Two-Particle Cluster Approximation For Ising Type Model With Arbitrary Value Of Spin. Correlation Functions Of Blume-Emery-Griffiths Model

The Ising type model with arbitrary value of spin is investigated within the two-particle cluster approximation. For this model on hy-percubic lattices the expressions for the pair correlation functions in ~ q-space are obtained. For Blume-Emery-Griiths model (S = 1) on a simple cubic lattice with bilinear K, biquadratic K 0 interactions and single-ion anisotropy D, a projection of the phase diagram on D=K ? K 0 =K plane is constructed. The temperature dependences of hS z i, h(S z) 2 i and pair correlation functions at ~ q = 0 are calculated at various values of parameters D=K and K 0 =K.


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-Gri ths model (BEG) corresponding to the Isingtype model with = 2 (hereafter S z = ?; ?+2; : : :; ?2; ) was proposed for the investigation of phase transitions (PT) in He 3 ?He 4 mixture 1].A general form of the Ising-type model with = 2 contains extra terms, as compared to the BEG model, like S z i (S z j ) 2 .It is used 2,3] for description of external pressure in uence on ferromagnetics ( = 1), for investigation of crystals with ferromagnetic ( = 1) impurities, three-component nonmagnetic alloys, two-component lattice liquids etc.The Ising-type model is proved to be useful in studies of tricritical behaviour of anisotropic ferromagnetics FeCl 2 and DAG, of pressure-induced structural phase transitions in NH 4 Cl and KH 2 PO 4 2,4].
For compounds described by pseudospin models with essential shortrange correlations, the cluster expansion method (CEM) [5][6][7] is the most natural many-particle generalization of the molecular eld approximation (MFA).It should be mentioned that the CEM gives much better results at temperatures far from transition point.Within CEM, an in nite lattice is replaced with a cluster with a xed number of pseudospins, the in uence 58 S.I.Sorokov, R.R.Levitskii, O.R.Baran of rejected sites is taken into account as a single external eld '(S), acting on boundary sites of a cluster.The rst consistent formulation of the CEM which allows to determine corrections to free energy related to cluster interaction, was made in 8].The rst order of CEM is called cluster approximation.
Despite a number of papers where CEM is applied to calculation of physical characteristics of various substances, we are aware only of a few ones where pseudospin models with a spin value > 1 are considered.Thus, in 9] two-particle cluster approximation (TPCA) was applied to Heisenberg model with spin > 1, with the dependence of the variational eld ' on spin value being neglected.This neglecting gave qualitatively wrong results even for Ising-type model with = 2 3].In 10] the BEG model was studied within TPCA.Even though the dependence '(S) was taken into account here, the coe cients of expansion of variational elds ' in powers of pseudospin ('(S) = ' (1) + ' (2) S) were taken to be equal, up to constant terms, to hSi, hS 2 i, what gave results close to those of MFA.
For investigation of thermodynamic properties of BEG model, other approximate methods also were widely used.In particular, in a series of papers [11][12][13][14][15], equation chains obtained on the basis of Callen identities are closed with the help of the simplest decoupling hS 1 S 2 : : :S k i hS 1 ihS 2 i : : :hS k i. Calculated in such an approximation transition temperature T c for the case of = 1 16] di ers from results of numerical methods more signi cantly than the T c calculated in TPCA does.
The PT of the BEG model at negative values of single-ion anisotropy and positive values of bilinear and biquadratic interaction were studied in 17] within the constant-coupling approximation, which results correspond to the TPCA results.The three-dimensional phase diagram was constructed, and temperature dependences of dipole and quadrupolar moments were obtained for some values of Hamiltonian parameters.In 18] the Bethe approximation (results of this approximation also correspond to the TPCA results) was used for investigation of the BEG model at arbitrary values of single-ion anisotropy and biquadratic interaction.Particular attention was paid the case of antiferro biquadratic interaction.Obtained results were compared with those of Monte Carlo method.We also should mention the results of studies of BEG model within the high-temperature expansion 19] and Monte Carlo methods 20,21].To our best knowledge, only thermodynamic properties of the considered models have been studied so far.
The goal of this work is to develope a two-particle cluster approximation for calculation of thermodynamic characteristics and correlations functions (CF) of Ising-type model with an arbitrary value of spin and investigate the Blume-Emery-Gri ths model within this approximation.

