Scale-free correlations in two-dimensional geometry-driven hyperuniformity

The collective arrangement of interacting agents often leads to the formation of polarized domains. Among a diverse range of biological and nonbiological systems, it has been generally understood that the absence of a characteristic scale of the aligned domains implies the readiness of a collective response to a sudden perturbation, which is known as a scale-free phenomenon. Here, we use a computational framework, namely Lloyd’s algorithm, to evolve a two-dimensional disordered point pattern to a hyperuniform pattern by using a simple interaction rule dictated by geometric and topological relations between each point and its nearest neighbors. Along the ordering transition, we show the development of hexagonal domains with aligned orientations, separated by nonhexagonal topological defects. As the system size increases, reflecting an increased number of interacting points therein, we find that the size of such orientationally correlated domains in a converged hyperuniform system scales linearly, indicating scale-free correlations. Furthermore, our research highlights the static geometric characteristics of ordered domains, offering a contrast to previous studies focused on dynamic variables such as velocities. This approach provides new insights into the collective dynamics and emergent behavior within complex systems.