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2 Blume–Capel model on the Bethe lattice using the recursion method
Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
The spin-3/2 Blume}Capel model on the Bethe lattice using the recursion method Erhan Albayrak, Mustafa Keskin* Department of Physics, Erciyes University, FEN Edebiyat Fakultesi, 38039 Kayseri, Turkey Received 3 December 1999; received in revised form 9 March 2000
Abstract The spin-3/2 Blume}Capel model is solved on the Bethe lattice using the exact recursion equations. The nature of the variation of the Curie temperature with the ratio of the single-ion anisotropy term to the exchange-coupling constant is studied and the phase diagrams are constructed on the Bethe lattice with the co-ordination numbers q"3 and 6. A comparison is made with the results of the other approximation schemes. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.70.Fh; 64.60.Cn; 75.10.Hk Keywords: Spin-3/2 Ising system; Bethe lattice; The exact recursion equations
1. Introduction A spin-3/2 Ising model is also known as the spin-3/2 Blume}Emery}Gri$ths (BEG) model, and with vanishing biquadratic interaction is called the spin-3/2 Blume}Capel (BC) model. The spin-3/2 BEG model was introduced earlier to explain the phase transition [1] in DyVO [2,3], and tricritical properties in ternary mixtures by using the mean"eld approximation (MFA) [4]. The model has also been studied within the MFA and Monte Carlo (MC) simulations [5], the renormalization group (RG) technique [6], and the e!ective "eld theory (EFT) [7,8]. Recently, the critical properties of the
* Corresponding author. Tel.: #90-352-437-4938; fax: #90352-437-4931. E-mail address: [email protected] (M. Keskin).
model on a square lattice were investigated by using the MC simulations and a density matrix RG method [9], and by a corner transfer matrix RG method [10]. On the other hand, the spin-3/2 BC model on the honeycomb and square lattices within the statistical accuracy of the Bethe}Peierls approximation (or the correlated e!ective "eld treatment) [11], and only on the honeycomb lattice within the framework of an e!ective "eld theory [12], are investigated in detail. The model on the simple cubic lattice is also studied by using the two-spin cluster approximation in the cluster expansion method [13]. Recently, the phase diagram of the spin-3/2 BC model in two dimensions is explored by the conventional "nite-size scaling, the conformal invariance, and the MC simulation [14]. The antiferromagnetic spin-3/2 BC model is investigated within the MFA [15], and the transfer matrix "nite-size scaling calculations and the MC
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 6 6 - 8
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E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
simulation [16]. Finally, we should also mention that a spin-S model on a Bethe lattice, which is based on the Katsura}Takizawa method [17], is studied by Tamashiro and Salinas [18] who considered the spin-3/2 BC model and presented bifurcation sets for the paramagnetic unstable "xed points in the (k¹ /J, D/J) plane for some values of coordination numbers. They obtained the secondorder phase transition line which they called the paramagnetic stability boundary, but they did not calculate the "rst-order phase transition line. The purpose of this work is to solve the spin-3/2 BC model on the Bethe lattice using the exact recursion equations [19], and to study the nature of variation of the Curie temperature with the ratio of the single-ion anisotropy to exchange coupling constants, as well as to construct the phase diagrams, which include the second- as well as the "rst-order phase transition lines. The plan of our presentation is as follows: in Section 2 the formulation of the problem is given. The Curie temperatures or the second-order phase transition temperatures are found in Section 3. Finally, the phase diagrams and conclusions are presented in the last section.
2. Formulation of the problem The spin-3/2 BC model is described by the Hamiltonian on the Bethe lattice G: H"!J p p !D p!H p , (1) G H G G 6G H7 G G where J is the exchange coupling, D is the single-ion anisotropy constant, H is the external magnetic "eld, and p can take on the values $3/2 and G $1/2. The "rst sum runs over nearest-neighbor pairs of G, and the second and third over all sites of G. The calculation on the Bethe lattice is done recursively [19]. The partition function of the system is given by Z" exp(!bH)" P(p) N
