Attractive effect of a strong electronic repulsion -- the physics of vertex divergences

While the breakdown of the perturbation expansion for the many-electron problem has several formal consequences, here we unveil its physical effect: Flipping the sign of the effective electronic interaction in specific scattering channels. By decomposing local and uniform susceptibilities of the Hubbard model via their spectral representations, we prove how entering the non-perturbative regime causes an enhancement of the charge response, ultimately responsible for the phase-separation instabilities close to the Mott MIT. Our analysis opens a new route for understanding phase-transitions in the non-perturbative regime and clarifies why attractive effects emerging from a strong repulsion can induce phase-separations, but not s-wave pairing or charge-density wave instabilities.