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arXiv:cond-mat/9912008 · Physical review. B, Condensed matter

Quantum effects on the BKT phase transition of two-dimensional Josephson arrays

Alessandro Cuccoli, Andrea Fubini, V. Tognetti, Ruggero Vaia

Published May 1, 2000• 19 citations
arXivHTMLPDF

Abstract

The phase diagram of two-dimensional Josephson arrays is studied by means of the mapping to the quantum $\mathrm{XY}$ model. The quantum effects on the thermodynamics of the system can be evaluated with quantitative accuracy by a semiclassical method, the pure-quantum self-consistent harmonic approximation, and those of dissipation can be included in the same framework by the Caldeira-Leggett model. Within this scheme, the critical temperature of the superconductor-metal transition, which is a Berezinskii-Kosterlitz-Thouless one, can be calculated in an extremely easy way as a function of the quantum coupling and of the dissipation mechanism. Previous quantum Monte Carlo results for the same model appear to be rather inaccurate, while a comparison with experimental data leads us to conclude that the commonly assumed dissipation model is not suitable to describe in detail a real system.

DOI
10.1103/physrevb.61.11289

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Progress of Theoretical and Experimental Physics•Sep 26, 2024•4 citations
The phase diagram of two-dimensional Josephson arrays is studied by means of the mapping to the quantum XY\mathrm{XY}XY model. The quantum effects on the thermodynamics of the system can be evaluated with quantitative accuracy by a semiclassical method, the pure-quantum self-consistent harmonic approximation, and those of dissipation can be included in the same framework by the Caldeira-Leggett model. Within this scheme, the critical temperature of the superconductor-metal transition, which is a Berezinskii-Kosterlitz-Thouless one, can be calculated in an extremely easy way as a function of the quantum coupling and of the dissipation mechanism. Previous quantum Monte Carlo results for the same model appear to be rather inaccurate, while a comparison with experimental data leads us to conclude that the commonly assumed dissipation model is not suitable to describe in detail a real system.
Earlier Monte Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY transition, i.e., one with a dynamical critical exponent z=1z=1z=1. The transition changes from z=1z=1z=1 to the class of criticality with $z\ensuremath{\rightarrow}\ensuremath{\infty}$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.
The two-dimensional quantum XY model, with a Caldeira-Leggett form of dissipation, is applicable to the quantum-critical properties of diverse experimental systems, ranging from superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, to the loop-current order transition in cuprates. We solve the reexpression of this model in terms of orthogonal topological excitations, vortices, and a variety of instantons, by renormalization group methods. The calculations explain the extraordinary properties of the model discovered in Monte Carlo calculations: the product form of the quantum-critical fluctuations in space and time, a spatial correlation length proportional to the logarithm of the temporal correlation length near the transition from a disordered to a fully ordered state, and the occurrence of a phase with spatial order without temporal order. They are intimately related to the flow of the metric of time in relation to the metric of space, i.e., of the dynamical critical exponent zzz. These properties appear to be essential in understanding the strange metallic phase found in a variety of quantum-critical transitions as well as the accompanying high-temperature superconductivity.
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 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 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 EJ∕kB{E}_{J}∕{k}_{B}EJ​∕kB​ with the Josephson coupling strength EJ{E}_{J}EJ​), the correlation length critical exponent $\ensuremath{\nu}=1.2$, and the dynamic critical exponent z=2.0z=2.0z=2.0 in the strong disorder regime agree with the existing studies of the 2D gauge-glass model.
This thesis presents seven research papers on topics in condensed-matter theory. Five of the papers report on Monte Carlo studies of quantum phase transitions in various (d+1)-dimensional statistical mechanics models featuring Caldeira-Leggettlike dissipation. The principal motivation for these studies was to investigate a particular bond-dissipative (2+1)-dimensional XY model of circulating currents in cuprate high-temperature superconductors. It has been proposed that quantum critical fluctuations associated with a local quantum critical point described by this model can explain the marginal-Fermi-liquid behaviour of the normal state of these compounds. We present simulation results for this model for both compact and noncompact phase variables and show unambiguously that the quantum critical point in the compact case is not local. If the phases are taken to be noncompact variables, the model is also a model of resistively shunted Josephson junction arrays. The results in this case reveal a more complicated phase diagram, but we have not been able to establish critical behaviour consistent with the scenario of local quantum criticality.\n\nThe study of extended quantum dissipative models is also motivated by the general effect on condensed-matter systems of the coupling to environmental degrees of freedom. Their influence on quantum critical phenomena is characterized by the dynamical critical exponent z, a measure of spatiotemporal anisotropy, the value of which can be estimated by naive scaling arguments. We confirm by numerical means that such scaling estimates give correct results to a good approximations (with a few reservations), irrespective of system dimensionality, order parameter symmetry, or whether the variables are compact or noncompact. Corrections to the naive scaling estimates have to be invoked for strongly super-Ohmic dissipation for d = 1 due to relatively large values of the anomalous scaling dimension η.\n\nThe two last research papers are concerned with the superconducting pairing state of the recently discovered class of iron-based high-temperature superconductors. Here, we calculate possible signatures of the proposed s±-wave pairing state in conductance-spectroscopy and Josephson-effect experiments.
In this paper we present a study of the phase diagram and critical properties of the square-lattice quantum XY model, with a single-ion anisotropy and spin S = 1, as a function of the anisotropy parameter D . For D less than a critical value D C , which is the critical point for a quantum phase transition at T = 0, the model presents a Berezinskii–Kosterlitz–Thouless (BKT) transition. This region is adequately described by the self-consistent harmonic approximation. We show that, if we want to use a Schwinger boson theory in this region, we should include fluctuations around the mean-field approximation (which leads to a gauge field) to describe correctly the BKT transition. However both methods are inadequate to describe the large- D phase, and we show that this phase can be studied using the bond operator formalism which gives the correct behaviour for the correlation length as a function of the temperature in the critical point and above. Our results agree with scaling arguments.
Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional XYXYXY model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature TBKT,HT_{\text{BKT,H}}TBKT,H​ is found to be 1/2{1}/{\sqrt{2}}1/2​ exactly. Surprisingly, the value of TBKT,HT_{\text{BKT,H}}TBKT,H​ obtained from our NN calculations is 0.572(3), which deviates significantly from 1/2{1}/{\sqrt{2}}1/2​. In addition, based on the helicity modulus, the TBKT,HT_{\text{BKT,H}}TBKT,H​ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.