Fig. 2

Dependence of the polarization components (a–c) and dielectric susceptibility (d–f) of BaTiO₃ crystal on temperature calculated using HMC, MMC and MD algorithms. Simulation were performed using a 16 × 16 × 16 supercell. At each temperature, 40,000 HMC (MMC) sweeps were performed, out of which the first 10,000 were considered as thermalization sweeps and the thermodynamic averages were computed over the remaining 30,000 sweeps. In MD simulations, we have 0.5 millions of thermalization and 1.0 million of avering steps. The sequence of structural phase transitions (corresponding transition temperatures are indicated by dashed lines) is successfully reproduced by all algorithms. However, HMC scheme yields higher accuracy of thermodynamic averages as can be seen from (g–i) that show temperature evolution of autocorrelation times of polarization components obtained for HMC, MMC and MD simulations respectively. Note that while the autocorrelation time in (g, h) is in the units of MC sweeps, for (i) the autocorrelation time units are 10³ MD steps. For each temperature, the total simulation cpu time for MMC, MD and HMC simulations was of 35.4, 19.5, and 19.6 min, respectively