Critical Properties of Mixed Ising Spin System with Different Trimodal Transverse Fields in the Presence of Single-Ion Anisotropy

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1 Commun. Theor. Phys. (Beijing, China) 45 (2006) pp c International Academic Publishers Vol. 45, No. 6, June 15, 2006 Critical Properties of Mixed Ising Spin System with Different Trimodal Transverse Fields in the Presence of Single-Ion Anisotropy CHEN Qiang 1, and YAN Shi-Lei 1,2,4, 1 Department of Physics, Suzhou University, Suzhou , China 2 Jiangsu Key Laboratory of Thin Films, Suzhou University, Suzhou , China Nantong Shipping College, Nantong , China 4 CCAST (World Loboratory), P.O. Box 870, Beijing , China (Received September 29, 2005) Abstract Within the framework of an effective field approximation, the effects of single-ion anisotropy and different trimodal transverse fields of two sublattices on the critical properties of the mixed spin-1/2 and spin-1 Ising system are investigated on the simple cubic lattice. A smaller single-ion anisotropy can magnify magnetic ordering phases and a larger one can depress magnetic ordering phase for T -Ω 1/2 space at low temperatures, while a smaller single-ion anisotropy can hardly change the value of critical transverse field for T -Ω 1 space. On the other hand, influences of two different trimodal transverse fields concentrations on tricritical points and magnetic ordering phases take on some interesting results in T -D space. The main reason comes from the common action of single-ion anisotropy, different transverse fields and two trimodal distributions. PACS numbers: Dg, Jm, Cx Key words: critical property, mixed spin system, trimodal transverse field, single-ion anisotropy 1 Introduction In recent years, there have been an increasing number of works dealing with critical properties of quantum spin systems. One of the simplest is the transverse Ising model (TIM), which has been introduced to explain the phase transition and the order-disorder phenomenon when tunneling effect is present. Many authors have investigated the single spin or the mixed spin MIT within different approximations. [1 6] Of course, a more general strategy in studying a single or mixed spin TIM considers anisotropy interaction and various disorder distributions simultaneously. In these cases, it is possible to gain richer and more fruitful results. Zhong et al. introduced firstly single-ion anisotropy interaction in single spin (S = 1) TIM. [7] Yan et al. generalized mixed spin system. [8] Benayad et al. have discussed the mixed transverse spin system with random single-ion anisotropy. [9] One of the present authors has studied the thermodynamic properties of mixed random transverse Ising spin system with singleion anisotropy on square lattice. [10] The results show that a smaller single-ion anisotropy can induce magnetic ordering phase at low temperatures in the presence of a uniform random transverse field, while a larger one can depress sharply magnetic ordering phase again. Moreover, the existence of a larger transverse field can lead to occurrence of the reentrant phenomena, while a smaller one cannot. Thus, the role of the random transverse field is very curious when there exists the single-ion anisotropy in the mixed spin system. In fact, random transverse field represents quantum fluctuation effect of the system. In general situation, the random transverse field satisfies independent trimodal transverse field probability distribution. It has widely been employed in ferromagnetic and ferrimagnetic systems, [11 1] quantum Hopfield neuralnetwork model [14] and so on. On the other hand, it is worth while mentioning that the mixed