Static Critical Properties Research Articles

Critical Exponents and Universality for Fractal Time Processes above the Upper Critical Dimensionality

We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point.

Applicability of the Generalized Stokes-Einstein Equation of Mode-Coupling Theory to Near-Critical Polyelectrolyte Complex Solutions.

We examine whether the mode-coupling theory of Kawasaki and Ferrell (KF) [Kawasaki, K. Kinetic Equations and Time Correlation Functions of Critical Fluctuations. Ann. Phys. 1970, 61 (1), 1-56; Ferrell, R. A. Decoupled-Mode Dynamical Scaling Theory of the Binary-Liquid Phase Transition. Phys. Rev. Lett. 1970, 24 (21), 1169-1172] can describe dynamic light scattering (DLS) measurements of the dynamic structure factor of near-critical polyelectrolyte complex (PC) solutions that have been previously shown to exhibit a theoretically unanticipated lower critical solution temperature type phase behavior, i.e., phase separation upon heating, and a conventional pattern of static critical properties (low angle scattering intensity and static correlation, ξs) as a function of reduced temperature. Good qualitative accord is observed between our DLS measurements and the KF theory. In particular, we observe that the collective diffusion coefficient Dc of the PC solutions obeys the generalized Stokes-Einstein equation (GSE), Dc = kBT/6πηξs, where ξs is specified from our previous measurements and where η is measured by capillary rheometry under the same thermodynamic conditions as in our previous study of these solutions, allowing for a no-free-parameter test of the GSE. We also find that even the wavevector (q)-dependent collective diffusion coefficient Dc(q), measured by varying the scattering angle in the DLS measurements over a large range, is also well-described by the mean-field version of the KF theory. We find it remarkable that the KF theory provides such a robust description of collective diffusion in these complex charged polyelectrolyte blends under near-critical conditions given that charge fluctuations and association of the polymers might be expected to lead to physical complications that would invalidate the standard model of uncharged fluid mixtures.

Kinetics of phase separation in thermally isolated critical binary fluids.

Spinodal decomposition in a near-critical binary fluid is examined for experimental scenarios in which the liquid is quenched abruptly by changing the pressure and the subsequent phase separation occurs with no heat flow from the outside, i.e., adiabatically. Equations of motion for the system volume and effective temperature are derived. It is shown for this case that the nonequilibrium decomposition process is well approximated as one of constant entropy, i.e., as thermodynamically reversible. Quantitative comparison, with no adjustable parameters, is made with the experimental light scattering data of Bailey and Cannell [Phys. Rev. Lett. 70, 2110 (1993)PRLTAO0031-900710.1103/PhysRevLett.70.2110]. It is found that including these adiabatic effects accounts for most of the discrepancies between these experiments and previous isothermal theory. The equilibrium static critical properties of the isothermal theory are also examined, this discussion serving to justify some approximations in the current theory.

Static critical behavior of the [formula omitted]-states Potts model: High-resolution entropic study

Here we report a precise computer simulation study of the static critical properties of the two-dimensional q-states Potts model using very accurate data obtained from a modified Wang–Landau (WL) scheme proposed by Caparica and Cunha-Netto (2012). This algorithm is an extension of the conventional WL sampling, but the authors changed the criterion to update the density of states during the random walk and established a new procedure to windup the simulation run. These few changes have allowed a more precise microcanonical averaging which is essential to a reliable finite-size scaling analysis. In this work we used this new technique to determine the static critical exponents β, γ, and ν, in an unambiguous fashion. The static critical exponents were determined as β=0.10811(77), γ=1.4459(31), and ν=0.8197(17), for the q=3 case, and β=0.0877(37), γ=1.3161(69), and ν=0.7076(10), for the q=4 Potts model. A comparison of the present results with conjectured values and with those obtained from other well established approaches strengthens this new way of performing WL simulations.

Finite-time scaling via linear driving

We propose systematic finite-time scaling and show that it can be used to determine both static and dynamic critical properties. The controllable finite time scale that affects the scaling of a system and that serves as an analogue of the system size in finite-size scaling is achieved simply but generally by manipulating the dynamics with an external field varying linearly with time. An analytic renormalization-group theory is developed to justify the scaling. Monte Carlo simulation results for two- and three-dimensional Ising models agree with existing ones and confirm the scaling and the method.

