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Showing papers on "Ising model published in 1992"


Book
01 Jan 1992
TL;DR: In this article, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

7,349 citations


Journal ArticleDOI
15 Jul 1992-EPL
TL;DR: In this article, the authors proposed a new global optimization method (Simulated Tempering) for simulating effectively a system with a rough free-energy landscape (i.e., many coexisting states) at finite nonzero temperature.
Abstract: We propose a new global optimization method (Simulated Tempering) for simulating effectively a system with a rough free-energy landscape (i.e., many coexisting states) at finite nonzero temperature. This method is related to simulated annealing, but here the temperature becomes a dynamic variable, and the system is always kept at equilibrium. We analyse the method on the Random Field Ising Model, and we find a dramatic improvement over conventional Metropolis and cluster methods. We analyse and discuss the conditions under which the method has optimal performances.

1,723 citations


Journal ArticleDOI
TL;DR: A renormalization-group analysis of the spin-1/2 transverse field Ising model with quenched randomness is presented; it become exact asymptotically near the zero temperature ferromagnetic phase transition.
Abstract: A renormalization-group analysis of the spin-1/2 transverse field Ising model with quenched randomness is presented; it become exact asymptotically near the zero temperature ferromagnetic phase transition. The spontaneous magnetization is found to vanish with an exponent \ensuremath{\beta}=1/2(3- \ensuremath{\surd}5 ), while in the disordered phase the typical and average spin correlations are found to decay with different correlation lengths, which diverge with exponents \ensuremath{ u}\ifmmode \tilde{}\else \~{}\fi{}=1 and \ensuremath{ u}=2, respectively.

480 citations


Journal ArticleDOI
TL;DR: The stationary critical properties of the isotropic majority vote model on a square lattice were calculated by Monte Carlo simulations and finite size analysis as mentioned in this paper, and the critical exponentsν, γ, and β were found to be the same as those of the Ising model and critical noise parameter was qc=0.075±0.001.
Abstract: The stationary critical properties of the isotropic majority vote model on a square lattice are calculated by Monte Carlo simulations and finite size analysis. The critical exponentsν, γ, andβ are found to be the same as those of the Ising model and the critical noise parameter is found to beqc=0.075±0.001.

241 citations


Journal ArticleDOI
TL;DR: In this paper, the authors analyze those integrable statistical systems which originate from some relevant perturbations of the minimal models of conformal field theories and show that the central charge of the original conformal theories can be recovered from the scattering data.

213 citations


Journal ArticleDOI
TL;DR: In this article, the conceptual foundations of the renormalization-group (RG) formalism were revisited, and rigorous theorems on the regularity properties and possible pathologies of the RG map were proved.
Abstract: We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.

207 citations


Journal ArticleDOI
TL;DR: An extensive Monte Carlo study of the density and energy fluctuations in a 2D Lennard-Jones fluid, within the grand-canonical ensemble, incorporating the mixed character of the scaling fields that manifests particle-hole asymmetry.
Abstract: We report an extensive Monte Carlo study of the density and energy fluctuations in a 2D Lennard-Jones fluid, within the grand-canonical ensemble. The results are analyzed within a finite-size-scaling theory incorporating the mixed character of the scaling fields that manifests particle-hole asymmetry. The limiting critical distribution of the density matches that of the magnetization in the 2D Ising model, illuminating the configurational basis of universality; the corrections to the limiting behavior contain an antisymmetric component which reflects the critical Ising energy operator, corroborating the anticipated structure of the scaling fields.

201 citations


Journal ArticleDOI
TL;DR: In this paper, a finite-size scaling theory was developed to describe the joint density and energy fluctuations in a near-critical fluid, where the energy operator features in the critical density distribution as an antisymmetric correction to the limiting scale-invariant form.
Abstract: The authors develop a finite-size-scaling theory describing the joint density and energy fluctuations in a near-critical fluid. As a result of the mixing of the temperature and chemical potential in the two relevant scaling fields, the energy operator features in the critical density distribution as an antisymmetric correction to the limiting scale-invariant form. Both the limiting form and the correction are predicted to be functions that are characteristic of the Ising universality class and are independently known. The theory is tested with extensive Monte Carlo studies of the two-dimensional Lennard-Jones fluid, within the grand canonical ensemble. The simulations and scaling framework together are shown to provide a powerful way of identifying the location of the liquid-gas critical point, while confirming and clarifying its essentially Ising character. The simulations also show a clearly identifiable signature of the field-mixing responsible for the failure of the law of rectilinear diameter.

