Hopfions in Anisotropic Heisenberg Magnets

We find exact static soliton solutions for the unit spin vector field of an inhomogeneous, anisotropic three-dimensional (3D) Heisenberg ferromagnet. Using Whitehead's integral expression, we calculate the corresponding Hopf invariant H analytically and obtain an integer, showing that these solitons are Hopfions. H is a product of two integers, the first being the usual winding number of a baby Skyrmion in two dimensions, while the second encodes the periodicity in the third dimension. We study the underlying geometry of H, by mapping the 3D unit vector field to tangent vectors of three appropriately defined space curves. Our analysis shows that a certain intrinsic twist is necessary to yield a nontrivial topological invariant. We illustrate our results using the invariant H of the exact Hopfions we have found.