- Email: [email protected]
Magnetic phase diagram around the multi-phase point of the ANNNI model with arbitrary spin
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1489-1490
~,~#~ Journalof magnetism and magnetic ~
ELSEVIER
m~rlals
Magnetic phase diagram around the multi-phase point of the ANNNI model with arbitrary spin Y. Muraoka a, *, M. Ochiai b, T. Idogaki b a Department of General Education, Ariake National College of Technology, Omuta, Fukuoka 386, Japan b
. . Department of Apphed Sctence, Faculty of Engineering, Kyushu University,Fukuoka 812, Japan
Abstract The three-dimensional ANNNI model with arbitrary spin is discussed by the molecular field approximation for two types of next-nearest layer interactions. In the S > 1 ANNNI model with J3SiS2+ 1Si+2, there appear direct transitions from the phase (2) to (3) or (2k3), and the devil's flower is partially destroyed around the multi-phase point.
The axial next-nearest neighbor Ising (ANNNI) model is one of several very attractive systems with competing interactions which represent the various types of phase transition and ordered states. Although ANNNI models seem to be too simple to describe specific materials quantitatively, they well reproduce qualitative features observed experimentally in ferroelectrics, magnetic systems, alloys etc. which exhibit modulated structures [1]. In previous papers [2,3], we have discussed the threedimensional S = 1 ANNNI model in which two types of competing antiferromagnetic interactions have been considered between next-nearest layers, i.e. ordinary two-site two-spin J2SiSj ([2-2] model) or three-site four-spin J3SiS2Sk ([3-4] model) interactions in addition to the ferromagnetic intra- and inter-layer interactions Jo and J r These models are described by the following Hamiltonian: n.n.
H = - Y] JijSiSj(i,j)
numerically determined for the system consisting of n( < 8) layers along the competing axis. The ground state changes from the ferromagnetic (F) (oo) to the antiphase (A) (2) 1 state at K 2 ~ 3 ) = - 3 with the increase of competition, where r 2 = J 2 / J 1 and K3 = S2J3/J1 . At the MP (K20), T ) = ( - ½ , 0), all ten phases such as (2}, (23), (24), (3), (25), (34), (26), (35), (4) and (oo) degenerate in this n( < 8) layer model. In the limit of n ~ 0% the degeneracy at the MP becomes infinite as well as the S = 1 ANNNI model [4,5]. The Hamiltonian in the molecular field approximation (MFA) is obtained as follows:
HMF = -- E [Syhy+S2h; - ½(Sy)hj- (S2)h7], J
(2)
where
fl.n.n.
E
JijkSiS2S~,
(1)
(i,j,k)
where S i = S , S - 1 , . . . , - ( S - 1 ) , - S , and Ju denotes J0, J1 and J2, while Jijk denotes the three-site four-spin interaction J3 along the >axis. For J3 = 0 or J2 = 0, this Hamiltonian reduces to the [2-2] or [3-4] models, respectively. In the present paper, detailed phase diagrams, especially around the multi-phase point (MP), are obtained by analytical and/or numerical method for both the S > 1 [2-2] and S > 1 [3-4] models. The ground state and the MP for the S = 1 model have been rigorously determined by means of the transfer matrix method [2], while those for the S > 1 models are
* Corresponding author. Fax: +81-944-53-8876; [email protected].
email:
hj = EJij
h7 = E Jijk(Si)(Sk) •
(3) (4)
(i,k) At low temperatures, (Sj) and (S 2) can be written as (Sj)=I-Sj and (S J2 ) = S 2 - ( 2 S L 1 ) 8 J with 8.<<1, J and it is easy to show that 1 N
F = - - ~ E [ ½ ( S + ~ j ) h j + ( 2 S - 1 ) S j h 7 + T8j]. j=l
(5) The Boltzmann constant is taken to be unity. To compare the free energies for the F, A and modulated (M) phases, we denote 8y as 8 F in the F phase, 8a in the A phase and 81 or 82 in the modulated (3) phase since there exist two kinds of spin in this phase. If we estimate the strength of hj and h~ for each phase, after some lengthy calculations
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(94)01354-3
.+ E J i j k ( S 2 > < S k > , i,k
Y. Muraoka et al. /Journal of Magnetism and MagneticMaterials 140-144 (1995) 1489-1490
1490
we find 82 >> 8 F = 8A >> 81 ([2-2] model) and 8 F > ~2 >> 8A >> 81 ([3-4] model). In Eq. (5), therefore, all other 8j except for 82 for the [2-2] and except for both 82 and 8 F for the [3-4] model can be neglected. Near the MP, K2(3) can be written as K2(3)= - 0 . 5 - A (A << 1), and comparing the free energies of each phase, the phase boundaries are determined as follows: [2-2] model:
(4 1)
[-
(k--4,5,6....)