Critical properties of the three-dimensional Ising model with quenched disorder

The static critical properties of the three-dimensional Ising model with quenched disorder are studied by the Monte-Carlo (MC) method on a simple cubic lattice, in which the quenched disorder is distributed as nonmagnetic impurities by the canonical manner. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.0; 0.95; 0.9; 0.8; 0.7; 0.6. The systems of non-linear sizes L× L× L, L=20–60 are researched. On the basis of the finite-size scaling (FSS) theory, the static critical exponents of specific heat α, susceptibility γ, magnetization β, and an exponent of the correlation radius in a studied interval of concentrations p are calculated. It is shown that the three-dimensional Ising model with quenched disorder has two regimes of the critical behavior universality in a dependence on nonmagnetic impurities.

Nonequilibrium critical relaxation of the order parameter and energy in the two-dimensional ferromagnetic Potts model

The static and dynamic critical properties of the ferromagnetic q -state Potts models on a square lattice with q=2 and 3 are numerically studied via the nonequilibrium relaxation method. The relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated, from which static and dynamic critical exponents can be obtained. We find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents, especially for the case of the q=2 (Ising) model, when care is taken in the choice of the initial states for the relaxation of the second moments. As for the case of q=3 , the estimates for the static exponents become less accurate, but still exhibit reasonable agreement with the exactly known static exponents. The dynamic critical exponent for the q=2 (Ising) model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z(q=2) approximately 2.1668(19) .

Critical properties of small magnetic particles of YFeO3

Different ways of modeling small magnetic particles of yttrium orthoferrite are proposed. Monte Carlo methods are used to study the static critical properties of these models. The conventional formalism of power-law functions is used to calculate the main static critical exponents for the proposed models, and their universality classes are determined. It is shown that models of small magnetic particles of YFeO3 are described mainly by the Ising universality class of critical behavior. The results can explain the contradiction between the experimentally observed and theoretically predicted pictures of critical behavior of small magnetic particles of yttrium orthoferrite.

Static and dynamic critical properties of the quasi-two-dimensional antiferromagnetMnPS3

A comprehensive study of the static and dynamic critical properties of the quasi-two-dimensional antiferromagnet MnPS3 is presented. The relatively large spin S=5/2 of the Mn ions ensure predominantly classical behavior, and the compound is believed to be a good example of a Heisenberg-type system on a honeycomb lattice. The sample has been measured using a variety of neutron-scattering techniques and instrumentation. The results are compared to the three-dimensional (3D) Heisenberg model, the two-dimensional classical O(3) rotator model, the two-dimensional XY model, and the related two-dimensional anisotropic Heisenberg model. The critical properties appear to be best described by the last model almost entirely over the measured temperature range, except just below the Neel temperature where the critical behavior seems to be 3D. There are some discrepancies, particularly in the rescaling of the spin stiffness and the energy widths of the structure factor. A possible explanation for the XY-like behavior is offered, involving the magnetic anisotropy in the system and its influence on the Hamiltonian.

Phase transition in fiber bundle models with recursive dynamics.

We study the phase transition in a class of fiber bundle models in which the fiber strengths are distributed randomly within a finite interval and global load sharing is assumed. The dynamics is expressed as recursion relations for the redistribution of the applied stress and the evolution of the surviving fraction of fibers. We show that an irreversible phase transition of second-order occurs, from a phase of partial failure to a phase of total failure, when the initial applied stress just exceeds a critical value. The phase transition is characterized by static and dynamic critical properties. We calculate exactly the critical value of the initial stress for three models of this kind, each with a different distribution of fiber strengths. We derive exact expressions for the order parameter, the susceptibility to changes in the initial applied stress, and the critical relaxation of the surviving fraction of fibers for all the three models. The static and dynamic critical exponents obtained from these expressions are found to be universal.

Invaded cluster simulations of the XY model in two and three dimensions.

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000(2) and 120(3), respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K(c)=0.454 12(2) for the three-dimensional XY model and eta=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are K(c)=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.

