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

. 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.