Effective field theory of dilute Ising systems

Journal of Magnetism and Magnetic Materials 102 (1991) 144-150 North-Holland

Effective field theory of dilute Ising systems J.W. Tucker Physics Department,

Sheffield biversity,

Sheffield S3 7RH, UK

Received 13 March 1991

An effective field theory that correctly accounts for all the single-site kinematic relations is applied to a study of a quenched site-diluted spin 1 Ising model, containing in addition to the usual biinear exchange, both biquadratic exchange and singleion anisotropy. In doing so, particular attention is paid to the approximation made in effecting the configurational averaging. As a .by-product, the nature of the configurational averaging adopted by previous authors in studies of the spin i Ising system is clarified with reference to a single formalism, and the reason for their differing results explained.

1. Introduction During the last decade, numerous publications dealing with Ising systems have appeared that employ an effective field theory which correctly accounts for the single-site kinematic relations between the spin operators. These include studies of the spin $ Ising system, with or without a transverse field; spin 1 models having biquadratic exchange and single-ion anisotropy; and mixed spin systems. Apart from pure systems, both site- and bond-diluted magnets have been studied, as well as random bond models, amorphous magnets, semi-infinite systems with a variety of surfaces and thin films. Although the method can be employed quite generally within the framework of a n-site cluster theory, a single-site approximation (n = 1) has most frequently been adopted, in order that the resulting analytic expressions have a manageable form. In this approximation, attention is focused on a cluster comprising just a single-spin, the central spin, and the neighbouring spins with which it directly interacts. The starting point in that case is a set of formal identities, of the type discussed by Suzuki [l] for classical systems, that exist for the thermal averages (S_&) of a single spin:

(S;)

= (trace,[exp( -~~~)S,p,]/traceg[eXP(-BHg)])r

where Hg is that part of the Hamiltonian containing the spin g. The evaluation of the trace, on the right-hand side of this equation leaves one with a thermal average of a transcendental function whose argument contains operators belonging to the spins with which the central spin, g, interacts. The use of the kinematic relations for the spin operators is then a crucial step in the theory as it allows the thermal average of the transcendental function to be expressed as an average over a finite polynomial of spin operators belonging to the neighbouring sites. This is achieved, either by solving the set of equations obtained for the coefficients in the polynomial expansion, when each spin connected to the central spin is given a set of fixed values, or by use of a differential operator technique introduced by Honmura and Kaneyoshi [2]. The former approach was introduced by Matsudaira as early as 1973, and used by him and by Takase [3-51 in studies of spin 4 Ising systems on a number of lattices. Later, the same method was proposed by Boccara [6], who was apparently unaware of this earlier work, and it has subsequently been used extensively by him and groups of researchers in Morocco (see for example, refs. [6-81). Clearly, as far 0304-8853/91/$03.50

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J. W. Tucker / Effective jieki theory of dilute Ising systems

145

as the physics is concerned it is immaterial whether one uses the combinatorial method, or the differential operator approach, for reducing the transcendental function to a polynomial form. We stress this point, because there appears to be an almost complete lack of cross referencing in the literature originating from the two groups of workers using one, or other, of these different techniques. Indeed, the two groups of workers even refer to their theories by different names, finite cluster approximation and effective field theory respectively. Owing to this lack of contact, the fact that results originating independently from these two groups have sometimes differed (see section 3), has gone unnoticed in the literature. Of the two methods, the differential operator technique of Honmura and Kaneyoshi has generally been the more favoured, because of the relative ease with which results applicable to general lattice coordination numbers may be obtained. In the combinatorial approach, each lattice structure has usually been considered separately, although general expressions for the polynomial coefficients have been obtained [9]. This method is, however, readily extendable to larger clusters (n > 1) [7,10], whereas, to our knowledge the differential operator method has not been used beyond a two-site cluster level [11,12].

