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Three-dimensional topological orbital Hall effect caused by magnetic hopfions

Börge Göbel* and Samir Lounis

  • *Contact author: boerge.goebel@physik.uni-halle.de
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Phys. Rev. B 112, 134426 – Published 15 October, 2025

DOI: https://doi.org/10.1103/sc9g-l9by

Abstract

Magnetic hopfions are noncollinear spin textures that are characterized by an integer topological invariant, called the Hopf index. The three-dimensional magnetic solitons can be thought of as a tube with a twisted magnetization that has been closed at both ends to form a torus. The tube consists of a magnetic whirl called an in-plane skyrmion or bimeron. Although hopfions have been observed by microscopy techniques, their detection remains challenging as they lack an electronic hallmark so far. Here we predict a three-dimensional orbital Hall effect caused by hopfion textures: When an electric field is applied, the hopfion generates a transverse current of orbital angular momentum. The effect arises due to the local emergent field that gives rise to in-plane and out-of-plane orbital Hall conductivities. This orbital Hall response can be seen as a hallmark of hopfions and allows us to distinguish them from other textures, such as skyrmioniums, that look similar in real-space microscopy experiments. While the two-dimensional topological invariant of a skyrmion determines its topological Hall transport, the unique three-dimensional topological orbital Hall effect can be identified with the three-dimensional topological invariant that is the Hopf index. Our results make hopfions attractive for spin-orbitronic applications because their orbital signatures allow for their detection in devices and give rise to large orbital torques.

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Phys. Rev. B 112, 134426– Published 15 October, 2025
  • Received 7 June 2025
  • Revised 18 August 2025
  • Accepted 1 October 2025
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