The transverse spin-1 Ising model with random interactions

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

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The transverse spin-1 Ising model with random interactions Touria Bouziane a,, Mohammed Saber a,b a b

Department of Physics, Faculty of Sciences, University of Moulay Ismail, B.P. 11201 Meknes, Morocco Dpto. Fı´sica Aplicada I, EUPDS (EUPDS), Plaza Europa, 1, San Sebastian 20018, Spain

a r t i c l e in fo

abstract

Article history: Received 13 March 2008 Received in revised form 2 July 2008 Available online 30 July 2008

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(Jij) ¼ pd(JijJ)+(1p)d(JijaJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter a. & 2008 Published by Elsevier B.V.

Keywords: Ising model Phase diagram Spin-1

1. Introduction Over several decades there has been considerable interest in the study of pure Ising models and their variants because they have been used to describe many physical situations in different fields of physics. The study of the two-state transverse Ising model has been of interest since the early work of De Gennes [1] in 1963, and was originally introduced as a valuable model of hydrogen-bounded ferroelectrics [2,3] such as the KH2PO4 type. Since then, it has successfully been used to investigate a number of problems of phase transitions associated with order–disorder phenomena in several other systems, for example, a cooperative Jahn–Teller system [4] (such as DyVO4 and TbVO4) and some real magnetic materials for which the crystal-field ground state is a singlet [5]. The wider applicability of the model has extensively been reviewed in the literature [6,7]. The model is described by the two-state Ising Hamiltonian with a term representing a field transverse to the Ising spins, namely, H¼

X

Ji;j szi szj  O

i;j

X

sxi

(1)

i

where siz and six are the components of a spin-12 operator at site i, Ji,j is the exchange interaction, O represents a transverse field, and the first sum is carried out only over the nearest-neighbor pair of spins of the lattice. The thermodynamic properties of the model Hamiltonian in Eq. (1) have been exactly obtained only for the one-dimensional lattice [8–10]. In order to study higher-dimen Corresponding author.

E-mail address: [email protected] (T. Bouziane). 0304-8853/$ - see front matter & 2008 Published by Elsevier B.V. doi:10.1016/j.jmmm.2008.07.028

sional lattices some sort of approximation has to be done, and the problem of finding a solution has generated a number of different approximation schemes [11–14]. However, all of these approaches consider the spin-12 two-state Ising model (TIM) and most of them have been restricted to analysis of particular regimes, both at a low and at a high temperature. The spin-12 Ising model has been studied extensively by many techniques, including an effective field treatment [14,15], based on a generalized, but approximate, Suzuki relation. In addition to the study of the pure systems, site-diluted systems [17–19], random bond models [17,18,20], amorphous magnets [21], semi-infinite systems with a variety of surfaces [22–25], thin films [26,27], superlattices [28], and multilayers [29] have been studied. The effective field theory has also been extended to the spin-1 system [30–34] and to the mixed spin-12 and spin-1 systems [35–38]. Elkouraychi et al. [39] have studied the transverse Ising model with an arbitrary spin within the framework of an effective field theory based on the use of approximate Van der Wearden identities. Mielnicki et al. [32] has developed a general method of magnetization calculations for the Ising ferromagnet with arbitrary spin values. Detailed calculations have dealt with the particular case of spin-1 and for the s.c. lattice (simple cubic lattice); hence, the critical temperature and the magnetization together with a quadrupolar moment versus temperature relation has been obtained. Jiang [33] found many phase diagrams of the bond diluted transverse spin-1 Ising model with a crystal-field interaction within the framework of the effective field theory proposed by Honmura and Kaneyoshi [40]. The phase diagrams containing first- and second-order phase transitions and the tricritical points

