High-throughput computational framework for high-order anharmonic thermal transport in cubic and tetragonal crystals

Accurate first-principles prediction of lattice thermal conductivity (κL) remains challenging in identifying materials with extreme thermal behavior. While the harmonic approximation with three-phonon scattering (HA + 3ph) is now routine, reliable κL prediction often requires higher-order anharmonic effects, including self-consistent phonon renormalization, three- and four-phonon scattering, and off-diagonal heat flux (SCPH + 3, 4ph + OD). We present a state-of-the-art high-throughput workflow that unifies these effects and apply it to 773 cubic and tetragonal crystals spanning diverse chemistries and structures. From 562 dynamically stable compounds, we assess the hierarchical impacts of higher-order anharmonicity. For around 60% of materials, HA + 3ph predictions closely match those from SCPH + 3, 4ph + OD. SCPH generally increases κL, by over 8 times in extreme cases, whereas four-phonon scattering universally suppresses κL, sometimes to 15% of the HA + 3ph value. Off-diagonal contributions are negligible in high-κL systems but can rival diagonal terms in highly anharmonic low-κL compounds. We highlight four case studies, Rb2TlAlH6, Cu3VSe4, CuBr, and KTlCl4, that exhibit distinct extreme behaviors. This work delivers not only a robust workflow for high-fidelity κL dataset but also a quantitative framework to determine when higher-order effects are essential. The hierarchy of κL results, from the HA + 3ph to SCPH + 3, 4ph + OD level, offers a scalable, interpretable route to discovering next-generation extreme thermal materials.