Sub-GHz breathing dynamics of magnetic hopfions.

Magnetic hopfions are three-dimensional topological solitons whose static stability has recently been confirmed in experiments, yet their dynamical modes remain largely unexplored. Here we combine micromagnetic simulations and analytical modelling to characterise the fundamental breathing excitation of hopfions. We show that the breathing mode corresponds to a coherent oscillation of both the hopfion core diameter and the shell width, while preserving the topological charge. An analytical domain-wall interaction model explains the weak field dependence of the shell thickness and yields a closed-form expression for the restoring stiffness. From this curvature and the collective-coordinate inertia, we derive an estimated breathing frequency in excellent agreement with micromagnetic spectra. The ability to capture the hopfion dynamics quantitatively from material constants highlights a direct route to experimental detection by ferromagnetic resonance or Brillouin light scattering, and establishes a framework for frequency-encoded control in reconfigurable spintronic devices.