Topological characterization of Hopfions in finite-element micromagnetics

Topological magnetic structures, such as Hopfions, are central to three-dimensional magnetism, but their characterization in complex geometries remains challenging. We introduce a robust finite-element method for calculating the Hopf index in micromagnetic simulations of three-dimensional nanostructures. By employing the Biot-Savart form for the vector potential, our approach ensures gauge-invariant results, even in multiply connected geometries like tori. A novel variance-based correction scheme significantly reduces numerical errors in highly inhomogeneous textures, achieving accurate Hopf index values with fast mesh-dependent convergence. We validate the method using an analytically defined Hopfion structure and demonstrate its ability to detect topological transitions through a simulation of a Hopfion's field-induced destruction into a toron, marked by an abrupt change in the Hopf index. This method enables precise quantification of topological features in complex three-dimensional magnetic textures forming in finite-element micromagnetic simulations, offering a powerful tool for advancing topological magnetism studies in general geometries.