  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.
