Dynamic Universality Class Research Articles

Fluctuation conductivity and the dynamical universality class of the superconducting transition in the high-T c cuprates

Systematic measurements of the fluctuation conductivity in several high-temperature superconducting cuprates show the occurrence of a genuine critical regime. The conductivity exponent is consistent with the predictions of the 3D-XY universality class and yields a dynamical critical exponent z = 1.5.

Dynamics of temporal learning rules.

The changes of synaptic strength are analyzed on two time scales: the fast local field dynamics, and the slow synaptic modification dynamics. The fast dynamics are determined by the synaptic strengths and background noise in the system. The slow dynamics are determined by the functional form of a temporal learning rule. Temporal learning rules are defined to be functions yielding state dependent changes in synaptic strengths depending on the timing of pre- and postsynaptic states in the network. The evolution of local field dynamics that result from various learning rules are analyzed for a stochastic, discrete time neural model with no relative refractory period that receives a series of delayed adaptive inputs. A fixed point is found in the learning dynamics, and conditions for two types of instabilities are analyzed. Four universality classes of dynamics are found that are independent of the details of the temporal learning rules. Examples are given of biological systems in which these temporal learning rules have been identified, and their functional consequences are discussed.

Viscous flow and coarsening of microdomains in diblock copolymer thin films

For thin block copolymer (BCP) films on a homogeneous substrate, a fast domain growth exponent has recently been observed in experiments. This growth exponent is larger than what one would expect from theoretical considerations of two-dimensional coarsening of small molecular liquid mixtures in the viscous regime. Thus it is not clear whether the growth kinetics is truly different for block copolymer films, or if the observed anomalous exponent is a result of a possible dimensional crossover from two to three dimensions. To address part of this question, we have carried out numerical simulations of ordering and domain growth in a two-dimensional system of BCP melts. The model calculations reported here explicitly include viscous, hydrodynamic flow and provide a scaling description of the growth of domains in a quenched BCP system. Our results indicate that the growth kinetics of BCP melts in two dimensions belong to the same dynamical universality class of small molecular liquid mixtures in the viscosity dominated regime.

Critical dynamics of the four-dimensional XY model

In order to examine the uniqueness of the dynamic critical exponents and the exponent's dependence on dimensionality for XY models with time-dependent Ginzburg–Landau (TDGL) and resistively shunted junction (RSJ) dynamics we use two recently proposed methods to calculate the dynamic exponents in four dimensions. We discuss the relation to the dynamic universality classes and earlier works.

Dynamical universality classes of vapor deposition models with evaporation

Growth models for vapor depositions in which evaporation and deposition can occur, both at a randomly chosen column and at its nearest neighbor columns (NNCs), are studied by Monte Carlo simulations. The growth processes in these models are determined by comparing local chemical potentials of the chosen column and its NNCs to the chemical potential of vapor. The universality classes and the characteristics of the models are studied by the scaling ansatz of kinetic roughening and by measurements of tilt-dependent currents and height step widths. Through measurements of the ratio of number of growth processes at NNCs to those at the chosen column, the key processes which are relevant to the determination of the universality class are also studied.

Deterministic Equations of Motion and Dynamic Critical Phenomena

Taking the two-dimensional $\phi^4$ theory as an example, we numerically solve the deterministic equations of motion with random initial states. Short-time behavior of the solutions is systematically investigated. Assuming that the solutions generate a microcanonical ensemble of the system, we demonstrate that the second order phase transition point can be determined already from the short-time dynamic behavior. Initial increase of the magnetization and critical slowing down are observed. The dynamic critical exponent z, the new exponent $\theta$ and the static exponents $\beta$ and $\nu$ are estimated. Interestingly, the deterministic dynamics with random initial states is in a same dynamic universality class of Monte Carlo dynamics.

Fluctuation specific heat in multilayered superconductors: Bilayered Gaussian-Ginzburg-Landau scenario for the thermal fluctuations of Cooper pairs aroundTcinYBa2Cu3O7−δsingle crystals

