Abstract:
We reconsider the issue of Berezinskii-Kosterlitz-Thouless (BKT) transition into an insulating state in the Coulomb-dominated Josephson junction arrays. We show that previously predicted picture of the Cooper-pair BKT transtion at T = T_2 is valid only under the condition that T_2 is considerably below the parity-effect temperature (which is usually almost 10 times below the value of superconductive transition temperature), and even in this case it is not a rigorous phase transition but only a crossover, whereas the real phase transition takes place at T_1 = T_2/4. Our theory is in agreement with available experimental data on Coulomb-dominated Josephson arrays and also sheds some light on the origin of unusual reentrant temperature dependence of resistivity in the array with nearly-criticial ratio of Coulomb to Josephson energies.

Abstract:
The quantum XY model shows a Berezinskii-Kosterlitz-Thouless (BKT) transition between a phase with quasi long-range order and a disordered one, like the corresponding classical model. The effect of the quantum fluctuations is to weaken the transition and eventually to destroy it. However, in this respect the mechanism of disappearance of the transition is not yet clear. In this work we address the problem of the quenching of the BKT in the quantum XY model in the region of small temperature and high quantum coupling. In particular, we study the phase diagram of a 2D Josephson junction array, that is one of the best experimental realizations of a quantum XY model. A genuine BKT transition is found up to a threshold value $g^\star$ of the quantum coupling, beyond which no phase coherence is established. Slightly below $g^\star$ the phase stiffness shows a reentrant behavior at lowest temperatures, driven by strong nonlinear quantum fluctuations. Such a reentrance is removed if the dissipation effect of shunt resistors is included.

Abstract:
We numerically investigate dynamic critical behaviors of two-dimensional (2D) Josephson-junction arrays with positional disorder in the scheme of the resistively shunted junction dynamics. Large-scale computation of the current voltage characteristics reveals an evidence supporting that a phase transition occurs at a nonzero critical temperature in the strong disorder regime, as well as in the weak disorder regime. The phase transition at weak disorder appears to belong to the Berezinskii-Kosterlitz-Thouless (BKT) type. In contrast, evidence for a non-BKT transition is found in the strong disorder regime. These results are consistent with the recent experiment %by Yun {\it et al.} in cond-mat/0509151 on positionally disordered Josephson-junction arrays; in particular, the critical temperature of the non-BKT transition (ranging from 0.265 down to the minimum 0.22 in units of $E_J/k_B$ with the Josephson coupling strength $E_J$), the correlation length critical exponent $\nu=1.2$, and the dynamic critical exponent $z=2.0$ in the strong disorder regime agree with the existing studies of the 2D gauge-glass model.

Abstract:
Experiments on one-dimensional small capacitance Josephson Junction arrays are described. The arrays have a junction capacitance that is much larger than the stray capacitance of the electrodes, which we argue is important for observation of Coulomb blockade. The Josephson energy can be tuned in situ and an evolution from Josephson-like to Coulomb blockade behavior is observed. This evolution can be described as a superconducting to insulating, quantum phase transition. In the Coulomb blockade state, hysteretic current-voltage characteristics are described by a dynamic model which is dual to the resistively shunted junction model of classical Josephson junctions.

Abstract:
We review experiments on small-capacitance Josephson junctions. When the Josephson junction is fabricated in the configuration of dc superconducting quantum interference device (SQUID), the Josephson coupling can be tuned IN SITU with an external magnetic field. The electrical transport properties of one-dimensional arrays of small-capacitance dc SQUIDs have been investigated in order to study the superconductor-insulator transition. The arrays have also been used to bias a single Josephson junction, and a clear Coulomb blockade of Cooper-pair tunneling has been observed in the single junction.

Abstract:
The properties of vortices in Josephson junction arrays are investigated in the quantum regime near the superconductor-insulator transition. We derive and study an effective action for vortex dynamics that is valid in the region where the charging energy is comparable to the Josephson coupling energy. In the superconducting phase the onset of quantum effects reduces the vortex mass and depinning current. In the case of long range Coulomb interaction between Cooper pairs we find that as the transition is approached, the velocity window in which ballistic vortex motion is possible grows. At the superconductor-insulator transition the vortex mass vanishes and vortices and spinwaves decouple. In the case of on-site Coulomb repulsion (which is of relevance for superconducting granular films) the vortex mass it is sample-size dependent in the superconducting phase, but stays finite at the critical point where it is scale invariant. The relation of our work to experiment is discussed.

Abstract:
We use driven Monte Carlo dynamics to study the resistive behavior of superconducting Josephson junction arrays on a honeycomb lattice in a magnetic field corresponding to $f$ flux quantum per plaquette. While for $f=1/3$ the onset of zero resistance is found at nonzero temperature, for $f=1/2$ the results are consistent with a transition scenario where the critical temperature vanishes and the linear resistivity shows thermally activated behavior. We determine the thermal critical exponent of the zero-temperature transition for $f=1/2$, from a dynamic scaling analysis of the nonlinear resistivity. The resistive behavior agrees with recent results obtained for the phase-coherence transition from correlation length calculations and with experimental observations on ultra-thin superconducting films with a triangular pattern of nanoholes.

Abstract:
the critical behavior of zero-temperature superconducting transitions which can occur in disordered two-dimensional josephson-junction arrays are investigated by monte carlo calculation of ground-state excitation energies and dynamical simulation of the current-voltage characteristics at nonzero temperatures. two models of arrays in an applied magnetic field are considered: random dilution of junctions and random couplings with half-ux quantum per plaquette f = 1/2. abovea critical value of disorder, finite-size scaling of the excitation energies indicates a zero-temperature transition and allows an estimate of the critical disorder and the thermal correlation length exponent characterizing the transition. current-voltage scaling is consistent with the zero-temperature transition. the linear resistance is nonzero at finite temperatures but nonlinear behavior sets in at a characteristic current density determined by the thermal critical exponent. the zero-temperature transition provides an explanation of the washing out of structure for increasing disorder at f = 1/2 while it remains for f = 0, observed experimentally in supercondoucting wire networks.

Abstract:
The critical behavior of zero-temperature superconducting transitions which can occur in disordered two-dimensional Josephson-junction arrays are investigated by Monte Carlo calculation of ground-state excitation energies and dynamical simulation of the current-voltage characteristics at nonzero temperatures. Two models of arrays in an applied magnetic field are considered: random dilution of junctions and random couplings with half-ux quantum per plaquette f = 1/2. Abovea critical value of disorder, finite-size scaling of the excitation energies indicates a zero-temperature transition and allows an estimate of the critical disorder and the thermal correlation length exponent characterizing the transition. Current-voltage scaling is consistent with the zero-temperature transition. The linear resistance is nonzero at finite temperatures but nonlinear behavior sets in at a characteristic current density determined by the thermal critical exponent. The zero-temperature transition provides an explanation of the washing out of structure for increasing disorder at f = 1/2 while it remains for f = 0, observed experimentally in supercondoucting wire networks.

Abstract:
We report the results of ac sheet conductance measurements performed on fully frustrated square arrays of Josephson junctions whose coupling energy is periodically modulated in one of the principal lattice directions. Such systems are predicted to exhibit two distinct transitions: a low-temperature Ising-like transition triggered by the proliferation of domain walls and a high-temperature transition driven by the vortex unbinding mechanism of the Beresinskii-Kosterlitz-Thouless (BKT) theory. Both the superfluid and dissipative components of the conductance are found to exhibit features which unambiguously demonstrate the existence of a double transition whose properties are consistent with the Ising-BKT scenario.