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arXiv:1212.1330 · Journal of Physics Condensed Matter

Quantum superconductor–insulator transition: implications of BKT critical behavior

T. Schneider, S. Weyeneth

Published Jul 3, 2013• 6 citations
arXivHTMLPDF

Abstract

We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical behavior on the two-dimensional (2D) quantum superconductor-insulator (QSI) transition driven by the tuning parameter x. Concentrating on the sheet resistance R(x,T) BKT behavior implies: an explicit quantum scaling function for R(x,T) along the superconducting branch ending at the nonuniversal critical value Rc = R(xc); a BKT-transition line T(c)(x) [proportionality] (x - x(c))(zν[overline]), where z is the dynamic exponent and ν[overline] the exponent of the zero-temperature correlation length; independent estimates of zν[overline], z and ν[overline] from the x dependence of the nonuniversal parameters entering the BKT expression for the sheet resistance. To illustrate the potential and the implications of this scenario we analyze the data of Bollinger et al (2011 Nature 472 458) taken on gate voltage tuned epitaxial films of La2-xSrxCuO4 that are one unit cell in thickness. The resulting estimates, z ~/= 3.1 and ν[overline] ~/= 0.52, indicate a clean 2D-QSI critical point where hyperscaling, the proportionality between d/λ(2)(0) and Tc, and the correspondence between the quantum phase transitions in D dimensions and the classical ones in (D + z) dimensions are violated.

