Joule heating in bad and slow metals

TL;DR

This paper develops a general theoretical framework to describe energy transfer from electrons to the lattice in bad and slow metals, applying it to models like the Hubbard model and experimental data from cuprates and ultracold atoms.

Contribution

It introduces a Kubo formula for energy relaxation in bad and slow metals, extending understanding of electron-phonon interactions in these regimes.

Findings

Derived a Kubo formula for energy relaxation rate.

Applied the formula to the Hubbard model and experimental data.

Estimated relaxation rates in cuprate strange metals.

Abstract

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.