Physica A 392 (2013) 2643–2651

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The two- and three-dimensional spin-3/2 randomBlume–Capel model by the position spacerenormalization groupMohammed El Bouziani ∗, Abou GayeLaboratoire L.P.M.C., Equipe de Physique Théorique, Département de Physique, Faculté des Sciences, Université Chouaib Doukkali, B.P.20, 24000 El Jadida, Maroc

a r t i c l e i n f o

Article history:Received 5 November 2012Received in revised form 7 February 2013Available online 14 February 2013

Keywords:Phase transitionsBlume–Capel modelRandom crystal fieldRenormalization groupPhase diagramsDimensional crossover

a b s t r a c t

We use the Migdal–Kadanoff renormalization group technique to study the spin-3/2Blume–Capel model under a random crystal field, in the two- and three-dimensional cases.Studying the fixed points and the phase diagrams established, we find interesting resultsallowing us to understand the critical behavior of the system. In the two-dimensional case,the randomness, even in small amounts, removes completely the first order transition be-tween the two ferromagnetic phases present, replacing it by a smooth continuation. Onlythe second order phase transitions occur. In the three-dimensional case, the first orderphase transition disappears only at a certain threshold of randomness. Below this thresh-old, we observe the presence of an end-point where finishes the first order transition lineinside the ferromagnetic phases. This end-point reaches T = 0 K at a critical value of prob-ability, beyond which only the second order transitions occur.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Many interesting studies have been devoted to the Blume–Capel model [1,2], initially introduced for the study of firstorder magnetic phase transitions. It is a spin-1 Ising model with a single anisotropic ion. Later, the Blume–Emery–Griffithsmodel [3], applied to the isotopic mixtures of helium 3He–4He, was proposed as a generalization of the Blume–Capel model.In these mixtures, the 3He and 4He atoms are respectively represented by the state S = 0 and S = ±1. The spin-1Blume–Capel and Blume–Emery–Griffiths systems have been investigated by a variety of approximation methods, suchas the variational methods [4,5], the effective field theory [6], the renormalization group techniques [7,8], the mean fieldapproximation [9] and Monte Carlo simulations [10,11].

It is possible to extend the spin-1 Blume–Capel and Blume–Emery–Griffiths models by including higher spin values. Thesimplest extensions are probably the spin-3/2 Blume–Capel and Blume–Emery–Griffiths models, proposed to explain thetricritical properties in ternary fluids mixtures [12] and the magnetic and crystallographic phase transitions in some rare-earth compounds such as DyVO4 [13]. Here also, severalmethods have been used, such as themean field approximation [14],the effective field theory [15], and the techniques of renormalization group [16,17].

Many studies have been devoted for decades to systems subject to random fields [14,18,19] and in the view to understandtheir effect on the phase transitions. The presence of randomness produces remarkable impacts on the critical behavior of thesystems. In two-dimensional systems, the first order phase transitions are removed totally for any amount of randomness. Inthree-dimensional systems, it is generally expected that the first order remains for weak disorder, disappearing completely

∗ Corresponding author. Tel.: +212 666881814; fax: +212 523 34 21 87.E-mail address: elbouziani.m@ucd.ac.ma (M. El Bouziani).

0378-4371/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physa.2013.02.003

2644 M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651

at a critical value of randomness, above which only the second order phase transitions occur. It has been noted that only thepresence of randomness is important; the exact form of the probability distribution does not play a fundamental role.

