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Physical review. B, Condensed matter

Universal conductivity of two-dimensional films at the superconductor-insulator transition

Min-Chul Cha, Matthew P. A. Fisher, S. M. Girvin, Mats Wallin, A. P. Young

Published Oct 1, 1991• 355 citations
Publisher

Abstract

The zero-temperature universal conductivity of two-dimensional (2D) films at the supeconductor-insulator transition is studied. The existence of a finite conductivity at T=0 and the universality class for this transition is discussed. Neglecting disorder as a first approximation, so the transition is from a commensurate Mott-Hubbard insulator to a superconductor, we calculate analytically the universal conductivity for the 2D pure boson Hubbard model up to the first order in a large-N expansion and numerically by Monte Carlo simulation of the (2+1)-D XY model. From the Monte Carlo results we find the universal conductivity to be ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$=(0.285\ifmmode\pm\else\textpm\fi{}0.02)${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$, where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}^{\mathrm{\ensuremath{-}}1}$==${\mathit{R}}_{\mathit{Q}}$==h/(2e${)}^{2}$\ensuremath{\approxeq}6.45 k\ensuremath{\Omega}. An analysis in 1D suggests that in the presence of disorder, the universal conductivity in films might be somewhat smaller than this value. The possible existence of universal dissipation in $^{4}\mathrm{He}$ films is also discussed briefly.

