Spin-resolved Yu-Shiba-Rusinov states and bistable phase caused by side-coupled Majorana bound states in quantum-dot Josephson junctions

We theoretically investigate the Josephson phase transition in a superconductor/quantum-dot/superconductor junction with side-coupled Majorana bound states (MBSs), by means of the zero-bandwidth approximation and numerical renormalization group (NRG) methods. It is found that the bistable phase of the Josephson current is allowed to arise in this system, when the dot level is of electron-hole symmetry. The NRG calculation shows that such a phase is induced by the appearance of the spin-dependent states in the quantum dot, due to the side coupling of MBSs. Through further research, we observe that the MBS leads to the formation of two distinct spin-resolved Yu-Shiba-Rusinov (YSR) states. The spin screening of the superconductors to the YSR states causes the occurrence of spin-dependent Kondo effect, and then the two-stage Kondo effect can be observed with the special and nontrivial contributions of the MBS. It also shows that the properties of the first-stage Kondo effect can be modulated in detail by adjusting the dot-MBS and inter-MBS couplings, which leads to the emergence of the new bistable phase of the Josephson current. All these results are helpful for understanding the Kondo physics and phase transition in a Josephson junction with side-coupled MBSs.