  While ferroelectric hafnia ($\mathrm{HfO_2}$) has become a technically
important material for microelectronics, the physical origin of its
ferroelectricity remains poorly understood. The tetragonal $P4_2/nmc$ phase is
commonly assigned as its paraelectric mother phase but has no soft mode at the
Brillouin zone center. In this work, we propose that the
paraelectric-ferroelectric transition in hafnia-like $Pca2_1$ ferroelectric
family can be described by a $Pcca$-$Pca2_1$ transition, where the $Pcca$
mother phase will evolve into either the $Pca2_1$ ferroelectric phase or the
centrosymmetric $P2_1/c$ monoclinic phase, depending on the strain conditions.
The $Pcca$ phase is directly linked to both phases in the context of continuous
phase transition. Hafnia is regarded as a special case of this family, in that
it has accidental atomic degeneracy because all anions are oxygen. The theory
is also correlated to the seven-coordination theory that explains the
ferroelectricity in hafnia from a chemical perspective. In addition, the strain
conditions to promote the ferroelectric phase in hafnia are discussed.
