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Journal of the Physical Society of Japan

One-Dimensional Ising Model with General Spin

Takeshi Obakata, Takehiko Oguchi

Published Aug 1, 1968• 53 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.25.322

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International Journal of Modern Physics B•Aug 18, 2011•5 citations
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.
Statistical mechanics of the Bethe lattice was developed. Properties of the Bethe lattice, i. e., (of the Bethe approximation) were summarized including low temperature susceptibilities of ferro- and antiferromagnets. The equivalence of the exact solution of the Bethe lattice to the Bethe approximation of the Ising model in the case of the antiferromagnetism was shown. The partition function of the Bethe lattice with first and second neighbor interactions was obtained and the critical temperature was calculated in the case where the first neighbor interaction is ferromagnetic.
In the Bethe lattice in which the Bethe approximation is exact, the Curie temperatures of the Ising model with spin S=12S=\frac{1}{2}S=21​, 1,…, 52\frac{5}{2}25​ are obtained exactly by generalizing the method of Katsura and Takizawa for the system with S=12S=\frac{1}{2}S=21​. This method is applied to the planar rotator model on the infinite Cayley tree, and the critical temperature and the susceptibility are rigorously calculated. It is pointed out that the susceptibility diverges below the critical temperature, on the other hand the spontaneous magnetization vanishes as the ground state is degenerated infinitely.
In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.
The square Ising lattice with unequal first- and second-neighbor interactions has been considered. Among the exact results discussed, we show that the lowest energy state can be ferromagnetic, antiferromagnetic, or superantiferromagnetic, and the transition temperature should vanish in some cases. It is also shown that this problem is a special case of a more general problem arising in the statistical consideration of the hydrogen-bonded crystals. A well-defined approximation procedure is then introduced to solve the latter problem and to derive the (approximate) critical condition and expressions for the thermodynamic functions. The critical temperature thus determined is exact for the regular Ising lattice and for the lattices with Tc=0{T}_{c}=0Tc​=0 while for the equivalent neighbor model the error is less than 2%. The specific heat possesses the usual logarithmic singularity in all cases.
Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.