DOI
10.1088/0953-8984/25/30/305701

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We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical behavior on the two-dimensional (2D) quantum superconductor-insulator (QSI) transition driven by the tuning parameter x. Concentrating on the sheet resistance R(x,T) BKT behavior implies: an explicit quantum scaling function for R(x,T) along the superconducting branch ending at the nonuniversal critical value Rc = R(xc); a BKT-transition line T(c)(x) [proportionality] (x - x(c))(zν[overline]), where z is the dynamic exponent and ν[overline] the exponent of the zero-temperature correlation length; independent estimates of zν[overline], z and ν[overline] from the x dependence of the nonuniversal parameters entering the BKT expression for the sheet resistance. To illustrate the potential and the implications of this scenario we analyze the data of Bollinger et al (2011 Nature 472 458) taken on gate voltage tuned epitaxial films of La2-xSrxCuO4 that are one unit cell in thickness. The resulting estimates, z ~/= 3.1 and ν[overline] ~/= 0.52, indicate a clean 2D-QSI critical point where hyperscaling, the proportionality between d/λ(2)(0) and Tc, and the correspondence between the quantum phase transitions in D dimensions and the classical ones in (D + z) dimensions are violated.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature TBKTT_{\rm BKT}TBKT​ vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent zzz as the quantum Griffiths transition but have nothing to do with disorder.
The Berezinskii–Kosterlitz–Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models, these are vortices: above a transition temperature TBKT, vortices are free; below this transition temperature, they get confined. In this work, we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent z as the quantum Griffiths transition but are not related to disorder.
There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high Tc{T}_{c}Tc​ superconductors. The conventional view is that a Kosterlitz-Thouless-Berezinskii transition occurs as long as finite size effects do not obscure the transition. However, there have been recent suggestions that a different transition actually occurs which incorporates aspects of both the dynamic scaling theory of Fisher, Fisher, and Huse and the Kosterlitz-Thouless-Berezinskii transition. Of general interest is that this modified transition apparently has a universal dynamic critical exponent. Some have countered that this apparent universal behavior is rooted in a newly proposed finite-size scaling theory; one that also incorporates scaling and conventional two-dimensional theory. To investigate these issues we study dc voltage versus current data of a 12-\AA{}-thick ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ film. We find that the newly proposed scaling theories have intrinsic flexibility that is relevant to the analysis of the experiments. In particular, the data scale according to the modified transition for arbitrarily defined critical temperatures between 0 and 19.5 K, and the temperature range of a successful scaling collapse is related directly to the sensitivity of the measurement. This implies that the apparent universal exponent is due to the intrinsic flexibility rather than some real physical property. To address this intrinsic flexibility, we propose a criterion which would give conclusive evidence for phase transitions in two-dimensional superconductors. We conclude by reviewing results to see if our criterion is satisfied.
It has been suggested [Pierson et al., Phys. Rev. B 60, 1309 (1999); Ammirata et al., Physica C 313, 225 (1999)] that for two-dimensional (2D) superconductors there exists a phase transition with the dynamic critical exponent $z\ensuremath{\approx}5.6.$ We perform simulations for the 2D resistively-shunted-junction model and compare the results with the experimental data in Repaci et al. obtained for an ultrathin YBCO sample [Phys. Rev. B 54, R9674 (1996)]. We then use a different method of analyzing dynamic scaling than in Pierson et al., and conclude that both the simulations and the experiments are consistent with a conventional Kosterlitz Thouless (KT) transition in the thermodynamic limit for which z=2.z=2.z=2. For finite systems, however, we find both in simulations and experiments that the change in the current-voltage $(I\ensuremath{-}V)$ characteristics caused by the finite size shows a scaling property with an exponent $\ensuremath{\alpha}\ensuremath{\approx}1/6,$ seemingly suggesting a vanishing resistance at a temperature for which $z={\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is pointed out that the dynamic critical exponent found in Pierson et al. corresponds to the exponent ${\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is emphasized that this scaling property does not represent any true phase transition since in reality the resistance vanishes only at zero temperature. Nevertheless, the observed scaling behavior associated with $\ensuremath{\alpha}\ensuremath{\approx}1/6$ appears to be a common and intriguing feature for the finite size caused change in the $I\ensuremath{-}V$ characteristics around the KT transition.
Recent experiments on 2D superconductors allow the characterization of the critical temperature and of the phase diagram across the BCS-BEC crossover as a function of density. We obtain from these experiments the microscopic parameters of the superconducting state at low temperatures by the BCS mean-field approach. For LixZrNCl{\mathrm{Li}}_{x}\mathrm{ZrNCl}Lix​ZrNCl, the extracted parameters are used to evaluate the superconducting phase stiffness and the Berezinskii-Kosterlitz-Thouless (BKT) critical temperature throughout the BCS-BEC crossover by implementing the corresponding renormalization group (RG) approach. In this way, we make a quantitative test of the predictive power of the BKT theory for evaluating the critical temperature. The RG flow equations turn out to give a sizable renormalization of the phase stiffness and of the critical temperature, which is crucial to obtain a satisfactory agreement between the BKT theory and the experiments, in particular, in the BCS-BEC crossover regime. We predict the temperature range where phase stiffness renormalization can be measured in LixZrNCl{\mathrm{Li}}_{x}\mathrm{ZrNCl}Lix​ZrNCl across the BCS-BEC crossover. Contrary to other microscopic theories of superconductivity, we find that the BKT theory can be exploited to evaluate quantitatively the critical temperature of 2D superconductors in different pairing regimes.
The disorder-induced quantum phase transition between superfluid and nonsuperfluid states of bosonic particles in one dimension is generally expected to be of the Berezinskii-Kosterlitz-Thouless (BKT) type. Here, we show that hard-core lattice bosons with power-law hopping decaying with distance as $1/{r}^{\ensuremath{\alpha}}$ with a finite integral over space---corresponding in spin language to an XYXYXY model with power-law couplings---undergo a non-BKT continuous phase transition instead. We use exact quantum Monte Carlo methods to determine the phase diagram for different values of the exponent $\ensuremath{\alpha}$, focusing on the regime $\ensuremath{\alpha}>2$. We find that the scaling of the superfluid stiffness with the system size is scale invariant at the transition point for any $\ensuremath{\alpha}\ensuremath{\le}3$---a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By a scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\ensuremath{\nu}\ensuremath{\ge}2$ and demonstrate a universal behavior of disordered bosons in one dimension. For $\ensuremath{\alpha}>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder.