  The qualitative reliability of the dynamical mean field theory (DMFT) is
investigated for systems in which either the actual carrier density or the
effective carrier density is low, by comparing the exact perturbative and
dynamical mean field expressions of electron scattering rates and optical
conductivities. We study two interacting systems: tight binding models in which
the chemical potential is near a band edge and Dirac systems in which the
chemical potential is near the Dirac point. In both systems it is found that
DMFT underestimates the low frequency, near-Fermi surface single particle
scattering rate by a factor proportional to the particle density. The
quasiparticle effective mass is qualitatively incorrect for the low density
tight binding model but not necessarily for Dirac systems. The dissipative part
of the optical conductivity is more subtle: in the exact calculation vertex
corrections, typically neglected in DMFT calculations, suppress the low
frequency optical absorption, compensating for some of the DMFT underestimate
of the scattering rate. The role of vertex corrections in calculating the
conductivity for Dirac systems is clarified and a systematic discussion is
given of the approach to the Galilean/Lorentz invariant low density limit.
Relevance to recent calculations related to Weyl metals is discussed.
