We were able to gain detailed numerical data showing the phase transition of the quantum dot from the 0-junction-regime to the
-junction-regime of
the Josephson current if the superconducting gap 
is increased. Our
data confirms the traditional interpretation that the Kondo-effect at small
corresponds to the 0-junction-regime, while the formation of a
magnetic moment on the quantum dot leads to the -phase-shift of the
Josephson current.
Our findings give a solid numerical foundation to the usual interpretation of the
-phase-shift of the Josephson current without relying on any
approximation. The application of the novel diagrammatic determinantal
quantum Monte Carlo algorithm shows its power even in models with a complex
phase factor.
The talk is divided in two parts: The first part is rather technical and gives a short introduction to the DDQMC-Algorithm. In the second part, the numerical results are presented along with the data gained from the study of two effective models valid in the limit
and
, 
.