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Journal of Mathematical Physics

One-Dimensional Ising Model with General Spin

Masuo Suzuki, Bunjiro Tsujiyama, Shigetoshi Katsura

Published Jan 1, 1967• 93 citations
Publisher

Abstract

The one-dimensional Ising model with general spin S has been formulated as an eigenvalue problem of order 2S + 1. Two methods to reduce the order to [S + 1] have been developed for calculating the energy and the susceptibility at zero external field. Exact solutions for S = 32 and S = 1 have been obtained. Numerical calculations of S = 32, 1, and ½ have been compared.

DOI
10.1063/1.1705089

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Exact eigenvalues of the Ising Hamiltonian in one-, two- and three-dimensions in the absence of a magnetic field

No author provided

The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies. (C) 2001 Elsevier Science B.V. All rights reserved.

•Sep 10, 2026•0 citations
The one-dimensional Ising model with general spin S has been formulated as an eigenvalue problem of order 2S + 1. Two methods to reduce the order to [S + 1] have been developed for calculating the energy and the susceptibility at zero external field. Exact solutions for S = 32 and S = 1 have been obtained. Numerical calculations of S = 32, 1, and ½ have been compared.
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.
The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.
As an illustration of a quasiclassical spin formalism developed in a recent paper a new method for solving the one-dimensional Ising model is presented. The method used is an extension to spin systems of the Wigner method familiar in computations involving position and momentum. The partition function, pair correlation function and the zero-field susceptibility for the one-dimensional Ising model with arbitrary spin are shown to be expressed in terms of the eigenfunctions and eigenvalues of a certain integral equation whose kernel approaches the classical kernel as the spin approaches infinity.
According to the traditional solutions, there is no magnetisation in the one-dimensional Ising model in the absence of external magnetic field. It is shown that there is a mistake within them. In them, local order of spins was not taken into account and the partition function and the correlation function were calculated wrongly. In this paper, that mistake is corrected and a new solution is presented. According to it, there is ferromagnetism in that model. Analogously, solutions of the two-dimensional and three-dimensional Ising models are obtained.
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.
The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies. (C) 2001 Elsevier Science B.V. All rights reserved.