Showing papers on "Ising model published in 1987"
TL;DR: In this paper, the equality of two critical points -the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially -is proven for all translation invariant independent per-colation models on homogeneous d-dimensional lattices.
Abstract: The equality of two critical points - the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially - is proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, h\ which for h = Q reduces to the percolation density P^ - at the bond density p = l—e~β in the single parameter case. These are: (1) M^hdM/dh + M2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a "φ3 structure" in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5 —1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.
457 citations
TL;DR: In this paper, the ergodic properties of stochastic Ising models were studied in terms of large deviations and convergence in distribution, using logarithmic Sobolev inequalities.
Abstract: We use logarithmic Sobolev inequalities to study the ergodic properties of stochastic Ising models both in terms of large deviations and in terms of convergence in distribution.
332 citations
TL;DR: There is no hidden order pattern for spin glasses at all T less than T(C), the ordered-phase spin correlations being chaotic functions of spin separation at fixed temperature or of temperature at scale lengths L greater than (T delta T) exp -1/zeta.
Abstract: The microscopic structure of the ordered phase of spin glasses is investigated theoretically in the framework of the T = 0 fixed-point model (McMillan, 1984; Fisher and Huse, 1986; and Bray and Moore, 1986). The sensitivity of the ground state to changes in the interaction strengths at T = 0 is explored, and it is found that for sufficiently large length scales the ground state is unstable against arbitrarily weak perturbations to the bonds. Explicit results are derived for d = 1, and the implications for d = 2 and d = 3 are considered in detail. It is concluded that there is no hidden order pattern for spin glasses at all T less than T(C), the ordered-phase spin correlations being chaotic functions of spin separation at fixed temperature or of temperature (for a given pair of spins) at scale lengths L greater than (T delta T) exp -1/zeta, where zeta = d(s)/2 - y, d(s) is the interfacial fractal dimension, and -y is the thermal eigenvalue at T = 0.
329 citations
TL;DR: In this paper, an Introduction to the Ising Model is presented, along with a discussion of its application in the context of algebraic geometry problems and its application to algebraic logic.
Abstract: (1987). An Introduction to the Ising Model. The American Mathematical Monthly: Vol. 94, No. 10, pp. 937-959.
325 citations
TL;DR: In this article, the critical properties of exactly soluble Ising model on a planar random dynamical lattice representing a regularization of the zero-dimensional string with internal fermions were investigated.
312 citations
TL;DR: The conformal anomaly number for new two-dimensional critical points obtained by adding a slightly relevant perturbation φ (renormalization group eigenvalue y ⪡ 1) to a given critical theory is obtained to lowest order in y to be c ′ = c − y 3 / b 2 + …, where b is the operator product expansion coefficient in φφ ∼ (− b ) φ.
257 citations
TL;DR: For a family of translation-invariant, ferromagnetic, one-component spin systems, including Ising and ϕ4 models, the phase transition is sharp in the sense that at zero magnetic field the high and low-temperature phases extend up to a common critical point as discussed by the authors.
Abstract: For a family of translation-invariant, ferromagnetic, one-component spin systems—which includes Ising and ϕ4 models—we prove that (i) the phase transition is sharp in the sense that at zero magnetic field the high- and low-temperature phases extend up to a common critical point, and (ii) the critical exponent β obeys the mean field bound β⩽1/2. The present derivation of these nonperturbative statements is not restricted to “regular” systems, and is based on a new differential inequality whose Ising model version isM⩽βhχ+M3+ βM2∂M/∂β. The significance of the inequality was recognized in a recent work on related problems for percolation models, while the inequality itself is related to previous results, by a number of authors, on ferromagnetic and percolation models.
230 citations
TL;DR: In this paper, Monte Carlo simulations are presented for a model of a symmetrical polymer mixture on the simple cubic lattice, modeling both polymers A, B by self-avoiding walks of NA=NB=N steps.