196 citations


Journal ArticleDOI
TL;DR: It is argued that the Heisenberg model has long-range antiferromagnetic order in the limit T 0, and the system appears to remain disordered for T 0.
Abstract: We examine the classical antiferromagnet on the Kagom\'e lattice with nearest-neighbor interactions and n-component vector spins. Each case n=1,2,3 and ng3 has its own special behavior. The Ising model (n=1) is disordered at all temperatures. The XY model (n=2) in the zero-temperature (T\ensuremath{\rightarrow}0) limit reduces to the three-state Potts model, which in turn can be mapped onto a solid-on-solid model that is o/Iat its roughening transition. Exact critical exponents are derived for this system. The spins in the Heisenberg model (n=3) become coplanar and more ordered than the XY model as T\ensuremath{\rightarrow}0. Thus we argue that the Heisenberg model has long-range antiferromagnetic order in the limit T\ensuremath{\rightarrow}0. For ng3 the system appears to remain disordered for T\ensuremath{\rightarrow}0.

190 citations


Journal ArticleDOI
TL;DR: In this article, an Ising-like model, in the mean-field approach, involving two "antiferromagnetically" coupled sublattices, is presented.
Abstract: We have analyzed an Ising-like model, in the mean-field approach, involving two “antiferromagnetically” coupled sublattices. This model simulates the so-called “two-step” spin-crossover transition, for which a precise definition is given. If both sublattices are equivalent, it implies a spontaneous breaking of symmetry which may occur within a temperature range limited by two “Neel temperatures”. It, also predicts a simultaneous reversal of the magnetization of the sublattices (if they are unequivalent) at a “characteristic” value of temperature. These features are analyzed simultaneously with some details. The present model fits and explains well the available experimental data concerning [ Fe(2-pic)3] Cl2- EtOH and FeII[ 5NO2 – sal – N(1, 4, 7, 10)] .

189 citations


Journal ArticleDOI
TL;DR: In this article, a review describes various attempts to develop a theoretical understanding for ordering and dynamics of randomly diluted molecular crystals, where quadrupole moments freeze in random orientations upon lowering the temperature, as a result of randomness and competing interactions.
Abstract: This review describes the various attempts to develop a theoretical understanding for ordering and dynamics of randomly diluted molecular crystals, where quadrupole moments freeze in random orientations upon lowering the temperature, as a result of randomness and competing interactions. While some theories attempt to model this freezing into a phase with randomly oriented quadrupole moments in terms of a bond-disorder concept analogous to the Edwards-Anderson model of spin glasses, other theories attribute the freezing to random field-like terms in the Hamiltonian. While models of the latter type have been studied primarily by microscopic molecular field-type treatments, the former models have been treated both in the Sherrington-Kirkpatrick-Parisi infinite-range limit, and in the short-range case. Among the surprising findings of these treatments we emphasize the first-order glass transition (though lacking a latent heat) of the infinite-range Potts glass, the suggestion that the short-range Pot...

Journal ArticleDOI
TL;DR: A new construction to obtain restricted solid-on-solid (RSOS) models out of loop models is reported, which finds a spin-1 Ising model, which is solvable not only at the critical point, but also in a fieldlike deviation away from it.
Abstract: In this Letter we report a new construction to obtain restricted solid-on-solid (RSOS) models out of loop models. The method is a generalization of ideas developed by Owczarek and Baxter, and by Pasquier. In particular we consider a solvable O(n) model and point out that some of the RSOS models thus obtained admit an off-critical extension. Among these models we find a spin-1 Ising model, which is solvable not only at the critical point, but also in a fieldlike deviation away from it. We calculate the critical exponent \ensuremath{\delta}=15 directly from the relation between the free energy and the field. This is the first determination of this exponent without the use of scaling relations.

