Two-spin cluster theory for the Blume-Capel model with arbitrary spin

PHYSICA ELSEVIER

Physica A 224 (1996) 684-696

Two-spin cluster theory for the Blume-Capel model with arbitrary spin M. Jur~igin, A. Bob~k, M. Jag6ur Departnwnt ~[" Theoretical Physics and Geophysics. Faculty of Natural Sciences, PJ. Safarik University 041 54 Ko.(ice, Slovakia

Received 24 July 1995

Abstract

A two-spin cluster effective field theory for the Blume-Capel model with arbitrary spin S is presented by making use of exact spin identities and taking advantage of the differential operator technique. The dependence of the transition temperature is studied as a function of the single-ion anisotropy field strength for the particular cases S = 1, 3 and 2. The results are compared to those of the single-spin cluster theory recently reported in the literature as well as to other methods. In particular, it is shown that the present approach correctly distinguishes the geometry of the lattice structure beyond its coordination number.

1. Introduction A spin-1 Ising model with single-ion anisotropy, known as the Blume-Capel model, was first introduced by Blume [ 1 ] and independently by Capel [2]. The model has played an important role in the development of the theory of multicritical phenomena associated with various physical systems and has extensively been investigated in the literature (for instance, see the review [ 3 ] ) . It is known that the role of single-ion anisotropy is of particular importance when such a term has the opposite sign to the exchange interactions, Then for sufficiently strong anisotropy value the tricritical point occurs on the phase diagram, separating second- and first-order phase transitions. On the other hand, the general effective-field theory for arbitrary spin-S Blume-Capel model has recently been presented by Kaneyoshi et al. [4]. The theory is based on the use of exact Ising-spin identities for the single-site cluster as a starting point and utilizes a differential operator technique [5]. Although it does not introduce mathematical complexities, this approach provides results of reasonable accuracy, leading, unlike 0378-4371/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSDI 0378-437 I (95)00306- I

M. Jur(i,qin et al./Physica A 224 (1996) 684-696

685

mean-field theory, to coordination-number-dependent predictions. However, one of the severest limitations of the single-site cluster theory is its inability to discern between systems of the same coordination number but different structure (e.g. plane triangular and simple cubic). In order to incorporate details of the geometry of the lattice beyond its coordination number, we now develop in this paper an improved theory based on an exact spin identity for a cluster of two nearest-neighbouring spins. With this approach, which follows the same strategy as that of [4], we obtain the phase diagrams for several lattices, especially for those having the same coordination number but with different geometry or space-dimensionality. The contens of the present paper is outlined as follows. In Section 2 we present the formalism. The general expressions for calculating the phase diagram in the BlumeCapel model with arbitrary spin value S are developed in Section 3. The numerical results and discussions are presented in Section 4. Finally, in Section 5 we comment on the results.

2. F o r m a l i s m

The Hamiltonian of the Blume-Capel model we investigate is H = - Z

JijSi,. .. S t,.,. - D ~

(i,j)

S 2,: ,

( I)

i

where Si: is the z-component of a spin-S (S > 1) operator at site i, Jst is the exchange interaction between sites i and j, and D is the single-ion anisotropy constant. In two-spin cluster theory, attention is focussed on the cluster comprising just two selected spins, labelled i and j, and the neighbouring spins with which they directly interact. To this end the Hamilton/an is split into two parts, H = H 0 + H t, where H o is that part of the Hamilton/an containing spins i and j, and H t represents the rest of the Hamilton/an and does not depend on sites i and j for spin operators. In the present case,

H i / = - Jij Si: St- - sir hi - Stz hj - D ( $2. + St2z ),

(2)

where h, =

a, ks,

,

k~t

ht =

(3)

tvq

For the lsing model, in which H(i and H' commute, the starting point of the two-spin cluster theory is the set of formal identities of the type

r

\

Tri.i [ exp ( - f l H i j ) ]

/ '

(4)

M. Jurdi,(in et al. IPhysica A 224 (1996) 684-696

686

where /3 = I/kBT and (.) denotes a thermal average. After performing the trace over the selected spins i and j on the right-hand side of (4), one finds that for r = 1

m =- 51(Si: + Siz) = (fs(hi, hi)),.

