Trilinear couplings and the multi-mode anti-ferroelectric transition of PbZrO$_{3}$: a first-principles investigation*

We have studied ab initio the phase transition in PbZrO$_{3}$, a perovskite oxide usually presented as the prototypic anti-ferroelectric material. Our work reveals the crucial role that anti-ferrodistortive modes--involving concerted rotations of the oxygen octahedra in the structure--play in the transformation, as they select the observed anti-ferroelectric phase, among competing structural variants, via a cooperative trilinear coupling. The resulting picture is that of a complex transition whose multi-mode character is essential to its very occurrence, and poses the provocative question of whether such an intricate behavior can be taken as representative of anti-ferroelectricity in perovskite oxides.