Diluted mixed-spin transverse Ising system with longitudinal crystal field interactions

*Corresponding author. Fax:#212-2-23-0674.

E-mail addresses: benayad@facsc-achok.ac.ma, nbenayad@

hotmail.com (N. Benayad).

The diluted mixed-spin transverse Ising system with longitudinal crystal "eld interactions

N. Benayad*, A. Fathi, R. Zerhouni

Groupe de Me&canique Statistique, Laboratoire de Physique The&orique, Faculte&des Sciences An(n Chock, Universite Hassan II Ain Chock, B.P. 5366 Maarif, Casablanca, Morocco

Received 18 May 2000; received in revised form 15 August 2000

Abstract

The diluted mixed-spin transverse Ising system consisting of spin-and spin-1 with crystal"eld interactions is studied by the use of an e!ective"eld method within the framework of a single-site cluster theory. The state equations are derived using a probability distribution method based on the use of Van der Waerden identities. The complete phase diagrams are investigated in the case of the honeycomb lattice. In particular, the in#uence of the crystal"eld interactions and the concentration of magnetic sites are examined in detail. We"nd that the system exhibits a variety of interesting features, especially near the percolation threshold. 2000 Elsevier Science B.V. All rights reserved.

PACS: 05.30.!d; 05.50#q; 05.70.!a

Keywords: Mixed spins; Site dilution; Transverse"eld; Crystal"eld interactions

1. Introduction

During several decades, there has been consider- able interest in the study of pure and disordered classical Ising models and their variants because they have been used to describe many physical situations in di!erent "elds of physics. Special at- tention has been focused on the study of two- and three-state spin systems. In the latter case, the in#u- ence of the crystal "eld interaction on the phase diagram has also been investigated.

The e!ect of quantum #uctuations in classical spin models has been investigated extensively for the last few decades. The simplest of such systems is the spin- Ising model in a transverse "eld, which was originally introduced by De Gennes [1] as a valuable model for the tunnelling of the proton in hydrogen-bonded ferroelectrics [2] such as potassi- um dihydrogen phosphate (KH PO ). Since then, it has been successfully applied to several physical systems, such as cooperative Jahn}Teller systems [3] (like D

W VO

and T VO

), ordering in rare- earth compounds with a singlet crystal "eld ground state [4], and also to some real magnetic materials with strong uniaxial anisotropy in a transverse "eld [5]. It has been extensively studied by the use of various techniques [6}10], including the e!ective

"eld treatment [11,12] based on a generalized but

0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 5 6 8 - 0

approximated Callen}Suzuki relation derived by Sa` Barreto et al. In addition to the works on the two-state spin systems, the spin-one transverse Is- ing models [13}19] have received some attention, as well as the quantum transverse spin higher than one [20}25]. The e!ects of the crystal "eld interac- tion on its phase diagram has also been investi- gated [26].

Recently, many e!orts have been devoted to the study of the magnetic properties of the two-sublat- tice mixed-spin- and spin-S Ising systems de- scribed by the Hamiltonian

H "!

6GH7

J GH p XG S XH !

G

C

G p VG !

H

C

H S VH , (1)

where p? G and S

?

H ( a " x, z) are components of spin- and spin-S operators at sites i and j, respectively.

J GH " J is the exchange interaction,

C

G and

C

H are transverse "elds, and the "rst summation is carried out only over nearest-neighbour pairs of spins. The Hamiltonian (1) is of interest because it has less translational symmetry than its single-spin counterparts. It shows spin reversal symmetry ( p X P ! p X , S X P ! S X , p V P # p V , S V P # S V ) which is spontaneously broken below a "eld-de- pendent critical temperature. In the absence of the transverse "elds (

C

G "

C

H " 0), the system is well adapted to study a certain type of ferrimagnetism [27]. Experimentally, it has been shown that the MnNi(EDTA)-6H

O complex is an example of a mixed-spin system [28]. The mixed-spin Ising system, in the case of S " 1, has been studied by renormalization group technique [29,30], by high- temperature series expansions [31], by free-fermion approximation [32] and by "nite-cluster approxi- mation [33]. The in#uence of dilution on the transition temperature of these kind of systems are investigated by performing various techniques [30,33,34]. The e!ects of single-ion crystal "eld interaction on their phase diagrams have also been investigated by renormalization group method [30], Monte-Carlo simulation in the case of square lattice [35], e!ective "eld theory with correlations [36], and "nite-cluster approximation [37]. The two latter methods predict a tricritical behaviour in pure systems with a coordination number z larger than three. It is important to note here that the

exact solution for the phase diagram in (temper- ature-crystal "eld) plane can be obtained analyti- cally if the structure of the system is chosen to be a honeycomb lattice [38,39]. In this case, the transition line is always of second order for any value of the crystal "eld.

