  We reveal that the origin of ferromagnetism caused by $sp$ electrons in
graphene with vacancies can be traced to electrons partially filling
$sp^{2*}$-antibonding and $p_z^*$-nonbonding states, which are induced by the
vacancies and appear near the Fermi level. Because the spatial wavefunctions of
the both states are composed of atomic orbitals in an antisymmetric
configuration, their spin wavefunctions should be symmetric according to the
electron exchange antisymmetric principle, leading to electrons partially
filling these states in spin polarization. Since this $p_z^*$ state originates
not from interactions between the atoms but from the unpaired $p_z$ orbitals
due to the removal of $p_z$ orbitals on the minority sublattice, the $p_z^*$
state is constrained, distributed on the atoms of the majority sublattice, and
decays gradually from the vacancy as $\sim$ $1/r$. According to these
characteristics, we concluded that the $p_z^*$ state plays a critical role in
magnetic ordering in graphene with vacancies. If the vacancy concentration in
graphene is large enough to cause the decay-length regions to overlap,
constraining the $p_z^*$ orbital components as little as possible on the
minority sublattice atoms in the overlap regions results in the vacancy-induced
$p_z^*$ states being coherent. The coherent process in the overlap region leads
to the wavefunctions in all the involved regions antisymmetrized, consequently
causing ferromagnetism according to the electron exchange antisymmetric
principle. This unusual mechanism concerned with the origin of $sp$-electron
magnetism and magnetic ordering has never before been reported and is
distinctly different from conventional mechanisms.
