Stability of hopfions in bulk magnets with competing exchange interactions

Magnetic hopfions are string-like three-dimensional topological solitons, characterised by the Hopf invariant. They serve as a fundamental prototype for magnetic quasi-particles and an inspiration novel device concepts in field of spintronics. Based on micromagnetic model without considering temperature, existence such has been predicted certain magnets with competing exchange interactions. However, physical realisation freely moving bulk have so far elusive. Here, we consider effective Heisenberg interactions study stability small toroidal number $Q_\text{H}=1$ finding first-order saddle points energy surface representing transition state decay via formation two coupled Bloch points. We combine geodesic nudged elastic band method adapted implementation dimer to resolve sharp profile reaction path near point. Our analysis reveals that barrier can reach substantial height is largely determined size hopfion relative lattice constant.