Determination of the Critical Behavior of the Antiferromagnetic Interaction Spin-1 System in an External Magnetic Field by Using the Constant-Coupling Approximation

Journal of the Korean Physical Society, Vol. 56, No. 4, April 2010, pp. 1046∼1050

Determination of the Critical Behavior of the Antiferromagnetic Interaction Spin-1 System in an External Magnetic Field by Using the Constant-coupling Approximation

Ahmet Erdinc¸∗ and Osman Canko Department of Physics, Erciyes University, Kayseri, 38039, Turkey

The critical behavior of the antiferromagnetic interaction spin-1 system model in an external magnetic field is studied by using the constant-coupling approximation for four different nearest- neighbors coupling constant. First, we have investigated the thermal variations of the sublattice magnetizations for different values of the interaction parameters in an external magnetic field. The model exhibits distinct critical region, including the first-order, the second-order tricritical point, and the zero critical point region. Second, we have calculated the phase diagrams. Finally, a discussion and a comparison of the phase diagrams are given.

PACS numbers: 05.50.+q, 05.70.Fh, 64.60.Cn, 75.10.Hk Keywords: Constant-coupling approximation, Phase diagram, BEG model, Antiferromagnetism DOI: 10.3938/jkps.56.1046

I. INTRODUCTION rochlore lattice with nearest- and next-nearest- neighbors intereactions and well site dilution by using the general- ized constant coupling (GCC) method. They calculated The constant-coupling approximation (CCA), intro- the magnetic properties and found them to be in excel- duced by Kasteleijn and van Kranendonk [1–3] and by lent agreement with the Monte Carlo (MC) calculations Radcliffe [4], provides a method for calculating equi- [10]. In a recent work, Rausch and Nolting calculated librium properties of spin systems. The method at- the thickness-dependent Curie temperature of thin fer- tempts to deal with correlations that are usually ne- romagnetic films by using the Oguchi cluster method [11] gleted by molecular field approximations, and some and the CCA [12]. results of the constant-coupling approximation, e.g., critical-temperature values, represent an improvement over molecular field results. In addition to the origi- II. FORMULATION IN TERMS OF THE nal work of Kateleijn and van Kranendonk, the general CONSTANT COUPLING cluster expansion for Heisenberg and Ising systems has APPROXIMATION been shown to provide a common framework for arriving at both the molecular field and the constant-coupling approximation. Then, Glauber proposed the time- The effective Hamiltonian in an external magnetic field dependent statistics of the one-dimensional Ising model for a single-spin on the k-th site can be written in the with spin-S = 1/2 [5]. Obokata extended Glauber’s the- form [6] ory to the case of S = 1, developed the time-dependent CCA, and studied the equilibrium spin system with S = H = −zλS − zµS2 − DS2 − hS , (1) 1 by using the static CCA [6]. Falk obtained the for- k k k k k mulation of the CCA for S = 1/2 [7]. Chen and Lee where Sk located at site k is spin-1 with three discrete studied the ferromagnetic exchange-interaction model for spin values, i.e. ±1 and 0, and z is the number of the general spins by using the CCA and found that a first- nearest neighbors, i.e., the coordination number. Two order transition occurred for all spin S ≥ 1 [8]. Fechner parameters λ and µ represent the exchange interaction and Pikula examined some thermodynamic properties with the nearest neighboring spins, D is a single-ion of an antiferromagnetic spin-1/2 system with two dis- anisotropy constant or crystal-field interaction, where h tinct anisotropic exchange interactions in each case for = gµBH and H is the applied magnetic field along the two different nearest-neighbor coupling constant by using z-axis, g Landéfactor, and µB is the Bohr magneton. the CCA [9]. Garcia-Adeva and Huber studied the py- Next, let us assume that the effective Hamiltonian for two spins or a pair Sk and Sl a nearest neighbors can be ∗E-mail: [email protected] written in form -1046- Determination of the Critical Behavior of the Antiferromagnetic Interaction ··· – Ahmet Erdinç and Osman Canko -1047-

The normalized density matrices belonging to each sublattice, because each spin only interacts with its Hkl = −2JSkSl − (z − 1)λ(Sk + Sl) nearest-neighbors, can be written, and the averages of 2 2 2 2 −(z − 1)µ(Sk + Sl ) the Sk,Sl,Sk , and Sl are, respectively, given as 2 2 −D(Sk + Sl ) − h(Sk + Sl), (2)

