Showing papers on "Ising model published in 1972"
TL;DR: It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
1,664 citations
TL;DR: In this paper, the partition function of the zero-field eight-vertices model on a square M by N lattice is calculated exactly in the limit of M, N large.
1,648 citations
TL;DR: In this article, it was shown that no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomerdensity is minimal (i.e., x = 0).
Abstract: We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.
530 citations
Book•
01 Jan 1972
TL;DR: These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Abstract: While most introductions to statistical mechanics are either too mathematical or too physical, Colin Thompson's book combines mathematical rigor with familiar physical materials. Following introductory chapters on kinetic theory, thermodynamics, the Gibbs ensembles, and the thermodynamic limit, later chapters discuss the classical theories of phase transitions, the Ising model, algebraic methods and combinatorial methods for solving the two-dimensional model in zero field, and some applications of the Ising model to biology. Originally published in 1979. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
464 citations
TL;DR: In this article, a scaling theory for thermodynamic functions and spin correlations near the surface is developed, and relations among the exponents of the half-space are found among the sparsification terms.
Abstract: Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.
320 citations
TL;DR: In this paper, it was shown that at low temperature, the line of separation between the two pure phases shows large fluctuations in shape, which implies the translation invariance of the correlation functions associated with some non-translation invariant boundary conditions and should be a peculiarity of the dimensionality of the model.
Abstract: We prove that, at low temperature, the line of separation between the two pure phases shows large fluctuations in shape. This implies the translation invariance of the correlation functions associated with some non translation invariant boundary conditions and should be a peculiarity of the dimensionality of the model.
167 citations
TL;DR: In this paper, a general perturbation series is developed for a random walk with a repulsive interaction w delta ij between the ith and jth steps using the generating function for returns to the origin.
Abstract: A general perturbation series is developed for a random walk with a repulsive interaction w delta ij between the ith and jth steps using the generating function for returns to the origin. The interaction w is equal to -1+exp(- beta J), so that for a lattice model, beta to infinity (w to -1) corresponds to a self avoiding wall. Expansions are obtained in powers of w for cN(w) the total number of walks of N steps, uN(w) the number of walks of N steps terminating at the origin and (RN2(w)) the mean square length of a walk of N steps. The analytic behaviour of the corresponding generating functions is examined, and it is suggested that they are analogous to Ising model expansions about a singular point which is associated with a change of exponent. Hence it is conjectured using the 'smoothness postulate' of Griffiths (1970), that a change of exponent occurs at w=0 but for no other value of w. A distinction is drawn between ladder and non-ladder contributions, following Chikahisa (1970). Using a method of Brout (1961), it is found that all ladder diagrams can be summed, and the result conforms to the conjectured pattern of behaviour. However non-ladder contributions give rise to logarithmic terms as envisaged by Chikahisa. It is also concluded that the two-parameter theory which assumes that (RN2(w))/N is a function of wN1/2 is an approximation of limited validity.
144 citations
TL;DR: In this paper, a spin-phonon coupled system was used to explain the phase transitions associated with the orientational order of NH 4 + ions. But the model was based on a microscopic hamiltonian and the properties of correlated fluctuations of displacement of Br - ions were not analyzed.
Abstract: CsCl type NH 4 Br crystal has been treated as an Ising spin-phonon coupled system to explain the phase transitions associated with the orientational order of NH 4 + ions. The appearance of two ordered phases is explained as essentially due to the fact that the direct interaction between NH 4 + ions stabilizes parallel ordering while the indirect interaction through phonons stabilizes the antiparallel ordering. Based on a microscopic hamiltonian describing the spin-phonon coupled system, several experimental results have been analized such as (i) anomalous lattice expansions (ii) orientational order parameters (iii) spontaneous displacement of Br - ions (iv) P - T phase diagram (v) X-ray critical diffuse scattering. Semiquantitative agreements are obtained. Especially the property of correlated fluctuations of displacement of Br - ions seems to give a support on the validity of the microscopic model used.
143 citations
TL;DR: In this article, the authors used a cluster model to describe the excitations near to the defect and compared the predictions of the various theories with the experimental results, showing that the theory is fairly satisfactory for a low concentration of defects and low temperatures.
Abstract: The introduction of a low concentration of defects into a magnetic insulator modifies the spectrum of the magnetic excitations. In general the spectrum consists of a set of impurity modes associated with the defect and its immediate neighbors. Impurity modes that occur outside the band of host excitations are localized in the neighborhood of the defect and at the same time perturb the host band, while modes lying within the band lead to resonant behavior of the excitations of the host. In recent years, optical, neutron scattering, and nuclear magnetic resonance techniques have been used to study mixed crystals of antiferromagnetic transition metal fluorides. Many of the features may be understood by using the molecular field or Ising model for the excitations. An improvement on this form of the theory is to use a cluster model to describe the excitations near to the defect. Some features may however be described only when the excitations of the host are treated adequately; this requires the use of Green's function theories that have been developed for antiferromagnets containing defects. A detailed comparison is presented of the predictions of the various theories with the experimental results. Although the theory is fairly satisfactory for a low concentration of defects and low temperatures, considerable complexities arise in extending it to higher temperatures and large concentrations.
