Energy relaxation due to two-phonon scattering of electrons

Recent THz spectroscopy of high-T_c cuprates (arXiv:2503.15646) and quantum paralelectric SrTiO₃ (arXiv:2501.15771) rekindled interest in energy relaxation processes in correlated electron systems. Motivated by these studies, we consider a situation when, for certain modes, single-phonon scattering is forbidden by symmetry or momentum conservation, while two-phonon scattering is allowed. By solving the Boltzmann equation, we show that in the quasi-elastic regime (above the Bloch-Grüneisen temperature), the two-phonon energy relaxation rate exceeds the single-phonon one: while the latter falls as 1/T, the former saturates at a T-independent value in the intermediate temperature window. At higher temperatures, two-phonon processes become strong enough to destroy quasiparticles, and the energy relaxation rate is obtained using an exact solution of the Dyson equation. In the inelastic regime (below the Bloch-Grüneisen temperature), the energy relaxation rate scales in the same way as just the single-particle scattering rate.