Violation of the Wiedemann-Franz Law in Hydrodynamic Electron Liquids

The Wiedemann-Franz law, connecting the electronic thermal conductivity to the electrical conductivity of a disordered metal, is generally found to be well satisfied even when electron-electron (e-e) interactions are strong. In ultraclean conductors in the hydrodynamic regime, however, large deviations from the standard form of the law are expected, due to the fact that e-e interactions affect the two conductivities in radically different ways. Thus, the standard Wiedemann-Franz ratio between the thermal and the electric conductivity is reduced by a factor 1+τ/τₜₕᵉᵉ, where 1/τ is the momentum relaxation rate and τₜₕᵉᵉ is the relaxation time of the thermal current due to e-e collisions. Here we study the density and temperature dependence of 1/τₜₕᵉᵉ of two-dimensional electron liquids. We show that at low temperature 1/τₜₕᵉᵉ is $8/5$ of the quasiparticle decay rate; remarkably, the same result is found in doped graphene and in conventional electron liquids in parabolic bands.