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Journal of Physics A Mathematical and General

High-temperature series expansion studies of mixed spin-1/2-spin-S Ising models

B Y Yousif, Roger Bowers

Published Dec 1, 1984• 42 citations
Publisher

Abstract

Mixed spin Ising models have less translations symmetry than their 'single spin' counterparts and are well adapted for the study of a certain type of ferromagnetism. In this articles the high-temperature series expansion work of Schofield and Bowers (1981)-who studied mixed spin- 1 / 2 -spin-1 Ising models-is generalised by replacing the spin-1 objects by arbitrary spin-S ones. The new series are used to investigate the spin dependence or independence of critical parameters in a novel setting. There is no evidence to suggest that the exponents studied are spin dependent.

DOI
10.1088/0305-4470/17/17/016

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Mixed spin Ising models have less translations symmetry than their 'single spin' counterparts and are well adapted for the study of a certain type of ferromagnetism. In this articles the high-temperature series expansion work of Schofield and Bowers (1981)-who studied mixed spin- 1 / 2 -spin-1 Ising models-is generalised by replacing the spin-1 objects by arbitrary spin-S ones. The new series are used to investigate the spin dependence or independence of critical parameters in a novel setting. There is no evidence to suggest that the exponents studied are spin dependent.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
Describes the development of high-temperature series expansions for general mixed spin-S-spin-S' Ising models on simple loose-packed lattices. It extends previous work on the simpler special case S'=1/2. Coefficients of the series for the initial susceptibility and specific heat of these models on the square, SC and BCC lattices are presented. A preliminary analysis of the results for the particular case S=1, S'=3/2 is given, and the critical ratio K c and the susceptibility exponent gamma are estimated.
The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.