Some physical properties of disorder Blume-Emery-Griffiths model

Some physical properties of Some physical properties of disorder Blume-Emery-Griffiths model disorder Blume-Emery-Griffiths model

S.L. Yan, H.P DongS.L. Yan, H.P DongDepartment of PhysicsDepartment of Physics

Suzhou UniversitySuzhou University

CCAST 2007-10-26CCAST 2007-10-26

OutlineOutliness

1. Introduction1. Introduction

2. Theory2. Theory

3. Results & discussions3. Results & discussions

4. Summary4. Summary


(1) Ising model and its development

Ising model (1925)

Transverse Ising model (Gennes, SSC, 1 (1963) 132)

Blume-Capel model (PR 141 (1966) 517; Physica 32 (1966) 966)

Blume-Emery-Griffiths model ( PRA 4 (1971) 1071) ………..


Multi component fluid liquid crystal mixtures,

Magnetic materials (PRB 71 (2005) 024434)

Microemulsion

Semiconducting alloy systems Ternary mixtures (PRL 93 (2004) 025701)

(2) Applications of the BEG model


(3) Methods of research the BEG model Mean field method

Real space renormalization group theory

Effective field theory (EFT)

Cluster variational method in pair approximation

Monte Carlo simulations

………

In this report, we give the phase diagrams and magnetic properties of the disorder BEG model

2. Theory2. Theory

The Hamiltonian for the spin-1 BEG model can be defined as:

i

zi

i

zii

ij

zj

zi

zj

ij

ziij ShSDSSKSSJH )()()(

222

zi

zii

j

zj

zi

j

zjij

zii hSSDSSKSJSH 222

The effective Hamiltonian is

and K is the exchange and biquadratic interaction between the nearest-neighbor pairs. Di is a crystal field parameter. h is a uniform magnetic field.

J ij

,/1 TkB ,0ijJ K>0 or K<0.

(1)

(2)

2. Theory2. Theory

)()1()()( JpJJpJP ijijij

)()1()()( DtDDtDP iii

By intruducing the ratio of biquadratic and exchange interaction, defined to be .JK /

0.1 ppc 0.10 twhere and .

i

i

Hi

Hziiz

i eTr

eSTrSm

i

i

Hi

Hziiz

i eTr

eSTrSq

2

2

(3)

(4)

(5)

(6)

2. Theory2. Theory

For 1/ JK case, it is a single lattice problem.

Within the EFT, by using differential operator technique and the van der Waerden identities, the average magnetization and the quadrupolar moment can be formulated by:

|0,0

2222),()(1)sinh()cosh()()(1)(

yx

z

j

zjxij

zjxij

zj

z

j

zj

Kzj

r

zi

ryxFSJSJSSeSSm y

|0,0

22222),()(1)sinh()cosh()()(1)(

yx

z

j

zjxij

zjxij

zj

z

j

zj

Kzj

r

zi yxGSJSJSSeSSq y

(7)

(8)

2. Theory2. Theory

and xx / ./ yy

DdhDyxfDPyxF iii ),,,()(),(

DdhDyxgDPyxG iii ),,,()(),(

ehxe

hxehDyxfDy

y

ii

)cosh(2

)sinh(2),,,(

ehxe

hxehDyxgDy

y

ii

)cosh(2

)cosh(2),,,(

where

(9)

(10)

(11)

(12)

zj

jijSJx .

