Quantum fluctuations in chains of Josephson junctions

We study the effect of quantum fluctuations of the phase on the low-temperature behavior of two models of Josephson junction chains with Coulomb interactions taken into account. The first model, which represents a chain of junctions close to a ground plane, is the Hamiltonian version of the two-dimensional $\mathrm{XY}$ model in one space and one time dimension. We demonstrate explicitly how the Nelson-Kosterlitz jump manifests itself in the conduction properties of this system at a critical value of the superconducting grain capacitance. In the second model, the charging energy for a single junction in the chain is just the parallel-plate capacitor energy $\frac{Q^2}{8C}$, where $Q$ is the charge difference across the junction and $C$ is its capacitance. We show that for any nonzero charging energy (i.e., $C<\mathrm{}\infty$) quantum fluctuations produce exponential decay of the order-parameter correlation function. Therefore, in contrast to the first model, the Coulomb interaction always succeeds in disrupting the phase coherence of the array.