Problem formulation
We consider a pseudospin system with S = S z = (? ; ?+ 2; : : :; ?2; ), described by the Hamiltonian H fh ( ) Ising type model with arbitrary value of spin 59 Here K (nm) and J (nm) ij are the constants of short-range and long-range interactions, respectively; the notation H fh ( ) g means that H is a function of (h (1)   1 ; : : :; h (1) N ; : : :; h ( ) 1 ; : : :; h ( ) N ).The factor = (k B T) ?1 occurring in h (n)  i , K (nm) , J (nm) ij will be written explicitly only in some nal formulas.As a particular case we consider Blume-Emery-Gri ths model ( = 2) with long-range interaction: Here ?i is an external eld, D i is a single-ion anisotropy, K and K 0 are the constants of bilinear and biquadratic short-range interactions; J ij and J 0 ij are those of long-range interactions.Within the molecular eld approximation in the long-range interaction, the Hamiltonian (2.1) can be expressed as H fh ( ) ij hS n i ihS m j i : (2. 3) The following notation for the reference Hamiltonian is used J (nm) ij hS m j i : (2.5) The function F fh ( ) g (logarithm of the partition function) within the MFA in the long-range interaction reads: where k F f ( ) g is a logarithm of the partition function of the reference system (2.4).Correlation functions (cumulant averages of spin operators calculated with the Gibbs' distribution with H) of the considered model will be evaluated as: h(S n1 i1 ) 1 : : : and CFs of the reference system as: From the expression for the F-function (2.6) we can easily derive some relations between single-site CFs of the reference system (2.4) and those of the general system within the MFA in long-range interactions (2.3): k F f ( ) g = k hS n i i : (2.9) For the sake of simplicity, we present relations between pair correlation functions for the BEG model (2.2) only.These relations can be obtained from (2.9) (n = 1; 2 ; J = J (11) ; J 0 = J (22) ; J (12) = J (21) = 0).Using matrix notations in the indices i; j and performing Fourier transformation, we get a system of four equations relating hS n S m i c q and k hS n 0 S m 0 i c q.In a (2.10) b b(q) = hSSi c q hSS 2 i c q hS 2 Si c q hS 2 S 2 i c q ; k b b(q) = k hSSi c q k hSS 2 i c q k hS 2 Si c q k hS 2 S 2 i c q : (2.11) From equation (2.10) one can easily nd the pair CFs of the BEG model expressed in terms of pair CFs of the reference system, the long-range interactions taken into account in the MFA.

