Optical Conductivity of the Two Dimensional Hubbard Model: Exact Perturbative vs Dynamical Mean Field Results

We compute the frequency dependent conductivity including the total spectral weight and Drude weight of the two dimensional square lattice Hubbard model at zero temperature as a function of density in the weak interaction limit using both exact perturbation and the dynamical mean field approximation. We find that dynamical mean field approximation, which ignores the vertex correction, fails to capture the correct low frequency behavior of the optical conductivity for low band filling when there is approximate Galilean invariance, and around half filling when nesting effect is important.