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Journal of Physics A Mathematical and General

The critical isotherm of the mixed spin Ising model

B Y Yousif, Roger Bowers

Published Oct 1, 1983• 1 citation
Publisher

Abstract

Series expansions are used to estimate the critical exponent delta for several two- and three-dimensional mixed spin Ising models. The evidence supports the conjecture of extended spin independence. This suggests that mixed models and the standard 'single spin' models share the same value of delta .
DOI
10.1088/0305-4470/16/14/027

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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.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
The high-temperature series expansions for the locus of the maximum susceptibility are calculated from the high-temperature series for the magnetic field for various two- and three-dimensional lattices of the spin- 1 / 2 Ising model. The authors examine their critical behaviour and determine the critical temperature nu e and the exponent Delta (= beta delta ) using the ratio method and the Pade approximant method. The values obtained for nu c are in good agreement with exact values and values obtained from standard series method and the Delta also agrees with beta delta obtained from other estimates. These results are another new supporting evidence for the universality hypothesis.