A B where J represents the bilinear or coupling constant be- hSki = T r(Skρk ), hSli = T r(Slρl ), tween the two magnetic atoms and is assumed to be neg- hS2i = T r(S2ρA), hS2i = T r(S2ρB). (6) ative (antiferromagnetic). In the ferromagnetic case, the k k k l l l magnetic interactions between magnetic atoms is posi- tive and favors a paralel alignment of the atomic mag- Now, let us consider the effective Hamiltonian for two netic moments. However, in the antiferromagnetic case, spins Sk and Sl that are nearest neighbors to other as the interactions cause a antiparallel arrangement of mag- two sublattices. Then, the effective Hamiltonian can be netic moments. Therefore, the lattice of magnetic atoms written in the form should be divided into two equivalent interpenetrating sublattice, i.e., A and B, such that A atoms have only B AB atoms as nearest-neighbor and vice versa. Hkl = −2JSkSl − (z − 1)(λASk + λBSl) The CCA, higher mean-field theory, is applied to ob- 2 2 −(z − 1)(µASk + µBSl ) tain a quantitative improvement of the result of the −D(S2 + S2) − h(S + S ). (7) mean-field theory. Furthermore, the mean-field theory k l k l is the best effective field that can be devised for a single- atom model and may not be suitable for interpenetrating sublattice models [11]. Now, let us consider a two- The normalized density matrix for the two sublattices sublattice, i.e. A and B, spin-1 system and apply the is constant coupling approximation method. For a two- sublattice system, the effective Hamiltonian in an exter- AB AB nal magnetic field for a single spin on the k-th and l-th AB −βHkl −βHkl ρkl = e /T re . (8) sites can be defined as follows: 2 2 The averages of the Sk,Sl,Sk, and Sl are then given A 2 2 by Hk = −zλASk − zµASk − DSk − hSk, (3) B 2 2 Hl = −zλBSl − zµBSl − DSl − hSl, (4) hS i = T r(S ρAB), hS i = T r(S ρAB), where λA, λB, µA, and µB are exchange interactions for k k kl l l kl 2 2 AB 2 2 AB the A and the B sublattices, respectively. The normal- hSki = T r(Skρkl ), hSl i = T r(Sl ρkl ). (9) ized density matrices for the k-th and the l-th spins as two sublattices are given by Both the single-particle Hamiltonian and the pair Hamiltonian predict the same order parameters [8]. A −βHA −βHA Hence, from the self-consistence conditions that the four ρ = e k /T re k , k expressions for S ,S ,S2 , and S2 in Eq. (6) must be B B k l k l B −βHl βHl ρl = e /T re . (5) equal to Eq. (9), respectively, we have

β(zµA+D) e sinh β(zλA + h) β(zµ +D) 2e A cosh β(zλA + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} sinh βt1 + eβ{z µA+D} sinh βta2 + e2β{−J+z µ +D} sinh βta3 = 0 0 0 0 0 eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βta3 + eβz (−λA+λB )]} + 2eβ{z µA+D} cosh βta4 + 2eβ{z µB +D} cosh βta5 + 1

β(zµB +D) e sinh β(zλB + h) β(zµ +D) , 2e B cosh β(zλB + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} sinh βt1 + eβ{z µB +D} sinh βtb2 + e2β{−J+z µ +D} sinh βta3 = 0 0 0 0 0 , eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βtb3 + eβz (λA−λB )]} + 2eβ{z µB +D} cosh βtb4 + 2eβ{z µB +D} cosh βtb5 + 1 -1048- Journal of the Korean Physical Society, Vol. 56, No. 4, April 2010

β(zµA+D) e cosh β(zλA + h) β(zµ +D) 2e A cosh β(zλA + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} cosh βt1 + eβ{z µA+D} cosh βta2 + e2β{−J+z µ +D} cosh βta3 = 0 0 0 0 0 eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βta3 + eβz (−λA+λB )]} + 2eβ{z µA+D} cosh βta4 + 2eβ{z µB +D} cosh βta5 + 1

β(zµB +D) e cosh β(zλB + h) β(zµ +D) 2e B cosh β(zλB + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} cosh βt1 + eβ{z µB +D} cosh βtb2 + e2β{−J+z µ +D} cosh βtb3 = 0 0 0 0 0 , eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βtb3 + eβz (λA−λB )]} + 2eβ{z µB +D} cosh βtb4 + 2eβ{z µB +D} cosh βtb5 + 1 (10)

