RKKY interaction in graphene from the lattice Green’s function

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We study the exchange interaction $J$ between two magnetic impurities in undoped graphene (the Ruderman-Kittel-Kasuya-Yosida [RKKY] interaction) by directly computing the lattice Green's function for the tight-binding band structure for the honeycomb lattice. The method allows us to compute $J$ numerically for much larger distances than can be handled by finite-lattice calculations as well as for small distances. In addition, we rederive the analytical long-distance behavior of $J$ for linearly dispersive bands and find corrections to the oscillatory factor that were previously missed in the literature. The main features of the RKKY interaction in half-filled graphene are that unlike the $J\ensuremath{\propto}(2{k}_{F}R){}^{\ensuremath{-}2}\mathrm{sin}(2{k}_{F}R)$ behavior of an ordinary two-dimensional metal in the long-distance limit, $J$ in graphene falls off as $1/{R}^{3}$, shows the $1+\mathrm{cos}[(K\ensuremath{-}{K}^{\ensuremath{'}})\ifmmode\cdot\else\textperiodcentered\fi{}R]$-type oscillations with additional phase factors depending on the direction, and exhibits a ferromagnetic interaction for moments on the same sublattice and an antiferromagnetic interaction for moments on the opposite sublattices as required by particle-hole symmetry. The computed $J$ with the full band structure agrees with our analytical results in the long-distance limit, including the oscillatory factors with the additional phases.