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A phase diagram for the S=1 BEG model
Journal of Magnetism and Magnetic Materials 104-107 (1992) 282-284 North-Holland
A phase diagram for the S = 1 B E G model Katsumi Kasono and Ikuo Ono Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan We have obtained a full phase diagram for the BEG model with an antiferromagnetic biquadratic interaction by the Bethe approximation. This diagram shows successive, re-entrant and double re-entrant phase transitions. In the special case of a zero bilinear interaction, this model can be mapped onto the spin-½ Ising model in a magnetic field. T h e B E G m o d e l [1] w a s s u g g e s t e d by B l u m e , E m e r y a n d G r i f f i t h s a n d t h e H a m i l t o n i a n for this m o d e l is
Z = - J , ~_, SiS j - J2 £ S?S? - D ~_, S,2, n .n.
rl ,n
i
( S i = O, + 1),
(1)
w h e r e J1 > 0 is t h e f e r r o m a g n e t i c b i l i n e a r i n t e r a c t i o n a n d t h e s u m n.n. is o v e r n e a r e s t n e i g h b o r pairs, a n d D is t h e s i n g l e - i o n a n i s o t r o p y e n e r g y . T h e B E G m o d e l h a s b e e n i n v e s t i g a t e d as a s p i n s y s t e m o f rich p h a s e s . T h e s e a r e a t t r i b u t e d to t h e c o m p e t i t i o n b e t w e e n J~, J2 a n d D (J2/Jl < 0). If J2 < 0, this m o d e l h a s t h e g r o u n d s t a t e w i t h t w o - s u b l a t t i c e o r d e r i n g , w h i c h is c a l l e d in
k T/Z J, 1.0
a
o.s-
(a) Q
-3.0
--2.0
J,
-1.0
0.0
j
1.0
(d)
(e)
2.0
k T/z d~ 1.0
b 0.8-
o.6-
Q
-o.2 ~ -0.1
-4-.0
-3.0
-2.0
-1.0
j
0.0
1 .0
Fig. 1. The phase diagram for the several values of d( =- D / z J 0 is plotted in J( ~- J 2 / J 1 ) and the reduced temperature plane. (a) d = - 1.0, - 0.5, 0.0, 0.5, 1.0; (b) d = - 0.4, - 0.25, - 0.2, - 0.1, - 0.05 and 0.0. The right-hand side lines indicate the the (Q) ~ (F) phase transition, and left hand lines the (Q) ~ (SQ) phase transition. When d > 0, both phases meet each other, and the first order (Q)--* (SQ) transition appears. A tricritical phase transition occurs on the (F) phase. Here the solid lines and the broken lines indicated the second order and first order transitions, respectively. 0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
283
K. Kasono, L Ono / A phase diagram for the S = 1 B E G model
the staggered q u a d r u p o l e ' (SQ) p h a s e [2,3]. In the case of J1 > 0, J2 < 0, we have f o u n d many types of p h a s e transitions such as simple second a n d first o r d e r p h a s e transitions, a n d a successive and r e - e n t r a n t p h a s e transitions. W e introduce the two-sublattice A, B a n d four o r d e r p a r a m e t e r s , m A =
(Si=0,
2,
_ 1),
n.n
~0.6
Y'~ri+const. i
I
* x
I
o . ,
qa L'=18 qb 18 m 18m m
14
10
Q O O O O e O O O 0 ~ ~ 0 0 0 0
~'0.4
-
2
o0.2 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
kT/d I Fig. 2. Temperature dependence of order parameters for the successive transition (Q)--+ (F)---, (SQ) by Monte Carlo simulation on the simple-cubic lattice L x L x L (L = 10, 14, 18).
H e r e it should be n o t e d that the state o-i = 1 corresponds doubly to the states S i = 1 and - 1 . T h u s the free energy F [ D , J2] for (2) can be associated with the free energy F I [ H I , Jn] for the spin - g1 Ising model with the exchange interaction Jl and the field H r A p a r t from the n o n s i n g u l a r part we have [4] In 2 + D + ½zJ2) , ¼J2].