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T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

are given for a honeycomb lattice. He also found that the reentrant phenomenon of second order occurs for the system in the appropriate region, especially near the critical region where the tricritical behavior disappears. The transverse spin-1 Ising model in a random field is an interesting problem. Theoretically the random-field Ising model has been widely investigated by the use of various techniques [51–55]. Sarmento and Kaneyoshi [34] obtained the phase diagram and magnetization curves of a diluted transverse spin-1 Ising model in a random field. The tricritical points were found and the reentrant phenomenon displayed by the system was investigated. Following the experimental observation of the reentrant magnetism by Maletta and Flesch [41], many authors [42–50] have studied theoretically the properties of the random-mixed bond spin 12, 1, and 32 Ising model in order to understand the system having both frustration and disorder. The common result of the previous investigation is that the phase diagrams predict the existence of reentrant behavior in a certain range of negative values of a. In this work, we investigate the transverse spin-1 Ising model with a random interaction. We are particularly interested in systems where a is a variable (cases where a takes on negative and positive values are studied). We outline this work as follows: in Section 2, we briefly present the effective field theory that is derived by using a probability distribution method [56] based on the generalized Van der Waerden identities [57]. The nearestneighbor exchange couplings are described by the probability distribution law: P(Ji,j) ¼ pd(Ji,jJ)+(lp)d(Ji,jaJ). The critical temperature at which the magnetization vanishes is studied as a function of p and the transverse field strength. The phase diagrams of the system in the (Tc, O, 1p) space are presented in Section 3. A brief conclusion is given in Section 4.

This equation may be viewed as a generalization of the more widely known Suzuki relation [58] * + Trg ½Sg g ebHg  hSg g i ¼ (7) Trg ½ebHg  that holds for ‘‘classical systems’’, where Hg and H0 commute (i.e. D ¼ 0). For the transverse spin 1 Ising model, where D is non-zero, Eq. (7) is no longer exact. However, on the basis of a simple approximate decoupling of the left-hand side of Eq. (4), and the observation that /DS ¼ 1, Sa` Barreto et al. [15,16] argued that it is nevertheless a reasonable approximation to the exact relation (4), and forms the starting point of their theory. The longitudinal disorder parameters in the system are given by * + A 2 sinh½bðA2 þ O2 Þ1=2 Þ hSgz i ¼ ðA2 þ O2 Þ1=2 1 þ 2 cosh½bðA2 þ O2 Þ1=2  ¼ f 1;z ðA; OÞ * hS2gz i ¼

(8)

O2

ð2A2 þ O2 Þcosh½bðA2 þ O2 Þ1=2 Þ

ðO2 þ A2 Þ1=2

1 þ 2 cosh½bðA2 þ O2 Þ1=2 

+ (9)

The transverse components are obtained by interchanging A and O in Eqs. (8) and (9). We obtain hSgx i ¼ f 1;x ðA; OÞ ¼ f 1;z ðO; AÞ

(10)

To recast these equations in a more convenient form, the probability distribution technique of Tucker et al. [56] may be employed to give 0 1 X @ (11) hSg g i ¼ f g xgj Sjz A j

where Sjz is a discrete random variable with distribution 2. Theory

PðSiz Þ ¼ 12½ðqz  mz ÞdSjz 1 þ 2ð1  qz ÞdSjz ;0 þ ðqz þ mz ÞdSjz þ1 

We consider the transverse spin-1 Ising model with random nearest-neighbor interactions. The Hamiltonian is given by X X H¼ Jij Siz Sjz  O Six (2)

2 f 1;z 4

X

3

where Sjg(g ¼ x, z) are the components of the spin-1 operator at site i. O represents the transverse field, and the exchange interactions Jij are assumed to be of the form Jij ¼ J xi,j, where J40 and xij is a random variable with distribution Pðxij Þ ¼ pdðxij  1Þ þ ð1  pÞdðxij  aÞ

(3)

2ð 2

P

j xgj Sjz Þ

P

2 1=2 j xgj Sjz Þ  + P sinh bJ½ð j xgj Sjz Þ2 þ ðO=JÞ2 1=2  P 1 þ 2 cosh bJ½ð j xgj Sgj Þ2 þ ðO=JÞ2 1=2

j

i

hi;ji

*

xgj Sjz 5 ¼ 

(12)

½O þ ð

(13)

where g ¼ x,z 2 3 X 4 xgj Sjz 5 f 2;z j

where 0ppp1. The effective field theory to be adopted is based on the exact identity introduced by Sa`-Barreto et al. [15,16] for the thermal average /SggS of a single spin * + ! Trg ½Sgg ebHg  Sg g  D Þ ¼0 (4) ð1  Trg ½ebHg  where 0

