Intrinsic non-adiabatic topological torque in magnetic skyrmions and vortices

We propose that topological spin currents flowing in topologically non-trivial magnetic textures, such as magnetic skyrmions and vortices, produce an intrinsic non-adiabatic torque of the form ${\bf T}_t\sim [(\partial_x{\bf m}\times\partial_y{\bf m})\cdot{\bf m}]\partial_y{\bf m}$. We show that this torque, which is absent in one-dimensional domain walls and/or non-topological textures, is responsible for the enhanced non-adiabaticity parameter observed in magnetic vortices compared to one-dimensional textures. The impact of this torque on the motion of magnetic skyrmions is expected to be crucial, especially to determine their robustness against defects and pinning centers.