  We demonstrate an approach for calculating temperature-dependent quantum and
anharmonic effects with beyond density-functional theory accuracy. By combining
machine-learned potentials and the stochastic self-consistent harmonic
approximation, we investigate the cubic to tetragonal transition in strontium
titanate and show that the paraelectric phase is stabilized by anharmonic
quantum fluctuations. We find that a quantitative understanding of the quantum
paraelectric behavior requires a higher-level treatment of electronic
correlation effects via the random phase approximation. This approach enables
detailed studies of emergent properties in strongly anharmonic materials beyond
density-functional theory.