In our present study, we are interested in the spin-3/2 Blume–Capel model, on which we apply a random crystal fieldobeying to a probability distribution of two peaks. We use a position space renormalization group technique, namely theMigdal–Kadanoff [20,21] one. In its pure version, the spin-3/2 Blume–Capel model presents a second order transitionline which separates the ferromagnetic and paramagnetic phases in both two- and three-dimensional cases; the twoferromagnetic phases are separated by a first order transition line which finishes at the second order transition line.We findthat the introduction of randomness affects considerably the critical behavior of the system. Indeed, in the two-dimensionalcase, the randomness removes totally the first order phase transition, leaving only those of second order. But in the three-dimensional case, the first order transition is present at small amount of randomness, disappearing completely only beyonda certain threshold of randomness. Below this threshold, we observe the presence of an end-point where finishes the firstorder transition inside the ferromagnetic phases. This end-point reach T = 0 K at a critical value of the probability, beyondwhich only the second order transition occurs. Thus,wenote the existence of a dimensional crossover indicating a qualitativedifference in the critical behavior of the two- and three-dimensional random spin-3/2 Blume–Capel models.

We organize our article as follows. In Section 2, we treat the formalism of our method and we establish theMigdal–Kadanoff recursion equations. In Section 3, we present our results and discuss important points. Finally, we givea conclusion in the last section.

2. Model and technique

In order to study the spin-3/2 Blume–Capel model, we consider the following Hamiltonian:

− βH = Ji,j

SiSj +

i

∆iS2i (1)

where the spins Si, located at the site i on a discrete d-dimensional lattice, can take the four values ±3/2 and ±1/2. J is thereduced bilinear interaction and ∆i is the crystal field at the site i. The first summation is over all nearest neighbor pairs ofthe lattice and the second one over all sites.

In the Blume–Capel model, the reduced biquadratic interaction K is equal to zero, but we will take it into account dueto the renormalization group technique we are using. We also introduce two additional interactions, C and F , to obtainself-consistent recursion relations. Thus, the Hamiltonian we will effectively use in the remainder of our work is as follows:

− βH = Ji,j

SiSj +

i

∆iSi + Ci,j

SiS3j + S3i Sj

+ F

i,j

S3i S3j . (2)

We are not concerned here about the physical meaning of the interactions C and F , they are added only for a purely technicalpurpose in order to preserve the parameters space renormalization. Indeed, the renormalization does not keep in general theparameters space of the Hamiltonian, what constitutes an anomaly being able to cause physical aberrations. For example,in the Blume–Emery–Griffiths model, the three parameters J , K and ∆ are insufficient to stabilize the ferrimagnetic phasewhich has been obtained by all the methods of effective fields.

The different phases of the Blume–Capel model can be characterized by two order parameters, the magnetizationm = ⟨Si⟩ and the quadrupolar momentum q = ⟨S2i ⟩. When m = 0, we have two paramagnetic phases, referred to asP3/2 and P1/2, characterized respectively by q > 5/4 and q < 5/4. Whenm = 0, we have two ferromagnetic phases labeledF3/2 and F1/2, distinguished respectively by q > 5/4 and q < 5/4.

The crystal field∆i is subject to randomness and obeys to a probability distribution P(∆i)with two peaks that is given by

P (∆i) = pδ (∆i + ∆) + (1 − p) δ (∆i − ∆) . (3)

To have a more reliable qualitative appreciation of the phase transitions characteristics, we use an approximation of thereal space renormalization group, namely the Migdal–Kadanoff one, which combines decimation and bond shifting and istractable in all space dimensionalities. In order to implement the renormalizationmachinery,we consider a one-dimensionalchain of four spins S1, S2, S3 and S4, coupled by the interactions J, K , C and F ; ∆1, ∆2, ∆3 and ∆4 are the crystal fields ateach site of the chain. The spatial factor rescaling, denoted by b, is chosen as an odd integer to keep the possible sublatticesymmetry breaking character of the system. In our present study, we take b = 3. The crystal field is a local interaction andmust be adaptedwith the renormalization procedure by transforming it into a bond, what is possible by performing an equalsharing of ∆i along the 2d bonds leading to the site i. Furthermore, we have to take into account the coordination number ofthe site i in the crystal field term.With these considerations,we canwrite the reducedHamiltonian of the four spins cluster as

− βH = J (S1S2 + S2S3 + S3S4) + KS21S

22 + S22S

23 + S23S

24

+

∆1S21 + ∆4S242d

+∆2S22 + ∆3S23

d

+ CS1S32 + S31S2 + S2S33 + S32S3 + S3S34 + S33S4

+ F

S31S

32 + S32S

33 + S33S

34

. (4)

M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651 2645

After performing the decimation on the twomiddle spins S2 and S3, we obtain a two spins cluster described by the followingreduced Hamiltonian:

− βH = JS1S4 + K S21S24 +

∆1S21 + ∆4S242d

+ CS1S34 + S31S4

+ F S31S

34 (5)

with J, K , ∆1, ∆4, C and F the interactions after decimation and functions of J, K , ∆1, ∆2, ∆3, ∆4, C and F and the dimen-sion d. By using the renormalization group equation, we can make the link between (4) and (5) to obtain

S2,S3

exp (−βH) = A0 exp−βH

(6)

with A0 a constant produced by the renormalization scheme. A0 is useful for the determination of the free energy and there-fore of all the thermodynamic quantities of system. We obtain, after bond shifting, the Migdal–Kadanoff renormalized in-teractions:

J ′ = bd−1 J K ′= bd−1K C ′

= bd−1CF ′

= bd−1F ∆′

1 = bd−1∆1.(7)

Although at the beginning only the crystal field was random, because of the renormalization procedure, randomness isintroduced in all the renormalized quantities of the system.We consider that the renormalized distributions are in the sameform as the initial ones. The interactions J ′, K ′, C ′ and F ′ and the renormalized constant A0 obey to a probability distributionof one peak. By averaging on the disorder, we find these expressions

J ′ = bd−1 (1 − p)2 ln

AA′

8132

.

B′

B

316

.

C1

C ′

1

1288

+ bd−1p2 ln

DD′

8132

.

E ′

E

316

.

F1F ′

1

1288

+ bd−1p (1 − p) ln

GG′

8132

.

H ′

H

316

.

LL′

1288

+ bd−1p (1 − p) ln

GG′

8132

.

I ′

I

316

.

LL′

1288

(8)

K ′=

bd−1

8(1 − p)2 ln

AA′C1C ′

1

(BB′)2

+

bd−1

8p2 ln

DD′F1F ′

1

(EE ′)2

+

bd−1

8p (1 − p) ln

GG′

HI ′7 LL′

(H ′I)9

+

bd−1

8p (1 − p) ln

GG′

H ′I7 LL′

(HI ′)9

(9)

C ′= bd−1 (1 − p)2 ln

A′

A

98

.

BB′

512

.

C1

C ′

1

172

+ bd−1p2 ln

D′

D

98

.

EE ′

512

.

F1F ′

1

172

+ bd−1p (1 − p) ln

G′

G

98

.

HH ′

512

.

L′

L

172

+ bd−1p (1 − p) n

G′

G

98

.

II ′

512

.

L′

L

172

(10)

F ′= bd−1 (1 − p)2 ln

AA′

12

.

B′

B

13

.

C1

C ′

1

118

+ bd−1p2 ln

DD′

12

.

E ′

E

13

.

F1F ′

1

118

+ bd−1p (1 − p) ln

GG′

12

.

H ′

H

13

.

LL′

118

+ bd−1p (1 − p) ln

GG′

12

.

I ′

I

13

.