(5)

where m is magnetization per site and the function fs(x,y) depends on the value of S. The explicit form of fs(x,y) for spin 1, ~ and 2 is given in the appendix. By using the exponential operator technique [ 5 ], namely exp (hDx + yDy) f(x, y) = f(x + ,l,y ÷ y), where Du = 0/0/z(/z = x,y) are differential operators, Eq. (5) may be rewritten in a more useful form:

m = ( l-I eJ'kSk=O"I I eJJ~S'~D") f s( x, Y) x_O~__.o. \ k~(i

t~

(6)

- '-'-

At this point one should notice that the two products over k and l appearing in Eq. (6) are over all the set of nearest-neighbour spins of sites i and j, respectively. For some lattices, the spins i a j forming the pair cluster share common nearest-neighbours. Thus, in the double product I-[k~(iI-It~, these spins must be selected out and one may write 1,l=

II~'eJitSk'D'teJJ'S"D"te s ~ ' ' ~- N__~ \ k~j

15¢i

.(Ji,,D.,+Jj,,Dr)

) fs(X,y) X--'Ov--O'

(7)

nsli,.j

where now the products 1T over k and l are, respectively, over the isolated nearestneighbour spins of sites i and j, while the product 1-/t over n is restricted to the sites which are, simultaneously, nearest-neighbours of both i and j spins. In fact, it is at this point that the theory, unlike more elementary effective field theories, discerns between lattices of the same coordination number but different structures (e.g. square and Kagom6 or simple cubic and plane triangular, see Fig. 1). To transform the right-hand side of Eq. (7) into a convenient form for the mathematical treatments, we can use the appropriate Van der Waerden identities for the value of the spin in question. However, for a general spin S greater than unity, the Van der Waerden identity has a complex form and the mathematical treatment becomes extremely complicated when it is introduced into (7). Moreover, the exact Van der Waerden identity has a form depending on S and hence the formulation must be given in a form depending on S. For this reason, the generalized but approximate Van der Waerden identity was introduced for tackling problems of general spin value [4], namely exp(aSk: ) = cosh(av/q) + -Skz ~ sinh(av/~) '

(8)

q = /S~..),

(9)

with

which is valid for any spin value S. It should be noted here that for S = t the relation (8) reduces to the exact Van der Waerden identity and for S ___ l the framework based

M. JurELgin et al. IPhysica A 224 (1996) 684-696

(a)

687

(a')

?

? i

I

,

0 .....

I

~ ,1

"-r" . . . . ,.l

I

N

'0 / I

j

xx

/

\

"b

d 0

(b)

0--

(b')

....

o"

9

9,

,I .'°

,, .o

~'" ..,1 _

',

q

. ' ~- .t "j o"

C--. . . .

I

I

6

,o

,, , "

/" tl o

',

6

~ ,.

, / \\

,--,,. . . . .

'

~

• \\ \

~

/J I

'

o

k

"o

Fig. 1. Two-spin clusters on lattices with coordination numbers: ( a ) and ( a ' ) z = 4 (square: z ' = 3, z " = 0) and (Kagom6: z' = 2, z " = 1), respectively; (b) and ( b ' ) z = 6 (simple cubic: z' = 5, z " = 0) and (plane triangular: .-/ = 3, z " = 2), respectively•

on (8) has given reasonable results in comparison with those based on the exact Van der Waerden identities [4]. If one now uses the relation (8), Eq. (7) takes the form

Skz sinh(Jikv~Dx)] x tI~ ' [c°sh ( 'lit v~Dy)

+ -'~Sl"sinh(Jflv~D,,)]

x I - [ ' {cosh[ v~(Ji.Dx ":Ad

+ .li.Dy) l

+ -~sinh[v/-q(Ji.Dx+JmDy)]

fs(x,y)

(lo)

To proceed further, one now has to approximate the thermal multiple correlation functions occurring on the right-hand side. The simplest approximation, and the one most frequently adopted, is to decouple them according to

(sk:s,: ... s,,:) ~ ( s ~ : ) ( s , : ) . . .

