The dynamics of a vortex in a thin-film ferromagnet resembles the motion of a charged massless particle in a uniform magnetic field. Similar dynamics is expected for other magnetic textures with a nonzero skyrmion number. However, recent numerical simulations revealed that skyrmion magnetic bubbles show significant deviations from this model. We show that a skyrmion bubble possesses inertia and derive its mass from the standard theory of a thin-film ferromagnet. Besides center-ofmass motion, other low energy modes are waves on the edge of the bubble traveling with different speeds in opposite directions.Dynamics of topological defects is a topic of longstanding interest in magnetism. The attention to it stems from rich basic physics as well as from its connection to technological applications [1]. Theory of magnetization dynamics in ferromagnets well below the critical temperature is based on the Landau-Lifshitz equation [2] for the unit vector of magnetization m(r) = M(r)/M ,where γ is the gyromagnetic ratio, α ≪ 1 is a phenomenological damping constant [3], and the effective magnetic field is a functional derivative of the free energy, B(r) = −δU/δM(r). The latter includes local (e.g., exchange and anisotropy) as well as long-range (dipolar) interactions, thus making Eq. (1) a nonlinear and nonlocal partial differential equation with multiple length and time scales solvable in only a few simple cases. For example, translational motion of a rigid texture, m = m(r − R(t)), is fully parametrized by the texture's "center of mass" R. For steady motion, R(t) = Vt, the velocity can be obtained from Thiele's equation [4] expressing the balance of gyrotropic, conservative, and viscous forces:Here G is a gyrocoupling vector, F = −∂U/∂R is the net conservative force, and D is a dissipation tensor. A rigid texture moves like a massless particle with electric charge in a magnetic field and an external potential through a viscous medium. If G = 0, the "Lorentz force" greatly exceeds the viscous drag. We thus ignore dissipation. Although Eq. (2) was derived for steady motion, Thiele anticipated that it could serve as a good first approximation in more general situations. Indeed, his equation describes very well the dynamics of vortices in thin ferromagnetic films [5][6][7][8][9]. In this case, the gyrocoupling vector G = (0, 0, G) is proportional to a topological invariant known as the skyrmion charge q = (1/4π) dx dy m · (∂ x m × ∂ y m), the film thickness t, and the density of angular momentum M/γ; to wit, G = 4πqtM/γ. A vortex has q = ±1/2 and thus G = 0. In a parabolic potential well, U (X, Y ) = K(X 2 + Y 2 )/2, it moves in a circle at a frequency ω = K/G. Similar behavior is expected for other topologically nontrivial textures, e.g., magnetic bubbles in thin films with magnetization normal to the plane of the film [10][11][12]. A bubble is a circular domain with m z < 0 surrounded by a domain with m z > 0, or vice versa, Fig.

At short time scales, the inertia term becomes relevant for the magnetization dynamics of ferromagnets and leads to nutation for the magnetization vector. For the case of spatially extended magnetic systems, for instance, Heisenberg spin chains with the isotropic spin-exchange interaction, this leads to the appearance of a collective excitation, the “nutation wave,” whose properties are elucidated by analytical arguments and numerical studies. The one-particle excitations can be identified as relativistic massive particles. These particles, the “nutatons,” acquire their mass via the Brout–Englert–Higgs mechanism, through the interaction of the wave with an emergent topological gauge field. This spin excitation would appear as a peak in the spectrum of the scattering structure factor in inelastic neutron scattering experiments. The high frequency and speed of the nutation wave can open paths for realizing ultrafast spin dynamics.