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Phase diagram and tricritical behavior of the spin-1 Heisenberg model with Dzyaloshinskii–Moriya interactions
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Physica A 370 (2006) 585–590 www.elsevier.com/locate/physa
Phase diagram and tricritical behavior of the spin-1 Heisenberg model with Dzyaloshinskii–Moriya interactions Guang-Hou Suna, Xiang-Mu Konga,b, a
Department of Physics, Qufu Normal University, Qufu 273165, China The Interdisciplinary Center of Theoretical Studies, Chinese Academy of Sciences, Beijing 100080, China
b
Received 24 October 2005; received in revised form 5 January 2006 Available online 27 April 2006
Abstract Using the two-spin cluster mean-field method, the spin-1 Heisenberg model with Dzyaloshinskii–Moriya (DM) ~ ¼ D^z (D is the DM interaction interactions is studied for the simple cubic lattice. For the case of the DM vector coupling D parameter and z^ is the unit vector of the z-axis direction), the phase diagram of this system and the thermal behavior of the magnetization are obtained, and it is found that the system exhibits the tricritical point. The critical behavior of the system may be interpreted as a result of a competition between the exchange interaction and the DM interaction. r 2006 Elsevier B.V. All rights reserved. Keywords: Spin-1 Heisenberg model; Dzyaloshinskii–Moriya interaction; Phase diagram; Tricritical point
1. Introduction The critical properties of quantum magnetic systems have been subjects of intense research [1]. The spin-1 Heisenberg Hamiltonian deserves special attention since it provides a model for many magnetic materials. In recent years, there have been many interesting works dealing with the spin-1 Heisenberg model. For instance, So´lyom and Timonen transformed the spin-1 Heisenberg chain to the one-dimensional fermion gas and obtained phase diagrams of the fermion system [2], and Bo¨hm et al. calculated the coefficients of the shorttime expansion of the spin-pair correlations in one-dimensional spin model [3]. Besides, the ground-state properties of the spin-1 Heisenberg ferromagnet with an arbitrary crystal-field potential have been studied using the linked-cluster series expansion [4]. On the other hand, the Heisenberg model with anisotropy has been attracting much attention since various types of anisotropy have a profound influence on the properties of the systems. There are many different ways to introduce anisotropies in the Heisenberg model and an important type of anisotropic interaction is the Dzyaloshinskii–Moriya (DM) interaction [5,6], which is the antisymmetric spin coupling. The DM interaction plays an important role in describing certain class of insulators [7,8], in studying spin glasses [9–12] and also in explaining the electron paramagnetic resonance [13–15]. The Heisenberg models with DM interactions have Corresponding author. Department of Physics, Qufu Normal University, Qufu 273165, China.
E-mail address: [email protected] (X.-M. Kong). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.025
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been intensively studied over past years. Cordeiro et al. have obtained the phase diagram and critical exponents of a spin-12 Heisenberg model with DM interactions by using the renormalization group technique [16]. Subsequently, the phase diagram of this system was also obtained within a two-spin cluster version of mean-field technique [17], as well as within the framework of a new correlated effective-field theory [18]. In Refs. [17,18], the tricritical point (TCP) of this system was found, which had improperly been overlooked in all previous calculations. The tricritical behavior is a very important critical phenomenon, so there has been increasing interest in the study of the behavior [19–23]. In Ref. [20], it was observed that the TCP of antiferromagnetic La2 CuO4 including the DM interaction was close to the Ne´el point. Recently, the effects of the DM interaction on the stability of the Ne´el phase and the energy gap for XXZ Heisenberg model have been studied using the linear spin-wave theory [24]. Pires has investigated the spin-12 alternating Heisenberg model with DM interactions in a fermion representation and calculated the ground state energy, the low-lying excitations and static nearest-neighbors correlation functions in the T ¼ 0 limit [25]. In addition, by the density matrix renormalization group, the physical effects of the DM interaction in copper benzoate were found to produce a gap in the spin excitations [26]. In this paper, the effects of the DM interaction on properties of the spin-1 Heisenberg model are investigated systemically within the framework of the two-spin cluster mean-field method, and it is found that the system exhibits the TCP. The outline of the remainder of this paper is as follows. In Section 2, we give the formulation of this problem. Section 3 is the numerical results and discussions, and Section 4 is the conclusion. 2. Formulation Let us describe the spin-1 Heisenberg model with DM interactions on the simple cubic lattice. The Hamiltonian is described as X X ~ij ðS~i S~j Þ, H ¼ J D ½ð1 DÞðS xi Sxj þ Syi Syj Þ þ S zi S zj (1) hi;ji
hi;ji
where the two sum terms are the ferromagnetic Heisenberg and DM interactions, respectively. The exchange coupling constant J is restricted to the nearest-neighbor pairs of spins. S ai (a ¼ x; y; z) are the components of ~ij are the exchange anisotropic parameter and the DM vector the spin-1 operator at site i. DðD 2 ½0; 1Þ and D ~ij ¼ D ~ji . coupling, respectively. And the DM vector coupling is antisymmetric, i.e., D ~ ~ For convenience, we shall take Dij ¼ D^z, i.e., Dij parallels to the z-axis direction which really is a special choice of the DM interaction term in Eq. (1). Thus, according to the two-spin cluster mean-field approximation, the two-spin cluster Hamiltonian H MFA can be written as [17] 12 H MFA ¼ J½ð1 DÞðS x1 Sx2 þ Sy1 Sy2 Þ þ S z1 S z2 DðS x1 S y2 S y1 S x2 Þ Jðq 1ÞmðS z1 þ Sz2 Þ, 12
(2)
where q is the coordination number of the every site of the lattice and the magnetization m is the averaged magnetic moment along a fixed direction z^ related to the cluster with two spins (i.e., m ¼ h12 ðSz1 þ S z2 Þi). In the representation of the direct product of Sz1 and S z2 , H MFA can be written as the form of 9 9 matrix. We can get 12 nine eigenvalues by diagonalizing the matrix of H MFA . Thus, the partition function Z ¼ Tr12 expðbH MFA Þ 12 12 has the following expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zðm; TÞ ¼ sinh½2Kðq 1Þm þ sinh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ sinh K ðq 1Þm D20 þ ð1 DÞ2 eK þ 2eK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh½2Kðq 1Þm þ 2 cosh K ðq 1Þm D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 cosh K ðq 1Þm þ D20 þ ð1 DÞ2 þ 2eK=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 8ðD20 þ ð1 DÞ2 Þ þ 1 , cosh ð3Þ 2
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where K ¼ bJ (b ðkB TÞ1 , kB is the Boltzmann constant) is the reciprocal of the reduced temperature and D0 ¼ D=J is the reduced DM parameter. Hence, with the present two-spin cluster mean-field method, the free energy of a bond as a function of the temperature T and the magnetization m is given by f ðm; TÞ ¼ ðq 1Þm2 K 1 ln Zðm; TÞ.
(4)
According to the equilibrium condition of the system, we can obtain the magnetization m from Eqs. (3) and (4), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K m ¼ 2e sinh½2Kðq 1Þm þ sinh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ sinh K ðq 1Þm D20 þ ð1 DÞ2 eK þ 2eK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh½2Kðq 1Þm þ 2 cosh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2 K=2 2 þ2 cosh K ðq 1Þm D0 þ ð1 DÞ 8ðD20 þ ð1 DÞ2 Þ þ 1 . þ 2e cosh ð5Þ 2 Thus, by expanding the right-hand side of Eq. (4) around m ¼ 0, one obtains the following expression of Landau order parameter series expansion for the reduced free energy f ðm; TÞ f ð0; TÞ þ 12ð1 aÞm2 14bm4 16cm6 , where the coefficients a, b and c are, respectively, 2ðq 1ÞK c eK c F 1 þ 1, a ¼ 2ðq 1Þ C
(6)
(7)
b ¼ f2ðq 1Þ4 K 3c eK c ½8eK c ðX coshðK c xÞ2 Þ þ coshðK c xÞðX 10e2K c Þg=ð3C2 Þ,
(8)
c ¼ f2ðq 1Þ6 K 5c e3K c ½480F2 60eK c FðF þ 6eK c ÞC þ e2K c ðF þ 30eK c Þg=ð15C3 Þ,
(9)
where F ¼ 2eK c þ coshðK c xÞ, X ¼ 1 4e2K c þ 2eK c =2 cosh
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kc 1 þ 8x2 , 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kc 1 þ 8x2 , C ¼ 1 þ 2e2K c þ 4eK c coshðK c xÞ þ 2eK c =2 cosh 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which x ¼ D20 þ ð1 DÞ2 , K 1 c ¼ k B T c =J is the reduced critical temperature and T c is the critical temperature. According to the stable condition of the system, the free energy satisfies qf ðm; TÞ ¼ m½ð1 aÞ bm2 ¼ 0 qm
(10)
and q2 f ðm; TÞ ¼ ð1 aÞ 3bm2 40. (11) qm2 By solving Eq. (10), in the vicinity of the second-order phase transition line, the magnetization m is expressed as m2 ¼
1a . b
(12)
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From Eqs. (10) and (11), we can obtain that a41 and bo0 for the ferromagnetic phase, ao1 and bo0 for the paramagnetic phase. Therefore, when a ¼ 1 and bo0, the phase transition is of the second order; when a ¼ 1 and b40, the phase transition is of the first order [21,22]. The above conclusions show that the TCP at which the phase transition changes from the second order to the first order is determined by the critical condition a ¼ 1 and b ¼ 0. 3. Results and discussions In this section, we study the magnetic properties of the spin-1 Heisenberg model with DM interactions on the simple cubic lattice (q ¼ 6). According to a ¼ 1 and b ¼ 0, within the framework of the present two-spin cluster mean-field method, the reduced tricritical temperature T t is found to be independent of the exchange anisotropy parameter D, which is the constant value of T t ¼ 1:71, while the reduced tricritical DM parameter qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt , as a function of D, is given by Dt ¼ 19:50 ð1 DÞ2 . Comparing above results with those of the spin-12 Heisenberg model with DM interactions [17], the reduced tricritical temperature is lower but the reduced tricritical DM parameter is larger. Now we investigate the temperature dependence of the magnetization in the case D ¼ 1 (Ising limit) for various values of D0 . As can be seen from Fig. 1, for values D0 ¼ 0:8 and 4:2, the magnetization falls smoothly to zero when temperature increases from zero to the critical temperature T c , characterizing a second-order phase transition. Note that for the case D0 ¼ 4:2, we also find unphysically unstable solutions of Eq. (12), which are described by dotted curve labelled c. On the other hand, for D0 4Dt , the magnetization drops discontinously from a finite mc ðD; D0 Þ to zero, at a temperature T c ðD; D0 Þ, which characterizes the first-order phase transition. The first-order phase transition temperature T c ðD; D0 Þ can be obtained from the free energy, when the local minimum at ma0 has the same value as the local minimum at m ¼ 0. Thus, when D0 ¼ 4:5 and 4:554Dt ðD ¼ 1Þ ¼ 4:42, the systems undergo the first-order phase transition, and the first-order phase transition reduced temperatures are 1.035 and 0.632, respectively. We have solved numerically the equation a ¼ 1 for bo0 which determines the second-order phase transition line. The critical boundaries separating the ferromagnetic phase from the paramagnetic one, are represented in Fig. 2, in which the variation of the reduced transition temperature kB T c =J are plotted as a function of the reduced DM parameter D0 for some typical values of the exchange anisotropy parameter D, where D ¼ 0, 0.3 and 1.0 represent the isotropic Heisenberg model, the anisotropic Heisenberg model and the spin-1 Ising
Fig. 1. Temperature dependence of the magnetization m for D ¼ 1 (Ising model) and the simple cubic lattice, when the value of D0 is different. The curves a–d correspond to D0 ¼ 4:55, 4.5, 4.2 and 0.8, respectively. Solid and dotted curves correspond to the stable and unstable solutions, respectively. The dashed-dotted curves represent the discontinuously of the magnetization for D0 4Dt ðD ¼ 1Þ.
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Fig. 2. Phase diagram of the spin-1 Heisenberg model with DM interaction for the simple cubic lattice, and for three typical values of D (D ¼ 0, 0.3 and 1.0). The solid and dotted curves represent the second-order and the first-order phase transitions. The black points refer to the TCP.
model, respectively. As seen, when D0 increases from zero, the second-order phase transition reduced temperatures fall from their values without the DM interaction, reaching TCPs at (T t , Dt ). Moreover, the firstorder phase transition lines (dotted ones) fall smoothly to zero when the parameter D0 increases from its qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values at the TCP to the zero temperature critical value D0 ¼ Dc ¼ 20:98 ð1 DÞ2 , which is larger than the value of the spin-12 Heisenberg model with DM interactions [17]. For the Ising limit, we obtain that the reduced transition temperature without DM interactions 1 (K 1 c ðD ¼ 1; D0 ¼ 0Þ) is 3.922. This result is consistent with the one (K c ¼ 4:0) obtained by the standard 1 mean-field approximation [27], but is not well in accord with the one (K c ¼ 3:5) obtained by the effective field theory [27]. 4. Conclusions In this paper, we have studied the magnetic and thermodynamics properties and phase diagram of the spin-1 Heisenberg model with DM interactions within the framework of the two-spin cluster mean-field approximation. A number of interesting phenomena for the system have been found, which are due to the DM interaction. Especially, the system exhibits the tricritical behavior. The critical behavior of the system may be interpreted as a result of a competition between the exchange interaction and the DM interaction. The exchange interaction tries to align the spins in the same direction, while the DM interaction tends to destroy this alignment. Acknowledgments This work was supported by the National Science Foundation for Post-doctoral Scientists of China (2005037442) and the Science Foundation of Qufu Normal University. One of the authors (Sun) thanks XiuXing Zhang, Zhong-Qiang Liu, Xun-Chang Yin, Xin Zhang and Xian-Ming Li for fruitful discussions. References [1] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, 1999.
ARTICLE IN PRESS 590 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
G.-H. Sun, X.-M. Kong / Physica A 370 (2006) 585–590 J. So´lyom, J. Timonen, Phys. Rev. B 40 (1989) 7150. M. Bo¨hm, H. Leschke, Physica A 199 (1993) 116. K.K. Pan, Y.L. Wang, Phys. Rev. B 51 (1995) 3610. I. Dzyaloshinskii, J. Phys. Chem. Solids 4 (1958) 241. T. Moriya, Phys. Rev. Lett. 4 (1960) 228. V. Bindilatti, T.Q. Vu, Y. Shapira, C.C. Agosta, E.J. McNiff, R. Kershaw, K. Dwight, A. Wold, Phys. Rev. B 45 (1992) 5328. See e.g., F. Keffer, in: S. Flu¨gge (Ed.), Encyclopedia of Physics, vol. 18/2, edited by H.P.J. Wijn, Springer, Berlin, 1966, p. 1. L. Yi, G. Bu¨ttner, K.D. Usadel, K.-L. Yao, Phys. Rev. B 47 (1993) 254 and references therein. G. Kotliar, H. Sompolinsky, Phys. Rev. Lett. 53 (1984) 1751. T.K. Kopec´, G. Bu¨ttner, Phys. Rev. B 43 (1991) 10853. Y.M. Shang, K.L. Yao, J. Magn. Magn. Mater. 277 (2004) 322. T. Ishii, I. Yamada, J. Phys.: Condens. Matter 2 (1990) 5771. I. Yamada, M. Nishi, J. Akimitsu, J. Phys.: Condens. Matter 8 (1996) 2625. V.N. Glazkov, A.I. Smirnov, O.A. Petrenko, D.M. Paul, A.G. Vetkin, R.M. Eremina, J. Phys.: Condens. Matter 10 (1998) 7879. C.E. Cordeiro, E.V. de Mello, M.A. Continentino, Z. Phys. B. (Condens. Matter) 85 (1991) 307. J. Ricardo de Sousa, D.F. de Albuquerque, I.P. Fittipaldi, Phys. Lett. A 191 (1994) 275. F. Lacerda, J. Ricardo de Sousa, I.P. Fittipaldi, J. Appl. Phys. 75 (1994) 5829. A. Benyoussef, N. Boccara, M. Saber, J. Phys. C 19 (1986) 1983. T. Thio, A. Aharomy, Phys. Rev. Lett. 73 (1994) 894. Q. Zhang, G.Z. Wei, Y.Q. Liang, J. Magn. Magn. Mater. 253 (2002) 45. W. Jiang, G.Z. Wei, A. Du, L.Q. Guo, Physica A 313 (2002) 503. K. Htoutou, A. Oubelkacem, A. Ainane, M. Saber, J. Magn. Magn. Mater. 288 (2005) 259. A. Benyoussef, A. Boubekri, H. Ez-Zahraouy, Physica B 266 (1999) 382. A.S.T. Pires, J. Magn. Magn. Mater. 223 (2001) 304. J.Z. Zhao, X.Q. Wang, T. Xiang, Z.B. Su, L. Yu, Phys. Rev. Lett. 90 (2003) 207204. I.P. Fittipaldi, E.F. Sarmento, T. Kaneyoshi, Physica A 186 (1992) 591.
Physica A 370 (2006) 585–590 www.elsevier.com/locate/physa
Phase diagram and tricritical behavior of the spin-1 Heisenberg model with Dzyaloshinskii–Moriya interactions Guang-Hou Suna, Xiang-Mu Konga,b, a
Department of Physics, Qufu Normal University, Qufu 273165, China The Interdisciplinary Center of Theoretical Studies, Chinese Academy of Sciences, Beijing 100080, China
b
Received 24 October 2005; received in revised form 5 January 2006 Available online 27 April 2006
Abstract Using the two-spin cluster mean-field method, the spin-1 Heisenberg model with Dzyaloshinskii–Moriya (DM) ~ ¼ D^z (D is the DM interaction interactions is studied for the simple cubic lattice. For the case of the DM vector coupling D parameter and z^ is the unit vector of the z-axis direction), the phase diagram of this system and the thermal behavior of the magnetization are obtained, and it is found that the system exhibits the tricritical point. The critical behavior of the system may be interpreted as a result of a competition between the exchange interaction and the DM interaction. r 2006 Elsevier B.V. All rights reserved. Keywords: Spin-1 Heisenberg model; Dzyaloshinskii–Moriya interaction; Phase diagram; Tricritical point
1. Introduction The critical properties of quantum magnetic systems have been subjects of intense research [1]. The spin-1 Heisenberg Hamiltonian deserves special attention since it provides a model for many magnetic materials. In recent years, there have been many interesting works dealing with the spin-1 Heisenberg model. For instance, So´lyom and Timonen transformed the spin-1 Heisenberg chain to the one-dimensional fermion gas and obtained phase diagrams of the fermion system [2], and Bo¨hm et al. calculated the coefficients of the shorttime expansion of the spin-pair correlations in one-dimensional spin model [3]. Besides, the ground-state properties of the spin-1 Heisenberg ferromagnet with an arbitrary crystal-field potential have been studied using the linked-cluster series expansion [4]. On the other hand, the Heisenberg model with anisotropy has been attracting much attention since various types of anisotropy have a profound influence on the properties of the systems. There are many different ways to introduce anisotropies in the Heisenberg model and an important type of anisotropic interaction is the Dzyaloshinskii–Moriya (DM) interaction [5,6], which is the antisymmetric spin coupling. The DM interaction plays an important role in describing certain class of insulators [7,8], in studying spin glasses [9–12] and also in explaining the electron paramagnetic resonance [13–15]. The Heisenberg models with DM interactions have Corresponding author. Department of Physics, Qufu Normal University, Qufu 273165, China.
