Bond Dilution Effects on Bethe Lattice the Spin-1 Blume Capel Model

Transcription

1 Commun. Theor. Phys. 68 (2017) Vol. 68, No. 3, September 1, 2017 Bond Dilution Effects on Bethe Lattice the Spin-1 Blume Capel Model Erhan Albayrak Erciyes University, Department of Physics, 38039, Kayseri, Turkey (Received April 5, 2017; revised manuscript received June 20, 2017) Abstract The bond dilution effects are investigated for the spin-1 Blume Capel model on the Bethe lattice by using the exact recursion relations. The bilinear interaction parameter is either turned on ferromagnetically with probability p or turned off with probability 1 p between the nearest-neighbor spins. The thermal variations of the order-parameters are studied in detail to obtain the phase diagrams on the possible planes spanned by the temperature (T ), probability (p) and crystal field (D) for the coordination numbers q = 3, 4, and 6. The lines of the second-order phase transitions, T c -lines, combined with the first-order ones, T t -lines, at the tricritical points (TCP) are always found for any p and q on the (T, D)-planes. It is also found that the model gives only T c -lines, T c -lines combined with the T t -lines at the TCP s and only T t -lines with the consecutively decreasing values of D on the (T, p)-planes for all q. PACS numbers: q, Rh, Cn DOI: / /68/3/361 Key words: bond dilution, spin-1, Bethe lattice, bilinear interaction, Blume Capel 1 Introduction The inclusion of any kind randomness into the spin systems can have significant effects on the critical properties, which may lead to many interesting phenomena such as the occurrence of a new ordered phase, possible changes in the universality class, the order of phase transitions, reentrant behavior, etc. They may be achieved by introducing randomness into the single ion anisotropy (D), magnetic field (h), bilinear (J) and biquadratic (K) exchange interactions and so on. Especially J is important in which the bimodal random distribution of ferromagnetic (F) and antiferromagnetic (AF) interactions leads to the spin glass phase, see Ref. [1] and the references therein. The other case of which may include the bond dilution in which J is either turned on or off randomly throughout the lattice. Thus, the investigation of bond dilution is the subject of this study. After the introduction of the Blume Capel (BC) model by Blume [2] and independently by Capel, [3] it has evolved to many different forms and many different techniques were employed to study them. Some of the common techniques that are used to study the spin-1 BC model may be given as follows: The mean-field approximation, [4] the effective-field theory, [5] the cluster variation method, [6] the cluster expansion method, [7] a real-space renormalizationgroup approximation, [8] the Monte Carlo simulation, [9] using a celular automation, [10] using an improved pair approximation within the framework of Bogoliubov variational approach for the free energy, [11] on the Voronoi Delaunay random lattices, [12] using the Wang Landau method, [13] linear chain approximation [14] and so on. In addition to these, the effects of bond dilution was also considered in a few works such as: The ferromagnetic properties of spin-1 system were considered in the albayrak@erciyes.edu.tr c 2017 Chinese Physical Society and IOP Publishing Ltd frame of bond dilution and random positive or negative anisotropy Blume Capel (BC) model in the effective field theory (EFT) and a cutting approximation. [15] The bond diluted BC model with random crystal field was studied in the framework of the EFT and expressions of magnetizations for honeycomb and square lattices were derived. [16] The critical properties of a random transverse crystal field Ising model with bond dilution were studied on a square lattice under both weak and strong bond dilution conditions. [17] A bond diluted spin-1 Ising model with crystal and transverse field interactions was examined for honeycomb lattice