Stable free-standing hopfions in bulk chiral magnets
In plain words
A hopfion is a three-dimensional knotted twist of spins in which the lines of equal spin direction form linked loops. In bulk crystals hopfions have so far been seen only attached to skyrmion strings, and a free one has not been observed.
Precise statement
In a bulk or thick-film chiral magnet (FeGe, other B20 compounds, Co-Zn-Mn alloys) or a frustrated magnet, stabilize an isolated hopfion with Hopf index $Q_{H}=1$ not linked to skyrmion strings or sample surfaces, and image it in three dimensions. An answer is the system, the stability window in H and T, and a tomographic reconstruction showing $Q_{H}=1$.
What would settle it
Electron or x-ray magnetic vector tomography resolving the closed, linked preimage loops of a single hopfion away from surfaces and skyrmion strings.
Status in the literature
Zheng et al. (Nature 2023) imaged hopfion rings in FeGe bound to skyrmion strings.