A theoretical study of the spin-32 Ising model in a random field with crystal field

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 265 (2003) 305–310

A theoretical study of the spin-32 Ising model in a random field with crystal field Ya-Qiu Lianga,b,*, Guo-Zhu Weia,c, Qi Zhanga, Zi-Hua Xina a

College of Sciences, Northeastern University, Shenyang 110006, China Department of Physics, Liaoning University, Shengyang 110036, China c International Center for Material Physics, Academia Sinica, Shengyang 110015, China b

Received 10 December 2002

Abstract A spin-32 random-field Ising model with a crystal field on the honeycomb lattice is studied within the framework of the effective-field theory with correlations. We have investigated the effect of the crystal field on the phase diagrams, magnetizations, quadrupolar moments of the system. The phase diagram exhibits a rich variety of behaviors: the reentrant phenomena and the existence of tricritical points. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.H; 75.10.D; 75.50.G Keywords: Ising model; Random field; Crystal field; Phase diagram; Magnetization

1. Introduction Considerable progress has been made in the understanding of the random-field Ising model (RFIM) (see Refs. [1–4] and references therein). One of the interesting phenomena in the RFIM is the occurrence of a tricritical behavior. Schneider and Pytte [5] studied the RFIM on the traditional mean-field approximation and they found that for a model with a Gaussian distribution of random fields, the phase transition remains of second order, down to zero temperature. On the other hand, Aharony [6] showed within the framework *Corresponding author. College of Sciences, Northeastern University, Shenyang 110006, China. E-mail address: [email protected] (Y.-Q. Liang).

of the same approximation that a tricritical point can be observed for a model with a two-peakdistribution function. Afterwards, the existence of a tricritical behavior has been examined by the use of various techniques, such as mean-field theory [7], Monte Carlo simulations [8], renormalizationgroup calculations [9], Bethe–Peierls approximation [10], and effective-field theories (EFT) [11]. Recently, some interest have been directed to the understanding of more complicated systems in the presence of random fields, i.e. the transverse Ising model [12–14], the amorphous Ising ferromagnet [15], the site-dilute Ising model [16,17], the semiinfinite Ising model [18], and the Blume–Capel model [19]. It has been shown that they can find in these systems very rich critical behaviors and many interesting phenomena can appear (i.e. the

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reentrance behavior or the existence of two tricritical points). As far as we know, at present there are no works dealing with the RFIM with a crystal field. Therefore, in this work we extend the study of the crystal field effects on the spin-32 of the RFIM on the honeycomb lattice. The problem is analyzed by using the EFT based on the exact spin-identities and differential operator technique [20]. The paper is organized as follows. In Section 2 the basic framework for the spin-32 RFIM with the presence of the crystal field is given. In Section 3, the phase diagram and temperature dependences of the magnetization are examined in detail for the honeycomb lattice. Section 4 contains the conclusion.

2. Formulations The Hamiltonian of a random-field spin-32 Ising model with a crystal interaction is given by X X 1X H¼ Jij Si Sj  Hi S i  D ðSi Þ2 ; ð1Þ 2 i;j j i where Si ¼ 732; 12 and Ji;j is the exchange interaction which couples nearest neighbors only, D represents the crystal-field interaction and Hi is the random field. We shall consider that each spin is subjected to a random magnetic field Hi according to the following distribution: PðHi Þ ¼ 12 ½dðHi  HÞ þ dðHi  HÞ:

ð2Þ

As usual, we shall assume that the random fields on different lattice sites are uncorrelated. The averaged lattice magnetization is given by m ¼ //Si SSr ;

ð3Þ

can be obtained. They are given as follows: *  + z Y Si cos hðZJrÞ þ sin hðZJrÞ /Si S ¼ Z i¼1 F ðx; HÞjx¼0 ; * q¼

z  Y i¼1

Si cos hðZJrÞ þ sin hðZJrÞ Z

ð5Þ +

Gðx; HÞjx¼0 :

