Spin excitation spectrum of a magnetic hopfion

Abstract

Magnetic hopfions are three-dimensional topological solitons with a nontrivial Hopf index. Here, we theoretically investigated the spin excitation spectrum and revealed corresponding spin-wave modes of a magnetic hopfion. Compared with skyrmion tubes, the hopfions have distinctly less resonance peaks due to the suppression of vertical spin-wave modes by the internal topological defect. We also found that breathing and rotating modes could hybridize in hopfions under z-direction excitations and, thus, characterized the five individual resonance modes by a set of number pair (b, r). The results provide a fundamental understanding of the spin-wave modes of magnetic hopfions and open a route to detect and manipulate 3D topological solitons using microwave magnetic fields.