On the other hand, the mixed-spin transverse Ising system (

C

G O 0) described by the Hamiltonian (1) have been studied by using di!erent approxim- ate schemes, such that the e!ective-"eld theory based on the generalized Van der Waerden identi- ties [40}42], the two-spin cluster approximation [19], and the discretized path-integral represen- tation [19,43]. Recently, we have investigated [44,45], using the "nite-cluster approximation [46], the e!ects of single-site crystal "eld interac- tion on the phase diagram of the mixed-spin- and spin-1 transverse Ising model.

Recently, the diluted mixed-spin Ising system consisting of spin- and spin-1 in a transverse "eld, has been investigated [41,47] by using an e!ective

"eld method. It has been found that the phase diagrams present outstanding features, especially near the percolation threshold. The purpose of this paper is to examine the e!ects of the single-site crystal "eld interaction on the phase diagram of the diluted mixed-spin transverse Ising system.

This system can be described by Eq. (1) in which we introduce a term corresponding to the single- site crystal "eld interaction and the site occupancy number m G which takes the value 0 or 1 depend- ing on whether the site is occupied or not. Thus, the Hamiltonian of such a system takes the form

H " !

6GH7

J GH m G m H p XG S XH !

C

G m G p VG

!

C

H m H S VH # D

H m H (S XH ) . (2)

In the present work, we limit our study to the case

S " 1. To this end, we used an e!ective method

within the framework of a single-site cluster theory

[48]. The e!ective "eld equations are derived using

a probability distribution method based on the use

of generalized Van der Waerden identities [49] that

account exactly for the single-site kinematic rela-

tions.

The outlines of this work is as follows: in Section 2, we describe the theoretical framework and calcu- late the state equations. In Section 3, we investigate and discuss the phase diagrams and "nally, we comment our results in Section 4.

2. Theoretical framework

The theoretical framework we adopt in the study of the diluted mixed-spin- and spin-1 transverse Ising model with longitudinal crystal "eld interac- tion, described by the Hamiltonian (2), is the e!ec-

tive "eld theory based on a single-site cluster

theory. In this approach, attention is focused on a cluster consisting of just a single selected spin, labelled o, and the neighbouring spins with which it directly interacts. To this end, the total Hamil- tonian given by Eq. (2) is split into two parts, H " H

# H , where H

includes all terms of H as- sociated with the lattice site o, namely

H

N

" ! H

J

H

m m H S XH ^pX

^!

^Cm

^pV

^{, (3)}

H 1 " ! G

J

G

m m G p XG ^{S X}

^!

^Cm

^{S V}

^{# D m}

^{(S X}

⁾

⁽⁴⁾

if the lattice site o belongs to p or S sublattice, respectively.

First, the problem consists in evaluating the sub- lattice longitudinal and transverse components of the magnetization and its quadrupolar moments.

Following Sa` Barreto et al. [11,12], the starting point of our approach, in the framework of the single-site cluster theory, is the set of the following identities:

1p? 2 " ^{Tr Tr}

^NN

^p?

exp( ^exp( ! ^! b ^b H ^H

N

)

^N

^{) (5)}

and

1 (S

?

) L2 " ^{Tr Tr 1}

^(S 1

^?

exp( ^{) L exp(} ! ^! b H ^b 1 ^H ) ¹

^{) , (6)}

where b " 1/ ¹ , a " x or z speci"es the components of the spin operators p? G and S

?

H , and n " 1, 2 corres-

ponds to the magnetization and the quadrupolar moment, respectively. Tr

N

(or Tr

1

) means the par- tial trace with respect to the p -sublattice site o (or S-sublattice site o) and 1 2 2 denotes the canonical thermal average.

