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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
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OutlineOutliness
1. Introduction1. Introduction
2. Theory2. Theory
3. Results & discussions3. Results & discussions
4. Summary4. Summary
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1. Introduction1. Introduction
(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) ………..
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1. Introduction1. Introduction
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
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1. Introduction1. Introduction
(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
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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)
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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)
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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)
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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
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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)
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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)
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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)
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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
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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’)
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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’)
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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’)
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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
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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
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3. Results & Discussions3. Results & Discussions
-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
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3. Results & Discussions3. Results & Discussions
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.
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3. Results & Discussions3. Results & Discussions
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
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3. Results & Discussions3. Results & Discussions
-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
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3. Results & Discussions3. Results & Discussions
-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
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(3) Magnetic properties of the BEG
model for
3. Results & Discussions3. Results & Discussions
1/ JK
Influence of positive and negative
ratio on Initial magnetizations
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3. Results & Discussions3. Results & Discussions
Influence of ratio and bond dilution on Initial magnetizations
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3. Results & Discussions3. Results & Discussions
Influence of positive and negative ratio on susceptibility
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3. Results & Discussions3. Results & Discussions
Influence of ratio and bond dilution on susceptibility
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3. Results & Discussions3. Results & Discussions
The temperature dependence of the magnetization under different parameters
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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
3. Results & Discussions3. Results & Discussions
(PRB 71 (2005) 024434) Our result
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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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