Ferroelectric incommensurate spin crystals.

Ferroics, especially ferromagnets, can form complex topological spin structures such as vortices 1 and skyrmions 2,3 when subjected to particular electrical and mechanical boundary conditions. Simple vortex-like, electric-dipole-based topological structures have been observed in dedicated ferroelectric systems, especially ferroelectric-insulator superlattices such as PbTiO 3 /SrTiO 3 , which was later shown to be a model system owing to its high depolarizing field 4-8 . To date, the electric dipole equivalent of ordered magnetic spin lattices driven by the Dzyaloshinskii-Moriya interaction (DMi) 9,10 has not been experimentally observed. Here we examine a domain structure in a single PbTiO 3 epitaxial layer sandwiched between SrRuO 3 electrodes. We observe periodic clockwise and anticlockwise ferroelectric vortices that are modulated by a second ordering along their toroidal core. The resulting topology, supported by calculations, is a labyrinth-like pattern with two orthogonal periodic modulations that form an incommensurate polar crystal that provides a ferroelectric analogue to the recently discovered incommensurate spin crystals in ferromagnetic materials 11-13 . These findings further blur the border between emergent ferromagnetic and ferroelectric topologies, clearing the way for experimental realization of further electric counterparts of magnetic DMi-driven phases.