Showing papers on "Ising model published in 1976"
TL;DR: In this paper, the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+ 1) -dimensional ising model at finite temperatures.
Abstract: The partition function of a quantal spin system is expressed by that of the Ising model, on the basis of the generalized Trotter formula. Thereby the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+ 1) -dimensional Ising model at finite temperatures. A general relationship is established between the two partition functions of a general quantal spin system and the corresponding Ising model with many-spin interactions, which yields some rigorous results on quantum systems. Some applications are given.
875 citations
TL;DR: In this article, the spin-spin correlation functions for the two-dimensional Ising model on a square lattice in zero magnetic field for T>Tc and T
Abstract: We compute exactly the spin-spin correlation functions 〈σ0,0σM,N〉 for the two-dimensional Ising model on a square lattice in zero magnetic field for T>Tc and T
683 citations
TL;DR: In this article, it was shown that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions.
Abstract: We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).
484 citations
TL;DR: In this paper, the present situation in theoretical and experimental studies on one-dimensional magnetic systems is fully discussed, including equal-time and dynamic properties with an emphasis on the latter.
Abstract: The present situation in theoretical and experimental studies on one-dimensional magnetic systems is fully discussed. Equal-time as well as the dynamic properties are included with an emphasis on the latter. Four model systems are examined in detail: TMMC (Heisenberg antiferromagnet), CsNiF3 (planar ferromagnet), CoCl2. 2NC5H5 (Ising ferromagnet), CuCl2. 2NC5H5 (Heisenberg antiferromagnet with S = ½). The equal-time properties are quite well understood in theory and in experiment but the dynamical properties much less so. The open questions and possible investigations for the future are discussed.
477 citations
TL;DR: In this paper, a renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems, and applied to the two-dimensional Ising model.
Abstract: A renormalization group transformation is introduced with the help of which critical properties of infinite systems can be related to finite systems. As a numerical example the method is applied to the two-dimensional Ising model. The critical point and critical point exponent are computed in addition to the amplitude of the logarithmic singularity in the specific heat.
370 citations
TL;DR: In this paper, it was shown that the critical exponents of a phase transition in a $d$-dimensional system with short-range exchange and a random quenched field are the same as those of a ($d\ensuremath{-}2$)-dimensional pure system.
Abstract: We prove that to all orders in perturbation expansion, the critical exponents of a phase transition in a $d$-dimensional ($4ldl6$) system with short-range exchange and a random quenched field are the same as those of a ($d\ensuremath{-}2$)-dimensional pure system. Heuristic arguments are given to discuss both this result and the random-field Ising model for $2ldl6$.
280 citations
TL;DR: In this article, an importance sampling Monte Carlo method is used to study finite lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions, and the internal energy, specific heat, order parameter, susceptibility, and near-nighbor spin-spin correlation functions are determined as a function of $N$ and extrapolated to the corresponding infinite system values.
Abstract: An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of $N$ and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.
270 citations
TL;DR: In this article, the behavior of the interface in 3D Ising and related lattice systems is modeled using a 2D array of columns of varying heights, and it is shown that the "roughening" transition (the transition from a localized to a delocalized interface) is directly related to the metal-insulator transition in a 2-D Coulomb gas.
Abstract: The behavior of the interface in the three-dimensional (3-D) Ising and related lattice systems is modeled using a 2-D array of columns of varying heights We show that the "roughening" transition (the transition from a localized to a delocalized interface) is directly related to the metal-insulator transition in a 2-D Coulomb gas Implications of this relationship are discussed
254 citations
TL;DR: The coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an exp(-As^4+Bs^2) spin-weight factor are expanded in the dimensionless, renormalized coupling constant.
Abstract: The coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an exp(-As^4+Bs^2) spin-weight factor are expanded in the dimensionless, renormalized coupling constant. These series are summed by the Pade-Borel method to yield the critical indices γ=1.241±0.002, η=0.02±0.02, ν=0.63±0.01, and Δ1=0.49±0.01.
