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arXiv:cond-mat/9907385 · Physical review. B, Condensed matter

Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type-II superconductors

Anh Kiet Nguyen, Asle Sudbø

Published Dec 1, 1999• 106 citations
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Abstract

We study the effect of critical fluctuations on the $(B,T)$ phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function $D(p)$ of vortex loops of perimeter p changes from an exponential function $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]$ to a power law distribution $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$ at the zero-field critical temperature ${T=T}_{c}.$ We find that the long-wavelength vortex-line tension vanishes as $\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.$ At ${T=T}_{c},$ an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global $\mathrm{U}(1)$-symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line ${T}_{L}(B),$ which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.

DOI
10.1103/physrevb.60.15307

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We study the effect of critical fluctuations on the (B,T)(B,T)(B,T) phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function D(p)D(p)D(p) of vortex loops of perimeter p changes from an exponential function D(p)\ensuremath∼p\ensuremath−\ensuremathαexp[\ensuremath−\ensuremathε(T)p/kBT]D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]D(p)\ensuremath∼p\ensuremath−\ensuremathαexp[\ensuremath−\ensuremathε(T)p/kB​T] to a power law distribution D(p)\ensuremath∼p\ensuremath−\ensuremathαD(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}D(p)\ensuremath∼p\ensuremath−\ensuremathα at the zero-field critical temperature T=Tc.{T=T}_{c}.T=Tc​. We find that the long-wavelength vortex-line tension vanishes as \ensuremathε(T)\ensuremath∼∣T\ensuremath−Tc∣\ensuremathγ;\ensuremathγ\ensuremath≈1.45,\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,\ensuremathε(T)\ensuremath∼∣T\ensuremath−Tc​∣\ensuremathγ;\ensuremathγ\ensuremath≈1.45, as T\ensuremath→Tc.\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.T\ensuremath→​Tc​. At T=Tc,{T=T}_{c},T=Tc​, an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global U(1)\mathrm{U}(1)U(1)-symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line TL(B),{T}_{L}(B),TL​(B), which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.
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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.
Monte Carlo simulations in conjunction with finite-size scaling analysis are used to investigate the (H,T)(H,T)(H,T)-phase diagram in uniaxial anisotropic high-Tc{T}_{c}Tc​ superconductors, both in zero magnetic field (B=0)(B=0)(B=0) and in intermediate magnetic fields $(0\ensuremath{\ll}B\ensuremath{\ll}{B}_{c2})$ for various mass anisotropies. The model we consider is the uniformly frustrated anisotropic Villain model, which is dual to the lattice London model with an infinite London penetration length. The quantities we consider are various helicity moduli, the structure function, the specific heat, and the distribution of closed non-field-induced vortex loops as a function of the loop size. In zero magnetic field, and for all anisotropies considered, we find one single second-order phase transition, mediated by an Onsager vortex-loop unbinding transition, or blowout. This is the superconductor--normal-metal transition. A comparison with numerical simulations and a critical scaling analysis of the zero-field loop transition yields the same exponent of the loop-distribution function at the critical point. In the intermediate magnetic-field regime, we find two anomalies in the specific heat. The first anomaly at a temperature Tm{T}_{m}Tm​ is associated with the melting transition of the flux-line lattice. The Lindemann ratio at the melting is given by ${c}_{L}\ensuremath{\approx}0.24.$ The second anomaly at a temperature Tz{T}_{z}Tz​ is one where phase coherence in the BCS order parameter across the sample along the field direction is destroyed. We argue that Tm=Tz{T}_{m}{=T}_{z}Tm​=Tz​ in the thermodynamic and continuum limit. Hence, there is no regime where the flux-line lattice melts into a disentangled flux-line liquid. The loss of phase coherence parallel to the magnetic field in the sample is argued to be due to the proliferation of closed non-field-induced vortex loops on the scale of the magnetic length in the problem, resulting in flux-line cutting and recombination. In the flux-line liquid phase, therefore, flux lines appear no longer to be well-defined entities. Above the melting temperature, the system always exhibits an incoherent vortex-liquid phase characterized by lack of phase coherence in the BCS order parameter parallel to the magnetic field. For increasing anisotropy, we resolve a \ensuremath{\delta}-function peak in the specific heat. A finite-size scaling analysis of the \ensuremath{\delta}-function peak specific-heat anomaly at the melting transition is used to extract the discontinuity of the entropy at the melting transition. This entropy discontinuity is found to increase rapidly with mass anisotropy, at least for not too layered compounds.
We present results from large-scale Monte Carlo simulations on the full Ginzburg-Landau (GL) model, including fluctuations in the amplitude and the phase of the matter field, as well as fluctuations of the noncompact gauge field of the theory. From this we obtain a precise critical value of the GL parameter ${\ensuremath{\kappa}}_{\mathrm{tri}}$ separating a first-order metal-to-superconductor transition from a second-order one, ${\ensuremath{\kappa}}_{\mathrm{tri}}=(0.76\ifmmode\pm\else\textpm\fi{}0.04)/\sqrt{2}.$ This agrees surprisingly well with earlier analytical results based on a disorder theory of the superconductor to metal transition, where the value ${\ensuremath{\kappa}}_{\mathrm{tri}}=0.798/\sqrt{2}$ was obtained. To achieve this, we have done careful infinite volume and continuum limit extrapolations. In addition we offer an interpretation of ${\ensuremath{\kappa}}_{\mathrm{tri}},$ namely, that it is also the value separating type-I and type-II behavior.
We investigate the effect of random columnar disorder on the superconducting phase transition of a type-II superconductor in zero applied magnetic field using numerical simulations of three dimensional XYXYXY and vortex loop models. We consider both an unscreened model, in which the bare magnetic penetration length is approximated as infinite, and a strongly screened model, in which the magnetic penetration length is of order the vortex core radius. We consider both equilibrium and dynamic critical exponents. We show that, as in the disorder free case, the equilibrium transitions of the unscreened and strongly screened models lie in the same universality class, however scaling is now anisotropic. We find for the correlation length exponent $\ensuremath{\nu}=1.2\ifmmode\pm\else\textpm\fi{}0.1$, and for the anisotropy exponent $\ensuremath{\zeta}=1.3\ifmmode\pm\else\textpm\fi{}0.1$. We find different dynamic critical exponents for the unscreened and strongly screened models.
Phase transitions in a number of models related to hightemperature superconductors are investigated, using scalingmethods and Monte Carlo simulations. This thesis considers twomain topics. The first topic is phase transitions, phase diagrams, andvortex motion in high temperature superconductors at finitetemperature, subject to magnetic fields and disorder. We studya vortex glass model at finite temperature, with stronguncorrelated vortex pinning and a magnetic field. We find thatthe vortex glass exists at finite temperature and calculate thecritical exponents of the transition. We also investigate hightemperature superconductors with columnar disorder in zero andapplied magnetic fields. Some of these studies are alsorelevant for the superfluid to Mott insulator transition ofbosons in two dimensions. We find that the unscreened Boseglass transition belongs to a new universality class. Wecalculate the critical exponents of the superconductingtransition with columnar defects in zero applied magneticfield. The transverse Meissner transition is studied, and wefind an exotic universality class with a correlation volumethat is infinitely anisotropic in all directions. The second topic is confinement-deconfinement transitions incompact Abelian Higgs models. We develop a new order parameter,related to a large Wilson loop for fractionalized charges, anduse it to study the concept of topological order. Thesetransitions may be relevant for strongly correlated electronsin two dimensions.