Hopfion

[1][2][3][4] It is a stable three-dimensional localised configuration of a three-component field

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.

[5] 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.

The terms of higher-order derivatives are required to stabilize the hopfions.

Stable hopfions were predicted within various physical platforms, including Yang–Mills theory,[6] superconductivity[7][8] and magnetism.

[9][10][11][4] Hopfions have been observed experimentally in chiral colloidal magnetic materials,[2] in chiral liquid crystals,[12][13] in Ir/Co/Pt multilayers using X-ray magnetic circular dichroism[14] and in the polarization of free-space monochromatic light.

[17] In recent years, the concept of a "fractional hopfion" has also emerged where not all preimages of magnetisation have a nonzero linking.