Lattice Thermal Transport Beyond the Quasiparticle Approximation: Nontrivial Spectral Competition between Three- and Four-Phonon Interactions

The breakdown of the quasiparticle approximation (QPA) for phonons in strongly anharmonic materials necessitates advanced first-principles frameworks for accurate lattice dynamics and thermal transport predictions. We develop a comprehensive beyond-quasiparticle approximation (BQPA) approach incorporating both three- (3ph) and four-phonon (4ph) interactions and apply it to investigate lattice thermal conductivity (k_L) in MgO, PbTe, and AgCl -- materials that span a broad spectrum of anharmonicity, from weak to severe anharmonic regimes with overdamped phonons. We reveal that while BQPA consistently increases k_L relative to QPA due to phonon softening when considering only 3ph interactions, the inclusion of additional 4ph interactions hardens the phonon spectrum and suppresses this enhancement, bringing BQPA and QPA predictions into close agreement via subtle spectral competition effects across all three compounds. These findings highlight that accurate modeling of k_L in strongly anharmonic materials requires treating both full phonon spectral function and higher-order anharmonicity on equal footing. Our work establishes a systematic framework for modeling thermal transport in systems with overdamped phonons and provides critical insights for materials design beyond the limits of conventional phonon transport theory.