Dynamic Universality Class Research Articles

Experimental Observations of Dynamic Critical Phenomena in a Lipid Membrane

Near a critical point, the time scale of thermally induced fluctuations diverges in a manner determined by the dynamic universality class. Experiments have verified predicted three-dimensional dynamic critical exponents in many systems, but similar experiments in two dimensions have been lacking for the case of conserved order parameter. Here we analyze the time-dependent correlation functions of a quasi-two-dimensional lipid bilayer in water to show that its critical dynamics agree with a recently predicted universality class. In particular, the effective dynamic exponent z(eff) crosses over from ~2 to ~3 as the correlation length of fluctuations exceeds a hydrodynamic length set by the membrane and bulk viscosities.

Role of weighting in the dynamics of front propagation

Non-equilibrium front propagation in a two-dimensional network modelling wildfire propagation was studied. The model includes deterministic long-range interactions due to radiation and a time weighting procedure. Three weight-dependent propagation regimes were found: dynamical, static, and non-propagative. The dynamical regime shows saturation for small weight values and a percolation transition area depending on the weight and size of the interaction domain. From the scaling interface exponents, the model seems to belong to the dynamical percolation universality class. In the limit of static regime it belongs to the random deposition class.

Dynamical behavior of the Niedermayer algorithm applied to Potts models

In this work, we make a numerical study of the dynamic universality class of the Niedermayer algorithm applied to the two-dimensional Potts model with 2, 3, and 4 states. This algorithm updates clusters of spins and has a free parameter, E0, which controls the size of these clusters, such that E0=1 is the Metropolis algorithm and E0=0 regains the Wolff algorithm, for the Potts model. For −1<E0<0, only clusters of equal spins can be formed: we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L, but eventually saturates at a given lattice size L˜, which depends on E0. For L≥L˜, the Niedermayer algorithm is in the same dynamic universality class of the Metropolis one, i.e, they have the same dynamic exponent. For E0>0, spins in different states may be added to the cluster but the dynamic behavior is less efficient than for the Wolff algorithm (E0=0). Therefore, our results show that the Wolff algorithm is the best choice for Potts models, when compared to the Niedermayer’s generalization.

Universality class of the depinning transition in the two-dimensional Ising model with quenched disorder

With Monte Carlo methods, we investigate the universality class of the depinning transition in the two-dimensional Ising model with quenched random fields. Based on the short-time dynamic approach, we accurately determine the depinning transition field and both static and dynamic critical exponents. The critical exponents vary significantly with the form and strength of the random fields, but exhibit independence of the updating schemes of the Monte Carlo algorithm. From the roughness exponents ζ, ζloc and ζs, one may judge that the depinning transition of the random-field Ising model belongs to the new dynamic universality class with ζ ≠ ζloc ≠ ζs and ζloc ≠ 1. The crossover from the second-order phase transition to the first-order one is observed for the uniform distribution of the random fields, but it is not present for the Gaussian distribution.

Density-Temperature-Softness Scaling of the Dynamics of Glass-Forming Soft-Sphere Liquids

We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter ν(-1)) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard-sphere fluid, constituted by the soft-sphere systems whose dynamic parameters depend on n, T, and ν only through the reduced density n*≡nσ(HS)(T*,ν). A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short-range interactions are found to be a direct manifestation of this general dynamic equivalence principle.

Study of the in-plane and c-axis fluctuation conductivity of melt-textured YBa2Cu3O7 under hydrostatic pressure

We have experimentally studied the pressure dependence of the fluctuation conductivity in melt-processed YBa2Cu3O7 (YBCO) for currents applied parallel or perpendicular to the Cu-O2 atomic layers. Results show that the asymptotic critical regime is described by the 3D-XY universality class for both in-plane and off-plane conductivity components. The dynamical exponent for the in-plane conductivity component is z = 1.5, independently of the applied pressure. However, a pressure induced crossover in the dynamical universality class is observed in the c-axis fluctuation conductivity, where the exponent changes gradually from z = 1.5 to z = 2. The Ginzburg number increases with pressure for both crystalline orientations. This behavior suggests that the in-plane and c-axis components of the G-L coherence length decrease upon pressure application. The fluctuation spectrum for the Gaussian regime in the planar conductivity keeps the three dimensional character. However, filamentary fluctuations are observed along the c-axis above the Ginzburg temperature. Our results for the out-of-plane fluctuation conductivity in YBCO suggest that the c-axis transport is predominantly coherent in this system.

Dynamic relaxation of topological defect at Kosterlitz–Thouless phase transition

With Monte Carlo methods we study the dynamic relaxation of a vortex state at the Kosterlitz–Thouless phase transition of the two-dimensional XY model. A local pseudo-magnetization is introduced to characterize the symmetric structure of the dynamic systems. The dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. To illustrate the dynamic effect of the topological defect, similar analysis for the dynamic relaxation with a spin-wave initial state is also performed for comparison. We demonstrate that a limited amount of quenched disorder in the core of the vortex state may alter the dynamic universality class. Furthermore, theoretical calculations based on the long-wave approximation are presented.