The effects around the superconducting transition of thermal fluctuations of Cooper pairs on the heat capacity in zero applied magnetic field $(H=0)$ are explicitly calculated in bilayered superconductors, with two superconducting layers and tunneling couplings per layer periodicity length. The calculations are performed on the grounds of a generalization to multilayered superconductors of the Lawrence-Doniach Ginzburg-Landau functional, and assuming Gaussian fluctuations. In addition to the fluctuation heat capacity ${c}_{\mathrm{fl}},$ we also obtain various useful relationships between ${c}_{\mathrm{fl}}$ and other fluctuation-induced observables experimentally accessible in multilayered copper oxide superconductors. It is then shown that if the effects of the multilaminarity are taken into account, the mean-field-like Gaussian-Ginzburg-Landau approach may explain simultaneously and at a quantitative level the available experimental data, both the amplitude and the \ensuremath{\epsilon} behavior, of the fluctuation specific heat, the in-plane paraconductivity $\ensuremath{\Delta}{\ensuremath{\sigma}}_{\mathrm{ab}},$ and the fluctuation-induced diamagnetism $\ensuremath{\Delta}{\ensuremath{\chi}}_{\mathrm{ab}},$ in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}(\mathrm{Y}\ensuremath{-}123)$ single crystals under zero or weak magnetic fields $(\stackrel{\ensuremath{\rightarrow}}{H}0),$ up to reduced temperatures of the order of $\ensuremath{\epsilon}\ensuremath{\equiv}|T\ensuremath{-}{T}_{c}|{/T}_{c}\ensuremath{\sim}{10}^{\ensuremath{-}2}.$ The corresponding coherence length amplitudes (at $T=0\mathrm{K})$ are ${\ensuremath{\xi}}_{\mathrm{ab}}(0)=1.1\mathrm{nm}$ and ${\ensuremath{\xi}}_{c}(0)=0.12\mathrm{nm}$ for the in-plane (ab) and transversal (c) directions, respectively. In contrast, the same data cannot be explained, in the same \ensuremath{\epsilon} region, in terms of the 3DXY theory for full-critical fluctuations with a value of the dynamic critical exponent of $z=\frac{3}{2},$ which corresponds to the same universality class as the superfluid-normal \ensuremath{\lambda} transition of ${}^{4}\mathrm{He}$ liquid, although these analyses do not exclude the applicability of such a scenario for $\ensuremath{\epsilon}\ensuremath{\lesssim}{10}^{\ensuremath{-}2},$ as suggested by previous measurements of $\ensuremath{\Delta}{\ensuremath{\sigma}}_{\mathrm{ab}}$ and $\ensuremath{\Delta}{\ensuremath{\chi}}_{\mathrm{ab}}$ in the same Y-123 single crystals. However, another dynamic universality class, with $z=2,$ makes the 3DXY full-critical behavior compatible with the experimental data for $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\ensuremath{\lesssim}\ensuremath{\epsilon}\ensuremath{\lesssim}{10}^{\ensuremath{-}1}.$

Universality and scaling for the structure factor in dynamic order-disorder transitions

The universal form for the average scattering intensity from systems undergoing order-disorder transitions is found by numerical integration of the Langevin dynamics. The result is nearly identical for simulations involving two different forms of the local contribution to the free energy, supporting the idea that the model A dynamical universality class includes a wide range of local free-energy forms. An absolute comparison with no adjustable parameters is made to the forms predicted by theories of Kawasaki-Yalabik-Gunton, Ohta-Jasnow-Kawasaki, and Mazenko. The numerical results are well described by the Ohta-Jasnow-Kawasaki theory, except in the crossover region between scattering dominated by domain geometry and scattering determined by Porod's law.

Dynamical universality classes of the superconducting phase transition

We present a finite temperature Monte Carlo study of the $\mathrm{XY}$ model in the vortex representation and study its dynamical critical behavior in two limits. The first neglects magnetic-field fluctuations, corresponding to the absence of screening, which should be a good approximation in high-${T}_{c}$ superconductors $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}\ensuremath{\infty})$ except extremely close to the critical point. Here, from finite-size scaling of the linear resistivity we find the dynamical critical exponent of the vortex motion to be $z\ensuremath{\approx}1.5$. The second limit includes magnetic-field fluctuations in the strong screening limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\kappa}}0)$ corresponding to the true asymptotic inverted $\mathrm{XY}$ critical regime, where we find the unexpectedly large value $z\ensuremath{\approx}2.7.$ We compare these results, obtained from dissipative dynamics in the vortex representation, with the universality class of the corresponding model in the phase representation with propagating (spin-wave) modes. We also discuss the effect of disorder and the relevance of our results for experiments.

Dynamic SU(2) lattice gauge theory at finite temperature

The dynamic relaxation process for the (2+1)--dimensional SU(2) lattice gauge theory at critical temperature is investigated with Monte Carlo methods. The critical initial increase of the Polyakov loop is observed. The dynamic exponents $\theta$ and $z$ as well as the static critical exponent $\beta/\nu$ are determined from the power law behaviour of the Polyakov loop, the auto-correlation and the second moment at the early stage of the time evolution. The results are well consistent and universal short-time scaling behaviour of the dynamic system is confirmed. The values of the exponents show that the dynamic SU(2) lattice gauge theory is in the same dynamic universality class as the dynamic Ising model.

