  Motivated by the recent upsurge in research of three-dimensional topological
semimetals (SMs), we theoretically study the RKKY interaction between magnetic
impurities in node-line SMs with and without the chirality and obtain the
analytical expressions. We find that unique toroidal Fermi surface (FS) in
nodal-line SMs, distinctly different from the spheroid FS in the SMs with the
point nodes, has significant influences on the RKKY interaction, leading to
strong anisotropic oscillation and unique decay features. In the direction
perpendicular to node-line plane, as usual, there is only one oscillation
period related to the Fermi energy. In contrast, in the node-line plane, the
RKKY interaction form a beating pattern and oscillates more rapidly with two
distinct periods: one is coming from the Fermi energy and the other is from the
radius of node-line. More importantly, inside nodal-line SMs bulk, the decay
rate of RKKY interaction manifests a typical two-dimensional feature for
impurities aligned along the direction perpendicular to nodal-line plane.
Furthermore, the magnetic interactions in nodal-line SMs with linear and
quadratic dispersions in the nodal-line plane are compared. We also discuss the
possible application of the present theory on realistic NLSM ZrSiSe. Our
results shed the light for application of magnetically doped node-line SMs in
spintronics.
