Inertial motion of a magnetic hopfion in the framework of internal local modes

Magnetic hopfions are a novel kind of topological solitons existing in three-dimensional (3D) magnets, including chiral magnets, frustrated magnets, and magnetic multilayers. Their rich dynamics is demonstrated to be promising in constructing nonconventional computing architectures. However, the mechanism of a hopfion's inertial motion, which is associated with finding its internal eigenmodes, is still being explored either by rigid-body approximation or by 3D micromagnetic simulation. The first neglects the excitation of internal eigenmodes, while the latter costs substantial computing resources but cannot exhaust the topic. In this paper, a framework based on local eigenmodes is proposed to calculate inertial motion. Utilizing the variational method, the complexity of solving inertial motion is significantly reduced to an eigenproblem with fewer dimensions, thereby fully revealing the modulation mechanism of a hopfion's inertia. We calculate two kinds of inertial motion: translation and rotation, within our framework. This demonstrates that our framework is a comprehensive tool for studying inertial motion. Our research highlights the fascinating dynamics of hopfions and is expected to facilitate the modulation and manipulation of hopfions for wider applications.