Quantum phase diagram of granular superconducting quantum dots array

We study the quantum phase diagram of granular superconducting quantum dots (GSQD) array. We implement the physics of granularity by considering site dependent Josephson couplings, on-site charging energies and the intersite interactions. We predict dimer density wave and staggered phases at the insulating state for higher order commensurability. Several parts of the quantum phase diagram of GSQD are in contrast with the clean superconducting quantum dots array. We also obtain the superconducting phase of GSQD. We develop the theory for weak tunneling conductance and the Coulomb energy is smaller than the superconducting gap.