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Journal of Applied Physics

General-temperature series expansion for Ising systems

Felix Lee, B. Westwański, Y. L. Wang

Published Mar 1, 1982• 5 citations
Publisher

Abstract

A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.

DOI
10.1063/1.330673

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A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
The linked-cluster series expansion is developed fur use on models of quantum spin systems with single-ion potentials. The linked-cluster series is defined as a statistical mechanical perturbation series in which the mean-field Hamiltonian and the single-ion potential are treated exactly with the perturbation representing the effects of fluctuation correlations. All terms of the free energy up to a given finite order are calculated, and estimates of thermodynamic functions are made based on extrapolation methods including the ratio method and Pade approximants.
We consider the generalized Ising model for a spin-1 system, i.e., the most general static Hamiltonian with pair interactions in a spin-1 subspace. This Hamiltonian has previously been used to model lattice-gas systems, ternary alloys, and normal liquid mixtures, as well as other threestate systems. As special cases one obtains the Blume-Emery-Griffiths model for 3He^{3}\mathrm{He}3He-4He^{4}\mathrm{He}4He mixtures and the Blume-Capel model for singlet-ground-state systems. Here we use exact hightemperature expansions in field, to the linear term in $\ensuremath{\beta}={({k}_{B}T)}^{\ensuremath{-}1}$ for the pair-correlation functions, to construct the generalized static susceptibility. The singularity in the susceptibility as the system is lowered in temperature yields an expression for the temperature at which the system becomes unstable with respect to fluctuations in the order parameters. This stability temperature is a function of the order parameters and the interactions. Our result is equivalent to the mean-field approximation (with its incumbent limitations) and yields a surface in thermodynamic space which separates regions of stability (or metastability) and instability. We look specifically at the stability surfaces and stability temperatures for a number of special cases as well as the general result.
Introduction Among pseudospin systems studied theoretically, a great attention is paid to Ising-type models, that is, to systems with the Hamiltonian containing only z-component of pseudospin. That is so due to relative simplicity with which approximate calculations for these models can be carried out and tested and possibility of their application to a wide class of real objects. Thus, the Blume-Emery-Griffiths model (BEG) corresponding to the Isingtype model with j = 2 (hereafter S z = \\Gammaj; \\Gammaj +2; : : : ; j \\Gamma 2; j) was proposed for the investigation of phase transitions (PT) in He 3 \\Gamma He 4 mixture [1]. A general form of the Ising-type model with j = 2 contains extra terms, as compared to the BEG model, like S z i (S z j<
Linked-cluster series have been obtained for the Blume-Capel model in an fcc lattice to the eighth order. We show that the linked-cluster series aided by the standard extrapolation techniques provide an effective method in the study of models which display first-order phase transitions as well as second-order phase transitions. The full phase diagram and the tricritical phenomena have been studied. Results are compared with those of the Monte Carlo simulation and the high-temperature and low-temperature series analysis.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.