Adequacy of the dynamical mean field theory for low density and Dirac materials

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- or Lorentz-invariant low-density limit. Relevance to recent calculations related to Weyl metals is discussed.