E-mail address: [email protected] (X.-M. Kong). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.025
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been intensively studied over past years. Cordeiro et al. have obtained the phase diagram and critical exponents of a spin-12 Heisenberg model with DM interactions by using the renormalization group technique [16]. Subsequently, the phase diagram of this system was also obtained within a two-spin cluster version of mean-field technique [17], as well as within the framework of a new correlated effective-field theory [18]. In Refs. [17,18], the tricritical point (TCP) of this system was found, which had improperly been overlooked in all previous calculations. The tricritical behavior is a very important critical phenomenon, so there has been increasing interest in the study of the behavior [19–23]. In Ref. [20], it was observed that the TCP of antiferromagnetic La2 CuO4 including the DM interaction was close to the Ne´el point. Recently, the effects of the DM interaction on the stability of the Ne´el phase and the energy gap for XXZ Heisenberg model have been studied using the linear spin-wave theory [24]. Pires has investigated the spin-12 alternating Heisenberg model with DM interactions in a fermion representation and calculated the ground state energy, the low-lying excitations and static nearest-neighbors correlation functions in the T ¼ 0 limit [25]. In addition, by the density matrix renormalization group, the physical effects of the DM interaction in copper benzoate were found to produce a gap in the spin excitations [26]. In this paper, the effects of the DM interaction on properties of the spin-1 Heisenberg model are investigated systemically within the framework of the two-spin cluster mean-field method, and it is found that the system exhibits the TCP. The outline of the remainder of this paper is as follows. In Section 2, we give the formulation of this problem. Section 3 is the numerical results and discussions, and Section 4 is the conclusion. 2. Formulation Let us describe the spin-1 Heisenberg model with DM interactions on the simple cubic lattice. The Hamiltonian is described as X X ~ij ðS~i S~j Þ, H ¼ J D ½ð1 DÞðS xi Sxj þ Syi Syj Þ þ S zi S zj (1) hi;ji
hi;ji
where the two sum terms are the ferromagnetic Heisenberg and DM interactions, respectively. The exchange coupling constant J is restricted to the nearest-neighbor pairs of spins. S ai (a ¼ x; y; z) are the components of ~ij are the exchange anisotropic parameter and the DM vector the spin-1 operator at site i. DðD 2 ½0; 1Þ and D ~ij ¼ D ~ji . coupling, respectively. And the DM vector coupling is antisymmetric, i.e., D ~ ~ For convenience, we shall take Dij ¼ D^z, i.e., Dij parallels to the z-axis direction which really is a special choice of the DM interaction term in Eq. (1). Thus, according to the two-spin cluster mean-field approximation, the two-spin cluster Hamiltonian H MFA can be written as [17] 12 H MFA ¼ J½ð1 DÞðS x1 Sx2 þ Sy1 Sy2 Þ þ S z1 S z2 DðS x1 S y2 S y1 S x2 Þ Jðq 1ÞmðS z1 þ Sz2 Þ, 12
(2)
where q is the coordination number of the every site of the lattice and the magnetization m is the averaged magnetic moment along a fixed direction z^ related to the cluster with two spins (i.e., m ¼ h12 ðSz1 þ S z2 Þi). In the representation of the direct product of Sz1 and S z2 , H MFA can be written as the form of 9 9 matrix. We can get 12 nine eigenvalues by diagonalizing the matrix of H MFA . Thus, the partition function Z ¼ Tr12 expðbH MFA Þ 12 12 has the following expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zðm; TÞ ¼ sinh½2Kðq 1Þm þ sinh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ sinh K ðq 1Þm D20 þ ð1 DÞ2 eK þ 2eK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh½2Kðq 1Þm þ 2 cosh K ðq 1Þm D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 cosh K ðq 1Þm þ D20 þ ð1 DÞ2 þ 2eK=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 8ðD20 þ ð1 DÞ2 Þ þ 1 , cosh ð3Þ 2
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where K ¼ bJ (b ðkB TÞ1 , kB is the Boltzmann constant) is the reciprocal of the reduced temperature and D0 ¼ D=J is the reduced DM parameter. Hence, with the present two-spin cluster mean-field method, the free energy of a bond as a function of the temperature T and the magnetization m is given by f ðm; TÞ ¼ ðq 1Þm2 K 1 ln Zðm; TÞ.
