  Weak-coupling phenomena of the two-dimensional Hubbard model is gaining
momentum as a new interesting research field due to its extraordinarily rich
behavior as a function of the carrier density and model parameters. Salmhofer
[{\it Commun. Math. Phys}. \textbf{194}, 249 (1998);{\it Phys. Rev. Lett}. {\bf
87}, 187004 (2001)] developed a new renormalization-group method for
interacting Fermi systems and Metzner [{\it Phys. Rev. B} {\bf 61}, 7364
(2000);{\it Phys. Rev. Lett}. {\bf 85}, 5162 (2000)] implemented this
renormalization group analysis of the two-dimensional Hubbard model. In this
work, we demonstrate the spin-wave dependent susceptibility behavior of model
graphene-phosphorene van der Waals heterostructure in the framework of
renormalization group approach. We implement signlet vertex response function
for the weakly interacting van der Waals Fermi system with nearest-neighbor
hopping amplitudes. This analytical approach is further correlated with {\it ab
initio} simulation results and extended for spin-wave dependent susceptibility
behavior with possible experimental protocols. We present the resulting
compressibility and phase diagram in the vicinity of half-filling, and also
results for the density dependence of the critical energy scale.
