Josephson current through a single Anderson impurity coupled to BCS leads

Abstract

We investigate the Josephson current $⟨J(\ensuremath{\phi})⟩$ through a quantum dot embedded between two superconductors showing a phase difference $\ensuremath{\phi}$. The system is modeled as a single Anderson impurity coupled to BCS leads, and the functional and the numerical renormalization group frameworks are employed to treat the local Coulomb interaction $U$. We reestablish the picture of a quantum phase transition occurring if the ratio between the Kondo temperature ${T}_{K}$ and the superconducting energy gap $\ensuremath{\Delta}$ or, at appropriate ${T}_{K}∕\ensuremath{\Delta}$, the phase difference $\ensuremath{\phi}$ or the impurity energy is varied. We present accurate zero- as well as finite-temperature $T$ data for the current itself, thereby settling a dispute raised about its magnitude. For small to intermediate $U$ and at $T=0$ the truncated functional renormalization group is demonstrated to produce reliable results without the need to implement demanding numerics. It thus provides a tool to extract characteristics from experimental current-voltage measurements.