Journal ArticleDOI
TL;DR: The phase behavior of a simple fluid or Ising magnet (at temperatures above its roughening transition) confined between parallel walls that exert opposing surface fields is studied in this paper.
Abstract: The phase behaviour of a simple fluid or Ising magnet (at temperatures above its roughening transition) confined between parallel walls that exert opposing surface fields h 2 = - h 1 is found to be markedly different from that which arises for h 2 = h 1 . Whereas critical wetting plays little role for confinement by identical walls, it is of crucial importance for opposing surface fields. Analysis of a Landau functional and other mean-field treatments show that if h 1 is such that critical wetting occurs at a single wall ( L = ∞) at a transition temperature T w , then phase coexistence, for finite wall separation L , is restricted to temperatures T T c , L , where the critical temperature T c , L lies below T w . In the temperature range T c , b > T > T w there is a single soft mode phase that is characterized, for zero bulk field and large L , by a +- interface located at the centre of the slit, a transverse correlation length ξ ∼≈ e L and a solvation force that is repulsive. For large h 1 , T w can lie arbitrarily far below the bulk critical temperature T c , b . Scaling arguments, whose validity we have confirmed in two dimensions by comparison with exact solutions for interfacial Hamiltonians, predict that such behaviour persists beyond mean-field for systems with short-ranged forces. They also predict similar phase behaviour for long-ranged forces, but with ξ ξ ∼ increasing algebraically with L in the soft mode phase. The solvation force tf s changes from repulsive to attractive (at large L ) as the temperature is reduced below T w , i.e. the sign of tf s reflects wetting characteristics.

Book
01 Jan 1992
TL;DR: In this article, Renormalization group methods for the Ising Model are presented. But they do not cover the real-space version of the model, which is the one we use in this paper.
Abstract: Self-Similarity and Scale Invariance. Renormalization Group Approach to Chaos. Renormalization Approach to Percolation. Renormalization Group and Critical Phenomena. The Ising Model. Renormalization Group for the Ising Model. Other Real-Space Renormalization Group Methods for the Ising Model. Mean Field Theory and the Gaussian Fixed Point. The epsis Expansion. The Spherical Model and the 1/n Expansion. The Two-Dimensional X-Y Model and the Kosterlitz-Thouless Transition. Appendices. Author Index. Subject Index.

Journal ArticleDOI
TL;DR: In this paper, the temperature dependence of the peak of the density distribution function near the critical point is studied, and Monte Carlo simulations for the simple special case of the two-dimensional lattice gas model are presented.
Abstract: In the Gibbs ensemble gas–liquid phase coexistence can be studied by obtaining the density distribution function in a finite system from the study of two subsystems exchanging particles. The temperature dependence of the peak of this distribution function is studied near the critical point, and Monte Carlo simulations for the simple special case of the two‐dimensional lattice gas model are presented. This case is a ‘‘restricted Gibbs ensemble’’ where the particle numbers of the two systems fluctuate but their volume fluctuations are suppressed. From formal analysis and physical arguments, we predict that the density difference of the peak positions vanishes according to a classical power law [1−T/Tc(L)]1/2, where Tc(L) is a shifted critical temperature of the finite system of linear dimension L, for temperatures within a regime where fluctuations are significant (L does not exceed the correlation length ξ there). This behavior is verified by Monte Carlo simulations for L×L lattices with periodic boundary ...

Journal ArticleDOI
TL;DR: In this paper, the authors developed a theory of self-assembly based on the concept of frustrated charges and showed that this theory can be used to derive useful analytical estimates for frustrated Ising systems.
Abstract: We develop a concept of frustrating charges to create a theory of self-assembly. In particular, we note that the constraints of stoichiometry frustrate ordinary phase equilibria and lead to self-assembly of systems such as oil-water-surfactant mixtures. Further we note that at long wavelengths, the constraints of stoichiometry are isomorphic to the constraints of charge neutrality in a specific electrostatic analogy. We expand upon this analogy, first noted by Stillinger, and show that it can be used to derive useful analytical estimates. In addition we use the analogy to create a new model for frustrated systems, and we present Monte Carlo results for this charge frustrated Ising system that exhibits varied behaviors of self-assembly. The Monte Carlo calculations are made possible through the development of an algorithm which permits cluster moves. 9 refs., 5 figs.

Journal ArticleDOI
TL;DR: It is argued that the picture based on the infinite-ranged Sherrington-Kirkpatrick model, with many non-congruent pure states, leads to a breakdown of the thermodynamic limit.
Abstract: We propose a test to distinguish, both numerically and theoretically, between the two competing pictures of short-ranged Ising spin glasses at low temperature: ``chaotic'' size dependence. Scaling theories in which at most two pure states (related by a global spin flip) occur require that finite-volume correlations (with, say, periodic boundary conditions) have a well-defined thermodynamic limit. We argue, however, that the picture based on the infinite-ranged Sherrington-Kirkpatrick model, with many noncongruent pure states, leads to a breakdown of the thermodynamic limit. The argument combines rigorous and heuristic elements; one of the fomer is a proof that in the infinite-ranged model itself, non-self-averaging implies chaotic size dependence. Numerical tests, based on chaotic size dependence, could provide a more sensitive measure than the usual overlap distribution P(q) in determining the number of pure states.