Critical phenomena inN=2*plasma

We use gauge theory/string theory correspondence to study finite temperature critical behavior of mass-deformed $\mathcal{N}=4$ $SU(N)$ supersymmetric Yang-Mills theory at strong coupling, also known as $\mathcal{N}={2}^{*}$ gauge theory. For a certain range of the mass parameters, $\mathcal{N}={2}^{*}$ plasma undergoes a second-order phase transition. We compute all the static critical exponents of the model and demonstrate that the transition is of the mean-field theory type. We show that the dynamical critical exponent of the model is $z=0$, with multiple hydrodynamic relaxation rates at criticality. We point out that the dynamical critical phenomena in $\mathcal{N}={2}^{*}$ plasma is outside the dynamical universality classes established by Hohenberg and Halperin.

Finite-size scaling analysis of the critical behavior of a general epidemic process in 2D

We investigate the critical behavior of a stochastic lattice model describing a General Epidemic Process. By means of a Monte Carlo procedure, we simulate the model on a regular square lattice and follow the spreading of an epidemic process with immunization. A finite size scaling analysis is employed to determine the critical point as well as some critical exponents. We show that the usual scaling analysis of the order parameter moment ratio does not provide an accurate estimate of the critical point. Precise estimates of the critical quantities are obtained from data of the order parameter variation rate and its fluctuations. Our numerical results corroborate that this model belongs to the dynamic isotropic percolation universality class. We also check the validity of the hyperscaling relation and present data collapse curves which reinforce the accuracy of the estimated critical parameters.

Critical Phenomena in the AdS/CFT Duality

We review black holes with second-order phase transition in string theory (R-charged black holes and holographic superconductors) and review their static and dynamic critical phenomena. Holographic superconductors have conventional mean-field values for static critical exponents, but R-charged black holes have unconventional ones. For dynamic universality class, holographic superconductors belong to model A, and R-charged black holes belong to model B in the classification of Hohenberg and Halperin. The analysis suggests that some black holes do obey the theory of critical phenomena in condensed matter physics.

Critical phenomena in [formula omitted] SYM plasma

Strongly coupled N = 4 supersymmetric Yang–Mills plasma at finite temperature and chemical potential for an R-symmetry charge undergoes a second order phase transition. We demonstrate that this phase transition is of the mean field theory type. We explicitly show that the model is in the dynamical universality class of ‘model B’ according to the classification of Hohenberg and Halperin, with dynamical critical exponent z = 4 . We study bulk viscosity in the mass deformed version of this theory in the vicinity of the phase transition. We point out that all available models of bulk viscosity at continuous phase transition are in conflict with our explicit holographic computations.

Nonequilibrium quantum criticality in bilayer itinerant ferromagnets

We present a theory of nonequilibrium quantum criticality in a coupled bilayer system of itinerant electron magnets. The model studied consists of the first layer subjected to an in-plane current and open to an external substrate. The second layer is subject to no direct external drive, but couples to the first layer via short-ranged spin-exchange interaction. No particle exchange is assumed between the layers. Starting from a microscopic fermionic model, we derive an effective action in terms of two coupled bosonic fields which are related to the magnetization fluctuations of the two layers. When there is no interlayer coupling, the two bosonic modes possess different dynamical critical exponents $z$ with $z=2$ $(z=3)$ for the first (second) layer. This results in multiscale quantum criticality in the coupled system. It is shown that the linear coupling between the two fields leads to a low-energy fixed point characterized by the larger dynamical critical exponent $z=3$. We compute the correlation length in the quantum disordered and quantum critical regimes for both the nonequilibrium case and in thermal equilibrium where the whole system is held at a common temperature $T$. We identify an effective temperature scale ${T}_{\text{eff}}$ with which we define a quantum-to-classical crossover that is in exact analogy with the thermal equilibrium case but with $T$ replaced by ${T}_{\text{eff}}$. However, we find that the leading correction to the correlation length in the quantum critical regime scales differently with respect to $T$ and ${T}_{\text{eff}}$. We also note that the current in the lower layer generates a drift of the magnetization fluctuations, manifesting itself as a parity-breaking contribution to the effective action of the bosonic modes. In this sense, the nonequilibrium drive in this system plays a role which is distinct from $T$ in the thermal equilibrium case. We also derive the stochastic dynamics obeyed by the critical fluctuations in the quantum critical regime and find that they do not fall into the previously identified dynamical universality classes.

Dynamic critical phenomena from spectral functions on the lattice

We investigate spectral functions in the vicinity of the critical temperature of a second-order phase transition. Since critical phenomena in quantum field theories are governed by classical dynamics, universal properties can be computed using real-time lattice simulations. For the example of a relativistic single-component scalar field theory in 2 + 1 dimensions, we compute the spectral function described by universal scaling functions and extract the dynamic critical exponent z. Together with exactly known static properties of this theory, we obtain a verification from first principles that the relativistic theory is well described by the dynamic universality class of relaxational models with conserved density (Model C).

