  Hafnia ferroelectrics combine technological promise and unprecedented
behaviors. Their peculiarity stems from the many active extrinsic mechanisms
that contribute to their properties and from a continuously growing number of
novel intrinsic features. Partly because of their unconventional nature, basic
questions about these materials remain open and one may wonder about the
pertinence of some frequent theoretical assumptions. Aided by first-principles
simulations, here we show that, by adopting an original high-symmetry reference
phase as the starting point of the analysis, we can develop a mathematically
simple and physically transparent treatment of the ferroelectric state of
hafnia. The proposed approach describes hafnia as a uniaxial ferroic, as
suggested by recent studies of (woken-up) samples with well developed
polarization. Also, it is compatible with the occurrence of polar soft modes
and proper ferroelectric order. Further, our theory provides a straightforward
and unified description of all low-energy polymorphs, shedding light into old
questions (e.g., the prevalence of the monoclinic ground state), pointing at
exciting possibilities (e.g., an antiferroelastic behavior) and facilitating
the future development of perturbative theories (from Landau to
second-principles potentials). Our work thus yields a deeper understanding of
hafnia ferroelectrics, improving our ability to optimize their properties and
induce new ones.