Static and dynamic critical properties of 3D systems with long-range correlated quenched defects

A field-theoretic description of static and purely relaxational dynamic critical behaviour of systems with quenched defects obeying power law correlations ~|x|-a for large separations x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 2 a3, a renormalization analysis of the scaling functions in the two-loop approximation is carried out, and the fixed points corresponding to the stability of various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by a double ,-expansion. The static and dynamic critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique.

Pair contact process in two dimensions.

We study the stationary properties of the two-dimensional pair contact process, a nonequilibrium lattice model exhibiting a phase transition to an absorbing state with an infinite number of configurations. The critical probability and static critical exponents are determined via Monte Carlo simulations, as well as order-parameter moment ratios and the scaling of the initial density decay. The static critical properties fall in the directed percolation universality class.

Finite-size scaling and critical exponents of the real antiferromagnetic model

The static critical properties of the Cr 2O 3 real antiferromagnetic model are investigated by the Monte-Carlo method. The systems with periodic boundary conditions containing N=500; 864; 1372; 2048; 2916; 4000 spins are studied. Using the theory of finite-size scaling, the values of critical exponents α, β, γ are calculated.The comparison of data with the results of theoretical and experimental research is carried out.

Critical behavior of O(n)-symmetric systems with reversible mode-coupling terms: Stability against detailed-balance violation

We investigate nonequilibrium critical properties of O(n)-symmetric models with reversible mode-coupling terms. Specifically, a variant of the model of Sasv\'ari, Schwabl, and Sz\'epfalusy (SSS) is studied, where violation of detailed balance is incorporated by allowing the order parameter and the dynamically coupled conserved quantities to be governed by heat baths of different temperatures ${\mathrm{T}}_{\mathrm{S}}$ and ${\mathrm{T}}_{\mathrm{M}}$, respectively. Dynamic perturbation theory and the field-theoretic renormalization group are applied to one-loop order, and yield two new fixed points in addition to the equilibrium ones. The first fixed point corresponds to \ensuremath{\Theta}=${\mathrm{T}}_{\mathrm{S}}$/${\mathrm{T}}_{\mathrm{M}}$=\ensuremath{\infty} and leads to model A critical behavior for the order parameter and to anomalous noise correlations for the generalized angular momenta; the second one is at \ensuremath{\Theta}=0 and is characterized by mean-field behavior of the conserved quantities, by a dynamic exponent z=d/2 equal to that of the equilibrium SSS model, and by modified static critical exponents. However, both these new fixed points are unstable, and upon approaching the critical point detailed balance is restored, and the equilibrium static and dynamic critical properties are recovered.

Critical Viscosity and Ising‐to‐Mean‐Field Crossover Near the Upper Consolute Point of an Ionic Solution

AbstractWe report on shear viscosity data near the upper consolute point of the ionic system tetrabutylammonium picrate+1‐tridecanol (Tc = 341.845 K) which shows a large mean‐field region in its static critical properties with crossover to Ising behavior close to the critical temperature. After suitable assessment of the non‐singular background viscosity we establish that the asymptotic scaling law η ∝ τ−y applies in the region τ < 5 ×10−3, where τ = (T‐Tc)/Tc is the reduced temperature. The value of the exponent is determined to y = 0.0434±0.001 which is close to the current best value for the 3d‐Ising model, y = 0.041. At higher reduced temperatures pronounced deviations from Ising criticality are observed. In agreement with the mean‐field prediction y = 0, the critical contribution is vanishing. This crossover is discussed in the framework of current theory.

Influence of disorder on the static critical behavior in the frustrated ferromagnetic system CdCr2(1-x)In2xS4

We present a study of the static critical properties of CdCr2(1-x)In2xS4 type frustrated ferromagnetic compounds. It was performed by means of both Small Angle Neutron Scattering and Neutron Diffraction techniques, which respectively allowed us to investigate the temperature dependence of the magnetization fluctuations in the paramagnetic phase, and of the order parameter close to but below the critical point. For T >Tc, the deduced values of the static critical exponents γ and ν appeared to be strongly increased by dilution. Surprisingly, the magnetization exponent β seems to be disorder independent, remaining at a value which is close to the 3D-Heisenberg result.

Dynamic critical phenomena in aqueous protein solutions.