2. Site-diluted BEG model Recently there has been renewed interest in the study of the BEG spin 1 Ising model that contains biquadratic exchange and single-ion anisotropy, in addition to the usual bilinear exchange. This model, originally introduced to describe the properties of 3He-4He mixtures, has also been extended to account for multicritical behaviour in magnetic systems, ternary alloys and multicomponent fluids. The effective field theory employing the differential operator technique was developed for this problem by Tucker [13,14] and applied to a study of the second order phase boundaries, the location of the tricritical points, the first order phase transitions, and the temperature yariation of the magnetization and the quadrupolar moment [15]. The purpose of the present paper is to extend the theory to the quenched site-diluted system. In doing so, particular attention will be paid to the approximation made in effecting the configurational averaging, as this has led in the past to conflicting results being obtained in studies of the spin $ Ising model - an issue to be discussed in section 3. The site-diluted S = 1 BEG model can be described by the Hamiltonian H = - c JiicicjSizSjz (ii>

-

c .$cicjS~S; (ij>

+ DC ciSi’, , i

(2)

where the site occupancy number, ci, takes values 1 or’0 depending on whether the site is occupied or not. The starting point is the the generalised Suzuki relation, eq. (l), with Hg = - Sgzc JgjcgcjSjz - S;zc Jg;cgcjS,; -I-DC&~. j i

(3)

Setting p = 1 and 2 in turn in eq. (l), one has on effecting the inner trace over the central spin,

(S&v)=

2 Sfi(PCJgjcgcjS'z) exp[ - Bc,( -D

+ ~,J~jcjS,~)]

+ 2 cosh( /l~,JgjcgcjSjz)

’

2 mSh( BCjJ,cgcjsj,) =

exp[ -&(

-D

+ C,J~C~S~)]

+ 2 cosh(/3~,JgjcgcjSjz)

*

(4

(5)

146

J. W. Tucker /

Effectivefield theory of dilute Ising system

With the aid of the differential operator technique these equations can be recast in the form, -D+

CJ$CjS; j

-D

with D, = a/ax, f(x,

Y)

(6)

Ix,y+09

d--G

(7)

Y> I x,y*o,

and

D,, = tl/ay

y)=2

+ CJ,‘iCjS’ i

f(xv

sinh(x)/[exp(-y)

+2cosh(x>],

(8)

g(x, y) = 2 cosh(x)/[exp( -y) + 2 cash(x)].

(9)

Using the fact that the occupation numbers satisfy the relations cf = ci and exp( ac,) = 1 - ci + ci exp( a) it follows that (S,,) = Cc, exp(-PoO,)n (S&3 = $0

(1 -Cj+CjexP[B(JsjSj,Dx+Jg;S~Dy)]))f(x,

Y)~x,~-+ov

(10)

- c,) + ( c,.x~(-BDD~)~~l-cj+cjeXP[~(J~jSj~D~+J~jS~D~)]))

xdx,

Y> I x,y-0,

(11)

where the fact that f(0, 0) = 0 and g(0, 0) = $ has been used. If one now uses the Van der Waerden identities for spin unity, these equations take the form &A

= (c, exp(-BDDY) X7 xf(x,

(1+ cjS’z sinh( BJgjR)

exP( SJi’,D,> + CjSi [COsh( PJgjDx)

exP( PJijD,)

- I]

)> (12)

Y) IX,Y~O,

(S,‘,) = +(I - cg) + (c, exP( -SDD,) X n

(I+

CjSjz sa(

SJgjDx)

exP( SJ,‘iD,> + CjSi [msh( BJgjDx)

exP( SJijD,)

- 11))

i xdx,

(13)

Y> I x,y+o

and are exact. These equations are for the site-diluted problem. To obtain the corresponding results pertaining to the situation where there is bond dilution described by the Hamiltonian, H = - C JijCijSicSjz

(ij>

-

C J,~CijS~S~ + DC Si’, 3

(ii)

(14)

i

one can simply make the replacement cg + 1 and cj -+ cij in these equations. To proceed further, one now has to make approximations as the right-hand side of eqs. (12) and (13) contain thermal averages of multiple correlation functions. The simplest approximation, and the one most frequently adopted, is to decouple these according to (S,S,S, . * . ) = (S,)( S,)(S,) for i #j Z k * - * , thus enabling the thermal averages in eqs. (12) and (13) to be taken inside the product sign. To simplify the

J. W. Tucker / Effectivefield theory of dilute Ising system

notation, only nearest neighbour interactions will be considered nearest-neighbour coordination number z, it follows that

henceforth.

