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The self-consistent harmonic approximation is an effective harmonic theory tocalculate the free energy of systems with strongly anharmonic atomicvibrations, and its stochastic implementation has proved to be an efficientmethod to study, from first-principles, the anharmonic properties of solids.The free energy as a function of average atomic positions (centroids) can beused to study quantum or thermal lattice instability. In particular thecentroids are order parameters in second-order structural phase transitionssuch as, e.g., charge-density-waves or ferroelectric instabilities. Accordingto Landau's theory, the knowledge of the second derivative of the free energy(i.e. the curvature) with respect to the centroids in a high-symmetryconfiguration allows the identification of the phase-transition and of theinstability modes. In this work we derive the exact analytic formula for thesecond derivative of the free energy in the self-consistent harmonicapproximation for a generic atomic configuration. The analytic derivative isexpressed in terms of the atomic displacements and forces in a form that can beevaluated by a stochastic technique using importance sampling. Our approach isparticularly suitable for applications based on first-principlesdensity-functional-theory calculations, where the forces on atoms can beobtained with a negligible computational effort compared to total energydetermination. Finally we propose a dynamical extension of the theory tocalculate spectral properties of strongly anharmonic phonons, as probed byinelastic scattering processes. We illustrate our method with a numericalapplication on a toy model that mimics the ferroelectric transition inrock-salt crystals such as SnTe or GeTe.

physical review. b./physical review. b

Second order structural phase transitions, free energy curvature, and  temperature-dependent anharmonic phonons in the self-consistent harmonic  approximation: theory and stochastic implementation