0

D ¼ 1  ebHg ebH ebðHg þH Þ

(5)

Hg is that part of the Hamiltonian containing the spin g, namely Hg ¼ ASgz  OSgx with X A¼ xij Sjz j

* ¼

O2

P ½ð j xgj Sjz Þ2 þ O2 

+ P P 2½ j xgj Sjz Þ2 þ O2  cosh bJ½ð j xgj Sjz Þ2 þ O2 1=2  P 1 þ 2 cosh bJ½ð j xgj Sjz Þ2 þ O2 1=2

(14)

The next step is to carry out the configurational averaging, denoted /ySr. One then obtains the following equations for magnetizations and quadrupolar moments 2 3+ * X   4 mg ¼ Sgg r ¼ f 1;g xgj Sjz 5 (15) j

r

2 3+ * DD EE X 2 Sg g ¼ f 2;g 4 xgj Sjz 5 qg ¼ r

(6) where g ¼ x, z.

j

(16) r

ARTICLE IN PRESS T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

It is now convenient now to rewrite the above equations in the form mg ¼

**Z

d½u 

j

** ¼ 1 2p

¼

X

1 2p ZZ

ZZ

il½u

++

P

e

1 2p

¼

P

P j

(17)

r

++ xgj Sjz f 2;g ðuÞ du r

DD EE eilxgj Sjz eilu f 2;g ðuÞ du dl

qr ¼

1 2p

hhe

ZZ

ilu iiN f 1;g ðuÞ du dl r e

(20)

hheilxgj Sjz iir ¼

PðSjz Þ

ðthe exponents of mz are even and pNÞ (25)

The term q0,z, is a solution of the equation m N X Nk X 1 X N

2

ðNkÞ Xm

N¼0 m¼0 m1 ¼0

(26)

The expression of q1z is given by q1z ¼

Pðxgj Þeilxgj Sjz dz

m N X N k X 1 X

2

N

ðNkÞ k X mX

N¼0 m¼0 m1 ¼0

ðNkÞm

m X i2 ðNkÞ i3 X X mX

i1 ¼0 i2 ¼0 j2 ¼0

b1 ¼0

Nk m C N C m1 C b k Cm

Sjz ¼1

m

i3 ¼0

ðNkÞm

C ki1 C i2 C ij2 C i3 2

1

1 ¼ f½peil þ ð1  pÞeila ðqg  mg Þ þ 2ð1  qg Þ 2 þ ½peil þ ð1  pÞeila ðqg þ mg Þg

Nk N ðNkÞm1 m1 þb1 2k C N C m1 C m1 p k Cm

b1 ¼0

 ð1  pÞðNkÞðm1 þb1 Þ f 2z ½m1  b1   a½ðN  kÞ  2m  Nk þðm1  b1 Þ ð1  q0z Þk q0z

where Z

(24)

Near the transition lines, we write in general Eq. (25) as an expansion in powers of mz

(19)

ilu f 2;g ðuÞ du dl hheilxgj Sjz iiN r e

þ1 X

p

3. Results and discussion

q0z ¼ ilxgj Sjz

1

Nkm m1 þb1

Thus, now one has a set of coupled equations for the order parameters mg and qg that can be solved directly by a numerical iteration without any further algebraic manipulation. The same equations hold for a general lattice with a coordination number N, and so results for different structures may be obtained without carrying out the detailed algebra found when employing other techniques.

qz ¼ q0z þ q1z m2z þ   

thus enabling the thermal average to be taken inside the product sign. For the simplest approximation of a lattice having only nearest-neighbor exchange interactions, with coordination number N, it follows that 1 mr ¼ 2p

m

(18)

r

       Siz Sjz Skz . . . r ¼ Siz r Sjz r Skz r . . . ; for iajak

ZZ

b1 ¼0

ð1  pÞ   f 2;g ðm1  b1 Þ  a½ðN  kÞ  2m þ ðm1  b1 Þ

f 1;g ðuÞ du

j

k¼0 m¼0 m1 ¼0

ðNkÞm

Nk m 2k C N C m1 C b K Cm

ðNkÞðm1 þb1 Þ

DD EE eilxgj Sjz eilu f 1;g ðuÞ du dl

X

ðNkÞ Xm

ð1  qg Þ ðqg þ mg Þ ðqg  mg Þ

where g ¼ x, z. To proceed further, one now has to approximate the thermal multiple correlation functions on the right-hand sides of Eqs. (17) and (18). The simplest approximation, and the one most frequently adopted, is to decouple them according to 