F~,[D, J21 = F , [ ½ ( k T
(4)
In the case of an a n t i f e r r o m a g n e t i c exchange interaction ( J l < 0), M i i l l e r - H a r t m a n n and Z i t t a r t z have shown [5] that a s e c o n d - o r d e r p h a s e transition a p p e a r s in the region of dHI [ < H c . T h e critical value H c was estim a t e d from the interface energy. In fig. 5, we p r e s e n t the p h a s e diagram for the spin-1 Ising model with negative J2 and vanishing Jl using the M H - Z solution a n d t r a n s f o r m a t i o n (4). T h e result clearly shows the
0.e
I
I
I
J=-31/30
I
I
d= 0 o qa L=20 x qb 20
~0.6 ~>
o o ~'
Q3
m m m
20 16 12
~ 0.4 ID O
x
1
I
J=-8/5 d= I / 2
m 0.8
0.0
w h e r e J2 < 0. W e introduce a new variable o-~ by ~ri = 2 S ~ - 1 . Substituting it into (2), we have a new Hamiltonian ,gU,,= - a J 2 Y ' ~ i ~ r ~ - ½ ( D + $ z l 2 )
@ @ @ @ ~iil~
(2)
i
1
1.0
(3)
0.0
J,,, ,, I
0.2
-
0.4
' i
0.6
0.8
""1|1
t .0
kT/dl Fig. 3. Temperature dependence of order parameters for the re-entrant transition (Q) ~ (F) --* (Q) ~ (SQ) by Monte Carlo simulation on the simple-cubic lattice L × L x L (L= 12, 16, 20).
284
K. Kasono, I. Ono / A phase diagram ]'or the S = 1 BEG model 1.0
l
I
o o
mO.B £QJ
° A
~O.G
l
d:-89/360
j=-1/2
q L=18 m 18 m 14 m
0.3
I
[
I
I
0.5
1.0
-~ 0.2 -
12
.5 L
0.1 -
0.4
Q 000000
000000 000000 000000 0.0 t - i .0 -0.5
"0
o0.2 0.0 0.4
O.G
0.B
t .0
1.2
kT/O~
Fig. 4. Temperature dependence of order parameters for the double reentrant transition (Q) ~ (F) ~ (Q)-~ (F) by Monte Carlo simulation (L = 12, 14, 18). The spin system has been heated gradually from a spin configuration of the ferromagnetic state.
a b s e n c e o f a r e - e n t r a n t p h a s e t r a n s i t i o n on t h e square-lattice. In t h e t h r e e d i m e n s i o n a l lattice we also have estimated the d dependence of the transition temperat u r e s just a b o v e t h e a b s o l u t e z e r o t e m p e r a t u r e . T h e s e are also b a s e d on t h e exact r e l a t i o n s h i p b e t w e e n the repulsive lattice-gas m o d e l a n d t h e a n t i f e r r o m a g n e t i c Ising m o d e l w h e n T ~ 0, H l ~ 0 for H I l T 4= 0 [10], A s s h o w n in fig. 5, this strongly suggests t h a t t h e r e - e n t r a n t p h a s e t r a n s i t i o n d o e s o c c u r in t h e t h r e e d i m e n sional lattices but not in t h e two d i m e n s i o n a l lattices. W e t h a n k P r o f e s s o r Y. T a m u r a for use o f t h e c o m p u t e r H I T A C M - 6 8 2 H at t h e I n s t i t u t e o f Statistical M a t h e m a t i c s . W e have partially u s e d t h e H I T A C S8 2 0 / 8 0 at t h e I n s t i t u t e for M o l e c u l a r Science. W e also w o u l d like to t h a n k Dr, K. Y a s u m u r a for h e l p f u l discussions.
be a
O.O
1.5
D/Z[ J21 Fig. 5. The phase diagram for the negative biquadratic interactoin and Jl = 0, and the spin configurations of the ground state. The solid lines, a, b, c, d and e are the tangential lines of each phase diagram at D - T = 0, and on the simple-cubic, body-centered-cubic, square, honeycomb and triangular lattice respectively. These results are obtained by lattice-gas studies [6] (a, b), [7] (c, d), [8] (c) and exact solution [9] (e). This analogy is also used in [10]. The highly accurate line c, which comes from series expansion by Baxter et al., is proved to be slightly different from the line of MH-Z. References
[1] M. Blume, V.J. E m e ~ and R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. [2] Y.L. Wang and C. Wentworth, J. Appl. Phys. 61 (1987) 411. [3] I. Ono, J. de Phys. 49 (1988) C8-1541. [4] R.G. Griffiths, Physica 33 (1967) 689. [5] E. Miiller-Hartmann and J. Zittartz, Z. Phys. B 27 (1977) 261. [6] D.S. Gaunt, J. Chem. Phys. 46 (1967) 3237. [7] D.S. Gaunt and M.E. Fisher, J. Chem. Phys. 43 (1965) 2480. [8] R.J. Baxter, I.G. Enting and S.K. Tsang, J. Stat. Phys. 22 (1980) 465. [9] R.J. Baxter, J. Phys. A 13 (1980) L61. [10] Z. Rficz and T. Vicsek, Phys. Rev. B 27 (1983) 2992.
