Emergent magnetic field and vector potential of the toroidal magnetic hopfions

TL;DR

This paper analytically derives the vector potential, magnetic field, and magnetization configuration of toroidal magnetic hopfions, revealing their topological properties and coordinate transformations, advancing understanding of magnetic solitons.

Contribution

It provides an explicit analytical calculation of the magnetic hopfion's vector potential, magnetic field, and magnetization for arbitrary Hopf indices using spinor representation.

Findings

Explicit formulas for toroidal magnetic hopfions' vector potential and magnetic field.

Demonstration of the connection between Hopf charge density and coordinate Jacobian.

Analysis of the role of toroidal coordinates in gauge and topological properties.

Abstract

Magnetic hopfions are localized magnetic solitons with non-zero 3D topological charge (Hopf index). Here I present an analytical calculation of the toroidal magnetic hopfion vector potential, emergent magnetic field, the Hopf index, and the magnetization configuration. The calculation method is based on the concept of the spinor representation of the Hopf mapping. The hopfions with arbitrary values of the azimuthal and poloidal vorticities are considered. The special role of the toroidal coordinates and their connection with the emergent vector po tential gauge are demonstrated. The hopfion magnetization field is found explicitly for the arbitrary Hopf indices. It is shown that the Hopf charge density can be represented as a Jacobian of the transformation from the toroidal to the cylindrical coordinates.