by introducing an effective field approximation that takes into account the correlations between different spins that emerge when expanding the identities. [18] Similarly, the ferrimagnetic properties of spin 1/2 and spin-1 system was studied by means of the EFT for the bond diluted mixed BC model with random single-ion anisotropy. [19] The bond diluted Ising model was studied by Monte Carlo method in which the simulation is carried out on a two-dimensional square lattice with missing bonds and free boundary conditions. [20] The study for critical concentration of mixed-bond Ising model with competitive interactions was considered by using a differential operator technique in the EFT. [21] The last work considers a quenched disordered Ising model with a distribution function that includes dilution and random bonds through the mean-field renormalization-group approach. [22] It should be mentioned that in Refs. [15 16] both random J z and D z effects, in Ref. [17] both random J z and D x effects, in Ref. [18] the transverse magnetic field effects were considered, so the details of the bond dilution in the spin-1 system with uniaxial crystal fields were not considered in detail. Therefore in this work, the bond dilution for the spin- 1 BC model is studied in terms of the exact recursion

2 362 Communications in Theoretical Physics Vol. 68 relations on the Bethe lattice (BL) for given coordination number q = 3, 4 and 6. The phase diagrams are calculated exactly on the (T, p) and (T, D) planes for given D and p values, respectively. The remaining part of this work is set up as follows: The formulation in terms of the exact recursion relations are obtained on the BL. The last section contains our results, i.e. the phase diagrams, discussions and a brief summary. with k > 0, which counts the shells and N k is the number of the spins in the k-th shell. Even if the BL is an artificial tree, it serves to be a fantastic tool easy to adapt to different problems and its results are exact. 2 Formulation The Hamiltonian for the spin-1 BC model is given as H = J ij S i S j D Si 2, (1) ij i where S i is spin-1 with the values of ±1 and 0, D is the crystal field and J ij is the site-dependent bilinear interaction parameter between the nearest-neighbor (NN) spins, which is either turned on ferromagnetically (F) (J > 0) or turned off (J = 0). The random distribution of J ij in the case of bond dilution is given as P (J ij ) = pδ(j ij J) + (1 p)δ(j ij ), (2) in which J ij is either turned on feromagnetically with probability p or turned off with probability 1 p throughout the lattice. The model reduces to the regular BC model for p = 1, the spins are only under the effect of D when p = 0 and it becomes bond diluted model when 0 < p < 1. Since we use a fictitious tree, i.e. Bethe lattice as shown in Fig. 1, we should say a few words about it. It consists of a central spin S 0, which is called the first generation spin. S 0 has q NN s, which form the second generation spins S 1. Each spins in the second generation is joined to (q 1) NN s. Therefore, in total, the second generation has [q(q 1)] NN s, which form the third generation spins S 2 and so on to infinity. Each spin generations is called as the shell of the BL and the number of generations goes to infinity in thermodynamic limit. Thus the Bethe lattice may be described by the formula such as N k = q(q 1) k 1 Fig. 1 The BL of coordination number q=3. The filled circles denote the spins and the dotted circles represent the shells of the BL. The phase diagrams are usually obtained by studying the thermal variations of the order-parameters, i.e. magnetization and quadrupolar moment. They are going to be obtained in here in terms of the exact recursion relations. In order to do so, one usually starts with the partition function, which is given as Z = e βh = P (Spc), (3) Spc All Config. where P (Spc) can be thought of as an unnormalized probability distribution and β = 1/(kT ), k is the Boltzmann