Oscillation modes of skyrmion strings in a ferromagnetic film

It is shown that a skyrmion string in a ferromagnetic film consisting of $N$ atomic layers exhibits flexural oscillations at eigenfrequencies $|\omega_{m}|=(4J'S/\hbar)\sin^{2}[\pi m/(2N)]$, where $J'$ is the interlayer exchange coupling, $S$ is the length of the atomic spin, and $m=1,...,N-1$. This result is confirmed numerically for individual skyrmion strings in a discrete microscopic spin-lattice model of up to ten atomic layers, as well as for skyrmion-string lattices at finite temperature.