Nonrelativistic Nambu-Goldstone modes propagating along a Skyrmion line

Abstract

We study Nambu-Goldstone (NG) modes or gapless modes propagating along a Skyrmion (lump) line in a relativistic and nonrelativistic $O(3)$ sigma model, the latter of which describes isotropic Heisenberg ferromagnets. We show for the nonrelativistic 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. The counting rule is consistent with the Nielsen-Chadha and Watanabe-Brauner relations only when we take into account quasi-NG modes.