Three-dimensional topological orbital Hall effect caused by magnetic hopfions

TL;DR

This work addresses identifying magnetic hopfions, 3D textures characterized by the Hopf index, by predicting a three-dimensional orbital Hall effect arising from their emergent field. Using a tight-binding sd model and a Kubo-based calculation of orbital Berry curvature, the authors show that hopfions exhibit multiple nonzero orbital Hall components (xy, yz, zx) whose magnitudes scale with the average squared emergent field, while the conventional topological Hall signal remains suppressed due to field compensation. The key insight is that the Hopf index imprints a distinct orbital transport signature, providing a practical electronic hallmark for hopfions and enabling their detection in device contexts. These results point to potential spin-orbitronic applications leveraging large orbital currents and torques in hopfion-hosting materials.

Abstract

Magnetic hopfions are non-collinear spin textures that are characterized by an integer topological invariant, called Hopf index. The three-dimensional magnetic solitons can be thought of as a tube with a twisted magnetization that has been closed at both ends to form a torus. The tube consists of a magnetic whirl called in-plane skyrmion or bimeron. Although hopfions have been observed by microscopy techniques, their detection remains challenging as they lack an electronic hallmark so far. Here we predict a three-dimensional orbital Hall effect caused by hopfion textures: When an electric field is applied, the hopfion generates a transverse current of orbital angular momentum. The effect arises due to the local emergent field that gives rise to in-plane and out-of-plane orbital Hall conductivities. This orbital Hall response can be seen as a hallmark of hopfions and allows us to distinguish them from other textures, like skyrmioniums, that look similar in real-space microscopy experiments. While the two-dimensional topological invariant of a skyrmion determines its topological Hall transport, the unique three-dimensional topological orbital Hall effect can be identified with the three-dimensional topological invariant that is the Hopf index. Our results make hopfions attractive for spin-orbitronic applications because their orbital signatures allow for their detection in devices and give rise to large orbital torques.

Figures (7)

• Figure 1: Magnetic hopfion. (a) Magnetic texture $\boldsymbol{m}(\boldsymbol{r})$ of a $\lambda=8a$ hopfion. (b) Corresponding emergent field $\boldsymbol{B}_\mathrm{em}(\boldsymbol{r})$. (c) A cut along the $xy$ plane reveals a magnetic skyrmionium. (d) A cut along the the $yz$ plane reveals two in-plane skyrmions also called bimerons. The color of the arrows in all panels encodes their orientation in the $xy$ plane. White and black arrows point along the positive and negative $z$ direction, respectively. Note that these arrows that surround the hopfion have been plotted translucently to allow for a clearer view of the non-collinear part of the texture.
• Figure 2: Electronic properties caused by a hopfion with $m=7t$. (a) Density of states [Eq. \ref{['eq:dos']}] (blue) and comparison to the density of states without the hopfion (gray). (b) Slices of the orbital Berry curvature $\Omega_{\nu,xy}^{L_z}(\boldsymbol{k})$ of the lowest band $\nu=1$.
• Figure 3: Three-dimensional orbital Hall effect caused by a hopfion in the strong-coupling limit. (a) The orbital conductivity tensor element $\sigma_{xy}^{L_z}$ as a function of energy. This orbital Hall conductivity characterizes a transport of orbital angular momentum $L_z$ in the Hopfion plane $xy$. Application of an electric field $\boldsymbol{E}$ along $y$ gives rise to an orbital current $\boldsymbol{j}^{L_z}$ along $x$; see schematic image above. The orbital angular momentum is polarized out of plane. (b) The equivalent results for the $\sigma_{yz}^{L_x}=\sigma_{zx}^{L_y}$ tensor element. The Hund's coupling is characterized by $m=7t$ so that the conduction electrons' spins almost completely align with the hopfion texture. The color of the curve indicates the temperature; see legend.
• Figure 4: Three-dimensional orbital Hall effect caused by a hopfion in the weak-coupling limit. The figure is analogue to Fig. \ref{['fig:hopfion7']} but with a weaker Hund's coupling of $m=2t$. (a) The orbital conductivity tensor element $\sigma_{xy}^{L_z}$ and (b) the tensor element $\sigma_{yz}^{L_x}=\sigma_{zx}^{L_y}$ are shown as a function of energy for various temperatures.
• Figure 5: Orbital Hall effect and topological Hall effect caused by a skyrmion tube. (a) Magnetic texture of the skyrmion tube. The color indicates the orientation of the magnetic moments as in Fig. \ref{['fig:overview']}. (b) Corresponding emergent field that is collinear and points along the tube direction. (c) Orbital Hall conductivity $\sigma_{xy}^{L_z}$ as a function of energy. (d) Topological Hall conductivity $\sigma_{xy}$ as a function of energy. The color of the curve indicates the temperature; see legend. The Hund's coupling is characterized by $m=7t$.