Understanding and simulating the thermodynamic and dynamical properties of materials affected by strong ionic anharmonicity is a central challenge in material science. Much interest is in material displaying critical displacive behavior, such as near a ferroelectric transition, charge-density waves, or in general displacive second-order transitions. In these cases, molecular dynamics suffers from a critical slowdown and emergent long-range fluctuations of the order parameter. Two prominent methods have emerged to solve this issue: self-consistent renormalization of the phonons like the self-consistent harmonic approximation and self-consistent phonons, and methods that fit the potential energy landscape from short molecular dynamics trajectories, like the temperature-dependent effective potential (TDEP). Despite their widespread use, the limitations of these methods are often overlooked in the proximity of critical points. Here, we establish a guiding rule set for the accuracy of each method on critical quantities: free energy for computing the phase diagrams, static correlation functions for inferring phase stability and critical behaviors, and dynamic correlation functions for vibrational spectra and thermal transport. Also, a new TDEP implementation is introduced to fix the calculation of dynamical spectra, restoring the correct perturbative limit violated by the standard TDEP approach. Results are benchmarked both against an exact one-dimensional anharmonic potential and two prototypical anharmonic crystals: the ferroelectric PbTe and the metal-halide perovskite ${\mathrm{CsSnI}}_3$.

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• Anharmonic lattice dynamics
• Entropy
• First-principles calculations
• Phonons
• Structural order parameter
• Thermal expansion
• Thermodynamics