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 $Δ_α=|Δ_α| e^{i θ_α}$, where $α= L, R$ for the lead at left and right. We show that, when one of the SC gaps is much larger than the others $|Δ_L| \gg |Δ_R|$, the starting model can be mapped exactly onto the single-channel model, which consists of the right lead of $Δ_R$ and the Anderson impurity with an extra onsite SC gap of $Δ_d \equiv Γ_L e^{i θ_L}$. Here $θ_L$ and $Γ_L$ are defined with respect to the starting model, and $Γ_L$ 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, $|Δ_L| \gg |Δ_R|$, using the numerical renormalization group (NRG) method. The results show that the phase difference of the SC gaps $φ\equiv θ_R -θ_L$, 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.