(4)
According to the equilibrium condition of the system, we can obtain the magnetization m from Eqs. (3) and (4), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K m ¼ 2e sinh½2Kðq 1Þm þ sinh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ sinh K ðq 1Þm D20 þ ð1 DÞ2 eK þ 2eK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh½2Kðq 1Þm þ 2 cosh K ðq 1Þm þ D20 þ ð1 DÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2 K=2 2 þ2 cosh K ðq 1Þm D0 þ ð1 DÞ 8ðD20 þ ð1 DÞ2 Þ þ 1 . þ 2e cosh ð5Þ 2 Thus, by expanding the right-hand side of Eq. (4) around m ¼ 0, one obtains the following expression of Landau order parameter series expansion for the reduced free energy f ðm; TÞ f ð0; TÞ þ 12ð1 aÞm2 14bm4 16cm6 , where the coefficients a, b and c are, respectively, 2ðq 1ÞK c eK c F 1 þ 1, a ¼ 2ðq 1Þ C
(6)
(7)
b ¼ f2ðq 1Þ4 K 3c eK c ½8eK c ðX coshðK c xÞ2 Þ þ coshðK c xÞðX 10e2K c Þg=ð3C2 Þ,
(8)
c ¼ f2ðq 1Þ6 K 5c e3K c ½480F2 60eK c FðF þ 6eK c ÞC þ e2K c ðF þ 30eK c Þg=ð15C3 Þ,
(9)
where F ¼ 2eK c þ coshðK c xÞ, X ¼ 1 4e2K c þ 2eK c =2 cosh
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kc 1 þ 8x2 , 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kc 1 þ 8x2 , C ¼ 1 þ 2e2K c þ 4eK c coshðK c xÞ þ 2eK c =2 cosh 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which x ¼ D20 þ ð1 DÞ2 , K 1 c ¼ k B T c =J is the reduced critical temperature and T c is the critical temperature. According to the stable condition of the system, the free energy satisfies qf ðm; TÞ ¼ m½ð1 aÞ bm2 ¼ 0 qm
(10)
and q2 f ðm; TÞ ¼ ð1 aÞ 3bm2 40. (11) qm2 By solving Eq. (10), in the vicinity of the second-order phase transition line, the magnetization m is expressed as m2 ¼
1a . b
(12)
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From Eqs. (10) and (11), we can obtain that a41 and bo0 for the ferromagnetic phase, ao1 and bo0 for the paramagnetic phase. Therefore, when a ¼ 1 and bo0, the phase transition is of the second order; when a ¼ 1 and b40, the phase transition is of the first order [21,22]. The above conclusions show that the TCP at which the phase transition changes from the second order to the first order is determined by the critical condition a ¼ 1 and b ¼ 0. 3. Results and discussions In this section, we study the magnetic properties of the spin-1 Heisenberg model with DM interactions on the simple cubic lattice (q ¼ 6). According to a ¼ 1 and b ¼ 0, within the framework of the present two-spin cluster mean-field method, the reduced tricritical temperature T t is found to be independent of the exchange anisotropy parameter D, which is the constant value of T t ¼ 1:71, while the reduced tricritical DM parameter qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt , as a function of D, is given by Dt ¼ 19:50 ð1 DÞ2 . Comparing above results with those of the spin-12 Heisenberg model with DM interactions [17], the reduced tricritical temperature is lower but the reduced tricritical DM parameter is larger. Now we investigate the temperature dependence of the magnetization in the case D ¼ 1 (Ising limit) for various values of D0 . As can be seen from Fig. 1, for values D0 ¼ 0:8 and 4:2, the magnetization falls smoothly to zero when temperature increases from zero to the critical temperature T c , characterizing a second-order phase transition. Note that for the case D0 ¼ 4:2, we also find unphysically unstable solutions of Eq. (12), which are described by dotted curve labelled c. On the other hand, for D0 4Dt , the magnetization drops discontinously from a finite mc ðD; D0 Þ to zero, at a temperature T c ðD; D0 Þ, which characterizes the first-order phase transition. The