Emergent Magnetic Field and Nonzero Gyrovector of the Toroidal Magnetic Hopfion

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 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 Thiele's approach.

Emergent magnetic field and vector potential of the toroidal magnetic hopfions

“…The details on the calculations of the emergent magnetic field components for the particular case (m=1, n=1) and their connection with the hopfion gyrovector are given in Ref. [35]. For such particular case the simple relation 𝑩(𝒓) = [4𝑎 (𝑎 2 + 𝑟 2 ) ⁄ ]𝑨(𝒓) holds.…”