  Carrier-mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction plays an
important role in itinerant magnetism. There have been intense interest on its
general trend on bipartite lattice with particle-hole symmetry. In particular,
recently fabricated graphene is well described by the honeycomb lattice within
tight-binding approximation. We use SUSY quantum mechanics to study the RKKY
interaction on bipartite lattices. The SUSY structure naturally differentiate
the zero modes and those paired states at finite energies. The significant role
of zero modes is largely ignored in previous literature because their measure
is often zero in the thermodynamic limit. Employing both real-time and
imaginary-time formalism, we arrive at the same conclusion: The RKKY
interaction for impurity spins on different sublattices is always
antiferromagnetic. However, for impurity spins on the same sublattice, the
carrier-mediated RKKY interaction is not always ferromagnetic. Only in the
absence of zero modes, the sign rule on the bipartite lattice holds true. Our
finding highlight the importance of the zero modes in bipartite lattices. Their
significance needs further investigation and may lead to important advances in
carrier-mediated magnetism.
