Tilt-driven antiferroelectricity in PbZrO$_3$

Antiferroelectricity is a state of matter that has so far eluded a clear-cut definition. Even in the best-known material realization, PbZrO$_3$, the physical nature of the driving force towards an antipolar order has not been settled yet. Here, by building a Landau-like continuum Hamiltonian from first-principles via an exact long-wave approach, we reconcile the existing theories in terms of a single physical mechanism. In particular, we find that a formerly overlooked trilinear coupling between tilts, tilt gradients and polarization provides a surprisingly accurate description of the energetics and structure of the antiferroelectric ground state of PbZrO$_3$. We discuss the relevance of our findings to other ferrielectric and incommensurate polar structures that were recently observed in perovskites.