TIM may have different quantum tunneling effects. Thus, different quantum effects of two sublattices can be represented by different inner transverse fields in Hamiltonian. [15 17] The research finds that the transverse fields of two different sublattices play a very distinct role when one combines the mixed spin-1/2 and spin-1 Ising system with different transverse fields and single-ion anisotropy in the same time. Transverse field of spin-1/2 sublattice induces reentrant phase transitions, while the other depresses them. [18] Meanwhile, the influence of the transverse fields of different sublattices on the normal and induced magnetic ordering displays dissimilar results. [19] In this work, we will study the phase diagrams of three-dimensional mixed spin-1/2 and spin-1 Ising system with different trimodal transverse fields in the presence of single-ion anisotropy. The global phase diagrams are presented on a simple cubic lattice. Phase diagrams take on some interesting results in T -Ω 1/2, T - Ω 1, and T -D spaces, respectively. To our knowledge, the The project supported partly by the Key Projects of Natural Science Foundation of Jiangsu Province of China under Grant No. 0KJA and the Open Foundation of Jiangsu Key Laboratory of Thin Films under Grant No. K slyan@suda.edu.cn

2 112 CHEN Qiang and YAN Shi-Lei Vol. 45 above subject has not been investigated yet. Additionally, the theoretical procedure of the present problem will be derived based on effective field theory (EFT). [20] The formulation is given in Sec. 2. In Sec., numerical results are obtained and a discussion on the possible physical reasons is presented. Finally, we give a brief summary about this work. 2 Theory We consider the mixed spin-1/2 and spin-1 Ising system in which the different trimodal transverse fields and single-ion anisotropy are involved. The Hamiltonian of the system can be described by the following equation, H = J i,j σ z i S z j i Ω i σ x i j Ω j S x j + D (Sj z ) 2. (1) j The underlying lattice is composed of two interpenetrating sublattices A and B, one of which is occupied by spin- 1/2 with moment σi z and σi x at site i, and the other one is occupied by spin-1 with moment Sj z and Sx j at site j. The first summation is carried out only on all the nearestneighbor pairs of spins. The second summation extends to all sites of sublattice A. The third and the fourth summations involve all sites of sublattice B. Here J defines the interaction between the spin at site i and its nearest neighbor located at site j. D is single-ion anisotropy, assumed to be negative. The quantities Ω i and Ω j represent the transverse field acting on the sublattices A and B and have independent trimodal probability distribution function, P (Ω i ) = p 1/2 δ(ω i ) (1 p 1/2)[δ(Ω i + Ω 1/2 ) + δ(ω i Ω 1/2 )], (2) P (Ω j ) = p 1 δ(ω j ) (1 p 1)[δ(Ω j + Ω 1 ) + δ(ω j Ω 1 )], () where 0 p α 1 (α = 1/2 or 1). The p α indicates the different trimodal transverse fields concentrations in the sublattice A and B, respectively. Thus, we can calculate the partition functions of sublattices A and B as basic statistic mechanics for the investigation of equilibrium thermodynamics. We obtain Z i = Tr i e βhi = 2 m=1 [ exp ( 1) m β 2 (E2 i + Ω i ) 1/2], (4) [ (n 1)2π + θ Z j = Tr j e βhj = exp 2βC cos n=1 + 2βD ]. (5) Here with θ = arccos A C, (6) C 6 = A 2 + B 2 (7) A = 1 27 D + 1 DE2 j 1 6 DΩ2 j, (8) B = 1 [ ( D 2 E j E ) ] 1/2 j D2 Ω 4 j + 15D 2 Ω 2 jej 2 + 9Ω 2 jej 4 + 9Ω 4 jej 2 + Ω 6 j, (9) where β = 1/k B T. The E i and E j are the local fields in sites i and j. They belong to sublattices A and B, respectively. Within the framework of effective field theory (EFT), the standard procedure