2j

zjSKy

2. Theory2. Theory

Since order parameter equations deal with multi-spin correlations, we cannot treat exactly all the spin-spin correlations. Therefore, a cutting approximation procedure shall be adopted:

SSSSSS zk r

zj r

zi r

zk

zj

zi r

)()(22 kji for

Then equations may be rewritten to be

|0,0

),()cosh()sinh(1

yx

xij rKxij r

Kz

yxFJeqJemqm yy

|0,0

),()cosh()sinh(1

yx

xij rKxij r

Kz

yxGJeqJemqq yy

(13)

(14)

2. Theory2. Theory

where z is the coordination number. By solving above-mention equations, the expression can be obtained:

mcmbamm 53

According to Laudau theory, the second-order phase transition equation is determined by:

(15)

(16) 1),(cosh1sinh |)()(0,000

1

yx

xijr

Kz

xijr

K yxFJeqqJeza yy

0),(cosh1sinh

!3

)2)(1(),(cosh1

1coshsinh)1(

|)()(

|)(

)()(

0,000

33

3

0,000

2

1

yxxij

r

Kz

xijr

K

yxxij

r

Kz

xijr

Kxij

r

K

yxFJeqqJ

ezzz

yxFJeqq

JeJqezzb

y

yy

yy

(17)

2. Theory2. Theory

where and are given by:q0 q1

|)(0,0000 ),(cosh1

yxxij

r

Kz

yxGJeqqq y

)1/(1 srq

|)()(0,000

1

),(cosh11cosh

yxxij

r

Kz

xijr

K yxGJeqqJezs yy

|)()(0,000

222 ),(cosh1sinh!2

)1(

yx

xijr

Kz

xijr

K yxGJeqqJezz

r yy

(18)

(19)

2. Theory2. Theory

The first-order phase transition equation is determined

by 1a .0band

1a .0b

The tricritical point at which the second-order phase

transition changes to the first-order one must satisfy

and

2. Theory2. Theory

For case, it is necessary to adopt two sublattices. We can discuss staggered quadrupolar (SQ) phase and bicritical point of the BEG model. The Hamiltonian is

1/ JK

2222)()()( B

jj

ji

Aii

ij

Bj

Ai

Bj

ij

Aiij SDSDSSKSSJH

|][0,021

),,(1

yxBBB

ZAiA DyxFCqCmqSm

|][0,021

),,(1

yxAAA

ZBjB DyxFCqCmqSm

|][0,021

2),,(1

yxBBB

ZAiA DyxGCqCmqSq

|][0,021

2),,(1

yxAAA

ZBjB DyxGCqCmqSq

(1’)

(2’)(3’)(4’)

(5’)

2. Theory2. Theory

rxijK JeC y )sinh(1

rxijK JeC y )cosh(2 where and

0 BA mm BA qq The SQ phase satisfies and conditions.

|][0,02

),,(1

yxBB

Z

A DyxGCqqq

|][0,02

),,(1

yxAA

Z

B DyxGCqqq

This is condition of the two-cycle fixed points. In addition, the paramagnetic phase corresponds 0 BA mm

qqq BA and

conditions.

(6’)

(7’)

2. Theory2. Theory

Thus, one has:

|][0,02

),,(1

yx

ZDyxGqCqq

Obviously, the paramagnetic phase satisfies the single fixed point condition. In order to obtain the SQ and the paramagnetic phase boundary, the single fixed point must bifurcate into two-cycle fixed points. The limit of stability for bifurcation of the fixed point is as follows:

1,,1 2

DyxGqCqq

Z

(8’)

(9’)

2. Theory

The solution of the SQ-P phase boundary is given by equations (8’) and (9’).

The F-P second-order phase boundary is still given by equations (16) and (17).

The intersection of the SQ-P line and the F-P one is a BCP. The coordinate of the BCP is solution of the set of coupled equations (8’)-(9’) and (16)-(17).

