Conductivity in the square lattice Hubbard model at high temperatures: importance of vertex corrections

Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results and the question of the exact solution for the conductivity in the Hubbard model remained open. In this letter we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity, and identify the best available result to be compared to experiment. We find that at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size $4\times 4$ and the corresponding Lanczos diagonalization result is therefore close to the exact result in the thermodynamic limit.