Foundations of Topology in Ferroelectrics

During the past decade, it has been realized that various topological structures, arising in ferroelectrics may significantly contribute to their physical properties. Even the simplest topological formation, domain walls (DWs), bring new functional complexity and open a new avenue in the operating of ferroelectric devices at the nanoscale. Here we generalize topological consideration of the nanostructured ferroelectrics to small nanoparticles, nanodots, films, and nanorods and show that the topologically stable <i>knotting</i> of the polarization field lines and<i> topological chirality</i> results in a wealth of useful functionalities such as optical activity and THz vibrations of DWs.<br/> Importantly, despite the seeming similarity, the origin of topological structures is very different in magnetic and ferroelectric systems. In magnetic systems, it is the local built-in Dzyaloshinskii Moriya interaction that induces the long-range chiral ordering of topological excitations. At variance, a distinct feature of ferroelectrics is that the topological excitations appear as a result of the spontaneous symmetry breaking due to an interplay of the confinement and depolarization effects where the topological excitations are generated by the long-range interaction of the depolarization charges, ρ =- divP. Upon confinement the polarization field swirls to keep its divergenceless structure, the situation being similar to the incompressible liquid flow in the whirlpools or to the magnetohydrodynamic flow of the plasma interior of the stars. The topological classification scheme of these formations, unified by the constraint of the divergenceless of the vector field is by far more complex than that for the magnetic systems.<br/> We demonstrate that the advanced 3D topological excitations, <i>vortices, skyrmions,</i><i> and Hopfions</i>, characterized by the nontrivial <i>linking numbers</i> and <i>helicity </i>arise inside the nanostructured ferroelectrics and mimic the “topology of the Universe”. Our consideration is in-line with the ongoing paradigmatic shift in physics, marking the change from deriving properties of the matter out of the local structural symmetries to the global topological paradigm, which describes the fundamental properties of matter on the basis of their invariance under continuous transformations or perturbations.