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physica status solidi (b)

On the Critical Point Behaviour of a Model of a 3‐d Ising Ferromagnet

A. K. Gupta

Published Jan 1, 1970• 0 citations
Publisher

Abstract

Abstract An analysis is made of the critical‐point behaviour of a 3‐d Ising ferromagnet using a magnetic equation of state based on an expression for the order‐parameter proposed by Frankel. The asymptotic distribution‐function of zeroes for the partition function on a unit‐circle in a complex‐fugacity plane is determined. The critical indices γ and α are found to be 1, and 0 which are inconsistent with high‐temperature series expansion results.

DOI
10.1002/pssb.19700380234

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Abstract An analysis is made of the critical‐point behaviour of a 3‐d Ising ferromagnet using a magnetic equation of state based on an expression for the order‐parameter proposed by Frankel. The asymptotic distribution‐function of zeroes for the partition function on a unit‐circle in a complex‐fugacity plane is determined. The critical indices γ and α are found to be 1, and 0 which are inconsistent with high‐temperature series expansion results.
The thermodynamic quantities of the Ising model are analyzed near Tc, using the asymptotic distribution function g(θ, t) of zeros for the partition function on a unit circle in the plane of the complex magnetic field. The distribution function is derived as g(θ, t) = tλf(θ/tλ + γ), assuming that the susceptibility behaves like χ0+ ∼ t-γ near Tc where t = T/Tc-1. It is shown that there are two types of the situation; type I, in which a critical angle θc(t) exists such that g(θ, t) = 0 for 0 ≦ |θ| ≦ θc, and type II, in which there is no critical angle. The critical indices are related to one another, using the above g(θ, t). The relation between the magnetic equation of state M = M(h, t) and the distribution function is given as follows: g(θ, t) = (1/2π)Re M(iθ, t), where h = 2mH/kT. g(θ, t) for an exactly soluble model near Tc is given by the equation θ2 ∼ t3 + a1t2g2 + a2tg4 + a3g6 (ai > 0). This is a good example for the general theory.
The theorem of Lee and Yang has been extended to the ferromagnetic Ising model with arbitrarily mixed spin values of Sj = ½, 1, and 32, including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic-field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling-law approach.
The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian [MATH] where the σ's are Pauli matrices defined at the sites i and j of a regular lattice. This model is appropriate for magnetic insulators for which g⊥ >> g|| temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and s.c. lattices and a susceptibility index γ ≈ 4/3. The fourth order fluctuation series yields the estimate Δ ≈ 5/3 from which by the scaling hypothesis all other indices are obtained. A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.
New series expansion data for the three-dimensional Ising model are analysed. A special study of the diamond and face-centred cubic lattices is made. Below the critical temperature convergence is found to be slow. It is concluded that all available data are consistent with beta = lambda 5/16 (magnetization) and not inconsistent with the 'scaling' value gamma '=5/4 (susceptibility) and that with the data and methods available at present it is not possible to draw more precise conclusions.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.