What controls the temperature of a soft mode-driven structural phase transition?

We have used an effective model of ferroelectric PbTiO$_{3}$, which displays a representative soft mode-driven phase transition, to investigate how different features of the potential-energy surface affect the transition temperature $T_{\rm C}$. We find that the energy difference between PbTiO$_{3}$'s high-symmetry (cubic) and low-symmetry (tetragonal) phases (which we call ground state energy $E_{gs}$) is the parameter that most directly and strongly determines $T_{\rm C}$. We have also found that other simple features of the energy landscape, such as the amplitude of the distortion connecting the high-symmetry and low-symmetry structures, can be used as a predictor for $T_{\rm C}$ only as long as they are correlated with the magnitude of $E_{gs}$. We discuss how our results relate to the expected behaviors that can be derived from simpler theoretical approaches, as well as to phenomenological studies in the literature. Our findings support the empirical rule for estimating $T_{\rm C}$ proposed by Abrahams et al. [Physical Review 172, 551 (1968)] and clarify its physical interpretation. The evidence also suggests that deviations from the expected behaviors are indicative of complex lattice-dynamical effects involving strong anharmonic interactions (and possibly competition) between the soft phonon driving the transition and other modes of the material.