Changes of Kondo effect in the junction with DIII-class topological and s-wave superconductors

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We discuss the change of the Kondo effect in the Josephson junction formed by the indirect coupling between a one-dimensional DIII-class topological and s-wave superconductors via a quantum dot. By performing the Schrieffer-Wolff transformation, we find that the single-electron occupation in the quantum dot induces various correlation modes, such as the Kondo and singlet-triplet correlations between the quantum dot and the s-wave superconductor and the spin exchange correlation between the dot and Majorana doublet. Moreover, it plays a nontrivial role in modifying the Josephson effect, leading to the occurrence of anisotropic and high-order Kondo correlation. In addition, due to the quantum dot in the Kondo regime, extra spin exchange correlations contribute to the Josephson effect as well. Nevertheless, if the DIII-class topological superconductor degenerates into D-class because of the destruction of time-reversal invariance, all such terms will disappear completely. We believe that this work shows the fundamental difference between the D- and DIII-class topological superconductors.