The ferroelectric phase transitions described by Landau-Devonshire's phenomenological theory can be statistically simulated using the effective Hamiltonian $\left(H_{\mathrm{eff}}\right)$ based on first-principles calculations. In a Monte Carlo (MC) simulation, we observe a statistical singularity energy caused by a subtle difference between the macroscopic phenomenological model and the microscopic statistical model in the mathematical space of the phase transition point. This statistical singularity energy is measured according to the parameters of the $H_{\mathrm{eff}}$ model and introduced by the transition probability of the MC algorithm, which greatly improves the accuracy of the prediction of the phase transition temperatures and restores the true fluctuations near the phase transition point. For the cubic-tetragonal phase transition for $\mathrm{BaTi}{\mathrm{O}}_3$ and $\mathrm{PbTi}{\mathrm{O}}_3$, we obtain a good estimation of the thermal hysteresis corresponding to the experiments. We further reproduce the difference of the phase transitions between heating and cooling sequences, which is verified by the experimental observations.

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• Critical phenomena
• Ferroelectricity
• Phase transitions
• Ferroelectrics
• Monte Carlo methods