ð6Þ

Here r ¼ q=qx is a differential operator and functions F ðx; HÞ and Gðx; HÞ are as follows: F ðx; HÞ "     1 3 sin h 32 bðx þ hÞ þ e2bD sin h 12 bðx þ hÞ     ¼ 2 2 cos h 32 bðx þ hÞ þ 2e2bD cos h 12 bðx þ hÞ    # 3 sin h 32 bðx  hÞ þ e2bD sin h 12 bðx  hÞ     ; þ 2 cos h 32bðx  hÞ þ 2e2bD cos h 12 bðx  hÞ ð7Þ Gðx; HÞ ¼

    1 9 sin h 32 bðx þ hÞ þ e2bD sin h 12 bðx þ hÞ 3  1  2 4 cos h 2 bðx þ hÞ þ 4e2bD cos h 2 bðx þ hÞ     ! 9 sin h 32 bðx  hÞ þ e2bD sin h 12 bðx  hÞ     ; þ 4 cos h 32 bðx  hÞ þ 4e2bD cos h 12 bðx  hÞ

ð8Þ where b  1=KB T: Substituting Eqs. (5)–(8) into Eqs. (3) and (4) and using the decoupling approximation, /Si ðSj Þ2 ySm S ¼ /Si S/ðSj Þ2 Sy/Sm S;

ð9Þ

for iaja?am; Eqs. (3) and (4) for a honeycomb lattice (Z ¼ 3) can be written as m ¼ A1 m þ A2 m3 ;

ð10Þ

q ¼ q0 þ q2 m 2 :

ð11Þ

and the parameter q is defined as q ¼ Z2 ¼ //Si2 SSr :

ð4Þ

Here, the inner angular brackets and the outer /?Sr denote a thermal averaging and a random configurational averaging, respectively. The latter one is to be carried out after the thermal averaging has been taken. By using the effective-field theory with correlation, the mean values /Si S; /ðSi Þ2 S

The coefficients A1 ; A2 ; A3 and q1 can be easily calculated by applying a mathematical relation ear f ðxÞ ¼ f ðx þ aÞ: Here A1 ¼

3 ½F ð3JZÞ þ F ðJZÞ; 4Z

ð12Þ

ARTICLE IN PRESS Y.-Q. Liang et al. / Journal of Magnetism and Magnetic Materials 265 (2003) 305–310

3 ½F ð3JZÞ  3F ðJZÞ; 4Z3

q0 ¼ 14 ½Gð3JZÞ þ 3GðJZÞ; q1 ¼

3 ½Gð3JZÞ  GðJZÞ: 4Z2

5

ð13Þ

ð15Þ

For determining the phase transition point, expanding the right-hand side of Eqs. (10) and (11), the lattice magnetization in the vicinity of the phase transition point can be written as 1a m2 ¼ : ð16Þ b The parameters a and b are obtained as follows: a ¼ A1 ðZ0 Þ; b¼

A01 ðZ0 Þq1 ðZ0 Þ þ A2 ðZ0 Þ; 2Z0  q00 ðZ0 Þ

ð17Þ ð18Þ

where Ai ði ¼ 1; 2Þ are defined as a function of Z0 ; which is the value of Z at T ¼ Tc ; and A01 denotes the first derivative of A1 with Z0 : According to Kaneyoshi [21], the second-order phase-transition line is determined by a ¼ 1; bo0;

0 2 O O

1

0 0

1

2

H/J

Fig. 1. Phase diagrams for the spin-32 RFIM with the crystal field (D=JX0) on the honeycomb lattice in the (H=J; KB Tc =J) planes. The solid and dashed lines represent the second- and first-order phase transitions, respectively. The white circle indicates the tricritical point. The numbers at the curves are the crystal field D=J:

3

-0.5

ð19Þ

0

-0.75 2

ð21Þ

KBTC/J

ð20Þ

and the tricritical point by a ¼ 1; b ¼ 0:

1

3

the first-order phase-transition line by a ¼ 1; b > 0;

5

4

ð14Þ

KBTC/J

A2 ¼

307

O

1 O

-1.5

Within the above framework, we can investigate magnetic properties for the honeycomb lattice system with crystal-field interaction.

