Mode-coupling formulation of heat transport in anharmonic materials

Temperature-dependent harmonic approximations generalize the ground-state phonon picture, and have found widespread use in computing the thermal properties of materials. However, applying these approaches to access the thermal conductivity still lacks a formal justification, in particular due to the use of perturbation theory. In this work, we derive a theory of heat transport in anharmonic crystals, using the mode-coupling theory of anharmonic lattice dynamics. Starting from the Green-Kubo formula, we develop the thermal conductivity tensor based on the system's dynamical susceptibility, or spectral function. Our results account for both the diagonal and off-diagonal contributions of the heat current, with and without collective effects. We implement our theory in the temperature-dependent effective potential (tdep) package, and have notably introduced a Monte Carlo scheme to compute phonon scattering due to third- and fourth-order interactions, achieving a substantial reduction in computational cost which enables full convergence of such calculations. We apply our methodology to systems with varying regimes of anharmonicity and thermal conductivity to demonstrate its universality. These applications highlight the importance of the phonon renormalizations, and their interactions beyond the harmonic order. Overall, our work advances the understanding of thermal conductivity in anharmonic crystals and provides a theoretically robust framework for predicting heat transport in complex materials.