Journal ArticleDOI
01 Apr 1992-EPL
TL;DR: In this paper, the critical behavior of the random field Ising model is discussed using techniques of replica symmetry breaking familiar from the theory of spin glasses and random manifolds, using an approximation that is valid in the limit of a large number of spin components, m, and obtain equations for solutions which break replica symmetry according to a natural scaling ansatz.
Abstract: We discuss the critical behavior of the random field Ising model, using techniques of replica symmetry breaking familiar from the theory of spin glasses and random manifolds. Using an approximation that is valid in the limit of a large number of spin components, m, we find that the replica symmetric solution is unstable and obtain equations for solutions which break replica symmetry according to a natural scaling ansatz. Although we are unable to solve these equations exactly, we show that they lead to exponents which are different to order 1/m from the replica symmetric solution.

Journal ArticleDOI
TL;DR: In this article, a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model was considered, where the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic nonincreasing fashion.
Abstract: We consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion.

Journal ArticleDOI
TL;DR: In this paper, a connection between the equilibrium Landau-type theories and kinetic rate theories is made, and it is shown that kinetic processes with continuous order parameters can be analyzed using a general rate equation which contains δG δQ (G is often a Landau potential, Q is the order parameter) as a driving force.

Journal ArticleDOI
TL;DR: A first-order, entropy-driven demixing transition in a simple lattice model for a hard-core mixture is proved and leads to a very simple interpretation of the entropic contribution of the solvent to the interaction parameter χ in the Flory-Huggins theory of polymer solutions.
Abstract: We prove the existence of a first-order, entropy-driven demixing transition in a simple lattice model for a hard-core mixture. The existence of this transition follows from the fact that this lattice model can be mapped onto an Ising model with nearest-neighbor interactions for which the phase behavior is known. The same mapping leads to a very simple interpretation of the entropic contribution of the solvent to the interaction parameter χ in the Flory-Huggins theory of polymer solutions.

Journal ArticleDOI
TL;DR: A finite-size-scaling analysis of several thermodynamic quantities strongly suggests that the critical exponents fall into the universality class of the two-dimensional Ising model.
Abstract: We have studied the effect of quenched, bond randomness on the nature of the phase transition in the two-dimensional eight-state Potts model. Through extensive Monte Carlo simulations, we confirm that the phase transition changes from first order to second order. A finite-size-scaling analysis of several thermodynamic quantities strongly suggests that the critical exponents fall into the universality class of the two-dimensional Ising model.

Journal ArticleDOI
TL;DR: In this article, an equation of state, expressed in the form of a recursion relation for the Helmholtz free energy density, is presented, which represents the free energy densities as an initial mean field free energy, plus a sum of density fluctuation correction terms which serve both to dress the initial MEF into a more accurate analytic MEF, and to provide the characteristic asymptotic singular behavior in the critical region.
Abstract: An equation of state, expressed in the form of a recursion relation for the Helmholtz free energy density, is presented. This equation of state represents the free energy density as an initial mean field free energy density, plus a sum of density fluctuation correction terms which serve both to dress the initial mean field equation of state into a more accurate analytic equation of state away from the critical region, and to provide the characteristic asymptotic singular behavior in the critical region. Numerical solutions of the recursion relations are described and presented as pressure isotherms and the calculated density coexistence curve. A comparison is made with simple fluid data for temperatures in the range 0.7≤T/Tc≤1.5 and densities in the range 0≤ρ/ρc≤2.5. In particular, good agreement is found near the critical point. Three critical exponents, δ, γ, and β, are calculated and found to be in good agreement with simple fluid data as well as with renormalization group results for the 3D Ising model.

