A hopfion is a

topological soliton A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...

. It is a stable three-dimensional localised configuration of a three-component field

\vec=(n_x,n_y,n_z)

with a knotted topological structure. They are the three-dimensional counterparts of

skyrmions In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological solito ...

, which exhibit similar topological properties in 2D. The soliton is mobile and stable: i.e. it is protected from a decay by an

energy barrier In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...

. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician,

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

. A model that supports hopfions was proposed as follows

H= (\partial )^2 + (\epsilon_\cdot\partial_i \times \partial_j)^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.

Experimental observation

Hopfions have been observed experimentally in Ir/Co/Pt multilayers using

X-ray magnetic circular dichroism X-ray magnetic circular dichroism (XMCD) is a difference spectrum of two X-ray absorption spectra (XAS) taken in a magnetic field, one taken with left circularly polarized light, and one with right circularly polarized light. By closely analyzing ...

and in the polarization of free-space monochromatic light.

References

Magnetism Quasiparticles {{electromagnetism-stub

Experimental observation

See also

References