AI 中文总结
研究完美准晶体中无序尺度，发现增加旋转对称性会抑制准周期性局部统计特征，保留长程有序，确定对称控制的几何临界点，还揭示有限\(N\)时的统计单胞大小及不同维度的长度尺度层次结构，提供连接晶体、准晶体和非晶物质的定量框架。
AI 中文摘要
缺乏传统晶体对称性却保留类晶体性质的新状态日益挑战晶体有序与非晶无序的经典二分法。准晶体具有长程有序但无平移周期性，能有任意\(N\)重旋转对称性。远离其独特对称中心时，高对称准晶体类似无序图案。研究表明，增加旋转对称性会抑制准周期性的局部统计特征，同时保留其精确长程有序。这种有序隐藏在随\(N\)线性增长的交叉长度之下。当\(N \to \infty\)，无序状区域无界扩展，定义了一个对称控制的几何临界点，此时确定性有序和随机性在任何有限观察窗口内统计上无法区分。对于有限\(N\)，准周期有序在交叉点之外可检测到，揭示了第二个出现的长度尺度即“统计单胞”大小。在一维中，统计单胞大小与交叉长度一致，二维中它随\(N^2\)增长，小于典型近似物大小，建立了出现长度尺度的层次结构。无序到有序的交叉和统计单胞提供了一个连接晶体、准晶体和非晶物质的定量框架，展示了明显无序如何从纯确定性几何中出现。
英文摘要
The classical dichotomy between crystalline order and amorphous disorder is increasingly challenged by novel states that lack conventional crystalline symmetries while retaining crystal-like properties. Quasicrystals occupy a distinctive position within this expanding framework by possessing long-range order without translational periodicity, thereby permitting arbitrary $N$-fold rotational symmetry. Paradoxically, far from their unique symmetry center, high-symmetry quasicrystals closely resemble disordered patterns, raising the question of how deterministic order can be detected. Here we show that increasing rotational symmetry progressively suppresses local statistical signatures of quasiperiodicity, while preserving its underlying exact long-range order. This order is thus concealed below an emergent crossover length that grows linearly with $N$. Therefore, as $N \rightarrow \infty$, the disorder-like regime expands without bound, defining a symmetry-controlled geometric critical point at which deterministic order and randomness become statistically indistinguishable over any finite observation window. For finite $N$, however, quasiperiodic order becomes detectable beyond this crossover, revealing a second emergent length scale that we identify as the size of a \textit{statistical unit cell} -- finite patches over which statistical properties recur despite the absence of conventional translational periodicity. In one dimension, the statistical-unit-cell size coincides with the crossover length, whereas in two dimensions it grows as $N^2$, remaining smaller than the size of typical approximants and establishing a hierarchy of emergent length scales. Together, the disorder-to-order crossover and statistical unit cells provide a quantitative framework connecting crystals, quasicrystals, and amorphous matter, showing how apparent disorder can emerge from purely deterministic geometry.
Comments22 pages, 13 figures
