Hopfion

A hopfion is a topological soliton.^[1][2][3][4] It is a stable three-dimensional localised configuration of a three-component field ${\displaystyle {\vec {n}}=(n_{x},n_{y},n_{z})}$ 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]

${\displaystyle H=(\partial {\bf {n}})^{2}+(\epsilon _{ijk}{\bf {n}}\cdot \partial _{i}{\bf {n}}\times \partial _{j}{\bf {n}})^{2}}$

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]}