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Quantum Monte Carlo Study of a Disordered 2-D Josephson Junction Array

We have studied the superconducting-insulating phase transition in a disordered two-dimensional Josephson junction array, using quantum Monte Carlo techniques. We consider disorder in both the capacitive energies and in the values of the offset charges. The calculated phase diagram shows that the lobe structure of the phase diagram disappears for sufficiently strong disorder in the offset charge. Our results agree quite well with previous calculations carried out using a mean-field approximation.


I. INTRODUCTION
A Josephson junction array (JJA) consists of a collection of superconducting islands connected by Josephson coupling. The coupling can arise from tunnel junctions through an insulating layer, or from the proximity effect. The arrays themselves can be produced experimentally in a wide range of geometries, and with a great variety of individual junction parameters. They can serve as valuable model systems for studying quantum phase transitions 1,2 under conditions such that the experimental parameters can be readily tuned 3 .
Recently, it has also been proposed that small groups of Josephson junctions may serve as quantum bits (qubits) in quantum information technology. In this case, the quantum logic operations are performed by manipulating experimental parameters such as gate voltages or magnetic fields 4 .
If the superconducting islands in a Josephson array are sufficiently large, the array is believed to become superconducting in two stages. First, at a temperature near the bulk transition temperature, each island becomes superconducting. Secondly, at a lower temperature, and provided that the array is at least two-dimensional, thermal fluctuations become sufficiently weak that a global phase coherence can be established throughout the array 2 .
If, however, the islands in the array are small, such phase coherence may not be established even at temperature T = 0. The reason is that there is a second energy which becomes important in small grains, namely the charging energy of the grains 5 . The crucial physics is then determined by the competition between this charging energy and the Josephson coupling energy. If the charging energy is sufficiently large, then it becomes prohibitively expensive energetically to transfer Cooper pairs from grain to grain, and the array becomes an insulator, even though each grain is in its superconducting state. If, on the other hand, the Josephson dominates, the array becomes phase coherent at T = 0. In this case, Cooper pairs can tunnel between neighboring grains and the array as a whole will be in the superconducting phase. Thus, by tuning the ratio of the charging and Josephson energies, one can cause the array to undergo a quantum phase transition between a superconducting and an insulating state 1,3 .
In the limit of very large number of Cooper pairs per grain, the Josephson junction array with diagonal charging energy is equivalent to the Bose Hubbard model (BHM), which describes soft core bosons hopping on a lattice with on-site Coulomb interactions. The BHM has previously been extensively studied by Fisher et al 8 , who constructed its T = 0 phase diagram. In the absence of disorder, the phase diagram shows two phases: a Mott insulating phase and a superfluid phase. In the presence of disorder, an additional phase, known as the Bose glass phase, emerges 9 . Recently, the phase diagram of the disordered BHM in 2-dimensions (2D) has been studied by Lee et al using QMC 10 .
The aim of this paper is to construct the phase diagram for the superconducting-insulating (SI) transition in a particular model for a JJA, including the effects of disorder. Such disorder is clearly unavoidable in most practical systems. The model Hamiltonian we study has previously been investigated by several authors in ordered and disordered arrays, using both mean-field theory (MFT) and quantum Monte Carlo (QMC) techniques. A central goal of our work is to compare the MFT of Refs. 6,7 and QMC results in a disordered array in order to check the accuracy of the MFT. We shall show that the MFT is generally satisfactory, even for disordered systems.
The remainder of this paper is organized as follows. In the next section, we present the model Hamiltonian and describe the QMC algorithm we use. Our numerical results are presented in Section III, followed by a brief concluding discussion in Section IV.

