Fig. 1: Topological structures of half-skyrmion strings and colloidal response.

a Schematic of the formation of skyrmion strings. The bottom substrate adopts alternating splay-bend distortions, and the top surface is along the y-axis. The skyrmion strings are produced in the splay regions (black). b Micrograph of colloids in skyrmion strings. The point defect of the colloid is on the right-hand side, and \(\hat{{{\bf{p}}}}\) points along the −y-axis (φ  = 180°). The black arrow points at the defect. The violet arrow represents \(\hat{{{\bf{p}}}}\). c\(\,\hat{{{\bf{p}}}}\) points along the y-axis (φ = 0°). d Schematic 3D half-skyrmion string. It is defined as a half-anti-skyrmion (1/2, − 1/2, 0, 0). e A half-skyrmion string is defined as a half-Néel-skyrmion (1/2, 1/2, 0, π). f Schematic vector field of a half-anti-skyrmion (1/2, − 1/2, 0, 0); Red (blue) region shows that the y-component of the director is positive(negative), while the white region shows that the director is in-plane. g Different perspectives of the vector field wrapping the unit sphere. h Schematic vector field of the half-Néel-skyrmion (1/2, 1/2, 0, π). i Corresponding unit sphere. j Simulated topological structures (blue frame from part (b)) of the two half-anti-skyrmions (1/2, − 1/2, 0, 0) formed in the splay regions at x = 0 and x = 2 L. The color represents the component of the director field on the z-axis. k Topological structures (red frame from part (c)) of adjacent half-anti-skyrmions (− 1/2, − 1/2, 0, π) at x = 0 and half-Néel-skyrmions (1/2, 1/2, 0, π) at x = 2L. l Experimental (Exp.) and simulated (Sim.) FCPM images in the vertical cross section through the half skyrmions. The linearly polarized excitation lights for FCPM are marked in yellow. m Mapping of the interaction potential between colloids and half-anti-skyrmions (1/2, − 1/2, 0, 0). The colloid with φ = 180° has minimized potential at location 1 (black dashed line). The arrow represents the locally preferred orientations of the dipoles. n Potential mapping between colloids and half-Néel skyrmions (1/2, 1/2, 0, π). The colloid with φ = 0° is at location 2 (black dashed line). Red solid lines indicate the center of the skyrmion string (x = 0). (Scale bar, 20 μm).