-2

0

1

2

H/J

3. Numerical results and discussions In this section we are interested in studying the transition temperature and thermal variations of magnetization on the honeycomb lattice. At first, we investigate the phase diagram of the system with different crystal fields by solving Eqs. (16)– (21) numerically. The results are depicted in Fig. 1 for the system with D=JX0 and in Fig. 2 for the system with D=Jp0: Figs. 1 and 2 show the phase diagrams in the ðH=J; KB Tc =JÞ plane with differ-

Fig. 2. Phase diagrams for the spin-32 RFIM with the crystal field (D=Jp0) on the honeycomb lattice in the (H=J; KB Tc =J) planes. The solid and dashed lines represent the second- and first-order phase transitions, respectively. The white circle indicates the tricritical point. The numbers at the curves are the crystal field D=J:

ent D=J values. In these figures, the solid and dashed lines represent the second- and first-order phase transitions, respectively. The white circle indicates the tricritical point. In these figures, we

ARTICLE IN PRESS Y.-Q. Liang et al. / Journal of Magnetism and Magnetic Materials 265 (2003) 305–310

308

see that the Tc curves of the system exhibit the reentrant phenomena when D=JX  0:41 and D=Jp  0:64: Further, the Tc curves of the system change from second- to first-order and a tricritical point lying in between the two transitions when 0:75pD=Jp3:58: For D=JX3:58 and D=Jp 0:75; we obtained a second-order transition line only and there is no tricritical point on it. In particular, the Tc curve for D=J ¼ 0 has similar form to that in Refs. [11,22]. The tricritical point ðH=J; KB Tc =JÞ; for D=J ¼ 0 is (1.981, 1.199), can be compared with the result (1.981, 1.199) in

Ref. [22] and with the result (2.072, 1.011) in Ref. [11]. Finally, we note that the tricritical point ðH=J; KB Tc =JÞ shifts towards the higher values H=J and lower temperatures with the decrease of D=J: Now, let us show typical temperature dependences of the order parameters m; q to illustrate the random-field and crystal-field effects in the system. In Figs. 3–5, the temperature dependence of the magnetization and the quadrupolar moment for the honeycomb lattice is given when random field H=J and crystal field D=J are selected at

2.5

2.5

q

q 2.0

2.0

m

1.5

1.5

m,q

m,q

m 1.0

1.0

D/J=1 H/J=1.78

0.5

0.5

D/J=1 H/J=1.36 0.0

0.0 0

1

2

0

3

H/J

(a)

1

2

3

H/J

(b)

2.5

q 2.0

m,q

1.5

1.0

D/J=1 H/J=1.90

0.5

m

0.0 0

(c)

1

2

3

H/J=1.90

Fig. 3. The temperature dependences of the magnetization m and the quadrupolar moment q for the spin-32 RFIM with the crystal fields on the honeycomb lattice, when D=J ¼ 1: The random field is selected as (a) H=J ¼ 1:36; (b) H=J ¼ 1:78; and (c) H=J ¼ 1:90:

ARTICLE IN PRESS Y.-Q. Liang et al. / Journal of Magnetism and Magnetic Materials 265 (2003) 305–310

q

309

0.50

2

m

m,q

m,q

m q 0.25

1

D/J=-2 H/J=0.355

D/J =5 H/J =1.30 0.00 0.0

0 0

1

2

3

4

T/J

(a)

0.5

1.0

T/J

(a) 0.50

q m

m,q

m,q

2

q 0.25

m 1

D/J=-2 H/J=0.60

D/J =5 H/J= 1.78

0.00 0.00

0.25

0.50

0 0

(b)

