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. 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 ImGn(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 Im GnRn(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.

Automated summary

Heat supplied to a metal is absorbed by the electrons and transferred to the lattice. In conventional metals, energy is released to the lattice through phonons emitted from the Lindhard continuum. However, in a 'bad' metal with a short mean free path, the low energy Lindhard continuum is destroyed. To describe energy transfer to the lattice in these cases, a general Kubo formula is obtained for the energy relaxation rate based on the electronic density spectral weight.