Electronic scattering off a magnetic hopfion

Abstract: We study scattering of itinerant electrons off a magnetic hopfion in a three-dimensional metallic magnet described by a magnetization vector $\mathbf S(\mathbf r)$. A hopfion is a confined topological soliton of $\mathbf S(\mathbf r)$ characterized by an {\it emergent} magnetic field $B_\gamma(\mathbf r) \equiv \epsilon_{\alpha\beta\gamma} \,\mathbf S\cdot(\nabla_\alpha \mathbf S\times \nabla_\beta \mathbf S)/4 \neq 0$ with vanishing average value $\langle \mathbf B(\mathbf r)\rangle = 0$. 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 topological Hall effect in non-collinear magnetic structures necessarily implies $\langle \mathbf B(\mathbf r)\rangle \neq 0$. 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.