Ab initio statistical mechanics of the ferroelectric phase transition in PbTiO₃

An effective Hamiltonian for the ferroelectric transition in PbTiO₃ is constructed from ab initio pseudopotential local-density-functional total-energy and linear-response calculations through the use of a localized, symmetrized basis set of ``lattice Wannier functions.'' Explicit parametrization of the polar lattice Wannier functions is used for subspace projection, addressing the issues of LO-TO splitting and coupling to the complementary subspace. In contrast with ferroelectric BaTiO₃ and KNbO₃, we find significant involvement of the Pb atom in the lattice instability. Monte Carlo simulations for this Hamiltonian show a first-order cubic-tetragonal transition at 660 K. The resulting temperature dependence of spontaneous polarization, c/a ratio, and unit-cell volume near the transition are in good agreement with experiment. Comparison of Monte Carlo results with mean-field theory analysis shows that both strain and fluctuations are necessary to produce the first-order character of this transition.