Book ChapterDOI
Kurt Binder1
TL;DR: In this article, the phenomenological theory of finite size scaling is reviewed with emphasis on the concept of probability distributions of order parameter and/or energy, and attention is also drawn to recent developments concerning anisotropic geometries and critical behavior, as well as crossover phenomena from one universality class to another.
Abstract: For many models of statistical thermodynamics and of lattice gauge theory computer simulation methods have become a valuable tool for the study of critical phenomena, to locate phase transitions, distinguish whether they are of first or second order, and so on. Since simulations always deal with finite systems, analysis of finite size effects by suitable finite size scaling concepts is a key ingredient of such applications. The phenomenological theory of finite size scaling is reviewed with emphasis on the concept of probability distributions of order parameter and/or energy. Attention is also drawn to recent developments concerning anisotropic geometries and anisotropic critical behavior, as well as to crossover phenomena from one “universality class” to another.

Journal ArticleDOI
TL;DR: In this article, integral equations for the exact finite-volume energies of the first and kth excited states in φ 1,3 -perturbed M 2k + 2 on the cylinder (of circumference R) were proposed.

BookDOI
01 Apr 1992
TL;DR: In this paper, the number of lattice points inside a random domain, D.V. Kertesz and D. Stauffer's model of the Lagrangian of a random lattice point inside a lattice domain, E. del Rio et al chaotic properties of the noncommutative 2-shift, W. Van den Broeck and T. Tel dynamics of growing self-affine surfaces, T. Kawakatsu and K. Vicsek and others.
Abstract: Anisotropic segregation in a gravitational field, K. Binder et al replica symmetry breaking in the ising spin glass in finite dimensions, C. De Dominicis and I. Kondor problems of high-Tc superconductors - the resistivity, C. Enz correction to dynamic scaling for the lambda transition in liquid HE4 III quasi-sealing at the natural boundary, R.A. Ferrell evolution of patterns in rotating Benard convection, H. Haken phase transitions in binary systems in the presence of amphiphilic molecules, T. Kawakatsu and K. Kawasaki droplet dynamics of ising ferromagnets at the critical point, J. Kertesz and D. Stauffer on the number of lattice points inside a random domain, D.V. Kosygin and Ya G. Sinai oscillations and onset of chaos in a driven nonlinear oscillator with possible escape to infinity, E. del Rio et al chaotic properties of the noncommutative 2-shift, W. Thirring Sinai disorder - intermittency in random maps, C. Van den Broeck and T. Tel dynamics of growing self-affine surfaces, T. Vicsek and others.

Journal ArticleDOI
TL;DR: The results are described of studies of a random-anisotropy Blume-Emery-Griffiths spin-1 Ising model using mean-field theory, transfer-matrix calculations, and position-space renormalization-group calculations that lead to a rich phase diagram with a variety of phase transitions and reentrant behavior.
Abstract: The results are described of studies of a random-anisotropy Blume-Emery-Griffiths spin-1 Ising model using mean-field theory, transfer-matrix calculations, and position-space renormalization-group calculations. The interplay between the quenched randomness of the anisotropy and the annealed disorder introduced by the spin-1 model leads to a rich phase diagram with a variety of phase transitions and reentrant behavior. The results may be relevant to the study of the phase separation of He-3 - He-4 mixtures in porous media in the vicinity of the superfluid transition.

Journal ArticleDOI
TL;DR: In this article, it is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory and the Pfaffian state is related to the 2D Ising model.
Abstract: It is argued that fractional quantum Hall effect wavefunctions can be interpreted as conformal blocks of two-dimensional conformal field theory. Fractional statistics can be extended to nonabelian statistics and examples can be constructed from conformal field theory. The Pfaffian state is related to the 2D Ising model and possesses fractionally charged excitations which are predicted to obey nonabelian statistics.

Journal ArticleDOI
TL;DR: The chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass is investigated and the chaos in the critical region is characterized by a new critical exponent ζ c that has hitherto escaped attention.
Abstract: We investigate the chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass. Some of our analytic and numerical results, obtained by the Migdal-Kadanoff renormalization scheme but argued to have general validity, differ from those of the scaling and droplet theory. First, chaos is present also at and above the critical temperature. Second, between dr=2.46 (the lower critical dimension) and d+=3.4, the chaos in the critical region is characterized by a new critical exponent ζ c that has hitherto escaped attention

Journal ArticleDOI
Paul Fendley1
TL;DR: In this article, it is shown how to modify the boundary conditions to project out the lowest energy state, which enables one to find excited-state energies, by calculating thermodynamic expectation values of operators which generate discrete symmetries.