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

Abstract

We present results from an extensive analytic and numerical study of a twodimensional model of a square array of ultrasmall Josephson junctions. We include the ultrasmall self and mutual capacitances of the junctions, for the same parameter ranges as those produced in the experiments. The model Hamiltonian studied includes the Josephson, E J , as well as the charging, E 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 non-perturbative quantum Monte Carlo 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 T QU IT as a function of the magnitude of imaginary time axis L τ . We find that for some relatively large values of α = E C E J (1 ≤ α ≤ 2.25), the L τ → ∞ limit does appear to give a non-zero T QU IT , while for α ≥ 2.5, T QU IT = 0. We use the SCHA to analytically understand the L τ 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 α −1 .

Phase and charge re-entrant phase transitions in two capacitively coupled Josephson arrays with ultrasmall junctions

“…This simplification cannot be used for the integer fields, except when C m = 0, in which case the m(τ, r) fields can be summed up exactly. This approach allows to build up look up tables that introduce an adequate effective potential [15].…”

Quantum phase transitions in superconducting arrays under external magnetic fields

“…The effects of external magnetic fields on phase transitions have been studied both in the classical arrays without charging energy [1] and in the quantum arrays [9][10][11][12]. However, most existing analytical works on the latter have employed mean-field-like approximations, which are not reliable in two dimensions.…”

“…In the opposite limit, on the other hand, it is well known that there exists an interesting duality between charges and vortices [3,4], and the system with only junction capacitance at zero temperature undergoes a chargeunbinding Berezinskii-Kosterlitz-Thouless (BKT) transition [5] from the insulating phase to the superconducting one as the ratio E J /E 1 with E 1 ≡ e 2 /2C 1 is increased. The critical value (E J /E 1 ) c , beyond which the array is superconducting, has been found 0.6 in experiment [6], 0.23 in the duality argument [4] and in the variational method [7], 0.51 in perturbation expansion [8], and 0.36 in quantum Monte Carlo simulations [9]. As C 0 is increased from zero in this system, the interactions between charges are screened, with the screening length given by Λ ≡ C 1 /C 0 [4], and the nature of the phase transition is expected to alter.…”

Berezinskii-Kosterlitz-Thouless Transition in Josephson Junction Arrays

“…A reentrance of the phase stiffness was already found for a related model in Ref. [10], but the authors concluded that the drop of the helicity modulus at lowest temperatures was probably due to the finiteness of the Trotter number P . Systematic extrapolations in the Trotter number and in the lattice size have been done, and presented in Figs.…”

“…[10], by derivation of the path-integral expression of the partition function (see Appendix). Kosterlitz's renormalization group equations provide the critical scaling law for the finite-size helicity modulus Υ L :…”