Intrinsic nonadiabatic topological torque in magnetic skyrmions and vortices

We propose that topological spin currents flowing in topologically nontrivial magnetic textures, such as magnetic skyrmions and vortices, produce an intrinsic nonadiabatic torque of the form ${T}ₜ{~}[({{∂}}ₓm×{}{{∂}}ym)·{}m]{{∂}}ym$. We show that this torque, which is absent in one-dimensional domain walls and/or nontopological textures, is responsible for the enhanced nonadiabaticity 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.