Mixed spin Ising model with four-spin interaction and random crystal field

Mixed spin Ising model with four-spin interaction and random crystal field

N. Benayad

^a,b,n

, M. Ghliyem

^a,b

aGroupe de Me´canique Statistique, Laboratoire de physique the´orique et applique´e, Faculte´ des sciences-Aı¨n Chock, Universite´ Hassan II-Casablanca, B.P 5366 Maarif, Casablanca 20100, Morocco

bLaboratoire de physique des hautes e´nergies et de la matiere condense´e, Faculte´ des sciences-Aı¨n Chock, Universite´ Hassan II-Casablanca, B.P 5366 Maarif, Casablanca 20100, Morocco

a r t i c l e i n f o

Article history:

Received 21 December 2010 Received in revised form 27 July 2011

Accepted 26 August 2011 Available online 22 September 2011 Keywords:

Mixed spin Ising model Four-spin interactions Random crystal field Re-entrant phenomenon

a b s t r a c t

The effects of fluctuations of the crystal field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions are investigated within the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the two-dimensional square lattice. It has been found that the system exhibits a variety of interesting features resulting from the fluctuation of the crystal field interactions. In particular, for low mean valueDof the crystal field, the critical temperature is not very sensitive to fluctuations and all transitions are of second order for any value of the four-spin interactions. But for relatively high D, the transition temperature depends on the fluctuation of the crystal field, and the system undergoes tricritical behaviour for any strength of the four-spin interactions. We have also found that the model may exhibit reentrance for appropriate values of the system parameters.

&2011 Elsevier B.V. All rights reserved.

1. Introduction

Over recent years, a particular interest has been devoted to the theoretical and experimental study of Ising models with multi- spin interactions. The origin of such interactions found its theoretical explanation in the theories of the superexchange interaction, the magnetoelastic effect, the perturbation expansion, and the spin–phonon coupling [1]. The investigation of models with higher-order interactions is important, since they may exhibit rich phase diagrams and can describe phase transitions in some physical systems. They also show physical behaviours not observed in the usual spin systems. For instance, they display the nonuniversal critical phenomena [2,3], and deviations from the bloch

T^3/2

law at low temperatures [4].

Theoretically, monoatomic Ising models, with multispin inter- actions have been studied within different methods, such as mean field approximation [5,6], effective field theory [7–9], series expan- sions [10,11], renormalization group methods [12], Monte Carlo simulations [13], and exact calculations [14]. From the experi- mental point of view, the models with multispin interactions can be used to describe various physical systems such as classical fluids [15], solid

³

He [16], lipid bilayers [17], and rare gases [18]. More- over, it has been shown that for certain materials, these interac- tions play a significant role and they are comparable or even much

important than the bilinear ones. Indeed, the models with pair and quartet interactions have been used to study and explain the existence of first-order phase transition in squaric acid crystal H

₂

C

₂

O

₄

[19]. Such models have been also applied to describe thermodynamical properties of hydrogen-bonded ferroelectrics PbHPO

4

and PbDPO

4

[20], copolymers [21], and optical conductiv- ity [22] observed in the cuprate ladder La

_x

Ca

_14x

Cu

₂₄

O

₄₁

. It is worthy to note here that the four spin interaction plays an important role in the two-dimensional antiferromagnet La

2

CuO

4

[23], the parent material of high-T

c

superconductors.

Recently, attention has been directed to the study of the magnetic properties of two-sublattice mixed spin Ising systems.

They are of interest for the following main reasons. They have less translational symmetry than their single-spin counterparts, and they are well adopted to study a certain type of ferrimagnetism [24].

Experimentally, it has been shown that the MnNi(EDTA) 6H

2

O complex [25] is an example of a mixed spin system. The mixed Ising spin model consisting of spin-1/2 and spin-1 with only two- bilinear interaction has been studied by the renormalization group technique [26], by high temperature series expansions [27], by free- fermion approximation [28], and by finite cluster approximation [29]. The influence of uniform crystal-field interactions on its transition temperature have been also investigated using exact calculations (for a honeycomb lattice) [30], Monte Carlo simulation [31], renormalization group method [32], cluster variation method [33], and finite cluster approximation [34]. The two latter methods predict a tricritical behaviour in systems with a coordination number larger than three. We have to note here that, one of us (N.B) has shown that the fluctuations of the crystal-field interaction

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journal homepage:www.elsevier.com/locate/physb

Physica B

0921-4526/$ - see front matter&2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.physb.2011.08.087

nCorresponding author. Tel.:þ212 5 22 23 06 84; fax:þ212 5 22 23 06 74.

