Stability of Hopfions in Bulk Magnets with Competing Exchange Interactions

TL;DR

This study investigates the stability of magnetic hopfions in bulk magnets with competing exchange interactions, revealing that they can have substantial energy barriers to decay, which depends on their size relative to the lattice.

Contribution

It introduces a detailed analysis of hopfion stability in an effective Heisenberg model, employing advanced computational methods to identify transition states and energy barriers.

Findings

Energy barriers for hopfion decay can be substantial.

Stability depends on hopfion size relative to the lattice.

First-order saddle points characterize decay transition states.

Abstract

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