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Journal of the Physical Society of Japan

Exactly Solved Two-Dimensional Ising Model with Spin S Greater than 1/2

Tohru Morita

Published Dec 15, 1993• 13 citations
Publisher

Abstract

The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.

DOI
10.1143/jpsj.62.4218

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The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.
We study a two-dimensional XXZ -Ising model on a square-hexagon (denoted for simplicity by 4-6) lattice with spin 1/2. The phase diagram at zero temperature is discussed, where five states are found, two types of ferrimagnetic states, two types of antiferromagnetic states, and one ferromagnetic state. To solve this model, we have mapped onto the eight-vertex model with union Jack interaction term, and it was verified that the model cannot be completely mapped onto eight-vertex model. However, by imposing an exact solution condition, we have found the region where the XXZ -Ising model on 4-6 lattice is exactly soluble with one free parameter, particularly for the case of symmetric eight-vertex model condition. In this manner we have explored the properties of the system and have analyzed the interacting competition parameters which preserve the region where there is an exact solution. Unfortunately the present model does not satisfy the free fermion condition of the eight-vertex model, unless for a trivial solution. Even so, we are able to discuss the critical point region, beyond the region of exact resolvability.
Although the two-dimensional spin- 1 / 2 Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin- 1 / 2 model, but appears to exhibit non-analytic corrections to scaling as expected for a phi 4 model and unlike the spin- 1 / 2 model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi 4 universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta 1 <1.3 is found. The validity of this result is enhanced by a comparison with the S= 1 / 2 case.
The effect of the uniaxial anisotropy in an Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice, is studied. The exact solution is determined by mapping the model onto an effective spin 1/2 Ising model on the triangular lattice. The transition temperature is obtained as a function of the parameters and it is shown that for s integer there is no long range order for the parameter values in a determined region which is independent of the value of s. The critical exponents are obtained and it is shown that they are the same as those of the two-dimensional spin 1/2 Ising model.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.
The one-dimensional Ising model is investigated by generalizing the Bethe approximation, which, in this case, gives exact solutions. The energy, specific heat and the zero field susceptibility for S =1, 3/2 and 2 except the susceptibility for S =2 are calculated exactly and compared with the results of Suzuki et al. . As a direct application of this method, the special lattice called the Bethe lattice is treated, and the Curie temperature of the lattice for S =1 is obtained. This method is also applied to solve the one-dimensional Ising model with the second neighbor interactions as well as the first neighbor ones, and the energy and specific heat for S =1/2 are calculated.