Competing magnetism in $\pi$ electrons in graphene with a single carbon vacancy

One intriguing finding in graphene is the vacancy-induced magnetism that highlights the interesting interaction between local magnetic moments and conduction electrons. Within density functional theory, the current understanding of the ground state is that a Stoner instability gives rise to ferromagnetism of $\pi$ electrons aligned with the localized moment of a $\sigma$ dangling bond and the induced $\pi$ magnetic moments vanish at low vacancy concentrations. However, the observed Kondo effect suggests that $\pi$ electrons around the vacancy should antiferromagnetically couple to the local moment and carry non-vanishing moments. Here we propose that a phase possessing both significant out-of-plane displacements and $\pi$ bands with antiferromagnetic coupling to the localized $\sigma$ moment is the ground state. With the features we provide, it is possible for spin-resolved STM, STS, and ARPES measurements to verify the newly proposed phase.