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The anti-symmetric and anisotropic symmetric exchange interactions betweentwo magnetic dipole moments - responsible for intriguing magnetic textures(e.g., magnetic skyrmions) - have been discovered since last century, whiletheir 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 exchangeinteractions between electric dipoles was proved to exist (with materialshosting such an interaction being still rare) and the existence of anisotropicsymmetric exchange interaction between electric dipoles remains to be revealed.Here, by symmetry analysis and first-principles calculations, we identify acandidate material in which our aforementioned exchange interactions betweenelectric dipoles are perceptible. More precisely, we find that various phasesof hafnia showcase non-collinear alignment of electric dipoles, which isinterpreted by our phenomenological theories. This gives evidence that hafniasimultaneously accommodates anti-symmetric and anisotropic symmetric exchangeinteractions between electric dipoles. Our findings can hopefully deepen thecurrent knowledge of electromagnetism in ferroelectrics, magnets andmultiferroics, and have a potential to guide the discovery of novel states ofmatter (e.g., electric skyrmions) in hafnia and related materials.

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