Planckian behaviour in the optical conductivity of the weakly coupled Hubbard model

We study the frequency and temperature dependence of the optical conductivity in the weakly coupled two-dimensional Hubbard model using a renormalized perturbative expansion. The perturbative expansion is based on the skeleton series for the current-current correlation function with a dressed Green`s function and the results are obtained directly on the real frequency axis using Algorithmic Matsubara Integration (AMI). The resulting conductivity shows a temperature-independent 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. The Planckian behaviour of the conductivity, observed in optimally doped cuprates for example, can thus be obtained from a different form of self-energy than the Planckian model, such as the weakly coupled Hubbard model at half-filling.