" exp b J p p #H p #D p G H G G N 6G H7 G G
(2)
P(p) can be thought of as an unnormalized probability distribution. If the Bethe lattice is `cuta in some central point with a spin p , then it splits up into q identical branches, i.e. disconnected pieces. Each of these is a rooted tree at the central spin p . This implies that P(p) can be written as O P(p)"exp[b(Hp #Dp )] Q (p "sH), (3) L H where sH indicates all the spins on the jth sub-tree other than the central spin p , the su$x n denotes the fact that the sub-tree has n-shells, i.e. n steps from the root to the boundary sites, and Q (p "sH)"exp L
Jp s #J s s #H p G H G 6G H7 G
#D p , (4) G G where s is the spin of the site i of the sub-tree (other G than the central spin p ). Site 1 is the site next to the central point 0. The "rst summation in Eq. (4) is over all edges of the sub-tree other than the edge (0, 1) and the summation over i is over all sites other than the central site. In addition, if the sub-tree, say the upper sub-tree, is cut at the site 1 next to 0, then it also decomposes into q pieces: one being the `trunka (0, 1) and the rest being the identical branches. Each of these branches is a sub-tree like the original, but with n!1 shells and q!1 neighbors. Thus Q (p "s)"exp[b+Jp s #Hs #Ds ,] L O\ ; Q (s "tH), (5) L\ H where tH denotes all the spins (other than s ) on the jth branch of the sub-tree. From these factorization relations in Eqs. (3) and (5), one can easily calculate the magnetization (dipolar) and quadrupolar order parameters. We de"ne g (p )" Q (p "s) (6) L L N and using this de"nition in Eq. (2) the partition function takes the form Z" exp[b(Hp #Dp )][g (p )]O. L N
(7)
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
123
If p is the spin at the central site 0, then the magnetization or the dipole moment and quadrupolar order parameters are given by the de"nition
In order to calculate M and Q explicitly from Eqs. (12) and (13), "rst we need to sum Eq. (6) over all the spins s, over s and tH to "nd
M"Z\ p P(p) and q"Z\ p P(p), N N
g (p )" exp[b+Jp s #Hs #Ds ,][g (s )]O\. L L\ Q (14)
(8)
respectively. Using Eqs. (3)}(8), one can easily calculate the magnetization M M"Z\ p exp[b(Hp #Dp )][g (p )]O, L N
(9)
and similarly the quadrupolar order parameter Q Q"Z\ p exp[b(Hp #Dp )][g (p )]O. L N
(10)
Since all the spins can have the values p"$3/2, $1/2, the partition function can be calculated using Eqs. (3), (6) and (7) as Z"exp[b(3/2H#9/4D)][g (3/2)]O L
Since p and s take four possible values, i.e. $3/2 and $1/2, one can obtain four di!erent g (p ) for L four possible values of p . If the central spin has p "$3/2, then g ($3/2)" exp[b+$3/2Js #Hs #Ds ,] L Q ;[g (s )]O\ (15) L\ "exp[b+$9/4J#3/2H#9/4D,] ;[g (3/2)]O\ L\ #exp[b+G9/4J!3/2H#9/4D,]
#exp[b(!3/2H#9/4D)][g (!3/2)]O L
;[g (!3/2)]O\#exp[b+$3/4J L\
#exp[b(1/2H#1/4D)][g (1/2)]O L
#1/2H#1/4D,][g (1/2)]O\ L\
#exp[b(!1/2H#1/4D)][g (!1/2)]O. (11) L
#exp[b+G3/4J!1/2H#1/4D,]
M and Q can be written by using Eqs. (3) and (9)}(11) as M"Z\[3/2 exp[b(3/2H#9/4D)][g (3/2)]O L !3/2exp[b(!3/2H#9/4D)][g (!3/2)]O L #1/2exp[b(1/2H#1/4D)][g (1/2)]O L !1/2 exp[b(!1/2H#1/4D)][g (!1/2)]O], L (12) Q"Z\[9/4 exp[b(3/2H#9/4D)][g (3/2)]O L
;[g (3/2)]O\, L\ and if the central spin has p "$1/2, then g ($1/2)" exp[b+$1/2Js #Hs #Ds ,] L Q ;[g (s )]O\ (16) L\ "exp[b+$3/4J#1/2H#9/4D,] ;[g (3/2)]O\ L\ #exp[b+G3/4J!3/2H#9/4D,]
#9/4 exp[b(!3/2H#9/4D)][g (!3/2)]O L
;[g (!3/2)]O\#exp[b+$1/4J L\
#1/4 exp[b(1/2H#1/4D)][g (1/2)]O L
#1/2H#1/4D,][g (1/2)]O\ L\
#1/4 exp[b(!1/2H#1/4D)][g (!1/2)]O]. L
#exp[b+G1/4J!1/2H#1/4D,]
(13)
;[g (!1/2)]O\, L\
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Now one can introduce new notations: g (3/2) , X " L L g (!1/2) L
[exp[b +!3/4#3/2h#9/4d,]XO\ L\
g (!3/2) g (1/2) >"L , Z" L L g (!1/2) L g (!1/2) L L (17)
and summing up over all possible values for the central spin p (i.e. $3/2, $1/2), we obtain a set of three recursion equations from which the dipole (magnetization) and quadrupolar order parameters can be found. Substituting Eqs. (15) and (16) into Eq. (17), the recursion equations are calculated as X "[exp[b +9/4#3/2h#9/4d,]XO\ L L\
#exp[b +3/4!3/2h#9/4d,]>O\ L\ #exp[b +!1/4#1/2h#1/4d,]ZO\ L\ #exp[b +1/4!1/2h#1/4d,]],
