We present theoretical results for the equilibrium Josephson current through an Anderson dot tuned into the magnetic regime, using Hirsch-Fye Monte Carlo simulations covering the complete crossover from Kondo-dominated physics to pi junction behavior in a numerically exact way. Within the "magnetic" regime, U/Gamma >> 1 and epsilon0/Gamma < or = 1, the Josephson current is found to depend only on Delta/TK, where Delta is the BCS gap and TK the Kondo temperature. The junction behavior can be classified into four different quantum phases. We describe these behaviors, specify the associated three transition points, and identify a local minimum in the critical current of the junction as a function of Delta/TK.
Copyright 2004 The American Physical Society