2

++r

xgj Sjz 

j

d½u 

ZZ

N

k

and qg ¼

m N X Nk X 1 X

r

j

**Z

and qg ¼

xgj Sjz f 1;g ðuÞ du

19

2k ð1Þði1 þi3 þj3 Þ

i1 ¼0

C ij3 pb1 þm ð1  pÞðNkÞðm1 þb1 Þ 3

i þj þj

ðNkÞði1 þi3 Þ 1 2 3 ð1  q0z Þki1 q0z q1z d2i1 þi2 þj2 þi3 þj3 ;2   f 2z ½m1  b1   a½ðN  kÞ  2m þ ðm1  b1 Þ

(21)

(27)

By substituting Eq. (26) into Eq. (24), we obtain By successive binomial expansions, hheilxgj Sjz iiN r may be written as

mz ¼

1 n il hheilxgj Sjz iiN ðpe þ ð1  pÞeila Þðqg  mg Þ r ¼ 2N þ2ð1  qg Þ þ ðpeil þ ð1  pÞeibla Þðqg þ mg Þ ¼

m N X Nk X 1 X N

2

k¼0 m¼0 m1 ¼0

ðNkÞ Xm

Nkm

Nk m 2k C N C m1 C b k Cm

1

b1 ¼0

oN

2

N

k¼0 m¼0 m1 ¼0

ðNkÞ Xm

b1 ¼0

m X i2 ðNkÞ i3 X X mX

2k ð1Þði1 þi3 þj3 Þ

i þj þj

2i þi þj þi þj

ðNkÞði1 þi3 Þ 1 2 3 ð1  q0z Þki1 q0z q1z mz 1 2 2 3 3   f 1;g ðm1  b1 Þ  a½ðN  kÞ  2m þ ðm1  b1 Þ

(28)

(22) The longitudinal magnetization mz can be expressed as

are the binomial coefficient: (m!/n!(nm)!). where Now substituting Eq. (22) into Eqs. (19) and (20) we obtain the following relations for the ordering parameters: m N X N k X 1 X

ðNkÞ k X mX

k¼0 m¼0 m1 ¼0 b1 ¼0 i1 ¼0 i2 ¼0 j2 ¼0 i3 ¼0 j3 ¼0 ðNkÞm k m i2 ðNkÞm i3 m1 þb1 Nk m C N C m1 C b C i1 C i2 C j C i3 Cj p ð1  pÞðNkÞðm1 b1 Þ k Cm 2 3 1

Cm n

mg ¼

m N X Nk X X

ðNkÞm

Nk m 2k C N C m1 C b k Cm

mz ¼ amz þ bm3z þ . . . with a¼

1

(29)

m N X N k X X k¼0 m¼0 m1 ¼0

ðNkÞ k X mX

b1 ¼0

m X i2 ðNkÞ i3 X X mX

i1 ¼0 i2 ¼0 j2 ¼0

ðNkÞm

m

i3 ¼0 ðNkÞm

2k ð1Þði1 þi3 þj3 Þ

j3 ¼0

ð1  qg Þk ðqg þ mg Þm ðqg  mg ÞNkm pm1 þb1

Nk m C N C m1 C b k Cm

ð1  pÞðNkÞðm1 þb1 Þ f 1;g    ðm1  b1 Þ  a½ðN  kÞ  2m þ ðm1  b1 Þ

ðNkÞði1 þi3 Þ 1 2 3 ð1  q0z Þki1 q0z q1z d2i1 þi2 þj2 þi3 þj3 ;1   f 1z ½m1  b1   a½ðN  kÞ  2m þ ðm1  b1 Þ

1

C ki1 C i2 C ij2 C i3 2

C ij3 pb1 þm1 ð1  pÞðNkÞðm1 þb1 Þ 3

i þj þj

(23)

(30)