constant, which is set equal to 1 for convenience. The formulation of the spin-1 BC model has already given in terms of recursion relations, [1] therefore, they are not going to be obtained in here. They are calculated as the ratios of partial partition functions, g n s, and are obtained as X (ij) n Y (ij) n = g n(+1) g n (0) = g n( 1) g n (0) = eβ(jij+d) [X (ij) n 1 ]q 1 + e β( Jij+D) [Y (ij) e βd [X (ij) n 1 ]q 1 + e βd [Y (ij) = eβ( Jij+D) [X (ij) n 1 ]q 1 + e β(jij+d) [Y (ij) e βd [X (ij) n 1 ]q 1 + e βd [Y (ij),. (4) Note that one has to carry out an averaging procedure over the bilinear interaction distribution P (J ij ) to get the correct recursion relations for the bond diluted model, i.e. X n = X n (ij) P (J ij )dj ij = X n (ij) [pδ(j ij J) + (1 p)δ(j ij )]dj ij, Y n = Y n (ij) P (J ij )dj ij = Y n (ij) [pδ(j ij J) + (1 p)δ(j ij )]dj ij. (5) Since all the sites of the BL are equivalent deep inside, one can pick a central spin, S 0, and calculate its order parameters accordingly. Therefore, the order-parameters, i.e. magnetization and quadrupolar moment, are given respectively in terms of the recursion relations as M = S 0 = eβd [X q n] e βd [Y q n ] e βd [X q n] + e βd [Y q n ] + 1, (6)

3 No. 3 Communications in Theoretical Physics 363 Q = S 2 0 = eβd [X q n] + e βd [Y q n ] e βd [X q n] + e βd [Y q n ] + 1. (7) The free energy of the system in terms of exact recursion relations may also be needed to classify the type of phase transitions, i.e. whether it is the second- or firstorder, and it is calculated as F = 1 β {ln[ e βd (X q n + Y q n ) + 1] + q 2 q ln[ e βd (X q 1 n 1 + Y q 1 n 1 ) + 1]}. (8) One should study the thermal changes of the orderparameters and free energy to calculate the phase diagrams of the model. This is carried out as follows: after the calculating the numerical values of the recursion relations by using an iteration procedure, their obtained values are inserted into the definitions of our thermodynamic functions, i.e. the order-parameters and free energy, to obtain their thermal variations for given D, q and p. Thus, in the next section we present the phase diagrams of the model in addition to our results, discussions and comparisons whenever possible. 3 Phase Diagrams of the Model After the detailed study of the thermal variations of the order parameters, the phase diagrams are calculated on the (T, D)-planes for given p and on the (T, p)-planes for given D when q = 3, 4, and 6. In the phase diagrams, the solid lines indicate the lines of the second-order phase transitions, T c -lines, and the dashed lines correspond to the lines of the first-order phase transitions, T t -lines. The gray circles show the places of the TCP s where the T c - and T t -lines coincide. These lines separate the F and paramagnetic phase (P) regions from each other. Figures 2(a) 2(c) show the phase diagrams on the (T, D)-planes for given p values when q = 3, 4, and 6, respectively. The phase transition lines are always consist of T c -lines merged with the T t -lines at the TCP s. The temperature of which decrease as p decreases in agreement with Refs. [15 16, 18]. It is clear that as q increases the temperature of the phase lines increase as in Ref. [16]. In addition, it seems that the TCP s stay on straight lines for each values of q. They move towards right to higher D s and decrease in temperature as p decreases. We can also compare our phase lines with the references but it is only possible when p = 1, which just corresponds to the BC model. Even if the shapes and temperatures are different, they are all very similar, i.e. the T c and T c -lines meet at the TCP s. Fig. 2 The phase diagrams on the (T, D) planes for given values of p for the coordination numbers (a) q = 3, (b) q = 4, and (c) q = 6 where the solid and dashed lines are the second- and first-order phase lines and the circles are tricritical points. The regions below each line are F phase regions and above each line are the P phase regions for each q.