LL′

118

(11)

A0 = (1 − p)2AA′81 C1C ′

1

(BB′)18+ p2

DD′

81 F1F ′

1

(EE ′)18+ 2p (1 − p)

GG′81 LL′

(HH ′II ′)9

(12)

with A, A′, B, B′, C1, C ′

1,D,D′, E, E ′, F1, F ′

1,G,G′,H,H ′, I, I ′, L and L′ functions of J, K , ∆, C, F and d. Let Ap be the value ofthe renormalization constant at the p-th iteration. The reduced free energy of the system is given by the relation

f =

∞p=1

Ap

bpd. (13)

Knowing that ∆1, ∆2, ∆3 and ∆4 are equal to ±∆, we find 16 peaks for ∆′

1, denoted by hi, i = 1, . . . , 16, given by theexpressions below:

h1 = h3 = bd−1∆ +bd−1d16

ln

BB′10

(AA′)9 C1C ′

1

h2 = h4 = −bd−1∆ +

bd−1d16

ln

EE ′10

(DD′)9 F1F ′

1

(14)

2646 M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651

h5 = h11 = bd−1∆ +bd−1d16

ln

HH ′

II ′9

(GG′)9 LL′

h6 = h12 = −bd−1∆ +

bd−1d16

ln

II ′HH ′

9(GG′)9 LL′

(15)

h7 = h13 = bd−1∆ +bd−1d16

ln

II ′HH ′

9(GG′)9 LL′

h8 = h14 = −bd−1∆ +

bd−1d16

ln

HH ′

II ′9

(GG′)9 LL′

(16)

h9 = h16 = −bd−1∆ +bd−1d16

ln

BB′10

(AA′)9 C1C ′

1

h10 = h15 = bd−1∆ +

bd−1d16

ln

EE ′10

(DD′)9 F1F ′

1

. (17)

That allows us to write the probability distribution after renormalization as follows:

P ′∆′

1

= (1 − p)4 δ

∆′

1 − h1+ p4δ

∆′

1 − h2+ p (1 − p)3 δ

∆′

1 − h3

+ p3 (1 − p) δ∆′

1 − h4+ p (1 − p)3 δ

∆′

1 − h5+ p3 (1 − p) δ

∆′

1 − h6

+ p (1 − p)3 δ∆′

1 − h7+ p3 (1 − p) δ

∆′

1 − h8+ p (1 − p)3 δ

∆′

1 − h9

+ p3 (1 − p) δ∆′

1 − h10+ p2 (1 − p)2 δ

∆′

1 − h11+ p2 (1 − p)2 δ

∆′

1 − h12

+ p2 (1 − p)2 δ∆′

1 − h13+ p2 (1 − p)2 δ

∆′

1 − h14+ p2 (1 − p)2 δ

∆′

1 − h15

+ p2 (1 − p)2 δ∆′

1 − h16. (18)

As one can see, the renormalization procedure has given rise to a broadening of the parameters space of the probabilitydistribution, making them pass from two before renormalization to sixteen after. Let us bring back this probabilitydistribution of 16 peaks into a distribution of two peaks:

P ′∆′

1

= p′δ

∆′

1 + ∆′+1 − p′

δ∆′

1 − ∆′. (19)

For this purpose, we use the Stinchcombe–Watson approximation [22], but its direct application gives aberrant values ofthe probability because the sign of each peak changes along the iteration process. To overcome this difficulty, we have toestimate the value of the 16 peaks at each iteration and to calculate the sum of the momenta of positive peaks on the onehand and the sum of the negative ones on the another hand. Let s1 and s2 be these sums. Thereby, we can determine therenormalized crystal field and probability by doing

∆′= s1 − s2 p′

=−s2

s1 − s2. (20)

This allows us to bring back the renormalized probability distribution into a one of two peaks and to obtain a renormalizedprobability without any aberration.

The recursion relations J ′, K ′, C ′, F ′, p′ and the free energy obtained permit us to study the fixed points, the phases andtransitions of our system. The analysis of these equations shows the existence of a physical symmetry. When we change ∆

to −∆ and p to 1− p, these equations remain invariant. This property of symmetry concerning the recursion equations willappear at the fixed points of the renormalization group transformation. If (J∗, K ∗, ∆∗, C∗, F∗, p∗) is a fixed point of (8)–(11)and (20), then (J∗, K ∗, −∆∗, C∗, F∗, 1− p∗) is also a fixed point of the transformation. This symmetry is also reflected in thephase diagrams in the space (1/J, ∆/J, p). We shall give only the types of diagrams between p = 0 and p = 1/2, the otherbeing obtained by changing ∆ to −∆ and p to 1 − p.

