Quantum critical point and scaling in a layered array of ultrasmall Josephson junctions

We have studied a quantum Hamiltonian that models an array of ultrasmall Josephson junctions with short range Josephson couplings, EJ , and charging energies, EC , due to the small capacitance of the junctions. We derive a new effective quantum spherical model for the array Hamiltonian. As an application we start by approximating the capacitance matrix by its self-capacitive limit and in the presence of an external uniform background of charges, qx. In this limit we obtain the zero–temperature superconductor–insulator phase diagram, Ecrit J (EC , qx), that improves upon previous theoretical results that used a mean field theory approximation. Next we obtain a closed–form expression for the conductivity of a square array, and derive a universal scaling relation valid about the zero–temperature quantum critical point. In the latter regime the energy scale is determined by temperature and we establish universal scaling forms for the frequency dependence of the conductivity. PACS numbers: 74.50.+r, 67.40.Db