where b "bJ, d"D/J and h"H/J. It should be mentioned that the values of X, Y, and Z have no direct physical sense, but one can express all thermodynamic functions of interest in terms of X, Y, and Z. Thus the order parameters can be written in terms of these recursion relations as M"[3/2 exp[b (3/2h#9/4d)]XO L
#exp[b +!9/4!3/2h#9/4d,]>O\ L\ #exp[b +3/4#1/2h#1/4d,]ZO\ L\
!3/2 exp[b (!3/2h#9/4d)]>O L
#exp[b +!3/4!1/2h#1/4d,]]/
#1/2 exp[b (1/2h#1/4d)]ZO L !1/2 exp[b (!1/2h#1/4d)]]/
[exp[b +!3/4#3/2h#9/4d,]XO\ L\ #exp[b +3/4!3/2h#9/4d,]>O\ L\
[exp[b (3/2h#9/4d)]XO L
#exp[b +!1/4#1/2h#1/4d,]ZO\ L\
#exp[b (!3/2h#9/4d)]>O L
#exp[b +1/4#1/2h#1/4d,]],
(18)
#exp[b (1/2h#1/4d)]ZO L #exp[b (!1/2h#1/4d)]],
> "[exp[b +!9/4#3/2h#9/4d,]XO\ L L\
#exp[b +!3/4#1/2h#1/4d,]ZO\ L\
#9/4 exp[b (!3/2h#9/4d)]>O L
#exp[b +3/4!1/2h#1/4d,]]/
#1/4 exp[b (1/2h#1/4d)]ZO L #1/4 exp[b (!1/2h#1/4d)]]/
[exp[b +!3/4#3/2h#9/4d,]XO\ L\
[exp[b (3/2h#9/4d)]XO L
#exp[b +3/4!3/2h#9/4d,]>O\ L\
#exp[b (!3/2h#9/4d)]>O L
#exp[b +!1/4#1/2h#1/4d,]ZO\ L\
Z "[exp[b +3/4#3/2h#9/4d,]XO\ L L\ #exp[b +!3/4!3/2h#9/4d,]>O\ L\ #exp[b +1/4#1/2h#1/4d,]ZO\ L\ #exp[b +!1/4!1/2h#1/4d,]]/
(21)
Q"[9/4 exp[b (3/2h#9/4d)]XO L
#exp[b +9/4!3/2h#9/4d,]>O\ L\
#exp[b +1/4!1/2h#1/4d,]],
(20)
(19)
#exp[b (1/2h#1/4d)]ZO L #exp[b (!1/2h#1/4d)]].
(22)
From the recursion relations Eqs. (18)}(20), X , L > and, Z can be obtained, using the values of L L these parameters in Eqs. (21) and (22) and varying the other system parameters one can study the behavior of the order parameters as a function
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
of temperature for various values of the coupling constant D/J. The metastable and unstable branches of order parameters and their phase transitions can also be obtained. Moreover, the in#uence of an external magnetic "eld on the thermal variation of order parameters, especially metastable and unstable branches can also be investigated. These works will be presented in our next paper.
3. Curie temperature In this section, we will obtain the exact expression for the Curie temperatures or the second-order phase transition temperatures of the spin-3/2 BC model on the Bethe lattice. First, setting the external magnetic "eld, H, to zero and searching for the temperature, i.e. Curie temperature, k¹ /J, at which the magnetization goes to zero, ! we obtain the following expression for the Curie temperature. M"0"[3/2 exp(9/4b d)XO !3/2 exp(9/4b d)>O L L #1/2 exp(1/4b d)ZO !1/2 exp(1/4b d)]. L (23) This expression can be rearranged and simpli"ed as exp(1/4b d)[3 exp(2b d)(XO !>O )#(ZO !1)]"0. L L L (24) It is obvious that Eq. (24) is satis"ed when X "> L L and Z "1, i.e. g (3/2)"g (!3/2) and g (1/2)" L L L L g (!1/2). The physical insight of these conditions, L i.e., g (3/2)"g (!3/2) and g (1/2)"g (!1/2) is L L L L following. At the Curie temperature the magnetization must be equal to zero, therefore the probability of spins being up and spins being down has to be equal to each other. This implies g (3/2)"g L L (!3/2) and g (1/2)"g (!1/2). These results can L L also be con"rmed from Eqs. (18)}(20) when the external magnetic "eld is set equal to zero. It should also be mentioned that far from the boundary sites, i.e. deep inside the Bethe lattice, all sites are equivalent. Therefore we can omit the su$x n, hence X"> and Z"1. Now one can "nd the equation
125
for the Curie temperature using Eqs. (18) or (19), exp(2b d) cosh(9/4b )XO\#cosh(3/4b ) . X" exp(2b d) cosh(3/4b )XO\#cosh(1/4b ) (25) The numerical solutions of this equation allow us to obtain the ¹ phase diagram in (k¹ /J, D/J) plane. The numerical calculation and phase diagrams will be given in the next section. Using Eq. (22) and setting H"0, X"> and Z"1, we obtained an equation to study the behavior of the quadrupolar order parameter at the critical temperature as
1 9 exp(2b d)XO#1 . Q" 4 exp(2b d)XO#1
(26)
4. Phase diagrams and conclusions The numerical solution of only Eq. (25) or Eqs. (18)}(22) allows us to obtain the ¹ phase diagrams in (k¹ /J, D/J) plane. The resulting phase diagram for the spin-3/2 BC model are constructed on the Bethe lattice with the coordination numbers q"3, as seen in Fig. 1a and q"6, as illustrated in Fig. 1b. In the phase diagrams, the solid lines represent the second-order phase transition and the dashed lines are the "rst-order phase transition. The secondorder phase transition is obtained by using Eqs. (21) and (25). On the other hand, the "rst-order phase transition is found by solving Eqs. (21) and (25) numerically. It is worthwhile