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T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

and b¼

m N X Nk X X k¼0 m¼0 m1 ¼0

ðNkÞ k X mX

b1 ¼0

i1 ¼0 i2 ¼0 j2 ¼0

ðNkÞm

Nk m C N C m1 C b k Cm

1

m X i2 ðNkÞ i3 X X mX

m

i3 ¼0 ðNkÞm

C ki1 C i2 C ij2 C i3 2

2k ð1Þði1 þi3 þj3 Þ

j3 ¼0

C ij3 pb1 þm1 ð1  pÞðNkÞðm1 þb1 Þ 3

i þj þj

ðNkÞði1 þi3 Þ 1 2 3 ð1  q0z Þki1 q0z q1z d2i1 þi2 þj2 þi3 þj3 ;3   f 1z ½m1  b1   a½ðN  kÞ  2m þ ðm1  b1 Þ

(31)

The second-order transition line is then given by a¼1

and

bo0

(32)

In the vicinity of the second order, the transition line mz is given by m2z ¼

1a b

(33)

The right-hand side of Eq. (33) must be positive. If this is not the case the transition is of first order, and hence the point at which a ¼ 1 and b ¼ 0 is the tricritical point [60]. We are now in a position to investigate the phase diagrams of the transverse spin-1 Ising model with random interactions. The above expressions are valid for any value of N. The phase diagrams of the system are presented as a function of the parameters T, p, a, and O for a simple cubic lattice N ¼ 6. Fig. 1 shows the phase diagram in the (1p, Tc/J) plane for different values of a. All the transition lines are of second order and have the typical behavior of the temperature as a function of the probability. With the decrease of a the phase region in which the ferromagnetic state is realizable gradually becomes small. If a ¼ 0.25 (case a40), for any value of (1p), the value of Tc/J is greater than Tc/J ¼ 0.8797 and the curve is above this value of Tc/J.

For a ¼ 0, we have the bond diluted model. At zero temperature this model exhibits a critical bond concentration at pc ¼ 0.2706 which confirms with the result of Tucker [59]. If ao0 (frustrated case), we can find a reentrant behavior for certain range of a (for example a ¼ 0.1). Moreover, there appear different thresholds as solutions of T/J ¼ 0 (namely: (1p)c ¼ 0.2495, 0.3676, 0.3904, and 0.5269 at a ¼ 1.0, 0.75, 0.5, and 0.1, respectively). Such a non-uniform convergence of Tc/J as a function of (1p) and for ao0 is also obtained in a random bond mixed spin-12 and spin-1 Ising ferromagnet [60–62]. It is seen from curve (e) that the phase diagram exhibits a reentrant behavior in the region 0.400opo0.473 which results from the competition between the fluctuations induced by the temperature and the exchange interaction terms. However, the reentrant ferromagnetic (RF) phase is a general characteristic of systems in which frustration and disorder are present [63], and EuxSr1xS is believed to be an example of the system [64]. Fig. 2 shows the behavior of Tc/J as a function of (1p) for a fixed value of a namely, a ¼ 0.25 and for different values of O/J. It is seen that when O/J increases the temperature decreases and the curves are below those corresponding to O/J ¼ 0. We also show that Tc/J declines from its value at (1p) ¼ 0.0 and reaches zero only if O/J is greater than a critical value. For example (1p)c ¼ 0.7724 and 0.9262 correspond to the cases O/J ¼ 2.0 and 1.50, respectively. Fig. 3 shows the phase diagram in the plane (Tc/J, O) for a fixed value of a ¼ 0.25 and several values of p, namely p ¼ 0.0, 0.5, 1.0. With the increase of p the phase region in which the ferromagnetic state is realizable gradually becomes greater. As it is seen in Fig. 1 for p ¼ 0.0, 0.5, 1.0, Tc/J declines from its value at O/J ¼ 0.0 (Tc/J ¼ 0.880, 2.070, 3.518, respectively), and reaches zero at the critical values of Oc/J (1.3146, 3.0849, 5.2587, respectively).

3.5

3.5

3.0

3.0

2.5

2.5

Tc/J

Tc/J

−1 2.0

1.5

2.0

1.5 +0.25

1.0

0.5

0

α = −0.75

1.0

0.5 α = −0.5 0.0

0.2

1.5

−0.1

0.0 0.4

0.6

1.0

2.0

0.5

α = −0.25

0.0 0.8

1.0

1-p Fig. 1. The phase diagram of the spin-1 Ising model with random interactions in zero transverse field for N ¼ 6, O/J ¼ 0 and for some typical values of a (the numbers at the curves).