Two particle cluster approximation in short-range interactions
In this section we consider the reference pseudospin system with Hamiltonian (2.4).Let us divide the lattice into two-particle clusters.As P n=1 r ' (n)  i S n i we denote an operator of the e ective eld created by the site r and acting on the site i, provided that the site r is a nearest neighbour of the site i (r 2 i ).Apparently, when the lattice is divided into the twoparticle clusters, the number of elds acting on the given site is equal to the number of the nearest neighbours z.We transfer from summing over lattice sites to summing over clusters 22]: 1 2 X i; Using (3.1), we can write the reference Hamiltonian (2.4) in the form Ising type model with arbitrary value of spin 61 Notation H 1 f~ ( ) 1 g means that H 1 is a function of (~ (1) 1 ; : : :; ~ ( ) 1 ).
For k F f ( ) g; f' ( ) g we have: (1;2) U 12 i 0 ; hAi 0 = Sp fSg 0 (fSg) A ; Here we introduced the following notations for the single-particle F 1 -function: Let us restrict ourselves to the rst order of the cluster expansion 22].
Then k F-function can be written as a sum of single-particle and two-particle intracluster F-functions where the two-particle F 12 -function reads: Let us consider now the k F-function.Using the method, proposed in 23] for the case = 1 we can obtain equations for hS n 1 i = k hS n 1 i and cluster elds r ' (n)   1 .From (2.8) and (2.9) we get: Taking into account the fact that k F-function (3.8) is a sum of single-particle and two-particle F-functions and making use of the notations (3.3) and (3.11), we get the following expressions for the partial derivatives of k Ffunctions: Here we use the notations for the partial derivatives of the single-particle and two-particle F-functions { the single-site and pair intracluster CFs: On the basis of (2.9), (3.3), (3.11), (3.12), (3.13) and (3.14) taking into account the condition of an extremum of k F-function with respect to r ' (n we can easily get the system of (z+1) N equations for r ' (n) i and hS n i i (n = 1; : : :; ; i = 1; : : :; N).
Let us note that in uniform elds h (n)  i ( h 0 hS m i ; J (nm) 0 = J (nm) (q = 0) ; relations (3.18) and (3.19) form the system of 2 equations for ' (n) and hS n i.When the long-range interactions are absent (J (nm) 0 = 0) and the elds are uniform, we have the system of equations (3.19) for the cluster elds ' (n) and expressions (3.18) for the single-site CFs hS n i.
Let us brie y consider the proposed by us 3] method of calculation of pair CFs of the reference system with an arbitrary value of , which is based on the method developed in 23] for the case = 1.From (2.8) and (3.18) one can obtain an expression for pair CFs of the reference system Since evaluation of F 1 ( 1 n j 1 k j) for a particular system is straightforward, we only have to nd an equation for ~ which can be transformed to h b F (20)   1r + b F (11)   1r b ~ r2 = b F (20)   1r r b ' 12 + b F (11)   1r 1 b ' r2 : (3.30) Here we use the notations: b F (20)   ir = 0 B B B @ F ir 2 1 j F ir 1 1 j 1 2 j : : : F ir 1 1 j 1 j . . .F ir 1 j 1 1 j F ir 1 j 1 2 j : : : b F (11)   ir = 0 B @ F ir ( 1 1 j 1 1 jj) : : : F ir ( 1 1 j 1 jj) . . .F (11)  1r ; (3.
where for a hypercubic lattice sin 2 q i a 2 : (3.41) d = z=2 is the lattice dimension, (q) is the Fourier transform of 1r .
Using the matrix form of (3.26) and performing a Fourier transformation, taking into account (3.40), we get the following relation for pair CFs of the reference system where b F (11) .

The Blume-Emery-Gri ths model
Let us consider the reference Blume-Emery-Gri ths model ( = 2), which is described by the Hamiltonian Here i = ?i + P N j=1 J ij hS j i ; 0 i = D i + P N j=1 J 0 ij hS 2 j i, ?i is an external eld.D i stands for single-ion anisotropy.Let us note that the factor = (k B T) ?1 is written explicitly hereafter.

