Fig. 5: Frequency splitting of spin-wave mode doublets in the Möbius strip.

a Mode frequencies for the Möbius strip (orange circles) and the bent ring (blue diamonds). While the bent ring exhibits nearly degenerate doublets, the Möbius strip shows a clear frequency splitting. For example, modes #7 and #8 in the bent ring both resonate at 15.68 GHz, whereas the corresponding Möbius modes occur at 15.41 GHz and 15.93 GHz, yielding a frequency difference of Δf ≈ 520 MHz. b Line scans of the dynamic magnetization projected along the central path of the Möbius strip, shown for several modes. The numbers above each scan indicate the number of wavelengths (in units of 2π phase) along the circular path. The scans exhibit odd half-integer mode numbers, consistent with the antisymmetric boundary conditions imposed by the Möbius topology. Clockwise and counterclockwise modes display visibly different wavelengths.