We propose an exactly soluble Ising model with two kinds of exchange integrals on a square lattice. We assume that the number of bonds on which the exchange integrals take a value is even around each elementary plaquette; we name such a plaquette an even-bonds plaquette. We find two phase transition temperatures in the Ising model. We have a magnetic phase transition at one of them and a phase transition relating to the ordering of a dual spins introduced by a dual transformation of bond variables at the other. The latter phase transition is investigated in detail and turned out to have a relation with the confinement of odd-bonds plaquettes in a generalized system. Both phase transitions are of Onsager type.
We propose an exactly soluble Ising model with two kinds of exchange integrals on a square lattice. We assume that the number of bonds on which the exchange integrals take a value is even around each elementary plaquette; we name such a plaquette an even-bonds plaquette. We find two phase transition temperatures in the Ising model. We have a magnetic phase transition at one of them and a phase transition relating to the ordering of a dual spins introduced by a dual transformation of bond variables at the other. The latter phase transition is investigated in detail and turned out to have a relation with the confinement of odd-bonds plaquettes in a generalized system. Both phase transitions are of Onsager type.
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
We investigate a fully frustrated Ising model with four nearest-neighbor interaction constants on a square lattice. We obtain an exact form of the partition function by using a decimation transformation and a dual transformation. The partition function is then expressed as a product of two partition functions each of which is the one of the Ising model with a nearest-neighbor interaction constant on a square lattice. Special systems with two interaction constants are studied in detail and phase transitions for these systems are discussed.
The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.
It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 < n < d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.
The applicability of the n-replica method to an Ising system with positive and negative bonds of equal strength is investigated. It is shown that this method does reproduce the correct low-temperature and low-(negative bond) concentration series for the ground-state energy. It is essential, however, to take the low-temperature limit after the n to 0 limit. For a square lattice, the slope of the critical line near the pure-system transition point is found to be 2 square root 2/ln( square root 2+1). The onset of spin-glass order is conjectured to be associated with a negative bond concentration at which unfrustrated plaquettes no longer percolate the lattice. The estimated value of 16% is in agreement with Monte Carlo results.
Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.