Superconducting arrays in a magnetic field: Quantum effects

We consider a Josephson-coupled superconducting array with finite quantum fluctuations, arising from a nonzero capacitive charging energy, placed in a transverse magnetic field. To estimate the superconducting transition temperature as a function of magnetic field, we introduce a Hartree-type mean-field approximation. With no applied magnetic field, this approximation is very similar to that of Simanek, but unlike the latter, it does not lead to a reentrant normal phase transition. Reentrance is absent because we include no 4 -periodic eigenstates of Mathieus equation in calculating quantum-statistical expectation values. We argue that these 4 -periodic functions are properly omitted because the original Hamiltonian does not include pair-breaking terms. With charging energies included, we find the transition temperature to be highly nonmonotonic in magnetic field, just as in the zero-capacitance limit. For every field B, there exists an upper critical charging energy Uc(B) above which the array is normal even at T=0; this charging energy is highly nonmonotonic in field. A brief comparison is made between our results and other recent calculations involving superconducting arrays in the presence of charging energies.