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Topological swarmalators

Jun-Bo Gou (勾俊博)1,2, Marc Timme3,4, Xiaozhu Zhang (张潇竹)1,2,3,*, and Gang Yan (严钢)1,2,†

  • 1MOE Key Laboratory of Advanced Micro-Structured Materials, and School of Physical Science and Engineering, Tongji University, Shanghai, People’s Republic of China
  • 2National Key Laboratory of Autonomous Intelligent Unmanned Systems, MOE Frontiers Science Center for Intelligent Autonomous Systems, and Shanghai Research Institute of Intelligence Science and Technology, Tongji University, Shanghai, People’s Republic of China
  • 3Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed), Institute for Theoretical Physics, and Center for Synergy of Systems, TUD Dresden University of Technology, 01062 Dresden, Germany
  • 4Lakeside Labs, Lakeside B04b, 9020 Klagenfurt, Austria

  • *Contact author: xiaozhu_zhang@tongji.edu.cn
  • †Contact author: gyan@tongji.edu.cn
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Phys. Rev. Research 8, 013060 – Published 20 January, 2026

DOI: https://doi.org/10.1103/kghb-vpst

Abstract

Swarmalators constitute a paradigmatic model for understanding the collective dynamics of coupled moving agents, integrating both internal and spatial degrees of freedom. Empirical evidence from systems such as bird flocks and living matter highlights the relevance of topological, metric-free coupling, but their impact on swarmalator dynamics remains largely unknown to date. Here, we present and analyze a topological swarmalator model in which the units interact topologically, on Delaunay networks. We find intriguing self-organized collective dynamics, including patterns with local vortices and unprecedented spatiotemporal patterns absent in metric-based models. Identifying three order parameters to quantify synchrony, spatial order, and vortex formation, we map the phase diagram that classifies these diverse patterns. Notably, we uncover a first-order transition even if the phases of all units are frozen, a dynamics inverted relative to the classical Kuramoto model. These insights not only advance our theoretical understanding of locally coupled systems of moving agents, but also offer key guidelines for their control.

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Phys. Rev. Research 8, 013060– Published 20 January, 2026
  • Received 9 July 2025
  • Accepted 16 December 2025
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