E-mail addresses:n.benayad@fsac.ac.ma, noureddine_benayad@yahoo.fr (N. Benayad).

modify qualitatively and quantitatively the phase diagram [35]. The introduction of multispin interactions in such systems certainly modifies their magnetic properties. Experimentally, it has been shown that the mixed spin Ising model with four-spin interactions can be used to describe all types of collinear metamagnets [36].

In a very recent work [37], we have investigated, using the finite cluster approximation [38,39], the mixed spin-1/2 and spin- 1 Ising model with four-spin interactions on the square lattice.

We also examined the influence of uniform crystal-field interac- tions on the obtained phase diagram. In particular, we found that for large negative values of the four-spin interaction

J4

, the second-order transition temperature increases for increasing

J4

. Whereas, for the remaining part of the phase diagram, the critical temperature increases with the increase of

J4

, passes through a maximum and then decreases to reach the tricritical point. In particular, such behaviour has also been observed in the absence of the crystal-field (D¼0). We have to mention that these results are qualitatively in agreement with those obtained recently by Monte Carlo simulation [40]. In these studies, we neglected the fluctuations of the crystal-field interactions.

The purpose of the present work is to examine the effects of the fluctuations of the crystal-field interactions on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four- spin interactions on the square lattice. Such a system can be described by the following Hamiltonian:

H ¼ J

₂X

/ijS

s

i

S

j

J

₄X

fi,j,k,lg

s

i

s

k

S

j

S

l

þ

X

i

D

i

S

²_i

ð1Þ

The underlying lattice is composed of two interpenetrating sublattices. One occupied by spins with spin moment s ¼

7

1/2 and the other one is occupied by spins with spin moment

S¼0, 7

1. The first summation is carried out only over nearest-

neighbour pair of spins. The second term represents the four-spin interaction, where the summation is over all alternate squares, shaded in Fig. 1.

Di

describes the random crystal field interactions with an independent probability distribution function

P

(D

i

) PðD

i

Þ ¼

¹₂

d D

i

Dð1þdÞ

þ d D

i

Dð1dÞ

ð2Þ with

d ¼ D D D

r1

where D

D

is the fluctuation from the mean value

D.

Our presentation is as follows: In Section 2, we describe the theoretical framework and calculate the state equations. In Section 3, we investigate and discuss the phase diagrams. Our concluding remarks are summarized in Section 4.

2. Theoretical framework

The theoretical framework that we adopt in the study of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions and random crystal field described by the Hamiltonian (1) is the finite cluster approximation (FCA) [38,39] based on a single-site cluster theory. We have to mention that this method has been successfully applied to a number of interesting pure and disordered spin Ising systems [34,41–43]. It has also been used for transverse Ising models [35,44–46] and semi-infinite Ising systems [47–50]. In all these applications, it was shown that the FCA improves qualita- tively and quantitatively the results obtained in the frame of the mean-field theory. In this approach, attention is focused on a cluster comprising just a single selected spin s

0

(S

0

) and its neighbour spins { s

1

, s

2

,S

1

,S

2

,S

3

,S

4

}({S

1

,S

2

, s

1

, s

2

, s

3

, s

4

}) with which it directly interacts (see Fig. 2). We split the total Hamiltonian (1) into two parts,

H¼H0

þH

⁰

, where

H0

includes all parts of

H

associated with the lattice site o. In the present system,

H0

takes the form

H

os

¼ J

₂X⁴

i¼1

S

i

þJ

₄

ðS

1

S

2

s

1

þS

3

S

4

s

2

Þ

" #

s

0

ð3Þ

H

os

¼ J

₂X⁴

j¼1

s

j

þJ

₄

ð s

1

s

2

S

1

þ s

3

s

4

S

2

Þ

2

4

3

5S₀

þD

0

S

²₀

ð4Þ

whether the lattice site

o

belongs to s or S-sublattice, respectively.

The problem consists in evaluating the sublattice magnetiza- tions and the quadrupolar moment. To this end, we denote by

/

s

0S_c

and,

/S0nS_c^e

( e ¼ þ or and

n¼1, 2), respectively, the mean

value of s

0

and

S0n

for a given configuration

c

of all other spins (i.e.

when all other spin s

i

and

Sj

(i,

ja

0) are kept fixed) and a fixed

Fig. 1.Part of the square lattice.Kandcorrespond tosand S-sublattice sites,

respectively.