to mention that the "rst-order phase transition can also be found using the `equal areasa Maxwell rule to the van der Waals-like isotherms in the (H, M) plane [8,12]. In Fig. 1, the second-order phase transition line separates the disordered phase, i.e. a paramagnetic phase (para ($)) from the ordered phase, namely a ferromagnetic phase (3/2) (ferro (3/2)) and ferromagnetic (1/2) (ferro (1/2)). One can see that the value of k¹ /J approaches the constant values when the ! value of "D" becomes large. Physically, the constant value for large negative D values (DP!R) comes from the fact that Q at ¹"¹ approaches the value of 1/4 (or the state p "$1/2) and the conG stant value for large positive D (DPR) results from
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of the spin-1 BEG model on the Bethe lattice for only the second-order phase line [21]. In conclusion, we have found that the ¹ phase diagrams calculated here, as illustrated in Fig. 1, are qualitatively similar to those obtained within the other approximation schemes [5}8,11}13]. Our ¹ phase diagram is also sim4ila-r to the phase diagram of the antiferromagnetic spin-3/2 BC model [15,16], compare Fig. 2a of Ref. [15] and Fig. 1 in Ref. [16] with Fig. 1 of this work. Moreover, the second-order phase transition lines in our phase diagrams which are based on the exact recursion equations [19] are the same with the secondorder phase transition lines of Tamashiro and Salinas [18] based on the Katsura}Takizawa method, compare Figs. 5 and 6 of Ref. [18] with our "gures. It is worthwhile to mention that Tamashiro and Salinas [18] did not obtain the "rst-order phase transition lines. However, the "rst-order phase transition lines are qualitatively similar to other approximation methods [5}8,11}13,15,16]. Acknowledgements
Fig. 1. Phase diagram of the spin-3/2 Blume}Capel model on the Bethe lattice in the (k¹ /J, D/J) plane. The solid line is the ! second-order phase transition line which separates the paramagnetic phase from the two ordered phases designated by Ferro(3/2) and Ferro(1/2). The dashed line is the "rst-order phase boundary which separates the Ferro(3/2) from the Ferro(1/2) at low temperature: (a) the coordination number q"3; (b) the coordination number q"6.
taking p "$3/2 state (or Q"9/4) at ¹"¹ . G This fact can also be seen by letting DP!R and R in Eq. (26). On the other hand, the dashed lines in Fig. 1 are the "rst-order phase transition lines that separate the two ferromagnetic phases at low temperature. The tricritical point (TCP) does not appear in the phase diagram. The non-existence of the TCP for semi-integer spins for spin-3/2 Ising model is explained by Plascak et al. [20]. For larger values of the coordination numbers, the critical temperatures occur for the bigger values and the "rst-order phase transition temperatures appear for larger negative values of D, as seen in Fig. 1. This fact has also been seen in the phase diagrams
This work was supported by the Research Fund of Erciyes University, Grant Numbers: 97-051-3 and 98-51-5. One of us (M.K) would like to thank the Royal Society of England and the Scienti"c and Technical Research Council of Turkey (TUG BI TAK) for awarding a Research Fellowship and Prof. Dr. D. Sherrington for hospitality in the Physics Department of Universty of Oxford where this work was initiated. References [1] J. SivardieH re, M. Blume, Phys. Rev. B 5 (1972) 1126. [2] A.H. Cooke, D.M. Martin, M.R. Wells, J. Phys. (Paris), Colloq. 32 (1971) C1}488. [3] A.H. Cooke, D.M. Martin, M.R. Wells, Solid State Commun. 9 (1971) 519. [4] S. Krinsky, D. Mukamel, Phys. Rev. B 11 (1975) 399. [5] F.C. Sa( Barreto, O.F. De Alcantara Bon"m, Physica A 172 (1991) 378. [6] A. Bakchich, A. Bassir, A. Benyouse!, Physica A 195 (1993) 188. [7] T. Kaneyoshi, M. Jas\ c\ ur, Phys. Lett. A 177 (1993) 172. [8] A. Bakkali, M. Kerouad, M. Saber, Physica A 229 (1996) 563.
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127 [9] N. Tsushima, Y. Honda, T. Horiguchi, J. Phys. Soc. Jpn. 66 (1997) 3053. [10] N. Tsushima, T. Horiguchi, J. Phys. Soc. Jpn. 67 (1998) 1574. [11] G. Le Gal, T. Kaneyoshi, A. Khater, Physica A 195 (1993) 174. [12] L. Peliti, M. Saber, Phys. Stat. Sol. (b) 195 (1996) 537. [13] V. Ilkovic\ , Physica A 234 (1996) 545. [14] J.C. Xavier, F.C. Alcaraz, D. Pena( Lara, J.A. Plascak, Phys. Rev. B 57 (1998) 11575. [15] A. Bakchich, S. Bekhechi, A. Benyouse!, Physica A 210 (1994) 415.