then leads to the following expectation expressions of averaged magnetizations in sublattices A and B: z σ = σi z [ = (S z j ) 2 cosh( J) + Sj z sinh( J) + 1 (Sj z ) 2] F (x) x=0, (10) j=1 z [ m = (Sj z ) = cosh( 1 ( 1 )] 2 J) + 2σz i sinh 2 J G(x) x=0, (11) while the quadrupolar moment q is given by z ) q = (Sj z ) 2 = cosh 2 J i=1 i=1 ( 1 )] + 2σi z sinh 2 J H(x) x=0, (12) where z is the coordination number of lattice and = / x is a differential operator. The functions F (x), G(x) and H(x) are defined by F (x) = f(x, Ω i )p 1/2 (Ω i )dω i, (1) G(x) = g(x, Ω j )p 1 (Ω j )dω j, (14) H(x) = h(x, Ω j )p 1 (Ω j )dω j, (15)

3 No. 6 Critical Properties of Mixed Ising Spin System with Different Trimodal Transverse Fields in 11 where f(x, Ω i ), g(x, Ω j ) and h(x, Ω j ) are defined as f(x, Ω i ) = 1 x [ β 2 (Ω 2 i + x2 ) tanh 1/2 2 (Ω2 i + x 2 ) 1/2], (16) { [ g(x, Ω j ) = exp 2βCE 1 (n) + 2βD ][ 2x C E 1(n) + 2 D x Dx +.5DΩ 2 j x ]} E 2 (n) Z 1 j, 27 BC (17) h(x, Ω j ) = n=1 n=1 { [ exp 2βCE 1 (n) + 2βD ][ 2D 9C E 1(n) where (n 1)2π + θ E 1 (n) = cos, (19) (n 1)2π + θ E 2 (n) = sin. (20) However, it is clear that if we try to treat exactly the multi-spin correlation presented in Eqs. (10) (12), the problem is mathematically intractable. Therefore, we shall take a decoupling approximation σ z i σ z j σ z l σ z i σ z j σ z l, (21) S z j (S z k) 2 S z m S z j (S z k) 2 S z m, (22) for i j = l. By applying the approximation, the averaged magnetizations σ, m, and quadrupolar moment q in the sublattice A and B can be evaluated from the following coupled equations, σ = [q cosh(j ) + m sinh(j ) + 1 q] z F (x) x=0, (2) 0.5Ω 2 j x2 D 2 x 2 + x 4 0.5Ω 4 j E 2 (n) + 2 ]} Z 1 j, (18) BC m = q = ) 1 )] zg(x) x=0 cosh 2 J + 2σ sinh( 2 J, (24) ) 1 )] zh(x) x=0 cosh 2 J + 2σ sinh( 2 J. (25) If we expand the right-hand side of equations (2) (25) and combine them, the self-consistent equation of the magnetization σ in the sublattice A is given by σ = aσ + bσ + cσ 5 + (26) Since σ is small enough near the second-order phase transition line, the critical ordering frontier is the second order phase transition line when a = 1 and b < 0 are satisfied in the same time. If a = 1 and b > 0, that is the first order one. Hence the point at which a = 1 and b = 0 determines the TCP on the critical ordering frontiers. The coefficients a and b of Eq. (26) can be given by a = 2z 2 L 1 [sinh(j )][Q 1 [cosh(j )] + 1 Q 1 ] z 1 F (x) x=0, (27) b = 4 z2 (z 1)(z 2)L 2 [sinh(j )][Q 1 [cosh(j )] + 1 Q 1 ] z 1 F (x) x=0 + 4 z4 (z 1)(z 2)L 1 [sinh(j )] [Q 1 [cosh(j )] + 1 Q 1 ] z F (x) x=0 + 4z (z 1) 2 L 1 Q 2 [sinh(j )][[cosh(j )] 1][Q 1 [cosh(j )] + 1 Q 1 ] z 2 F (x) x=0, (28) where L 1, L 2, Q 1 and Q 2 are of the following forms: )][ 1 )] z 1G(x) x=0 L 1 = sinh cosh( 2 J 2 J, (29) L 2 = Q 1 = Q 2 = )] [ 1 )] z G(x) x=0 sinh cosh( 2 J 2 J, (0) [ cosh( 1 2 J )] zh(x) x=0, (1) )] 2 [ 1 )] z 2H(x) x=0 sinh cosh( 2 J 2 J. (2) It should be noted here that we have not touched the sign of the J. When J is positive, the ground state is ferromagnetic, while when J is negative, the system is ferrimagnetic. We have derived the expressions of the mixed spin-1/2 and spin-1 system with different trimodal transverse fields distribution in the presence of single-ion anisotropy. It is obvious that the above expressions are suitable for the lattice with an arbitrary coordination number. Here, a simple cubic lattice is selected as the three-dimensional version mainly because the three-dimensional system is experimentally the most relevant dimension. Results and Discussions In this section, by solving equation a = 1 and b < 0 numerically, the phase diagrams of the present system can be obtained.

