Enhanced anti-ferromagnetic exchange between magnetic impurities in a superconducting host*

It is generally believed that superconductivity only weakly affects the indirect exchange between magnetic impurities. If the distance $r$ between impurities is smaller than than the superconducting coherence length $(r<\xi)$, this exchange is thought to be dominated by RKKY interactions, identical to the those in a normal metallic host. This perception is based on a perturbative treatment of the exchange interaction. Here, we discuss a non-perturbative analysis and demonstrate that the presence of Yu-Shiba-Rusinov bound states induces a strong $1/r^2$ anti-ferromagnetic interaction that can dominate over conventional RKKY even at distances significantly smaller than the coherence length $(r\ll\xi)$. Experimental signatures, implications and applications are discussed.