σ

³

S

2

S

¹

S

⁴

S

2

S

³

σ

2

S

O

σ

4

S

1

σ

¹

σ

⁰

σ

¹

σ

²

Fig. 2.(a) Neighbours of spinsowith which it directly interacts. (b) Neighbours of spinSowith which it directly interacts.

configuration {D

i

¼(1

7d)D} of the random crystal field. /

s

0S_c

and

/S0nS_c^e

are given by

/

s

oS_c

¼ Tr

so

s

o

expð b H

os

Þ Tr

so

expð b H

os

Þ ð5Þ

/

S

ⁿ₀S^e

c

¼ Tr

So

S

ⁿ_o

expð b H

os

Þ Tr

So

expð b H

os

Þ ð6Þ

where Tr

so

(or Tr

So

) means the trace performed over s

o

(or

So

) only.

As usual

ß¼1/T, whereT

is the absolute temperature. According to the probability distribution (2), where the crystal field

Di

on any site

i

is assumed to take on two values

D(17d) with equal

probability, the sublattice magnetizations m ,

m, and the quad-

rupolar moment

q

are then given by

m

//

s

oS

cS

¼ Tr

so

s

o

expð b H

os

Þ Tr

so

expð b H

os

Þ

* +

ð7Þ

m 1 2

//

S

oS^þ

c

þ

/

S

oS

cS

¼ 1 2

Tr

So

S

o

expð b H

os

Þ Tr

So

expð b H

os

Þ

Do¼Dð1þdÞ

þ Tr

So

S

o

exp ð b H

os

Þ Tr

So

exp ð b H

os

Þ

Do¼Dð1dÞ

* +

ð8Þ q 1

2

//

S

²_oS^þ

c

þ

/

S

²_oS

cS

¼ 1 2

Tr

So

S

²_o

expð b H

os

Þ Tr

So

expð b H

os

Þ

Do¼Dð1þdÞ

þ Tr

So

S

²_o

expð b H

os

Þ Tr

So

expð b H

os

Þ

Do¼Dð1dÞ

* +

ð9Þ where

/yS

denotes the average over all spin configurations.

Performing the inner traces in Eq. (7)–(9) over the states of the selected spin s

0

(S

0

), we obtain the following exact relations:

m ¼ 1 2 tanh K

2 ðS

1

þS

2

þS

3

þS

4

Þ þ a ðS

1

S

2

s

1

þS

3

S

4

s

2

Þ

ð10Þ

m ¼

sinh½Kfðs1þs2þs3þs4Þ þaðs1s2S₁þs3s4S₂Þg expðbDð1þdÞÞ þ2 cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S1þs3s4S2Þg

þ

sinh½Kfðs1þs2þs3þs4Þ þaðs1s2S₁þs3s4S₂Þg expðbDð1dÞÞ þ2 cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S1þs3s4S2Þg

* +

ð11Þ

q ¼

cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S1þs3s4S2Þg expðbDð1þdÞÞ þ2 cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S1þs3s4S2Þg

þ

cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S1þs3s4S2Þg expðbDð1dÞÞ þ2 cosh½Kfðs1þs2þs3þs4Þ þaðs1s2S₁þs3s4S₂Þg

* +

ð12Þ

where

K¼ßJ2

and a ¼J

4

/J

2

.

It is a formidable task to calculate the average on the right- hand-side of Eqs. (10)–(12) over all spin configurations. We can easily observe that any function such as

e(

s ,S) of s and

S

can be written as the linear superposition

f ð s

,SÞ ¼

f

1

þf

2

s þf

3

Sþf

4

S

²

þf

5

s Sþf

6

s S

²

ð13Þ

with appropriate coefficients

fi

(i ¼1,

y

,6). Applying this to all spins s

i

and

Sj

in expressions between brackets in Eqs. (10)–(12), we obtain:

/

s

oS

c

¼

X²

p¼0

X⁴

n¼0

X⁴ⁿ

l¼0

X²

i1¼1

X⁴

i2¼1

X⁴

i3ai2¼1

A

^p,n,l_i

1,i₂,i₃

ðK, a Þ½ s

i₁

S

i₂

S

²_i

3

ð14Þ

/

S

oS

c

¼

X⁴

p¼0

X²

n¼0

X²ⁿ

l¼0

X⁴

i1¼1

X²

i2¼1

X²

i3ai2¼1

B

^p,n,l_i

1,i₂,i₃

ðK, a

,D,dÞ½

s

i1

S

_i2

S

²_i

3

ð15Þ

/

S

²_oS_c

¼

X⁴

p¼0

X²

n¼0

X²ⁿ

l¼0

X⁴

i₁¼1

X²

i₂¼1

X²

i₃ai₂¼1

C

^p,n,l_i

1,i2,i3

ðK, a

,D,dÞ½

s

i1

S

i2

S

²_i3

ð16Þ where [ s

i1Si2Si32

] denotes the term containing

p

different factors of s

i1

,

n

different factors of

Si2

, and

l

different factors of

Si32

, with

i3ai2

. These factors are selected from sets { s

1

, s

2

,S

1

,S

2

,S

3

,S

4

} and {S

₁

,S

₂

, s

1

, s

2

, s

3

, s

4

} for

/

s

0S

c

and

/S₀ⁿS

c

, respectively. For example:

if

p¼1,n¼2,l

¼0, then A

^p,n,l_i

1,i2,i3

ðK, a Þ s

i1

S

i2

S

²_i3

¼ A

^1,2,0_2,ð3,4Þ

ðk, a Þ s

2

S

3

S

4

if

p¼1,n¼0,l

¼4, then A

^p,n,l_i

1,i2,i3

ðK, a Þ s

i1

S

i2

S

²_i3

¼ A

^1,0,41,ð1,2,3,4Þ

ðk, a Þ s

1

S

²₁

S

²₂

S

²₃

S

²₄

if

p¼0,n¼1,l

¼2, then A

^p,n,l_i

1,i2,i3

ðK, a Þ s

i1

S

i2

S

²_i3

¼ A

^0,1,2_3,ð1,2Þ

ðk, a ÞS

3

S

²₁

S

²₂

if

p¼0,n¼1,l

¼0, then A

^p,n,l_i

1,i2,i3

ðK, a Þ s

i1

S

i2

S

²_i3

¼ A

^0,1,0₄

ðk, a ÞS

4

The sublattice magnetizations m ,

m, and the quadrupolar

moment

q

are given by Eqs. (10)–(12) using the expansions (14)–(16). They constitute a set of exact relations according to which we can study the present system. However, in order to carry out the thermal average over all spin configurations, we have to deal with multispin correlations appearing via the right- hand-side of Eqs. (14)–(16). The problem becomes mathemati- cally untractable if we try to treat them exactly in the spirit of the FCA. In this paper, we use the simplest approximation in which we treat all spin self-correlations exactly while still neglecting correlations between quantities pertaining to different sites. This leads to the following coupled equations:

m ¼ m ½2A

1

q

²

þ4A

2

q

³

þ2A

3

q

⁴

þ m½4A

4

þ4A

5

qþ4A

6

q

²

þ4A

7

q

³

þ m m

²

½4A

8

q

³

þ m m

²

½2A

9

þ4A

10

q

þ2A

11

q

²

þ m

³

½4A

12

þ4A

13

q þ m m

⁴

½2A

14

þ m

²

m

³

½4A

15

q ð17Þ m ¼ m ½4B

1

þ8B

2

qþ4B

3

q

²

þm½2B

4

þ2B

5

q

þ m

³

½4B

6

þ8B

7

qþ4B

8

q

²

þ m

²

m½2B

9

þ2B

10

q þ m m

²

½4B

11

þ m

⁴

m½2B

12

þ2B

13

q þ m

³

m

²

½4B

14

ð18Þ q ¼ ½C

1

þ2C

2

qþC

3

q

²

þ m

²

½6C

4

þ12C

5

qþ6C

6

q

²

þm

²

½C

7

þ m m½4C

8

þ4C

9

q þ m

⁴

½C

10

þ2C

11

qþC

12

q

²

þm m

³

½4C

13

þ4C

14

q þ m

²

m

²

½6C

15

þ m

²

m

⁴

½C

16

ð19Þ The non-zero coefficients quoted in Eqs. (17)–(19), are listed in the Appendix A.

If we replace

m

and

q

in (17) by their expressions taken from (18) and (19), we obtain an equation for m of the form

m ¼ a m þb m

³

þ ð20Þ

In order to obtain the second-order transition temperature, we neglect higher-order terms in the magnetizations in Eqs. (17)–(19).Therefore, the critical temperature is analytically obtained through a determinantal equation, i.e.

1 ¼ 2A

1

q

²₀

þ4A

2

q

³₀

þ2A

3

q

⁴₀

þ ð4B

1

þ8B

2

q

0

þ4B

3

q

²₀

Þð4A

4

þ4A

5

q

0

þ4A

6

q

²₀

þ4A

7

q

³₀

Þ

1ð2B

4

þ2B

5

q

₀

Þ ð21Þ

where

q0

is the solution of

q

₀

¼ C

1

þ2C

2

q

₀

þC

3

q

²₀:

ð22Þ Eq. (21) means that its right-hand-side corresponds to the coefficient

a

in Eq. (20).

In the vicinity of the second-order transition, the sublattice magnetization m is given by

m

²

¼ 1a

b

:

ð23Þ

The right hand side of Eq. (23) is positive since we are in the long-ranged ordered regime. This means that the signs of 1a and

b

are the same.

