Fig. 3. Mechanism for realizing omnidirectional motion of a hopfion in the 3D space.

(A) Proposed experimental setup for omnidirectional motion. The cyan glass is the magnet, in the center of which a violet torus is embedded, representing the hopfion. The red wires are the antennae responsible for inducing exciting magnetic fields from currents. The two antennae above the magnet are used to excite LX waves, while the two below the magnet are used to excite LY waves. The four antennae are distributed on both sides of the magnet and constitute a pair of spin-wave sources. The spin waves are released using sub-terahertz spin oscillators (45–48). (B) Non-coplanar Hall planes of the two spin-wave sources. The cyan cones denote the cones of imaginary displacements. The red planes are the Hall planes. (C) Trajectories of hopfions driven by spin waves of different kinds of polarization emanating from each of the two spin-wave sources. These results confirm the prediction in (B).