Showing papers on "Ising model published in 1973"
TL;DR: In this paper, the equivalence between the zero-field eight-vertex model and an Ising model with four-spin interaction was established, where each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L).
337 citations
TL;DR: In this article, a brief review of the spin-iIsing model in a transverse field provides a useful description (insulating magnetic systems, order-disorder ferroelectrics, cooperative Jahn-Teller systems and other systems with 'pseudo-spin'- phonon interactions).
Abstract: A brief review is first made of systems for which the spin-iIsing model in a transverse field provides a useful description (insulating magnetic systems, order-disorder ferroelectrics, cooperative Jahn-Teller systems and other systems with 'pseudo-spin'- phonon interactions). A perturbation expansion is then developed which provides for all temperatures an approximate description of the model. The perturbation series is classified with respect to the small parameter l/z, where z is the number of spins interacting with a given spin; this generates the molecular-field approximation and the random-phase approxi- mation as the lowest-order description of the thermodynamic functions and correlation functions respectively. The leading-order approximations for the correlation functions, susceptibilities and spectral functions are discussed in detail. The formalism, which is the first application of such techniques to systems with varying magnetization direction, provides a basis for subsequent higher-order calculations of the free energy and correlation functions.
250 citations
TL;DR: In this article, the Ising model on a triangular lattice with three-spin interactions is solved exactly by solving an equivalent coloring problem using the Bethe Ansatz method, which is given in terms of a simple algebraic relation.
Abstract: The Ising model on a triangular lattice with three-spin interactions is solved exactly. The solution, which is obtained by solving an equivalent coloring problem using the Bethe Ansatz method, is given in terms of a simple algebraic relation. The specific heat is found to diverge with indices $\ensuremath{\alpha}={\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{2}{3}$.
237 citations
TL;DR: In this paper, the Monte Carlo sampling technique is used to calculate the equilibrium thermodynamics of fluids and magnets, and the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model.
Abstract: By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.
214 citations
207 citations
TL;DR: In this article, an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions was used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (φ4)2 theory.
Abstract: We approximate a (spatially cutoff) (φ4)2 Euclidean field theory by an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions. This approximation is used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (φ4)2 theory. Application of these results are presented.
203 citations
181 citations
TL;DR: In this paper, a spin-1 Hamiltonian with arbitrary bilinear and biquadratic pair interactions has been studied in the molecular-field approximation by using one-and two-sublattice models.
Abstract: A spin-1 Hamiltonian with arbitrary bilinear and biquadratic pair interactions has been studied in the molecular-field approximation by using one- and two-sublattice models. Various types of orderings and transitions are found for the Hamiltonians with different symmetries. For a system described by a one-sublattice Hamiltonian with Ising, isotropic, or cubic symmetry, only one phase transition is found; either from the paramagnetic phase to a ferromagnetic or to a ferroquadrupolar phase. (Parallel alignment of quadrupoles is denoted ferroquadrupolar.) For the one-sublattice Hamiltonian with axial symmetry, we find two separate phase transitions; first from the paramagnetic phase to a ferroquadrupolar phase and then to a ferromagnetic phase. A two-sublattice system described by an Ising or isotropic Hamiltonian can have a transition either from the paramagnetic phase to a ferromagnetic phase, or to a ferriquadrupolar phase. Additional transitions are also found from the ferromagnetic to a ferrimagnetic phase and then back to the ferromagnetic phase. The system can have as many as three successive transitions. For a two-sublattice Hamiltonian with cubic symmetry, besides the transitions found in Ising systems, there are "reorientation" phase transitions; i.e., transitions from a quadrupole ordering along the cube edge to an ordering along the cube diagonal. This system can have more than three successive phase transitions.
180 citations
TL;DR: In this article, the authors studied the interface between regions of opposite spin in the simple cubic Ising model and found that the interface width diverges at a temperature about half the critical temperature.
Abstract: We have studied the interface between regions of opposite spin in the simple cubic Ising model. Low-temperature expansions of moments of the gradient of the density profile and of the slope at its midpoint suggest that the interface width diverges at a temperature ${T}_{R}$ about half the critical temperature. We discuss the physical significance of this transition and implications for general theories of interfacial properties.
159 citations
TL;DR: In this paper, the authors apply the Kikuchi approximation to the order-disorder transformations in an fcc binary alloy that is assumed to conform to a generalized Ising model, taking formally into account all interactions between 2, 3 and 4 nearest neighbours.
TL;DR: In this paper, the correlation function of the two-dimensional Ising model was studied near the T √ c bound of the Ising correlation function, where c is the number of vertices in the graph.
Abstract: We study near ${T}_{c}$ the correlation function $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{MN}}〉$ of the two-dimensional Ising model.
