Emergent Magneto-multipoles and Nonlinear Responses of a Magnetic Hopfion

The three-dimensional emergent magnetic field ${\mathbf{\text{B}}}^e$ 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 $T_z^e=\frac{1}{2}\int (\mathbf{\text{r}}\times {\mathbf{\text{B}}}^e{)}_{z}dV$ and emergent magnetic octupole component ${\Gamma }^e=\int [(x^2+y^2)B_z^e-xz{B}_x^e-yz{B}_y^e]dV$. 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 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.