Marginal Effect of Structural Defects on the Nonequilibrium Critical Behavior of the Two-Dimensional Ising Model

The influence of various initial magnetizations m0 and structural defects on the nonequilibrium critical behavior of the two-dimensional Ising model is numerically simulated by Monte Carlo methods. Based on analysis of the time dependence of magnetization and the two-time dependences of autocorrelation function and dynamic susceptibility, we revealed the influence of logarithmic corrections and the crossover phenomena of percolation behavior on the nonequilibrium characteristics and the critical exponents. Violation of the fluctuation–dissipation theorem is studied, and the limiting fluctuation–dissipation ratio is calculated for the case of high-temperature initial state. The influence of various initial states on the limiting fluctuation–dissipation ratio is investigated. The nonequilibrium critical dynamics of weakly disordered systems with spin concentrations p ≥ 0.9 is shown to belong to the universality class of the nonequilibrium critical behavior of the pure model and to be characterized by the same critical exponents and the same limiting fluctuation–dissipation ratios. The nonequilibrium critical behavior of systems with p ≤ 0.85 demonstrates that the universal characteristics of the nonequilibrium critical behavior depend on the defect concentration and the dynamic scaling is violated, which is related to the influence of the crossover effects of percolation behavior.     

Critical behavior of the two-dimensional thermalized bond Ising model

A suitably modified Wolff single-cluster Monte Carlo simulation has been performed to investigate the critical behavior of a two-dimensional Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model, which is intended to simulate the thermal excitations of electronic bond degrees of freedom as in covalently bonded network liquids. A finite-size scaling analysis of the susceptibility and the fourth-order cumulant, results in a reliable estimation of the critical exponents in the thermodynamic limit. The exponents are found to be consistent with those predicted by the Fisher renormalization relations, despite the well known violations of the renormalization relations when approximate methods such as real space renormalization group are employed to investigate two-dimensional Ising model with annealed bond dilution, and the temperature variation of the bond concentration in thermalized bond model system.     

Random-bond Ising chain in a transverse magnetic field: A finite-size scaling analysis

We investigate the zero-temperature quantum phase transition of the randombond Ising chain in a transverse magnetic field. Its critical properties are identical to those of the McCoy-Wu model, which is a classical Ising model in two dimensions with layered disorder. The latter is studied via Monte Carlo simulations and transfer matrix calculations and the critical exponents are determined with a finite-size scaling analysis. The magnetization and susceptibility obey conventional rather than activated scaling. We observe that the order parameter and correlation function probability distribution show a nontrivial scaling near the critical point, which implies a hierarchy of critical exponents associated with the critical behavior of the generalized correlation lengths.     

Supersymmetry and its spontaneous breaking in the random field Ising model

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ? 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.     

Phase transitions and critical properties in the antiferromagnetic Ising model on a layered triangular lattice with allowance for intralayer next-nearest-neighbor interactions

The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.     

Correlation functions in the multiple Ising model coupled to gravity

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar φ³ graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.     

On the critical behavior of the Ising model with mixed two- and three-spin interactions

A study is made of a two-dimensional Ising model with staggered three-spin interactions in one direction and two-spin interactions in the other. The phase diagram of the model and its critical behavior are explored by conventional finite-size scaling and by exploiting relations between mass gap amplitudes and critical exponents predicted by conformal invariance. The model is found to exhibit a line of continuously varying critical exponents, which bifurcates into two Ising critical lines. This similarity of the model with the Ashkin-Teller model leads to a conjecture for the exact critical indices along the nonuniversal critical curve. Earlier contradictions about the universality class of the uniform (isotropic) case of the model are clarified.     

Wang-Landau study of the 3D Ising model with bond disorder

We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes L in the range L=8-64L=8{-}64. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the nature of the phase transition of the disordered model, departing gradually from the fixed point of the pure system. Our analysis (based on the determination of the critical exponents) shows that the 3D random-bond Ising model belongs to the same universality class with the site- and bond-dilution models, providing a single universality class for the 3D Ising model with these three types of quenched uncorrelated disorder.     

The critical behavior of φ 1 4

The eigenvalues, eigenfunctions, and Schwinger functions of the ordinary differential operator $$H(\lambda ,m) = \tfrac{1}{2}\{ p^2 + \lambda q^4 + (m^2 - \lambda m^{ - 2} )q^2 \} $$ are studied as λ → ∞. It is shown that the scaling limit of the Schwinger functions equals the scaling limit of a one dimensional Ising model. Critical exponents ofH(λ,m) are shown to equal critical exponents of the Ising model, while critical exponents of the renormalized theory are shown to agree with those of a harmonic oscillator.     