II. MODEL HAMILTONIAN AND QMC ALGORITHM
Our model of a JJA Hamiltonian involves two types of degrees of freedom: the number of excess Cooper pairsn i on the ith grain, and the phaseφ i of the superconducting order parameter on the ith grain.n i andφ i are taken to be quantum-mechanically conjugate variables with commutation relations [n i ,φ j ] = −iδ i,j . We consider the following model Hamiltonian on a square lattice in 2D: Here E J is the Josephson coupling strength between nearest neighbors denoted by i, j (assumed to be the same for all nearest neighbor pairs), and U ii is the charging energy of the ith grain. We expect that U ii = q 2 /C ii , where C ii represents the capacitance of the ith grain with respect to ground, and q = 2e is the charge of a Cooper pair. In a more general model, the charging part of the energy would be written 1 2 U ij (n i −n i )(n j −n j ), where U ij is an element of the charging energy matrix. In earlier calculations using MFT, nearest-neighbor and next-nearest-neighbor terms have been included, but these are numerically more difficult to include in QMC than the diagonal terms. Finally, the quantityn i represents the "offset charge."n i is related to the voltage between the ith grain and a common ground plane.
In an ordered array,n i should be independent of i. We omit any dissipative terms arising from Ohmic shunts in the junctions, and we assume that amplitude fluctuations of the gap on the individual grains can be neglected; this neglect of amplitude fluctuations should be reasonable if T ≪ T c0 , the single-grain transition temperature. Thus tunneling of charge between grains involves only Cooper pairs, not single electrons.
In our calculations, we include disorder in two terms in the Hamiltonian: the diagonal charging element U ii and the offset chargen i . Disorder in U ii may arise from randomness in the size of the individual grains, while disorder inn i could arise from random offset charges near the superconducting grains. Such disorder is unavoidable in practical realizations of Josephson arrays.
It is worthwhile to discuss the effects of the offset chargesn i qualitatively. Such charges can arise in two ways: (i) from a voltage applied between the array and the substrate, and (ii) from random charges in the substrate. In the latter case,n i is a random variable. In arrays with only a few grains,n i can, in principle, be tuned individually to desired values.
However, such tuning, although possibly still achievable in principle, may be difficult to attain in practice in arrays with many junctions.
In ordered arrays, the most important effect of the offset charge is to reduce the region of the phase diagram where the Mott insulating phase is stable. The reason is that, ifn i = 0, charge fluctuations become energetically less expensive; it is therefore correspondingly easier to establish phase coherence and to destroy the Mott insulator.
For large values of x, f (x) ≈ 1.
Finally, the partition function Z can be expressed as Z = Tre −S , where the action S is defined as In Eq. (2), the new degrees of freedom are integer-valued current loops J To evaluate S, we use the standard Metropolis algorithm to generate configurations of the currents J i,τ at inverse temperature β. It is convenient to work in the grand canonical ensemble. The system size is assumed to be a parallelepiped with dimensions (L, L, L τ ) along the space and imaginary time axes respectively, with periodic boundary conditions in all three directions.
We locate the superconducting-insulating (SI) phase transition by examining the superfluid density ρ, which is proportional to the stiffness of the system against a twist in the phase. As has been shown previously 11 , ρ can be expressed in terms of the current variables Here 2 is the so-called winding number in the x-direction, and · · · denotes both a grand canonical and a disorder average. If ρ is non-zero in the thermodynamic limit of a large system, then there is long-range phase coherence and the system will be in the superconducting phase.

III. RESULTS
We have carried out our simulations for three different system sizes: 8 3 , 10 3 , and 14 3 , and have usually averaged over ∼ 100 − 500 realizations of the disorder. For system with no disorder we studied system with a size of 20 3 . For each disorder realization, the system is equilibrated by slowly annealing in temperature, over about 2000 × L 2 L τ passes through the entire system for each value of E J considered, followed by approximately twice as many steps over which the grand canonical averages are computed. Figure 1 shows the phase diagram for a JJA at k B T = 0.3U with diagonal charging energy and no disorder, as calculated using QMC as described above. For reference, we also show the same phase diagram as calculated using MFT 6,7 . The regions of the phase diagram denoted S or I correspond to the superconducting and Mott insulating regions, respectively.
The two curves show the phase boundaries as calculated using MFT (squares) and QMC We have considered two types of disorder: randomness in the offset chargesn i , and randomness in the diagonal charging energies U ii . Disorder inn i is the analog of chemical potential disorder in the BHM, while disorder in the U ii 's is analogous to randomness in the mean-field on-site Coulomb energy in the BHM.
We begin by describing our results for offset charge randomness. We have carried out calculations with two different random distributions ofn's. In the first case, we choose the offset charge at each lattice site so thatn i =n + σ i , where σ i is a random number uniformly  in the U ii always increases the region of the phase diagram in which the S phase is stable, whatever the value ofn. This trend is not surprising, because fluctuations in the U ii 's lead to greater fluctuations in the number of Cooper pairs on each island. Thus, by the uncertainty principle, the phase fluctuations should be relatively reduced, and therefore the S (phase-ordered) state should become more favorable, as seen in our calculations.

IV. DISCUSSION AND SUMMARY
In this paper, we have studied the phase diagram of a model Josephson junction array at low temperatures using quantum Monte Carlo techniques. Our goal was to analyze the effects of disorder in both the offset chargesn i and grain sizes (which affect the diagonal elements of the charging energy matrix, U ii ). For disorder in then i 's, we find that disorder favors the S phase when the average offset chargen is close to an integer number of Cooper pairs, but that it favors the I phase nearn = 1/2. We also find that for large disorder, the average valuen has little effect on the phase diagram at all. As for the second kind of disorder, we find that disorder in the U ii 's tends to favor the S phase for all values of n, as shown in Fig. 5. Finally, we have found that the critical value of E J /U at the phase boundary, as calculated from the MFT of Grignani et al 6 , agrees well with that calculated using QMC (almost perfectly for small disorder and within 10% even for large disorder).
Thus, the MFT gives good results for most of the phase diagram.

V. ACKNOWLEDGMENTS.
This work has been supported by NSF Grant DMR01-04987. The calculations were carried out on the COE machines of the Ohio Supercomputer, with the help of a grant of time.