  To better understand the electronic structure of a single vacancy in
graphene, we study the ground state property of an effective Anderson model,
consisting of three dangling $sp^2$ orbitals of the surrounding carbon atoms
around the vacancy and the $\pi$ orbitals of carbon atoms that form the
honeycomb lattice with a single vacancy. Employing the block-Lanczos
density-matrix renormalization group method, we show that there are two phases
in the relevant parameter space, i.e., a nonmagnetic phase in the weak coupling
region and a free magnetic moment phase in the realistic parameter region. The
systematic analysis finds that, in the free magnetic moment phase, local
multiplets of the doubly degenerate irreducible representation of the
$\mathcal{C}_3$ point group with spin 1 become dominant in the ground state,
and approximately half of this local spin 1 is screened by electrons in the
surrounding $\pi$ orbitals, indicating the emergence of the residual spin-1/2
free magnetic moment. Furthermore, we find that the emergence of the free
magnetic moment is robust against carrier doping, which is in sharp contrast to
the case of graphene with an adatom, thus explaining the qualitative difference
observed experimentally in these two classes of systems. We also find the
enhancement of the spin correlation function between $\pi$ electrons around the
vacancy and those in the conduction band away from the vacancy in the undoped
case, as compared to that in the doped case, while the spin correlation
function between the $\sigma$ electrons in the $sp^2$ dangling orbitals around
the vacancy and the $\pi$ electrons in the conduction band remains large in
both undoped and doped cases. Our calculations thus support qualitatively the
previous experiment that suggests the emergence of free magnetic moment with
two distinct origins.
