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arXiv:cond-mat/9903247 · Physical review. B, Condensed matter

Scaling critical behavior of superconductors at zero magnetic field

C. de Calan, Flavio S. Nogueira

Published Aug 1, 1999• 20 citations
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Abstract

We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where ${m}_{h,0}$ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with ${m}_{h,0}$ having the scaling behavior of the renormalized photon mass.

DOI
10.1103/physrevb.60.4255

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We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form \ensuremathρs\ensuremath∼t\ensuremathν;{\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};\ensuremathρs​\ensuremath∼t\ensuremathν; that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of YBa2Cu3O7\ensuremath−\ensuremathδ.{\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.YBa2​Cu3​O7\ensuremath−\ensuremathδ​. We argue that the measured mean-field-like behavior of the penetration depth exponent \ensuremathν\ensuremath′{\ensuremath{\nu}}^{\ensuremath{'}}\ensuremathν\ensuremath′ is possibly associated with a nontrivial critical behavior and we predict the exponents \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 and \ensuremathα=\ensuremath−1\ensuremath{\alpha}=\ensuremath{-}1\ensuremathα=\ensuremath−1 for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter \ensuremathζ\ensuremath{\zeta}\ensuremathζ introduced such that mh,02\ensuremath∼t\ensuremathζ,{m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},mh,02​\ensuremath∼t\ensuremathζ, where mh,0{m}_{h,0}mh,0​ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with \ensuremathν\ensuremath′=\ensuremathν\ensuremath≈2/3{\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3\ensuremathν\ensuremath′=\ensuremathν\ensuremath≈2/3 is having \ensuremathζ\ensuremath≈4/3,\ensuremath{\zeta}\ensuremath{\approx}4/3,\ensuremathζ\ensuremath≈4/3, that is, with mh,0{m}_{h,0}mh,0​ having the scaling behavior of the renormalized photon mass.
We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The Josephson's relation for a charged superfluid is proved without assuming the hyperscaling relation. On the other hand we discuss the dual Ginzburg-Landau model. In this dual model, due to the presence of two mass scales, a continuous family of non equivalent scalings can be defined. The relevant critical regimes are identified, and the corresponding critical exponents are predicted.
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We consider the effects of critical superconducting fluctuations on the scaling of the linear ac conductivity, $\ensuremath{\sigma}(\ensuremath{\omega}),$ of a bulk superconductor slightly above Tc{T}_{c}Tc​ in zero applied magnetic field. The dynamic renormalization-group method is applied to the relaxational time-dependent Ginzburg-Landau model of superconductivity, with $\ensuremath{\sigma}(\ensuremath{\omega})$ calculated via the Kubo formula to $O({\ensuremath{\epsilon}}^{2})$ in the $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion. The critical dynamics are governed by the relaxational XY\mathrm{XY}XY-model renormalization-group fixed point. The scaling hypothesis $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{2\ensuremath{-}d+z}S(\ensuremath{\omega}{\ensuremath{\xi}}^{z})$ proposed by Fisher, Fisher, and Huse is explicitly verified, with the dynamic exponent $z\ensuremath{\approx}2.015,$ the value expected for the d=3d=3d=3 relaxational XY\mathrm{XY}XY model. The universal scaling function S(y)S(y)S(y) is computed and shown to deviate only slightly from its Gaussian form, calculated earlier. The present theory is compared with experimental measurements of the ac conductivity of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ near Tc,{T}_{c},Tc​, and the implications of this theory for such experiments is discussed.
We discuss some problems related to quantum critical phenomena in superfluids and superconductors. In Ch. 1, we apply generalizations of hyperuniversality to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. In Ch. 2, we apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial fixed point. The onset of mean-field behavior for dimensions d > d[subscript]c = 4 is unconventional, yielding discontinuous exponents. Including positive temperatures, we give a clear picture of various crossover regimes. In Ch. 3, We give a detailed derivation of the critical thermodynamics of O(n) spin models, correct to O([epsilon] = 4 - d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique introduced by Rudnick and Nelson. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n [...] 2), deriving detailed expressions for the renormalized spin-wave stiffness and longitudinal susceptibility.
The anomalous scaling in the Ginzburg-Landau model for the superconducting phase transition is studied. It is argued that the negative sign of the $\ensuremath{\eta}$ exponent is a consequence of a special singular behavior in momentum space. The negative sign of $\ensuremath{\eta}$ comes from the divergence of the critical correlation function at finite distances. This behavior implies the existence of a Lifshitz point in the phase diagram. The anomalous scaling of the vector potential is also discussed. It is shown that the anomalous dimension of the vector potential ${\ensuremath{\eta}}_{A}=4\ensuremath{-}d$ has important consequences for the critical dynamics in superconductors. The frequency-dependent conductivity is shown to obey the scaling $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{z\ensuremath{-}2}.$ The prediction $z\ensuremath{\approx}3.7$ is obtained from existing Monte Carlo data.
Using the scaling theory of quantum critical phenomena we explore the occurrence of universal critical behavior at the insulator-to-superconductor and superconductor-to-normal state transitions at zero temperature.Experimentally, these phase transitions are driven by doping and correspond to critical end points of the phase transition łine in the temperature-hole concentration plane.Provided that the order parameter is a complex scalar in two dimensions, and that the London relation between superfluid number density and magnetic penetration depth holds, the scaling theory predicts universal behavior close to the insulator-to-superconductor transition.In particular, transition temperature and zero temperature penetration depth are universally related and the sheet resistance adopts a universal value.These predictions agree remarkably well with available experimental data and provide useful constraints for a microscopic theory.