Emergent nearest-neighbor attraction in the fully renormalized interactions of the single-band repulsive Hubbard model at weak coupling

We compute the perturbative expansion for the effective interaction $W$ of the half-filled 2-dimensional Hubbard model. We derive extensions of standard RPA resummations that include arbitrarily high order contributions in the $W_{\uparrow \uparrow }$ and $W_{\uparrow \downarrow }$ basis. Using algorithmic tools we explore the static $Q$-dependent interaction as well as the same-time quantity both in momentum- and real-space. We emphasize the absence of screening in the Hubbard interaction where we find an enhanced repulsive local $W_{\uparrow \downarrow }$ with a non-zero attractive $W_{\uparrow \uparrow }$. Finally, starting from only a locally repulsive bare interaction find an emergent non-local nearest-neighbor attraction for low temperatures at sufficiently large values of $U\mathrm{/}t$ which may be key to understanding pairing processes in the model.