  Skyrmion crystals (SkX) are periodic alignment of magnetic skyrmions, i.e., a
type of topologically protected spin textures. Compared with ordinary crystals,
they can be drastically deformed under anisotropic effects because they are
composed of field patterns whose deformation does not cause any bond-breaking.
This exotic ductility of SkX bring about great tunability of its collective
excitations called emergent phonons, which are vital for magnonics application.
The question is how to quantitatively determine the emergent phonons of
distorted SkX. Here we systematically study the long wavelength emergent
phonons of SkX distorted by (a) a negative exchange anisotropy, and (b) a
tilted magnetic field. In both cases, deformation and structural transitions of
SkX thoroughly influence the frequency, anisotropy of vibrational pattern and
dispersion relation, and coupling between lattice vibration and in-lattice
vibration for all modes. Tilted magnetic fields are very effective in tuning
the emergent phonons, such that all modes except the Goldstone mode can be
excited by AC magnetic fields when a tilted bias field is presented.
