multicritical phase diagrams of the blume–emery–griffiths model with repulsive biquadratic...

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Journal of Magnetism and Magnetic Materials 283 (2004) 392–408

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Multicritical phase diagrams of the Blume–Emery–Griffithsmodel with repulsive biquadratic coupling including

metastable phases: the pair approximation and the pathprobability method with pair distribution

Mustafa Keskin�, Ahmet Erdinc-

Department of Physics, Erciyes University, 38039 Kayseri, Turkey

Received 11 May 2003

Available online 2 July 2004

Abstract

As a continuation of the previously published work, the pair approximation of the cluster variation method is applied

to study the temperature dependences of the order parameters of the Blume–Emery–Griffiths model with repulsive

biquadratic coupling on a body centered cubic lattice. We obtain metastable and unstable branches of the order

parameters besides the stable branches and phase transitions of these branches are investigated extensively. We study

the dynamics of the model by the path probability method with pair distribution in order to make sure that we find and

define the metastable and unstable branches of the order parameters completely and correctly. We present the

metastable phase diagram in addition to the equilibrium phase diagram and also the first-order phase transition line for

the unstable branches of the quadrupole order parameter is superimposed on the phase diagrams. It is found that the

metastable phase diagram and the first-order phase boundary for the unstable quadrupole order parameter always exist

at the low temperatures which are consistent with experimental and theoretical works.

r 2004 Elsevier B.V. All rights reserved.

Pacs: 05.70.Fh; 64.60.Cn; 64.60.Kw; 64.60.My; 75.10.Hk

Keywords: The Blume–Emery–Griffiths model; Pair approximation; Path probability method; Metastable and unstable phases

- see front matter r 2004 Elsevier B.V. All rights reserve

/j.jmmm.2004.06.012

onding author. +90-352-437490133105; fax: +90-

1.

ddress: [email protected] (M. Keskin).

1. Introduction

This paper constitutes the continuation of thetreatment previously published paper [1] (andreferred to as paper I in the following) dealingwith multicritical phase diagrams of the Blume–

d.

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M. Keskin, A. Erdinc- / Journal of Magnetism and Magnetic Materials 283 (2004) 392–408 393

Emery–Grifftihs (BEG) model [2] with the repul-sive biquadratic coupling. The emphases in paper Iwas mainly to investigate the thermal variations ofthe order parameters of the BEG model with therepulsive biquadratic interaction by using thelowest approximation of the cluster-variationmethod [3] (LACVM) which is identical to themean-field approximation (MFA). Besides thestable branches of the order parameters, themetastable and unstable parts of these curvesand the phase transitions of the metastablebranches of the order parameters were determined.Furthermore, the dynamics of the model wasstudied by the path probability method (PPM)with point distribution [4] in order to make surethat the metastable and unstable branches of theorder parameters were obtained and definedcompletely and correctly. Finally, the metastablephase diagrams in addition to the equilibriumphase diagrams in the (kT/J, D/J) and (kT/J, K/J)planes were presented. The LACVM, in spite of itslimitations, is an adequate starting point. It offersa very practical and simple tool to solve mostcollective phenomena. On the other hand, while itis completely universal and generic, associatedquantitative results are unusually poor. Therefore,one should check that whether or not the results ofpaper I, especially branches of the metastable andunstable order parameters and as well as themetastable phase diagrams are an artifact of theLACVM which is identical to the MFA. Hence,the purpose of the current work is to address thisquestion. For this aim, we have chosen to use thecluster-variation method in pair approximation(CVMPA) [3] (the results of which coincide withthe results of the constant-coupling, Bethe and thetwo-particle cluster approximations), the methodthat has been used the spin-1 [5–9], spin-3

2[10] and

mixed spin-12–spin-1 [11] Ising systems, success-

fully. We also study the dynamics of the model byusing the PPM with pair distribution [4] in order tomake sure that we have found and classified themetastable and unstable branches of the orderparameters completely and correctly.The outline of the remaining part of this paper is

as follows. In Section 2, we define the model brieflyand obtain its solutions at equilibrium within thePACVM. The equilibrium properties of the system

are investigated in Section 3. The dynamics of themodel is studied by the path probability method inSection 4. In Section 5, transition temperatures arecalculated precisely and metastable phase dia-grams are presented in addition to the equilibriumphase diagrams. Section 6 contains the summaryand conclusion.

2. Model and method

The spin-1 Ising model Hamiltonian with bi-linear (J) and biquadratic (K) nearest-neighborpair interaction and single-ion potential or crystal-field interaction (d) is known as the Blume–Emery–Griffths (BEG) model [2] which is one of the mostextensively studied models in the statistical me-chanics and condensed matter physics. The BEGmodel is defined by the Hamiltonian

H ¼ �JXoij4

SiSj � KXoij4

ðSiÞ2ðSjÞ

2þ d

Xi

ðSiÞ2;

ð1Þ

where Si ¼ �1, 0 at each lattice site i, and hiji

indicates summation over all pairs of nearest-neighbor.The BEG model for K=J40 has been globally

analyzed by a variety of techniques in thestatistical physics: The MFA, high temperatureseries expansion, Monte Carlo (MC) methods,renormalization group (RG) techniques, effectivefield theory (EFT), cluster-variation method(CVM) and its modified versions, linear chainapproximation (LCA) among others [1,12,2]. Onthe other hand, the BEG model with repulsivebiquadratic coupling, i.e., K=Jo0 has not beenthoroughly explored. An early attempt to studythe BEG model with K=Jo0 was made by Chenand Levy [13] for d ¼ 0 and Oguchi et al. [14] byusing the MFA and the effective Hamiltonianmethod (EHM), respectively. Later on it has beenstudied by the MC method [15], the Monte CarloRenormalization group [16], the MFA [17], theRG [18], the CVM [1,6] and the two-particlecluster approximation [19]. The BEG model withrepulsive biquadratic coupling was also solved onthe Bethe lattice using the exact recursion equa-tions [20].

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M. Keskin, A. Erdinc- / Journal of Magnetism and Magnetic Materials 283 (2004) 392–408394

The BEG model is defined as a two-sublatticemodel, with spin variables Si ¼ �1, 0 andSj ¼ �1, 0 on sites of sublattices A and B,

respectively. The average value of each of the spin

states will be denoted by XA1 , XA

2 and XA3 on the

sites of sublattice A and XB1 , XB

2 and XB3 on the

sublattice B, which are also called the state or

point variables. XA1 and XB

1 are the fractions of the

spin value +1 on A and B sublattices, respectively,

and XA2 and XB

2 are the fractions of the spins that

have value 0 on A and B sublattices, respectively,

and XA3 and XB

3 are the fractions of the spins that

have the value �1 on A and B sublattices,respectively. These variables obey the followingtwo normalization relations for A and B sub-lattices,

X3i¼1

XAi ¼ 1 and

X3j¼1

XBj ¼ 1: ð2Þ

In order to consider the pair correlation, we

introduce the internal variables YAij and YB

ij ,

indicating the average number of the states inwhich the first member of the nearest neighborpair is in state i and the second member in states j

in sublattice A and B, respectively. These will becalled pair or bond variables interchangeably. The

relations between XAi , XB

i and YAij , YB

ij are also

follows:

XAi ¼

X3j¼1

YAij and XB

j ¼X3i¼1

YBji : ð3Þ

If we assume the symmetry, i.e., YAij ¼ YA

ji , then

nine bond variables reduces to six. If the number

of pairs in sublattice A is NAp , the number of

(+, +) bonds in the sublattice A is YA11N

Ap , (0,0)

bonds, YA22N

Ap , (�, �) bonds, YA

33NAp , (+, 0) or

(0, +) bonds, YA12N

Ap ð¼ YA

21NAp Þ, (+,�) or (�, +)

bonds, YA13N

Ap ð¼ YA

31NAp Þ, and finally (0, �) or

(�, 0) bonds in the system is YA23N

Ap ð¼ YA

23NAp Þ.