Early-time critical dynamics of lattices of coupled chaotic maps

The early-time critical dynamics of continuous, Ising-like phase transitions is studied numerically for two-dimensional lattices of coupled chaotic maps. Emphasis is placed on obtaining accurate estimates of the dynamic critical exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $z$. The critical points of five different models are investigated, varying the mode of update, the coupling, and the local map. Our results suggest that the nature of update is a relevant parameter for dynamic universality classes of extended dynamical systems, generalizing results obtained previously for the static properties. They also indicate that the universality observed for the static properties of Ising-like transitions of synchronously updated systems does not hold for their dynamic critical properties.

Phase Structure of Systems with Infinite Numbers of Absorbing States

Critical properties of systems exhibiting phase transitions into phases with infinite numbers of absorbing states are studied. We analyze a non-Markovian Langevin equation recently proposed to describe the critical behavior of such systems, and also introduce and study a non-Markovian discrete model, which is argued to present the same critical features. On the basis of mean-field analysis, Monte Carlo simulations, and theoretical arguments, we conclude that the phenomenology of the non-Markovian models closely parallels that of systems with many absorbing states in one and two dimensions. The “bulk” or “static” critical properties of these systems fall in the directed percolation (DP) universality class. By contrast, the critical properties associated with the spread of an initially localized seed exhibit a more complex behavior: Depending on parameter values they can, both in one and two dimensions, fall either in the dynamical percolation or DP universality class, or else exhibit apparently nonuniversal exponents. In contrast to previous results, however, the nonuniversal exponents in 2D are found to satisfy a scaling law which implies that a particular linear combination of them is universal and assumes DP values. These results demonstrate the efficacy of the non-Markovian approach for understanding systems with many absorbing states, which are difficult to analyze in their original microscopic formulation.

Determination of the Universality Class of Gadolinium

We resolve a longstanding puzzle for the static and dynamic critical behavior of Gadolinium by a combined theoretical and experimental investigation. It is shown that the spin dynamics of a three dimensional ferromagnet with hcp lattice structure and a spin-spin interaction given by both exchange and dipole-dipole interaction belongs to a new dynamic universality class, model J$^*$. Comparing results from mode coupling theory with results from three different hyperfine interaction probes we find quantitative agreement. The crossover scenario for the wavevector dependence of the hyperfine relaxation rate is determined by a subtle interplay between three length scales: the correlation length, the dipolar and the uniaxial wave vector.

Magnetic-field dependence of the critical dynamics of a superconductingYBa2Cu3O7−δdetwinned single crystal

The in-plane longitudinal resistivities ${\ensuremath{\rho}}_{\mathrm{aa}}$ and ${\ensuremath{\rho}}_{\mathrm{bb}}$ of a single-crystal of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ have been measured as a function of magnetic field $H$ (less than 6.75 T) and temperature $T$ near ${T}_{c}.$ We found that the ${\ensuremath{\rho}}_{\mathrm{aa}}(T,H)$ data in the mixed state collapse into a single curve as a function of ${\ensuremath{\rho}}_{\mathrm{bb}}(T,H).$The extracted fluctuation conductivity exhibits dynamic scaling over a wide range of $T$ and $H:$ assuming that the static universality class is the three-dimensional $\mathrm{XY}$ model yields the observed dynamic critical exponent $z=1.5\ifmmode\pm\else\textpm\fi{}0.1(H>0),$ apparently consistent with the dynamic universality class of superfluid ${}^{4}\mathrm{He}.$ In zero field, the data are inconclusive, and a reliable determination of $z$ is not possible.

Crossover from Isotropic to Directed Percolation

Directed percolation is one of the generic dynamic universality classes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model. A parameter $r$ controls the spontaneous birth of new forest fires. We obtain the exact crossover exponent ${y}_{\mathrm{DP}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}y}_{T}\ensuremath{-}1$ at $r\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ using Coulomb gas methods. Isotropic percolation is stable in 2D. Our numerical finite-size scaling results confirm this. An intuitive argument suggests that directed percolation at $r\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ does not change stability for $D\ensuremath{\ge}3$. It remains unstable, such that forest fires at $0<r<1$ have the same scaling properties as isotropic percolation all dimensions.

Dynamical universality class of Brownian motion and exact results for a single-impurity s=1/2 XY chain.

Relaxation phenomena in a class of nondissipative systems with two highly disparate time scales (i.e., Brownian systems) have unique commonalities quantifiable via two scalars. It is shown that the exactly solvable problem of the dynamics of a weakly linked impurity spin in a s=1/2 XY chain belongs to this dynamical universality class and so does that of a heavy mass in an infinite harmonic oscillator chain and a spinless quasi two-dimensional attractive Fermi gas in the long-wavelength limit. The case of the strongly linked impurity in the XY chain is also discussed along with the corresponding limits in the harmonic oscillator chain and the electron-gas problems. \textcopyright{} 1996 The American Physical Society.

