Charge instabilities near a Van Hove singularity

The charge instabilities of electron systems in the square lattice are analyzed near the Van Hove singularity by means of a Wilsonian renormalization group approach. We show that the method preserves the spin rotational invariance at all scales, allowing a rigorous determination of spin and charge instabilities of the $t-t^{\prime }$ Hubbard model. Regarding the latter, repulsive interactions fall into two different universality classes. One of them has nonsingular response functions in the charge sector, while the other is characterized by the splitting of the Van Hove singularity. At the level of marginal perturbations, the Hubbard model turns out to be at the boundary between the two universality classes, while extended models with nearest-neighbor repulsive interactions belong to the latter class. In the case of open systems allowed to exchange particles with a reservoir, we show the existence of a range of fillings forbidden above and below the Van Hove singularity. This has the property of attracting the Fermi level in the mentioned range, as the system reaches its lowest energy when the Fermi energy is at the singularity.