The scales of disorder in perfect quasicrystals

Abstract

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.