Fig. 4: Deterministic skyrmion molecule manipulation via boundary engineering.

a Boundary modulation by injecting measured amounts of silicone oil into the boundary cavities, tuning their depth. b Band structure evolution under cavity depth change \(\delta h\). As \(\delta h\) increases, the even band shifts toward high frequencies, detaching from the Dirac points and causing deformation and annihilation of the even skyrmion molecules. Concurrently, the odd band emerges and intersects the Dirac points at \(\delta h=1.2\) mm, signaling the odd skyrmion molecules. c, d Measurements analogous to Fig. 3c, d, but with boundary cavities modulated by \(\delta h=1.2\) mm. Observation of skyrmion molecules \(\left\{{{{{\rm{S}}}}}_{-},{{{{\rm{S}}}}}_{+}\right\}\) and their vortex-like scalar features reveals polarizability inversion relative to the even molecules, validating the boundary engineering control. The Fourier spectrum retains valley-locking, affirming symmetry-protected robustness under boundary engineering.