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