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arXiv:1905.07586 · physica status solidi (b)

Pair Correlation Functions of the Ising Type Model with Spin 1 within Two-Particle Cluster Approximation

O. R. Baran, Р. Р. Левицкий

Published Jun 1, 2000• 11 citations
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Abstract

The Blume-Emery-Griffiths model on hypercubic lattices within the two-particle cluster approximation is investigated. The expressions for the pair correlation functions in q-space are derived. On the basis of obtained results (at q = 0) the static susceptibility of this model on the simple cubic lattice is calculated at various values of the single-ion anisotropy and biquadratic interaction.

DOI
10.1002/1521-3951(200006)219:2<357::aid-pssb357>3.0.co;2-b

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The Blume-Emery-Griffiths model on hypercubic lattices within the two-particle cluster approximation is investigated. The expressions for the pair correlation functions in q-space are derived. On the basis of obtained results (at q = 0) the static susceptibility of this model on the simple cubic lattice is calculated at various values of the single-ion anisotropy and biquadratic interaction.
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.
The spin-1 Ising model with bilinear (J) and biquadratic (K) nearest-neighbor exchange interactions and a single-ion potential (\ensuremath{\Delta}), a model introduced by Blume, Emery, and Griffiths (BEG), is studied on a honeycomb lattice. The exact solution of the BEG model is obtained for a certain relation K=K(J). Also an approximate treatment of the BEG model outside the surface, in the space of coupling constants, given by the relation above is discussed. Finally a digression is made on the critical temperature of the spin-1 Ising model.
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<
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.
I study the one-dimensional spin-1 Blume-Emery-Griffiths model with bilinear and biquadratic exchange interactions and single-ion crystal field under an applied magnetic field. This model can be exactly mapped into a tight-binding Hubbard model - extended to include intersite interactions - provided one renormalizes the chemical and the on-site potentials, which become temperature dependent. After this transformation, I provide the exact solution of the Blume-Emery-Griffiths model in one dimension by means of the Green's functions and equations of motion formalism. I investigate the magnetic variations of physical quantities - such as magnetization, quadrupolar moment, susceptibility - for different values of the interaction parameters and of the applied field, focusing on the role played by the biquadratic interaction in the breakdown of the magnetization plateaus.