The Universality of the Critical Dynamics of the Kinetic Ising Model on the Regular Fractals

The form of the universal scaling law of the critical dynamic exponent, z = D_ƒ + 2/υ, is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents υ₀ = ∞ and υ = 1. So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, z = D_ƒ + R_max/2υ, on the site models of the regular fractals with R_min = 2. Meanwhile, we discuss Daniel Kandal's correction to the formula of the,critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.     

A JUSTIFICATION FOR THE SCALING OF THE THOM-SYSTEM

In the present paper we discuss the critical behavior of Thornsystem using Catastrophe Theory. The universal critical asymptotic form of the family of free energy functions for Thomsystem with one order parameter and two field parameters is obtained. The expressions of critical exponents, the scaling laws, and the scaling hypotheses are all derived from this universal asymptotic form.     

Delocalization transition of a rough adsorption-reaction interface

We introduce a kinetic interface model suitable for simulating adsorption-reaction processes which take place preferentially at surface defects such as steps and vacancies. As the average interface velocity is taken to zero, the self-affine interface with Kardar-Parisi-Zhang-like scaling behavior undergoes a delocalization transition with critical exponents that fall into a different universality class. As the critical point is approached, the interface becomes a multivalued, multiply connected self-similar fractal set. The scaling behavior and critical exponents of the relevant correlation functions are determined from Monte Carlo simulations and scaling arguments.     

A lattice gas coupled to two thermal reservoirs: Monte Carlo and field theoretic studies

We investigate the collective behavior of an Ising lattice gas, driven to non-equilibrium steady states by being coupled to two thermal baths. Monte Carlo methods are applied to a two-dimensional system in which one of the baths is fixed at infinite temperature. Both generic long range correlations in the disordered state and critical properties near the second order transition are measured. Anisotropic scaling, a key feature near criticality, is used to extract and some critical exponents. On the theoretical front, a continuum theory, in the spirit of Landau-Ginzburg, is presented. Being a renormalizable theory, its predictions can be computed by standard methods of -expansions and found to be consistent with simulation data. In particular, the critical behavior of this system belongs to a universality class which is quite different from the uniformly driven Ising model. Received 4 October 2000     

Corner exponents in the two-dimensional Potts model

The critical behavior at a corner in two-dimensional Ising and three-state Potts models is studied numerically on the square lattice using transfer operator techniques. The local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite-size scaling results at the critical point for isotropic or anisotropic couplings. The scaling dimensions compare quite well with the values expected from conformal invariance, provided the opening angle is replaced by an effective one in anisotropic systems.     

Novel scaling behavior of the Ising model on curved surfaces

We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat C(T) and magnetic susceptibility χ(T) show sharp peaks. The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.     

Thom系统标度性的证明

本文应用突变论(catastrophe theory)证得了Thom系统自由能函数族的普适的临界渐近形式,并从它推导了临界指数公式,标度律和标度假设。 关键词：     

Critical properties of the three-dimensional frustrated Ising model on a cubic lattice

The critical properties of the three-dimensional fully frustrated Ising model on a cubic lattice are investigated by the Monte Carlo method. The critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length), as well as the Fisher exponent η, are calculated in the framework of the finite-size scaling theory. It is demonstrated that the three-dimensional frustrated Ising model on a cubic lattice forms a new universality class of the critical behavior.     

Relaxation phenomena at criticality

The collective behaviour of statistical systems close to critical points is characterized by an extremely slow dynamics which, in the thermodynamic limit, eventually prevents them from relaxing to an equilibrium state after a change in the thermodynamic control parameters. The non-equilibrium evolution following this change displays some of the features typically observed in glassy materials, such as ageing, and it can be monitored via dynamic susceptibilities and correlation functions of the order parameter, the scaling behaviour of which is characterized by universal exponents, scaling functions, and amplitude ratios. This universality allows one to calculate these quantities in suitable simplified models and field-theoretical methods are a natural and viable approach for this analysis. In addition, if a statistical system is spatially confined, universal Casimir-like forces acting on the confining surfaces emerge and they build up in time when the temperature of the system is tuned to its critical value. We review here some of the theoretical results that have been obtained in recent years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the Ising model with Glauber dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting in a film is discussed within the Gaussian model.     

EXACT DECIMATION APPROACH WITH MEAN-FIELD APPROXIMATION FOR RANDOMLY DILUTED ISING MODELS

A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.     

Universality Under Conditions of Self-tuning

We study systems with a continuous phase transition that tune their parameters to maximize a quantity that diverges solely at a unique critical point. Varying the size of these systems with dynamically adjusting parameters, the same finite-size scaling is observed as in systems where all relevant parameters are fixed at their critical values. This scheme is studied using a self-tuning variant of the Ising model. It is contrasted with a scheme where systems approach criticality through a target value for the order parameter that vanishes with increasing system size. In the former scheme, the universal exponents are observed in naïve finite-size scaling studies, whereas in the latter they are not.