A dissipative phase transition (DPT) is found in resistively shunted (4-75
k Omega) mesoscopic Josephson junctions owing to suppression of macroscopic
quantum tunneling of phase by the interaction with dissipative quantum-mec
hanical environment. We have studied the behavior of this transition as a f
unction of magnetic held. Instead of a field-induced insulator-superconduct
or transition, we find a hysteretic insulator-insulator (I-I) transition. T
he phase diagram for the DPT in a weak magnetic field is outlined. (C) 2000
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