We study the quantum phase transitions in two-dimensional arrays of Josephson-couples junctions with short range Josephson couplings (given by the Josephson energy $E_J$) and the charging energy $E_C$. We map the problem onto the solvable quantum generalization of the spherical model that improves over the mean-field theory method. The arrays are placed on the top of a two-dimensional electron gas separated by an insulator. We include effects of the local dissipation in the presence of an external magnetic flux $f=\Phi ∕{\Phi }_0$ in square lattice for several rational fluxes $f=0,\frac{1}{2},\frac{1}{3},\frac{1}{4}$, and $\frac{1}{6}$. We also have examined the $T=0$ superconducting-insulator phase boundary as a function of a dissipation ${\alpha }_0$ for two different geometry of the lattice: square and triangular. We have found a critical value of the dissipation parameter independent on geometry of the lattice and presence magnetic field.

• Received 14 October 2004

DOI:https://doi.org/10.1103/PhysRevB.72.014509