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Carrier-mediated exchange coupling, known as Ruderman-Kittel-Kasuya-Yosida(RKKY) interaction, plays a fundamental role in itinerant ferromagnetism andhas great application potentials in spintronics. A recent theorem based on theimaginary-time method shows that the oscillatory RKKY interaction becomescommensurate on bipartite lattice and predicts that the effective exchangecoupling is always ferromagnetic for the same sublattice but antiferromagneticfor opposite sublattices. We revisit this important problem by real- andimaginary-time methods and find the theorem misses important contributions fromzero modes. To illustrate the importance of zero modes, we study the spinsusceptibility in graphene nanoribbons numerically. The effective exchangecoupling is largest on the edges but does not follow the predictions from thetheorem.

Ruderman-Kittel-Kasuya-Yosida interactions on a bipartite lattice