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Hyperuniformity of Weighted Particle Systems

Salvatore Torquato1,2,3,*, Jaeuk Kim1,4,5, Michael A. Klatt6,7,8, Roberto Car1,3, and Paul J. Steinhardt2

  • *Contact author: torquato@electron.princeton.edu
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Phys. Rev. X 16, 011042 – Published 2 March, 2026

DOI: https://doi.org/10.1103/fr99-qh7h

Abstract

Hyperuniform particle arrangements are characterized by a local number variance within a spherical window of radius R that grows more slowly than the volume of the window, i.e., Rd, in d-dimensional Euclidean space. We generalize this concept to describe the large-scale behavior of particle systems in which particles carry weights: internal degrees of freedom such as scalars (charges and masses), vectors (electric dipole moments, velocities, and torques), pseudovectors (spins and angular momenta), directors (bond orientations), tensors (quadrupole moments), or extrinsic local attributes (Voronoi-cell characteristics). The underlying hyperuniform arrangement may be ordered (crystals and quasicrystals) or disordered, the latter of which has been extensively studied for its novel properties. Our generalization extends hyperuniformity from fluctuations in particle positions to fluctuations in the spatial distribution of weights and examines how weighted fluctuations compare to their unweighted counterparts. We derive generalized weighted pair correlation functions, autocovariance functions, and spectral functions and show their relation to formulas for the local variance in weighted many-particle systems. Then, we apply our formalism to determine the hyperuniformity or nonhyperuniformity of bond-orientational ordered phases, dipolar liquid water, Voronoi-cell volumes, and certain ionic liquids in various Euclidean space dimensions. We demonstrate that hyperuniformity in the particle system does not necessarily translate to hyperuniformity of the weighted system. In fact, cases exist where a hyperuniform particle system becomes antihyperuniform when weighted and others where nonhyperuniform or antihyperuniform particle systems yield hyperuniform weighted systems. This theoretical framework provides a road map for quantifying large-scale fluctuations in weighted many-particle systems, offering a powerful tool for identifying systems with novel physical properties.

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Popular Summary

Hyperuniform many-particle systems, including crystals, quasicrystals, and exotic disordered states, are characterized by an anomalous suppression of large-scale density fluctuations, which grow more slowly than the volume of an observation window, i.e., the fluctuations are nonextensive. Such systems have attracted great interest across the physical, mathematical, and biological sciences, particularly in their disordered varieties, which exhibit novel physical properties. We generalize this concept to describe the large-scale behavior of particle systems in which particles carry weights: internal degrees of freedom such as scalars (charges and masses), vectors (electric dipole moments, velocities, and torques), pseudovectors (spins and angular momenta), directors (bond orientations), tensors (quadrupole moments), or extrinsic local attributes (Voronoi-cell characteristics).

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Phys. Rev. X 16, 011042– Published 2 March, 2026
  • Received 23 October 2025
  • Accepted 30 January 2026
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