Exact Hopfion Vortices in a 3D Heisenberg Ferromagnet

TL;DR

This paper presents exact static soliton solutions called hopfion vortices in a 3D Heisenberg ferromagnet, revealing their topological properties, energy characteristics, and stability conditions influenced by inhomogeneity.

Contribution

It provides the first exact solutions for hopfion vortices in an inhomogeneous 3D Heisenberg ferromagnet, including their topological invariants and stability analysis.

Findings

Hopf invariant of the soliton is an integer H=nm.

Preimages form unknots or knots, linking H times.

The energy has a sublinear dependence on H.

Abstract

We find exact static soliton solutions for the unit spin vector field of an inhomogeneous, anisotropic three-dimensional Heisenberg ferromagnet. Each soliton is labeled by two integers $n$ and $m$. It is a (modified) skyrmion in the $z=0$ plane with winding number $n$, which twists out of the plane $m$ times in the $z$-direction to become a 3D soliton. Here $m$ arises due to the periodic boundary condition at the $z$-boundaries. We use Whitehead's integral expression to find that the Hopf invariant of the soliton is an integer $H =nm$. It represents a hopfion vortex. Plots of the preimages of this topological soliton show that they are either unknots or nontrivial knots, depending on $n$ and $m$. Any pair of preimage curves links $H$ times, corroborating the interpretation of $H$ as a linking number. We also calculate the exact energy of the hopfion vortex, and show that its topological lower bound has a sublinear dependence on $H$. Using Derrick's scaling analysis, we demonstrate that the presence of a spatial inhomogeneity in the anisotropic interaction, which in turn introduces a characteristic length scale in the system, leads to the stability of the hopfion vortex.