Hopfions

This review presents theoretical and brief experimental results for a new and intriguing class of string-like solitons in magnets, called hopfions. The importance of using topological methods to describe localized structures in condensed matter physics is emphasized. The Hopf theory for three-dimensional topological solitons is presented. The results of numerical studies of stationary and dynamic precessional topological solitons with a nonzero Hopf invariant in a uniaxial ferromagnet are presented. The results of numerical studies in chiral magnets of limited volume, where the Dzyaloshchinskii-Moriya interaction significantly changes the hopfion structure, are described. The structure of a hopfion in a helical spin spiral, called a helicton, and its stability in films of varying thickness are considered. The structure of hopfions is presented in models that, in addition to the standard exchange interaction, include additional terms with higher derivatives. The first experimental detection of hopfions is described, where twisted skyrmion strings can be bent into rings in magnetic crystals, resulting in the formation of hopfions.