Wave fields with spiral phase dislocations carrying orbital angular momentum (OAM) have been realized in many branches of physics, such as for photons, sound waves, electron beams, and neutrons. However, the OAM states of magnons (spin waves)$-$the building block of modern magnetism$-$and particularly their implications have yet to be addressed. Here, we theoretically investigate the twisted spin-wave generation and propagation in magnetic nanocylinders. The OAM nature of magnons is uncovered by showing that the spin-wave eigenmode is also the eigenstate of the OAM operator in the confined geometry. Inspired by optical tweezers, we predict an exotic magnetic tweezer effect by showing skyrmion gyrations under twisted magnons in exchange coupled nanocylinder$|$nanodisk heterostructure, as a practical demonstration of magnonic OAM to manipulate topological spin defects. Our study paves the way for the emerging magnetic manipulations by harnessing the OAM degree of freedom of magnons.
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