Abstract: Monte Carlo simulations are presented for a model of a symmetrical polymer mixture on the simple cubic lattice, modeling both polymers A, B by self‐avoiding walks of NA=NB=N steps. If a pair of nearest‐neighbor sites is taken by monomers of the same species, an energy e is won. In the Monte Carlo algorithm local motions of the chains are considered (allowing for 20% vacancies to ensure enough chain mobility) as well as transformations of A chains into B chains and vice versa, since the simulation applies the grand‐canonical ensemble where the chemical potential difference rather than the volume fraction is fixed. The phase diagram, the excess specific heat, and the structure factor in the long‐wavelength limit are obtained for N=4, 8, 16, and 32 using finite L×L×L lattices with L ranging from 8 to 20. Analyzing these results with finite size scaling techniques, both critical exponents and critical amplitudes are estimated. Although the exponents are consistent with those of the three‐dimensional Ising mod...
221 citations
TL;DR: In this paper, it was shown that the Ising model with a small random magnetic field has two phases at low temperatures, i.e., its lower critical dimension is at most two.
Abstract: We show that the Ising model in three dimensions with a small random magnetic field has two phases at low temperatures, i.e., that its lower critical dimension is at most two. This is shown by our devising an exact renormalization-group flow which takes the theory to the zero-temperature, zero-field fixed point.
217 citations
TL;DR: In this article, the authors consider the time evolution of two Ising systems that differ at time t = 0 in the orientation of only one spin, and calculate detailed time development from two algorithms: (i) Glauber dynamics and (ii) Q2R dynamics (a deterministic cellular automaton).
Abstract: We consider the time evolution of two Ising systems that differ at time t=0 in the orientation of only one spin. The detailed time development is calculated from two algorithms: (i) Glauber dynamics and (ii) Q2R dynamics (a deterministic cellular automaton). We find that for both algorithms spreading of ``damaged regions'' is greatly hindered below a threshold temperature ${\mathrm{T}}_{\mathrm{s}}$ (or energy), which agrees numerically with the Curie point. For Glauber dynamics ${\mathrm{T}}_{\mathrm{s}}$ is found to be a sharp phase transition point; for Q2R dynamics we find a kinetic slowing down which is reminiscent of a (spin-) glass transition.
161 citations
TL;DR: In this article, a correspondence between Eigen's model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system is developed, where the free energy landscape of random Ising systems with the Hopfield Hamiltonian as a special example is applied to the replication rate coefficient landscape.
Abstract: The correspondence between Eigen's model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system is developed. The free energy landscape of random Ising systems with the Hopfield Hamiltonian as a special example is applied to the replication rate coefficient landscape. The coupling constants are scaled with 1/l, since the maxima of any landscape must not increase with the length of the macromolecules. The calculated error threshold relation then agrees with Eigen's expression, which was derived in a different way. It gives an explicit expression for the superiority parameter in terms of the parameters of the landscape. The dynamics of selection and evolution is discussed.
TL;DR: In this article, explicit expressions for all correlation functions of spin, disorder and energy operators of the critical Ising model in the plane or on the torus are given for all correlations.
TL;DR: The mean-field theory of dilute spin-glasses is studied in the limit where the average coordination number is finite and the zero-temperature phase diagram is calculated and the relationship between the spin-glass phase and the percolation transition is discussed.
Abstract: The mean-field theory of dilute spin-glasses is studied in the limit where the average coordination number is finite. The zero-temperature phase diagram is calculated and the relationship between the spin-glass phase and the percolation transition is discussed. The present formalism is applicable also to graph optimization problems.
TL;DR: In this paper, the authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary.
Abstract: The authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple Their analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1-p, the critical temperature scales as exp(-J/(kTc(p))) approximately mod p-pc mod , implying that the crossover exponent is Phi =1 If the constant couplings are replaced by a distribution F(J) with mass near J=0, quite different crossover behaviour is observed For example, if F(J) approximately Jalpha then, for p near pc, Tc(p) approximately mod p-pc mod 1 alpha /
TL;DR: In this article, the specific heat singularity of the 2D q-state Potts model with weak quenched bond randomness is obtained to two-loop order in an expansion in the parameter ( q − 2) around the Ising model, in analogy to the e-expansion for φ 4 scalar field theory around the gaussian model.
TL;DR: The critical fluctuations of a polymer blend (PVME/d-PS) were investigated by neutron small angle scattering, and a mean-field behavior was observed, except in a region very close to the critical temperature where a transition to an Ising-like behavior occurs.