For simplification, a simple cubic lattice is selected as the three-dimensional version

3. Results & Discussions3. Results & Discussions

-1 0 1 2 3 4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

6 8 10 12 14 16 18 20 22 24 260.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-D/J

KBT/Jt=0.718p=1.0

1.0

1.5

2.0

3.0

4.0

-D/J

KBT

/J

t=0.718p=1.0

-1.0-0.8

-0.6-0.4

00.3

0.50.7

(1) Critical behaviors of the BEG model for

• Ferromgnetic region;

• Reentrant phenomenon;

• Finite critical temperature;

• Double TCPs problem

1/ JK


-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

-D/J

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.5

0.75

1.0

KBT

/J

t=1.0p=0.6

1.5

0.25

Shrink of Double TCPs


0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

2.5

-D/J

p=1.0

1.0

0.9

0.8

0.761

0.728 0.72

0.717 0.7

0.6

0.5

0.1

0.735

(a)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

-D/JK

BT/

J

p=1.0

1.0

0.8

0.78

0.75

0.740.73

0.72

0.718

0.719

0.7

0.6

0.1(b)

Fig. (a) apparent reentrant phenomenon, Fig.(b) weak reentrant phenomenon

The TCP is depressed monotonically with decreasing crystal field concentration.


1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

p

KBT

/J

t=1.0-D/J=2.0

-0.3 -0.2 -0.1 0 0.2 0.3 0.5

1.0

0.8

0.8

0.5

(a)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

p

KBT

/J

t=1.0-D/J=0

1.0

0.5

0.2

0

-0.2-0.4-0.6-0.8-1.0

(b)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

p

-1.0

-0.5

0

1.0

0.5

t=1.0-D/J=-2.0

(c)• The bond percolation threshold is

for

• The bond percolation threshold has many different values for

• There exist two double bond percolation threshold in Fig. 4(a)

2929.0p

.0

.0


-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

KBT

/J t=1.0

0.80.6

0.4

0.2

p=1.0D/J=1.0

FSQ

P

0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

t=1.0D/J=1.0

KBT

/J

p=1.0

0.8

0.6

0.4

0.3

P

SQ

F

(2) Staggered quadrupolar phase and bicritical point

• SQ phase, SQ-P boundary, F-P boundary bicritical point

• different effect of two dilution factors

1/ JK

PRB 69 (2004) 064423


-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

p

KBT

/J

-D/J

t=1.0p=1.0

-1.3

-1.1

-1.5

F

SQ

-1.7

-0.8

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

p=1.0=-1.5

-D/J

KBT

/J

t=1.00.8

0.6

0.4

0.3

F SQ

P

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

t=1.0=-1.5

-D/J

KBT

/J

p=1.0

0.9

0.80.4

P

F

SQ

Influence of different parameters

on the SQ phase and the BCP

(3) Magnetic properties of the BEG

model for


1/ JK

Influence of positive and negative

ratio on Initial magnetizations


Influence of ratio and bond dilution on Initial magnetizations


Influence of positive and negative ratio on susceptibility


Influence of ratio and bond dilution on susceptibility


The temperature dependence of the magnetization under different parameters

0 2 4 6 8 100

5

10

15

20

25

30

1.0

H/J=0

t=1.0p=0.5D/J=-2.0

kBT/J


(PRB 71 (2005) 024434) Our result

4. Summary4. Summary

Phase diagrams for positive ratio are compared with those for negative one. There exist double TCPs. The first-order phase transition is enlarged with increasing of ratio at a fixed random crystal field concentration, while the first-order phase transition will shrink due to a fixed bond dilution concentration. Bond percolation threshold is always p=0.2929 for and has many different values. The reentrant phenomenon is shown.

The system has shown the SQ phase, the ferromagnetic phase and the BCP. A large negative ratio and two different dilution factors magnify the range of the SQ phase and reduce that of in or plane. These parameters can assist the reentrant behavior of the SQ-P lines and suppress that of the F-P lines. The influence of bond dilution on the BCP is dissimilar to that of anisotropy dilution.

The initial magnetization curves and susceptibility curves exhibit an irregular behavior in the region of low temperature when the crystal field takes a smaller value. The peak of the susceptibility curve has a explicit decline. The magnetization curves transform from ferromagnetism to paramagnetism at high temperatures. The magnetization curves have a remarkable fluctuation process for a negative ratio and show a discontinuity due to larger crystal field and a positive ratio.

0

T DT

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Some physical properties of disorder Blume-Emery-Griffiths model