Fig. 5: Pinning of polarization vortices by the local chemistry.

a Single atomic layer of Pb in PMN, with vectors indicating projections of the filtered Pb displacements onto the layer’s plane. Color – Q-criterion of vorticity. The color scale is selected to emphasize the displacements with Q > 0.6. b An overlay of the Pb atoms having Q > 0.6 (red symbols) and locations of the electric charge boundaries (black symbols) in this layer arising from the distribution of the neighboring Mg and Nb. This image reveals a clear preference for the Pb atoms acquiring large Q values to be located in the vicinity of these boundaries. c 3D rendering of the charge boundaries (highlighted in yellow) in the refined configuration of PMN, with the asterisks marking the centers of Pb clusters having Q > 0.6 (identified using HDBSCAN). d Statistical distributions of distances from the centers of clusters with large vorticity to the charge boundaries. Black—cluster centers as found in the refined configuration. Red—randomized locations of the cluster centers. The Matt-Whitney test confirms that the actual distances are significantly shorter than those for randomly distributed cluster centers, in line with the visual inference from Fig. 5b.