(s°~),

(11)

with k : / / 5 / • • • -7' n. Based on this approximation, Eq. (10) reduces, for the nearestneighbour interaction = J, to

Jq

M. Jur(i~inet al./PhysicaA 224 (1996)684-696

688 {[ 11l

=

]~'[~

Ax(q) 4-~qBx(q)

m

]'z

Ay(q) 4 - T B y ( q )

x [Axy(q) 4-._v~Bxy(q ) m ] j:

"}

(12)

fs(x,y) x~,).~

where

A,,(q) = cosh(Jv/-qDu),

(v=x,y),

B~ = sinh(Jv~Dv)

A,,.(q) = cosh[Jv/q(Dx + Dy)],

Bxy(q) = sinh[ Jv/q(Dx + Dy)].

(13) (14)

In (12) z' denotes the number of sites that are nearest neighbours of site i (or j), but are not neighbours of site j (or i) and z" denotes the number of sites that are simultaneously nearest neighbours of both sites i and j. Hence, z' 4- z" = z - 1, where z is the lattice coordination number (see Fig. 1). We note that, in order to solve the problem, we do need to evaluate the parameter q. It can be derived in the same way as m by the use of (4) for r = 2:

q=- ~(S~: + Siz) = [

Ax(q) +

]x/qm

× LA,,,(q) +--~Bxy(q)j

z" }

Ay(q) + ~qqBy(q)]:'

Bx(q) gs(x,y)

x=O v----O"

(15)

Here, the function gs(x, y) also depends on the spin value S and for S = 1,3 and 2 is given in the appendix.

3. Transition temperature and tricritical point

In this section, let us study the transition temperature (or the phase diagram) and the tricritical point of the system where the transition changes from second-order to first-order. For this, expanding the right-hand side of Eqs. (12) and (15), we obtain

m = KI (q)m + K3(q)m 3 + ....

(16)

q = Ko(q) + K2(q)m 2 + ....

(17)

with

Ki (q) = 7

1

[

I g/--I

2Z A].

,, + z ax

t

tt

(q)Bx(q)aZ.¢ (q)aZx,y(q)

(q)X~,, (q)AZj,,:_, (q)Bxy(q )] fs(x,y) x=o,y=o'

[ Zt ' 2 2 z' I K3(q)= (v~)3 {[2Zt(2)AZa.(q)Bx(q)Ay(q)By(q)

(18)

M. Jur(L~in el a l . / P h y s i c a A 224 (1996) 684-696

+ 2

3 Ax

(q)Bx(q)A~

(q)

689

Axy (q)

2 :' (q) + z " [2 ( 2 ' ) A Zx' - 2 (q)Bx(q)A;. .'-+ z 'eA?i. I ( q ) B x ( q ) A ~ . -~ I ( q ) B y ( q ) ] A xz"y - t ( q ) B x y ( q ) J /Z"~ ' I "' "" 2 2 + 2 z ' ~ 2 )a]. - (q)Bx(q)A~. (q)a'~.:.- (q)Bxy(q)

+

A] (q)A~. :"-3 • . (q)A.%. (q)R~.y(q)

x=O ,.=o'

(19)

and Ko(q) = a]. (q)a~. (q)aZ~.y(q)gs(x, y) ~

,

(20)

x=0 v=0

"

'{[ (2')

K2(q) = q

2

+z

'2

_~z.~

I

A].'-2 (q) Bx(q)Ay 2 z (q)

" A~.z~-I (q)B.,(q)A.,.z ' - I (q)B:,(q) ] A~y(q) J IIA~I--[

t

tt

zz Z /~'~. (q)Bx(q)a~. (q)aZxy -I (q)Bxy(q) ~ll

+ .

I

.

( q ) A i, (q)A~y

( q ) B x Y ( q ) ~ g s ( x ' y ) x--o,.--o'

(21)

where (~) are the binomial coefficients k!/n!(k - n)!. By substituting Eq. (17) into Eq. (16), one obtains in general an equation for m of the form m = am + b m 3 + . . . .

