Extended Data Fig. 8: Fractional and integer topological defects on the PDM state.

a,b, PDM modulation phase at the ±P_X and ±P_Y, calculated by the two-dimensional lock-in method (Supplementary information section 4). c,d, Separate visualization of the modulation at ±P_X and ±P_Y, calculated by FT filtering with only unidirectional vectors selected (see Methods). Several topological defects are identified (#1–#3). e,f, Phase winding around fractional and integer topological defects. For clarity, the curves are shifted to match their phase minima at zero. In f, a dislocation–antidislocation pair is demonstrated by their opposite phase winding (see also a simulation in Supplementary Fig. 28). g–i, Zoom-in of a half-integer quantized topological defect. g, A zoom-in image of the dashed blue square area in c, which contains the topological defect #1. h,i, Simulation of the fractional topological defect #1 (see Supplementary information section 7). A misaligned half dislocation on the ±P_X modulation of the PDM state (h) appears as an equivalent half vortex in its phase field (i). j–l, Same as g–i, showing an integer topological defect #2. A single dislocation on the ±P_X modulation (k) appears as an equivalent single vortex in its phase field (l). The Gaussian cut-off length (σ) of FT was set to 1.3 nm in a–d.