0.0

0.2

0.4

0.6

0.8

1.0

1-p Fig. 2. The phase diagram of the random bond transverse spin-1 Ising model in the (1p, Tc/J) plane for N ¼ 6 and a ¼ 0.25. The numbers at the curves are the transverse field O/J.

ARTICLE IN PRESS T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

for O/J ¼ 1.5, 1.0, 0.0, the curves declines from Tc/J ¼ 3.4105, 3.4712, 3.518 and reach zero at a critical value of (1p) namely, 0.5985, 0.6675, and 0.7294, respectively. In Fig. 5, we show the changes of Oc/J with (1p). For a ¼ 0 the critical line decreases with the increase of (1p). In Fig. 6, the critical temperature Tc/J is plotted as a function of (1p), for a ¼ 0.1 and for different values of O/J. It is seen that at zero field, the curves exhibit a bulge at a low value for Tc/J and O/J. The bulge disappears gradually with the increase of O/J. The curves for O/J ¼ 0.0, 0.1 are similar. They decrease from their value at p ¼ 1 (Tc/J ¼ 3.519) and disappear at different critical values of (1p), 0.5268, 0.5426, respectively. Fig. 7 shows the phase diagram in the (Tc/J, O/J) plane for the system with a ¼ 0.1 and for selected values of p, namely 0.400, 0.401, 0.4015, 0.403, 0.417, and 0.422. We investigate in detail the phase diagram in the region 0.400opo0.473, and we pay more attention to the region p40.400. From this figure, we can see that the critical line exhibits some characteristic behaviors; the first is that the curves exhibit a bulge which is larger when p increases, and the second is that if p increases the bulge disappears gradually and the curves decline to zero at two critical values of O/J. We have now two critical temperature and two ferromagnetic areas restricted by two phase

4.0

3.5

3.0 p = 0.0 2.5

Tc/J

21

2.0 0.5 1.5

1.0 1.0 0.5

0.0

3.5 0

1

2

3 Ω/J

4

5 3.0

Fig. 3. The changes of Tc/J with O/J for the system with a ¼ 0.25. The numbers at the curves are the value of p.

Ω/J

2.5 2.0 1.5

3.5

1.0 3.0

0.5 0.0

Tc /J

2.5

0.0

0.1

0.2

0.3

0.4 1-p

0.5

0.6

0.7

Fig. 5. The change of the critical transverse field Oc/J as Tc/J reduces to zero with (1p) for the bond diluted transverse spin-1 Ising model for N ¼ 6 and a ¼ 0.

2.0

1.5

4

1.0

3 Ω/J = 1.5

0 Tc/J

0.5 1.0

Ω/J = 2

2

1.75

0.0 0.0

0.2

0.4 1-p

0.6

0.8

Fig. 4. The phase diagram of the bond diluted (a ¼ 0) transverse spin-1 Ising model in the (1p, Tc/J) plane for N ¼ 6. The numbers at the curves are the transverse field O/J.

1

1.5 0.1

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

1-p Fig. 4 shows the phase diagram in (1p, Tc/J) plane in the case of bond dilution a ¼ 0.0 with selected values of O/J. When O/J increases the value of Tc/J decreases. As it is seen from the figure,

Fig. 6. The phase diagram of the random bond transverse spin-1 Ising model with a ¼ 0.1 in the (1p, Tc/J) plane for N ¼ 6. The numbers at the curves are the transverse field O/J.

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T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

0.8

1.0 qz

0.6

0.8

paramagnetic

The order parameters

p = 0.422

Tc/J

ferromagnetic

0.4

0.400 0.401 0.4015 0.417

0.2

qx 0.6 mz

0.4

0.403

p = 0.422

0.2

paramagnetic

mx

0.0 0.0

0.1

0.2

0.3

0.4

0.5 0.6 Ω/J

0.7

0.8

0.9

1.0

1.1

0.0

Fig. 7. The variation of Tc/J versus O/J of the system with a ¼ 0.1 and N ¼ 6. The number accompanying each curve are the parameter p.

