  The accurate description of the structural and thermodynamic properties of
ferroelectrics has been one of the most remarkable achievements of Density
Functional Theory (DFT). However, running large simulation cells with DFT is
computationally demanding, while simulations of small cells are often plagued
with non-physical effects that are a consequence of the system's finite size.
To avoid these finite-size effects one is thus often forced to use empirical
models that describe the physics of the material in terms of effective
interaction terms, that are fitted using the results from DFT. In this study we
use a machine-learning (ML) potential trained on DFT, in combination with
accelerated sampling techniques, to converge the thermodynamic properties of
Barium Titanate (BTO) with first-principles accuracy and a full atomistic
description. Our results indicate that the predicted Curie temperature depends
strongly on the choice of DFT functional and system size, because of emergent
long-range directional correlations in the local dipole fluctuations. Our
findings demonstrate how the combination of ML models and traditional bottom-up
modeling allow one to investigate emergent phenomena with the accuracy of
first-principles calculations over the large size and time scales afforded by
empirical models.
