Interacting Phonon Green's Function in Strontium Titanate via Self-Consistent Perturbation Theory and Irreducible Derivatives

Despite extensive first-principles analysis of strontium titanate (SrTiO₃), a systematic study of the irreducible phonon interactions is still needed to fully understand the vibrational Hamiltonian within a given first-principles framework. We present an in-depth study of the irreducible phonon interactions and strain-phonon coupling in SrTiO₃, focusing on the ferroelectric (FE) and anti-ferrodistortive (AFD) phase instabilities. Using the lone and bundled irreducible derivative (LID and BID) approaches, we compute the irreducible phonon interactions up to fourth order and derive the leading-order strain-phonon derivatives of the vibrational Hamiltonian. The interacting phonon hamilontian is solved within the Hartree-Fock approximation for phonons, in both the classical and quantum cases. These results are further refined using both the 3rd and 4th-order phonon interactions in conjunction with self-consistent perturbation theory, which evaluates the 3-phonon bubble and 4-phonon sunset diagrams using the Hartree-Fock Green's function. We also compute the classical phonon Green's function using molecular dynamics, offering a benchmark comparison for the classical diagrammatic calculations. This work provides a more advanced understanding of the phonon behavior of SrTiO₃.