0

1

2

3

4

T/J Fig. 9. The temperature dependence of the longitudinal magnetization mz. Quadrupolar moment qz, transverse magnetization mx, and quadrupolar moment qx, for a ¼ +0.25, p ¼ 0.8. The solid and dashed lines correspond to transverse field strengths of O/J ¼ 0.0 and 1.5.

3.5

3.0 Paramagnetic 1.0

2.5

qz Tc/J

2.0 0.8 The order parameters

Ferromagnetic 1.5 Ω//J= 1 1.0 0.5 0.5 0.1

0.1

mz

0.4

0

0.0 0.0

qx

0.6

0.2

0.3

0.4

mx

0.2

1-p Fig. 8. The phase diagram of the random bond transverse spin-1 Ising model a ¼ 0.75 in the (1p, T/J) plane for N ¼ 6. The numbers at the curves are the transverse field O/J.

0.0 0

1

2

3

4

T/J boundaries: curves p ¼ 0.403, 0.417, and 0.422. The ferromagnetic areas are marked by two critical lines. Fig. 8 shows the phase diagram in the plane (1p, Tc/J) for the system with a fixed value of a (a ¼ 0.75) and for selected values of O/J. For this system Tc/J decreases if O/J increases. The discussion here is similar to the case a ¼ 0.0 in Fig. 4. The typical behavior of the longitudinal magnetization and quadrupolar moment (solid lines) (transverse magnetization and transverse quadrupolar moments (dashed lines)) as a function of

Fig. 10. The temperature dependence of the longitudinal magnetization mz and quadrupolar moment qz, transverse magnetization mx, and quadrupolar moment qx for a ¼ 0.0, p ¼ 1.0. The solid and dashed lines correspond to transverse field strengths of O/J ¼ 0.0 and 1.5.

temperature, respectively, is shown in Figs. 9, 10 and 13 for selected values of a. In Fig. 9, the behaviors of mz and qz as a function of T/J are shown for p ¼ 0.8 and a ¼ 0.25; the longitudinal magnetization

ARTICLE IN PRESS T. Bouziane, M. Saber / Journal of Magnetism and Magnetic Materials 321 (2009) 17–24

23

1.0

1.0

qz

qz

0.8

0.8

qx

0.6

0.4

The order parameters

The order parameters

qx 0.6

mx

0.4

mz mz 0.2

0.2

0.0 mx

0.2

0.4

0.0 0.0

0.2

0.4

0.6

0.8

T/J

0.6

0.8

1.0

Ω/J Fig. 12. The variation of the components: mz, qz, mx, qx as a function of the transverse field O/J, for a ¼ 0.1, T/J ¼ 0, and p ¼ 0.422.

Fig. 11. The variation of the components: mz, qz, mx, qx as a function of the temperature T/J, for a ¼ 0.1, O/J ¼ 0, and p ¼ 0.422.

1.0 qz 0.8 The order parameters

declines with increasing T/J and reaches zero at the critical value of temperature Tc/J ¼ 2.928 at zero field. The longitudinal component of the quadrupolar moment qz decreases at a low temperature. It is also observed that at the critical temperature the decrease in the quadrupolar moment qz changes abruptly, and the curve has an asymptotic value at the highest-temperature limit. At zero field the transverse component mx is zero for all values of T/J (for example curve mx in Fig. 9). If O/J is non-zero (we set O ¼ 1.5), we can show the characteristic behavior of the transverse component mx (dashed curve mx), the value of mx increases with the increase of T/J passes through a cusp at the transition temperature and declines with increasing temperature. The longitudinal magnetization mz becomes smaller. Focusing our attention on the behavior of the transverse components mx and qx, we can see clearly an interesting behavior of mx and qx. The dashed curves of mx can be separated into two parts with different slopes. It accounts for the fact that the equation is satisfied by mx: the first is the nonmagnetic region in which mz ¼ 0 and the second is the region in which mz6¼0. Consequently, the curves of mx display a cusp at T ¼ Tc. However, this behavior disappears for higher values of O/J (Fig. 10). In Figs. 11 and 12, we examine the case when a ¼ 0.1. We see the behavior of mz and qz, mx and qx as a function of T/J in Fig. 11 (O/J in Fig. 12), respectively, and for a fixed value of p (p0.422). As is predicted in the phase diagram in Fig. 7, the reentrant phenomenon is observed in the magnetization curve of mz as a function of T/J in Fig. 11 (O/J in Fig. 12), respectively. The magnetization mz reaches zero at the critical value of Tc/J Oc/J) for which the phase diagram in Fig. 7 reaches zero namely, Tc/J ¼ 1.7801, 0.67117 in Fig. 11, (at Oc/J ¼ 0.2571, 0.9486 in Fig. 12). The thermal behavior of the longitudinal (transverse) component of quadrupolar moment qz (qx) in Fig. 11 displays a cusp

