Hopfion

A hopfion is a topological soliton. 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.

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:

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

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

Stable hopfions were predicted within various physical platforms, including Yang–Mills theory, superconductivity and magnetism.