Therefore in the last three cases there is adegeneracy g=2 indicating the number of differentconfigurations having the same probability.The similar definitions can be done for the sub-

lattice B. The YAij and YB

ij are normalized by the

equations

X3i;j¼1

Y Aij ¼ 1 and

X3i;j¼1

Y Bij ¼ 1 ð4Þ

On the other hand, in order to account for thepossible two-sublattice structure we need fourlong-range order parameters which are introduced

as follows: SA ¼ hSiiA, QA ¼ hS2i iA, SB ¼ hSjiB,

QB ¼ hS2j iB for A and B sublattices, respectively.

SA and SB are the average magnetizations which isthe excess of one orientation over the otherorientation, called magnetizations, and QA andQB are the quadrupolar moments which is theaverage squared magnetizations for A and Bsublattices, respectively. The values of these stablebranches of the order parameters define fourphases with different symmetry. These are (i)paramagnetic phase (p) with SA=SB=0, QA=QB,(ii) ferromagnetic phase (f) with SA=SB 6¼0,QA=QB, (iii) antiquadrupolar or staggered quad-rupolar phase (a) with SA=SB=0, QA 6¼QB, and(iv) ferrimagnetic phase (i) with SA 6¼SB 6¼0,QA 6¼QB.The order parameters can be expressed in terms

of the internal variables and are given by

SA hSAi i ¼ XA1 � XA

3 ;

QA ðSAi Þ2

� �¼ XA

1 þ XA3 ;

SB hSBj i ¼ XB1 � XB

3 ;

QB ðSBj Þ2

D E¼ XB

1 þ XB3 ; (5)

Using Eqs. (2) and (5), the internal variables canbe expressed as linear combinations of the orderparameters

XA1 ¼ 1

2ðSA þ QAÞ; XA

2 ¼ 1� QA;

XA3 ¼ 1

2ðQA � SAÞ

XB1 ¼ 1

2ðSB þ QBÞ;

XB2 ¼ 1� QB;

XB3 ¼ 1

2ðQB � SBÞ (6)

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The order parameters in terms of the bondvariables are found using Eqs. (3) and (5)

SA ¼ hSAi i ¼ YB11 þ YB

12 � ðYB23 þ YB

33Þ;

QA ¼ hQAi i ¼ YB11 þ YB

12 þ 2YB13 þ YB

23 þ YB33;

SB ¼ hSBj i ¼ Y A11 þ YA

12 � ðYA23 þ YA

33Þ;

QB ¼ hQBj i ¼ YA11 þ YA

12 þ 2YA13 þ YA

23 þ YA33; (7)

The Hamiltonian of such a two-sublattice BEGmodel is

H ¼ �JXhiji

SiSj � KXhiji

S2i S2j þ dX

i

S2i þX

j

S2j

!:

ð8Þ

The interaction energy E can be written in terms ofthe pair variables

bE12gN

¼X3i;j¼1

eijðYAij þ YB

ij Þ; E ¼ hHi; ð9Þ

where N is the number of lattice points and g is thecoordination number, and eij are

eij ¼ JSAi SBj þ KQAi QBj � DðQAi þ QBj Þ ð10Þ

and eij can be written explicitly as

e11 ¼ J þ K � 2D; e22 ¼ 0;

e33 ¼ J þ K þ 2D; e12 ¼ e22 ¼ 0;

e13 ¼ � J þ K � 2D; e23 ¼ e32 ¼ �D; (11)

where D ¼ d=g, g is the coordination number. Wehave taken gA ¼ gB, i.e., the coordination numberon A lattice is equal to the coordination numberon B lattice.The weight factorsWA andWB can be expressed

in terms of the internal variables for the A and Bsublattices, respectively, as

WA� 1=NA

¼P3i¼1ðX

Ai LAÞ!

� gA�1LA!

gA=2 P3i¼1ðYAij LAÞ!

h igB=2and

WB� 1=NB

¼P3j¼1ðX

Bj LBÞ!

j kgB�1LB!

gB=2 P3i¼1ðYBij LBÞ!

h igB=2 ; (12)

where NA and NB are the number of lattice pointson the A and B sublattices, respectively. LA and LB

are the number of the system on the A and Bsublattices, respectively.Using the definition of the entropy Se

(Se=k lnW) with the Stirling approximation(lnN!ffiN lnN�N), the free energy F (F=E�TS)per site can now be found as

dF ¼ dbF

N

� �¼ d

g2

X3i;j¼1

eijðYAij þ YB

ij Þ

"

þg2

X3i;j¼1

YAij ln YA

ij þX3

i;j

YBij ln YB

ij

( )

� ðg� 1ÞX3i¼1

XAi lnðX

Ai Þ þ

X3j¼1

XBj lnðX

Bj Þ

( )

þ blA 1�X3i;j¼1

Y Aij

( )þ blB 1�

X3i;j

YBij

( )#;

(13)

where lA and lB are introduced to maintain thenormalization condition, b ¼ 1=kT , T is theabsolute temperature and k is the Boltzmannfactor. For a system at equilibrium the free energyis minimum. The minimization of Eq. (13) withrespect to YA

ij and YBij gives

qF

qYAij

¼ 0; andqFqYB

ij

¼ 0; ð14Þ

which leads to the 18 self-consistent equations

YAij ¼ ðXA

i XAj Þ

g expð�eijÞ=ZA eA

ij

ZA;

YBij ¼ ðXB

i XBj Þ

g expð�eijÞ=ZB eBijZB

;ð15Þ

where

g ¼g� 1g

; ZA ¼ expð2blA=gÞ ¼X3ij¼1

eAij

and

ZB ¼ expð2blB=gÞ ¼X3ij¼1

eBij :

This set of 18 self-consistent equations reducesto 12 self-consistent equations, since YA

ij ¼ YAji and

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kT/J0.0 0.5 1.0 1.5 2.0 2.5 3.0

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

QA3,QB3

QA1,QB1

QA2,QB2

SA3,SB3

QA3,QB3

SA2,SB2

(c)

Tt3Tt2

1.60 1.61 1.62

0.72

0.73

0.74

0.75

SA2,SB2

QA2

,QB2

0 1 2 3 4

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

SA1,SB1

QA1,QB1

QA3,QB3

0 1 2 3 4

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

QA1,QB1

QA2,QB2

SA1,SB1

QA3,QB3

QA3,QB3

(a)

(b)

Tt3 Tc

Tc

Tt3

Fig. 1. The temperature dependences of the order parameters.

