Topological characterization of Hopfions in finite-element micromagnetics

TL;DR

This paper presents a finite-element method for accurately calculating the Hopf index in three-dimensional micromagnetic simulations, enabling better analysis of complex topological magnetic structures like Hopfions in various geometries.

Contribution

The authors introduce a gauge-invariant finite-element approach with a variance-based correction scheme for precise topological characterization in complex 3D magnetic textures.

Findings

Validated method with an analytical Hopfion structure.

Detected topological transition via abrupt change in Hopf index.

Achieved fast, mesh-dependent convergence in simulations.

Abstract

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.