RKKY interaction in graphene from the lattice Green's function

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 proportional to (2k(F)R)(-2) sin(2k(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 + cos[(K - K') . 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.