Subscript 1 indicates the stable state (solid lines), 2 the

metastable state (dashed–dotted lines), and 3 the unstable state

(dashed lines). (a) Exhibiting a second-order phase transition

for the stable branch of the order parameters and first-order

phase transition for the unstable branches of the quadrupole

order parameters for D=J ¼ 0:3 and K=J ¼ 0:0. Tc represents

the second-order phase transition temperature for the stable

branches of order parameters and Tt3 represents the first-order

phase transition temperature for the unstable branches of

quadrupolar order parameters. (b) The same as (a), but D=J ¼

0:4 and K=J ¼ 0:0. (c) Exhibiting a first-order phase transitionfor the metastable and unstable branch of the order parameters

for D=J ¼ 0:5 and K=J ¼ 0:0. Tt2 and Tt3 represent the first-

order phase transition temperature for the metastable branches


YBij ¼ YB

ji , can be written explicitly as

YA11 ¼ ðXA

1 XA1 Þ

g expð�e11Þ=ZA eA11ZA

;

YA22 ¼ ðXA

2 XA2 Þ


;

YA33 ¼ ðXA

3 XA3 Þ


;

YA12 ¼ ðXA

1 XA2 Þ


YA13 ¼ ðXA

1 XA3 Þ


;

YA23 ¼ ðXA

2 XA3 Þ


;

YB11 ¼ ðXB

1XB1 Þ

g expðe11Þ=ZB eB11ZB

;

YB22 ¼ ðXB

2XB2 Þ

g expð�e22Þ=ZB eB22ZB

;

YB33 ¼ ðXB

3XB3 Þ


;

YB12 ¼ ðXB

1XB2 Þ


;

YB13 ¼ ðXB

1XB3 Þ


;

YB23 ¼ ðXB

2XB3 Þ


: (16)

These 12 nonlinear algebraic equations aresolved by using the Newton–Raphson methodfor a fixed kT, K/J, D/J and g ¼ 8 whichcorresponds to the body centered cubic lattice(BCC). After establishing the YA

ij and YBij values,

the SA, QA, SB and QB values can be obtainedeasily using the Eq. (7). In the following section,we shall examine the thermal variations of thesystem.

of order parameters and the unstable branches of the

quadrupole order parameter, respectively.

3. Thermal variations

In this section, we shall study the temperaturedependence of the order parameters SA, QA, SBand QB. Thermal variations of the order para-meters for several values of D/J and K/J areplotted in Figs. 1–6. In the figures, subscript 1

denotes the stable states (solid lines), subscript 2corresponds to metastable states (dashed–dottedlines) and 3 unstable states (dashed lines). Thisclassification is done by matching the free energyvalues and investigating solution of the dynamic or

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

TtTc Tc'

TuTt3

QA3,QB3

SA3,SB3

QA3,QB3

SA1,SB1QA2,QB2

QA1,QB1

QA2,QB2SA2,SB2

QA1,QB1SA1,SB1

1.46 1.48 1.50

0.72

0.73

0.74

0.75

QA2

,QB2

SA2

,SB2

kT/J

Fig. 2. The temperature dependence of the order parameters,

exhibiting a first-order phase transition and the two successive

second-order phase transitions for the stable branches of the

order parameters for D=J ¼ 0:413 and K=J ¼ �0:15. Tt, Tt3and Tu represent the first-order phase transition for the stable

states and unstable state and the upper limit stability

temperature, respectively. Tc and Tc0 represent the second-

order phase transition temperature for the stable branches of

order parameters.

kT/J0 1 2 3 4

S A,Q

A,S

B,Q

B

0.0

0.5

1.0QA1

QA2

QA1,QB1

TcTc2

QB3,QA3SA1,SB1

QB2

TtQB1

Fig. 3. The temperature dependence of the order parameters,

exhibiting the first-order antiquadrupolar phase transition and

the second-order ferromagnetic phase transitions for D=J ¼

�0:3 and K=J ¼ �1:5. The metastable quadrupolar orderparameters also undergo a second-order antiquadrupolar phase

transition at Tc2.

kT/J0.0 0.5 1.0 1.5 2.0 2.5

S A,Q

A,S

B,Q

B

0.00

0.25

0.50

QA1,QB1

SA1,SB1

QB2

QA2

QA3,QB3

Tc'Tc2'

Tc2

Tc

Fig. 4. The temperature dependences of the order parameters

for D=J ¼ 0:12 and K=J ¼ �1:0. Tc and Tc0 are the second-

order ferromagnetic phase transition for the stable branches of

order parameters, and Tc2 and Tc20 are the second-order

antiquadrupolar phase transition temperatures for the meta-

stable quadrupolar order parameters.


rate equations of the system which is given inSection 4. Tc or Tc0 and Tt are the critical or thesecond-order phase transition and the first-orderphase transition temperatures for the stablebranches of the order parameters, respectively.

Tfi and Tif are the second-order phase transitiontemperatures from the ferromagnetic phase to theferrimagnetic phase and from the ferrimagneticphase to the ferromagnetic phase, respectively, forthe stable branches of order parameters. Tt2 andTc2 or Tc20 are the first- and second-order phasetransition temperatures for the metastablebranches of the order parameters, respectively.Tt2 and Tt3 are the first-order phase transitiontemperatures where the discontinuity occurs firstfor the metastable and unstable branches of theorder parameters, respectively. Finally, Tu is theupper limit of the stability temperature in whichthe discontinuity occurs first for the stablebranches of the order parameters. Therefore, thetransitions at Tt2 and Tu are based on the samemechanism.The behavior of the temperature dependence of

the order parameters depends on D/J and K/J

values and, by matching the free energy values ofthe solutions of the order parameters, the follow-ing six main topological different types of beha-viors are found and illustrated in Figs. 1–6. If onestudies these behaviors and compares the beha-viors found in paper I, one can see immediately

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0 1 2 3 4 5 6

S A,Q

A, S

B,Q

B

0.0

0.5

1.0

Tc

QA1,QB1

SA1,SB1

QB2

QA3,QB3

Tc2

0 1 2 3 4 5 6

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

Tc2TifTfi

QA2

SA1,SB1QB2

QA1,QB1QA3,QB3

kT/J0 1 2 3 4 5

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

Tc'

Tc QA3,QB3

QB1

QA1

(a)

(b)

(c)

Tc

Fig. 5. Temperature dependence of the order parameters for

constant K=J ¼ �3:0 and various values of the parameter D=J.

