Ferromagnetic domain wall as a nonreciprocal string

The collective coordinate method has been a great success in analyzing the dynamics of point-like topological defects, such as vortices in two dimensions and a domain wall in one dimension. To study a domain wall in a two-dimensional ferromagnetic film, we generalize the discrete collective coordinates into a continuous field describing the displacement of the wall. The domain wall moves like a string. However, because of the precessional nature of the spins, its dynamics is nonreciprocal: the waves traveling left and right have different speeds, differing from the familiar case of an elastic string. Given an initial deformation, the nonreciprocal string with free ends exhibits a uniform transverse motion in addition to oscillations, which is allowed by the broken time-reversal symmetry. We provide an intuitive explanation of this phenomena by deriving the conserved momentum with a geometric contribution. This framework also leads to a good understanding of how a domain wall evolves in the presence of external perturbations and dissipation.