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arXiv:cond-mat/9909361 · Physical review. B, Condensed matter

Cluster variation method and disorder varieties of two-dimensional Ising-like models

Alessandro Pelizzola

Published May 1, 2000• 9 citations
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Abstract

I show that the cluster variation method, long used as a powerful hierarchy of approximations for discrete (Ising-like) two-dimensional lattice models, yields exact results on the disorder varieties which appear when competitive interactions are put into these models. I consider, as an example, the plaquette approximation of the cluster variation method for the square lattice Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, and, after rederiving known results, report simple closed-form expressions for the pair and plaquette correlation functions.

DOI
10.1103/physrevb.61.11510

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I show that the cluster variation method, long used as a powerful hierarchy of approximations for discrete (Ising-like) two-dimensional lattice models, yields exact results on the disorder varieties which appear when competitive interactions are put into these models. I consider, as an example, the plaquette approximation of the cluster variation method for the square lattice Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, and, after rederiving known results, report simple closed-form expressions for the pair and plaquette correlation functions.
The phase diagram of the Ising square ferromagnet with nearest-neighbor, next-nearest-neighbor, and four-spin (plaquette) interactions is investigated using the cluster variation method in the square approximation. A complete analysis in temperature is performed for the ferromagnetic, superantiferromagnetic, and frustrated region of the ground state. Particular attention is devoted to the role of the four-spin interaction. The most remarkable effect is the occurrence, in several regions of the interaction parameter space, of first-order transitions and sometimes reentrance phenomena. Critical and tricritical points are almost analytically determined.
The spin-pair correlation functions in the Ising model with nearest-neighbor interaction J>0 and next-nearest-neighbor interaction J′ on the square lattice are obtained in three approximations of the simplified cluster variation method. We estimate the critical indices ην, γ and ν, by applying the coherent-anomaly method to the obtained spin-pair correlation functions. We conclude that the system belongs to the Onsager universality class when J′/J > −0.5 and does not when J′/J < −0.5.
This paper deals with 2d Ising model in the scope of cluster perturbation theory.Ising model is defined on a two-dimensional square lattice, the amount of nearest neighbors z=4.Lattice is divided into clusters of a given size and a complete set of energy eigenvalues and eigenvectors of the cluster is defined by the diagonalization method.On the basis of this, Hubbard operators are constructed and Green function is calculated, taking into account intercluster interactions according to perturbation theory, it allows us to obtain the dependence of the magnetization on the temperature in the Hubbard-I approximation.Obtained results are compared with the exact solution of the two-dimensional Ising model.
A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation--Pad\'eapproximant method. Particular attention is paid to the line of critical end points of the ferromagnetic-paramagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a different universality class. A rough estimate of the crossover exponent is also given.