147

For a lattice having a

(S,,> = cg exp(-PW X ,fir (I+ xf(x, (Sk> =

S(l -

cjCSz> sinh(BJoX) exp(PJ’o,)

+ c~(S~>[cosh(PJQ)

exp(BJ’D,)

- I]>

Y) I x,y+o,

W)

cg> + cg exp( -PDDy)

exp( SJ’D,,) x ,el(l + cj(sjz> sinh(PJDx)

xdx,

Y)

+ cj(Si)[cosh(

&JO,) exp( BJ’D,) - l] )

06)

I .&y-o.

It should be remembered that these equations are for a fixed configuration of occupied sites so the thermal averages are configurationally dependent. The next step is to carry out the configurational averaging, to be denoted by ( . . . )r. To make progress, the simplest approximation of neglecting the correlations between quantities pertaining to different sites will be adopted. That is, (X,X,X,), = (X,),( Xj)l( X,), * - . , from whence it follows that ((S~.Z)),=~~~[~

+ (cj(sj,))rDZ

Y) Ix,y-rO,

+ (cj($Z>>rD~]f(x~

((SgZr>>r= St1 -c> + cDi[l + (cj(sjz>>rD*

+ (cj(s,l))rD~]zg(x,

07) Y) Ix,y-+O~

(18)

where the abbreviated notation D, = exp( -/3DD,)

;

D2 = sinh( /?JO,)

exp( PJ’Dy

);

D3 = cosh( /3JDx) exp( PJ'Dy

)- 1

09)

has been introduced. The quantity c is the average site concentration defined by c = (c,)~. Now, a most important point arises. Initially, one might be tempted to decouple (c~(S~~))~ as c((Sjz)),. Indeed, throughout the extensive literature on effective field theories applied to dilute Ising systems (examples are discussed below) there is, to our knowledge, only one instant when this approximation has not been explicitly included (or tacitly assumed) within the decoupling approximation invoked to effect the configurational averaging. In the discussion that follows this decoupling will be termed “the conventional decoupling”. However, it is immediately clear from eq. (15) that cg(S,,) is in fact equal to (S,,), and hence m = (cj(Sj=)), = ((Sjz))r. Similarly, if both sides of eq. (16) are first multiplied by cg before the configurational average is taken, it follows from the resulting equation and eq. (18) that q = ((c&“))~ = ((Siz))r - <(l - c). The quantities m and q defined above are the physically meaningful average magnetization and quadrupolar moment per site respectively, and can thus be determined from the coupled equations m=cD,(I+mD,+qD3)‘f(x, q=

4(1+

m4

+ d4hdxy

Y> Ix,y+~p

(20)

Y) I x,y-o.

(21)

It is observed that these equations are exactly those obtained for the undiluted system [14] apart from the presence of the overall factor c on the right-hand side. Thus, if the binomial expansions on the right-hand side of eqs. (20) and (21) are performed, it follows that m=c[A(q)m+B(q)m3+C(q)m5+

. ..I.

(22)

q=c[A’(q)+B’(q)m2+C’(q)m4+

*..I,

(23)

J. W. Tucker /

148

Effectivefield theoryof diluteZsingsystems

where the coefficients are the same as those in ref. [14]. In deriving these equations, use has ,been, made of the fact that D2 is an odd function of D,, and also that g(x, y) and f(x, y) are even and .odd..in x respectively. The highest power of m occurring on the right-hand side of eqs. (22) and (23) will be the highest odd and even integer s z respectively. Of particular interest is the value of the critical temperature for the second order transition. By taking the limit, m + 0, in eqs. (20) and (21) it follows that the critical temperature is determined by the simultaneous solution of I=

cQ(I

+ @,)‘-‘4f(x,

4 = c&(1

+ q&)=g(x,

Y) lx,Y-o,

(24)

v> I x,y-.o-

(29

At this stage in the discussion, it is interesting to compare the results with those for the quenched dilute system treated within the framework of the conventional decoupling approximation for the configurational averaging. Results from a finite cluster calculation, using this approximation, were recently reported by Saber [16] for the particular example of the BEG model on the honeycomb lattice in the absence .of single-ion anisotropy. Benayad et al. [17] had earlier reported results for the Blume-Cape1 model (i.e. J’ = 0) on the same lattice. In the general case within this configurational approximation, one has in the place of eqs. (20) and (21) for the magnetization and the quadrupolar moment (now defined by m = ((S,,)), and q = ((Sgz))r, respectively), m= q =

cDr[l + cm4 ;(l -

c) +

+ cqDslZf(X,

cD,[l

Y) I x,y+o,

+ cmD2 + cqDh(x9

(26) Y) I x,Y-to-

(27)

