  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-Gr\"uneisen 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.