Eqs. (5) and (6) neglect the fact that H

and H do not commute. Therefore, they are not exact for an Ising system in a transverse "eld. Nevertheless, they have been successfully applied to a number of inter- esting transverse Ising systems. We emphasize that in the Ising limit (

C

" 0), the Hamiltonian contains only p XG and S XH . Then, relations (5) and (6) become exact identities. One notes that since H

N

and H 1 depend on m ( m " 0 or 1), Eqs. (5) and (6) can be written as

1p? 2 " 1 ! m 2 p # 1 Tr

( p? )

# m ^{Tr Tr}

^NN

^p?

exp( ^exp( ! ^! b ^b H

M

^H

N^M

)

^N

^{) , (7)}

1 (S

?

) L2 " 1 ! m

2S # 1 Tr ((S

?

) L )

# m ^{Tr Tr 1}

^(S 1

^?

exp( ^{) L exp(} ! ^! b H ^b

M

1 ^H )

^M

¹

^{) (8)}

which imply

1m p? 2 " m ^{Tr Tr}

^NN

^p?

exp( ^exp( ! ^! b ^b H

M

^H

N^M

)

^N

^{) , (9)}

1m (S

?

) L2 " m ^{Tr Tr 1}

^(S 1

^?

exp( ^{) L exp(} ! ^! b H ^b

M

1 ^H )

^M

¹

^{) (10)}

with

H

M N

"! H ^X

J

H

m H S XH ^pX

^!

^CpV

and

H

M

1 "! G ^X

J

G

m G p XG ^{S X}

^!

^C

^{S V}

^{# D(S X}

⁾

^,

where z is the nearest-neighbour coordination

number of the lattice.

Now we have to evaluate the partial traces on the right-hand side of Eqs. (9) and (10) over the states of the selected spins, labelled o. To do this, "rst one has to "nd the eigenstates and eigenvalues of H

M N

and H

M

1 in a representation in which pX and S X are diagonal. For H

M N

, it is more convenient to write it in a diagonal form by using the following rotation transformation:

pX " cos

u

pXY ! sin

u

pVY , (11a) pV " sin

u

pXY # cos

u

pVY (11b) with

cos

u

" E E 1

, sin

u

"

C

E

, (12)

where E 1 " X

H

J

H

m H S XH , E " (E 1 #

C

) . (13) Then, for a "xed con"guration of the site occupa- tional numbers m G , we obtain

1m p? 2 " m 1 f

?(E

1 ,

C

) 2 (14) with

f X (E

1 ,

C

) " E 2E 1

tanh ^{b E 2}

^{, (15)}

f V (E 1 ,

C

) "

C

2E

tanh ^{b E 2}

^{. (16)}

H

M

1 can also readily be diagonalized. Its eigenvalues c I are given by

c I " 2

3 [D #

(g

cos( h I )] with k " 1, 2, 3 (17) with

h I " 1

3 ar cos ^{! 2 g 27 r # 2 3 (k ! 1)}

^p

^{, (18)}

g " 3

(

3

2 [27r # " 4r # 27r

"

] (19)

and

r "! (E

N

#

C

) ! D 3 , r "

! D

3 ^2E

^N

^{! 2 9 D}

^!

^C

^{, (20)}

where E

N

" X

G J

G

m G p XG .

The corresponding eigenvectors are

"s2 I " a I " # 2 # b I " ! 2 # c I " 0 2 (21) with

a I "

"C ( c I ! D ! E

N

) "

(

2 +C [( c I ! D) # E

N

] # [( c I ! D) ! E

N

]

,

(22) b I " c I ! D # E

c I ! D ! E

N N

a I , c I "

(

2

C

( c I ! D # E

N

)a

I . (23)

Using the above eigenvalues and eigenvectors, to perform the inner traces in Eq. (10) and setting n " 1 and 2, we obtain

1m (S

?

) L2 " m 1 F

?

L (E

N

, D,

C

) 2 , (24) where

F X (E

N

, D,

C

) " I (a I ! b I ) exp( ! bc I )

I exp ( ! bc I ) , (25) F X (E

N

, D,

C

) " I (a I # b I ) exp( ! bc I )

I exp ( ! bc I ) , (26) F V (E

N

, D,

C

) "

(

2 I (a I # b I )c I exp( ! bc I )

I exp( ! bc I ) ,

(27) F V (E

N

, D,

C

) " I [ (a

I # b

I ) # c I ] exp( ! bc I )

I exp( ! bc I ) .

(28)

The next step is to carry out the con"gurational averaging over the site occupational numbers m G , to be denoted by 1 2 2 P .