with λA = JMA, λB = JMB, µA = KQA, µB = KQB, -1.0 and h/|J| = 1.5. As seen in the figure, sublattice 0 0 0 z = z − 1, µ = µA + µB, t1 = z (λA + λB) + 2h, ta2 = magnetizations undergo only a second-order phase tran- 0 0 0 2J + 2z λA + h, ta3 = z (λA − λB), ta4 = z (−λA + λB), sition at the Néel temperature (TN ), at which the an- 0 0 0 ta5 = z λB +h, tb2 = 2J+2z λB +h, tb3 = z (−λA+λB), tiferromagnetic ordering disappears for values of z = 3, 0 0 tb4 = z (λA − λB), and tb5 = z λA + h, K is the bi- 4, 6, and 8. As seen in the figure for all z, the sublat- quadratic exchange interaction parameter. The bilinear tice magnetizations start from their maximum possible (J), the biquadratic (K), and the crystal-field (D) inter- values at zero temperature, i.e., ±1, and as the tempera- action parameters were introduced by Blume et al. [13] ture is increased, they combine at the second-order phase to describe the phase separation and the superfluid or- transition or the Néel temperature TN . dering in He3-He4 mixtures. This model is known as (b) Type II: In Fig. 1(b) for D/|J| = -2.18 and the Blume-Emery-Griffiths (BEG) model. With disap- h/|J| = 0.2, the behaviors of the sublattice magnetiza- pearing biquadratic exchange interaction the model, is tions include first-order phase transition for z = 3 and known as the Blume-Capel (BC) model [14]. The model 4 at temperature Tt, and second-order phase transitions was subsequently re-interpreted to describe the phase for z = 6 and 8 at temperature TN . It should be men- transitions in simple and multicomponent fluids [15]. tioned that the first-order phase transition temperature MA ≡ hSki and MB ≡ hSli are the average magnetiza- is the temperature at which the sublattice magnetizations, which represent the excess of one orientation over tions show a discontinuity. The system gives first-order the other orientation, called magnetizations for the A phase transitions at the temperature Tt, as shown in this 2 and the B sublattices, respectively, and QA ≡ hSki and figure. 2 QB ≡ hSl i are the quadrupolar moments, which are the (c) Type III: In Fig. 1(c), we found that the sub- square of the average magnetizations for the A and the lattice magnetization had two successive second-order B sublattices, respectively. phase transitions at two different temperatures, e.g., for We are now able to examine the behaviors of the or- D/|J| = -4 and h/|J| = 13.17 for z = 8 only. These behav- der parameters of the two sublattice antiferromagnetic iors imply that the system exhibits a reentrant behavior. interaction (J < 0) spin-1 system with the bilinear interaction (J), the crystal-field intereaction (D), and a dis- appearing biquadratic exchange interaction, namely the III. PHASE DIAGRAMS BC model, in an external magnetic field (h) by solving the self-consistent equations, i.e., Eq. (10), numerically. First, we have to study the thermal variations of the In this section, we present the phase diagram of the two-sublattice order-parameters. At this stage, it should antiferromagnetic spin-1 system with coordination num- be mentioned that the thermal variations of the sublat- bers z = 3, 4, 6, and 8 in an external magnetic field. tice quadrupolar order-parameters are not included in The second-order phase transition temperatures for the the figures because the system does not present any sub- sublattice magnetizations in the case of a second-order lattice quadrupolar phase and because the locations of phase transition are calculated numerically; i.e., for the the critical temperatures are found by using the sublat- investigation of the behaviors of the order parameters as tice magnetization. As a result, the temperature change functions of the temperature, the sublattice magnetiza- of the sublattice dipolar order parameters, i.e., the mag- tions become equal as the temperature is lowered, and netization, are studied for z = 3, 4, 6, and 8, and some the temperature at which the sublattice magnetizations different topological results are found: become equal is the second-order phase transition tem- (a) Type-I: Figure 1(a) shows the temperature de- perature. On the other hand, the first-order phase tran- pendence of the sublattice magnetizations for D/|J| = sition temperatures found by using the sublattice magnetizations show a discontinuity. Determination of the Critical Behavior of the Antiferromagnetic Interaction ··· – Ahmet Erdinç and Osman Canko -1049-

Fig. 2. Phase diagrams in the (h/|J|, kT/|J|) plane for z = Fig. 1. Thermal variations of the sublattice magnetizations 3, 4, 6, and 8: (a) D/|J| = -1.0, (b) D/|J| = -2.5, and (c) D/|J| MA and MB for z = 3, 4, 6, and 8. Tt, and TN are the first- = -4.0. The AF and P denote the antiferromagnetic and the and second-order phase transition temperatures. (a) Only disordered phases, respectively. Dashed and solid lines, re- the second-order phase transitions for D/|J| = -1.0 and h/|J| spectively, indicate first- and second-order phase transitions. = 1.5. (b) The first- and second-order phase transitions for The specials points are zero-temperature critical (Z) and tri- D/|J| = -2.18 and h/|J| = 0.2. (c) Two successive second- critical points, shown as filled triangles ( ). order phase transitions for D/|J| = -4.0, h/|J| = 13.17 for H only z = 8.

