Critical properties of two-dimensional Josephson-junction arrays with zero-point quantum fluctuations

We present results from an extensive analytic and numerical study of a two-dimensional model of a square array of ultrasmall Josephson junctions. We include the ultrasmall self-capacitance and mutual capacitance of the junctions, for the same parameter ranges as those produced in the experiments. The model Hamiltonian studied includes the Josephson, ${\mathit{E}}_{\mathit{J}}$, as well as the charging, ${\mathit{E}}_{\mathit{C}}$, energies between superconducting islands. The corresponding quantum partition function is expressed in different calculationally convenient ways within its path-integral representation. The phase diagram is analytically studied using a WKB renormalization group (WKB-RG) plus a self-consistent harmonic approximation (SCHA) analysis, together with nonperturbative quantum Monte Carlo (QMC) simulations. Most of the results presented here pertain to the superconductor to normal (S-N) region, although some results for the insulating to normal (I-N) region are also included. We find very good agreement between the WKB-RG and QMC results when compared to the experimental data. To fit the data, we only used the experimentally determined capacitances as fitting parameters. The WKB-RG analysis in the S-N region predicts a low-temperature instability, i.e., a quantum-induced transition (QUIT). We carefully analyze the possible existence of the QUIT via the QMC simulations and carry out a finite-size analysis of ${\mathit{T}}_{\mathrm{QUIT}}$ as a function of the magnitude of the imaginary-time axis ${\mathit{L}}_{\mathrm{\tau }}$. We find that for some relatively large values of α=${\mathit{E}}_{\mathit{C}}$/${\mathit{E}}_{\mathit{J}}$ (1⩽α⩽2.25), the ${\mathit{L}}_{\mathrm{\tau }}$→∞ limit does appear to give a nonzero ${\mathit{T}}_{\mathrm{QUIT}}$, while for α≥2.5, ${\mathit{T}}_{\mathrm{QUIT}}$=0. We use the SCHA to analytically understand the ${\mathit{L}}_{\mathrm{\tau }}$ dependence of the QMC results with good agreement between them. Finally, we also carried out a WKB-RG analysis in the I-N region and found no evidence of a low-temperature QUIT, up to lowest order in ${\mathrm{\alpha }}^{\mathrm{-}1}$. © 1996 The American Physical Society.