Phonon collisional broadening and heat transport beyond the Boltzmann equation

Abstract: In crystals, macroscopic technological properties such as thermal conductivity originate from the microscopic drift and scattering of phonons, commonly described by the Boltzmann Transport Equation (BTE). Despite its widespread use, the most general space-time nonlocal form of the BTE still lacks a rigorous derivation of its collisional part based on Fermi's Golden Rule (FGR), and becomes inadequate in several regimes, including when the energy-variation scale set by phonon dispersion approaches that of collisional broadening. A hallmark of this issue is the poor numerical convergence of conductivity with respect to the smearing used to evaluate FGR rates. This is often circumvented using adaptive schemes, which however violate detailed balance and allow unphysical negative eigenvalues in the collision operator. Here, we overcome these limitations by rigorously deriving the space-time-dependent BTE from the Kadanoff-Baym Equations (KBE), and introduce a linearized generalized BTE (LGBTE) that goes beyond the FGR framework, incorporating self-consistent, physically derived, fully anharmonic, and mode-resolved collisional broadening and energy-nonconserving scattering. More generally, we establish a hierarchy of ansätze on Green's functions, enabling controlled extensions of the semiclassical BTE and a roadmap toward quantum KBE accuracy. Finally, using first-principles simulations complemented by analytical arguments, we show that this approach addresses two long-standing problems of the FGR-based linearized BTE across crystal dimensionalities: (i) the lack of conductivity convergence, common to heat conductors like diamond; and (ii) its universal failure in all 2D systems, rooted in FGR predicting an unphysical overdamping for scattering channels involving flexural vibrations, as shown in the insulating α-GeSe monolayer.