F - M : - A = 2 - - ~ l e x p - ---7- ,
(6)
T
<2> b
(7)
MP
-- K3
Fig. 2. The schematic phase diagram near the MP for the S = 2 [3-4] model.
[3-4] model: (8S + 1 ) J 1 F-M: A = ~
-
exp
f
2T
/ 'x°s2r Xexp
r _ A - M : A - S-~JI exp(
T
F-A: A = 2--~lexp
"
(8)
(10S2.T-1),] 1 ) ,
(9)
((8S+1)J1)(S>1).
(10)
2T
For the S > 1 [2-2] model and the S = 1 [3-4] model, the 1 M phase springs directly from the MP as well as the S = [2-2] model [4,5]. For the S > 1 [3-4] model, however,
<3>
<2 3>
[-. -<2~3> (k=4,5,6, ...) <2>
MP
- K2
Fig. 1. The schematic phase diagram near the MP for the S = 2 [2-2] model.
there appears a direct transition from the A to F phase since there exists an additional F - A phase boundary (Eq. (10)). Therefore, the M phase appeares from the bifurcation point at finite temperature and does not spring directly from the MP. For the S = 2 [2-2] and [3-4] models, detailed phase diagrams near the MP are obtained by means of numerical calculation of a coupled equation obtained from MFA. This calculation takes account of the (2k3) phases in addition to the (2) and (3) phases. Figs. 1 and 2 show the schematic phase diagrams for the [2-2] and [3-4] models, respectively. In the [2-2] model, all (2k3) phases spring directly from the MP (Fig. 1). On the other hand, in the [3-4] model, the (2k3) phases appear from bifurcation points at finite temperature (Fig. 2). From analytical and numerical calculations, we found that there appear direct transitions from the phase (2) to the phases (3), (23) or (223) etc., and the devil's flower is partially destroyed around the MP in the S > 1 [3-4] model. References
[1] W. Selke, Phys. Rep. 170 (1988) 213. [2] Y. Muraoka, M. Ochiai, T. Idogaki and N. Ury~, J. Phys. A 26 (1993) 1811. [3] Y. Muraoka, M. Ochiai and T. Idogaki, J. Phys. A 27 (1994) 2675. [4] M.E. Fisher and W. Selke, Phys. Rev. Lett. 44 (1980) 1502. [5] W. Selke and P.M. Duxbury, Z. Phys. B 57 (1984) 49.
~,~#~ Journalof magnetism and magnetic ~
ELSEVIER
m~rlals
Magnetic phase diagram around the multi-phase point of the ANNNI model with arbitrary spin Y. Muraoka a, *, M. Ochiai b, T. Idogaki b a Department of General Education, Ariake National College of Technology, Omuta, Fukuoka 386, Japan b
. . Department of Apphed Sctence, Faculty of Engineering, Kyushu University,Fukuoka 812, Japan
Abstract The three-dimensional ANNNI model with arbitrary spin is discussed by the molecular field approximation for two types of next-nearest layer interactions. In the S > 1 ANNNI model with J3SiS2+ 1Si+2, there appear direct transitions from the phase (2) to (3) or (2k3), and the devil's flower is partially destroyed around the multi-phase point.
The axial next-nearest neighbor Ising (ANNNI) model is one of several very attractive systems with competing interactions which represent the various types of phase transition and ordered states. Although ANNNI models seem to be too simple to describe specific materials quantitatively, they well reproduce qualitative features observed experimentally in ferroelectrics, magnetic systems, alloys etc. which exhibit modulated structures [1]. In previous papers [2,3], we have discussed the threedimensional S = 1 ANNNI model in which two types of competing antiferromagnetic interactions have been considered between next-nearest layers, i.e. ordinary two-site two-spin J2SiSj ([2-2] model) or three-site four-spin J3SiS2Sk ([3-4] model) interactions in addition to the ferromagnetic intra- and inter-layer interactions Jo and J r These models are described by the following Hamiltonian: n.n.
H = - Y] JijSiSj(i,j)
numerically determined for the system consisting of n( < 8) layers along the competing axis. The ground state changes from the ferromagnetic (F) (oo) to the antiphase (A) (2) 1 state at K 2 ~ 3 ) = - 3 with the increase of competition, where r 2 = J 2 / J 1 and K3 = S2J3/J1 . At the MP (K20), T ) = ( - ½ , 0), all ten phases such as (2}, (23), (24), (3), (25), (34), (26), (35), (4) and (oo) degenerate in this n( < 8) layer model. In the limit of n ~ 0% the degeneracy at the MP becomes infinite as well as the S = 1 ANNNI model [4,5]. The Hamiltonian in the molecular field approximation (MFA) is obtained as follows:
HMF = -- E [Syhy+S2h; - ½(Sy)hj- (S2)h7], J
(2)
where
fl.n.n.
