Exact hopfion vortices in a 3D Heisenberg ferromagnet

• DOI: 10.1016/j.physleta.2023.128975
• Publication Date: 2023-06-21T21:23:09Z
• Authors: Radha Balakrishnan; Rossen Dandoloff; Avadh Saxena
• Keywords: Condensed Matter - Materials Science; Nonlinear Sciences - Exactly Solvable and Integrable Systems; Materials Science (cond-mat.mtrl-sci); FOS: Physical sciences; Exactly Solvable and Integrable Systems (nlin.SI)

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.<br/>6 pages, 2 figures<br/>