Electronic hydrodynamics and the breakdown of the Wiedemann-Franz and Mott laws in interacting metals

Abstract

We present the theory of thermoelectric transport in metals with long-lived quasiparticles, carefully addressing the interplay of electron-electron scattering as well as electron-impurity scattering, but neglecting electron-phonon scattering. In Fermi liquids with a large Fermi surface and weak electron-impurity scattering, we provide universal and simple formulas for the behavior of the thermoelectric conductivities across the ballistic-to-hydrodynamic crossover. In this regime, the electrical conductivity is relatively unchanged by hydrodynamic effects. In contrast, the thermal conductivity can be parametrically smaller than predicted by the Wiedemann-Franz law. A less severe violation of the Mott law arises. We quantitatively compare the violations of the Wiedemann-Franz law arising from (i) momentum-conserving electron-electron scattering in the collision integral, (ii) hydrodynamic modifications of the electron-impurity scattering rate, and (iii) thermal broadening of the Fermi surface, and show that (i) is the largest. We present simple formulas for electrical and thermal magnetoconductivity across the ballistic-to-hydrodynamic limit, along with a more complicated formula for the thermoelectric magnetoconductivity. In a finite magnetic field, the Lorenz number may be smaller or larger than predicted by the Wiedemann-Franz law, and the crossover between these behaviors is a clear prediction for experiments. The arbitrarily strong violation of the Wiedemann-Franz law found in our work arises entirely from electron-electron interaction effects within the Fermi liquid paradigm, and does not imply any non-Fermi liquid behavior. We predict clear experimental signatures of bulk hydrodynamics in high-mobility 2D GaAs semiconductor structures, where failure of the Wiedemann-Franz law should persist down to very low temperatures in high-quality, low-density samples.