In order to perform the thermal and con"gura- tional averaging on the right-hand side of Eqs. (14) and (24), we expand the functions f

?

(E

1 ,

C

) and F

?

L (E

N

, D,

C

) as "nite polynomials of S XH and p XG , re-

spectively, that correctly account for the single-site kinematic relations. This can conveniently be done by employing the Van der Waerden operators [49]

f

?

(E

1 ,

C

) " “ H

O

1

(S XH , m H ) f

?

(E

1 ,

C

), (29)

F

?

L (E

N

, D,

C

) " “ G

O

N

( p XG , m G ) F

?

L (E

N

, D,

C

), (30) where

O

N

( p XG , m G ) " ^{p XG # 1 2}

^dNXG

# ^{! p XG # 1 2}

^dNXG

;

[ m G

dm

G # (1 ! m G )

dK^G

,

], (31) O

1

(S XH , m H ) " ^{1 2 (S XH # (S XH )}

⁾

^d

¹

^XH

# 1

2 ( ! S XH # (S XH ) )

d

1

^XH\

(32)

# (1 ! (S XH ) )

d

1

^XH

^,

;

[ m H

dK^XH

# (1 ! m H )

dK^H

],

where

d

? is a forward Kronecker delta-function substituting any operator A to the right by its eigenvalue a. In order to carry out the thermal and con"gurational averaging, we have to deal with correlation functions. In this work, we consider the simplest approximation by neglecting correlations between quantities pertaining to di!erent sites, but we include the correlation between the site disorder and the local con"gurational-dependent thermal averages of the spin operators [50] and use the

exact identities

11 (1 ! m ) (S

?

) L22 P " 1 ! c

2S # 1 Tr ((S

?

) L ), (33) 11 (1 ! m ) p? 22 P " 1 ! c

2 p # 1 Tr ( p? ) (34) which are directly derived from Eqs. (7)}(10). c de- notes the average site concentration de"ned by c " 1 m G 2 P . Doing this, we "nd

11 f

?

(E

1 ,

C

) 22 P "

“ X

H 1X

^H

^>

\ K^H

P(S XH , m H ) ^f

^?

^{(E 1 ,}

^C

^),

(35) 11 F

?

L (E

N

, D,

C

) 22 P "

“ X

G

N

X

G

^>

\ K^G

R( p XG , m G ) ^F

^?

^{L (E}

^N

^{, D,}

^C

^{) (36)}

with

P(S XH , m H ) " >

'

\

'

a(I , I ) d 1

^XH'

d

K^H'

, (37) R( p XG , m G ) " >

I

\

I

b(k , k

) d

N^XGI

d

K^GI

, (38) where

a( $ 1, 1) " 1

2 ( $ m X # m X ), (39a) a(0, 1) " (c ! m X ), (39b) a(I , 0) " 1

3 (1 ! c), (39c)

b ^{$ 2 1 , 1 " 2 c $ kX , (40a)}

b ^{$ 2 1 , 0 " 1 2 (1 ! c), (40b)}

where,

kX " 11m G p XG 22 P , m XL " 11m H (S XH ) L22 P . (41)

Thus, using the probability distributions, we obtain the following set of coupled equations for k? and m

?

L : k? " c >

'

\

2 >

'

^X\

K

2

K^X

H ^{“ X}

a(I H , m H )

;

f

?

( m S X (I

),2, m X S XX (I

X );

C

), (42)

m

?

L " c >

I

\

2 >

I

^X\

K

2

K^X

G ^{“ X}

b(k G , m G )

;

F

?

L ( m pX (k

),2, m X p XX (k

X ); D;

C

) (43)

with S XH (I) " I and p XG (k) " k. We like to note that these equations can be solved directly by numerical iteration without further algebraic calculations.

This treatment has successfully been used in the study of other systems [51]. Since the total number of loops 2z is relatively large, the combined sums in Eqs. (42) and (43) extend over large numbers ([2(2S # 1)] and [2(2 p # 1)] , respectively) of terms, leading to quite long computational time, particularly near second-order phase transition.

Therefore, it is advantageous to carry out further algebraic manipulations in Eqs. (35) and (36) em- ploying the di!erential operator technique f

?

(E 1 ,

C

) " exp(E 1 D V ) f

?

(x,

C

) " V , (44) F

?