fination similar to that of Hui has been used [16]. The We can now obtain the phase diagrams of the sys- obtained phase diagrams show three different topological tem the calculated phase diagrams in the (h/|J|, kT/|J|) behaviors: plane are presented in Fig. 2. In the phase diagrams, 1. The first phase diagram on the (h/|J|, kT/|J|) solid and dashed lines represent the second- and the first- plane, Fig. 2(a), shows only the second-order phase tran- order phase transition lines, respectively. Z and filled sitions, which occur at the Néel temperatures, obtained triangles are the special points that denote the zero- for D/|J| = -1.0 for z = 3, 4, 6, and 8. As seen in the temperature critical points and the tricritical point, re- figure, the Néel temperatures occur at high values of the spectively. Two phase regions are observed in the phase external magnetic field for all coordination numbers. For diagrams. The first phase is the disordered (or param- all z, as may be seen from Fig. 1, at higher values of agnetic) phase. The first is MA = MB 6= 0, shown with h/|J| the system shows a second-order phase transition. the letter P, and the second one is the antiferromagnetic The antiferromagnetic (AF) phase is separated from the phase (AF) with MA > MB or MA < MB, where a de- disordered phase (P) by the second-order phase transi- -1050- Journal of the Korean Physical Society, Vol. 56, No. 4, April 2010 tion line that terminates at the zero-temperature critical REFERENCES point. It should be mentioned that for the Ising model on a Cayley tree [17], a similar phase diagram is obtained. 2. The second topological phase diagram is given in [1] P. W. Kasteleijn and J. van Kranendonk, Physica 22, Fig. 2(b) for D/|J| = -2.5. This figure show the second- 317 (1956). [2] P. W. Kasteleijn and J. van Kranendonk, Physica 22, order phase lines for z = 6 and 8 as well as first-order 367 (1956). phase lines for z = 3 and 4 that we have given previously [3] P. W. Kasteleijn, Physica 22, 387 (1956). for the thermal variations of the sublattice magnetiza- [4] J. M. Radcliffe, Phys. Rev. 165, 635 (1968). tions (see Fig. 2(b)). The AF phase is separated from [5] R. J. Glauber, J. Math. Phys. 4, 294 (1963). the P phase by the second-order phase transition line for [6] T. Obokata, J. Phys. Soc. Jpn. 26, 895 (1969). z = 6 and 8, but the AF phase is separated from the P [7] H. Falk, Phys. Rev. B 5, 3638 (1972). phase by the first-order phase transition line for z = 3 [8] H. H. Chen and F. Lee, Phys. Rev. B 48, 9456 (1993). and 4. [9] B. J. Fechner and R. Pikula, Physica A 79, 43 (1975). 3. Another topological phase diagram is given in Fig. [10] A. J. Garcia-Adeva and D. L. Huber, Phys. Rev. B 65, 2(c) for D/|J| = -4.0, where the system exhibits a tricriti- 184418 (2002); 63, 140404(R) (2001); 63, 174433 (2001); cal point shown with filled triangles ( ) for z = 8, but for 64, 172403 (2001). H [11] T. Oguchi, Prog. Theor. Phys. 13, 148, (1955); J. H. z = 6 only the second-order phase transitions are seen. Van Vleck, Phys. Rev. 52, 1178 (1937); J. S. Smart, However, for D/|J| = -4.0, there are no phase transitions Effective Field Theories of Magnetism (W. B. Saunders for z = 3 and 4 with an external magnetic field. Co. Philadelphia, PA, 1966) p. 35. In the summary, the critical behavior of the anti- [12] R. Rausch and W. Nolting, J. Phys. Condens. Matter ferromagnetic interaction spin-1 system in an external 21, 376002 (2009). magnetic field has been studied by using the constant- [13] M. Blume, V. J. Emery and R. B. Griffiths, Phys, Rev. coupling approximation. We have studied the thermal A 4, 1071 (1971). variations of the two-sublattice magnetizations in or- [14] M. Blume, Phys. Rev. 141, 517 (1966); H. W. Capel, der to obtain the phase diagrams on the (h/|J|, kT/|J|) Physica 31, 966 (1966); 33, 295 (1967); 37, 423 (1967). plane. As a result, we have found that the system [15] J. Lajzerowicz and J. Sivardière, Phys. Rev. A 11, 2079 presents second- and first-order phase transition, and the (1975); J. Sivardièreand J Lajzerowicz, ibid. 11, 2090 (1975); 11, 2101 (1975); D. Mukamel and M. Blume, tricritical points for appropriate values of D/|J|, h/|J|, Phys. Rev. A 10, 610 (1974); D. Furman, S. Dattagupta and z. We have also found that the second-order transi- and R. B. Griffiths, Phys. Rev. B 15, 441 (1977). tion lines exhibit reentrant phenomena. As a final word [16] K. Hui, Phys. Rev. B 38, 802 (1988). on this work, we should also emphasize that our results [17] C. Ekiz, Phys. Lett. A, 332, 121 (2004). are in over-all agreement with three of other works, as mentioned above.