Local dissipation effects in two-dimensional quantum Josephson junction arrays with a magnetic field

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, 1/2, 1/3, 1/4, and 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.