Abstract: The critical fluctuations of a polymer blend (PVME/d-PS) were investigated by neutron small angle scattering. A mean-field behavior was observed, except in a region very close to the critical temperature where a transition to an Ising-like behavior occurs. The width of this region is T-${\mathrm{T}}_{\mathrm{c}}$\ensuremath{\simeq}2.4 K for a mixture with average molecular weights of 89 000 (PVME) and 232 000 (d-PS).
TL;DR: In this paper, the authors reviewed the recent developments in the theory of phase transitions in KH2PO4-type crystals and some relevant experiments and discussed the nature of the isotope effect in the static and dynamic properties of these systems on replacing hydrogen by deuterium.
Abstract: Recent developments in the theory of phase transitions in KH2PO4-type crystals are reviewed together with some relevant experiments. The nature of the isotope effect in the static and dynamic properties of these systems on replacing hydrogen by deuterium is discussed. Expressions for the static and dynamic properties derived by four-cluster analysis are compared with those found from a mean field treatment of the Ising model.
TL;DR: In this paper, the critical behavior of a mixed ferromagnetic Ising spin system consisting of spin-1 and spin-2 with a crystal-field interaction is investigated by the use of the effective field theory with correlations.
Abstract: The critical behavior of a mixed ferromagnetic Ising spin system consisting of spin\(-\tfrac{1}{2}\) and spin-1 with a crystal-field interaction is investigated by the use of the effective-field theory with correlations. The general expressions for evaluating the Curie temperature and the tricritical point are obtained. We find that the tricritical point exists in the system with Z >3, where Z is the coordination number.
TL;DR: It is shown how the temperature-composition phase diagrams and thermodynamic properties of noble-metal alloys can be accurately reproduced by solving the three-dimensional nearest-neighbor fcc Ising model with volume-dependent interaction energies determined from the properties of the ordered phases alone.
Abstract: It is shown how the temperature-composition phase diagrams and thermodynamic properties of noble-metal alloys can be accurately reproduced by solving the three-dimensional nearest-neighbor fcc Ising model with volume-dependent interaction energies determined from the properties of the ordered phases alone. It is found that lattice relaxation effects are essential in determining order-disorder critical temperatures. This approach enables the understanding of phase diagrams in terms of the electronic properties and atomic-scale structure of the constituent ordered phases.
TL;DR: It is shown that a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude but are same within each now allows for the interesting case of frustration.
Abstract: We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{
u}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.
TL;DR: The Gibbs-Markov equivalence theorem (Preston 1974) parameterizes Markov fields via their neighborhood structures, yielding exponential families in canonical form as discussed by the authors, and the requisite normalizing constants, however, are obstreperous.
Abstract: Markov fields provide a general context for describing the strength and structure of spatial interactions. The Gibbs—Markov equivalence theorem (Preston 1974) parameterizes Markov fields via their neighborhood structures, yielding exponential families in canonical form. Likelihood inference is, therefore, apparently straightforward. The requisite normalizing constants, however, are obstreperous. Even when asymptotic characterizations can be obtained, substantial location errors arise during implementation. Moreover, Markov fields can exhibit phase transitions and long-range interactions, thereby creating identifiability problems. These issues are illustrated in the simplest nontrivial case—the classical Ising model of ferromagnetism.
TL;DR: In this article, the critical spin-spin correlation function in the Z -invariant inhomogeneous Ising model was studied and various explicit representations for the correlation function were derived using quadratic difference equations.
Abstract: We study the critical spin-spin correlation function in the Z -invariant inhomogeneous Ising model, which includes the rectangular, triangular, honeycomb, and checkerboard lattices as special cases. Using quadratic difference equations we derive various explicit representations for the correlation function, confirming and simplifying also a miraculous result of Baxter.
TL;DR: In this article, the decay of connectivities in the low-temperature phase of the two-dimensional Ising model is studied and bounds on the rate at which the spin at the origin converges to the spontaneous magnetization are obtained.
Abstract: We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.
01 Jan 1987
TL;DR: In this article, it was shown that the semi-group of the stochastic Ising model converges to equilibrium exponentially fast in the uniform norm of the Gibbs state under the conditions of the Dobrushin Shlosman theorem.