(22)

The second-order phase transition line is then determined by a = I and b < 0, i.e. Ki (q0) = I,

(23)

where q0 is the solution of Ko(qo) = qo.

(24)

In the vicinity of the second-order phase transition line, the magnetization m is given by m2

_

I --

b

a

(25)

The right-hand side of Eq. (25) must be positive. If this is not the case, the transition is of the first order, and hence the point at which a= I

and

b=0

(26)

M. Jur(i,(in et aL /Physica A 224 (1996) 684-696

690

is the tricritical point [6]. At this point, in order to obtain the expression for b, let us substitute q = qo

+

qJ

m2

(27)

into Eq. (17). The expression of ql is then given by e

(28)

qm- i - f ' with 1 e

~ q

{[2(2')Ax'-2(qo)Ba2(qo)A~.'(qo)

qo

+

z'2A~.'-I (qo)Bx(qo)A~'-I(qo)By(qo)] azxii(qo) f

+2z

~t

t

tt

z"A~ -I (qo)Bx(qo)a~. (qo)aZxy-] (qo)Bxy(qo)

+(z2')Ax'(qo)A~.'(qo)AZ.i~-2(qo)B~y(qo)}gs(x,Y)x_O,y__O,

J

[

l

t

s

(29)

//

f = - - ~ Z A~ -I (qo)Bx(qo)A:~. (qo)Axy (qo)Dx + z"m'a:'(qo)A~;'(qo)AZj.i:-' (qo)Bxy(qo)Dx] gs(x, y) x--O,.---o'

(30)

Substituting Eq. (27) into Eq. (16), the expression of b in Eq. (22) is given by b=

Jql {zr [(Z~ - 1) Axz'-2 (qo)Bx(qo)Av 2 z' (qo) +AZx' (qo)A,,z' (qo) ] Ax~.(qo)Dx z" qo " " " '2 ' I ~-I z" +z A~. - (qo)Bx(qo)A~. (qo)By(qo)Axy (qo)Dy t

tt

1

t

tt

|

+z'z"A:~. '-1 (qo)Bx(qo)A~. (qo)AZy - (qo)Bxy(qo)(Dx + Dy) +z'z"Ar~ '-I (qo)Bx(qo)A~. (qo)Axy- (qo)Bxy(qo)Dx +z II A~."¢ (qo)A~, I (qo) [ ( z " k

1)A ~.>, zJ/ -2 (qe) B2xy(qo)

+ AZji:(qo)]Dx } fs(x,y)x--O,,.--o " .

ql

2q0

K1 ( q 0 ) + K3(q0).

(31)

It is clear from the above expressions that the two-spin cluster approximation incorporates details of the geometry of the lattice beyond its coordination number z, through the numbers z r and z". In the following, five lattices, namely honeycomb (hc.) (z = 3, z t = 2, z" = 0), square (sq.) (z = 4, z ~ = 3, z" = 0), Kagom6 (Ka.) (z = 4, z' = 2, z" = 1), triangular (tr.) (z = 6, z ~ = 3, z" = 2) and simple cubic (s.c.) (z = 6, z ~ = 5, z" = 0) are investigated.

M. Jur(i.gin et al./Physica A 224 (1996) 684-696

5

i

i

i

'

I

~

i

i

i

691

,

4 3

0

~

-,

I

-2

J

I

~

0

2

I

4

J

I

~

6

I

8

10

B/J Fig. 2. Phase diagrams in the T-D-plane for the spin-l Blume-Capel model for different lattices: curve a, simple cubic lattice; curve d , triangular lattice; curve b, square lattice; curve b ~, Kagom6 lattice; curve c, h o n e y c o m b lattice. The white circles denote the tricritical points. Table I Transition temperatures kBTc/J and the coordinates of the tricritical points in the spin-I B l u m e - C a p e l model for several lattices hc.

