  We compute the frequency dependent conductivity of the two dimensional square
lattice Hubbard model at zero temperature as a function of density to second
order in the interaction strength, and compare the results to the predictions
of single-site dynamical mean field theory computed at the same order. We find
that despite the neglect of vertex corrections, the single site dynamical mean
field approximation produces semiquantitatively accurate results for most
carrier concentrations, but fails qualitatively for the nearly empty or nearly
filled band cases where the theory exhibits an emergent Galilean invariance.
The theory also becomes qualitatively inaccurate very near half filling if
nesting is important.