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[16] S. Bekhechi, A. Benyouse!, Phys. Rev. B 56 (1997) 13594. [17] S. Katsura, M. Takizawa, Prog. Theoret. Phys. 51 (1974) 82. [18] M.N. Tamashiro, S.R.A. Salinas, Physica A 211 (1994) 124. [19] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982. [20] J.A. Plascak, J.G. Moreira, F.C. Sa` Barreto, Phys. Lett. A 173 (1993) 360. [21] K.G. Chakraborty, J.W. Tucker, Physica A 137 (1986) 122.
The spin-3/2 Blume}Capel model on the Bethe lattice using the recursion method Erhan Albayrak, Mustafa Keskin* Department of Physics, Erciyes University, FEN Edebiyat Fakultesi, 38039 Kayseri, Turkey Received 3 December 1999; received in revised form 9 March 2000
Abstract The spin-3/2 Blume}Capel model is solved on the Bethe lattice using the exact recursion equations. The nature of the variation of the Curie temperature with the ratio of the single-ion anisotropy term to the exchange-coupling constant is studied and the phase diagrams are constructed on the Bethe lattice with the co-ordination numbers q"3 and 6. A comparison is made with the results of the other approximation schemes. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.70.Fh; 64.60.Cn; 75.10.Hk Keywords: Spin-3/2 Ising system; Bethe lattice; The exact recursion equations
1. Introduction A spin-3/2 Ising model is also known as the spin-3/2 Blume}Emery}Gri$ths (BEG) model, and with vanishing biquadratic interaction is called the spin-3/2 Blume}Capel (BC) model. The spin-3/2 BEG model was introduced earlier to explain the phase transition [1] in DyVO [2,3], and tricritical properties in ternary mixtures by using the mean"eld approximation (MFA) [4]. The model has also been studied within the MFA and Monte Carlo (MC) simulations [5], the renormalization group (RG) technique [6], and the e!ective "eld theory (EFT) [7,8]. Recently, the critical properties of the
* Corresponding author. Tel.: #90-352-437-4938; fax: #90352-437-4931. E-mail address: [email protected] (M. Keskin).
model on a square lattice were investigated by using the MC simulations and a density matrix RG method [9], and by a corner transfer matrix RG method [10]. On the other hand, the spin-3/2 BC model on the honeycomb and square lattices within the statistical accuracy of the Bethe}Peierls approximation (or the correlated e!ective "eld treatment) [11], and only on the honeycomb lattice within the framework of an e!ective "eld theory [12], are investigated in detail. The model on the simple cubic lattice is also studied by using the two-spin cluster approximation in the cluster expansion method [13]. Recently, the phase diagram of the spin-3/2 BC model in two dimensions is explored by the conventional "nite-size scaling, the conformal invariance, and the MC simulation [14]. The antiferromagnetic spin-3/2 BC model is investigated within the MFA [15], and the transfer matrix "nite-size scaling calculations and the MC
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 6 6 - 8
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simulation [16]. Finally, we should also mention that a spin-S model on a Bethe lattice, which is based on the Katsura}Takizawa method [17], is studied by Tamashiro and Salinas [18] who considered the spin-3/2 BC model and presented bifurcation sets for the paramagnetic unstable "xed points in the (k¹ /J, D/J) plane for some values of coordination numbers. They obtained the secondorder phase transition line which they called the paramagnetic stability boundary, but they did not calculate the "rst-order phase transition line. The purpose of this work is to solve the spin-3/2 BC model on the Bethe lattice using the exact recursion equations [19], and to study the nature of variation of the Curie temperature with the ratio of the single-ion anisotropy to exchange coupling constants, as well as to construct the phase diagrams, which include the second- as well as the "rst-order phase transition lines. The plan of our presentation is as follows: in Section 2 the formulation of the problem is given. The Curie temperatures or the second-order phase transition temperatures are found in Section 3. Finally, the phase diagrams and conclusions are presented in the last section.
