Numerical Calculation of the Hopf Index for 3D Magnetic Textures

TL;DR

This paper reviews and compares numerical methods for calculating the Hopf index in 3D magnetic textures, addressing accuracy, efficiency, and common computational challenges to enhance topological analysis.

Contribution

It introduces and evaluates various numerical techniques for computing the Hopf index, providing insights into their accuracy and robustness for 3D magnetic structures.

Findings

Different methods have varying accuracy and computational efficiency.

Common pitfalls in numerical Hopf index calculation are identified and addressed.

Recommendations for improving robustness of topological computations are provided.

Abstract

To gain deeper insight into the complex, stable, and robust configurations of magnetic textures, topological characterisation has proven essential. In particular, while the skyrmion number is a well-established topological invariant for 2D magnetic textures, the Hopf index serves as a key topological descriptor for 3D magnetic structures. In this work, we present and compare various methods for numerically calculating the Hopf index, provide implementations, and offer a detailed analysis of their accuracy and computational efficiency. Additionally, we identify and address common pitfalls and challenges associated with the numerical computation of the Hopf index, offering insights for improving the robustness of these techniques.