Phase transition of dissipative Josephson arrays in a magnetic field

The phase diagram of an array of Josephson junctions in a transverse magnetic field is investigated. The capacitive interactions of charges on the superconducting islands and the associated quantum-mechanical effects, as well as the dissipation due to the flow of normal Ohmic currents, are taken into account. The mean-field approximation of this system can be mapped onto the tight-binding Schr\"odinger equation for Bloch electrons in a magnetic field, which had been analyzed by Hofstadter. We show how the transition temperature depends on the dissipation and the charging energy.