Tensor Learning and Compression of N-phonon Interactions

Phonon interactions from lattice anharmonicity govern thermal properties and heat transport in materials. These interactions are described by n-th order interatomic force constants (n-IFCs), which can be viewed as high-dimensional tensors correlating the motion of n atoms, or equivalently encoding n-phonon scattering processes in momentum space. Here, we introduce a tensor decomposition to efficiently compress n-IFCs for arbitrary order n. Using tensor learning, we find optimal low-rank approximations of n-IFCs by solving the resulting optimization problem. Our approach reveals the inherent low dimensionality of phonon-phonon interactions and allows compression of the 3 and 4-IFC tensors by factors of up to $10^3-10^4$ while retaining high accuracy in calculations of phonon scattering rates and thermal conductivity. Calculations of thermal conductivity using the compressed n-IFCs achieve a speed-up by nearly three orders of magnitude with >98% accuracy relative to the reference uncompressed solution. These calculations include both 3- and 4-phonon scattering and are shown for a diverse range of materials (Si, HgTe, MgO, TiNiSn and monoclinic ZrO$_2$). In addition to accelerating state-of-the-art thermal transport calculations, the method shown here paves the way for modeling strongly anharmonic materials and higher-order phonon interactions.