Optical Conductivity of Low Density and Dirac Systems: Exact Perturbative vs Dynamical Mean Field Results

We compute the electron self energy and the optical conductivity of both Hubbard models in the limit of low density and Dirac systems with the chemical potential near the Dirac point using both exact perturbative and the dynamical mean field approximation when the interaction is weak. The strong momentum dependence of the self energy implies that the dynamical mean field approximation is a fundamentally inaccurate representation of the low frequency electron scattering dynamics; furthermore, vertex corrections, neglected in DMFT-based calculations, change the low frequency conductivity qualitatively, but in different ways in the low density cubic lattice and Dirac systems.