4 114 CHEN Qiang and YAN Shi-Lei Vol. 45 Figures 1(a) 1(c) express the dependence of the Curie temperature on the transverse field values of Ω 1/2 /J corresponding to the single-ion anisotropy parameters D/J = 0.0, 0.5, and 1.0, respectively, when the values of trimodal random concentration p 1/2 in the sublattice A are changed. In Fig. 1(a), when D/J = 0.0, the range of magnetic ordering phase will enlarge with the increase of the random concentration p 1/2 in the sublattice A. The existence of the trimodal random transverse field distribution in sublattice A weakens the role of transverse field. In this case, the existence of a large transverse field will compensate for the influence of trimodal distribution. When p 1/2 = p 1/2c = 0.216, there exists a critical value of transverse field. From our calculations, value of critical transverse field Ω 1/2 /J is The second transition line extends to Ω 1/2 /J for all p 1/2 > This means that the critical temperature is still finite. In other words, the system is always magnetic ordering state at low temperatures. We introduce a small value (D/J = 0.5) of single-ion anisotropy in Fig. 1(b). We find that the magnetic ordering region is magnified at low temperatures comparing with Fig. 1(a). For example, when temperature extends to zero for p 1/2 = 0.1, the transverse value is Ω 1/2 /J = , while the transverse value is Ω 1/2 /J = in Fig. 1(a). The second order transition lines do not touch the ( Ω 1/2 /J ) axis for p 1/2 > Figure 1(c) shows that the range of magnetic ordering is depressed due to a larger single-ion anisotropy. The transverse value is Ω 1/2 /J = for p 1/2 = 0.1, while the second transition line extends to Ω 1/2 /J for p 1/2 > From the comparison between Figs. 1(a) 1(c), we find that the role of single-ion anisotropy is very unusual when there exists trimodal transverse field distribution in the sublattice A. A smaller value of single-ion anisotropy can enlarge magnetic ordering, while a larger value of single-ion anisotropy can depress magnetic ordering again. That is to say, the small one is very advantageous for magnetic ordering of system at low temperatures, while the large one is opposite. Fig. 1 Transverse field Ω 1/2 in sublattice A depends on the Curie temperature for selected values of single-ion anisotropy (a) D/J = 0.0; (b) D/J = 0.5; (c) D/J = 1.0 with various values of 0.0 p 1/2 < 1.0. Fig. 2 Transverse field Ω 1 in sublattice B depends on the Curie temperature for selected values of single-ion anisotropy (a) D/J = 0.0; (b) D/J = 0.5; (c) D/J = 1.0 with various values of 0.0 p 1 < 1.0. In Figs. 2(a) 2(c), we plot the dependence of the Curie temperature on the transverse field Ω 1 /J in sublattice B for D/J = 0.0, 0.5, and 1.0, respectively, when the values of trimodal random concentration p 1 in sublattice B are changed. We notice that the transverse value is Ω 1 /J = in Fig. 2(a) and Ω 1 /J = in Fig. 2(b) when the

5 No. 6 Critical Properties of Mixed Ising Spin System with Different Trimodal Transverse Fields in 115 temperature goes to zero for p 1 = 0.1. This indicates that the existence of a smaller single-ion anisotropy can hardly change transverse field value. Obviously, this situation is different from that of Figs. 1(a) and 1(b). The physical reason comes from the fact that there is only the single-ion anisotropy in sublattice B. Thus, the different influences of small single-ion anisotropy on phase diagrams in T -Ω 1/2 space and T -Ω 1 space are reasonable. Of course, the transverse value is Ω 1 /J = for p 1 = 0.1 and D/J = 1.0 in Fig. 2(c). This means a larger