Joule heating in bad and slow metals

Heat supplied to a metal is absorbed by the electrons and then transferred to the lattice. In conventional metals energy is released to the lattice by phonons emitted from the Lindhard continuum. However in a `bad' metal, with short mean free path, the low energy Lindhard continuum is destroyed. Furthermore in a `slow' metal, with Fermi velocity less than the sound velocity, particle-hole pairs are kinematically unable to emit phonons. To describe energy transfer to the lattice in these cases we obtain a general Kubo formula for the energy relaxation rate in terms of the electronic density spectral weight $\text{Im} \, G^R_{nn}(\omega_k,k)$ evaluated on the phonon dispersion $\omega_k$. We apply our Kubo formula to the high temperature Hubbard model, using recent data from quantum Monte Carlo and experiments in ultracold atoms to characterize $\text{Im} \, G^R_{nn}(\omega_k,k)$. We furthermore use recent data from electron energy-loss spectroscopy to estimate the energy relaxation rate of the cuprate strange metal to a high energy optical phonon. As a second, distinct, application of our formalism we consider `slow' metals. These are defined to have Fermi velocity less than the sound velocity, so that particle-hole pairs are kinematically unable to emit phonons. We obtain an expression for the energy relaxation rate of a slow metal in terms of the optical conductivity.