A phase diagram for the S = 1 B E G model Katsumi Kasono and Ikuo Ono Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan We have obtained a full phase diagram for the BEG model with an antiferromagnetic biquadratic interaction by the Bethe approximation. This diagram shows successive, re-entrant and double re-entrant phase transitions. In the special case of a zero bilinear interaction, this model can be mapped onto the spin-½ Ising model in a magnetic field. T h e B E G m o d e l [1] w a s s u g g e s t e d by B l u m e , E m e r y a n d G r i f f i t h s a n d t h e H a m i l t o n i a n for this m o d e l is
Z = - J , ~_, SiS j - J2 £ S?S? - D ~_, S,2, n .n.
rl ,n
i
( S i = O, + 1),
(1)
w h e r e J1 > 0 is t h e f e r r o m a g n e t i c b i l i n e a r i n t e r a c t i o n a n d t h e s u m n.n. is o v e r n e a r e s t n e i g h b o r pairs, a n d D is t h e s i n g l e - i o n a n i s o t r o p y e n e r g y . T h e B E G m o d e l h a s b e e n i n v e s t i g a t e d as a s p i n s y s t e m o f rich p h a s e s . T h e s e a r e a t t r i b u t e d to t h e c o m p e t i t i o n b e t w e e n J~, J2 a n d D (J2/Jl < 0). If J2 < 0, this m o d e l h a s t h e g r o u n d s t a t e w i t h t w o - s u b l a t t i c e o r d e r i n g , w h i c h is c a l l e d in
k T/Z J, 1.0
a
o.s-
(a) Q
-3.0
--2.0
J,
-1.0
0.0
j
1.0
(d)
(e)
2.0
k T/z d~ 1.0
b 0.8-
o.6-
Q
-o.2 ~ -0.1
-4-.0
-3.0
-2.0
-1.0
j
0.0
1 .0
Fig. 1. The phase diagram for the several values of d( =- D / z J 0 is plotted in J( ~- J 2 / J 1 ) and the reduced temperature plane. (a) d = - 1.0, - 0.5, 0.0, 0.5, 1.0; (b) d = - 0.4, - 0.25, - 0.2, - 0.1, - 0.05 and 0.0. The right-hand side lines indicate the the (Q) ~ (F) phase transition, and left hand lines the (Q) ~ (SQ) phase transition. When d > 0, both phases meet each other, and the first order (Q)--* (SQ) transition appears. A tricritical phase transition occurs on the (F) phase. Here the solid lines and the broken lines indicated the second order and first order transitions, respectively. 0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
283
K. Kasono, L Ono / A phase diagram for the S = 1 B E G model
the staggered q u a d r u p o l e ' (SQ) p h a s e [2,3]. In the case of J1 > 0, J2 < 0, we have f o u n d many types of p h a s e transitions such as simple second a n d first o r d e r p h a s e transitions, a n d a successive and r e - e n t r a n t p h a s e transitions. W e introduce the two-sublattice A, B a n d four o r d e r p a r a m e t e r s , m A =
(Si=0,
2,
_ 1),
n.n
~0.6
Y'~ri+const. i
I
* x
I
o . ,
qa L'=18 qb 18 m 18m m
14
10
Q O O O O e O O O 0 ~ ~ 0 0 0 0
~'0.4
-
2
o0.2 0.0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
kT/d I Fig. 2. Temperature dependence of order parameters for the successive transition (Q)--+ (F)---, (SQ) by Monte Carlo simulation on the simple-cubic lattice L x L x L (L = 10, 14, 18).
H e r e it should be n o t e d that the state o-i = 1 corresponds doubly to the states S i = 1 and - 1 . T h u s the free energy F [ D , J2] for (2) can be associated with the free energy F I [ H I , Jn] for the spin - g1 Ising model with the exchange interaction Jl and the field H r A p a r t from the n o n s i n g u l a r part we have [4] In 2 + D + ½zJ2) , ¼J2].