2. Formulation of the problem The spin-3/2 BC model is described by the Hamiltonian on the Bethe lattice G: H"!J p p !D p!H p , (1) G H G G 6G H7 G G where J is the exchange coupling, D is the single-ion anisotropy constant, H is the external magnetic "eld, and p can take on the values $3/2 and G $1/2. The "rst sum runs over nearest-neighbor pairs of G, and the second and third over all sites of G. The calculation on the Bethe lattice is done recursively [19]. The partition function of the system is given by Z" exp(!bH)" P(p) N
" exp b J p p #H p #D p G H G G N 6G H7 G G
(2)
P(p) can be thought of as an unnormalized probability distribution. If the Bethe lattice is `cuta in some central point with a spin p , then it splits up into q identical branches, i.e. disconnected pieces. Each of these is a rooted tree at the central spin p . This implies that P(p) can be written as O P(p)"exp[b(Hp #Dp )] Q (p "sH), (3) L H where sH indicates all the spins on the jth sub-tree other than the central spin p , the su$x n denotes the fact that the sub-tree has n-shells, i.e. n steps from the root to the boundary sites, and Q (p "sH)"exp L
Jp s #J s s #H p G H G 6G H7 G
#D p , (4) G G where s is the spin of the site i of the sub-tree (other G than the central spin p ). Site 1 is the site next to the central point 0. The "rst summation in Eq. (4) is over all edges of the sub-tree other than the edge (0, 1) and the summation over i is over all sites other than the central site. In addition, if the sub-tree, say the upper sub-tree, is cut at the site 1 next to 0, then it also decomposes into q pieces: one being the `trunka (0, 1) and the rest being the identical branches. Each of these branches is a sub-tree like the original, but with n!1 shells and q!1 neighbors. Thus Q (p "s)"exp[b+Jp s #Hs #Ds ,] L O\ ; Q (s "tH), (5) L\ H where tH denotes all the spins (other than s ) on the jth branch of the sub-tree. From these factorization relations in Eqs. (3) and (5), one can easily calculate the magnetization (dipolar) and quadrupolar order parameters. We de"ne g (p )" Q (p "s) (6) L L N and using this de"nition in Eq. (2) the partition function takes the form Z" exp[b(Hp #Dp )][g (p )]O. L N
(7)
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
123
If p is the spin at the central site 0, then the magnetization or the dipole moment and quadrupolar order parameters are given by the de"nition
In order to calculate M and Q explicitly from Eqs. (12) and (13), "rst we need to sum Eq. (6) over all the spins s, over s and tH to "nd
M"Z\ p P(p) and q"Z\ p P(p), N N
g (p )" exp[b+Jp s #Hs #Ds ,][g (s )]O\. L L\ Q (14)
(8)
respectively. Using Eqs. (3)}(8), one can easily calculate the magnetization M M"Z\ p exp[b(Hp #Dp )][g (p )]O, L N
(9)
and similarly the quadrupolar order parameter Q Q"Z\ p exp[b(Hp #Dp )][g (p )]O. L N
(10)
Since all the spins can have the values p"$3/2, $1/2, the partition function can be calculated using Eqs. (3), (6) and (7) as Z"exp[b(3/2H#9/4D)][g (3/2)]O L
Since p and s take four possible values, i.e. $3/2 and $1/2, one can obtain four di!erent g (p ) for L four possible values of p . If the central spin has p "$3/2, then g ($3/2)" exp[b+$3/2Js #Hs #Ds ,] L Q ;[g (s )]O\ (15) L\ "exp[b+$9/4J#3/2H#9/4D,] ;[g (3/2)]O\ L\ #exp[b+G9/4J!3/2H#9/4D,]
#exp[b(!3/2H#9/4D)][g (!3/2)]O L
;[g (!3/2)]O\#exp[b+$3/4J L\
#exp[b(1/2H#1/4D)][g (1/2)]O L
#1/2H#1/4D,][g (1/2)]O\ L\
#exp[b(!1/2H#1/4D)][g (!1/2)]O. (11) L
#exp[b+G3/4J!1/2H#1/4D,]
M and Q can be written by using Eqs. (3) and (9)}(11) as M"Z\[3/2 exp[b(3/2H#9/4D)][g (3/2)]O L !3/2exp[b(!3/2H#9/4D)][g (!3/2)]O L #1/2exp[b(1/2H#1/4D)][g (1/2)]O L !1/2 exp[b(!1/2H#1/4D)][g (!1/2)]O], L (12) Q"Z\[9/4 exp[b(3/2H#9/4D)][g (3/2)]O L
;[g (3/2)]O\, L\ and if the central spin has p "$1/2, then g ($1/2)" exp[b+$1/2Js #Hs #Ds ,] L Q ;[g (s )]O\ (16) L\ "exp[b+$3/4J#1/2H#9/4D,] ;[g (3/2)]O\ L\ #exp[b+G3/4J!3/2H#9/4D,]
#9/4 exp[b(!3/2H#9/4D)][g (!3/2)]O L
;[g (!3/2)]O\#exp[b+$1/4J L\
#1/4 exp[b(1/2H#1/4D)][g (1/2)]O L
#1/2H#1/4D,][g (1/2)]O\ L\
#1/4 exp[b(!1/2H#1/4D)][g (!1/2)]O]. L
#exp[b+G1/4J!1/2H#1/4D,]
(13)
;[g (!1/2)]O\, L\
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E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
Now one can introduce new notations: g (3/2) , X " L L g (!1/2) L
[exp[b +!3/4#3/2h#9/4d,]XO\ L\
g (!3/2) g (1/2) >"L , Z" L L g (!1/2) L g (!1/2) L L (17)
