The three-dimensional emergent magnetic field ${\mathbf{B}}^e$ of a magnetic hopfion gives rise to emergent magnetomultipoles 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 $T_z^e=\frac{1}{2}\int (\mathbf{r}\times {\mathbf{B}}^e{)}_{z}dV$ and emergent magnetic octupole component ${\mathrm{\Gamma }}^e=\int [(x^2+y^2)B_z^e-xz{B}_x^e-yz{B}_y^e]dV$. The hopfion exhibits nonreciprocal dynamics (nonlinear hopfion Hall effect) under an ac driving current applied along (perpendicular to) the direction of $T_z^e$. 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 Letter reveals the vital roles of emergent magnetomultipoles in nonlinear hopfion dynamics and could stimulate further investigations on the dynamical responses of topological spin textures induced by emergent electromagnetic multipoles.

• Received 20 June 2022
• Revised 13 October 2022
• Accepted 6 December 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.267201