L (E

N

, D,

C

) " exp(E

N

D

V )F

?

L (x, D,

C

) " V (45) or the integral representation

f

?

(E

1 ,

C

) " ^{dx d (x ! E 1 ) f}

^?

^(x,

^C

^{), (46)}

F

?

L (E

N

, D,

C

) " ^{dx d (x ! E}

^N

^{) F}

^?

^{L (x, D,}

^C

^{) (47)}

with the delta-function d (x) " 1

2

p

^{dy exp(iyx). (48)}

Choosing the di!erential operator approach, we obtain from Eqs. (42)}(45)

k? " c '

^>

\

>

'

a(I , I )

;

exp(I I

b JD

V )] X f

?

(x,

C

) " V , (49)

m

?

L " c I

^>

\

>

I

b(k , k )

;

exp(k k

b JD

V )] X F

?

L (x, D,

C

) " V (50) which can be reduced to

k? " c ^{1 2 (m X}

^{# m X}

^{) exp( b JD V ) # 1 2 ( ! m X}

^{# m X}

⁾

;

exp( ! b JD

V ) # (1 ! m X )] X; f

?

(x,

C

) " V . (51) m

?

L " c ^{c 2 # kX exp b JD 2 V # c 2 ! kX}

;

exp ^{! b JD 2 V # (1 ! c) X}

^;

^F

^?

^{L (x, D,}

^C

^{) " V .}

(52) Using the multinomial expansion, we "nd

k? " c X

L

X\L

L

2

\L\L

C L X

C L X\L

;

(m X # m X ) L

( ! m X # m X ) L

;

(1 ! m X ) X\L

\L

f

?

( b J(n ! n

),

C

), (53) m

?

L " c X

L

X\L

L

C L X

C L X\L

^{c 2 # kX L}

;

^{c 2 ! kX L}

^{(1 ! c) X\L}

^\L

;

F

?

L ^{b 2 J (n}

^{! n}

^{), D,}

^C

^{, (54)}

where C NL are the binomial coe$cients n!/

[p!(n ! p)!]. The iteration process of these equa- tions becomes suitable for the study of the present system even in the vicinity of the critical temper- ature.

3. Results and discussions

In this paper, we are interested in investigating

the phase diagram of the system described by the

Fig. 1. The phase diagram in¹}D}Cspace of the mixed-spin- and spin-1 transverse Ising system on honeycomb lattice. The number accompanying each curve denotes the value ofD/J.

Hamiltonian (2). At high temperature, the longitu- dinal magnetization moments kX and m XL are both equal to zero. Below a transition temperature ¹

, we have spontaneous ordering ( kX O 0, m X O 0), while the corresponding transverse magnetizations kV and m V are unequal zero at all temperatures. To calculate ¹

, it is preferable to expand the right- hand sides of Eqs. (53) and (54) with respect to m X (or kX ). Doing this, we "nd

k? " c X

L

X\L

L

L

G

L

G

2

\L\L

C L X

C L X\L

;

C G L

C G L

( ! 1) G

(m X ) G

>G

(m X ) L

>L

\G\G

;

(1 ! m X ) X\L

\L

f

?

( b J(n ! n ),

C

) (55) and

m

?

L " c X

L

X\L

L

L

G

L

G

2

\L\L

>G

>G

;

C L X

C L X\L

C G L

C G L

( ! 1) G

(c) L

>L

\G\G

( kX ) G

>G

;

(1 ! c) X\L

\L

F

?

L ^{b 2 J (n}

^{! n}

^{), D,}

^C

^{. (56)}

For the z-components ( a " z), they can be written in the following form:

kX " A

( b J, c,

C

, m X ) m X # B

( b J, c,

C

, m X )

;

(m X ) #2 , (57)

m X " A

( b J, D, c,

C

) kX # B

( b J, D, c,

C

)

;

( kX ) #2 , (58)

where A G , B

G ,2(i " 1,2) are obtained from Eqs. (55) and (56) by choosing the appropriate correspond- ing combinations of indices i H ( j " 1, 2). Retaining only terms linear in kX and mX , the second-order transition temperature is then obtained from the equation

1 " A

( b J, c,

C

, m X

A

) A

( b J, D, c,

C

), (59) where m X

A

is the solution of Eq. (56) for kX P 0, namely

m X

A

" c X

L

X\L

L

C L X

C L X\L

2

\L\L

(c) L

>L

;