1

2

3

4

T/J

Fig. 4. The temperature dependences of the magnetization m and the quadrupolar moment q for the spin-32 RFIM with the crystal fields on the honeycomb lattice, when D=J ¼ 5: The random field is selected as (a) H=J ¼ 1:30 and (b) H=J ¼ 1:78:

some typical values, respectively. At first, in Figs. 3a–c the magnetization m and the quadrupolar moment q are plotted versus temperature for H=J ¼ 1:36; 1.78, 1.90 and D=J ¼ 1: From Fig. 3a, we can see that the magnetization curve is monotonically decreasing from the saturation value m ¼ 1:5 at T ¼ 0 K and continuously vanishing at critical temperature Tc ¼ 2:932: In fact, for 0pH=Jp1:53 when D=J ¼ 1 only second-order phase transition can be observed in Fig. 1. In Fig. 3b the same dependences as in Fig. 3a are shown, but for H=J ¼ 1:78 the nonzero magnetization is reduced. And for

(b)

T/J

Fig. 5. The temperature dependences of the magnetization m and the quadrupolar moment q for the spin-32 RFIM with the crystal fields on the honeycomb lattice, when D=J ¼ 2: The random field is selected as (a) H=J ¼ 0:355 and (b) H=J ¼ 0:60:

1:53pH=Jp1:884 both first- and second-order phase transitions are clearly visible on the transition line of D=J ¼ 1: Finally, a different case is presented for H=J ¼ 1:90 in Fig. 3c, and a peak appears between the two critical temperatures when two phase transitions of second order occur on the transition line D=J ¼ 1: Figs. 4a and b show the temperature dependence of the magnetization and the quadrupolar moment for H=J ¼ 1:30; 1.78 and D=J ¼ 5: From our investigation we know that the magnetization curve in Fig. 4a is the same as the magnetization curve in Fig. 3a. This is due to second order for

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H=J ¼ 1:30 and D=J ¼ 5 observed in Fig. 1. Fig. 4b shows the same form of the magnetization curve as the magnetization curve in Fig. 3c. This is due to the reentrant phenomenon of second-order phase transition for H=J ¼ 1:78 and D=J ¼ 5 seen in Fig. 1. For the negative crystal field, the temperature dependence of the magnetization and the quadrupolar moment for H=J ¼ 0:355; 0.60 and D=J ¼ 2 is exhibited in Figs. 5a and b. Fig. 5a shows the same behavior of the magnetization as in Fig. 4a. And the same form of the magnetization curve as the magnetization curve in Fig. 4b is shown in Fig. 5b. This is due to the same features of the phase transition for D=J ¼ 2 and 5: On the other hand, different behaviors of the magnetization and quadrupolar moment can be found. The value of the saturation magnetization is shown in Fig. 4a, while the value of the saturation magnetization is m ¼ 0:5 in Fig. 5a for D=J ¼ 2: Moreover, the temperature dependence of the quadrupolar moment displays different behaviors for the positive crystal fields and the negative crystal fields. From Figs. 3 and 4, we can see that with the increase of the temperature, the quadrupolar moment decreases slowly from q ¼ 2:25 at low temperatures and approaches a constant value in the vicinity of critical temperature. With the increase of the temperature, the quadrupolar moment increases very slowly and the value of the quadrupolar moment qE0:25 at low temperatures and appears increasing obviously in the vicinity of critical temperature in Figs. 5a and b. Physically, the results of the magnetization and the quadrupolar moment for D=J ¼ 2 come from the fact that the state approaches the Si 712 state for large negative D=J: While the results of the magnetization and the quadrupolar moment for D=J ¼ 5 come from the fact that the state approaches the Si 712 state for large positive D=J:

4. Conclusion In this work we have studied the spin-32 Ising model in the random field with crystal field within the framework of the EFT. The numerical results are presented in Section 3 especially for the

ferromagnetic Ising system with the crystal field on honeycomb lattice. As discussed in Section 3, the behavior of the system is strongly influenced by the crystal field. A number of interesting phenomena have been found. The reentrant phenomenon appears when D=JX  0:41 and D=Jp  0:64: The phase transitions change from second- to firstorder and a tricritical point lying in between the two transitions when 0:75pD=Jp3:58: For D=JX3:58 and p  0:75; we obtained second-order transition line only and there is no tricritical point on it. Moreover, we studied the temperature dependence of the magnetization and the quadrupolar moment in detail on the basis of our analysis with the phase diagram. Finally, we hope that the present method will be potentially useful for studying more complicated RFIM systems in the presence of crystal field.

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