  Many recent calculations have been performed to study a Co atom adsorbed on
graphene, with significantly varying results on the nature of the bonding. We
use auxiliary-field quantum Monte Carlo (AFQMC) and a size-correction embedding
scheme to accurately calculate the binding energy of Co on graphene. We find
that as a function of the distance h between the Co atom and the six-fold
hollow site, there are three distinct ground states corresponding to three
electronic configurations of the Co atom. Two of these states provide binding
and exhibit a double-well feature with nearly equal binding energy of 0.4 eV at
h = 1.51 and h = 1.65 angstroms, corresponding to low-spin $^2$Co (3d9 4s0) and
high-spin $^4$Co (3d8 4s1), respectively.
