One- and two-particle properties of the weakly interacting two-dimensional Hubbard model in proximity to the van Hove singularity

We study the weak-coupling limit of $t-t^\prime-U$ Hubbard model on a two-dimensional square lattice using direct perturbative approach. Aided by symbolic computational tools, we compute longitudinal density-density correlation functions in $\chi_{\uparrow \uparrow}$ and \downarrow}$ basis from which can obtain dynamical spin charge susceptibilities at arbitrary doping temperature. find that for non-zero $t^\prime$, zero frequency commensurate $\mathbf{q} = (\pi, \pi)$ excitations are each strongest different densities observe clear behavioral change appears tied to van Hove singularity non-interacting dispersion upon expansion is built. strongly reduced compressibility vicinity as well double occupancy. For finite observed occurs away half-filling leading us conclude this reduction distinct Mott insulating physics one expects strong-coupling regime. full structure electron hole doped scenarios agreement with experiments cuprate materials. Finally, peculiar splitting singularity, origin traced near bottom band.