(a) Exhibiting the second-order ferromagnetic phase transition

for the stable branches of the magnetization, SA1 and SB1, and

the second-order antiquadrupolar phase transition for the

metastable branches of the quadrupolar order parameters,

QA2 and QB2, for D=J ¼ �3:0. Tc and Tc2 are the second-order

phase transition temperatures for SA1, SB1, QA2 and QB2,

respectively. (b) Exhibiting two successive second-order ferri-

magnetic phase transitions and one ferromagnetic phase

transition of stable order parameters for D=J ¼ �2:15. Themetastable quadrupolar order parameters, QA2 and QB2, also

undergo a single second-order antiquadrupolar phase transition

at Tc2. Tfi and Tif are the second-order phase transition

temperatures from the ferromagnetic phase to the ferrimagnetic

phase and from the ferrimagnetic phase to the ferromagnetic

phase, respectively, for the stable branches of order parameters.

(c) Exhibiting two successive second-order antiquadrupolar

phase transitions of stable quadrupolar order parameters, QA1and QB1, for D=J ¼ 0:2.

kT/J

0 1 2 3

S A,Q

A,S

B,Q

B

0.00

0.25

0.50

0.75

SA1,SB1

QA1,QB1QA2

QB2

QA3,QB3

Tc'Tc Tc2

Fig. 6. The temperature dependences of the order parameters

for D=J ¼ 0:0 and K=J ¼ �1:01. Tc and Tc0 are the second-

order ferromagnetic phase transition for the stable branches of

order parameters, and Tc2 is the second-order antiquadrupolar

phase transition temperatures for the metastable quadrupolar

order parameters.


only the first main type, illustrated in Fig. 1, showsdifferent behavior from the Fig. 1 of paper I. Moreexplicitly, the general form of Fig. 1(a) is differentfrom Figs. 1(b) and (c) and also Figs. 1(a) and (b)in paper I, but the value of D/J is lower. Weshould also mention that in paper I the coordina-tion number (g) was scaled into the J and K, so onehas to multiply the horizontal axis of Figs. 1–6 byeight (or the horizontal axis should be kT=gJ) forcomparison. The other five main topologicaldifferent types of behavior are similar to paper I,compare Figs. 2–6 in this paper with Figs. 2–6 inpaper I. The behavior of Fig. 1 is also similar tothe behavior of Fig. 1 of paper I, except in lowtemperatures in which one or two more unstablebranches of the quadrupolar order parameters,namely QA3 and QB3 are also present, compareFig. 1 with Fig. 1 in paper I. These branchesundergo a first-order phase transition for smallvalues of D/J, at low temperatures QA3 and QB3are zero, as the temperature increases QA3 and QB3also increase, then start to decrease discontinu-ously, hence a first-order phase transition occurs.Below Tt3, which is a first-order phase transitiontemperature for QA3 and QB3, one more unstablebranches of the quadrupolar order parameters

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exist, seen in Fig. 1(a) (D=J ¼ 0:3 and K=J ¼ 0:0),explicitly. Moreover, below Tc one more QA3 andQB3 are also present, seen in the figure. For a littlebe bigger values of D=J, these unstable branchesare one at zero temperature and decrease as thetemperature increases then start to decreasediscontinuously, hence a first-order phase transi-tion also occurs, seen in Figs. 1(b) and (c). On theother hand, since we discussed the behaviors of theother five main topological different types, namelyFigs. 2–6, in paper I and since these behaviors aresimilar to the paper I we will not repeat thediscussion here. We should also mention that therewas a numerical error in Fig. 6 of paper I. Thisfigure must have been similar to Fig. 6 of presentpaper. Therefore Fig. 6 i.e., type 6, is alike Fig. 4,that is type 4, but only differ from Fig. 4 in whichthe metastable branches of the quadrupolar orderparameters, QA2 and QB2, undergo a singlesecond-order antiquadrupolar or staggered quad-rupolar phase transition at Tc2=0.288.

4. Dynamics of the system

In this section, we study dynamics of the BEGmodel by the path probability method (PPM) withpair distribution [4], since the metastable behavioris a dynamical behavior. In this study we havechecked all the solutions, which were obtainedwithin the PACVM, and as well as their classifica-tions. The PPM is the natural extension into thetime domain of the cluster variation method(CVM) and provides a systematic derivation ofthe rate equations for successive approximationwhich are well known in the equilibrium statisticalmechanics. It has been successfully applied todescribe the nonequilibrium behavior of a numberof homogeneous and inhomogeneous stationarysystems such as substitutional diffusion in orderedsystems [21] diffusion and ionic conductivity insolid electrolytes [22], the kinetics of the order-disorder transformation in BCC (body-centered-cubic) alloys [23], a binary alloy [24] and a spin-1

2

Ising model [25] spin-1 Ising systems [1,7,26,27],phonon and atomic diffusion systems [28] aternary system [29] and the microscobic mechan-ism of the current-induced domain conversion

phenomena on the Si (0 0 1) vicinal surface [30].We should also mention that efforts have beenmade to show how the PPM can be used toevaluate atomistic parameters combining withexperiments [31]. Moreover the PPM has beenapplied to study the influence of the interfacedisorder on the electronic properties of thesemiconductor heterostructers [32]. In addition, ithas been employed to study the configurationalkinetics of the disorder-L12 transition [33]. Thekinetic evolution processes for a disorder-B2transition has been also obtained by the PPM [34].In the PPM with pair distribution the rate

change of the pair variables is given by

dYAij

dt¼�

Xkl¼Trans

k1½ðY ij;kl � Y kl;ijÞA þ ðY ij;lk � Y lk;ijÞA�

�X

kl¼Rot

k2½ðY ij;kl � Y kl;ijÞA þ ðY ij;lk � Y lk;ijÞA�;

dYBij

dt¼�

Xkl¼Trans

k1½ðY ij;kl � Y kl;ijÞB þ ðY ij;lk � Y lk;ijÞB�

�X

kl¼Rot

k2½ðY ij;kl � Y kl;ijÞB þ ðY ij;lk � Y lk;ijÞB�;

(17)

where Yij,kl is the path probability rate for thesystem to go from state ij to kl. The coefficientsYij,kl are the product of the two factors: (i) Atemperature dependent factor which guaranteesthat equilibrium state is the time independentstate, and (ii) the probability which the system is inthe state ij; e.g. Yij. Detailed balancing requiresthat

ðY ij;klÞA ¼ ðY kl;ijÞA and ðY ij;klÞB

¼ ðY kl;ijÞB ðall ij; klÞ: (18)

Following the method of Kikuchi [4] we caneither make the assumption that the temperaturedependent factor is given by the exponential of b=2times the energy increase in the transition which iscalled recipe I or b times the total energy ofactivation, namely recipe II. The formulas arebased on the second choice because the generalbehavior of the flow diagrams is not drasticallychanged, i.e., the lines follow more or less the samepattern using either recipe I or recipe II (e.g. seeRef. [26]). It should be mentioned that k1 and k2

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most likely are functions of the temperaturefollowing some kind of Arrhenius law. Theexpression for Yij,kl is

ðY ij;klÞA ¼ exp �2bgA

lA

� �eAkl

and

ðY ij;klÞBYAij ¼ exp �

2bgB

lB

� �eB

klYBij ; (19)

where eAkl and eBkl are given by Eq. (16), and k1 andk2 are the rate constants. k1 expresses insertion orremoval of particles from the lattice. K2 isassociated with reorientation. It is assumed thatdouble processes, the simultaneous insertion orremoval or rotations of two particles do not takeplace, i.e., only single jumps are allowed.Inserting Eq. (19) into Eq. (17), the set of

dynamic or the rate equations for the bondvariables are obtained:

ZAdYA

11

dt¼ 2k1½e

A11Y

A12 � eA12Y

A11�

þ 2k2½eA11Y

A13 � eA13Y

A11�;

ZAdYA

22

dt¼ 2k1½e

A12Y

A22 � eA22Y

A12�

� 2k2½eA23Y

A22 � eA22Y

A23�;

ZAdYA

33

dt¼ 2k1½e

A33Y

A23 � eA23Y

A33�

þ 2k2½eA33Y

A13 � eA12Y

A33�;

ZAdYA

12

dt¼ k1½e

A12ðY

A11 þ YA

22 þ YA13Þ

� YA12ðe

A11 þ eA22 þ eA13Þ�

þ k2½eA12Y

A23 � eA23Y

A12�;

ZAdYA

13

dt¼ k1 eA13ðY

A12 þ YA

23Þ�

�YA13 eA12 þ eA23� �

þ k2 eA13 YA11 þ YA

33

� ��YA

13 eA11 þ eA33� �

ZAdYA

23

dt¼ k1 eA23 YA

13 þ YA22 þ YA

33

� ��YA

23 eA13 þ eA22 þ eA33� �

þ k2 eA23YA12 � eA12Y

A23

�

ZBdYB

11

dt¼ 2k1½e

B11Y

B12 � eB12Y

B11�

þ 2k2½eB11Y

B13 � eB13Y

B11�

ZBdYB

22

dt¼ 2k1½e

B12Y

B22 � eB22Y

B12�

� 2k2½eB23Y

B22 � eB22Y

B23�

ZBdY B

33

dt¼ 2k1½e

B33Y

B23 � eB23Y

B33�

þ 2k2½eB33Y

B13 � eB12Y

B33�

ZBdYB

12

dt¼ k1½e

B12ðY

B11 þ YB

22 þ YB13Þ

� YB12ðe

B11 þ eB22 þ eB13Þ�

þ k2½eB12Y

B23 � eB

23YB12�

ZBdYB

13

dt¼ k1½e

B13ðY

B12 þ YB

23Þ

� YB13ðe

B12 þ eB23Þ�

þ k2½eB13ðY

B11 þ YB

33Þ

� YB13ðe

B11 þ eB33Þ�

ZBdYB

23

dt¼ k1½e

B23ðY

B13 þ YB

22 þ YB33Þ

� YB23ðe

B13 þ eB22 þ eB33Þ�

þ k2½eB23Y

B12 � eB12Y

B23� (20)

where eAij , eBij and YAij , YB

ij and ZA and ZB are givenin Eq. (15). Using Eqs. (7) and (20), the dynamicor the rate equations for the order parameters arefound to be

dSA

dt¼

dYB11

dtþdYB

12

dt

� ��

dYB23

dtþdYB

33

dt

� �;

dQAdt

¼dYB

11

dtþdYB

33

dtþ 2

dYB13

dt

� �

�dYB

12

dtþdYB

23

dtþ 2

dYB22

dt

� �;

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dSB

dt¼

dYA11

dtþdYA

12

dt

� ��

dYA23

dtþdYA

33

dt

� �;

dQBdt

¼dYA

11

dtþdYA

33

dtþ 2

dYA13

dt

� �

�dYA

12

dtþdYA

23

dtþ 2

dYA22

dt

� �; (21)

where dY ij=dt’s are given in Eq. (20).Solutions of these dynamic equations are given

by two different methods: the first one is theRunge–Kutta method. We use this method tostudy relaxation curves of order parameters and tosee the flatness property of the metastable state

Time0 10 20 30 40 50

S A, Q

A, S

B, Q

B

0.0

0.5

1.0

SA,s, SB,s

(a)

(b)

QA,s, QB,s

SA,m, QA,m, SB,m, QB,m

Time0 20 40 60 80

S A,Q

A,S

B,Q

B

0.0

0.5

1.0

SA,s

QA,s

SB,s

QB,s

QB,m

QA,m

Fig. 7. Relaxation curves of the order parameters SA, QA, SBand QB for different set of values of the rate constants: k1 ¼

k2 ¼ 1 (solid), k1 ¼ 1 and k2 ¼ 10 (dashed). Subscript i

indicates the initial value, s the stable state, and m metastable

state. (a) For K=J ¼ 0:0, D=J ¼ 0:5, and kT=J ¼ 1:5. SA,i=

SB,i=0.2, QA,i=QB,i=0.9; SA,i=SB,i=0.1, QA,i=QB,i=0.9 (b)

For K=J ¼ �3:0, D=J ¼ �2:15 and kT=J ¼ 3:1. SA,i=

SB,i=0.01, QA,i=QB,i=0.915; SA,i=SB,i=0.0, QA,i=0.59,

QB,i=0.09.

and as well as the overshooting phenomenon.Relaxation curves of order parameters for severalvalues of D=J, K=J, ki and kT=J are plotted inFigs. 7(a) and (b). The second one is to express thesolution of the equations by means of the flowdiagram [35], which shows the solution of theseequations in a two-dimensional phase space of S

and Q, starting with initial values very close toboundaries. As time progresses by given smallsteps, the values of S and Q are computed and thepoint representing them moves in the plane. A setof solution curve is obtained by considering alldifferent initial values. The results are presented,for fixed values of D=J, K=J, ki and kT=J, inFigs. 8(a) and (b). In these figures, the opencircle is the stable equilibrium solution whichcorresponds to the lowest values of the free energy

SA, SB

0.00 0.25 0.50 0.75 1.00

QA

,QB

0.0

0.2

0.4

0.6

0.8

1.0

SA,SB

0.00 0.25 0.50 0.75 1.00

QA

,QB

0.0

0.2

0.4

0.6

0.8

1.0

(a)

(b)

Fig. 8. The flow diagram of the system for two different sets of

rate constants (solid) k1 ¼ k2 ¼ 1 and (dashed) k1 ¼ 1, k2 ¼ 10.

The open circle corresponds to the stable state, the filled square

to the metastable state, and the filled circle is the unstable state.

(a) For K=J ¼ 0:0, D=J ¼ 0:5, and kT=J ¼ 1:5. (b) ForK=J ¼ �0:15, D=J ¼ 0:413, and kT=J ¼ 1:7.