When b changes sign (1a keeping its sign), m

² becomes negative. It means that we are not in the vicinity of a second-

order transition line. So, the transition is of the first order.

Therefore the point at which

aðK, a

,D,dÞ ¼

1 and bðK, a

,D,dÞ ¼

0 ð24Þ

characterizes the tricritical points.

To obtain the expression for

b, one has to solve Eqs. (17)–(19)

for small m and

m. The solution is of the form

q ¼ q

₀

þq

₁

m

²

þq

₂

m

²

þq

₃

m m ð25Þ where

q1

,

q2

, and

q3

are given by

q

₁

¼ 6C

4

þ12C

5

q

₀

þ6C

6

q

²₀

12C

2

2C

3

q

₀ ,

q

₂

¼ C

7

12C

2

2C

3

q

₀,

q

₃

¼ 4C

8

þ4C

9

q

₀

12C

2

2C

3

q

₀:

After some algebraic manipulations, Eqs. (18) and (25) can be written in the following forms:

m¼A

B

m

þ 8B2q₄

B þ8B3q₀q₄ B þC

BþAD B²þ2AB5q₄

B² þ4B11A² B³

!

m

³þ

ð26Þ

q ¼ q

₀

þq

₄

m

²

ð27Þ

where

A ¼ 4B

1

þ8B

2

q

0

þ4B

3

q

²₀,

B ¼ 1ð2B

4

þ2B

5

q

0

Þ, C ¼ 4B

6

þ8B

7

q

₀

þ4B

8

q

²₀

D ¼ 2B

9

þ2B

10

q

₀

q

₄

¼ q

₁

þ A B

2

q

₂

þ A B q

₃:

By substituting

m

and

q

in Eq. (17), with their expressions taken from Eqs. (26) and (27), we obtain the Eq. (20), where b is given by b ¼ 4A

1

q

₀

q

₄

þ12A

2

q

₄

q

²₀

þ8A

3

q

₄

q

³₀

þ Að4A

5

q

₄

þ8A

6

q

₄

q

₀

þ12A

7

q

₄

q

²₀

Þ

B þ 4A

8

q

³₀

A B þ EA

²

B

²

þ FA

³

B

³

þ ð4A

4

þ4A

5

q

₀

þ4A

6

q

²₀

þ4A

7

q

³₀

Þ 8B

2

q

₄

þ8B

3

q

₀

q

₄

þC

B þ AD þ2AB

5

q

₄

B

²

þ 4B

11

A

²

B

³

" #

ð28Þ with

E ¼ 2A

9

þ4A

10

q

0

þ2A

11

q

²₀

and F ¼ 4A

12

þ4A

13

q

0:

3. Results and discussions

First, we have to point for the system, when we ignore the fluctuation of the crystal field (d¼0) [37], that the transition is of

first order for relatively small values of the four-spin interaction

J4

(0.901oJ

4

/J

2o

a

c

¼1.114) and

D

near its critical value

Dc

¼2J

2

. The remaining part of the transition surface in (T,

D,J4

) space is always of second order for any values of the four-spin interaction and the crystal field.

In order to have an idea on the effects of the fluctuations of the crystal field on the mixed spin-1/2 and spin-1 Ising model with four-spin interaction, we have plotted in Fig. 3 the phase dia- grams in the

T–D

plane, for a fixed value of

d

(d¼0.5) and selected values of the four-spin interaction, when it belongs to the range 4

oJ4

/J

2o

1.114. In this latter region, the system undergoes a second order transition for

D¼0. As seen in

Fig. 3a for positive values of the four spin interactions, the transition temperature

Tc

decreases with the increase of

D, and ends in a tricritical point.

We have to note that the transition temperature becomes less and less sensitive to the strength of the four-spin interaction when it approaches its tricritical value (J

4c

¼1.114J

2

). We have repre- sented in Fig. 3b, the second order transition lines for negative values of

J4

, and we observe that the tricritical point still exists;

which means that the system undergoes a tricritical behaviour for any value of the four-spin interaction. Moreover for low

D, the

system exhibits an ‘‘horizontal’’ re-entrant behaviour when the strength of the four-spin interaction belongs to the region

Fig. 3.The phase diagrams in theD–Tplane. Second order transition lines are plotted whend¼0.5 for different positive (a) and negative (b) values ofJ4/J2. The number accompanying each curve denotes the value ofJ4/J2. The black point denotes the tricritical point.

4J

2oJ4o

3.305J

2

; while for high values of

D, a ‘‘vertical’’ re-

entrant phenomenon appears when

J4

belongs to the range 4J

2oJ4o

0.400J

2

.