kt~Tc/J (at D = 0) kt~Tt/J Dt/J

1.4660 0.7130 - 1.4355

Ka. 2.1046 1.0279 - 1.8949

sq. 2.1605 1.0521 - 1.8888

(kBT/J, D/J)-plane of the

tr. 3.4619 1.6939 -2.8126

s.c. 3.5084 1.7172 -2.8088

4. Numerical results and discussion

By using the general tbrmulation given in Section 3, let us examine the phase diagrams of the Blume-Capel model with spin values S = 1, S = 3 and S = 2. Our results are shown in Figs. 2-4 for s.c. (curves a), tr. (curves a ' ) , sq. (curves b), Ka. (curves b r) and hc. (curves c) lattices. Fig. 2 shows the phase diagram in the T-D-plane for the spin-1 Blume-Capel model. As expected, for each lattice the critical temperature decreases gradually from its twolevel Ising model value, T~.(oc), at D --~ o~, to the tricritical point (white circle) at some negative value of the single-ion anisotropy strength, Dt. The coordinates of the tricritical points and the results of kBTc/J at D = 0 are collected in Table 1. These values can be compared to those of the effective-field theory for the single site cluster [4], and those of other approximative methods (see tables I and II in Ref. [7] ). A number of points can be made concerning the results presented in Table 1 (or Fig. 2). First, one should note that in all cases the present approach represents only

692

M. Jur(i.(inet al./Physica A 224 (1996)684-696 'S=2

~ '

i

0

2

'

i

'

i

'

~

_

8

I0

16

12

0

-2

4

6

D/J Fig. 3. The same as in fig 2., for the spin S = 2. a slight improvement in the quantity o f keTc/J at D = 0 over the single-spin cluster approximation [4]. However, it is practically corroborated that ksTc/J correctly depends on the particular symmetry of the crystallographic lattice structure under consideration. For the z = 4 case, we find distinct values of keTc/J = 2.1046 and 2.1605 for Ka. and sq. lattices, respectively, in contrast with the unique value of 2.200 predicted by the single-spin cluster theory. Similarly, for the case of z = 6, we get the values of kBTc/J = 3.4619 and 3.5084 for tr. and s.c. lattices, whereas a single value of 3.528 is predicted by the one-spin cluster theory. Second, the initial difference of the kBTc/J for lattices having the same coordination number but different geometries is decreased when the value of D decreases from D --~ + o o to D < 0. This is reasonable, since for large negative D we have only the Si: = 0 state and the long-range ferromagnetic ordering disappears for D,.//J = -z//2. Accordingly, it is expected that the phase boundaries for the lattices with the same coordination number tend to a common value of critical anisotropy Dc at which Tc reduces to zero. In Fig. 3 we show the phase diagrams in the T-D-plane for the spin-2 Blume-Capel model. From this figure we can see that the effect of an anisotropy field on the critical temperature is very similar to that of the spin-1 Blume-Capel model (see Fig. 2) The coordinates of the tricritical points for the various lattices as well as the values of transition temperature kBTc/J at D = 0 are tabulated in Table 2. Here, one should notice that for both S = 1 and S = 2 the tricritical points occur at temperatures lower than those predicted within one spin-cluster approximation [4] and at fractionally higher negative anisotropy field strengths. Finally, in Fig. 4 there are plotted the variations of T~. versus D for the spin-~ Blume-Capel model. In contrast with an integer-spin Blume-Capel model, we have found that the spin-~ Blume-Capel model does not show the tricritical behaviour. This

M. Jur(igin et al./Physica A 224 (1996) 684-696

693

Table 2 Transition temperatures koTc/J and the coordinates of the tricritical points in the (kBT/J, D/J)-plane of the spin-2 Blume-Capel model for several lattices he.

kI~7~//.1 (at D = 0) kj~7"~/ J 1): / J

Ka.

4.5876 1.0433 - 1.4981

6.5183 1.5383 - 1.9951

I0

i

8

sq. 6.6680 1.5838 - 1.9943

I

[

tr.

s.c.

