Quantum phase transition in a minimal model for the Kondo effect in a Josephson junction

We propose a minimal model for the Josephson current through a quantum dot in a Kondo regime. We start with the model that consists of an Anderson impurity connected to two superconducting (SC) leads with the gaps Delta(alpha) = \Delta(alpha)\(ithetaalpha), where alpha = L, R for the lead at left and right. We show that, when one of the SC gaps is much larger than the others \DeltaL\ much greater than \Delta(R)\, the starting model can be mapped exactly onto the single-channel model, which consists of the right lead of Delta(R) and the Anderson impurity with an extra onsite SC gap of Delta(d) equivalent to Gamma(L)e(ithetaL). Here theta(L) and Gamma(L) are defined with respect to the starting model, and Gamma(L) is the level width due to the coupling with the left lead. Based on this simplified model,we study the ground-state properties for the asymmetric gap, \Delta(L)\ much greater than \Delta(R)\, using the numerical renormalization group (NRG) method. The results show that the phase difference of the SC gaps phi equivalent to theta(R) - theta(L), which induces the Josephson current, disturbs the screening of the local moment to destabilize the singlet ground state typical of the Kondo system. It can also drive the quantum phase transition to a magnetic doublet ground state, and at the critical point the Josephson current shows a discontinuous change. The asymmetry of the two SC gaps causes a re-entrant magnetic phase, in which the in-gap bound state lies close to the Fermi level.