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Journal of the Physical Society of Japan

One-Dimensional Ising System with Arbitrary Interactions. I. Derivation of the Dual Transformation and Application to Simple Examples

Mituru Tanimura, Norikiyo Ury

Published Oct 15, 1985• 0 citations
Publisher

Abstract

For the one-dimensional Ising model in which the interactions are usually confined in the limited range, an extension of the concept of exchange interaction has been made. In the case of unusual Ising models with arbitrary interactions between arbitrary spins some dual formula of the partition function is obtained in a closed form. In the present paper, part I, the derivation of its general formula and the application to some simple examples are shown.

DOI
10.1143/jpsj.54.3718

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For the one-dimensional Ising model in which the interactions are usually confined in the limited range, an extension of the concept of exchange interaction has been made. In the case of unusual Ising models with arbitrary interactions between arbitrary spins some dual formula of the partition function is obtained in a closed form. In the present paper, part I, the derivation of its general formula and the application to some simple examples are shown.
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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.
An exact result for the partition function for a general one-dimensional Ising chain of N spin-1/2 particles described by the Hamiltonian HN=−∑i=1NJiσiσi+1−∑i=1NHiσi is given. For an open strand, JN = 0; for a closed chain, σN+1 = σ1, JN ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When Ji and Hi are treated as random variables, the above Hamiltonian describes a one-dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.
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.
Abstract We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m . Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H . The 2D system, with size m × N / m , has the topology of a cylinder with helical BC. In the thermodynamic limit N / m → ∞ , m → ∞ , a 2D critical singularity develops on the self-duality line, sinh 2 K sinh 2 H = 1 .