Emergent Magnetic Field and Nonzero Gyrovector of the Toroidal Magnetic Hopfion

TL;DR

This paper analytically calculates the gyrovector of magnetic hopfions, revealing it remains nonzero even in infinite samples, which is crucial for understanding hopfion dynamics.

Contribution

It introduces an analytical method based on emergent magnetic fields to compute the gyrovector of magnetic hopfions, including the toroidal case with Hopf index 1.

Findings

Gyrovector of hopfions is nonzero in infinite samples.

Dependence of gyrovector components on magnetic dot size is established.

Analytical approach aids in understanding hopfion dynamics.

Abstract

Magnetic hopfions are localized magnetic solitons with a nonzero 3D topological charge (Hopf index). Herein, an analytical calculation of the magnetic hopfion gyrovector is presented and it is shown that it does not vanish even in an infinite sample. The calculation method is based on the concept of the emergent magnetic field. The particular case of the simplest nontrivial toroidal hopfion with the Hopf index $\|Q_H\|=1$ in the cylindrical magnetic dot is considered and dependencies of the gyrovector components on the dot sizes are calculated. Nonzero hopfion gyrovector is important in any description of the hopfion dynamics within the collective coordinate Thieles approach.