  Different from the skyrmion driven by the Dzyaloshinskii-Moriya interaction
in non-centrosymmetric materials, the skyrmionic texture in centrosymmetric
magnet may possess the extra internal degrees of freedom, which greatly enrich
its morphologies, imply the continuous deformation allowed by topological
protection, and enable flexible tunability and potential functionality. To
describe the internal degree of freedom to full extent, the conventional
integer-valued scalar vorticity is extended into a vector vorticity with
continuous rotation allowed. The further simplified vorticity angle, together
with helicity angle, represents the whole rotation freedom of spin space. The
centrosymmetric magnet with frustration provides a perfect platform for
realizing a controllable manipulation on these internal degrees of freedom,
where the magnetic field can be applied to tune vorticity continuously in both
crystal and isolated forms of skyrmionic texture, and the helicity can be
further controlled by electric field. Moreover, the simulation reveals the
distinctive dynamic effects related to the vorticity modulation, namely, a
straight motion can be generated by rotating magnetic field to tune vorticity,
and the vorticity can also be controlled to modulate the dynamics induced by
the spin polarized current.