Abstract: We show that, under the conditions of the Dobrushin Shlosman theorem for uniqueness of the Gibbs state, the reversible stochastic Ising model converges to equilibrium exponentially fast on the L2 space of that Gibbs state. For stochastic Ising models with attractive interactions and under conditions which are somewhat stronger than Dobrushin’s, we prove that the semi-group of the stochastic Ising model converges to equilibrium exponentially fast in the uniform norm. We also give a new, much shorter, proof of a theorem which says that if the semi-group of an attractive spin flip system converges to equilibrium faster than 1/td where d is the dimension of the underlying lattice, then the convergence must be exponentially fast.
TL;DR: The Monte Carlo simulation technique is used to study the phase diagrams of a two-dimensional spin-1 Ising model with bilinear and biquadratic nearest-neighbor pair interactions and a single-ion potential to distinguish the most probable phase diagram from the several possible ones based on the Monte Carlo data.
Abstract: The Monte Carlo simulation technique is used to study the phase diagrams of a two-dimensional spin-1 Ising model with bilinear and biquadratic nearest-neighbor pair interactions and a single-ion potential. A staggered quadrupolar phase appears at low temperatures with the competing bilinear and biquadratic interactions. The phase boundary line of the staggered quadrupolar phase and that of the ferromagnetic phase are extremely close to each other at low temperatures. An argument is used to distinguish the most probable phase diagram from the several possible ones based on the Monte Carlo data.
TL;DR: A quantitative theory of the scaling properties of Julia sets is presented, using them as a case model for nontrivial fractal sets off the borderline of chaos and showing that generally the theory has a "macroscopic" part which consists of the generalized dimensions of the set, or its spectrum of scaling indexes.
Abstract: We present a quantitative theory of the scaling properties of Julia sets, using them as a case model for nontrivial fractal sets off the borderline of chaos. It is shown that generally the theory has a ``macroscopic'' part which consists of the generalized dimensions of the set, or its spectrum of scaling indexes, and a ``microscopic'' part which consists of scaling functions. These two facets are formally and computationally equivalent to thermodynamics and statistical mechanics in the theory of many-body systems. We construct scaling functions for the Julia sets and argue that basically there are two different approaches to this construction, which we term the Feigenbaum approach and the Ruelle-Bowen-Sinai approach. For the cases considered here the two approaches converge, meaning that we can map the theory onto Ising models with finite-range interactions. The largest eigenvalue of the appropriate transfer matrix furnishes the thermodynamic functions.
TL;DR: It is found that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of pararneters; it strikingly changes into a power law for antiperiodic boundary conditions and is suggested that this puzzling phenomenon is associated with topological excitations.
Abstract: Boundary conditions monitor the finite-size dependence of scaling functions for the Ising model. We study the low-temperature phase for the extremely anisotropic limit, or quantum version of the 2D classical Ising model, by means of combined exact results and large-size numerical calculations. The mass gap (inverse of correlation length) is the suitable order parameter for the finite system, and its finite-size behavior is studied as a function of variable boundary conditions. We find that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of pararneters; it strikingly changes into a power law for antiperiodic boundary conditions. We suggest that this puzzling phenomenon is associated with topological excitations.
TL;DR: In this article, the effect of the finite lattice size on physical quantities, like masses and coupling constants, was numerically investigated in the 4-dimensional Ising model, and the feasibility to obtain numerical information about low energy scattering from finite volume effects in a lattice Monte Carlo calculation was demonstrated.
TL;DR: In this paper, the temperature dependences of the z and x y components of sublattice magnetizations and also measured the both components of staggered susceptibilities were investigated. And the character of the transitions is orderings and divergences of z and y components at independent temperatures.
Abstract: Successive phase transitions of triangular lattice Heisenberg antiferromagnet with a small Ising anisotropy, CsNiCl 3 , are studied by neutron scattering We have reexamined the temperature dependences of the z and x y components of sublattice magnetizations and also measured the both components of staggered susceptibilities The character of the transitions is orderings and divergences of the z and x y components at independent temperatures
TL;DR: A new method to find numerically the density of states of discrete statistical systems using the zero-field, three-dimensional Ising model on a 53 lattice yields an excellent approximation to the partition function and can accurately predict its zeros near the critical points.