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  • Letter
  • Open Access

Construction of hopfion crystals

Wen-Tao Hou1,*, Zhuoxian Xiang1, Yizhou Liu2,†, and Jiadong Zang3,‡

  • *Contact author: houwentao@tiangong.edu.cn
  • †Contact author: yzliu@hmfl.ac.cn
  • ‡Contact author: jiadong.zang@unh.edu
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Phys. Rev. Research 7, L042036 – Published 19 November, 2025

DOI: https://doi.org/10.1103/rr4k-x9m5

Abstract

Hopfions, three-dimensional topological solitons characterized by nontrivial Hopf indices, represent a fundamental class of field configurations that emerge across diverse areas of physics. Despite extensive studies of isolated hopfions, a framework for constructing spatially ordered arrays of hopfions, i.e., hopfion crystals, has been lacking. Here, we present a systematic approach for generating hopfion crystals with cubic symmetry by combining the Hopf map with rational mapping techniques. By superposing helical waves in R4, we construct hopfion crystals with tunable Hopf indices and controllable topology. We demonstrate simple cubic, face-centered cubic, and body-centered cubic hopfion crystals and extend our framework to create crystals of more complex topological structures, including axially symmetric tori, torus links, and torus knots with higher Hopf indices. Our results provide a foundation for searching hopfions in real materials and studying their collective phenomena.

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Phys. Rev. Research 7, L042036– Published 19 November, 2025
  • Received 8 April 2025
  • Accepted 17 October 2025
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