Wiedemann-Franz law and Fermi liquids

We consider in depth the applicability of the Wiedemann-Franz (WF) law, namely that the electronic thermal conductivity ($kappa$) is proportional to the product of the absolute temperature ($T$) and the electrical conductivity ($sigma$) in a metal with the constant of proportionality, the so-called Lorenz number $L_0$, being a materials-independent universal constant in all systems obeying the Fermi liquid (FL) paradigm. It has been often stated that the validity (invalidity) of the WF law is the hallmark of an FL (non-Fermi-liquid (NFL)). We consider, both in two (2D) and three (3D) dimensions, a system of conduction electrons at a finite temperature $T$ coupled to a bath of acoustic phonons and quenched impurities, ignoring effects of electron-electron interactions. We find that the WF law is violated arbitrarily strongly with the effective Lorenz number vanishing at low temperatures as long as phonon scattering is stronger than impurity scattering. This happens both in 2D and in 3D for $T<T_{BG}$, where $T_{BG}$ is the Bloch-Gruneisen temperature of the system. In the absence of phonon scattering (or equivalently, when impurity scattering is much stronger than the phonon scattering), however, the WF law is restored at low temperatures even if the impurity scattering is mostly small angle forward scattering. Thus, strictly at $T=0$ the WF law is always valid in a FL in the presence of infinitesimal impurity scattering. For strong phonon scattering, the WF law is restored for $T> T_{BG}$ (or the Debye temperature $T_D$, whichever is lower) as in usual metals. At very high temperatures, thermal smearing of the Fermi surface causes the effective Lorenz number to go below $L_0$ manifesting a quantitative deviation from the WF law. Our work establishes definitively that the uncritical association of an NFL behavior with the failure of the WF law is incorrect.