Leveraging the power of algorithmic Matsubara integration (AMI) we can generate results for virtually any diagrammatic expansion. These solutions can be stored symbolically and analytically continued to the real frequency axis. These advancements provide unprecedented access to correlation functions for both single and many-particle properties and have the potential to be applied to extension of density functional theory. However, the path to the general application of these ideas is fraught with technical hurdles. We will review progress in diagrammatic methods as a whole and present calculations from our group in that context for the Jellium and 2D Hubbard models and discuss extensions to generalized multiband problems.

In particular our methodology provides access to response functions such as the optical conductivity. We will present a study of the frequency and temperature dependence of the optical conductivity in the weakly coupled two-dimensional Hubbard model using a renormalized perturbative expansion. The resulting conductivity shows a linear power law behaviour in the intermediate frequency regime. Moreover, the associated transport scattering time and renormalized mass exhibit a Planckian behaviour. We show that the self-energy of the Hubbard model, however, is distinct from existing Planckian models yet gives rise to the same features in the conductivity.