Intrinsic polar vortex crystals in A-site layer-ordered perovskites

Topological phases, as characterized by their topological invariants, have been considered as distinct states from the raw phases and hold great promise as tiny yet robust information carriers for the era of artificial intelligence^1,2. However, these nontrivial states are typically found under non-equilibrium conditions, or stabilized by extrinsic electrical or mechanical boundary constraints^3-6, which limit their applications. Particularly in ferroelectrics, it usually entails a maximized depolarization field produced by interfacial bound charges to balance the large elastic and gradient energies as dipole whirling at the atomic scale^7-10. Despite substantial attempts, achieving highly ordered topological polar crystals in bulk ferroelectrics still remains a challenge^11-14. Here we show that a two-dimensional polar hedgehog lattice with a period down to 4 nm can crystallize spontaneously free from any external boundary constraints in a family of A-site layer-ordered perovskites. Using advanced scanning transmission electron microscopy, we observe the polar hedgehog vortices in real space and disclose the physical nature as the cooperative assembly of modulated in-phase and out-of-phase octahedral rotations, further underpinned by hybrid improper ferroelectricity. Theoretical calculations show that the exchange interaction of phonons describing the octahedral rotations is the primary driving force of this intriguing dipole topology. Our findings not only clarify the ambiguity in the structure and origin of the widespread superstructure in layer-ordered perovskites but also demonstrate a viable framework for designing nontrivial structures and functionalities beyond perovskites.