Persistent homology and topological statistics of hyperuniform point clouds

TL;DR

This paper investigates the topological properties of hyperuniform point clouds using persistent homology, revealing how global hyperuniformity influences local structure and providing new tools for detecting hyperuniformity in finite systems.

Contribution

It introduces a topological analysis of hyperuniform configurations, linking global hyperuniformity to local topological features and demonstrating their invariance in subsets of the system.

Findings

Topological properties vary systematically with structure factor.

Topological features are conserved in subsets of hyperuniform point clouds.

Hyperuniform arrangements form a one-dimensional manifold reflecting order-to-disorder transition.

Abstract

Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity only utilizes information at the largest scales, hyperuniform configurations have distinctive local characteristics. However, the influence of global hyperuniformity on local structure has remained largely unexplored; establishing this connection can help uncover long-range interaction mechanisms and detect hyperuniform traits in finite-size systems. Here, we study the topological properties of hyperuniform point clouds by characterizing their persistent homology and the statistics of local graph neighborhoods. We find that varying the structure factor results in configurations with systematically different topological properties. Moreover, these topological properties are conserved for subsets of hyperuniform point clouds, establishing a connection between finite-sized systems and idealized reference arrangements. Comparing distributions of local topological neighborhoods reveals that the hyperuniform arrangements lie along a primarily one-dimensional manifold reflecting an order-to-disorder transition via hyperuniform configurations. The results presented here complement existing characterizations of hyperuniform phases of matter, and they show how local topological features can be used to detect hyperuniformity in size-limited simulations and experiments.