Planckian scaling of the optical conductivity in the 2D Hubbard model

The optical conductivity of systems with strong interactions is one of the most studied quantities experimentally, yet its computation from microscopic models remains challenging. In the context of linear response theory and of the Kubo formula for conductivity, this difficulty is embedded in the momentum and energy dependence of the electron self-energy and of the vertex corrections. In this work, we present a study of the optical conductivity using Algorithmic Matsubara Integration which allows for the evaluation of diagrammatic series up to a fixed order directly on the real frequency axis without relying on analytic continuation tools.

The resulting optical conductivity is obtained at different temperatures at the leading order and its non-trivial frequency dependence is analysed through the lens of an extended Drude model by introducing a frequency-dependent scattering time and mass enhancement. We recover the linear temperature dependence of the dc resistivity that has been observed in previous work. Additionally, we show that the optical conductivity presents a regime of power-law scaling at intermediate frequencies. This scaling satisfies a "Planckian"-like behaviour similar to the one expected in the strange metal regime of non-fermi liquids and reported in experiments. Extensions to include higher-order effects such as the vertex corrections to the longitudinal conductivity do not destroy this behaviour and are also discussed.