Ising models on the lattice Sierpinski gasket

Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for pure frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for “pure” frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.     

Layered inhomogeneous Ising models with frustration on a square lattice

Generalizing a previous treatment we study inhomogeneous Ising models on a square lattice. In particular, we consider models with a layered structure, i.e. translational invariance in the horizontal direction. Otherwise, couplings are allowed to be of arbitrary strength and sign. The partition function and free energy are calculated excatly for a random coupling distribution of finite period. The phase transition is universally of Ising type. The critical temperature can be calculated from a remarkably simple formula and depends only on the mean couplings, averaged over the period. Generally ground state properties do not reflect the singular structure of the free energy at finite temperature. This is shown by a special example.Work performed within the research program of the Sonder-forschungsbereich 125 Aachen-Jülich-Köln     

Particles and scaling for lattice fields and Ising models

The conjectured inequality ⁽⁶⁾0 leads to the existence of _d ⁴ fields and the scaling (continuum) limit ford-dimensional Ising models. Assuming ⁽⁶⁾0 and Lorentz covariance of this construction, we show that ford6 these _d ⁴ fields are free fields unless the field strength renormalizationZ ^–1 diverges. Let be the bare charge and the lattice spacing. Under the same assumptions (⁽⁶⁾0, Lorentz covariance andd6) we show that if ^4–d is bounded as 0, thenZ ^–1 is bounded and the limit field is free.Supported in part by the National Science Foundation under Grant MPS 74-13252Supported in part by the National Science Foundation under Grant MPS 75-21212     

Applications of the interface approach to some frustrated Ising models on a simple cubic lattice

Three 3-dimensional frustrated Ising models are studied by the interface method. We calculate the interface free energies by Monte Carlo simulation, and estimate the critical temperatures with a size-dependent analysis of the interface free energies.     

Study of Ising model on the rectangular-triangular lattice

The Ising model is studied on a new type of lattice which is named the rectangular-triangular lattice. The critical temperature for the ferromagnetic lattice is calculated exactly and it is shown that the antiferromagnetic lattice does not order at any temperature. Ground state properties are investigated and some features of frustration on the antiferromagnetic Ising lattice outlined.     

A spin-one Ising model on the Bethe lattice

We study the statistical mechanics of spin-one Ising models on the Bethe lattice assuming that the spins interact by dipolar and quadrupolar interactions. An exact calculation of the properties of the system is performed on the basis of the general formulation of Morita. An exact expression for the Curie temperature is derived and the results are found to be in agreement with those of Obokata and Oguchi who utilized a generalized Bethe approximation to a spin-one Ising system. The nature of variation of the Curie temperature with respect to the change of quadrupolar exchange is discussed for various coordination numbers and the results agree qualitatively with the earlier works. The temperature variation of both dipolar and quadrupolar moments is studied.     

Diagrammatic decompositions on lattice for correlation functions of Ising spin models by iterative method

In the present paper the iterative method enables us to calculate correlation functions of Ising spin models by two approximate ways. The equations resulting from two models A and B, based on different physical considerations concerning correlations in surroundings of the reference spin, are solved for one, two, and three-dimensional Ising models with nearest neighbour interactions. Except of anomalies occuring in low-dimensional systems the model A leads to the critical pointA _c=0.191 for the three-dimensional cubic Ising model.     

Ising models on the Lobachevsky plane

We consider the Ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane. For large values of the inverse temperature we construct an uncountable number of mutually singular Gibbs states.     

An iterative method of the solution of Ising lattice models in a magnetic field

A special re-writing of the sum of δ-functions gives a new graphical representation of thermodynamic quantities. The lowering of the number of points on the reference lattice spin in this representation leads to an iterative scheme by use of which equations for correlation functions and for the order parameter are obtained.     

Point defects in the honeycomb Ising lattice

A plane isotropic honeycomb Ising lattice is considered with randomly distributed defects, namely missing lattice spins (including the three adjacent bonds). The impurities are in thermodynamic equilibrium through a chemical potential. We find a rescaled temperature and a finite cusp-like specific heat at the critical point.     

Ising models on hyperbolic graphs

We consider Ising models on a hyperbolic graph which, loosely speaking, is a discretization of the hyperbolic planeH ² in the same sense asZ ^d is a discretization ofR ^d . We prove that the models exhibit multiple phase transitions. Analogous results for Potts models can be obtained in the same way.     

Magnetization of the Ising model on the generalized checkerboard lattice

We consider the Ising model on the generalized checkerboard lattice. Using a recent result by Baxter and Choy, we derive exact expressions for the magnetization of nodal spins at two values of the magnetic field,H=0 andH=i1/2kT. Our results are given in terms of Boltzmann weights of a unit cell of the checkerboard lattice without specifying its cell structures.     

Universality in the partially anisotropic three-dimensional Ising lattice

Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths \(\mathop J\limits^ \to = (J',J',J)\). The universality of both fundamental critical exponents y _t and y _h is confirmed. It is shown that the critical finite-size scaling amplitude ratios \(U = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chi\), and \(Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }\) are independent of the lattice anisotropy parameter Δ=J′/J. For the Y² invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y²≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).     

Theory of invariant transformations of the Ising lattice

Operators of creation and annihilation of atoms in the Ising lattice that conserve the partition function are considered. A change in the geometrical and topological characteristics of the lattice may be accompanied by the replacement of many-particle interactions in the lattice by two-particle interactions and vice versa. The construction of the operators makes essential use of the theory of characters.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 5–11, November, 1980.     

Three-spin correlation of the Ising model on the generalized checkerboard lattice

The Ising model on the generalized checkerboard lattice is studied and the three-spin correlation function is obtained for the three nodal spins surrounding a unit cell of the checkerboard lattice. As an application of this result, the spontaneous magnetization of the internal spin within a unit cell is calculated.     

Gauge-invariant lattice gas for the microcanonical Ising model

We introduce a lattice gas for particles with discrete momenta (1, 0, –1) and local deterministic microdynamics, which exactly reproduces Creutz's microcanonical algorithm for the ferromagnetic Ising model. However, because of the manifest gauge invariance of our variables, both the Ising ferromagnetic and spin-glass systems share precisely the same dynamics with different initial conditions. Additional conservation laws in the 1D Ising case result in a completely integrable system in the limit of zero or unbounded demon energy cutoff. Numerical investigations of ergodicity are presented for the pure Ising lattice gas in one and two dimensions.     

Susceptibility of the Kagomé lattice Ising model

We explicitly calculate the zero-field magnetic susceptibility of the anisotropic Kagomé lattice Ising model on two different varieties of the parameter space. One of them is the limitH=0 of the solubility condition, obtained in a previous paper by Giacomini, for the model with magnetic field. The other one is the disorder variety of the model, for which a dimensional reduction occurs. These varieties do not contain any nontrivial critical behavior of the model. A functional relation is also established, which relates the zero-field susceptibility for ferromagnetic and competing interactions.