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We study a quantum dot Josephson junction inside an Aharonov-Bohmenvironment. The geometry is modeled by an Anderson impurity coupled to twodirectly-linked BCS leads. We illustrate that the well-established picture ofthe low-energy physics being governed by an interplay of two distinct (singletand doublet) phases is still valid for this interferometric setup. The phaseboundary depends, however, non-monotonically on the coupling strength betweenthe superconductors, causing the system to exhibit re-entrance behavior andmultiple phase transitions. We compute the zero-temperature Josephson currentand demonstrate that it can become negative in the singlet phase by virtue ofthe Coulomb interaction U. As a starting point, the limit of largesuperconducting energy gaps \Delta=\infty is solved analytically. In order totackle arbitrary \Delta<\infty and U>0, we employ a truncated functionalrenormalization group scheme which was previously demonstrated to givequantitatively reliable results for the quantum dot Josephson problem.

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