Supercurrent and multiple singlet-doublet phase transitions of a quantum dot Josephson junction inside an Aharonov-Bohm ring

We study a quantum dot Josephson junction inside an Aharonov-Bohm environment. The geometry is modeled by an Anderson impurity coupled to two directly linked BCS leads. We illustrate that the well-established picture of the low-energy physics being governed by an interplay of two distinct (singlet and doublet) phases is still valid for this interferometric setup. The phase boundary depends, however, nonmonotonically on the coupling strength between the superconductors, causing the system to exhibit re-entrance behavior and multiple phase transitions. We compute the zero-temperature Josephson current and demonstrate that it can become negative in the singlet phase by virtue of the Coulomb interaction U. As a starting point, the limit of large superconducting energy gaps Delta=infinity is solved analytically. In order to tackle arbitrary Delta 0, we employ a truncated functional renormalization-group scheme which was previously demonstrated to give quantitatively reliable results for the quantum dot Josephson problem.