(1 ! c) X\L

\L

F X ^{b 2 J (n}

^{! n}

^{), D,}

^C

^{. (60)}

First, we study the undiluted version (c " 1) of the system described by the Hamiltonian (2) with m G " 1,

∀

i. In Fig. 1, we represent the phase diagram in the ¹ }

C

}D space for a coordination number appropriate to the honeycomb lattice (z " 3). For a given value of the crystal "eld lower than D " 1.5 J, the critical temperature decreases grad- ually from its value in the mixed Ising system

¹ /J(

C

" 0), to vanish at some D-dependent criti- cal value

C

of the transverse "eld (for instance

C

(D " 0) " 1.42 J). It is important to note that this system does not exhibit a tricritical behaviour for any value of the parameters ¹ , D and

C

. This result is supported by the exact solution [38,39] obtained for

C

" 0.

Now, let us investigate the phase diagrams of the diluted mixed spin- and spin-1 transverse Ising system with longitudinal crystal "eld on honey- comb lattice (z " 3) by solving numerically Eq. (59).

First of all, we study the system in the absence of the transverse "eld and the crystal "eld interaction (

C

" 0, D " 0). For the honeycomb lattice, the phase diagram is represented in Fig. 2 and it expresses the standard result of a diluted magnetic system [33,34,47]. The critical temperature ¹ decreases linearly from its value in the mixed-Ising system ¹

(c " 1) to reduce rapidly to zero at the

percolation threshold c* " 0.5378 which is in

moderate agreement with the value 0.698 $ 0.003

obtained by series expansion [52].

Fig. 2. The phase diagram of the diluted mixed-spin- and spin-1 Ising system on honeycomb lattice in absence of the transverse"eld and crystal"eld (D"C"0).

Fig. 3. The phase diagram in¹}Dplane of the diluted mixed- spin-and spin-1 Ising system with longitudinal crystal"eld on honeycomb lattice. The number accompanying each curve de- notes the value of the dilution parameterc.

Secondly, the e!ects of the crystal "eld interac- tion on the diluted mixed spin Ising system are summarized in Fig. 3 which gives, in the absence of the transverse "eld (

C

" 0), the sections of the criti- cal surface ¹

(c, D) with planes of "xed values of the dilution parameter c. As seen in the "gure when c* ( c ( 1, the transition temperature decreases from its value ¹

/J(D " 0) to vanish at a critical value D

/J of the crystal "eld which depends on the

value of c. Moreover, a reentrant phenomenon is observed for certain ranges of c. An important behaviour of the system is found with c " 0.5 (c less than c*): ¹

reduces to zero at D " 0 but, in a certain range of D (0 ( D ( D

), the system exhibits a second-order transition at a "nite value of

¹ which vanishes at D " D

. These results indicate that for small crystal "eld strength, the system may have a magnetic ordering even if c is less than c*.

Such behaviour may be obtained in the system when the dilution parameter belongs to the narrow range 0.4975 ( c ) c*. These results have a form similar to those found by other authors [53,54].

Next, we investigate the e!ects of the crystal "eld interaction on the diluted mixed-spin transverse Ising model (

C

O 0). We select three values of the dilution parameter c which corresponding to the three behaviours observed in Fig. 3. Thus, in Fig.

4(a) (with c " 0.9), the general variation of the criti- cal temperature ¹

(

C

, D) falls with increasing

C

and D, and vanishes at a critical value D

of the crystal

"eld which depends on the value of

C

. In Fig. 4(b),

we have plotted, in ¹ }D plane, the phase diagram for selected values of the transverse "eld when c " 0.8. For such value of c, the system exhibits, in a narrow range of D, a reentrant behaviour only when

C

is relatively small (0 )

C

/J ) 0.36) and disappears when

C

is larger than

C

/J " 0.36.