TL;DR: In this article, the Ising of spin 1/2 with the nearest-neighbor interaction, in which nonmagnetic impurity atoms are dissolved, is investigated based on the identity =, decoupling approximation is introduced for treating the many-spin correlation functions.
Abstract: The Ising of spin 1/2 with the nearest-neighbor interaction, in which nonmagnetic impurity atoms are dissolved, is investigated. Based on the identity = , decoupling approximation is introduced for treating the many-spin correlation functions. The dependences of the critical temperature and of the spontaneous magnetization on the impurity concentration are calculated for two- and three-dimensional lattices. The results for the critical temperature and the critical concentration are better than the results of the results of the extended molecular-field approximations of Mamada and Takano, or of Oguchi and Obokata. Especially for the three-dimensional lattices the critical obtained agrees with that of Sykes and Essam, estimated from the series expansion.
TL;DR: Barber's scaling theory for surface properties is reformulated in terms of a gap exponent Δ 1 for a surface field H 1 as mentioned in this paper for a range of exactly soluble models Δ 1 equals 12; this is probably always a good approximation.
Abstract: The theory of critical phenomena in films (and general systems of restricted geometry) is reviewed, introducing the critical point shift exponent λ and the rounding or crossover exponent θ=1/ν, which describes the changeover from bulk behavior. The exponents α×, β×, and γ× for the surface corrections to bulk behavior are defined and discussed. The deficiencies of the “extrapolation” length concept, for representing the surface boundary conditions on the order parameter, are explained. Barber's recent scaling theory for surface properties is reformulated in terms of a gap exponent Δ1 for a surface field H1. In a range of exactly soluble models Δ1 equals 12; this is probably always a good approximation. The scaling relation β1=2−α−ν−Δ1 then predicts the critical behavior of the surface order. Comparisons of the theory are made with analytical and numerical work on Ising, Heisenberg, and spherical models and on ideal Bose fluid films. A table of exponents is presented.
TL;DR: In this article, a homogeneity assumption for the free energy, involving a new exponent to scale the surface field, is introduced, and exponent relations are derived for the surface exponents for the two-dimensional Ising model, the spherical model, and mean field theory.
Abstract: The critical behavior of the surface properties of magnets is discussed A homogeneity assumption for the free energy, involving a new exponent ${\ensuremath{\varphi}}_{1}$ to scale the surface field, is introduced New exponent relations are derived for the surface exponents These are satisfied for the two-dimensional Ising model, the spherical model, and mean-field theory The existing estimates for the three-dimensional Ising model, however, appear to be possibly inconsistent
TL;DR: For the Ising model with nearest neighbor interaction, it was shown in this article that the spin correlations decrease exponentially asd(A, B) → ∞ in a pure phase when the temperature is well below Tc.
Abstract: For the Ising model with nearest neighbour interaction it is shown that the spin correlations 〈σAσB〉 -〈σA〉〈σB〉decrease exponentially asd(A, B) → ∞ in a pure phase when the temperature is well belowTc. This is used to prove that the free energyF(β,h) is infinitely differentiable in β and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(β,h).
TL;DR: In this paper, a transformation in classical lattice statistics which generalizes the weak-graph theorem and includes the duality transformation is described, where the energy of the configurations is expressed by a function of "quantum numbers" which are subject to certain constraints.
TL;DR: In this article, the derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two-dimensional Ising model of a ferromagnetic and antiferromagnet was studied as a field grouping.
Abstract: The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two‐dimensional Ising model of a ferromagnet and antiferromagnet is studied as a field grouping. New results are given for the high field polynomials for the triangular lattice to order 10, the simple quadratic lattice to order 15, and the honeycomb lattice to order 21.
TL;DR: A functional form for the spontaneous order M0 of the eight-vertex model is conjectured from an investigation of the structure of a low-temperature expansion of M0.
Abstract: A functional form for the spontaneous order M0 of the eight-vertex model is conjectured from an investigation of the structure of a low-temperature expansion of M0 The precise expression then follows from the known result for the Ising case The exponent beta is then obtained and three-exponent scaling used to predict the remaining exponents of the eight-vertex model from the known exponents
TL;DR: In this article, the problem of diagonalizing the transfer matrix for the two dimensional Ising model with all boundary spins equal to + 1 is solved by use of the spinor method, and a simple proof that spontaneous magnetization is actually given by the well known formula for the long range order with torodial boundary conditions, and this means that the critical temperature is precisely that temperature above which the state is unique and below which it is non unique.
Abstract: The problem of diagonalizing the transfer matrix for the two dimensional Ising model with all boundary spins equal to +1 is solved by use of the spinor method. This provides a simple proof that the spontaneous magnetization is actually given by the well known formula for the long range order with torodial boundary conditions, and this means that the critical temperature is precisely that temperature above which the state is unique and below which it is non unique. An expression for the magnetization at finite distance from the boundary is also given, and a simple derivation of the formula for the surface tension between two coexisting phases is presented. Finally the relation between the degeneracy of the spectrum and the phase transition is discussed.