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or the deepest minimum, the filled square is themetastable state because the system relaxes into itand it does not correspond to the deepestminimum but corresponds to the secondaryminimum, and the filled circle is the unstablesolution or state which correspond to the peak orsaddle point. The stable and metastable states arefound by the dynamic study are exactly the samewith the equilibrium study, but to obtain theunstable states by the dynamic study is a tediousprocedure nevertheless one can still find close theequilibrium study. If one investigates Fig. 7(a) and8(a) (K=J ¼ 0:0, D=J ¼ 0:5 and kT=J ¼ 1:5) onecan see that the system relaxes into only twodifferent states. One is the stable state(SA1=SB1=0.0; QA1=QB1=0.1542) which corre-sponds to the lowest value of free energy or thedeepest minimum and the other is the metastablestate (SA2=SB2=0.8302; QA2=QB2=0.8309)which does not correspond to the deepest mini-mum but corresponds to the second lowest valueof free energy, i.e., the secondary minimum.Moreover, the unstable solution, marked with afilled circle, can be seen explicitly in Fig. 8(a),because it is seen as a saddle point. If onecompares Fig. 7(a) and 8(a) withFig. 1(c) for kT=J ¼ 1:5, one can see that thestable and metastable solutions coincide exactlywith each other. Moreover, if one compares onlyFig. 8(a) with Fig. 1(c) for kT=J ¼ 1:5, he can seethat the unstable point, as a separator between thestable and metastable points, coincide with eachother fairly well. On the other hand, the systemrelaxes only one state, i.e., the stable state inFig. 8(b) (K=J ¼ �0:15, D=J ¼ 0:413 andkT=J ¼ 1:7), because there do not exist metastablesolutions. If one compares this figure with Fig. 2for kT=J ¼ 1:7, one can see that the stablesolutions coincide with each other exactly andthe unstable solution coincides with each otherfairly well. If there is any metastable state in thesystem at this temperature, i.e., kT=J ¼ 1:7, thesystem should relax into it, because all the possibleinitial values are taken. Whereas, we have seen thesystem always relaxes into one state, i.e., the stablestate, hence there do not exist any metastable statein this case. Furthermore, Fig. 7(b) (K=J ¼ �3:0,D=J ¼ �2:15 and kT=J ¼ 3:1) shows the relaxa-

tion of the order parameters in which for theseK=J and D=J values the stable branches of theorder parameters experience three successive sec-ond-order phase transitions and the metastablebranches of quadrupolar order parameters under-go a single second-order antiquadrupolar phasetransition, see and compare Fig. 5(b). In this case,since the metastable solutions exist besides thestable solutions, the system may also relax into themetastable states. If the initial values close to themetastable solutions, the system relaxes into it,otherwise into the stable states, seen in Fig. 7(b)explicitly. Moreover, if k2>k1 and initial valuesare not very close to the stable state, the systemrelaxes into the metastable more than k1=k2, seenin Fig. 8(a). This fact has been also observedexperimentally in the rapid solidification technol-ogy in which if one cools some liquid alloys veryrapidly, one can obtain amorphous metallic alloysor metallic glasses [36]. In this stage, it isworthwhile to mention that if the systems stay intheir metastable state or phase, their propertieschange drastically. For example, the rapid coolingof alloys or metals leads to amorphous structures,namely metastable phases, and it is known that theproperties of alloys or metals improve signifi-cantly, such as strength, stiffness, fatigue behavior,corrosion, toughness, density modules [36]. On theother hand, the metastability is becoming a seriousproblem in high-performance very large scaleintegration (VLSI) in complimentary metal-oxide-semiconducter (CMOS) dynamic D-latch,mainly due to the relative high probability of errorwhen a bistable circuit operates at high frequencies[37]. Recently, Bouabci and Carrieiro [38] pre-sented a Monte Carlo cluster algorithm and calledthe mixed cluster algorithm (MCA) which elim-inates metastability in the first-order phase transi-tions of spin-models and they also applied it to theBEG model. Later, Rachadi and Benyousef [39]modified to the MCA to simulate the BEG modelin different regions of the parameter space includ-ing K=J43 where MCA does not work. Finally, ifone compares Figs. 7(b) and 8(b) with Fig. 5(b) atkT/J=3.1, one can see that the stable andmetastable solutions coincide with each otherexactly and the unstable solution coincides witheach other fairly well.

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In conclusion, these facts show us that thesolutions and their classifications obtained in thePACVM are complete and correct. We should alsomention that since the dynamic behaviors seen inFigs. 7 and 8 are similar to the dynamic behaviorsobtained by the PPM in point distribution,compare Figs. 7 and 8 with Figs. 7 and 8 in paperI, we will not discuss Figs. 7 and 8 any more in thispaper.

5. Phase diagrams

In this section, we present the metastable phasediagrams in addition to the equilibrium phasediagrams of the BEG model for K=Jp0 and alsothe first-order phase transition line for the unstablebranches of the quadrupolar order parameters issuperimposed on the phase diagrams since wemake sure that the metastable and unstablebranches of the order parameters obtained com-pletely and correctly in Section 4. The critical orsecond-order phase transition temperatures for thestable and metastable branches of the orderparameters in the case of a second-order phasetransition are calculated numerically, i.e., theinvestigation of the behavior of the order para-meters as a function of the temperature in whichthe order parameters decrease to zero continuouslyas the temperature increases and the temperaturewhere the order parameters becomes zero is thecritical or second-order phase transition tempera-ture [40]. The critical temperature can also befound using the Hessian determinant which is thedetermination of the second derivative of the freeenergy with respect to internal or spin variables,namely Y ij . The change in sign of the determinantcorresponds to the critical temperature. On theother hand, the first-order phase transition tem-peratures for the stable branches of order para-meters are found by matching the values of thetwo branches of the free energy followed whileincreasing and decreasing the temperature. Thetemperature at which the free energy values equaleach other is the first-order phase transitiontemperature (Tt) for the stable order parameters.Furthermore, the first-order phase transitiontemperature (Tt2) for the metastable and (Tt3) for

the unstable branches of the order parameters arethe temperature where the discontinuity occursfirst for SA2, QA2, SB2, QB2 and QA3 and QB3,respectively.We can now obtain the metastable and equili-

brium phase diagrams and as well as the first-orderphase transition line for the unstable branches ofthe quadrupole order parameter of the BEG modeland the calculated phase diagrams are presented inFigs. 9(a)–(f). In these phase diagrams, thick solidand thin dashed lines represent the second- andfirst-order phase transition for the stable branchesof the order parameters and thin and thickdashed–dotted lines indicate the first- and sec-ond-order phase transition of the metastablebranches of the order parameters and thick dashedlines illustrate the first-order phase transition forthe unstable branches of the quadrupolar orderparameter, respectively. The dotted line separatesthe different subphases.Fig. 9(a) shows the phase diagram in the (kT=J,

D=J) plane for K ¼ 0:0, which is called theBlume–Capel model. As is seen in the figure, threedifferent ferromagnetic (ordered) and paramag-netic (disordered) phases are found by includingthe phase transitions of the metastable andunstable branches of the order parameters, namelySA2, SB2, QA2, QB2, and QA3, QB3. (i) The pureferromagnetic (f) phase with the only stablebranches of the order parameters, SA1, QA1, SB1and QB1. (ii) The dense ferromagnetic 1 (df1)phase, the metastable order parameters occurbesides SA1, QA1, SB1 and QB1. (iii) The denseferromagnetic 2 (df2) phase, the unstable branch oforder parameters occur besides SA1, QA1, SB1, QB1,SA2, QA2, SB2 and QB2. Similarly, in the para-magnetic phase region following three differentsubregions have been found: (i) The pure para-magnetic (p) phase, only the stable branch of thequadrupolar order parameters exist, namely QA1and QB1. (ii) The dense paramagnetic 1 (dp1)phase, the metastable branches of order para-meters occur besides QA1 and QB1. (iii) The denseparamagnetic 2 (dp2) phase, the unstabalebranches of the quadrupole order parametersoccur besides QA1, QB1, QA2 and QB2. Similarlybehavior, except df2 and dp2 regions, has also beenfound in the LACVM, compare this figure with

