Lattice thermal conductivity in the anharmonic overdamped regime

In crystalline materials, low lattice thermal conductivity is often associated with strong anharmonicity, causing significant deviations from the expected Lorentzian lineshape of phonon spectral functions. These deviations, occurring in an overdamped regime, raise questions about the applicability of the Boltzmann transport equation. Furthermore, strong anharmonicity can trigger structural phase transitions with temperature that cannot be adequately described by the standard harmonic approximation. To address these challenges, we propose an approach for computing lattice thermal conductivity. Our method combines the Green-Kubo linear response theory with the stochastic self-consistent harmonic approximation. The latter describes the temperature-dependent evolution of the crystal structure, including first- and second-order phase transitions, as well as the vibrational properties in highly anharmonic materials. The Green-Kubo method considers the full lineshapes of phonon spectral functions in the calculation of lattice thermal conductivity, thus eliminating the questionable use of phonon lifetimes in the overdamped regime and naturally including coherent transport effects. Additionally, we extend our theory to model complex dynamical lattice thermal conductivity, enhancing understanding of time-dependent thermoreflectance experiments. As a practical application, we employ this approach to calculate lattice thermal conductivity of ${\mathrm{CsPbBr}}_{3}$, a complex crystal known for its anomalous thermal transport behavior and a complex phase diagram. Our method determines the thermal conductivity across different phases in good agreement with experiments.