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, its dynamics differs significantly from the familiar case of a taut string thanks to an additional Berry-phase term reflecting the precessional nature of the spins. The Berry phase gives rise to nonreciprocal dynamics, where the waves traveling left and right have different speeds. Given an initial deformation, the nonreciprocal string with free ends exhibits uniform translational motion in addition to oscillations, as if it is self-driven, which is allowed by the broken time-reversal symmetry. This can be understood from the view of the conserved transverse momentum derived from Noether’s theorem. It is then evident how the domain wall evolves in the presence of an external magnetic field and dissipation.