212 citations
208 citations
TL;DR: In this paper, a lower bound on the free energy of the Ising model and the Wilson-Fisher model is given for both models, and a simple lower bound which preserves the symmetry of the Hamiltonian (the one hypercube approximation) is described.
Abstract: Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by ɛ expansion in agreement with the results of Wilson and Fisher to leading order in ɛ. The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.
TL;DR: Theoretical information from series and ϵ-exponential expansions on the values of these ratios for short-ranged Ising, Heisenberg, and spherical models, and for dipolar systems is presented in this paper.
Abstract: The hypothesis of universality implies that there are four universal ratios among the six usually defined thermodynamic critical amplitudes. Theoretical information from series and $\ensuremath{\epsilon}$ expansions is presented on the values of these ratios for short-ranged Ising, Heisenberg, and spherical models, and for dipolar systems. A number of real materials are discussed (Xe, ${\mathrm{CO}}_{2}$, Ni, EuO, and ${\mathrm{LiTbF}}_{4}$), and the present state of our understanding of the thermodynamic ratios for these systems is found to be rather crude.
TL;DR: In this article, it was shown that scaling assumptions for the cluster concentration p (l, s ) imply that the critical behavior cannot be attributed to fully "ramified" clusters as suggested by Domb.
CERN1
TL;DR: In this article, the authors studied the asymptotic behavior of the scattering amplitude when the pomeron has intercept α (0) larger than one and showed that the spectrum of such a system shows a degenerate ground state for α(0) > α c >~ 1 and a continuum with vanishing excitation gap at α ( 0) = α c.
TL;DR: In this paper, a convergent expansion for nearly Gaussian quantum field theory in the multiphase region is given, which combines an expansion in phase boundaries, a cluster expansion, and a perturbation expansion to isolate dominant behavior.
TL;DR: In this paper, the Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin-1/2 Ising model and the technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to spin-glass type ground state at T = 0 for a two-dimensional square lattice.
Abstract: The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin-1/2 Ising model The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0 For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random Jij matrix whereas the latter has at least some localized eigenvectors
TL;DR: In this paper, optical measurements of the equations of state of Xe, S${\mathrm{F}}_{6}$, and C${O}}_{2} were reported very near their critical points.
Abstract: We report precise optical measurements of the equations of state of Xe, S${\mathrm{F}}_{6}$, and C${\mathrm{O}}_{2}$ very near their critical points ($\frac{|T\ensuremath{-}{T}_{c}|}{{T}_{c}}l5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$) We find that the critical exponents of these fluids in this region are close to the exponents calculated from the three-dimensional Ising model
TL;DR: In this paper, the authors give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins.
Abstract: We give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins. The picture is more complete in the two-dimensional Ising model, where we can also use a generalized version of a result by Miyamoto.
TL;DR: In this article, a unified derivation of Griffiths, Ginibre, Percus, Lebowitz, and Ellis and Monroe correlation inequalities for Ising ferromagnets with continuous spins is given.
Abstract: We investigate correlation inequalities for Ising ferromagnets with continuous spins, giving a simple unified derivation of inequalities of Griffiths, Ginibre, Percus, Lebowitz, and Ellis and Monroe. The single-spin measure and Hamiltonian for which an inequality may be proved become more restricted as the inequality becomes more complex. However, all results hold for a model with ferromagnetic pair interactions, positive (nonuniform) external field, and single-spin measureν eitherv(σ) = [1/(l + 1)] xΣ
f=0/l
δ(−l +2j +σ) (spinl/2) ordv(σ) = exp [−P(σ)]dσ, whereP is an even polynomial all of whose coefficients must be positive except the quadratic, which is arbitrary. The Lebowitz correlation inequality is a corollary of the Ellis-Monroe inequality. As an application, we generalize the method of van Beijeren to establish a sharp phase interface at low temperature in nearest neighbor ferromagnets of at least three dimensions with arbitrary (symmetric) single-spin measure.
TL;DR: In this paper, the interface profile of the two-dimensional Ising ferromagnet is obtained for all temperatures in the thermodynamic limit, and the width of the interface depends on its length as (length) 1/2.
