The anti-symmetric and anisotropic symmetric exchange interactions between electric dipoles in hafnia

The anti-symmetric and anisotropic symmetric exchange interactions between two magnetic dipole moments - responsible for intriguing magnetic textures (e.g., magnetic skyrmions) - have been discovered since last century, while their electric analogues were either hidden for a long time or still not known. As a matter of fact, it is only recently that the anti-symmetric exchange interactions between electric dipoles was proved to exist (with materials hosting such an interaction being still rare) and the existence of anisotropic symmetric exchange interaction between electric dipoles remains to be revealed. Here, by symmetry analysis and first-principles calculations, we identify a candidate material in which our aforementioned exchange interactions between electric dipoles are perceptible. More precisely, we find that various phases of hafnia showcase non-collinear alignment of electric dipoles, which is interpreted by our phenomenological theories. This gives evidence that hafnia simultaneously accommodates anti-symmetric and anisotropic symmetric exchange interactions between electric dipoles. Our findings can hopefully deepen the current knowledge of electromagnetism in ferroelectrics, magnets and multiferroics, and have a potential to guide the discovery of novel states of matter (e.g., electric skyrmions) in hafnia and related materials.