The corresponding equations for bond dilution are m=Dr[I

+cm4+cqDJZf(X,

q=D,[l+cmD*+cqD,lzg(x,

Y) lx,Y+o,

(28)

Y)I~,~-o,

(29)

where c is now the bond concentration, ( c~~)~.The results in refs. [16,17] for site and bond dilution are in accord with these general results, apart from the fact that Saber has mised the extra factor 3(1- c) that should have occurred on the right-hand side of his eq. (22) in the case of site dilution. (There is also a misprint in the last term of E2 in ref. [17].) Within the conventional decoupling approximation for the bond diluted model the critical temperature of the second order transition is determined by the coupled equations I = D,(I 4=

+ cqQ)‘-‘cD,f(x,

D,(l + cqD&dx,

Y> I x,Y-+o, Y>

I x,y-o.

PO)

(31)

This, interestingly, gives the same solutions as eqs. (24) and (25) that apply for the site diluted magnet treated within the improved configurational decoupling scheme. However, this correspondence does not carry over to the magnetization and the quadrupolar moment which are a factor c smaller in the latter case, as can be seen on comparing the equations of state, eqs. (20), (21) and eqs. (28), (29) for the two situations. Eqs. (22) and (23) can be analysed in the same way as was done for the corresponding equations in the case of the undiluted lattice [14,15]. Indeed, some results for the tricritical points of the BEG model on a honeycomb lattice have very recently been reported by us [18], although unfortunately, the statement we made there that they differ from those of the bond model treated within the conventional decoupling approximation was incorrect. The fact that the tricritical points in the two cases should agree can be shown

J. W. Tucker / Effective field theory of dilute Ising systems

149

directly from a small m expansion of eqs. (20), (21) and eqs. (28), (29). For the honeycomb lattice we reported that the tricritical temperature fell gradually with concentration, reaching about one third of its value corresponding to the undiluted system, at c = 0.7. Extending the calculations to lower concentrations, we find that below this point the tricritical temperature falls very rapidly, approaching zero at c = f.

3. The dilute spin f Ising model The discussion above concerning the decoupling approximation applied equally well to the spin idilute Ising model. However, because different authors have used different formalisms, essential differences in the results they obtain for the equation of state of the magnetization have gone unnoticed In this section the decoupling procedures and approximations used by various authors are clarified with reference to a single formalism. To our knowledge, all authors apart from those of ref. [19] who worked with the quantity ( ci(Si,)),, used a decoupling method that decoupled the site occupation variable from the thermal average of spin variables, even when both quantities referred to the same site. For the spin 5 problem ( Sj, = f 1) defined by H = - c .JjcicjSirSjr, (ij)

(32)

the generalised Suzuki relation gives

(33) Using the differential operator technique and proceding as in the case of the spin 1 model, it follows that the analogue of eq. (15) is

(sgz)=cg,fI[lj C + Cj

COsh(~~~~) + Ci(Sjr)

sinh(gJD,)]

tanh(X)

(x~o.

Boccara [6] and Benyoussef and Boccara [7], did not use the differential operator technique to express the transcendental function occurring in eq. (33) in polynomial form, and did not present the theory for a general lattice coordination number, but only for z = 3. However, it can be seen that their configurational averaging technique is equivalent to taking the average of eq. (34) as, m=c[l-c+ccosh(j?JD,)+cm

sinh(~~~~)]‘tanh(x)l,,,.

(35)

Their result for z = 3 follows directly from this equation. On the other hand, Balcerzak et al. [19] used the much more realistic decoupling method alluded to above, to give ,=c[l-c+ccosh(PJD,)+msinh(~~~~)]’tanh(x)I,,,.

