuniformity state

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 $\sigma^2\sim O(R^{2-\alpha})$ with $\alpha<0$. 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 $0<\alpha<1$. 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.