The computation of the frequency-dependent conductivity in interacting systems is still an open problem while being one of the most widely used experimental techniques. Even restricting the question to the zero-frequency limit, there remain unsolved questions such as the nature of the linear dependence of the dc resistivity in multiple strongly-correlated materials ranging from cuprates to magic-angle graphene .
The problem of computing the optical conductivity in systems with electron-electron interactions is made difficult due to the importance of vertex corrections  which include strong momentum and frequency dependence of scattering events and thus require additional care when evaluating the conductivity through numerical methods. One example of such vertex correction appears when performing a perturbative expansion for the current-current correlation in the Kubo formula for the conductivity.
We present here an alternative way of obtaining the current-current correlation function, and thus the optical conductivity, through the computation of the force-force correlation function . These two quantities are linked but have different perturbative expansions with different convergence properties, especially regarding the analytical continuation parameter when performing the computation for real frequencies. Moreover, the lowest order term in the perturbative series for the force-force correlation function encompasses all terms up to second order, thus taking into account the first vertex correction of the current-current correlation function. We present here the computation of the force-force correlation function in the Hubbard model using the Algorithmic Matsubara Integration  technique. These results are compared to the corresponding current-current correlation and we discussed the optical conductivity obtained in both cases.
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