3. Results and discussions

3.1. Pure Blume–Capel model

When the probability p is equal to 0 or 1, our system is equivalent to the pure Blume–Capel model. We study this purecase because it will be very useful for us to understand the effect of randomness we will introduce in the system. The tableof fixed points of this pure model has been given in the two-dimensional case in a study realized by one of us [17]. Wecomplete this table in the present work by determining the fixed points of the three-dimensional case. These fixed pointsare obtained by following the flow of the iterations in the subspace K = C = F = 0. The fixed points that control thissubspace are eight and are presented in Table 1 below in the two-dimensional case as well as in the three-dimensionalone, what allows us to define the phases and phase transitions governing our system. These results will help us to betterunderstand the randomness effect in this system.

The fixed point F3/2 and F1/2 define two ferromagnetic phases separated by a transition line of first order described by thefixedpointN1. To determine this first order,wehave calledupon theNienhuis–Nauenberg [23] criterion. The fixedpoints P3/2and P1/2 define two paramagnetic phases separated by a smooth continuation governed by the point S1. We can thereforetreat these two phases as a single paramagnetic phase, denoted by P . Regarding the points C3/2 and C1/2, they describerespectively the second order transition between the phases F3/2 and P3/2 and the phases F1/2 and P1/2. Let us note that the

M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651 2647

Table 1Fixed points of pure (p = 0) Blume–Capel model for d = 2 and d = 3.

Fixed point Type Coordinates (J∗, K ∗, ∆∗, C∗, F∗)

d = 2 d = 3

F3/2 Phase (+∞, −∞, −∞, −∞, +∞) (+∞, −∞, −∞, −∞, +∞)

F1/2 Phase (+∞, −∞, +∞, −∞, +∞) (+∞, −∞, +∞, −∞, +∞)

P3/2 Phase (0, 0, +∆, 0, 0) (0, 0, +∆, 0, 0)

P1/2 Phase (0, 0, −∆, 0, 0) (0, 0, −∆, 0, 0)

N1 Discontinuous surface (+∞, +∞, −∞, −∞, +∞) (+∞, +∞, −∞, −∞, +∞)

J ≈ ∆ J ≈ ∆

S1 Smooth continuation (0, 0, 0, 0, 0) (0, 0, 0, 0, 0)C3/2 Critical surface (0.005, −0.021, +∞, −0.020, 0.080) (1.662, 0, +∞, −0.511, 0.115)C1/2 Critical (3.653, −0.021, −∞, −1.622, 0.7206) (1.793, −0.0017, −∞, −0.796, 0.354)

Fig. 1. Phase diagram for the pure (p = 0) two-dimensional Blume–Capel model.

two-dimensional and three-dimensional cases present the same qualitative characteristics. They differ only by quantitativeconsiderations.We illustrate our words by Figs. 1 and 2, in which the full and dashed lines represent respectively the secondorder and first order transition lines.

Thus, we remark that the first order transition line ends up in the second order transition line at a tetracritical point,in the two-dimensional case as well as in the three-dimensional one. However, this result is in disagreement with resultsfound by certain studies realized bymeans of othersmethods, inwhich one observes a double critical end-point. It is notablythe case in Ref. [24] using the mean field approximation, in Ref. [25] based on the finite-size scaling renormalization groupand in Ref. [26] using the pair approximation and the Monte Carlo simulation. So, one can ask a question: Is the absence ofthe double critical end-point due to a deficiency of theMigdal–Kadanoff approach? It is difficult to answer affirmatively thisquestion because other studies in addition to ours [27] and based on different methods find a tetracritical point instead of adouble critical end-point. We give as example Refs. [28,29] using respectively the finite-size scaling renormalization groupand the Monte Carlo simulation. We should also note that the critical behavior at low temperatures of the Blume–Capelmodel, concerning the systems of spin-3/2 and of greater spins, is not still well controlled, what explains the disparity inthe results observed in the scientific literature.