TL;DR: In this paper, the derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two-dimensional Ising model of a ferromagnet and antiferromagnetic magnet was studied as a temperature grouping.
Abstract: The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two‐dimensional Ising model of a ferromagnet and antiferromagnet is studied as a temperature grouping. New results are given for the ferromagnetic polynomials for the triangular lattice to order 16, for the ferromagnetic and antiferromagnetic polynomials for the simple quadratic lattice to order 11, and for the honeycomb lattice to order 16.
TL;DR: In this paper, a leading-order approximation for an additive physical quantity in the melting region is introduced; it is intimately connected with the microscopic statistical herteropolymer structure, which allows one to derive the heteropolymer distribution function (over the length and composition of the sections) directly from experimental data.
Abstract: Heteropolymer melting corresponds to a low-temperature phase transition in a one-dimensional two-component Ising model A leading-order approximation for an additive physical quantity in the melting region is introduced; it is intimately connected with the microscopic statistical herteropolymer structure This allows one to derive the heteropolymer distribution function (over the length and composition of the sections) directly from experimental data
TL;DR: In this paper, one-dimensional dilute systems with nearest neighbor interactions were studied in terms of the power series of the concentration of magnetic atoms, and the specific heats, the susceptibilities and the magnetizations of the ferro magnets and antiferromagnets were obtained.
Abstract: One-dimensional dilute systems with nearest neighbor interactions are studied in terms of power series of the concentration of magnetic atoms. Recurrence relations between coefficients of the power series of the Ising systems are obtained. The Ising systems of S = 1/2 are studied. The specific heats, the susceptibilities, and the magnetizations of the ferro magnets and antiferromagnets are obtained. At low temperatures, three steps in the magnetization of antiferromagnetic systems are found. An Ising system of S = 1/2 with nearest and second nearest neighbor interactions is also studied by extending the method, and the specific heats are obtained when the magnetic field is zero.
TL;DR: In this article, the enumeration problem that arises in the derivation of low-temperature and high-field expansions for the Ising model of a ferromagnet and antiferromagnetic magnet.
Abstract: The enumeration problem that arises in the derivation of low‐temperature and high‐field expansions for the Ising model of a ferromagnet and antiferromagnet is studied. The method of partial generating functions (complete codes) is developed and a principle of complete code balance is explicitly stated. The detailed application of the method to a number of lattices is described and substitutions given that interpret the generating functions of certain lattices on the corresponding shadow lattice. It is shown that in zero‐field and two dimensions some of these substitutions reduce to the well‐known star triangle and magnetic‐moment results.
TL;DR: In this paper, a semi-infinite two-dimensional Ising model with its spins on the boundary row having a different interaction energy E′1 from the ferromagnetic bulk was considered, and it was shown that the boundary specific heat has two divergent terms: one of which diverges linearly at the bulk critical temperature Tc, and the other, logaritmically.
Abstract: We consider a semi‐infinite two‐dimensional Ising model with its spins on the boundary row having a different interaction energy E′1 from the ferromagnetic bulk. We find that the boundary specific heat has two divergent terms: one of which diverges linearly at the bulk critical temperature Tc, and the other, logaritmically. The linearly divergent term is independent of E′1, and the coefficient of the logaritmically divergent term is a decreasing function of E′1. There is a boundary latent heat at Tc, which is identical to McCoy and Wu's result. The boundary spins, which can be either ferromagnetic or antiferromagnetic, are aligned for temperature lower than Tc. The boundary spontaneous magnetization approaches zero in the form of A(E′1)(1−T/Tc)1/2, and the boundary zero field magnetic susceptibility diverges at Tc in the form −B (E′1)ln|1−T/Tc|, where A(E′1) and B(E′) are increasing functions of E′1.
TL;DR: In this paper, a theory for spin waves in a dilute antiferromagnet has been developed which uses the coherent potential approximation (CPA) to treat the randomly varying Ising interactions between nearest Antiferromagnetic neighbours but treats the off-diagonal parts of the Heisenberg interactions in a more approximate fashion.
Abstract: For pt. I see abstr. A78044 of 1972. A theory for spin waves in a dilute antiferromagnet has been developed which uses the coherent potential approximation (CPA) to treat the randomly varying Ising interactions between nearest antiferromagnet neighbours but treats the off-diagonal parts of the Heisenberg interactions in a more approximate fashion that is consistent with the Goldstone theorem. Although similar to the theory that was found to be successful for K(Co, Mn)F3 and (Co, Mn)F2 in paper I, the problem of the magnetic vacancy requires special treatment. A fictitious spin is placed on the Zn atoms but large potentials are introduced there to prevent the spin waves from propagating via these sites. The Green functions, neutron scattering and Raman scattering have been calculated.