Emergent Magneto-multipoles and Nonlinear Responses of a Magnetic Hopfion

TL;DR

This paper investigates the emergent magneto-multipoles of magnetic hopfions and their role in nonlinear electromagnetic responses, revealing nonreciprocal dynamics and potential for advanced spintronic applications.

Contribution

It introduces the concept of emergent magneto-multipoles in hopfions and analyzes their influence on nonlinear responses and nonreciprocal dynamics under ac currents.

Findings

Hopfions exhibit emergent magnetic toroidal and octupole moments.

Nonlinear hopfion Hall effect depends on hopfion polarity and chirality.

Emergent multipoles significantly affect nonlinear electromagnetic responses.

Abstract

The three-dimensional emergent magnetic field $\textbf{B}^e$ of a magnetic hopfion gives rise to emergent magneto-multipoles in a similar manner to the multipoles of classical electromagnetic field. Here, we show that the nonlinear responses of a hopfion are characterized by its emergent magnetic toroidal moment ${T}^e_z = \frac{1}{2}\int (\textbf{r}\times \textbf{B}^e)_z dV$ and emergent magnetic octupole component ${\it \Gamma}^e =\int [(x^2+y^2)B^e_z - xz B^e_x - y z B^e_y] dV$. The hopfion exhibits nonreciprocal dynamics (nonlinear hopfion Hall effect) under an ac driving current applied along (perpendicular to) the direction of ${T}^e_z$. The sign of nonreciprocity and nonlinear Hall angle is determined by the polarity and chirality of hopfion. The nonlinear electrical transport induced by a magnetic hopfion is also discussed. This work reveals the vital roles of emergent magneto-multipoles in nonlinear hopfion dynamics and could stimulate further investigations on the dynamical responses of topological spin textures induced by emergent electromagnetic multipoles.