3.2. Random Blume–Capel model

3.2.1. The two-dimensional caseWith all the necessary results of the pure Blume–Capel model, we are better equipped to understand the effect of

randomness presence in our system.We introduce this randomness by taking a probability p such as 0 < p < 1.We establishthe fixed points of the randomBlume–Capelmodel in the subspace K = C = F = 0 in the two-dimensional case. These fixedpoints, determined in a parameters space of six dimensions are of coordinates (J∗, K ∗, ∆∗, C∗, F∗, p∗). We find 8 fixed pointsincluding 7 which present their coordinates J∗, K ∗, ∆∗, C∗ and F∗ identical to those of 7 fixed points of the pure model. If wedenote by X the fixed point of the pure model, that of the random Blume–Capel model will be denoted by (X, p∗). These 7fixed points we are talking about are: (P3/2, p∗

= 0), (P1/2, p∗= 1), (F3/2, p∗

= 0), (F1/2, p∗= 1), (S1, p∗

= 0), (C3/2, p∗=

2648 M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651

Fig. 2. Phase diagram for the pure (p = 0) three-dimensional Blume–Capel model.

Fig. 3. Global phase diagram corresponding to different values of probability for the two-dimensional random Blume–Capel model.

0) and (C1/2, p∗= 1). Only the fixed point N1 of the pure model has not been found. At its place, we find the fixed point

(S2, p∗= 0) of coordinates (+∞, 0.269, −1.346, −∞, +∞, 0) and it corresponds to that in Ref. [17], which describes the

smooth continuation between the ferromagnetic phases F3/2 and F1/2 of the pure Blume–Emery–Griffiths model.The points (P3/2, p∗

= 0) and (P1/2, p∗= 1) define the paramagnetic phases of the random model, phases separated

by a smooth continuation described by the point (S1, p∗= 0). We can therefore represent these two phases by a single

paramagnetic phase denoted by P . The points (F3/2, p∗= 0) and (F1/2, p∗

= 1) describe the ferromagnetic phases separatedby a line governed by the point (S2, p∗

= 0). Between these paramagnetic and ferromagnetic phases, the system undergoesa second order transition characterized by the fixed point (C1/2, p∗

= 1) for the transition F1/2 − P1/2 and by (C3/2, p∗= 0)

for the transition F3/2 − P3/2.As previously stated, the ferromagnetic phases F3/2 and F1/2 are separated by a line controlled by the fixed point

(S2, p∗= 0). To define the nature of this line, the Nienhuis–Nauenberg criterion has been used and has shown that the

first order is absent. So, the separation line between the phases F3/2 and F1/2, controlled by the fixed point (S2, p∗= 0),

is a smooth continuation. We represent these phases by a single ferromagnetic phase denoted by F . We conclude that theintroduction of randomness in the system, even in small amounts, eliminates totally the first order transitions, replacingthem by a smooth continuation. Only the second order transitions between the ordered and disordered phases occur.

We give the phase diagrams of the random Blume–Capel model, for several values of probability (see Fig. 3).Comparing our results with those of other studies, we find that they are in agreement with those in Refs. [30,18] which

study the spin-1 random Blume–Capel model by the position space renormalization group respectively on a square latticeand a hexagonal one. Indeed, these references find that the randomness eliminates completely the first order transition,whatever the value of probability. Furthermore, Cardy and Jacobsen [31] show, by a mapping between the random-field

M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651 2649

Isingmodel and the large q-state Pottsmodel, that the first order transition is absent in the two-dimensional case. In general,it is recognized that randomness, even in small amounts, removes the first order transition in two-dimensional systems.

