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Condensed Matter Physics

TWO-PARTICLE CLUSTER APPROXIMATION FOR ISING TYPE MODEL WITH ARBITRARY VALUE OF SPIN. CORRELATION FUNCTIONS OF BLUME-EMERY-GRIFFITHS MODEL

Sorokov, Levitskii, O. R. Baran

Published Jan 1, 1997• 4 citations
Publisher

Abstract

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<

DOI
10.5488/cmp.9.57

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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<
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 method for calculation of the correlation functions of the Ising-type systems with short-range interaction and with arbitrary value of spin is developed within cluster approximation. For the Ising model (spin Sz=±1S^z=\pm1Sz=±1) the expressions for pair and ternary correlation functions within two-particle approximation in { q }\set{q}{q}-space are obtained for the hypercubic Bravais lattices. In the 1D case the exact expressions for them in the site space is obtained as well. On the basis of the Glauber equation within two-particle cluster approximation the longitudinal dynamical susceptibility χ({ q },E)χ(\set{q},E)χ({q},E) is found. In the 1D case and in the absence of external field the expression for χ({ q },E)χ(\set{q},E)χ({q},E) is exact. For the Emery-Blume-Griffiths model (Sz=−2,0,2S^z=-2,0,2Sz=−2,0,2) within two-particle approximation the pair correlation functions are calculated. The four-particle cluster approximation is used for calculation of static susceptibility χ({ q })χ(\set{q})χ({q}) of KD2PO4KD_2PO_4KD2​PO4​ ferroelectrics.
In this paper we discuss transformation properties of the spin-1 Blume-Emery-Griffiths (BEG) model arising in statistical physics. It is shown that the BEG model can always be transformed into either a spin-(1/2) Ising model or a 3-state Potts model. It is also shown that there exist other transformations relating the BEG model with spin-(1/2) king models, valid in special subspaces of the parameter space. Physical properties of the BEG model that can be drawn from the equivalence are discussed. We also present a formulation of the BEG model relating it to a 3-state vertex model.
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.
We develop a transfer matrix method to compute exactly the spin-spin correlation functions 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 of Bethe lattice spin models in the external magnetic field hhh and for any temperature T.T.T. We first compute 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 for the most general spin-SSS Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-SSS Ising model includes the spin-12\frac{1}{2}21​ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and TTT approaches the critical temperature Tc.{T}_{c}.Tc​. Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.