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

TL;DR

This paper models the Josephson current through a quantum dot in a Kondo regime, revealing how asymmetric superconducting gaps induce a quantum phase transition and magnetic re-entrance, analyzed via the numerical renormalization group method.

Contribution

It introduces a simplified exactly solvable model for the system with asymmetric gaps and studies the quantum phase transition using NRG, highlighting the effects of phase difference and gap asymmetry.

Findings

Phase difference destabilizes the singlet ground state.

Quantum phase transition to a magnetic doublet occurs at critical points.

Asymmetry causes re-entrant magnetic phases with in-gap states near the Fermi level.

Abstract

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 $\Delta_{\alpha}=|\Delta_{\alpha}| e^{i \theta_{\alpha}}$, 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 $|\Delta_L| \gg |\Delta_R|$, 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 $\Delta_d \equiv \Gamma_L e^{i \theta_L}$. 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, $|\Delta_L| \gg |\Delta_R|$, using the numerical renormalization group (NRG) method. The results show that the phase difference of the SC gaps $\phi \equiv \theta_R -\theta_L$, 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.