Computing dynamical response functions in interacting lattice models is a long-standing challenge in condensed matter physics. In view of recent results, the dc resistivity ${\rho }_{\mathrm{dc}}$ in the weak-coupling regime of the Hubbard model is of great interest, yet it is not fully understood. The challenge lies in having to work with large lattices while avoiding analytical continuation. The weak-coupling ${\rho }_{\mathrm{dc}}$ results were so far computed at the level of the Boltzmann theory and at the level of the Kubo bubble approximation, which neglects vertex corrections. Neither theory was so far rigorously proven to give exact results even at infinitesimal coupling, and the respective dc resistivity results differ greatly. In this Letter we develop, cross-check and apply two state-of-the-art methods for obtaining dynamical response functions. We compute the optical conductivity at weak coupling in the Hubbard model in a fully controlled way, in the thermodynamic limit, and without analytical continuation. We show that vertex corrections persist to infinitesimal coupling, with a constant ratio to the Kubo bubble. We connect our methods with the Boltzmann theory, and show that the latter applies additional approximations that lead to quantitatively incorrect scaling of ${\rho }_{\mathrm{dc}}$ with respect to the coupling constant.

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• Electrical conductivity
• Optical conductivity
• Strongly correlated systems
• Boltzmann theory
• Diagrammatic methods
• Hubbard model
• Monte Carlo methods
• Nonequilibrium Green's function
• Perturbation theory