The quantum-dot Josephson junctions (QD-JJ) have received a great theoretical and experimental attention because they can exhibit an interesting competition between two celebrated many-body correlations in condensed matter physics: the superconductivity and the Kondo effect. The physics of the QD-JJ is more exciting when one takes into account spin interactions that exist in real molecules. We have investigated the Josephson effect through single- or two-level quantum dot with spin exchange interaction by using the numerical renormalization group method supplemented with the perturbation theory and the scaling approach. The spin exchange coupling is observed to strongly affect or even suppress the Kondo correlation in the system. Consequently, depending on the characteristics of the spin exchange interaction (whether it is ferromagnetic or antiferromagnetic), we can observe different kinds of Kondo-driven 0-Pi transition and interesting states, among which are 0-Pi-0 or Pi-0-Pi double transition and a Kondo-assisted Pi-junction. In this talk we present several representative results that highlight the effect of the spin interaction in the Josephson effect in QD-JJ.
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