Formation of a Post-Perovskite Phase and Absence of High-Pressure Reentrant Ferroelectricity in PbTiO$_3$

We study ferroelectricity in the classic perovskite ferroelectric PbTiO$_3$ to high pressures with density functional theory (DFT) and experimental diamond-anvil techniques. We use second harmonic generation (SHG) spectroscopy to detect lack of inversion symmetry, if present. Consistent with early understanding and experiments, we find that ferroelectricity disappears at moderate pressures. However, the DFT calculations show that the disappearance arises from the overtaking of zone boundary instabilities, and not from the squeezing out of the off-centering ferroelectric displacements with pressure, as previously thought. We find a new dense centrosymmetric post-perovskite phase with P$2_1/m$ symmetry that becomes stable at about 70 GPa. Our computations predict a dramatic reentrance of ferroelectricity 90 GPa in the distorted R3c structure, but this phase is metastable relative to the post-perovskite phase.