single-ion anisotropy can reduce apparently magnetic ordering region. The result in Fig. 2(c) is very similar to that depicted in Fig. 1(c). On the other hand, we notice that p 1 = in sublattice B leads to Ω 1 /J in Fig. 2(a) and p 1/2 = in sublattice A leads to Ω 1/2 /J in Fig. 1(a). This means that trimodal concentration p 1 in sublattice B affects easily the change of the magnetic ordering regions. The situation is closely related to site spin features of different sublattices A and B. This is because the increase of trimodal concentration p 1 in sublattice B turns some sites spin S = 0 state into S = +1 or S = 1 one. The Curie temperatures versus negative single-ion anisotropy for given Ω 1/2 /J = 1.5 and.0 with Ω 1 /J = 0.0 are depicted in Figs. (a) and (b), when the values of trimodal concentration p 1/2 in sublatice A are changed. For p 1/2 = 1.0, they correspond to the pure mixed Blume-Capel model. From Fig. (a), the trajectory of the TCP in T -D space is changed with the increase of trimodal concentration p 1/2. There exists the TCP for all 0 p 1/2 1.0 and the trajectory displays an unusual arc curve. There exists only the TCP in the range of < p 1/2 1.0 and the trajectory is monotonous decline in Fig. (b). The trajectory shows rich variation with trimodal concentration p 1/2 when values of transverse field Ω 1/2 are different. Of course, the TCP is suppressed completely for 0.0 p 1/ Here, we note that the reentrant phenomena can be observed apparently no matter how the transverse field Ω 1/2 of sublattice A is weak or strong. In other words, the transverse field Ω 1/2 of sublattice A does not contribute to suppression of the reentrant phenomena. Fig. Curie temperature is plotted as a function of the single-ion anisotropy with various trimodal transverse field concentrations p 1/2 in sublattice A for (a) Ω 1/2 /J = 1.5 and (b) Ω 1/2 /J =.0 in the absence of Ω 1. Fig. 4 Curie temperature is plotted as a function of the single-ion anisotropy with various trimodal transverse field concentrations p 1 in sublattice B for (a) Ω 1/J = 1.5 and (b) Ω 1/J =.0 in the absence of Ω 1/2. In Figs. 4(a) and 4(b) the Curie temperature dependence of the single-ion anisotropy is plotted for Ω 1 /J = 1.5 and.0 with Ω 1/2 /J = 0.0, when the values of trimodal concentration p 1 in sublattice B are changed. In fact, the present

6 116 CHEN Qiang and YAN Shi-Lei Vol. 45 phase diagrams have large variation by comparing with Fig.. As seen from Figs. 4(a) and 4(b), the TCP can appear in the ranges of trimodal concentration 0.80 < p and < p 1 1.0, respectively. This is to say, it is easier for Ω 1 than for Ω 1/2 to depress the TCP. We here notice that for p 1 = 0.80 and Ω 1 /J = 1.5 as well as p 1 = and Ω 1 /J =.0, the TCP just disappears completely. This means that, under trimodal distribution condition in sublattice B, a smaller value of transverse field Ω 1 can easily suppress the TCP in Fig. 4(a), while it is difficult for a larger one to suppress the TCP in Fig. 4(b). The result is very unusual and does not agree with traditional recognition. Secondly, the reentrant phenomena shall be suppressed gradually till they vanish at all (see curve p 1 = 0.0) with the decrease of trimodal concentration p 1 (the decrease of p 1 means the increase of Ω 1 ). Clearly, the role of transverse field Ω 1 in sublattice B will destroy the reentrant tendency. Therefore, the influences of transverse field Ω 1/2 or Ω 1 in different sublattices A or B on the reentrant phenomena are different. Thirdly, we find that there is an intersection in Figs. 4(a) and 4(b). The law on the