Ferroelectric incommensurate spin crystals

Ferroics, especially ferromagnets, can form complextopological 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 PbTiO3/SrTiO3, 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 PbTiO3 epitaxial layer sandwiched between SrRuO3 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 ferroelectrictopologies, clearing the way for experimental realization of further electric counterparts of magnetic DMi-driven phases.