4 364 Communications in Theoretical Physics Vol. 68 The next figures, i.e. Figs. 3(a) 3(f), show the phase diagrams on the (T, p)-planes for given D values. When D = 0.0, we see that the T c -lines are almost straight lines except around p = 0 where they abruptly tend to zero for each q. When D = 0.5, we still see the almost straight T c -lines, but now they combine with T t -lines at TCP s for each q at very close temperatures to each other. They then also abruptly go to zero. The D = 1.0 case is similar with the previous case, but now the separation of the TCP s are clear, which move towards right further for small q, i.e. the T t -lines become longer. For D = 1.5, we still see the T c -lines combined with the T t -lines at the TCP s for q = 4 and 6, but now only a T t -line is seen for q=3. When D = 2.0, the T c -line combined with the T t -line at the TCP is only seen for q = 6, only T t -line is seen for q=4 and the lines disappear for q = 3. Finally, for D = 3.0, only a T t -line is left for q = 6. For further negative values of D, our model does not give anymore phase transitions. It should be noted that these phase diagrams can not be compared with the references given here but, of course, in full agreement with Figs. 2. Fig. 3 The phase diagrams on the (T, p) planes for given crystal field D values for each q when (a) D = 0.0, (b) D = 0.5, (c) D = 1.0, (d) D = 1.5, (e) D = 2.0 and (f) D = 3.0. In conclusion, we have studied the bond dilution effects for the spin-1 BC model on the Bethe lattice by using the exact recursion relations. The bilinear interaction parameter between the NN spins is either turned on ferromagnetically with the probability p or turned off with 1 p, respectively. The thermal variations of the order-parameters are studied

5 No. 3 Communications in Theoretical Physics 365 in detail to obtain the phase diagrams on the possible planes for given values of the coordination numbers q = 3, 4 and 6. The T c -lines combined with T t -lines at TCP is always found for any p and q on the (T, D)-planes. The model either gives only T c -lines, T c -lines combined with the T t -lines at the TCP s and only T t -lines on the (T, p)-planes with the decreasing values of D, respectively, for any q. References [1] E. Albayrak, J. Magn. Magn. Mater. 18 (2014) 355. [2] M. Blume, Phys. Rev. 141 (1966) 517. [3] H. W. Capel, Physica (Utrecht) 32 (1966) 966. [4] Y. L. Wang and J. D. Kimel, J. Appl. Phys. 69 (1991) 6167; M. Badehdah, S. Bekhechi, A. Benyoussef, and M. Touzani, Eur. Phys. J. B 4 (1998) 431; E. Albayrak and M. Keskin, J. Magn. Magn. Mater. 206 (1999) 83. [5] M. Jaščur and T. Kaneyoshi, Phys. Stat. Sol. B 174 (1992) 537; L. Liu and S. L. Yan, Commun. Theor. Phys. 44 (2005) 743; Y. F. Zhang and S. L. Yan, Phys. Lett. A 372 (2008) [6] C. Buzano and A. Pelizzola, Physica A 216 (1995) 158; T. Balcerzak and J. W. Tucker, J. Magn. Magn. Mater. 278 (2004) 87. [7] V. Ilkoviç, Phys. Stat. Sol. B 192 (1995) K7-K10; M. Jurcisin, A. Bobák, and M. Jaščur, Physica A 224 (1996) 684. [8] N. S. Branco and B. M. Boechat, Phys. Rev. B 56 (1997) [9] D. P. Lara and J. A. Plascak, Int. J. Mod. Phys. B 12 (1998) 2045; Y. Yüksel, Ü. Akinci, and H. Polat, Phys. Scr. 79 (2009) ; A. Jabar, N. Tahiri, K. Jetto, and L. Bahmad, Superlattices Microstruct. 104 (2017) 46. [10] B. Kutlu, Int. J. Mod. Phys. C 12 (2001) [11] D. C. Carvalho and J. A. Plascak, Physica 432 (2015) 240. [12] F. P. Fernandes, D. F. de Albuquerque, F. W. S. Lima, and J. A. Plascak, Phys. Rev. E 92 (2015) [13] W. Kwak, J. Jeong, J. Lee, and D. H. Kim, Phys. Rev. E 92 (2015) [14] E. Albayrak and M. Keskin, J. Magn. Magn. Mater. 213 (2000) 201. [15] H. X. Zhu and S. L. Yan, Commun. Theor. Phys. 789 (2004) 42. [16] S. L. Yan and L. L. Deng, Physica A 301 (2002) 308. [17] W. Ling and S. L. Yan, Cent. Eur. J. Phys (2011) 9. [18] Ü. Akinci, Y. Yüksel, and H. Polat, Physica A 541 (2011) 390. [19] L. Liu and S. L. Yan, Commun. Theor. Phys. 743 (2005) 44. [20] I. Zergoug, R. Bouamrane, and D. Addi, EPJ Web of Conferences 44 (2013) [21] J. B. Santos-Filho, D. F. de Albuquerque, and N. O. Moreno, Rev. Mex. Fis. S 210 (2012) 58. [22] E. Mina, A. Bohorquez, L. E. Zamora, and G. A. Perez Alcazar, Phys. Rev. B 7925 (1993) 47.