(36)

Other authors, for example, Matsudaira [3] using the combinatorial approach for specific lattices, Kaneyoshi et al. [20] and Taggart [21] using the differential operator technique, give results (in the version of their theories where the correlations between the different spins are neglected) that are equivalent to (or can be obtained from), m=[l-c+ccosh(/LJDJ+cm

sinh(j?JDX)]ztanh(x)),,,.

(37)

J. W. Tucker /

150

Effectivefield theory of diluteIsing systems

It is of interest to note that this result is just that which is obtained if the conventional decoupling approximation is used for bond dilution. This can be got from eq. (34) when the replacement cg + 1 and ci ---)cii is made, as discussed for the spin 1 model above. Indeed, it is the result obtained for the bond diluted Ising model by Honmura et al. [22] (when the correlations between the spins are neglected, see e.g. their eq. (6)). This equation also gives the results pertaining to bond dilution reported in refs. [6,7] for z = 3. Thus, even though the conventional decoupling method (or the equivalent thereof) has been used in refs. [3,20,21], eq. (37) differs from eq. (35). The reason for this is that those authors have only considered the equation for an occupied central site. In the language of the present paper, this amounts to putting cg = 1 on the right-hand side of eq. (34). Thus the overall factor c, as occurs in eq. (35) does not appear. However, there is evidently an inconsistency in this, as (S,,) on the left-hand side of eq. (34) and the (S,) on the right-hand side are then not the same quantity. The former is conditional on the spin being present, the latter is not. Thus both quantities cannot be replaced by m. Finally, by an argument similar to that presented above for the spin 1 model, it can be shown that the values of the transition temperature for the second order transition determined from eqs. (36) and (37), agree. Thus, though the equations of state are different, the improved decoupling method used in ref. [19] for site-dilution gives a transition temperature equal to that for bond-dilution, obtained within the conventional decoupling approximation. The fact that refs. [3,20,21] arrived at the same transition temperature for the site-diluted model was fortuitous. In conclusion, it should be pointed out that an improvement in the configurational averaging, beyond the conventional method, should be sought for the bond-diluted model. Attempts by us to do this have so far failed, in that they have yielded unsatisfactory results for the percolation limit.

References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [ll]

M. Suzuki, Phys. Lett. 19 (1965) 267. R. Homnura and T. Kaneyoshi, J. Phys. C 12 (1979) 3979. N. Matsudaira, J. Phys. Sot. Japan 35 (1973) 1593. N. Matsudaira and S. Takase, J. Phys. Sot. Japan 36 (1974) 305. S. Takase, J. Phys. Sot. Japan 36 (1974) 636. N. Boccara, Phys. Lett. A 94 (1983) 185. A. Benyoussef and N. Boccara, J. de Phys. 44 (1983) 1143. A. Benyoussef, N. B-a and M. Saber, J. Phys. C 18 (1985) 4275. S. Kobe, AR. Ferchmin and A. Szlaferek, Acta Phys. Pol. A 55 (1979) 707. P. Tomczak, E.F. Sarmento, A.F. Siqueira and A.R. Ferchmin, Phys. Stat. Sol. (b) 142 (1987) 551. A. Bob&k and M. JaXur, Phys. Stat. Sol. (b) 135 (1986) K 9.

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

J.W. Tucker, J. Magn. Magn. Mater. 87 (1990) 16. J.W. Tucker, J. Phys. C 21 (1988) 6215. J.W. Tucker, J. Phys. Condens. Matter 1 (1989) 485. J.W. Tucker, J. Magn. Magn. Mater. 80 (1989) 203. M. Saber, J. Magn. Magn. Mater. 72 (1988) 88. N. Benayad, A. Benyoussef and N. Boccara, J. Phys. C 18 (1985) 1899. J.W. Tucker, J. Appl. Phys. 69 (1991) 6164. T. Balcerzak, A. Bob&k, J. Miehncki and V.H. Trttong, Phys. Stat. Sol. (b) 130 (1985) 183. T. Kaneyoshi, I. Tamura and R Homnura, Phys. Rev. B 29 (1984) 2769. G.B. Taggart, Physica A 116 (1982) 34. R. Honmura, E.F. Sarmento and C. TsaBis, Z. Phys. B 51 (1983) 355.