10.5902 2.5742 -2.9895

10.6916 2.6087 -2.9892

'

S=3/2

0

-fO

,

t

-5

,

I

0

,

I

5

,

10

D/J Fig. 4. Phase diagrams in the T-D-plane for the spin-~ Blume-Capel model for different lattices: curve a, simple cubic lattice; curve a ~, triangular lattice; curve b, square lattice; curve b~, Kagom6 lattice; curve c, honeycomb lattice. When D < 0 the results for the lattices with the same coordination number but different geometries lie within the thickness of the solid line.

is not surprising because the system does not include the Si: = 0 state. In fact, the tricritical b e h a v i o u r o f an integer-spin B l u m e - C a p e l model results from the transition into the Si- = 0 state at the tricritical point. Further, it is seen from Fig. 4 that the value o f k B T c / J of

IDI

for each lattice approaches the two constant values when the value

b e c o m e s large. Physically, the constant value for D ---, - o c

fact that the Si- = ± 3

c o m e s from the

states are suppressed and the B l u m e - C a p e l m o d e l reduces to

the usual two-level Ising m o d e l with Si: = ±½. On the other hand, the constant value for D ----, + o c results from taking the Si: = ±-}_ states at T = To. For this reason the value o f k B T , . / J in the limit o f large negative D is nine times smaller than for large positive D. C o n s e q u e n t l y , the initial difference o f the k B T ~ / J for lattices having the same c o o r d i n a t i o n n u m b e r but different geometries in this limit is nine times decreased too (see curves a , a I and b, b t in Fig. 4). In particular, the values o f k B T c / J at D = - o c and D = 0 for all the lattices we have studied, are collected together in Table 3.

M. Jur(i.~inet al./Physica A 224 (1996) 684-696

694

Table 3 Transition temperatures kBTc/J of the spin-~ Blume-Capel model for several lattices. The values of ksTc/J at D = -cx~ are equivalent to that of the spin-½ lsing model

kBTc/J (at D = 0) kBTc/J (at D = -cx~)

hc.

Ka.

sq.

tr.

s.c.

2.8309 0.4967

4.0350 0.7309

4.1320 0.7563

6.5801 1.2376

6.6601 1.2598

5. Concluding remarks In the present paper, by making use of exact spin identities for a cluster of a two nearest-neighbouring spins and taking advantage of the differential operator technique, a two-spin cluster theory for the arbitrary spin-S Blume-Capel model is developed. It is shown that this framework provides results that are quite superior to those obtained within spin-one cluster theory [4]. In particular, the phase diagrams reflect the details of a lattice geometry beyond its coordination number and the values of the transition temperature kBTc/J are closer to the exact results on a Bethe lattice [8]. However, it should be noted here that a change from the single-site theory to the two-site theory produces a larger improvement in the kBT,./J for the honeycomb lattice than for other lattices. This result arises because, besides neglecting the correlations between nextnearest neighbours, the present treatment based on the approximation ( 11 ) also neglects (except for the honeycomb lattice) the correlations between nearest neighbours. Therelore, further improvement to the theory will be possible if at least nearest-neighbour correlations are treated more accurately, as has been accomplished in a corresponding study of the two-state Ising system [9]. Finally, it should be pointed out that the method described in this paper together with the one-spin cluster formalism [4] can be used to construct the effective-field renormalization group scheme [ 10] in a straight forward manner. Indeed, we have just completed a study of the Blume-Capel model with a high spin value using this approach, which is based on the comparison of clusters of different sizes in the presence of symmetry-breaking fields.

