Exact Hopfion Vortices in a 3D Heisenberg Ferromagnet

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.

6 pages, 2 figures

References in corpus (4)

• Advances in the Physics of Magnetic Skyrmions and Perspective for Technology
• Creation and confirmation of Hopfions in magnetic multilayer systems
• Skyrmion knots in frustrated magnets
• Hopf solitons and elastic rods

Cited by in corpus (4)

• Pseudo-spectral Landau-Lifshitz description of magnetization dynamics
• Twisted curve geometry underlying topological invariants
• Hopfions in screw chiral magnets
• Construction of Hopfion Crystals