In Fig. 4a, we plot the variation of

Tc

as a function of

D, for a fixed

positive value of the four-spin interaction (J

4

¼0.5J

2

) and different values of

d. When the fluctuations of the crystal field are neglected

(d¼0), the critical temperature

T_c

decreases gradually with the increase of

D

and ends in a tricritical point. As seen in the figure, the tricritical behaviour is kept by the system even when d takes a finite value. It is worthy to note here that for small

d, theT-component of

the tricritical point decreases with increase of

d, but for high values

of

d

the tricritical temperature seems to be insensitive to the fluctuations of the crystal field. The detailed investigation reveals that such qualitative behaviour is exhibited by the system for any positive value of the four-spin interaction less than a

c

.

On the other hand, we show in Fig. 4b the phase diagrams in

T–D

plane for a fixed negative value of

J4

and various values of

d. It

can be clearly seen from the figure, that with the increase of

D, the

transition temperature

Tc

decreases from its value at

D¼0, and

the ferromagnetic order domain becomes more and more large with increasing values of

d. As shown in

Fig. 4b and c, a new behaviour takes place in the phase diagram. Indeed, the system loses its tricritical behaviour for small values of

d

and therefore all

transition lines are of second order. Moreover, a re-entrant phenomena appears and becomes more and more pronounced for high fluctuations of the crystal field (high values of

d) and

large negative values of the four-spin interactions when this latter belongs to the region 4

oJ4

/J

2o0 (see for instance

Fig. 4c).

In this latter range, the detailed investigation shows that the phase diagrams are qualitatively similar, and when the system exhibits a tricritical behaviour, the

T-component of the tricritical

temperature seems to be independent of the fluctuations of the crystal field and depends only on the mean value of the crystal field and the strength of the four-spin interactions. On the other hand, as is clearly shown in Fig. 4, we note that for low mean values

D, the critical temperature is not very sensitive tod; and

around

DE

1.55J

2

,

Tc

seems to be independent of the value

d,

while it remarkably depends on it for higher

D.

It is worth mentioning that in the absence of the fluctuations (d¼0) [37], the transition surface

Tc

( a ,D) is completely located in the range 0rD/J

2r2, (i.e. the system does not exhibit a transi-

tion beyond

Dc

¼2J

2

for any value of a ). In order to illustrate more precisely the influence of

d

on the critical temperature

T_c

, we display in Fig. 5 the changes of

Tc

with

d

for fixed

D, and selected

values of the four-spin interaction

J4

. In Fig. 5a, one can observe that for low values of

D

(D

oD_c

), the critical temperature

T_c

is not

Fig. 4.The phase diagrams in theD–Tplane. Second order transition lines are plotted whenJ4/J2is kept fixed: ((a)J4/J2¼0.5; (b)J4/J2¼ 1; (c)J4/J2¼ 3). The number accompanying each curve denotes the value ofd. The black point denotes the tricritical point.

very sensitive to the value of

d; and all transition lines are of

second order for any value of the four spin interaction

J4

. But for relatively high mean values

D

of the crystal field (D

c4

2), the system behaves differently from the previous case. Indeed, as shown in Fig. 5b with

D¼3J2

, the transition temperature is sensitive to the value of

d. The system has two behaviours:

(i) For low fluctuations of the crystal field, as can be expected, the system does not undergo a transition for relatively high mean value

D

of the crystal field (D

ZDc

). This holds particularly for

d¼0, as is shown in

Fig. 5(b) for

D/J₂

¼3. (ii) For high fluctuations (high

d), the system may exhibit transition which depends on the

strength of the four-spin interactions and the value of

D/J2

even if

D

is greater than

D_c

, as illustrated in Fig. 5b. In this latter case, we are interested only by second order transition and the location of the tricritical point using Eqs. (21) and (24). Moreover, when the four- spin interaction belongs to the range 4o a

r

0.664, a re- entrant phenomenon appears and becomes more and more important when a approaches 4.

4. Conclusion

In this work, we have studied the effects of the fluctuations of the crystal field on the mixed spin-1/2 and spin-1 Ising model with four-spin interactions on the square lattice. Using the finite cluster approximation, we have found that the system undergoes a tricritical behaviour but its temperature component seems to be

independent of the high fluctuations (high

d) of the crystal field

and depend only the mean value

D

of the crystal field and the strength of the four-spin interactions. On the other hand, for low

D, the critical temperature is not very sensitive to the value ofd

and all transitions are of second order for any value of the four- spin interactions. But for relatively high

D, the transition tem-

perature depends on d and the system keeps its tricritical behaviour for any strength of the four-spin interactions. Finally, the critical behaviour of the system also showed a reentrance for relatively high

D

and large negative values of the four-spin interactions.