An aqueous protein solution is a binary mixture of large particles (protein molecules) and small particles (water molecules). In contrast, a binary liquid mixture is a mixture of two different kinds of particles that are approximately the same size. Thus, a comparison of the critical behavior in these two systems allows one to determine the effect on critical behavior of a relative difference of size in the two kinds of particles in a binary mixture. The correlation length j, osmotic compressibility kT , and coexistence curve of aqueous protein solutions in the vicinity of the critical point for liquid-liquid phase separation have been reported recently [1‐3]. These experimental results established that the static critical properties of these solutions are consistent with the behavior of members of the static universality class of three-dimensional systems with short range interactions and scalar order parameters. This universality class includes binary liquid mixtures. In contrast to the equilibrium properties, the dynamic properties of aqueous protein solutions in the vicinity of the critical point have not been investigated as thoroughly. In this Letter, we report our use of quasielastic light scattering (QLS) to investigate the time decay of spontaneous concentration fluctuations, as a function of both wave number q of the fluctuations and temperature T, along the critical isochore of aqueous solutions of the bovine eye lens protein gII-crystallin. The only previous study of critical dynamics in a protein solution of which we are aware is the work of Ishimoto and Tanaka who made QLS measurements at a single fixed wave number of the fluctuations using a 19 channel digital correlator on aqueous solutions of lysozyme [4] along the critical isochore for sT 2 TcdyTc $ 5 3 10 23 (q j, 0.1), where Tc is the critical temperature. The present study is a significant experimental advance over the work of Ishimoto and Tanaka in that (i) we obtain data for fluctuations at up to 12 different wave numbers, (ii) we obtain data using a 144 channel digital correlator, and (iii) we obtain data much closer to the critical point [ sT 2 TcdyTc $ 1.56 3 10 24 and q j# 3.18]. The more thorough investigation of critical dynamics presented here has revealed new and unexpected behavior. Specifically, we find that the wave number and temperature dependence of the average rate of decay of the concentration fluctuations are consistent with the theory for critical dynamics in binary liquid mixtures [5] only if we allow both the background viscosity and the background contribution to the decay rate to be unusually large. Furthermore, in contrast to the behavior seen in binary liquid mixtures we find that the concentration fluctuations exhibit very significant deviation from exponential decay with time. Of course, these protein solutions may be viewed as

Ferromagnetic Transition of Heisenberg Ferromagnetic Metal of CoS2–Static Critical Properties–

The static magnetic properties near the Curie temperature in CoS 2 have been thoroughly studied by using various experimental techniques from high quality single crystals, which were grown by the chemical transport technique. A very small anisotropy in magnetization confirms that CoS 2 is the best system to apply the Heisenberg Hamiltonian. The critical indices obey the static scaling hypothesis, which indicates negligible effect of the temperature-induced local moment on the ferromagnetic transition. However, the static critical properties suggest that the magnetic transition is close to the tricritical point, although the magnetic transition in CoS 2 is continuous.

Static critical properties of disordered ferromagnets studied by superconducting quantum interference device magnetometry and small-angle neutron-scattering techniques

CdCr(2−2x)In(2x)S4 is a highly frustrated magnetic system based on CdCr2S4, a well-known Heisenberg 3D ferromagnet. A study is presented of its static critical properties as a function of concentration, performed by low-field superconducting quantum interference device SQUID magnetometry and small-angle neutron-scattering SANS techniques. Samples were prepared from isotopic cadmium 114Cd with a low absorption cross section for thermal neutrons. Magnetic measurements performed with low dc fields in the temperature range 5–300 K allow one to determine the magnetic state of the samples, the reentrant temperature TG, the Curie temperature TC, and the critical exponent γ of the susceptibility above TC as a function of dilution in the domain 0<x<0.15. By directly measuring the critical magnetic fluctuations in zero field in SANS experiments, access is gained to the magnetic correlation length ξ, its exponent ν above TC, and another independent determination of γ. Whereas the pure compound exhibits exponents (γ=1.37, ν=0.70) compatible with a 3D Heisenberg ferromagnetic model, a pronounced increase of measured exponents with dilution was found, namely γ=1.97, ν=1.03. The results are compared to the ones of similar studies in the related system Eu(1−x)Sr(x)S and to the renormalization group treatment of strongly random ferromagnets, and possible mechanisms to explain how a small dilution can provoke large alterations in the critical behavior and move the system in the strong disorder region are discussed.