F~,[D, J21 = F , [ ½ ( k T
(4)
In the case of an a n t i f e r r o m a g n e t i c exchange interaction ( J l < 0), M i i l l e r - H a r t m a n n and Z i t t a r t z have shown [5] that a s e c o n d - o r d e r p h a s e transition a p p e a r s in the region of dHI [ < H c . T h e critical value H c was estim a t e d from the interface energy. In fig. 5, we p r e s e n t the p h a s e diagram for the spin-1 Ising model with negative J2 and vanishing Jl using the M H - Z solution a n d t r a n s f o r m a t i o n (4). T h e result clearly shows the
0.e
I
I
I
J=-31/30
I
I
d= 0 o qa L=20 x qb 20
~0.6 ~>
o o ~'
Q3
m m m
20 16 12
~ 0.4 ID O
x
1
I
J=-8/5 d= I / 2
m 0.8
0.0
w h e r e J2 < 0. W e introduce a new variable o-~ by ~ri = 2 S ~ - 1 . Substituting it into (2), we have a new Hamiltonian ,gU,,= - a J 2 Y ' ~ i ~ r ~ - ½ ( D + $ z l 2 )
@ @ @ @ ~iil~
(2)
i
1
1.0
(3)
0.0
J,,, ,, I
0.2
-
0.4
' i
0.6
0.8
""1|1
t .0
kT/dl Fig. 3. Temperature dependence of order parameters for the re-entrant transition (Q) ~ (F) --* (Q) ~ (SQ) by Monte Carlo simulation on the simple-cubic lattice L × L x L (L= 12, 16, 20).
284
K. Kasono, I. Ono / A phase diagram ]'or the S = 1 BEG model 1.0
l
I
o o
mO.B £QJ
° A
~O.G
l
d:-89/360
j=-1/2
q L=18 m 18 m 14 m
0.3
I
[
I
I
0.5
1.0
-~ 0.2 -
12
.5 L
0.1 -
0.4
Q 000000
000000 000000 000000 0.0 t - i .0 -0.5
"0
o0.2 0.0 0.4
O.G
0.B
t .0
1.2
kT/O~
Fig. 4. Temperature dependence of order parameters for the double reentrant transition (Q) ~ (F) ~ (Q)-~ (F) by Monte Carlo simulation (L = 12, 14, 18). The spin system has been heated gradually from a spin configuration of the ferromagnetic state.
a b s e n c e o f a r e - e n t r a n t p h a s e t r a n s i t i o n on t h e square-lattice. In t h e t h r e e d i m e n s i o n a l lattice we also have estimated the d dependence of the transition temperat u r e s just a b o v e t h e a b s o l u t e z e r o t e m p e r a t u r e . T h e s e are also b a s e d on t h e exact r e l a t i o n s h i p b e t w e e n the repulsive lattice-gas m o d e l a n d t h e a n t i f e r r o m a g n e t i c Ising m o d e l w h e n T ~ 0, H l ~ 0 for H I l T 4= 0 [10], A s s h o w n in fig. 5, this strongly suggests t h a t t h e r e - e n t r a n t p h a s e t r a n s i t i o n d o e s o c c u r in t h e t h r e e d i m e n sional lattices but not in t h e two d i m e n s i o n a l lattices. W e t h a n k P r o f e s s o r Y. T a m u r a for use o f t h e c o m p u t e r H I T A C M - 6 8 2 H at t h e I n s t i t u t e o f Statistical M a t h e m a t i c s . W e have partially u s e d t h e H I T A C S8 2 0 / 8 0 at t h e I n s t i t u t e for M o l e c u l a r Science. W e also w o u l d like to t h a n k Dr, K. Y a s u m u r a for h e l p f u l discussions.
be a
O.O
1.5
D/Z[ J21 Fig. 5. The phase diagram for the negative biquadratic interactoin and Jl = 0, and the spin configurations of the ground state. The solid lines, a, b, c, d and e are the tangential lines of each phase diagram at D - T = 0, and on the simple-cubic, body-centered-cubic, square, honeycomb and triangular lattice respectively. These results are obtained by lattice-gas studies [6] (a, b), [7] (c, d), [8] (c) and exact solution [9] (e). This analogy is also used in [10]. The highly accurate line c, which comes from series expansion by Baxter et al., is proved to be slightly different from the line of MH-Z. References
[1] M. Blume, V.J. E m e ~ and R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. [2] Y.L. Wang and C. Wentworth, J. Appl. Phys. 61 (1987) 411. [3] I. Ono, J. de Phys. 49 (1988) C8-1541. [4] R.G. Griffiths, Physica 33 (1967) 689. [5] E. Miiller-Hartmann and J. Zittartz, Z. Phys. B 27 (1977) 261. [6] D.S. Gaunt, J. Chem. Phys. 46 (1967) 3237. [7] D.S. Gaunt and M.E. Fisher, J. Chem. Phys. 43 (1965) 2480. [8] R.J. Baxter, I.G. Enting and S.K. Tsang, J. Stat. Phys. 22 (1980) 465. [9] R.J. Baxter, J. Phys. A 13 (1980) L61. [10] Z. Rficz and T. Vicsek, Phys. Rev. B 27 (1983) 2992.