Appendix A

The coefficients appearing in Eqs. (17)–(19) are given by:

For abbreviation we define new functions:

f ðxÞ 1

2 tanh kx 2

gðxÞ sinhðKxÞ

expð b Dð1 þdÞÞ þ2 coshðKxÞ þ sinhðKxÞ expð b Dð1dÞÞ þ2 coshðKxÞ

hðxÞ coshðKxÞ

expð b Dð1þdÞÞ þ2 coshðKxÞ þ coshðKxÞ expð b Dð1dÞÞ þ2 coshðKxÞ

A

1

¼ 1

2 f 2þ a

2

þf 2þ a

2

þ2f a

2

n o

A

2

¼ 1

4 f 3þ a

2

f 3 a

2

f 1þ a

2

þf 1 a

2

n o

1

2 f 2 þ a

2

f 2 a

2

2f a

2

n o

A

3

¼ 1

16 f ð4þ a Þf ð4 a Þ 1

2 f 3þ a

2

f 3 a

2

n o

þ 1

2 f 2þ a

2

f 2 a

2

n o

þ 1

2 f 1þ a

2

f 1 a

2

n o

f a

2 1 8 f ð a Þ A

4

¼ f ð1Þ

A

5

¼ 1

4 f 2þ a

2

þf 2 a

2

n o

3fð1Þ þf ð2Þ

A

6

¼ 3

8 f 3þ a

2

þf 3 a

2

þf 1þ a

2

þf 1 a

2

n o

2f ð2Þ þ3fð1Þ 1

2 f 2þ a

2

þf 2 a

2

n o

A

7

¼ 1

32 f ð4þ a Þ þ fð4 a Þ 3

8 f 3 þ a

2

þf 3 a

2

n o

þ 1

16 f ð2þ a Þ þfð2 a Þ þ 1

4 f 2þ a

2

þf 2 a

2

n o

3

8 f 1 þ a

2

þf 1 a

2

n o

þ 1 16 f ð4Þ þ 9

8 f ð2Þfð1Þ A

8

¼ 1

8 fð4 þ a Þ þ fð4 a Þ 1

4 f ð2þ a Þ þ fð2 a Þ 1

4 f ð4Þ þ 1 2 fð2Þ A

9

¼ 1

2 f 2þ a

2

f 2 a

2

n o

þf a

2 A

10

¼ 3

4 f 3þ a

2

f 3 a

2

n o

1

2 f 2þ a

2

f 2 a

2

o n

þ 1

4 f 1þ a

2

f 1 a

2

n o

f a

2

Fig. 5.The variation of the transition temperature withd, when the mean valueDof

the crystal field is kept fixed ((a)D¼1.5J2; (b)D¼3J2). The number accompanying each curve denotes the value ofJ4/J2. The black point denotes the tricritical point.

