Model of magnetic hopfion in a solid. Bem is emergent magnetic field (orange arrows); in a hopfion, it does not align to the external magnetic field (black arrow).
A hopfion is a topological soliton.[1][2][3][4] It is a stable three-dimensional localised configuration of a three-component field with a knotted topological structure. They are the three-dimensional counterparts of 2D skyrmions, which exhibit similar topological properties in 2D. Hopfions are widely studied in many physical systems over the last half century, as summarized here http://hopfion.com
The soliton is mobile and stable: i.e. it is protected from a decay by an energy barrier. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician, Heinz Hopf.
A model that supports hopfions was proposed as follows[1]
The terms of higher-order derivatives are required to stabilize the hopfions.
Stable hopfions were predicted within various physical platforms, including Yang–Mills theory,[5] superconductivity[6][7] and magnetism.[8][9][10][4]