Reentrance in this sense can be explained by a com- petition of energy E and entropy S in the free energy F " E} ¹ S. The entropy (measuring the dis- order of the state) is usually less important at low temperatures. For pure systems the energy term becomes dominant and an ordered state of the physical system is preferred. However, for a mag- netic system with site dilution and crystal "eld interaction an e!ective reduction of the energy E as compared with a disordered state may not be pos- sible for all interaction parameters. First of all, the crystal "eld D by itself energetically disfavours the state with " S

H " " 1 for positive values of D. Second- ly, the site-dilution induces inherent disorder. For a given set of D and c this conspires to a thermo- dynamic state with 1 S

H 2 " 0 for su$ciently low and su$ciently high temperature, but with 1 S

H 2 ' 0 for some intermediate temperature and

hence 1p G 2 ' 0. The transversal "eld is expected to

order the system in transversal direction and to

Fig. 4. The phase diagram of the diluted mixed-spin-1/2 and spin-1 transverse Ising system on honeycomb lattice when (a)c"0.9, (b) c"0.8, and (c)c"0.5. The number accompanying each curve denotes the value ofC/J.

reduce the e!ect of disorder. This qualitative expec- tation is indeed veri"ed by our quantitative calcu- lations.

Fig. 4(c) shows how the magnetic ordering in- duced by small values of D, behaves when the concentration c of magnetic sites is equal or very near c*. Thus, for instance c " 0.5, the region which corresponds to the long-range ferromagnetic order, decreases with increasing values of the transverse

"eld

C

and disappears at a c-dependent value of

C

.

Therefore, when c is less than 0.4975, no magne- tic ordering exists in the system for any values of D and

C

.

Furthermore, it is interesting to investigate the phase diagram of the system in the ¹ }c plane when D and

C

are kept "xed. This allows us to know the in#uence of the crystal "eld D on the site-dilution model, particularly on the dilution curve depicted in Fig. 2. The results are presented in Fig. 5 for various values of D when the transverse "eld is chosen to be zero. As shown in the "gure, some curves present a reentrant behaviour in appropriate ranges of D and c as it was observed in Fig. 3. We also note the existence of di!erent thresholds as solutions of the equation ¹

(c, D,

C

" 0) " 0. It in-

dicates that these thresholds in zero transverse

Fig. 5. The phase diagram in¹}cplane of the diluted mixed- spin-and spin-1 Ising system with longitudinal crystal"eld on honeycomb lattice. The number accompanying each curve de- notes the value of the crystal"eldD/J.

Fig. 6. The dependence of the dilution parametercas a function of D/J with zero-transverse"eld (C/J"0), when the critical temperature¹

is kept"xed at or very near zero.

Fig. 7. The variation of the critical value ofcwithD/Jat¹ "0 for the honeycomb lattice in non-zero transverse "eld. The number accompanying each curve denotes the value ofC/J.

"eld, depend strongly on the strength of the crystal

"eld. Therefore, it is interesting to clarify the role of D on the value of c at which the critical temperature

¹ goes to zero. Thus, in Fig. 6, we plot the vari- ation of the concentration c of magnetic sites in the system with D at very low critical temperatures ( ¹

" 0.05 J and ¹

" 0.02 J). As is clearly ex-

pressed in the "gure, the critical concentration varies in a stepwise way at ¹

" 0. This latter behaviour of c change dramatically in non-zero transverse "eld (

C

O 0). This is shown in zero- temperature ( ¹

" 0) phase diagram plotted in Fig.

7. We note that the outstanding feature obtained for

C

" 0 disappears and then the critical value of the concentration c change continuously with D/J.

Now, let us investigate for the system under study, the zero-temperature phase diagram in the

C

}c plane, since it is of considerable interest. It is obtained from the solution of the Eq. (59) keeping

¹ " 0. Fig. 8(a) shows the dependence of the criti-

cal value

C

on the concentration c when D/J takes

di!erent values. As seen from this "gure, the phase

diagrams represent an outstanding feature,

specially near the percolation threshold c*. In par-

ticular, for D/J " 0 the critical transverse "eld

C

takes a "nite value at c " c* and falls vertically

from a "nite value to zero at c " 0.5277 very near

c* " 0.5378. This result may support the conjecture

made by Harris [8] for the diluted transverse Ising

model (DTIM), which can be summarized as fol-

lows: at percolation threshold c*, the critical trans-

verse "eld should display a discontinuity. It is

worthy of notice here that the investigation of the

DTIM by series expansion techniques [55], CPA

treatments and e!ective "eld theory [56] led to

Fig. 8. (a) The zero-temperature phase diagram of the diluted mixed-spin-and spin-1 transverse Ising system with longitudinal crystal