DOI
10.1103/physrevb.44.6883

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The zero-temperature universal conductivity of two-dimensional (2D) films at the supeconductor-insulator transition is studied. The existence of a finite conductivity at T=0 and the universality class for this transition is discussed. Neglecting disorder as a first approximation, so the transition is from a commensurate Mott-Hubbard insulator to a superconductor, we calculate analytically the universal conductivity for the 2D pure boson Hubbard model up to the first order in a large-N expansion and numerically by Monte Carlo simulation of the (2+1)-D XY model. From the Monte Carlo results we find the universal conductivity to be \ensuremathσ∗{\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}\ensuremathσ∗=(0.285\ifmmode\pm\else\textpm\fi{}0.02)\ensuremathσQ{\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}\ensuremathσQ​, where \ensuremathσQ\ensuremath−1{\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}^{\mathrm{\ensuremath{-}}1}\ensuremathσQ\ensuremath−1​==RQ{\mathit{R}}_{\mathit{Q}}RQ​==h/(2e)2{)}^{2})2\ensuremath{\approxeq}6.45 k\ensuremath{\Omega}. An analysis in 1D suggests that in the presence of disorder, the universal conductivity in films might be somewhat smaller than this value. The possible existence of universal dissipation in 4He^{4}\mathrm{He}4He films is also discussed briefly.
The universal conductivity at the zero-temperature superconductor-insulator transition of the two-dimensional boson Hubbard model is studied for cases both with and without magnetic field by Monte Carlo simulations of the (2+1)-dimensional classical XY model with disorder represented by random bonds correlated along the imaginary time dimension. The effect of the magnetic field is characterized by the frustration f. From the scaling behavior of the stiffness, we determine the quantum dynamical exponent z, the correlation length exponent \ensuremath{\nu}, and the universal conductivity ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$. For the disorder-free model with f=1/2, we obtain z\ensuremath{\approxeq}1, 1/\ensuremath{\nu}\ensuremath{\approxeq}1.5, and ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=0.52\ifmmode\pm\else\textpm\fi{}0.03, where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$ is the quantum conductance. We also study the case with f=1/3, in which we find ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=0.83\ifmmode\pm\else\textpm\fi{}0.06. The value of ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$ is consistent with a theoretical estimate based on the Gaussian model. For the model with random interactions, we find z=1.07\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{\nu}\ensuremath{\approxeq}1, and ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=0.27\ifmmode\pm\else\textpm\fi{}0.04 for the case f=0, and z=1.14\ifmmode\pm\else\textpm\fi{}0.03, \ensuremath{\nu}\ensuremath{\approxeq}1, and ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the case f=1/2.
We study the critical behavior of the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ at the zero-temperature superconductor-Mott-insulator transition in ddd spacetime dimensions for a model of bosons with short-range interaction and no disorder. We obtain $\ensuremath{\sigma}({\ensuremath{\omega}}_{n})=(4\frac{{e}^{2}}{\ensuremath{\hbar}}){\ensuremath{\sigma}}_{\ensuremath{\epsilon}}{\ensuremath{\omega}}_{n}^{1\ensuremath{-}\ensuremath{\epsilon}}$, as predicted by the scaling theory, and the prefactor ${\ensuremath{\sigma}}_{\ensuremath{\epsilon}}$ is calculated in the $\ensuremath{\epsilon}$ expansion, to order ${\ensuremath{\epsilon}}^{2}(\ensuremath{\epsilon}=4\ensuremath{-}d)$. In two spatial dimensions (d=3d=3d=3), we find a value of the universal conductance ${\ensuremath{\sigma}}^{*}=0.315(4\frac{{e}^{2}}{h})$, in good agreement with the known Monte Carlo results.
We compute the universal conductivity of the (2+1)-dimensional XY universality class, which is realized for a superfluid-to-Mott insulator quantum phase transition at constant density. Based on large-scale Monte Carlo simulations of the classical (2+1)-dimensional J-current model and the two-dimensional Bose-Hubbard model, we can precisely determine the conductivity on the quantum critical plateau, σ(∞) = 0.359(4)σQ with σQ the conductivity quantum. The universal conductivity curve is the standard example with the lowest number of components where the bottoms-up AdS/CFT correspondence from string theory can be tested and made to use [R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011)]. For the first time, the shape of the σ(iω(n)) - σ(∞) function in the Matsubara representation is accurate enough for a conclusive comparison and establishes the particlelike nature of charge transport. We find that the holographic gauge-gravity duality theory for transport properties can be made compatible with the data if temperature of the horizon of the black brane is different from the temperature of the conformal field theory. The requirements for measuring the universal conductivity in a cold gas experiment are also determined by our calculation.
We investigate the superconductor-insulator phase transition in the two-dimensional boson Hubbard system with short-range interactions. Fluctuations of both the phase and the amplitude of the superfluid order parameter are included in the determination of the phase diagram at zero and finite temperatures. The mean-field phase boundary is compared to quantum Monte Carlo results. We also calculate the frequency-dependent conductivity in the vicinity of the phase transition and find it universal at the multicritical point.
We study numerically the superconductor-insulator transition in two-dimensional inhomogeneous superconductors with gauge disorder, described by four different quantum rotor models: a gauge glass, a flux glass, a binary phase glass, and a Gaussian phase glass. The first two models describe the combined effect of geometrical disorder in the array of local superconducting islands and a uniform external magnetic field, while the last two describe the effects of random negative Josephson-junction couplings or $\ensuremath{\pi}$ junctions. Monte Carlo simulations in the path-integral representation of the models are used to determine the critical exponents and the universal conductivity at the quantum phase transition. The gauge- and flux-glass models display the same critical behavior, within the estimated numerical uncertainties. Similar agreement is found for the binary and Gaussian phase-glass models. Despite the different symmetries and disorder correlations, we find that the universal conductivity of these models is approximately the same. In particular, the ratio of this value to that of the pure model agrees with recent experiments on nanohole thin-film superconductors in a magnetic field, in the large disorder limit.
In contrast to the case of free fermions which have all states localized in two dimensions, dirty, repulsively-interacting bosons exhibit a zero-temperature localization phase transition between superconducting and insulating states as disorder is increased. The conductivity σ* = g (2 e ) 2 / h at the quantum critical point is a universal critical amplitude analogous to the universal jump in superfluid density at the classical Kosterlitz-Thouless transition. Using numerical Monte Carlo methods we compute the value of g for disorder-tuned transitions with and without long-range Coulomb interactions. We argue that this model is relevant to experiments on superconducting thin films and Josephson junction arrays.