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  • Letter
  • Open Access

Hierarchy of collective ordering in swarmalator systems

Hyunsuk Hong1,*, Junghyo Jo2,3,4, and Daniel D. Lee4,5

  • *Contact author: hhong@jbnu.ac.kr
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Phys. Rev. Research 8, L042001 – Published 5 October, 2026

DOI: https://doi.org/10.1103/8p2h-fzt8

Abstract

We introduce an exactly solvable multispecies spherical spin model that unifies noisy Kuramoto dynamics and equilibrium swarmalator systems within a single mean-field framework. The hierarchy of collective ordering in swarmalator systems is elucidated by analytically mapping the full three-dimensional phase diagram, classifying the various collective phases, and determining their corresponding first- and second-order transition boundaries. Our results reveal that the diverse states of swarmalator systems are driven by the competition and cooperation between internal and cross-species couplings. This establishes the spherical spin model as a mathematically tractable foundation for understanding collective behavior in active phase systems and their higher-dimensional generalizations.

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Phys. Rev. Research 8, L042001– Published 5 October, 2026
  • Received 12 July 2026
  • Accepted 2 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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