"eld interaction on honeycomb lattice. The number accompanying each curve denotes the value ofD/J; (b) The zero-temperature phase

diagram of the diluted mixed-spin-1/2 and spin-1 transverse Ising system with longitudinal crystal"eld interaction on honeycomb lattice, on an enlarged scale for the values ofD/Jwhich leads to the discontinuity ofC

a critical transverse "eld

C

which reduces continu- ously to zero at c " c*. However, the position space renormalization group methods [9,57,58] showed the existence of a discontinuity of

C

at c " c*, and therefore they veri"ed the Harris conjecture. It is worth noting here that the above observed discon- tinuity for D/J " 0 is similar to that found in our very recent work [47] on the diluted mixed spin in a transverse random "eld. On the other hand, the presence of the crystal "eld interactions do not usually induce a discontinuity of

C

for any value of D. In fact and as seen in Fig. 8(a), the

C

presents a discontinuity only for few values of D/J namely D/J " 0.25, 0.5, 0.75, 1, and 1.25. Each correspond- ing discontinuity of

C

is located on a (D-depen- dent) well de"ned value of c. We can also note that the height of the observed discontinuities do not depend on the strength of the crystal "eld as is clearly shown in Fig. 8(b).

4. Conclusions

In this paper, we have studied the e!ects of the crystal "eld on the diluted mixed-spin transverse Ising system consisting of spin- and spin-1. We have used an e!ective "eld method within the

framework of a single-site cluster theory. In this approach, we have derived the state equations us- ing a probability distribution method based on the use of Van der Waerden identities accounting exact- ly for the single-site kinematic relations [48,49]. We have also included the correlation between the site disorder and local con"gurational-dependent ther- mal average of spin operators [50]. Let us summar- ize by stating the main results of this investigation.

First, we have studied the in#uence of the crystal

"eld D on the undiluted mixed-spin transverse Ising

system. We have examined, in the case of the honeycomb lattice, the full phase diagram and found that the system does not exhibit a tricritical behaviour for any values of D. This result is sup- ported by the exact solution [38,39] obtained in zero-transverse "eld (

C

" 0). However, the e!ective

"eld theory used here is far from perfect in its predictions for a mixed spin }1 system. Even for the undiluted square lattice in the absence of a transverse "eld it incorrectly predicts a tricritical point, and for ferrimagnetic exchange coupling a compensation point is wrongly predicted for both z " 3 and 4 [59].

For the site-diluted case, we have "rst investi-

gated the zero-transverse "eld phase diagrams of

the system for di!erent values of the dilution

parameter c. We have found that for the values of c greater than the percolation threshold c* "

0.5378, the dependence of transition temperature

¹ with D is, either qualitatively similar to that obtained for the pure system or we observe a reen- trant behaviour at low temperatures in certain ranges of c and D. A third behaviour of

¹ (D,

C

" 0) has been found when 0.4975 ( c ) c*

predicts that for small crystal "eld interaction, the system may have a magnetic order which disap- pears when c approaches 0.4975. In Section 3 we have given a possible explanation of the reentrant behaviour. However, some caution is required be- cause similar reentrances have been predicted in other contexts using e!ective "eld theory for which a simple explanation is lacking. For example, it was found that such behaviour occurs in a simple Ising ferromagnets even in the absence of single ion an- isotropy [60]. Next, we have investigated the e!ects of D on the system when a transverse "eld

C

acts on the two sublattices. We have found that the reen- trant behaviour, observed for certain D and c ' c*, persists in the system only for relatively small values of

C

. On the other hand, for a given value of c(0.497 ( c ) c*) less than c*, the induced magnetic order (in a well-de"ned range of D) disappears at a c-dependent value of the transverse "eld. Further- more, we have study the phase diagram, at ¹

" 0 in the c}D plane for various values of

C

. We have noted that the critical concentration varies in a stepwise way with D at

C

" 0 and remarkably depends on D when

C

' 0. We have also plotted the zero-temperature phase diagram in the

C

}c plane for di!erent values D of the crystal "eld. The phase diagrams represent an outstanding feature.

First, in the case of D " 0,

C

exhibits a discontinu- ity change from a "nite value to zero at a value of c below c* which may support the Harris conjec- ture [8]. Secondly, when D O 0,

C

exhibits a dis- continuity only for few discretized values of D at well-de"ned values of c.

Acknowledgements

The authors would like to thank the referee for interesting comments and for suggesting the Ref. [58}60].

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