Phonon Boltzmann transport equation beyond the semiclassical regime

A detailed understanding of phonon-phonon scattering is essential for predicting key properties like thermal conductivity in crystals. The semiclassical Boltzmann transport equation (BTE) is a well-established tool for this task, but fails in the overdamped regime where the concept of phonon quasiparticles breaks down. Additionally, BTE calculations often suffer from poor convergence with respect to smearing used to numerically evaluate phonon's scattering rates. To overcome these limitations, we derive from the Kadanoff-Baym equation (KBE) an extended BTE-like equation that includes energy non-conserving scattering events and replaces the numerical smearing with a physical Lorentzian collisional broadening. Exploiting the relation between repumping and depumping scattering, we avoid resummation issues and preserve global energy conservation. Most importantly, we establish a hierarchy of ansätze on Green’s and spectral functions, exposing and controlling the approximations needed to ensure smooth transitions from the KBE to the BTE. This approach can naturally incorporate the fundamental physics at play in 2D systems, where numerical smearing schemes lack clear convergence and out-of-plane flexural modes may lead to overdamped phonon lifetimes. Finally, we present a rigorous derivation of a time-dependent, non-homogeneous BTE for phonons, and highlight the potential of the developed framework for wider applicability and future extensions.