Controlling Knot Topology in Magnetic Hopfions via Spin-orbit Torque

Abstract

Knots, characterized by topological invariants called the Hopf number $H$, arise from the intertwining of strings and exhibit diverse configurations. The knot structures have recently been observed in condensed matters, as examplified by a magnetic hopfion, sparking interest in controlling their topology. Here, we show that spin-orbit torque (SOT) enables dynamic manipulation of the Hopf number of magnetic hopfions. We investigate the SOT-driven evolution of hopfions, revealing the splitting of a high-$H$ hopfion into multiple lower-$H$ ones, a process that can be quantified by an effective tension picture. Comparative analysis across different $H$ uncovers a hierarchy of instabilities that dictates these dynamical topological transitions. These findings establish SOT as a powerful tool for controlling hopfion topology, paving the way for potential applications in topological memory devices.

Figures (3)

• Figure 1: (a) Schematic illustrations of magnetic hopfions with $H = 1, 2$, and $4$, shown by the isosurfaces of constant $S^z$, along with their corresponding preimages that satisfy $S^x = 1$ (blue) and $S^x = -1$ (red). The color code in the inset is used throughout this paper to depict spin orientations. (b) A typical setup for this study. A spin current injected from the lower heavy metal layer works on magnetic moments as a SOT, thereby inducing hopfion dynamics in the upper magnetic layer. (c,d) Snapshots of the $H = 2$ (c) and $H = 4$ (d) hopfions under $(B, \zeta) = (0.003, 0.002)$ and $(B, \zeta) = (0.003, 0.0021)$, respectively. The lower panels show their preimages to support the observation of changes in the knot topology.
• Figure 2: Time evolution of the absolute sum of the spatial gradient of energy for the result of Fig. \ref{['schematic']}(c). Splitting occurs around the white time region, during which the value rapidly increases. The horizontal gray dashed line is the reference corresponding to twice the value of the gradient sum of a relaxed $H=1$ hopfion in the absence of the SOT. The insets shows the dynamics around the splitting. The colored dots represent the spins at each site. The arrows shown in grayscale represent the gradient vectors $\nabla_{\mathbf{r}} \mathcal{H}_i$ and their norms.
• Figure 3: SOT-driven steady-state phase diagrams for (a) $H = 1$, (b) $H = 2$, and (c) $H = 4$ hopfions, with plots of the local Hopf number $\tilde{H}$ as a function of $\zeta$. The lowerleft inset of (c) shows an enlarged plot for $0.0020 \lesssim \zeta \lesssim 0.0025$. We take $B=0.003$.