Using a new cluster Monte Carlo algorithm, we study the phase diagram and critical properties of an interacting pair of resistively shunted Josephson junctions. This system models tunnelling between two electrodes through a small superconducting grain, and is described by a double sine-Gordon model. In accordance with theoretical predictions, we observe three different phases and crossover effects arising from an intermediate coupling fixed point. On the superconductor-to-metal phase boundary, the observed critical behaviour is within error-bars the same as in a single junction, with identical values of the critical resistance and a correlation function exponent which depends only on the strength of the Josephson coupling. We explain these critical properties on the basis of a renormalization group (RG) calculation. In addition, we propose an alternative new mean-field theory for this transition, which correctly predicts the location of the phase boundary at intermediate Josephson coupling strength.
We numerically study the superconductor-insulator phase transition in a model disordered two-dimensional (2D) superconductor as a function of applied magnetic field. The calculation involves quantum Monte Carlo calculations of the
(2+1)D XY model in the presence of both disorder and magnetic field. The
XY coupling is assumed to have the form $\ensuremath{-}J\text{ }\text{cos}({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j}\ensuremath{-}{A}_{ij})$, where
Aij has a mean of zero and a standard deviation $\ensuremath{\Delta}{A}_{ij}$. In a real system, such a model would be approximately realized by a 2D array of small Josephson-coupled grains with slight spatial disorder and a uniform applied magnetic field. The different values $\ensuremath{\Delta}{A}_{ij}$ then correspond to an applied field such that the average number of flux quanta per plaquette has various integer values
N: larger
N corresponds to larger $\ensuremath{\Delta}{A}_{ij}$. For any value of $\ensuremath{\Delta}{A}_{ij}$, there appears to be a critical coupling constant ${K}_{c}(\ensuremath{\Delta}{A}_{ij})=\sqrt{{[J/(2U)]}_{c}}$, where
U is the charging energy, below which the system is a Mott insulator; there is also a corresponding critical conductivity ${\ensuremath{\sigma}}^{\ensuremath{\ast}}(\ensuremath{\Delta}{A}_{ij})$ at the transition. For $\ensuremath{\Delta}{A}_{ij}=\ensuremath{\infty}$, the order parameter of the transition is a renormalized coupling constant
g. Using a numerical technique appropriate for disordered systems, we show that the transition at this value of $\ensuremath{\Delta}{A}_{ij}$ takes place from an insulating (I) phase to a Bose glass (BG) phase, and that the dynamical critical exponent characterizing this transition is $z\ensuremath{\sim}1.3$. By contrast,
z=1 for this model at $\ensuremath{\Delta}{A}_{ij}=0$. We suggest that the superconductor-to-insulator transition is actually of this I to BG class at all nonzero $\ensuremath{\Delta}{A}_{ij}$'s, and we support this interpretation by both numerical evidence and an analytical argument based on the Harris criterion [A. B. Harris, J. Phys. C 7, 1671 (1974)].
Kc is found to be a monotonically increasing function of $\ensuremath{\Delta}{A}_{ij}$. For certain values of
K, a disordered Josephson array may undergo a transition from an ordered, Bose glass phase to an insulator with increasing $\ensuremath{\Delta}{A}_{ij}$.
We have used Monte Carlo simulations, combined with finite-size scaling and two different real-space renormalization group approaches, to study a fully frustrated three-dimensional
XY model on a simple cubic lattice. This model corresponds to a lattice of Josephson-coupled superconducting grains in an applied magnetic field $\mathbf{H}=({\ensuremath{\Phi}}_{0}∕{a}^{2})(1∕2,1∕2,1∕2)$. We find that the model has a continuous phase transition with critical temperature
Tc=0.681J∕kB, where
J is the
XY coupling constant, and critical exponents $\ensuremath{\alpha}∕\ensuremath{\nu}=0.87\ifmmode\pm\else\textpm\fi{}0.01$, $v∕\ensuremath{\nu}=0.82\ifmmode\pm\else\textpm\fi{}0.01$, and $\ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.07$, where $\ensuremath{\alpha}$,
v, and $\ensuremath{\nu}$ describe the critical behavior of the specific heat, helicity modulus, and correlation length. We briefly compare our results with other studies of this model, and with a mean-field approximation.
The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n 0 (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY -model—and approaches the latter as n 0 increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n 0 <10, while quantum fluctuations in the phase are significant when n 0 <8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY -model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/ n 0 using the method of functional integration.
It is well known that the metal-insulator transition in two dimensions for non-interacting fermions takes place at infinitesimal disorder. In contrast, the superconductor-to-insulator transition takes place at a finite critical disorder (on the order of Vc ~ 2t), where V is the typical width of the distribution of random site energies and t is the hopping scale. In this article we compare the localization/delocalization properties of one and two particles. Whereas the metal-insulator transition is a consequence of single-particle Anderson localization, the superconductor-insulator transition (SIT) is due to pair localization – or, alternatively, fluctuations of the phase conjugate to pair density. The central question we address is how superconductivity emerges from localized single-particle states. We address this question using inhomogeneous mean field theory and quantum Monte Carlo techniques and make several testable predictions for local spectroscopic probes across the SIT. We show that with increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long-range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT despite a robust single-particle gap.
In two dimensions there is a direct superconductor-to-insulator quantum phase transition driven by increasing disorder. We elucidate, using a combination of inhomogeneous mean field theory and quantum Monte Carlo techniques, the nature of the phases and the mechanism of the transition. We make several testable predictions specifically for local spectroscopic probes. With increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT, despite a robust single-particle gap.