Phase phonon spectrum and melting in a quantum rotor model with diagonal disorder

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We study the zero-temperature (T = 0) quantum rotor model with on-site disorder in the charging energy. Such a model may serve as an idealized Hamiltonian for an array of Josephson-coupled small superconducting grains, or superfluid ⁴He in a disordered environment. In the approximation of small-amplitude phase fluctuations, the Hamiltonian maps onto a system of coupled harmonic oscillators with on-site disorder. We study the effects of disorder in this harmonic regime, using the coherent potential approximation (CPA), obtaining the density of states and the lifetimes of the spin-wave-like excitations for several choices of the parameters which characterize the disorder. Finally, we estimate the parameters characterizing the T = 0 quantum melting of the phase order, using a suitable Lindemann criterion.