Fig. 6: Phase diagram of BaTiO₃ at ambient pressure.

Panels a–c show the R-O, O-T, and T-C clusters in the (s₁, s₂) plane across the coexistence regions. Each configuration generated in the MD is a point in this plane, colored according to its probability P. The latter is computed via the probabilistic analysis of molecular motifs (PAMM)¹³⁹ algorithm (see Methods, subsection “Physically-inspired order parameters” for additional details). P is smoothly increasing in the [0, 1] interval while going from R to O (panel a), then from O to T (panel b), and finally from T to C (panel c). Panels d–f show the temperature-dependent chemical potential differences across the R-O (d), O-T (e), and T-C (f) transitions. Blue error bars represent standard deviations computed across multiple MD runs, the cyan line the best linear fit across the coexistence regions, and the purple error bars the propagated errors on the critical temperatures. In addition, the orange triangles in panel f show the free-energy perturbed chemical potentials, using 50 reference tetragonal and cubic structures just below (170 K) and above (194 K) the Curie point. 6× magnified insets corresponding to these two temperatures show how the FEP-corrected chemical potentials consistently fall within the error bars due to the MD sampling, confirming the DFT accuracy of the GAP (see subsection “Thermodynamics of BaTiO₃”).