first-order phase transition temperature T c ðD; D0 Þ can be obtained from the free energy, when the local minimum at ma0 has the same value as the local minimum at m ¼ 0. Thus, when D0 ¼ 4:5 and 4:554Dt ðD ¼ 1Þ ¼ 4:42, the systems undergo the first-order phase transition, and the first-order phase transition reduced temperatures are 1.035 and 0.632, respectively. We have solved numerically the equation a ¼ 1 for bo0 which determines the second-order phase transition line. The critical boundaries separating the ferromagnetic phase from the paramagnetic one, are represented in Fig. 2, in which the variation of the reduced transition temperature kB T c =J are plotted as a function of the reduced DM parameter D0 for some typical values of the exchange anisotropy parameter D, where D ¼ 0, 0.3 and 1.0 represent the isotropic Heisenberg model, the anisotropic Heisenberg model and the spin-1 Ising
Fig. 1. Temperature dependence of the magnetization m for D ¼ 1 (Ising model) and the simple cubic lattice, when the value of D0 is different. The curves a–d correspond to D0 ¼ 4:55, 4.5, 4.2 and 0.8, respectively. Solid and dotted curves correspond to the stable and unstable solutions, respectively. The dashed-dotted curves represent the discontinuously of the magnetization for D0 4Dt ðD ¼ 1Þ.
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Fig. 2. Phase diagram of the spin-1 Heisenberg model with DM interaction for the simple cubic lattice, and for three typical values of D (D ¼ 0, 0.3 and 1.0). The solid and dotted curves represent the second-order and the first-order phase transitions. The black points refer to the TCP.
model, respectively. As seen, when D0 increases from zero, the second-order phase transition reduced temperatures fall from their values without the DM interaction, reaching TCPs at (T t , Dt ). Moreover, the firstorder phase transition lines (dotted ones) fall smoothly to zero when the parameter D0 increases from its qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values at the TCP to the zero temperature critical value D0 ¼ Dc ¼ 20:98 ð1 DÞ2 , which is larger than the value of the spin-12 Heisenberg model with DM interactions [17]. For the Ising limit, we obtain that the reduced transition temperature without DM interactions 1 (K 1 c ðD ¼ 1; D0 ¼ 0Þ) is 3.922. This result is consistent with the one (K c ¼ 4:0) obtained by the standard 1 mean-field approximation [27], but is not well in accord with the one (K c ¼ 3:5) obtained by the effective field theory [27]. 4. Conclusions In this paper, we have studied the magnetic and thermodynamics properties and phase diagram of the spin-1 Heisenberg model with DM interactions within the framework of the two-spin cluster mean-field approximation. A number of interesting phenomena for the system have been found, which are due to the DM interaction. Especially, the system exhibits the tricritical behavior. The critical behavior of the system may be interpreted as a result of a competition between the exchange interaction and the DM interaction. The exchange interaction tries to align the spins in the same direction, while the DM interaction tends to destroy this alignment. Acknowledgments This work was supported by the National Science Foundation for Post-doctoral Scientists of China (2005037442) and the Science Foundation of Qufu Normal University. One of the authors (Sun) thanks XiuXing Zhang, Zhong-Qiang Liu, Xun-Chang Yin, Xin Zhang and Xian-Ming Li for fruitful discussions. References [1] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, 1999.
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