Commensurate Antiferromagnetic Fluctuations with Small Electron Doping

We investigate the presence of antiferromagnetic fluctuations in the longitudinal and transversal spin susceptibilities of a square lattice. The inclusion of both first and second neighbour hopping terms, along with exchange coupling, induces anti-ferromagnetic fluctuations over a finite range of fillings in both longitudinal and transversal static spin susceptibilities. In the absence of on-site Hubbard interaction, we observe incommensurate antiferromagnetic fluctuations between the two Van Hove fillings. Beyond the second Van Hove singularity at $n=1.03$, commensurate antiferromagnetic fluctuations dominate in both longitudinal and transversal spin susceptibilities. When incorporating a finite Hubbard interaction strength, $U$, we find that the commensurate anti-ferromagnetic fluctuations are preserved in both longitudinal and transversal dressed spin susceptibilities. However anti-ferromagnetic fluctuations vanish beyond a critical value of the Hubbard coupling strength, $U_{c}$. Despite the exchange field can be induced by a ferromagnetic substrate, we do not observe any ferromagnetic fluctuations in the system.