We study scattering of itinerant electrons off a magnetic hopfion in a three-dimensional metallic magnet described by a magnetization vector $\mathit{S}\left(\mathit{r}\right)$. A hopfion is a confined topological soliton of $\mathit{S}\left(\mathit{r}\right)$ characterized by an emergent magnetic field $B_{\gamma }\left(\mathit{r}\right)\equiv {\epsilon }_{\alpha \beta \gamma }\mathit{S}·({\nabla }_{\alpha }\mathit{S}\times {\nabla }_{\beta }\mathit{S})/4\ne 0$ with vanishing average value $\langle \mathit{B}\left(\mathit{r}\right)\rangle =0$. We evaluate the scattering amplitude in the opposite limits of large and small hopfion radius $R$ using the eikonal and Born approximations, respectively. In both limits, we find that the scattering cross section contains a skew-scattering component giving rise to the Hall effect within a hopfion plane. That conclusion contests the popular notion that the Hall effect in noncollinear magnetic structures necessarily implies $\langle \mathit{B}\left(\mathit{r}\right)\rangle \ne 0$. In the limit of small hopfion radius $pR\ll 1$, we expand the Born series in powers of momentum $p$ and identify different expansion terms corresponding to the hopfion anisotropy, toroidal moment, and skew scattering.

• Received 14 April 2021
• Accepted 22 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.075102