0.6

qx

0.4

mx

mz

0.2

0.0 0

2

1

3

T/J Fig. 13. The temperature dependence of the longitudinal magnetization mz and quadrupolar moments qz and the transverse magnetization mx, and the transverse component of quadrupolar moment qx for a ¼ 0.75 and p ¼ 0.8. The solid and dashed lines correspond to transverse field strengths of O/J ¼ 0.0 and 1.5.

(pick) at Tc/J ¼ 1.7801 and decreases (increases) with the increase of T/J, respectively. In Fig. 12, the component qz (qx) displays a cusp (pick) at Oc/J ¼ 0.2571 and decreases (increases) with the increase of O/J, respectively. The curves in Figs. 10 and 13 exhibit the same characteristic behavior as that obtained for the case a ¼ 0.25. The discussion here is similar to the case for aX0 in Fig. 9.

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4. Conclusion In this work, we have discussed the spin-1 Ising model with random interactions in a transverse field by the use of the effective field theory. The phase diagrams have been examined by selecting some characteristic cases in the system with O/J ¼ 0. The effect of a transverse field on the phase diagram was shown and it is also shown that a number of interesting phenomena such as the two critical lines and reentrant phenomenon may occur. The main important conclusion of our study is the influence of the parameter a in the absence of the transverse field, but also for their presence as is shown in Figs. 6 and 7. We especially found a reentrant phenomenon for a ¼ 0.1 and 0.400opo0.473. When p increases (higher to 0.400) the reentrant phenomenon disappears and we found two ferromagnetic areas marked by two critical lines. The reentrant phenomenon discussed in Fig. 7 is also shown in thermal variation of mz and in the variation of mz with O/J (Figs. 12 and 13). Acknowledgments One of the authors, Mohammed Saber, is grateful to Prof. Julian Gonza´lez for his hospitality and help. This work has been completed at the San Sebastian University. M.S. has been awarded the Ikerbasque Fellowship 2008. References [1] P.G. De Gennes, Solid State Commun. 132 (1963). [2] R. Blinc, B. Zeks, Soft Modes in Ferroelectric and Antiferroelectrics, NorthHolland, Amsterdam, 1947. [3] R.J. Elliott, A.P. Young, Ferroelectrics 7 (1974) 23. [4] R.J. Elliott, G.A. Gehring, A.P. Malogemoff, S.R.P. Smith, N.S. Staude, R.N. Tyte, J. Phys. C4 L (1971) 179. [5] Y.L. Wong, B. Cooper, Phys. Rev. 172 (1968) 539. [6] B. Blinc, B. Zeks, Adv. Phys. Z. 1 (1972) 693. [7] R.B. Stinchcombe, J. Phys. C 6 (1973) 2459. [8] S. Kastura, Phys. Rev. 127 (1962) 1501. [9] P. Pfeuty, Ann. Phys. (N.Y.) 57 (1970) 79. [10] M. Suzuki, Phys. Lett. 34CA (1974) 94. [11] M.E. Fisher, J. Math. Phys. Status Solidi B 113 (1982) 367. [12] R.J. Elliott, C. Wood, J. Phys. C 4 (1971) 2359. [13] P. Pfeuty, R.J. Elliott, J. Phys. C 4 (1971) 2370. [14] J.A. Plascakand, S.S. Salinas, Phys. Status Solidi B 113 (1982) 367. [15] F.C. Sa` Barreto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39 (1981) 1103.

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