3.3. The three-dimensional case

By taking a value of probability p such as 0 < p < 1, we introduce randomness in the three-dimensional Blume–Capelmodel. We determine the fixed points by following the flow of iterations in the subspace K = C = F = 0 in a six-dimensional space of parameters (J∗, K ∗, ∆∗, C∗, F∗, p∗). Between the ten fixed points we find, eight of them presenttheir coordinates J∗, K ∗, ∆∗, C∗ and F∗ identical to those of the eight fixed points of the pure three-dimensional model.Adopting the same notation as for the random two-dimensional model, we can present as follows these eight fixed pointsof the random three-dimensional model: (P3/2, p∗

= 0), (P1/2, p∗= 1), (F3/2, p∗

= 0), (F1/2, p∗= 1), (C3/2, p∗

=

0), (C1/2, p∗= 1), (S1, p∗

= 0) and (N1, p∗= 0). The remaining two fixed points, whose coordinates have been newly

identified in our present study, are (S2, p∗= 0) and (G2, p∗

= 0) with S2 = (+∞, 1.552, −11.696, −∞, +∞) andG2 = (+∞, −0.495, −∞, −∞, +∞). The points (F3/2, p∗

= 0), (F1/2, p∗= 1), (P3/2, p∗

= 0) and (P1/2, p∗= 1) describe

the ferromagnetic and paramagnetic phases, whilst the points (C3/2, p∗= 0) and (C1/2, p∗

= 1) govern the second ordertransitions occurring between these phases. The paramagnetic phases are separated by a smooth continuation governed bythe fixed point (S1, p∗

= 0). Regarding the points (N1, p∗= 0), (S2, p∗

= 0) and (G2, p∗= 0), we will see below their role.

As in the two-dimensional random case, the introduction of randomness produces a clear impact on the critical behaviorof the system. But, unlike this two-dimensional random case, the first-order transition does not completely disappear sinceit subsists at low amount of randomness. It disappears only above a certain threshold of randomness, above which only thesecond-order transitions occur. Thus, we note the existence of a critical value of the probability that reveals two differentcritical behaviors. This critical value is pc = 0.08. For a probability p such as 0 < p ≤ pc , the first order transition lineis present, but it reduces progressively as the probability increases. Thus, this line, which met the second order line in thepure model, finishes now at lower temperature by an end-point inside the ferromagnetic phase due to the influence ofrandomness. Below this end-point, the ferromagnetic phases are separated by a transition of first order described by thefixed point (N1, p∗

= 0). Above this end-point, these two phases are separated by a smooth continuation governed by thefixed point (S2, p∗

= 0). Concerning the end-points, they are described by the fixed point (G2, p∗= 0). As the probability

increases, the first order transition line is reduced until the end-point reaches T = 0 K at the critical probability pc = 0.08.Above this critical value, only the second order transition occurs, the first order transition being completely converted in asmooth continuation.

We can therefore speak of a dimensional crossover, given that there is a qualitative difference between the criticalbehaviors of the two- and three-dimensional random Blume–Capel models.

Concerning the universality class, we find that in the two-dimensional case as well as in the three-dimensional one, thefixed points governing the second order phase transition of the randommodel are identical to those of the pure model. Thisfact indicates that the second order transition line of the randommodel belongs to the universality class of the pure modelsecond order transition line, which is the Ising universality class. Indeed, it is possible, according to the Migdal–Kadanoffmethod, to extract the critical exponents, which are related to the fixed points, and one finds that they are the same as theIsing ones [32].

Below, we illustrate our findings by giving some examples of phase diagrams plotted in the plane (∆/J, 1/J). We presentin Figs. 4 and 5 the phase diagrams for values of probability lower than pc = 0.08 in order to illustrate the coexistence of thefirst and second order transitions and the reduction of the first order transition line. In Fig. 6, the diagram for a probabilitygreater than pc is represented to see the absence of the first order transition. In all these figures, the second order and firstorder transitions are represented respectively by the full and dashed lines, whilst the symbol (°) is set for the end-point.

