Nonequilibrium dynamics of magnetic hopfions driven by spin-orbit torque

TL;DR

This paper explores the nonequilibrium behavior of magnetic hopfions with different Hopf numbers under spin-orbit torque, revealing their controllability and potential for topological spintronic applications.

Contribution

It provides a theoretical analysis of hopfion dynamics driven by spin-orbit torque, including splitting, recombination, and hierarchical behavior across Hopf numbers.

Findings

SOT induces translational and precessional motion in H=1 hopfions.

Intermediate SOT can split H=2 hopfions into two H=1 hopfions.

Controlled SOT scheduling enables topology switching of hopfions.

Abstract

Hopfions--three-dimensional topological solitons with knotted spin texture--have recently garnered attention in topological magnetism due to their unique topology characterized by the Hopf number $H$, a topological invariant derived from knot theory. In contrast to two-dimensional skyrmions, which are typically limited to small topological invariants, i.e., skyrmion numbers, hopfions can, in principle, be stabilized with arbitrary Hopf numbers. However, the nonequilibrium dynamics, especially interconversion between different Hopf numbers, remain poorly understood. Here, we theoretically investigate the nonequilibrium dynamics of hopfions with various Hopf numbers by numerically solving the Landau-Lifshitz-Gilbert equation with spin-orbit torque (SOT). For $H=1$, we show that SOT induces both translational and precessional motion, with dynamics sensitive to the initial orientation. For $H=2$, we find that intermediate SOT strengths can forcibly split the hopfion into two $H = 1$ hopfions. This behavior is explained by an effective tension picture, derived from the dynamics observed in the $H=1$ case. By comparing the splitting dynamics across different $H$, we identify a hierarchical structure governing SOT-driven behavior and use it to predict the dynamics of hopfions with general $H$. Furthermore, we show that by appropriately scheduling the time dependence of the SOT, it is possible to repeatedly induce both splitting and recombination of hopfions. These results demonstrate the controllability of hopfion topology via SOT and suggest a pathway toward multilevel spintronic devices based on topology switching.