Igor Lukyanchuk1
University of Picardie1
Past decade marked breakthrough discoveries of novel topological polarization structures in nanostructured ferroelectrics. These findings lie at the crossroad of the forefront of the physics of nanostructured materials, advanced topological concepts, and cutting-edge industrial nanotechnologies. The observed polarization structures have been interpreted in terms of the real-space vector field topology as multiple combinations of vortices, skyrmions, merons, and other topological formations. This topological approach was successfully used in the physics of dislocation, liquid crystals, and magnetism. Ferroelectricity has been viewed as an electric dual of magnetism; however, this visible analogy misses ferroelectricity's cornerstone, the dominant role of the long-range electrostatic forces. We establish a topological foundation of ferroelectricity arising from its electrostatic essence and discuss the observed topological phenomena. We point out at the shortcomings of the existing topological approaches widely utilized for the description of the polarization vector fields. We establish that the physics of ferroelectrics can be constructed in the framework of topological hydrodynamics, an advanced formalism of the topology of an incompressible flow of the vector fields, introduced by Arnold. Based on the introduced topological identity of the electrostatics of ferroelectrics and hydrodynamics of the incompressible liquid, we set up a unified approach for the explanation of topological states in ferroelectrics. We give a comprehensive description of the observed polarization topological structures and show that their rich variety can be exhaustively described by the foundational Arnold theorem in terms of fundamental formations of topological hydrodynamics, vortices, and Hopfions. We outline the prospects of implications of topological hydrodynamics to the physics of ferroelectrics and their diverse applications to the related emergent technologies.