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-0.4 -0.2 0.0 0.2

0

1

2

3

4

5

p

Z

f

df

D/J-0.8 -0.6 -0.4 -0.2 0.0 0.2

kT/J

0

1

2

3

4

5

df

i

a

Z

E

f

B p

S

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

kT/J

0

1

2

3

4

5

T

p

dp1

f

df2 dp2

0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

p

dp1

f

df2 dp2

D/J

-3 -2 0 0 1

0

1

2

3

4

5

6

p

a

i

df

f

Z

A

S

(a) (b) (c)

(d) (e) (f)

df1df1

T

D/J D/J D/J

K /J

-2.5 -2.0 -1.5 -1.0 -0.5 0.0kT

/J0

1

2

3

4

5

fp

a

df

B

Z

K /J-1.2 -1.0

kT/J

0 .0

0.5

1.0

1.5

f

df

B

Za

p

Fig. 9. The equilibrium phase diagrams (thick solid and thin dashed lines) and metastable phase diagrams (thin and thick

dashed–dotted lines) of the BEGmodel for K=J and D=J. Thin dashed line, thin dashed–dotted line and thick dashed line represent the

first-order phase transition for the stable, metastable and unstable branches of the order parameters, respectively. Thick solid line and

thick dashed–dotted line represent the second-order phase transition for the stable and metastable branches of the order parameters,

respectively. Ferromagnetic (f), dense ferromagnetic 1 and 2 (df1 and df2), ferrimagnetic (i), antiquadrupolar (a), paramagnetic (p) and

dense paramagnetic 1 and 2 (dp1 and dp2) phases are found. The dotted line separates f phase from df1. The special points are tricritical

(T), bicritical (B), multicritical (A), critical end point (E), zero-temperature highly degenerate (S), and zero-temperature critical (Z).

(a) K=J ¼ 0:0. (b) K=J ¼ �0:15. (c) K=J ¼ �1:0. (d) K=J ¼ �1:5. (e) K=J ¼ �3:0, (f) D=J ¼ 0:0.


Fig. 9(a) in paper I. We should also point out thatin paper I the coordination number (g) was scaledinto the J. If this were not done the axes in Fig. 9would be kT=gJ and D=gJ. The f, df1 and df2 canbe defined as fellows: f is the region where the onlysolution is the ferromagnetic solution (sometimesunstable solutions also occur), i.e., SA, SB=0, QA,QB 6¼0 while in the df1 and df2 regions, theseferromagnetic phases can coexist on sufficientlyshort-time scales with another less symmetricallocally stable phase. Moreover, the first-orderphase line for QA3 and QB3 also separates df2from df1. On the other hand, p, dp1 and dp2 canalso be defined as follows: p is a region where theonly solution is the paramagnetic solution, i.e.

SA=SB=0, while in the dp1 and dp2 regions, theseparamagnetic phases can coexist on sufficientlyshort time scales with another less symmetricallocally stable phase, i.e., metastable phase. Theboundary of the pure (p) and dense paramagnetic1 (dp1) phases (thin dashed–dotted line) is the first-order line for the metastable branches of the orderparameters that starts at D=J ¼ 1:0 and ends at T

in which is the tricritical point where the second-order phase transition line turns to a first-orderone. Similarly, the boundary of the dense para-magnetic 1 (dp1) and dense paramagnetic 2 (dp2)phases (thick dashed line) is the first-order line forthe unstable branches of the quadrupolar orderparameters. Therefore, we conclude that in f and p

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K/J

-3 -2 -1 0 1

kT/J

0

1

2

3

4

p

Z

a

f

Fig. 10. Same as Fig. 9(f) but g ¼ 5:0.

D/J

0.0 0.1 0.2 0.3 0.4 0.5 0.6

kT/J

0

1

2

3

T

dp1

pf

df2

df1

dp2

Fig. 11. Same as Fig. 9(a) but g ¼ 5:0.


phases the system always stays in the stable phase,but in df1, df2 and dp1, dp2 phases the systemeither stays in the stable phase or the metastablephase. We should also mention that since unstablestates of order parameters also exist in df1, df2 anddp1 and dp2 phases, the system tries to go and stayan unstable state via one order parameters, butafter some time the relaxation curves make a sharpturn (a ‘‘u turn’’, so to speak, or an inverse ‘‘uturn’’) and relaxes to either stable or the meta-stable state. Hence, the unstable states play a role aseparator between stable and metastable phase.These facts are seen in Figs. 7 and 8(b).We turn to the very interesting case when the

biquadratic (K) and bilinear exchange interactions(J) have opposite signs. Fig. 9(b) illustrates themetastable and unstable phase diagram in additionto the equilibrium phase diagram forK=J ¼ �0:15. This phase diagram is similar tothe phase diagram given in Fig. 9(a), except adouble re-entrant behavior takes place in thesystem. For the range of 0.401pD/Jp0.415, thesystem exhibits the f–p–f–p successive phasetransitions as the temperature is increased. Asvalue of K=J increases in the negative value,eventually the tricritical point disappears seen inthe Fig. 9(c). Since the phase diagrams presentedin Figs. 9(c)–(f) are qualitatively similar to theFigs. 9(c) and (f) in paper I, we will not discussthese figures again. In this point we should alsomention that there was a numerical error inFig. 9(f) of paper I. This phase diagram, i.e.,Fig. 9(f) of paper I must have been similar toFig. 9(f) of the present paper. Hence, it isworthwhile to discuss to Fig. 9(f) in whichillustrates the phase diagram of BEG model in(kT=J, K=J) plane for D=J ¼ 0:0. In this phasediagram, the four different regions have been seen,namely, (p), (a), (df), and (f) phases, and thebicritical (B) and zero-temperature critical (Z)special points occur. The second-order phasetransition line for stable branches of order para-meters separates the (f) phase from the (p) phaseand the second-order antiquadrupolar phasetransition line separates the (p) phase from the(a) phase above the bicritical point. The boundaryof the (f) and (df) phases (thick dashed–dottedline) is the second-order line for the metastable

branch of the quadrupolar order parameters andthe (df) phase is separated from the (a) phase bythe second-order phase transition line for thestable branches of the order parameters below thebicritical point. While with the decrease of thecoordination number (g), the bicritical point alsodecreases and for g ¼ 5:0, the bicritical point (B)occurs at zero temperature, hence the metastablephase line disappears, seen in Fig. 10. If onecompares Fig. 10 with Fig. 11 of Ref. [5], one cansee a very good agreement between these phasediagrams (as it should be), taking into account ofthe difference of notations, namely we defined theD as d=g, hence the horizontal axis of ourphase diagram has to be multiply by g, i.e., 5.0.