A

11

¼ 3

8 fð4 þ a Þfð4 a Þ 3

2 f 3 þ a

2

f 3 a

2

o n

þ 1

2 f 2 þ a

2

f 2 a

2

n o

1

2 f 1 þ a

2

f 1 a

2

n o

þf a

2 þ 1 4 fð a Þ A

12

¼ 1

8 f 3þ a

2

þf 3 a

2

n o

3

8 f 1þ a

2

þf 1 a

2

n o

A

13

¼ 1

32 fð4þ a Þ þ fð4 a Þ 1

8 f 3þ a

2

þf 3 a

2

n o

1

16 f ð 2 þ a Þ þf ð 2 a Þ þ 3

8 f 1 þ a

2

þf 1 a

2

n o

þ 1 16 fð4Þ 1

8 fð2Þ

A

14

¼ 1

16 fð4þ a Þf ð4 a Þ 1

8 f ð a Þ A

15

¼ 1

8 fð4 þ a Þ þf ð4 a Þ þ 1

4 f ð2þ a Þ þf ð2 a Þ 1

4 f ð Þ 4 1 2 f ð2Þ B

1

¼ 1

4 gð2Þ þ 1 2 gð1Þ B

2

¼ 1

8 g 2 þ a

4

þg 2 a

4

n o

þ 1

4 g 1þ a

4

þg 1 a

4

o n

1 4 gð2Þ 1

2 gð1Þ

B

3

¼ 1

16 g 2þ a

2

þg 2 a

2

n o

þ 3 8 gð2Þ þ 1

8 g 1þ a

2

þg 1 a

2

o n

þ 3 4 gð1Þ 1

4 g 2 þ a

4

þg 2 a

4

o

1

2 g 1 þ a

4

þg 1 a

4

n o

B

4

¼ 1

16 g 2þ a

4

g 2 a

4

n o

1 8 g a

4 B

5

¼ 1

32 g 2þ a

2

g 2 a

2

n o

1

16 g 2 þ a

4

g 2 a

4

o n

þ 1 8 g a

4 1 16 g a

2 B

6

¼ gð2Þ2gð1Þ B

₇

¼ gð2Þ þ2gð1Þ þ 1

2 g 2þ a 4

þg 2 a 4

o

g 1þ a 4

þg 1 a 4

n o

n

B

8

¼ 1

4 g 2 þ a

2

þg 2 a

2

n o

þ 3 2 gð2Þ 1

2 g 1 þ a

2

þg 1 a

2

o n

3gð1Þ g 2 þ a

4

þg 2 a

4

o

þ2 g 1þ a

4

þg 1 a

4

n o

n

B

9

¼ 3

2 g 2 þ a

4

g 2 a

4

n o

þg a

4 B

10

¼ 3

4 g 2 þ a

2

g 2 a

2

n o

3

2 g 2þ a

4

g 2 a

4

o n

g a

4 þ 1 2 g a

2 B

11

¼ 1

16 g 2þ a

2

þg 2 a

2

n o

1

8 g 1þ a

2

þg 1 a

2

o n

1 8 gð2Þ þ 1

4 gð1Þ B

12

¼ g 2þ a

4

g 2 a

4

2g a

4

B

13

¼ 1

2 g 2þ a

2

g 2 a

2

n o

g 2þ a

4

þg 2 a

4

þ2g a

4 g a

2

B

14

¼ 1

4 g 2þ a

2

þg 2 a

2

n o

1 2 gð2Þ þ 1

2 g 1þ a

2

þg 1 a

2

o n

gð1Þ C

1

¼ 1

8 hð2Þ þ 3 8 hð0Þ þ 1

2 hð1Þ C

2

¼ 1

16 h 2þ a

4

þh 2 a

4

n o

þ 3 8 h a

4 þ 1

4 h 1 þ a

4

þh 1 a

4

n o

3 8 hð0Þ 1

8 hð2Þ 1 2 hð1Þ C

3

¼ 3

16 h a

2 þ 9

16 hð0Þ þ 3

16 hð2Þ þ 1

32 h 2 þ a

2

þh 2 a

2

o n

þ 3 4 hð1Þ þ 1

8 h 1þ a

2

þh 1 a

2

n o

1

8 h 2þ a

4

þh 2 a

4

n o

3 4 h a

4 1

2 h 1 þ a

4

þh 1 a

4

n o

C

4

¼ 1 2 hð0Þ þ 1

2 hð2Þ C

5

¼ 1

4 h 2þ a

4

þh 2 a

4

n o

1 2 h a

4 1 2 hð2Þ þ 1

2 hð0Þ C

6

¼ 1

4 h a

2 þ 3 4 hð2Þ 3

4 hð0Þ þ 1

8 h 2þ a

2

þh 2 a

2

o n

þh a

4 1

2 h 2þ a

4

þh 2 a

4

n o

C

7

¼ 3 16 h a

2 3

16 hð0Þ þ 1

32 h 2þ a

2

þh 2 a

2

o

þ 1 4 h ð Þ 1

1

8 h 1þ a

2

þh 1 a

2

n o

1 16 hð2Þ C

8

¼ 1

4 h 2þ a

4

h 2 a

4

n o

C

9

¼ 1

8 h 2þ a

2

h 2 a

2

n o

1

4 h 2 þ a

4

h 2 a

4

n o

C

10

¼ 2hð2Þ þ 6hð0Þ8hð1Þ C

11

¼ h 2 þ a

4

þh 2 a

4

n o

þ6h a

4 4 h 1þ a

4

þh 1 a

4

n o

2hð2Þ6hð0Þ þ8hð1Þ

C

12

¼ 3hð2Þ þ 9hð0Þ12hð1Þ2 h 2þ a

4

þh 2 a

4

n o

12h a

4 þ8 h 1 þ a

4

þh 1 a

4

n o

þ 1

2 h 2þ a

2

n

þh 2 a

2

o

þ3h a

2 2 h 1 þ a

2

þh 1 a

2

n o

C

13

¼ h 2 þ a

4

h 2 a

4

C

14

¼ 1

2 h 2þ a

2

h 2 a

2

n o

h 2 þ a

4

h 2 a

4

n o

C

15

¼ 1 4 h a

2 þ 1 4 hð0Þ 1

4 hð2Þ þ 1

8 h 2þ a

2

þh 2 a

2

o n

C

16

¼ 3h a

2 3hð0Þhð2Þ þ 1

2 h 2þ a

2

þh 2 a

2

o

4hð1Þ

n

þ2 h 1þ a

2

þh 1 a

2

n o

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