Universality class of holographic superconductors

We study "holographic superconductors" in various spacetime dimensions. We compute most of the static critical exponents in the linear perturbations and show that they take the standard mean-field values. We also consider the dynamic universality class for these models and show that they belong to model A with dynamic critical exponent z=2.

Experimental Test of Curvature-Driven Dynamics in the Phase Ordering of a Two Dimensional Liquid Crystal

We study electric field driven deracemization in an achiral liquid crystal through the formation and coarsening of chiral domains. It is proposed that deracemization in this system is a curvature-driven process. We test this prediction using the recently obtained exact result for the distribution of hull-enclosed areas in two-dimensional coarsening with nonconserved scalar order parameter dynamics [J. J. Arenzon et al., Phys. Rev. Lett. 98, 145701 (2007)]. The experimental data are in very good agreement with the theory. We thus demonstrate that deracemization in such bent-core liquid crystals belongs to the Allen-Cahn universality class, and that the exact formula, which gives us the statistics of domain sizes during coarsening, can also be used as a strict test for this dynamic universality class.

Field theory of bicritical and tetracritical points. II. Relaxational dynamics

We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .

Relaxational dynamics in 3D randomly diluted Ising models

We study the purely relaxational dynamics (model A) at criticality in three-dimensionaldisordered Ising systems whose static critical behaviour belongs to the randomly dilutedIsing universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We performMonte Carlo simulations at the critical point using the Metropolis algorithm andstudy the dynamic behaviour in equilibrium at various values of the disorderparameter. The results provide a robust evidence of the existence of a uniquemodel-A dynamic universality class which describes the relaxational critical dynamicsin all considered models. In particular, the analysis of the size dependence ofsuitably defined autocorrelation times at the critical point provides the estimatez = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxationaldynamics following a quench from to Tc. In agreement with the field-theory scenario, the analysis of theoff-equilibrium dynamic critical behaviour gives an estimate ofz that is perfectly consistent with the equilibrium estimatez = 2.35(2).

Conductivity fluctuation in the high temperature superconductor with planar weight disparity Y0.5Sm0.5Ba2Cu3O7−δ

AbstractThe synthesis of the Y0.5Sm0.5Ba2Cu3O7–δ superconducting material by the standard solid state reaction is reported. DC resistivity measurements reveal the improvement of the critical temperature (Tc) when substitution of exact 50‐50 mix of Yttrium and Samarium is performed. A bulk Tc = 101 K was determined by the criterion of the maximum in the temperature derivative of electrical resistivity. Structure characterization by means the x‐ray diffraction technique shows the crystalline appropriated distribution of Yttrium and Samarium to create substantial planar weight disparity (PWD) in alternating layers. This PWD increases Tc in copper‐oxide superconductors. In order to examine the effect of PWD on the pairing mechanism close to Tc, conductivity fluctuation analysis was performed by the method of logarithmic temperature derivative of the conductivity excess. We found the occurrence of Gaussian and genuinely critical fluctuations. Our results are in agreement with reports on YBa2Cu3O7–δ, but an enhancement of the Gaussian fluctuation regimes was experimentally detected as a result of the PWD. The correlations of the critical exponents with the dimensionality of the fluctuation system for each Gaussian regime were performed by using the Aslamazov‐Larkin theory. The genuinely critical exponent is interpreted by the 3D‐XY model as corresponding with the dynamical universality class of the E‐model. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Field-theoretic approach to fluctuation effects in neural networks

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.

Universality class of triad dynamics on a triangular lattice

We consider triad dynamics as it was recently considered by Antal [Phys. Rev. E 72, 036121 (2005)] as an approach to social balance. Here we generalize the topology from all-to-all to the regular one of a two-dimensional triangular lattice. The driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state. The dynamics is parametrized by a so-called propensity parameter p that determines the tendency of negative links to become positive. As a function of p we find a phase transition between different kinds of absorbing states. The phases differ by the existence of an infinitely connected (percolated) cluster of negative links that forms whenever p<or=p(c). Moreover, for p<or=p(c), the time to reach the absorbing state grows powerlike with the system size L, while it increases logarithmically with L for p>p(c). From a finite-size scaling analysis we numerically determine the static critical exponents beta and nu(perpendicular) together with gamma, tau, sigma, and the dynamical critical exponents nu(parallel) and delta. The exponents satisfy the hyperscaling relations. We also determine the fractal dimension d(f) that satisfies a hyperscaling relation as well. The transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents. We generalize the triad dynamics to four-cycle dynamics on a square lattice. In this case, again there is a transition between different absorbing states, going along with the formation of an infinite cluster of negative links, but the usual scaling and hyperscaling relations are violated.