Indirect exchange of magnetic impurities in zigzag graphene ribbon

We use quantum Monte Carlo method to study the indirect coupling between two magnetic impurities on the zigzag edge of graphene ribbon. If the two adatoms are located on the sites belonging to opposite sublattices, their antiferromagnetic spin-spin correlation is drastically suppressed at the zero-energy, and is first enhanced and then decreased as we lower the chemical potential. If the two adatoms are adsorbed on the same sublattice, we find similar behavior of spin-spin correlation except for a crossover from ferromagnetic to antiferromagnetic correlation as we lower the chemical potential. We also calculate the weight of different components of d-electron wave function and local magnet moment for various values of parameters, and all the results are consistent with those of spin-spin correlation between two magnetic impurities.