Numerical analysis results
In this section we discuss the results of numerical calculations (? = 0) of thermodynamic characteristics and pair correlation functions (at q = 0) of BEG model on a simple cubic lattice (z = 6).
In the two-particle cluster approximation the system of equations for ', ' 0 (4.10) has several solutions, the number of which depends on values of parameters d, k 0 and temperature.Solution corresponding to P-phase exists at t 2 t P1 ; 1] (t P1 0, its value depends on d, k 0 ).Solutions corresponding to the F-phase and Q-phase exist at t 2 t F1 ; t F2 ] and t 2 t Q 1 ; t Q 2 ], respectively.The values of t F1 , t F2 and t Q 1 , t Q 2 depend on d, k 0 and are nite.The projection of the phase diagram on (d; k 0 ) plane at d < 0, k 0 > ?0:1 17] and d > 0, k 0 > ?1 ? 1  6 d (see gure 1) consists of seven regions: I { the phase transition Q $ P of the rst order (QP1), II { the PT FP2, III { the PT FP1, IV { the PT is absent, V { the PTs QF1 and FP2, VI { the PTs QF1 and FP1, VII { the PTs FQ1 and QP1.
In the present paper we restrict our consideration to the regions d < 0, k 0 > ?0:1 and d > 0, k 0 > ?1? 1  6 d, since for the regions d > 0, k 0 < ?1? 1 6 d and d < 0, k 0 < ?1 a two-sublattice model should be considered 18].Let us note that in the region d < 0, ?0:1 > k 0 > ?1 in the vicinity of the line k 0 = ?1 ? 1  3 d the regions with di erent numbers of various PTs exist.Construction of a projection of a phase diagram on (d; k 0 ) plane and study of thermodymamic properties of the model in these regions are subject of a separate paper.
Let us brie y consider the pair CFs.
Let us explore now the temperature behaviour of G ?1 22 (t > t c ) when k 0 increases and d decreases (approaching the region d > 0, k 0 < ?1 ? 1  6 d; the region III, the region V).Far from the regions III, V (see gures 4, 5, 6(a), 18(a)) G ?1 22 (t > t c ) decreases, and this decrease is not always related to a decrease in q(t).For instance, at small negative k 0 in the vicinity of  the region d > 0, k 0 < ?1 ? 1 6 d q(t) increases, but G ?1 22 (t) decreases (see gure 5(b)).At small positive k 0 with decreasing d (approaching the region III) G ?1 22 (t > t c ) and q(t > t c ) start to increase with temperature (see gure 6(b)), although at some values of parameters q decreases but G ? 1   22 increases (see gure 9(b)).At larger values of k 0 the decreasing of d (approaching the regions III, V and in the region V) lead to the fact that G ?1 22 has a minimum in the P-phase (see gures 18(b), 22,25).At FP1, the inverse correlation function G ?1 12 (t) has an in nite discontinuity (G ?1 12 (t c ? 0) > 0, G ?1 12 (t c + 0) = 1), whereas CFs G ?1 11 (t) and G ?1 22 (t)   have nite discontinuities and decrease in the F-phase.Let us analyse the behaviour of G ?1 11 (t) and G ?1 22 (t) in the vicinity of FP1 phase transition and in the P-phase in the region III for k 0 =0.0, 1.0, 2.0, 2.6, 2.88, 2.95 when the value of the single-ion anisotropy decreases (moving from the region II towards the region IV for k 0 =0.0, 1.0, 2.0, 2.6; moving from the region II towards the region VII for k 0 =2.88, 2.95; between the regions VII and IV for k 0 =2.88), in regions III, VI for k 0 =3.2 (moving from the region II towards the region I), and in region VI for k 0 =3.44 (moving from the region V towards the region I).In the region III and near to the region II G ?1 22 (t c ? 0) < G ?1 22 (t c + 0), G ?1 11 (t > t c ) and G ?1 22 (t > t c ) increase at all considered values of k 0 except for k 0 =3.2 (see gure 19), at which G ?1 22 (t) has a mimimum in the P-phase.Let us consider the case k 0 = 0.When d decreases, minimums of G ?1 11 (t) (see gure 7(b)), and, then, of G ?1 22 (t)  m, q G ?1 11 G ?1  That is, in the vicinity of the regions IV at k 0 =0.0, 1.0, 2.0 the behaviour m, q G ?1 11 G ?1     ).However, it is possible that even if q(t) increases in P, G ?1 11 (t > t c ) increases has no minimum (see gure 16(a)).The inverse CF G ?1 22 (t) increases in the paramagnetic phase at small k 0 (see gures 13,16,17,21).At large values of k 0 in the close proximity to the region V, the G ?1 22 (t) decreases before the cusps at the transition point and after it and starts to increase only at higher temperatures (see gures 24, 26).

Figure 1 .
Figure 1.The projection of the phase diagram onto d ?k 0 plane.