  We compute the thermoelectric and thermal transport in the weakly disordered
non-Fermi liquid phase of the Luttinger semimetals at zero doping, where the
decay rate associated with the (strong) Coulomb interactions is much larger
than the electron-impurity scattering rate. To this end, we implement the
Mori-Zwanzig memory matrix method, that does not rely on the existence of
long-lived quasiparticles in the system. We find that the thermal conductivity
at zero electric field scales as $\bar{\kappa}\sim T^{-n}$ (with $0\lesssim
n\lesssim 1)$ at low temperatures, whereas the thermoelectric coefficient has
the temperature dependence given by $\alpha\sim T^{p}$ (with $1/2\lesssim
p\lesssim 3/2)$. These unconventional properties turn out to be key signatures
of this long sought-after non-Fermi liquid state in the Luttinger semimetals,
which is expected to emerge in strongly correlated spin-orbit coupled materials
like the pyrochlore iridates. Finally, our results indicate that these
materials might be good candidates for achieving high figure of merit for
thermoelectric applications.
