Fig. 1. Propagating excitation modes on skyrmion strings.

a Schematic illustration of skyrmion strings. b, c Top view optical image of device structure and side view illustration of a coplanar waveguide, where the AC current injected from network analyzer (NA) generates oscillating magnetic field H^ν and causes spin excitation in the neighboring Cu₂OSeO₃. d H–T magnetic phase diagram for Cu₂OSeO₃, obtained with the cooling path shown by the arrows. SkX, C, H, FM, and P represent the skyrmion lattice, conical, helical, ferromagnetic, and paramagnetic states, respectively. e The corresponding H-dependence of magnetic resonance spectra ΔL₁₁ at 25 K. f–h Schematic illustration of CCW, breathing, and CW excitation modes on skyrmion strings. The central part represents the local oscillation manner of skyrmion at the z = 0 plane. The upper and lower parts are the snapshot images describing how the spin excitation launched at z = 0 propagates on the skyrmion strings, along the  ±z direction parallel and antiparallel to H, respectively. The cross-sectional images describing the size and position of skyrmion at selected z-planes (shown by red layers) are also indicated. i The direction of local magnetic moment at the core and edge position of skyrmion in each z layer. Black rounded arrows denote the sense of local moment precession in the time domain, and ± symbols indicate the sign of local DM energy gain ϵ_DM. j–m Calculated spatial distribution of local magnetization direction m₀(r) in the ground state and local amplitude of elliptical magnetization precession $\mathcal{A}\left(\mathbf{r}\right)$ for the CCW, B, and CW excitation modes in the SkX phase. In j, the arrows and background color ($m_0^z\left(\mathbf{r}\right)$) represent the in-plane and out-of-plane component of m₀(r), respectively. n Schematic illustration of local magnetization dynamics, described by $\mathbf{m}={\mathbf{m}}_0+\mathrm{Re}\left[\left(\delta m_{\mathrm{a}}{\mathbf{e}}_{\mathrm{a}}+i\delta m_{\mathrm{b}}{\mathbf{e}}_{\mathrm{b}}\right)\exp \left[-i2\pi \nu t\right]\right]$ with e_a and e_b being the orthogonal unit vectors normal to m₀. Here we define the local elliptical precession amplitude $\mathcal{A}\equiv \delta m_{\mathrm{a}}\delta m_{\mathrm{b}}$, whose spatial distribution for each mode is plotted in k–m.