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arXiv:cond-mat/0005418 · Physical review. B, Condensed matter

Anomalous dimensions and phase transitions in superconductors

Flavio S. Nogueira

Published Dec 1, 2000• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.62.14559

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The anomalous scaling in the Ginzburg-Landau model for the superconducting phase transition is studied. It is argued that the negative sign of the \ensuremathη\ensuremath{\eta}\ensuremathη exponent is a consequence of a special singular behavior in momentum space. The negative sign of \ensuremathη\ensuremath{\eta}\ensuremathη 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 \ensuremathηA=4\ensuremath−d{\ensuremath{\eta}}_{A}=4\ensuremath{-}d\ensuremathηA​=4\ensuremath−d has important consequences for the critical dynamics in superconductors. The frequency-dependent conductivity is shown to obey the scaling \ensuremathσ(\ensuremathω)\ensuremath∼\ensuremathξz\ensuremath−2.\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{z\ensuremath{-}2}.\ensuremathσ(\ensuremathω)\ensuremath∼\ensuremathξz\ensuremath−2. The prediction z\ensuremath≈3.7z\ensuremath{\approx}3.7z\ensuremath≈3.7 is obtained from existing Monte Carlo data.
For High-Tc superconductors the fluctuation regime is much wider than conventional superconductors due to the short coherence length and high transition temperature. AC measurements can probe the conductivity at a wide range of frequencies, and therefore provide a test of the scaling properties predicted from theory. The measured critical exponents and width of the fluctuation peak are sample dependent, which suggests that the sample inhomogeneities may play an important role in the critical region. In this thesis we study the effects of disorder on the fluctuation conductivity in the critical regime of zero-field normal-superconducting transition by the time-dependent Ginzburg-Landau theory. We set up a discretized model of the superconductor and calculate the two-dimensional and three-dimensional scaling function without disorder above Tc. The result of the discretized model deviates slightly from previous theoretical studies of the continuous model, which can be explained by a finite short wavelength cutoff in Ginzburg-Landau theory. Our results agree well with other theoretical investigations on cutoff effects from an analytical approach. Disorder in a superconductor is modeled by a distribution of Tc's at the lattice sites. We add random disorder to a two-dimensional lattice and calculate the scaling functions averaged over 1000 disorder configurations. When disorder is weak, the scaling functions are increased when T is close Tc . At strong disorder anomalous behavior may occur at the region close to Tc.
We consider the scaling properties of the linear ac and nonlinear dc conductivities of a superconductor near the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​. We first review a scaling theory of the conductivity near a critical point recently proposed by Fisher, Fisher, and Huse. By combining these scaling assumptions with causality arguments, we show that the phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$ of the linear ac conductivity takes on a universal value near the critical point, ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$=\ensuremath{\pi}(2-d+z)/2z, where d is the spatial dimension and z is the dynamic critical exponent. Hence, a measurement of the phase lag between the current and the voltage near the critical point provides a measurement of the exponent z. Then, using relaxational dynamics, we calculate these conductivities for 2d4, using (a) a Gaussian approximation, in which the quartic term in the Ginzburg-Landau Hamiltonian is neglected, and (b) an O(2n) generalization of the Ginzburg-Landau Hamiltonian, which for n\ensuremath{\rightarrow}\ensuremath{\infty} produces a Hartree approximation for the quartic term. These results agree with the scaling theory, and provide explicit expressions for the universal scaling functions. In addition, it is possible to study the crossover behavior from Gaussian to critical fluctuations as the critical temperature is approached. The prospects for going beyond the mean-field level by using renormalization-group methods are discussed, along with possible applications to the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors.
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 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 ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with mh,0{m}_{h,0}mh,0​ having the scaling behavior of the renormalized photon mass.
Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences.
The critical dynamics of superconductors in the charged regime is reconsidered within field-theory. For the dynamics, the Ginzburg-Landau model with complex order parameter coupled to the gauge field suggested earlier [Lannert et al. Phys. Rev. Lett. 92, 097004 (2004)] is used. Assuming relaxational dynamics for both quantities, the renormalization group functions within one loop approximation are recalculated for different choices of the gauge. A gauge independent result for the divergence of the melectric conductivity is obtained only at the weak scaling fixed point unstable in one loop order where the timescales of the order parameter and the gauge field are different.
We analyze the quantum phase transition between a semimetal and a superfluid in a model of attractively interacting fermions with a linear dispersion. The quantum critical properties of this model cannot be treated by the Hertz-Millis approach since integrating out the fermions leads to a singular Landau-Ginzburg order parameter functional. We therefore derive and solve coupled renormalization-group equations for the fermionic degrees of freedom and the bosonic order parameter fluctuations. In two spatial dimensions, fermions and bosons acquire anomalous scaling dimensions at the quantum critical point, associated with non-Fermi-liquid behavior and non-Gaussian order parameter fluctuations.