and summing up over all possible values for the central spin p (i.e. $3/2, $1/2), we obtain a set of three recursion equations from which the dipole (magnetization) and quadrupolar order parameters can be found. Substituting Eqs. (15) and (16) into Eq. (17), the recursion equations are calculated as X "[exp[b +9/4#3/2h#9/4d,]XO\ L L\
#exp[b +3/4!3/2h#9/4d,]>O\ L\ #exp[b +!1/4#1/2h#1/4d,]ZO\ L\ #exp[b +1/4!1/2h#1/4d,]],
where b "bJ, d"D/J and h"H/J. It should be mentioned that the values of X, Y, and Z have no direct physical sense, but one can express all thermodynamic functions of interest in terms of X, Y, and Z. Thus the order parameters can be written in terms of these recursion relations as M"[3/2 exp[b (3/2h#9/4d)]XO L
#exp[b +!9/4!3/2h#9/4d,]>O\ L\ #exp[b +3/4#1/2h#1/4d,]ZO\ L\
!3/2 exp[b (!3/2h#9/4d)]>O L
#exp[b +!3/4!1/2h#1/4d,]]/
#1/2 exp[b (1/2h#1/4d)]ZO L !1/2 exp[b (!1/2h#1/4d)]]/
[exp[b +!3/4#3/2h#9/4d,]XO\ L\ #exp[b +3/4!3/2h#9/4d,]>O\ L\
[exp[b (3/2h#9/4d)]XO L
#exp[b +!1/4#1/2h#1/4d,]ZO\ L\
#exp[b (!3/2h#9/4d)]>O L
#exp[b +1/4#1/2h#1/4d,]],
(18)
#exp[b (1/2h#1/4d)]ZO L #exp[b (!1/2h#1/4d)]],
> "[exp[b +!9/4#3/2h#9/4d,]XO\ L L\
#exp[b +!3/4#1/2h#1/4d,]ZO\ L\
#9/4 exp[b (!3/2h#9/4d)]>O L
#exp[b +3/4!1/2h#1/4d,]]/
#1/4 exp[b (1/2h#1/4d)]ZO L #1/4 exp[b (!1/2h#1/4d)]]/
[exp[b +!3/4#3/2h#9/4d,]XO\ L\
[exp[b (3/2h#9/4d)]XO L
#exp[b +3/4!3/2h#9/4d,]>O\ L\
#exp[b (!3/2h#9/4d)]>O L
#exp[b +!1/4#1/2h#1/4d,]ZO\ L\
Z "[exp[b +3/4#3/2h#9/4d,]XO\ L L\ #exp[b +!3/4!3/2h#9/4d,]>O\ L\ #exp[b +1/4#1/2h#1/4d,]ZO\ L\ #exp[b +!1/4!1/2h#1/4d,]]/
(21)
Q"[9/4 exp[b (3/2h#9/4d)]XO L
#exp[b +9/4!3/2h#9/4d,]>O\ L\
#exp[b +1/4!1/2h#1/4d,]],
(20)
(19)
#exp[b (1/2h#1/4d)]ZO L #exp[b (!1/2h#1/4d)]].
(22)
From the recursion relations Eqs. (18)}(20), X , L > and, Z can be obtained, using the values of L L these parameters in Eqs. (21) and (22) and varying the other system parameters one can study the behavior of the order parameters as a function
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
of temperature for various values of the coupling constant D/J. The metastable and unstable branches of order parameters and their phase transitions can also be obtained. Moreover, the in#uence of an external magnetic "eld on the thermal variation of order parameters, especially metastable and unstable branches can also be investigated. These works will be presented in our next paper.
3. Curie temperature In this section, we will obtain the exact expression for the Curie temperatures or the second-order phase transition temperatures of the spin-3/2 BC model on the Bethe lattice. First, setting the external magnetic "eld, H, to zero and searching for the temperature, i.e. Curie temperature, k¹ /J, at which the magnetization goes to zero, ! we obtain the following expression for the Curie temperature. M"0"[3/2 exp(9/4b d)XO !3/2 exp(9/4b d)>O L L #1/2 exp(1/4b d)ZO !1/2 exp(1/4b d)]. L (23) This expression can be rearranged and simpli"ed as exp(1/4b d)[3 exp(2b d)(XO !>O )#(ZO !1)]"0. L L L (24) It is obvious that Eq. (24) is satis"ed when X "> L L and Z "1, i.e. g (3/2)"g (!3/2) and g (1/2)" L L L L g (!1/2). The physical insight of these conditions, L i.e., g (3/2)"g (!3/2) and g (1/2)"g (!1/2) is L L L L following. At the Curie temperature the magnetization must be equal to zero, therefore the probability of spins being up and spins being down has to be equal to each other. This implies g (3/2)"g L L (!3/2) and g (1/2)"g (!1/2). These results can L L also be con"rmed from Eqs. (18)}(20) when the external magnetic "eld is set equal to zero. It should also be mentioned that far from the boundary sites, i.e. deep inside the Bethe lattice, all sites are equivalent. Therefore we can omit the su$x n, hence X"> and Z"1. Now one can "nd the equation
125
for the Curie temperature using Eqs. (18) or (19), exp(2b d) cosh(9/4b )XO\#cosh(3/4b ) . X" exp(2b d) cosh(3/4b )XO\#cosh(1/4b ) (25) The numerical solutions of this equation allow us to obtain the ¹ phase diagram in (k¹ /J, D/J) plane. The numerical calculation and phase diagrams will be given in the next section. Using Eq. (22) and setting H"0, X"> and Z"1, we obtained an equation to study the behavior of the quadrupolar order parameter at the critical temperature as
1 9 exp(2b d)XO#1 . Q" 4 exp(2b d)XO#1
(26)