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Moreover, we defined D with the plus sign, but inRef. [5] it was defined with the minus sign.We have also presented the phase diagram of the

model for g ¼ 5 and K=J ¼ 0:0, see in Fig. 11, inorder to make a direct comparison with Fig. 3 ofRef. [5] and a very agreement between theequilibrium phase diagram is found (as it shouldbe), taking into account of the difference ofnotations, explained above.Finally, we should also mention that the type of

Fig. 1(e) of Hoston and Berker [17] (K=J ¼ �0:5)was not presented in paper I and as well as in thispaper, because we have not observed the meta-stable phase diagram and the first-order phaseboundary for the unstable quadrupolar orderparameters in this type. Moreover, the equilibriumphase diagram obtained by CVMPA is alsoqualitatively similar with the equilibrium phasediagram of Hoston and Berker [17].

D/J

0.0 0.1 0.2 0.3 0.4 0.5 0.6

kT/J

0.0

0.5

1.0

1.5

T

pf

dp1

dp2

df1

df2

Fig. 12. Same as Fig. 9(a) but g ¼ 3:0.

6. Summary and conclusion

In this paper, we have investigated the thermalvariations of the order parameters of the spin-1Ising BEG model with the repulsive biquadraticinteraction by using the PACVM [3]. Besides thestable branches of the order parameters, we obtainthe metastable and unstable parts of these curvesand phase transitions of the metastable andunstable branches of the order parameters arealso found. The classification of the stable,metastable and unstable states is made by match-ing the free energy values of these solutions. Wealso studied the dynamics of the model by the pathprobability method with pair distribution [4] inorder to make sure that we have found and definedthe metastable and unstable branches of the orderparameters completely and correctly. Then, wehave presented the multicritical phase diagrams ofthe BEG model including the phase transitions ofmetastable branches of the order parameters.Therefore, we presented the metastable phasediagram in addition to the equilibrium phasediagram for the BEG model with the repulsivebiquadratic interaction. The calculated first-orderphase line of the unstable branches of the quadru-polar order parameters is also superimposed

on the equilibrium phase diagram and themetastable phase diagram for K=J ¼ 0 andK=J ¼ �0:15, in Figs. 9(a) and (b) respectively.We found that the equilibrium phase diagrams ofthe system are qualitatively similar to with theequilibrium phase diagrams of Hoston and Berker[17]. However, in general, the main difference isabout the paramagnetic and ferromagnetic phasesas follows: We found two different paramagneticand ferromagnetic phases which we called a pureparamagnetic (p) phase with SA1=SB1=0,QA1=QB1 and pure ferromagnetic (f) phase withthe stable branches of the order parameters, i.e.,SA1=SB1 6¼0, QA1=QB1 and dense paramagneticphase (dp) with the metastable and unstablebranches of order parameters existing besidesQA1=QB1 and dense ferromagnetic phase (df)with the stable, metastable and unstable branchesof the order parameters. Moreover, for K=J ¼ 0and K=J ¼ �0:15 two different dense ferromag-netic and paramagnetic phases df1 and df2; dp1and dp2 respectively exist, and a first-order phasetransition line for the unstable branches of thequadrupolar order parameters separates thesedense ferromagnetic and paramagnetic phases,seen in Figs. 9(a) and (b). This first-order phasetransition line is not present in the LACVM,compare these figures with Figs. 9(a) and (b) inpaper I. We should also mention that in the phasediagrams, the first-order phase lines (thin dashed-dotted lines) for the metastable branches of the

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order parameters separate (p) phase from (dp1)phase and the first-order phase transition lines(thick dashed lines) for the unstable branches ofthe quadrupole order parameter separate df1 fromdf2; dp1 from dp2, seen in Figs. 9(a) and (b). Weshould also point out that the new phase transi-tions i.e., a first-order phase transition boundariesfor the unstable quadrupolar order parameterobserved in Fig. 9(a) and (b) are not thecoordination number (g) dependent, because thesenew phase transitions are also observed all thevalues of g, e.g. see Fig. 12 for g ¼ 3:0 andK=J ¼ 0:0. Again, they are not observed withinthe LACVM even the higher values of g, i.e.,g ¼ 12, due to a simplicity of the LACVM. On theother hand, the second-order phase lines (thickdashed-dotted lines) for the metastable branchesof the order parameters separate (f) phase from(df) phase, seen in Figs. 9(c)–(e). We should alsomention that we also calculated the phase dia-grams (kT=J, K=J) plane for g ¼ 5:0 and D=J ¼

0:0 (Fig. 10) and in (kT=J, D=J) plane for g ¼ 5and K=J ¼ 0:0 (Fig. 11) for the comparison ofFig. 11 and Fig. 3 of Ref. [5], respectively and thegood agreement between the equilibrium phasediagrams is found, (as it should be). In thiscomparison one should multiply the horizontalaxis of Figs. 10 and 11 by the coordinationnumber, namely g ¼ 5:0, because we have definedthe D as d=g in Eq. (11). We should also mentionthat we defined D with a plus sign, but in Ref. [5] itwas defined with a minus sign. Moreover, inFig. 11, we also calculated the metastable phasediagram and the first-order phase boundary for theunstable quadrupole order parameters in whichthey were not calculated in Ref. [5]. It should bealso mention that we have found that themetastable phase diagrams of the BEG model forK=Jp0, which has served as a paradigm for alarge number of physically important phenomena,always exists at the low temperatures that areconsistent with experimental and theoretical workson some alloys [41–43], semiconductors [44,45]polymers [46], water [47] and the ternary system[48]. On the other hand, unstable phase transitionsoccur at low temperatures, which is also consistentwith the work on ðAIIBVÞI�XC

IV2X semiconductors

alloys [49].

In conclusion, the metastable and unstablebranches of the order parameters that exist in theBEG model are not an artifact of the LACVM, anidentical to the MFA in which we found in paper Iand they are exist in the model because thesebranches are also obtained by the PACVM.Therefore, we conclude that one can use thismodel to obtain the metastable phase diagrams ofthe some real systems such as alloys, semiconduc-tors, polymers, water, etc. We should also mentionthat the BEG model has been already used tocalculate the metastable phase diagram of theCu–Al–Mn shape-memory alloys [43], semicon-ductor [45] and the ternary system [48] so far.Finally, we should also mention that the meta-stable phase diagrams of the BEG model Hamil-tonian with the arbitrary bilinear, biquadratic andcrystal field interaction in addition to the equili-brium phase diagram are presented within to theLACVM by Keskin and Ekiz [50]. They alsoobtained the phase transition for the unstablebranches of the order parameters and the calcu-lated first- and second-order phase boundaries ofthe unstable branches of the order parameters andsuperimposed on the equilibrium phase andmetastable phase diagrams.

Acknowledgements

This work was supported by the Research Fundof Erciyes University, Grant Numbers: 00-12-2.

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multicritical phase diagrams of the blume–emery–griffiths model with repulsive biquadratic...

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