E
JijkSiS2S~,
(1)
(i,j,k)
where S i = S , S - 1 , . . . , - ( S - 1 ) , - S , and Ju denotes J0, J1 and J2, while Jijk denotes the three-site four-spin interaction J3 along the >axis. For J3 = 0 or J2 = 0, this Hamiltonian reduces to the [2-2] or [3-4] models, respectively. In the present paper, detailed phase diagrams, especially around the multi-phase point (MP), are obtained by analytical and/or numerical method for both the S > 1 [2-2] and S > 1 [3-4] models. The ground state and the MP for the S = 1 model have been rigorously determined by means of the transfer matrix method [2], while those for the S > 1 models are
* Corresponding author. Fax: +81-944-53-8876; [email protected].
email:
hj = EJij
h7 = E Jijk(Si)(Sk) •
(3) (4)
(i,k) At low temperatures, (Sj) and (S 2) can be written as (Sj)=I-Sj and (S J2 ) = S 2 - ( 2 S L 1 ) 8 J with 8.<<1, J and it is easy to show that 1 N
F = - - ~ E [ ½ ( S + ~ j ) h j + ( 2 S - 1 ) S j h 7 + T8j]. j=l
(5) The Boltzmann constant is taken to be unity. To compare the free energies for the F, A and modulated (M) phases, we denote 8y as 8 F in the F phase, 8a in the A phase and 81 or 82 in the modulated (3) phase since there exist two kinds of spin in this phase. If we estimate the strength of hj and h~ for each phase, after some lengthy calculations
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(94)01354-3
.+ E J i j k ( S 2 > < S k > , i,k
Y. Muraoka et al. /Journal of Magnetism and MagneticMaterials 140-144 (1995) 1489-1490
1490
we find 82 >> 8 F = 8A >> 81 ([2-2] model) and 8 F > ~2 >> 8A >> 81 ([3-4] model). In Eq. (5), therefore, all other 8j except for 82 for the [2-2] and except for both 82 and 8 F for the [3-4] model can be neglected. Near the MP, K2(3) can be written as K2(3)= - 0 . 5 - A (A << 1), and comparing the free energies of each phase, the phase boundaries are determined as follows: [2-2] model:
(4 1)
[-
(k--4,5,6....)
F - M : - A = 2 - - ~ l e x p - ---7- ,
(6)
T
<2> b
(7)
MP
-- K3
Fig. 2. The schematic phase diagram near the MP for the S = 2 [3-4] model.
[3-4] model: (8S + 1 ) J 1 F-M: A = ~
-
exp
f
2T
/ 'x°s2r Xexp
r _ A - M : A - S-~JI exp(
T
F-A: A = 2--~lexp
"
(8)
(10S2.T-1),] 1 ) ,
(9)
((8S+1)J1)(S>1).
(10)
2T
For the S > 1 [2-2] model and the S = 1 [3-4] model, the 1 M phase springs directly from the MP as well as the S = [2-2] model [4,5]. For the S > 1 [3-4] model, however,
<3>
<2 3>
[-. -<2~3> (k=4,5,6, ...) <2>
MP
- K2
Fig. 1. The schematic phase diagram near the MP for the S = 2 [2-2] model.
there appears a direct transition from the A to F phase since there exists an additional F - A phase boundary (Eq. (10)). Therefore, the M phase appeares from the bifurcation point at finite temperature and does not spring directly from the MP. For the S = 2 [2-2] and [3-4] models, detailed phase diagrams near the MP are obtained by means of numerical calculation of a coupled equation obtained from MFA. This calculation takes account of the (2k3) phases in addition to the (2) and (3) phases. Figs. 1 and 2 show the schematic phase diagrams for the [2-2] and [3-4] models, respectively. In the [2-2] model, all (2k3) phases spring directly from the MP (Fig. 1). On the other hand, in the [3-4] model, the (2k3) phases appear from bifurcation points at finite temperature (Fig. 2). From analytical and numerical calculations, we found that there appear direct transitions from the phase (2) to the phases (3), (23) or (223) etc., and the devil's flower is partially destroyed around the MP in the S > 1 [3-4] model. References
[1] W. Selke, Phys. Rep. 170 (1988) 213. [2] Y. Muraoka, M. Ochiai, T. Idogaki and N. Ury~, J. Phys. A 26 (1993) 1811. [3] Y. Muraoka, M. Ochiai and T. Idogaki, J. Phys. A 27 (1994) 2675. [4] M.E. Fisher and W. Selke, Phys. Rev. Lett. 44 (1980) 1502. [5] W. Selke and P.M. Duxbury, Z. Phys. B 57 (1984) 49.