Curie temperatures versus negative single-ion anisotropy is in qualitative agreement with Figs. (a) and (b) above the intersection and is opposite below the intersection. Fourthly, it is well known that, with the increase of transverse field, Curie temperature and magnetic ordering phase will be reduced. Here, our results show that when Curie temperature goes to zero, the critical values of the single-ion anisotropy extend to D/J =.59 and D/J =.814 at p 1 = 0.0, respectively. Thus, an interesting result is that although the Curie temperature with the increase of transverse field Ω 1 is reduced, the scope of magnetic ordering phase is magnified at low temperatures by comparison between Figs. 4(a) and 4(b). Hence, the role of transverse field Ω 1 and its trimodal distribution in T -D space is also curious. These variations between Figs. and 4 can be understood by a simple interpretation. The different transverse fields and trimodal distributions in sublattices A and B represent different quantum effects and quantum fluctuations. Another important factor follows from the fact that the single-ion anisotropy occurs only in sublattice B. A deeper understanding of the critical properties is of practical importance for the present system. In summary, selecting a simple cubic lattice within the EFT and making use of a decoupling approximation performs this work. We have given the effects of single-ion anisotropy, different sublattice transverse fields and the corresponding trimodal distributions on the critical properties of the mixed spin-1/2 and spin-1 Ising system. Our results show that a smaller single-ion anisotropy can magnify magnetic ordering phase for the T -Ω 1/2 space at low temperatures, while a larger one can depress magnetic ordering phase. In the T -Ω 1 space, a smaller single-ion anisotropy can hardly change the value of critical transverse field as depicted in Figs. 2(a) and 2(b). On the other hand, we have clarified that the different sublattice transverse fields and their fluctuations play a magical role in T -D space. The phase diagrams display many new results. Their behaviors are explicitly shown in Figs. and 4. The important physical ingredients are the trimodal distributions in different sublattice transverse fields and the existence of single-ion anisotropy. References [1] T. Kaneyoshi, E.F. Sarmento, and I.P. Fittipaldi, Phys. Rev. B 8 (1988) [2] G.M. Zhang and C.Z. Yang, Phys. Rev. B 48 (199) [] S.L. Yan and C.Z. Yang, Sol. Stat. Commun. 100 (1996) 851. [4] G.M. Buendia and E. Machado, Phys. Rev. B 61 (2000) [5] I. Puha and H.T. Diep, J. Magn. Magn. Mater. 224 (2001) 85. [6] S.L. Yan and C.Z. Yang, J. Magn. Magn. Mater. 27 (2001) 104. [7] X.F. Jiang, J.L. Li, J.L. Zhong, and C.Z. Yang, Phys. Rev. B 47 (199) 827. [8] S.L. Yan and C.Z. Yang, Z. Phys. B 10 (1997) 9. [9] N. Benayad, R. Zerhouni, and A. Klumper, Euro. Phys. J. B 5 (1998) 687. [10] Y.N. Zhang and S.L. Yan, Commun. Theor. Phys. (Beijing, China) 40 (200) 75. [11] T.F. Cassol, W. Figuriredo, and J.A. Plascak, Phys. Lett. A 160 (1991) 518. [12] X.M. Weng and Z.Y. Li, Phys. Stat. Sol. (b) 197 (1996) 487. [1] B. Laaboudi, M. Saber, and M. Kerouad, Phys. Stat. Sol. (b) 212 (1999) 15. [14] X. Qin and Y.Q. Ma, Commun. Theor. Phys. (Beijing, China) 4 (1999) 217. [15] T. Kaneyoshi, J. Appl. Phys. 64 (1988) [16] X.M. Weng and Z.Y. Li, Phys. Rev. B 5 (1996) [17] S.L. Yan and C.Z. Yang, Commun. Theor. Phys. (Beijing, China) 1 (1999) 51. [18] S.L. Yan and C.Z. Yang, Euro. Phys. J. B 1 (2000) 625. [19] S.L. Yan, Chin. Phys. 11 (2002) [20] T. Kaneyoshi, Z. Phys. B 71 (1988) 109.