Non-relativistic Nambu-Goldstone modes propagating along a skyrmion line

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We study Nambu-Goldstone (NG) modes or gapless modes propagating along a skyrmion (lump) line in a relativistic and non-relativistic O(3) sigma model, the latter of which describes isotropic Heisenberg ferromagnets.

• We show for the non-relativistic case that there appear two coupled gapless modes with a quadratic dispersion.
• In addition to the well-known Kelvin mode consisting of two translational zero modes transverse to the skyrmion line, we show that the other consists of a magnon and dilaton, that is, a NG mode for a spontaneously broken internal U(1) symmetry and a quasi-NG mode for a spontaneously broken scale symmetry of the equation of motion.
• We find that the commutation relations of Noether charges admit a central extension between the dilatation and phase rotation, in addition to the one between two translations found recently.