Universality versus nonuniversality of critical transport properties in liquid mixtures

The critical behavior near consolute points and plait points in mixtures and along lines connecting such points in the phase diagram belongs to the universality class of gas-liquid transitions in pure liquids. We give a survey of the results for the temperature dependence of transport coefficients, thermal conductivity, mass diffusion, and thermal-diffusion ratio, in mixtures within a non-asymptotic renormalization-group theory of critical dynamics. The observable critical behavior in some cases is nonuniversal and may be strongly concentration dependent. This is explained by different crossover temperatures in the singular Onsager coefficient of the order parameter and in the hydrodynamic transport coefficients. At the plait point the value of (ie1363-01) determines the crossover to the asymptotic behavior in the transport coefficients, and its smallness explains the situation in3He-4He mixtures. We also consider ionic solutions, where long-range forces may be present. The dynamical universality class in this case is different from that of mixtures with short-range interaction. As well as the “classical” static behavior for sufficient long-range interaction potentials, the dynamical critical behavior depends on the exponent of the power law for the spatial decrease in this interaction. This offers an additional possibility to determine this exponent by measuring the temperature dependence of the hydrodynamic transport coefficients.

Dynamic surface critical behavior of systems with conserved bulk order parameter: Detailed RG analysis of the semi-infinite extensions of modelB with and without nonconservative surface terms

The dynamic surface critical behavior of macroscopic systems whose dynamic bulk critical behavior is described by model $B$ is investigated. The semi-infinite extensions of bulk model $B$ introduced in a previous treatment [Phys. Rev. B 49, 2846 (1994)] and called models $B_A$ and $B_B$ are studied in detail by means of field-theoretic RG methods. The distinctive feature of these models is the presence or absence of relevant nonconservative surface terms. The earlier results on the structure of the required counterterms, the flow equations, and the representation of the surface critical exponents of dynamic quantities at the ordinary and special phase transitions in terms of static bulk and surface exponents are corroborated by means of an explicit two-loop calculation for $4-\epsilon$ dimensions. For parameter values corresponding to a given static surface universality class, models $B_A$ and $B_B$ represent distinct dynamic surface universality classes. Differences between these manifest themselves in the behavior of scaling functions for dynamic surface susceptibilities. RG-improved perturbation theory is used to compute some of these susceptibilities to one-loop order.

Width distribution for (2+1)-dimensional growth and deposition processes.

Nonequilibrium growth processes are frequently characterized by the width w(L,t) of the active zone, where t is the time elapsed since the start of the process and L is the spatial interval over which the measurement is carried out. Quite generally, w(L,t) obeys a scaling form w(L,t)\ensuremath{\sim}${\mathit{L}}^{\mathrm{\ensuremath{\zeta}}}$f(${\mathit{tL}}^{\mathrm{\ensuremath{-}}\mathit{z}}$), and many workers have attempted to determine the dynamic universality class of such processes by a measurement of the exponents \ensuremath{\zeta} and z. In this paper, we calculate the steady-state width distribution P(${\mathit{w}}^{2}$) for several three-dimensional growth processes and show that, expressed in a suitable form, P(${\mathit{w}}^{2}$) can be used to distinguish between different possible universality classes. We also reanalyze experimental data obtained by scanning-tunneling or atomic-force microscopy and show that P(${\mathit{w}}^{2}$) provides valuable information on the nature of a growth process.

Remarkably small critical exponent for the viscosity of a polymer solution

We have measured the apparent critical exponent y characterizing the divergence of the viscosity η∝(T−Tc)−y near the liquid–liquid critical point of the mixture polystyrene in diethyl malonate. The data span the range in reduced temperature of 10−4<(T−Tc)/Tc<10−1. The sample was prepared from the same materials used by Gruner et al. in their capillary viscometer [Macromolecules 23, 510 (1990)]; however our torsion oscillator viscometer had a shear rate 80 times lower. This increased the range of reduced temperatures where shear effects could be neglected. In spite of the large reduction in shear rate and the different viscometry technique, the parameters fitted to our data and those of Gruner et al. are in agreement. For this polymer solution, y is in the range 0.028±0.003, close to recent results for two other polymer solutions measured in capillary viscometers. However, it is significantly smaller than the exponent for pure fluids (0.041± 0.001) and simple binary mixtures (0.042±0.002). It appears that polymer solutions are in a dynamic universality class different from that of simpler fluids.