Dzyaloshinskii-Moriya-like interaction in ferroelectrics and antiferroelectrics

The Dzyaloshinskii-Moriya interaction (DMI) between two magnetic moments m_i and m_j is of the form [Formula: see text]. It originates from spin-orbit coupling, and is at the heart of fascinating phenomena involving non-collinear magnetism, such as magnetic topological defects (for example, skyrmions) as well as spin-orbit torques and magnetically driven ferroelectricity, that are of significant fundamental and technological interest. In sharp contrast, its electric counterpart, which is an electric DMI characterized by its [Formula: see text] strength and describing an interaction between two polar displacements u_i and u_j, has rarely been considered, despite the striking possibility that it could also generate new features associated with non-collinear patterns of electric dipoles. Here we report first-principles simulations combined with group theoretical symmetry analysis which not only demonstrate that electric DMI does exist and has a one-to-one correspondence with its magnetic analogue, but also reveals a physical source for it. These findings can be used to explain and/or design phenomena of possible technological importance in ferroelectrics and multiferroics.