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Journal of Mathematical Physics

Infinite-Spin Ising Model in One Dimension

Colin J. Thompson

Published Feb 1, 1968• 36 citations
Publisher

Abstract

The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.

DOI
10.1063/1.1664574

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The partition function z, the pair correlation function ρ, and the zero-field susceptibility χ for the one-dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high- and low-temperature expansions are given for z, ρ, and χ. It is shown that the low-temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.
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.
We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.
In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.
Transfer matrix methods are used to obtain the partition function and correlation length of Ising chains at low temperatures. For the infinite-spin chain, asymptotically exact eigenvalues and eigenvectors are obtained from an iteration scheme, and from a secular equation method. It is also shown how the Ising chain for general spin S (including S to infinity ) may be treated using a 2*2 transfer matrix. For low temperatures (K identical to J/K B T>>1) the inverse correlation length, in units of the spin spacing, is xi -1 =4S e -2K +O(e -4K ) for finite S, and xi -1 =8K e -2K (ln 4K+ gamma +O((lnK)/K)) for infinite spin where gamma is Euler's constant.
The partition functions of ferromagnetic Ising models of square lattices in a finite magnetic field is deduced using topological considerations within a heuristic graph-theoretical approach. These equations are derived separately for low and high temperature regimes while the exact solution of Onsager is obtained therefrom when the magnetic field is zero. The derived partition function equations here are almost similar to those given by Onsager, thus indicating a straight-forward protocol, even when the magnetic field is present. The spontaneous magnetization derived here using the Helmholtz free energy is identical with that arising from the exact solution. The partition functions lead to the known series expansions of the magnetization and zero-field susceptibility.