Abstract: The interface profile of the two-dimensional Ising ferromagnet is obtained for all temperatures in the thermodynamic limit. The width of the interface depends on its length as (length)1/2.
TL;DR: In this article, the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory was proved for spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models.
Abstract: We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models. The proofs are based on the properties of a classG
− of probability measures which contains all measures of the form const exp(−V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ∞). A new proof of the GKS inequalities using similar ideas is also given.
TL;DR: In this article, an Ising model with an arbitrary distribution of exchange interactions is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature.
Abstract: An Ising model with an arbitrary distribution $P(J)$ of exchange interactions $J$ is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature. The solution is expressed in terms of an Ising model on the same lattice with a single exchange parameter, allowing an exact solution to be obtained in one and two dimensions. Some feeling for the effect of annealing can be found by examining correlations between exchange interactions on neighboring bonds. Some special distributions $P(J)$ are examined in detail.
TL;DR: In this article, the Bethe lattice was applied to the site and the bond problems in the quenched Ising spin system and the free energy and the susceptibility of the site was obtained.
Abstract: (1974), 120] to obtain the specific heat and the susceptibility of the random mixture of magnets, is applied for the low·field expansion of the free energy and the magnetization. The quartic terms of the free energies of the linear chain and of the infinite Bethe lattice for the site and the bond problems are obtained. The exact solution of the infinite Bethe lattice is equivalent to the Bethe approximation of the ordinary lattices. A divergence of the quartic term of the free energy of the bond problem is discussed in connection with a phase transition relating to the glass-like phase. Transparent formal similarity (which serves as a check and an outlook) between the site and the bond problems is found, and a relation (which serves as an approximation) between the quenched and the annealed systems is discussed. § 1. Introduction and conclusion In a previous paperv a method using projection operators to obtain thermo dynamic and magnetic properties of the random mixture of the magnets (the site and the bond problems in the quenched Ising spin systems) was presented. The method was applied to the linear chain and to the infinite Bethe lattice (of which the exact solution is equivalent to the Bethe approximation) giving the free energy and the susceptibility at zero field. A remarkable distinction in the phase diagrams between the site and the bond problems was clarified. The method was also applied to the quenched classical Heisenberg model.") For the free energy and the magnetization at a finite magnetic field, a concentration expansion was carried out and anomalous behavior in the magnetization process of the dilute linear chain at low temperatures was explained.') In this paper the method of Ref. 1) is applied to the low-field expansion of the free energy and the magnetization of the site and the bond problems for the linear chain and for the infinite Bethe lattice, and the quartic terms with respect to the magnetic field is obtained. The divergence of the second derivative of the susceptibility (a 2x/(JH2 ) of the bond problem characterizes the appearance of the glass-like phase introduced by Matsubara and Sakata.4l Transparent formal simi larity between the site and the bond problems is found and it serves as a good check. An approximation for the bond model and a relation to the annealed system
TL;DR: In this article, the ionic conductivity for a solid in which the ions are undergoing an order-disorder phase transition is calculated for a lattice gas with a weak hopping term.
Abstract: The ionic conductivity is calculated for a solid in which the ions are undergoing an order-disorder phase transition. The interacting ion system is described by a lattice gas with a weak hopping term. Exact results are obtained in one and two dimensions by using the exact results known for the Ising model. The calculated theoretical conductivity agrees very well with the temperature dependence found experimentally.
TL;DR: In this article, the authors proved analyticity of the correlation functions for continuous-spin lattice systems, including continuous spin systems, at high temperatures and in strong external fields.
Abstract: We prove analyticity of the correlation functions for classical lattice systems, including "continuous-spin" systems, at high temperatures and in strong external fields. For systems whose configuration spaces are homogeneous spaces for compact groups (e.g. Ising, plane rotator and classical Heisenberg models), improved estimates on the region of analyticity are obtained by generalizing an integral equation of Gruber and Merlini. Ex- ponential cluster properties are also obtained for such systems with a finite- range interaction.