Appendix The functions f s ( x , y ) and g s ( x , y ) in Eqs. (5) and (15), respectively, are defined by

As(x, y) f s ( x , y ) - Cs(x,y---~'

Bs(x, y) g s ( x , y ) - Cs(x,y~)'

where tor S = 1, AI (x, y) =2e 13~J+2D)sinh[/3(x + y) ] + e/J°[sinh(~x) + sinh(/3y) ], Bl (x, y) = 2e2C~O{e¢~Jc o s h [ / J ( x + y) ] + e -#J cosh[/3(x - y) ] }

M. Jurdigin et al./Physica A 224 (1996) 684-696

695

+ e #° [cosh(flx) + cosh(fly) ], C1 (x, y) = 2e2~°{e #g cosh[fl(x + y) ] + e -BJ cosh[fl(x - y) ] } +2e/3° [cosh(flx) + cosh(fly) ] + 1, for S = ~-, A3/2(x y) = 3e (9/4)B(J+2D) sinh[ ~/3(x + y) ] +2e ~ua)#~3J+lOO) {sinh[ i /3( 3x + y ) ] + sinh[ ½/3(x + 3v) ] } +e
B3/2(x, y) = 9 e(9/2)BD{e(9/4)~J cosh[ 3 fl(x q- y)] --I-e-(9/4)BJ cosh[ 3 ~ ( x - y ) ] } q-5e( I/4)[~(3J+l°D){cosh[//~(3x + y)] --]--cosh[//~(x + 3y) ] } q-5e(I/4)l~(-3J+l°D){cosh[ ½/3(3x -- y)] + cosh[ ½/3(x - 3y)]} +½e(J/2)~°{e(U4)#J cosh[ ½fl(x + y)] + e - ( I/4)l~J cosh[ ½fl(x - y) ] },

C3/2(x, y) = 2e~9/2)3D(e(9/4)3J cosh[ 3/3(x + y) ] + e -~9/4)~'' cosh[ 3/3(x - y) ] } -]--2e( I/4)~(3J+IOD)(cosh[//~(3X"-]'-y) ] + cosh[ ½/3(x + 3y) ] } --~-2e( I/4)3(-3J+IOD){cosh[ ½/~(3x -- y)] -Jr-cosh[ ½~(x - 3y)] } +2e ~I/2)3O{e< I/4)PJ cosh [ ½/3(x + y) ] + e -~ I/4)3J cosh[ ½/3(x - y) ] }

and for S = 2 , A2(x, y) = 4e 4fl(J+2D) sinh[ 2/3(x + y) ] +3e#~2g+5O){sinh[fl(2x + y) ] + sinh[/3(x + 2y) ] +e#<-2g+sD){sinh[fl(2x -- y)] -- sinh[/3(x - 2 y ) ] } +2e 4/3D [ sinh(2/3x) + sinh(2/3y) 1 +e #° [sinh(/3x) + sinh(/3y) ] + 2e ~g+2°) sinh[/3(x + y) ],

B2 (x, y) = 8e8#°{e4tJ" cosh [ 2/3(x + y) ] + e -413J cosh[2fl(x - y) ] } +5eC~t2g+5°){cosh[fl(2x + y) ] + cosh[fl(x + 2y) ] }

696

M. Jur(i.~in et al./Physica A 224 (1996) 684-696 + 5 e #< - 2 J + 5 ° ) { c o s h [ / 3 ( 2 x - y ) ] + c o s h [ / 3 ( x - 2 y ) ] } + 2 e 2 # ° { e ~J c o s h [ / 3 ( x -+- y ) ] + e - # J c o s h [ / 3 ( x - y ) ] } + 4 e 4/3D [ c o s h ( 2 / 3 x ) + c o s h ( 2 / 3 y ) ] + e # ° [ c o s h ( / 3 x ) + c o s h ( / 3 y ) ], C: (x, y ) = 2e8'aO{e 4/3J c o s h [ 2 f l ( x + y ) ] + e -4'eJ c o s h [ 2 f l ( x - y ) ]} +2e/3t2J+sO){cosh[fl(2x + y ) ] + c o s h [ f l ( x + 2 y ) ] + 2 e # t - 2 J + S o ) { c o s h [ f l ( 2 x - y) ] + c o s h [ f l ( x - 2 y ) ] } + 2 e 4'ao [ c o s h ( 2 / 3 x ) + c o s h ( 2 f l y ) ] +2e ~° [cosh(flx) + cosh(/3y) ] +2e2~O{e ~J c o s h [ f l ( x + y ) ] + e -By c o s h [ f l ( x - y ) ] } + I.

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