Functional renormalization group and variational Monte Carlo studies of the electronic instabilities in graphene near 1/4 doping

We study the electronic instabilities of near 1/4 electron doped graphene using the functional renormalization group (FRG) and variational Monte-Carlo method. A modified FRG implementation is utilized to improve the treatment of the von Hove singularity. At 1/4 doping the system is a chiral spin density wave state exhibiting the anomalous quantized Hall effect, or equivalently a Chern insulator. When the doping drops below 1/4, the $d_{x^2-y^2}+i d_{xy}$ Cooper pairing becomes the leading instability. Our results suggest near 1/4 electron- or hole-doped graphene is a fertile playground for the search of Chern insulators and superconductors.