Now it is time to compare our results with those existing in the literature. Let us note, however, that there are not yetenough studies concerning the influence of random crystal field on the spin-3/2 Blume–Capel model. Since the methods ofmean field approximation give coherent results mainly for higher dimensions, we shall do the corresponding comparisonwith the three-dimensional case we have treated here. In Ref. [33], the authors, studying the spin-3/2 random Blume–Capelby themean field approximation, find like us a second order transition line at higher temperature between the ferromagneticand paramagnetic phases. But they observe a new ferromagnetic phase in addition to the phases F3/2 and F1/2 presentedin our current study. It is the phase denoted by F1 of magnetization m = 1, occurring between F3/2 and F1/2. All theordered phases are separated at low temperature by first order transition lines finishing by end-points above which thereare smooth continuations. The first order transition is always present, whatever the value of randomness. These results arein contradiction with ours because we find a single first order transition line between the only two ferromagnetic phasesF3/2 and F1/2 and this first order disappears for probabilities greater than the critical value pc = 0.08. This contradiction iscertainly due to the fact that the mean field approximation often finds additional results not recovered by other methods.In Ref. [34], based on the pair approximation, the two- and three-dimensional randommodels present the same qualitativebehavior and the first order transition line is always present. That is in contrast with our findings, given that we observea difference in the critical behavior between the two- and three-dimensional random models; in addition, the first ordertransition line is not always present.

On the other hand, Albayrak [35], studying the spin-3/2 Blume–Capel model on a Bethe lattice, shows that the singlefirst order transition line, occurring at low temperature, is only present for higher values of probability p; when p decreases,

2650 M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651

Fig. 4. Phase diagram corresponding to the probability p = 0.001.

Fig. 5. Phase diagram corresponding to the probability p = 0.03.

Fig. 6. Phase diagram corresponding to the probability p = 0.2.

M. El Bouziani, A. Gaye / Physica A 392 (2013) 2643–2651 2651

this line becomes shorter and finally disappears. This result is similar to ours for the phase diagrams obtained by changing∆ to −∆ and p to 1 − p. This agreement is probably due to the fact that the Bethe lattice calculations offer a qualitativedescription better than those of the conventional mean field theories [36].

We also compare our results with those of studies on other values of spins. In Ref. [37], the authors investigate the spin-2random Blume–Capel model by the mean field approximation and they find that the first order transition is always present.However, in Ref. [30], a position space renormalization group study on the spin-1 Blume–Capel model find that randomnessremoves completely the first order phase transition whatever the value of probability, in the two-dimensional case as wellas in the three-dimensional one; there is therefore no dimensional crossover, contrary towhat is found in this present study.

As one can easily see, there is still a controversy concerning the effect of randomness on critical behavior of the phasetransitions because of the absence of an exact result.

4. Conclusion

During this work, we have studied the spin-3/2 Blume–Capelmodel under a random crystal field obeying to a probabilitydistribution of two peaks, in the two- and three-dimensional cases. We have used the Migdal–Kadanoff renormalizationgroup technique and have established the recursion equations governing our system. By following the flow of iterations inthe subspace K = C = F = 0, we found the fixed points table of the pure Blume–Capel model in the two-dimensional caseand we complete it in the three-dimensional one; we have also established the fixed points of the random Blume–Capelmodel. The study of these fixed points and the phase diagrams reveals us that the presence of randomness in the two-dimensional model removes completely the first order phase transition whatever the value of probability, replacing it bya smooth continuation; only the second order phase transitions are present. But in the three-dimensional model, the firstorder transition resists to small amounts of randomness and the first order transition line finishes in the ordered phase byan end-point. Above a certain threshold of randomness, defined by a critical value of probability pc = 0.08, the first ordertransition disappears totally and is converted into smooth continuation between the ferromagnetic phases.

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