4. Phase diagrams and conclusions The numerical solution of only Eq. (25) or Eqs. (18)}(22) allows us to obtain the ¹ phase diagrams in (k¹ /J, D/J) plane. The resulting phase diagram for the spin-3/2 BC model are constructed on the Bethe lattice with the coordination numbers q"3, as seen in Fig. 1a and q"6, as illustrated in Fig. 1b. In the phase diagrams, the solid lines represent the second-order phase transition and the dashed lines are the "rst-order phase transition. The secondorder phase transition is obtained by using Eqs. (21) and (25). On the other hand, the "rst-order phase transition is found by solving Eqs. (21) and (25) numerically. It is worthwhile to mention that the "rst-order phase transition can also be found using the `equal areasa Maxwell rule to the van der Waals-like isotherms in the (H, M) plane [8,12]. In Fig. 1, the second-order phase transition line separates the disordered phase, i.e. a paramagnetic phase (para ($)) from the ordered phase, namely a ferromagnetic phase (3/2) (ferro (3/2)) and ferromagnetic (1/2) (ferro (1/2)). One can see that the value of k¹ /J approaches the constant values when the ! value of "D" becomes large. Physically, the constant value for large negative D values (DP!R) comes from the fact that Q at ¹"¹ approaches the value of 1/4 (or the state p "$1/2) and the conG stant value for large positive D (DPR) results from
126
E. Albayrak, M. Keskin / Journal of Magnetism and Magnetic Materials 218 (2000) 121}127
of the spin-1 BEG model on the Bethe lattice for only the second-order phase line [21]. In conclusion, we have found that the ¹ phase diagrams calculated here, as illustrated in Fig. 1, are qualitatively similar to those obtained within the other approximation schemes [5}8,11}13]. Our ¹ phase diagram is also sim4ila-r to the phase diagram of the antiferromagnetic spin-3/2 BC model [15,16], compare Fig. 2a of Ref. [15] and Fig. 1 in Ref. [16] with Fig. 1 of this work. Moreover, the second-order phase transition lines in our phase diagrams which are based on the exact recursion equations [19] are the same with the secondorder phase transition lines of Tamashiro and Salinas [18] based on the Katsura}Takizawa method, compare Figs. 5 and 6 of Ref. [18] with our "gures. It is worthwhile to mention that Tamashiro and Salinas [18] did not obtain the "rst-order phase transition lines. However, the "rst-order phase transition lines are qualitatively similar to other approximation methods [5}8,11}13,15,16]. Acknowledgements
Fig. 1. Phase diagram of the spin-3/2 Blume}Capel model on the Bethe lattice in the (k¹ /J, D/J) plane. The solid line is the ! second-order phase transition line which separates the paramagnetic phase from the two ordered phases designated by Ferro(3/2) and Ferro(1/2). The dashed line is the "rst-order phase boundary which separates the Ferro(3/2) from the Ferro(1/2) at low temperature: (a) the coordination number q"3; (b) the coordination number q"6.
taking p "$3/2 state (or Q"9/4) at ¹"¹ . G This fact can also be seen by letting DP!R and R in Eq. (26). On the other hand, the dashed lines in Fig. 1 are the "rst-order phase transition lines that separate the two ferromagnetic phases at low temperature. The tricritical point (TCP) does not appear in the phase diagram. The non-existence of the TCP for semi-integer spins for spin-3/2 Ising model is explained by Plascak et al. [20]. For larger values of the coordination numbers, the critical temperatures occur for the bigger values and the "rst-order phase transition temperatures appear for larger negative values of D, as seen in Fig. 1. This fact has also been seen in the phase diagrams
This work was supported by the Research Fund of Erciyes University, Grant Numbers: 97-051-3 and 98-51-5. One of us (M.K) would like to thank the Royal Society of England and the Scienti"c and Technical Research Council of Turkey (TUG BI TAK) for awarding a Research Fellowship and Prof. Dr. D. Sherrington for hospitality in the Physics Department of Universty of Oxford where this work was initiated. References [1] J. SivardieH re, M. Blume, Phys. Rev. B 5 (1972) 1126. [2] A.H. Cooke, D.M. Martin, M.R. Wells, J. Phys. (Paris), Colloq. 32 (1971) C1}488. [3] A.H. Cooke, D.M. Martin, M.R. Wells, Solid State Commun. 9 (1971) 519. [4] S. Krinsky, D. Mukamel, Phys. Rev. B 11 (1975) 399. [5] F.C. Sa( Barreto, O.F. De Alcantara Bon"m, Physica A 172 (1991) 378. [6] A. Bakchich, A. Bassir, A. Benyouse!, Physica A 195 (1993) 188. [7] T. Kaneyoshi, M. Jas\ c\ ur, Phys. Lett. A 177 (1993) 172. [8] A. Bakkali, M. Kerouad, M. Saber, Physica A 229 (1996) 563.
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