139 citations
TL;DR: In this paper, the possibility of a phase transition in an infinite simple random-cluster model (0 ≤ p ≤ 1, k ≥ 1), including the percolation model and the ferromagnetic Ising and Ashkin-Teller-Potts model, is studied by means of several criteria for large range connectivity.
119 citations
TL;DR: In this paper, the authors investigated the singularities observable at second-order phase transitions (e.g., the divergence of the specific heat) can only occur if the size of the considered system is very large (strictly speaking).
TL;DR: In this paper, the general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution function in the formalism of the cluster variation method.
Abstract: The general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution functions in the formalism of the cluster variation method. The parameters or the Lagrange multipliers occurring in the distribution functions must be determined by the reducibility condition of the distribution functions or by the stationariness condition of the free energy.
TL;DR: The first temperature-dependent term in a systematic low-temperature expansion of the free energy of the Heisenberg-Ising ring has been obtained in this paper, using the equations proposed by Gaudin.
Abstract: We use the equations proposed by Gaudin for the free energy of the Heisenberg-Ising ring for $|\ensuremath{\Delta}|\ensuremath{\ge}1$ to obtain the first temperature-dependent term in a systematic low-temperature expansion of the free energy.
TL;DR: Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v16, for the square lattice and for the honeycomb lattice, inclusive.
Abstract: Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v16, for the square lattice to v21 and for the honeycomb lattice to v32, inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
TL;DR: The scaling equation of state of an Ising-like ferromagnet is derived by an expansion in this article, where the scaling equation is expressed as a function of the dimension of space.
Abstract: The scaling equation of state of an Ising-like ferromagnet is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the dimension of space. The result is compared with numerical calculations on the three-dimensional Ising model. It is also established that the "linear model" is exact up to order ${\ensuremath{\epsilon}}^{2}$.
TL;DR: In this paper, it is shown how one may apply the Wilson theory to a liquid-vapour critical point and a fixed point, related to that of the Ising model and giving the same indices, is located.
TL;DR: In this paper, high temperature expansion techniques are used to study the changes in the critical behaviour of the susceptibility of the Ising ferromagnet arising from the introduction of nonmagnetic impurities.
Abstract: High temperature expansion techniques are used to study the changes in the critical behaviour of the susceptibility of the Ising ferromagnet arising from the introduction of nonmagnetic impurities.
TL;DR: The Ising model in 1 and 2 dimensions and zero field has been studied in this paper, where the separation ion and coexistence of phases has been investigated in a wider range of temperature.
Abstract: 133 1. Int roduct ion. 134 2. The model. Grand canonical and canonical ensembles. Their inequivalence. 136 3. Boundary conditions. Equil ibrium states. 138 4. The Ising model in 1 and 2 dimensions and zero field. 141 5. Phase transit ions. Definitions. 142 6. Geometric description of the spin configurations. 145 7. Phase t ransi t ion. Existence. 147 8. Microscopic description of the pure phases. 150 9. Results on phase t ransi t ions in a wider range of temperature . 153 10. Separat ion and coexistence of pure phases. Phcnomenological considerations. 155 11. Separat ion and coexistence of phases. Results. 156 12. Surface tension in two dimensions. Al ternat ive descriptions of the separat ion phenomena. 158 13. The s t ructure of the line of separation. Wha t a s t raight line really is. 159 14. Phase separat ion phenomena and boundary conditions. Fur the r results. 161 15. Conclusions and open problems. 163 API'E~I)IX: Transfer matr ix in the Ising model.
TL;DR: In this article, the authors studied the number of selfavoiding walks and self-avoiding returns to the origin on two and three dimensional lattices and studied numerically by the ratio method.
Abstract: New data for the number of selfavoiding walks and selfavoiding returns to the origin on two and three dimensional lattices are presented and studied numerically by the ratio method. Estimates for the critical attrition and critical indices are given. For a loose-packed lattice the selfavoiding walk generating function appears to have a singularity on the negative real axis. This singularity is at the same distance from the origin as the physical singularity, and is found to be cusp-like with an exponent of 1/2 in two dimensions and 3/4 in three dimensions. This behaviour enables a close analogy to be drawn between the behaviour of the Ising model high temperature susceptibility and the walk generating function.
