General principle of ferroelectric topological domain formation.

We propose a general principle for the formation of topological structures in ferroelectrics, demonstrating that the fundamental formation mechanism of ferroelectric vortex is the superposition of two orthogonal dipole waves, which has also been validated by the mathematical deduction, phase-field simulations, and angle-resolved piezoelectric force microscopy. Moreover, it is demonstrated that this principle can also be extended to a range of nontrivial topological structures, including Ising, Néel, and Bloch domain walls, merons, skyrmions, Hopf rings, Solomon rings, and others. These findings not only improve our understanding of the fundamental formation mechanism of existing topological structures but also enable the prediction of topological structures, such as Star of David rings, in ferroelectric/ferromagnetic materials, liquid crystals, and Bose-Einstein condensate states (superconductors and superfluids).