TL;DR: In this paper, a numerical representation of the specific heat in the range TC
Abstract: For pt. I see abstr. A37614 of 1967. High temperature series expansions for the specific heat of the Ising model of a ferromagnet are given for the face-centred cubic, body-centred cubic, and simple cubic lattices. From a numerical study it is concluded that the critical index ( alpha ) is lattice independent and that in three dimensions alpha approximately=1/8. A numerical representation of the specific heat in the range TC
TL;DR: In this article, an exact solution for three-dimensional Ising models with four-spin interaction is obtained for all temperatures, where spontaneous magnetization is vanishing for all temperature derivatives.
Abstract: An exact solution is obtained for some "three-dimensional" Ising models with a four-spin interaction. The spontaneous magnetization is vanishing for all temperatures. The temperature derivative of the susceptibility shows a divergence of the form ${({T}_{c}\ensuremath{-}T)}^{\frac{\ensuremath{-}7}{8}}$ below the critical point. The specific heat shows a logarithmic singularity. Several other related models are introduced with new types of singularities.
TL;DR: In this article, the authors studied the infinite system equilibrium states in the statistical mechanics of classical lattice gases and showed that breakdown of the translation invariance occurs if and only if the derivative dP(Φ + λΨ) dλ is discontinuous at λ = 0 for some Ψ.
TL;DR: In this paper, it was shown that the self dual property for the case J2 not=0, J3 not = 0 also holds for J2 = 0 and J3 = 0, thereby yielding the likely result that the transition point for both of these cases is given by exp(2J2,3/kT)= square root 2-1.
Abstract: An Ising model is defined on a simple quadratic lattice which has the usual nearest neighbour pair potential terms (-J2 sigma k sigma l) and also triplet potential terms (-J3 sigma k sigma j sigma l) coupling the spins over the elementary triangular mesh cycles formed by connecting half of the next nearest neighbour sites, such that when J2=0 the model reduces to a triangular lattice with only the triplet potentials present. It is shown that the well known self dual property for the case J2 not=0, J3=0 also holds for the case J2=0, J3 not=0, thereby yielding the likely result that the transition point for both of these cases is given by exp(-2J2,3/kT)= square root 2-1.
TL;DR: In this article, it was shown that at low enough temperature all translationally invariant equilibrium states for the Ising ferromagnet are a superposition of only two extremal states, i.e., the positively and negatively magnetized pure phases.
Abstract: We prove that at low enough temperature all translationally invariant equilibrium states for the Ising ferromagnet are a superposition of only two extremal states, i.e., the positively and negatively magnetized pure phases. In particular this proves, at low temperature and in two dimensions, the identity of the spontaneous magnetization and the Onsager's value ${M}_{0}={[1\ensuremath{-}{(sh\ensuremath{\beta})}^{\ensuremath{-}4}]}^{\frac{1}{8}}$.
TL;DR: In this paper, high temperature series expansions of the susceptibility and second moment of the 1/2, 1, 2 and infinite spin Ising models on a triangular lattice are obtained and the spin dependence of the critical indices gamma and nu investigated.
Abstract: High temperature series expansions of the susceptibility and second moment of the 1/2, 1, 2 and infinite spin Ising models on a triangular lattice are obtained and the spin dependence of the critical indices gamma and nu investigated.
TL;DR: In this article, the thermodynamic properties of the antiferromagnetic Ising simple cubic and square lattices with nearest-neighbor and next-nearest neighbor interactions have been studied using a Monte Carlo technique.
Abstract: The thermodynamic properties of the antiferromagnetic Ising simple cubic and square lattices with nearest-neighbor and next-nearest-neighbor interactions have been studied using a Monte Carlo technique. Although both systems were found to possess magnetic tricritical points, their behavior in the "tricritical region" differed from that found in ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures.
TL;DR: In this paper, it was shown that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix.
Abstract: In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contrast this prediction with the exactly known results for the two-dimensional Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Fr\"obenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.
TL;DR: In this paper, it was shown that the Wilson theory applies to the true 3D ferromagnetic Ising model, predicting critical indices independent of lattice, spin and interactions provided the latter are short-range and ferrous.
TL;DR: In this article, it was shown that the transfer matrices for various Ising lattices in two dimensions commute with certain linear operators and the problem of finding an explicit form for the largest eigenvector is considerably simplified.
Abstract: It is shown that the transfer matrices for various Ising lattices in two dimensions commute with certain linear operators. The problem of finding an explicit form for the largest eigenvector is considerably simplified. The expansion coefficients appearing in the eigenvectors found as the solution of a set of nonlinear difference equations are Pfaffians. The connection between this type of solution and other solutions is clarified. This form for the eigenvector also simplifies the calculation of correlation functions. Some geometrical aspects of the Ising model are discussed.