Journal of Statistical Physics

, Volume 174, Issue 1, pp 40–55 | Cite as

Finite Temperature Phase Diagrams of the Mixed Spin-1 and Spin-2 Blume–Capel Model by Renormalization Group Approach

  • A. Lafhal
  • N. Hachem
  • H. Zahir
  • M. El BouzianiEmail author
  • M. Madani
  • A. Alrajhi


The Blume–Capel model with mixed spins S = 1 and S = 2, on d-dimensional hypercubic lattice, is studied using the real space renormalization group approximation and specifically the Migdal–Kadanoff technique. We give the phase diagrams on the (Δ1/|J|, 1/|J|) and (Δ2/|J|, 1/|J|) planes which are studied for selected values of Δ2/|J| and Δ1/|J| respectively, with first and second order phase transitions and tricritical points. Also, the associated fixed points are drawn up in a table, and by linearizing the transformation at the vicinity of these points, we determine the critical exponents for d = 2 and d = 3. In particular, we find that the system under study may exhibit reentrant phenomenon.


Blume–Capel Mixed spins Renormalization Tricritical point Critical exponent 


  1. 1.
    Blume, M.: Theory of the first-order magnetic phase change in UO2. Phys. Rev. 141, 517 (1966)ADSCrossRefGoogle Scholar
  2. 2.
    Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 32, 966 (1966)ADSCrossRefGoogle Scholar
  3. 3.
    Ez-Zahraouy, H., Kassou-Ou-Ali, A.: Phase diagrams of the spin-1 Blume–Capel film with an alternating crystal field. Phys. Rev. B 69, 064415 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Yüksel, Y., Akıncı, Ü., Polat, H.: Critical behavior and phase diagrams of a spin-1 Blume–Capel model with random crystal field interactions: an effective field theory analysis. Phys. A 391, 2819 (2012)CrossRefGoogle Scholar
  5. 5.
    Berker, A.N., Wortis, M.: Blume–Emery–Griffiths–Potts model in two dimensions: phase diagram and critical properties from a position-space renormalization group. Phys. Rev. B 14, 4946 (1976)ADSCrossRefGoogle Scholar
  6. 6.
    de Oliveira, S.M., de Oliveira, P.M.C., de Sa Barreto, F.C.: The spin-S Blume–Capel RG flow diagram. J. Stat. Phys. 78, 1619 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Malakis, A., Berker, A.N., Fytas, N.G., Papakonstantinou, T.: Universality aspects of the d = 3 random-bond Blume–Capel model. Phys. Rev. E 85, 061106 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Zierenberg, J., Fytas, N.G., Janke, W.: Parallel multicanonical study of the three-dimensional Blume–Capel model. Phys. Rev. E 91, 032126 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    Zierenberg, J., Fytas, N.G., Weigel, M., Janke, W., Malakis, A.: Scaling and universality in the phase diagram of the 2D Blume–Capel model. Eur. Phys. J. Special Topics 226, 789 (2017)ADSCrossRefGoogle Scholar
  10. 10.
    Blume, M., Emery, V.J., Griffiths, R.B.: Ising model for the λ transition and phase separation in He3–He4 mixtures. Phys. Rev. A 4, 1071 (1971)ADSCrossRefGoogle Scholar
  11. 11.
    Lajzerowicz, J., Sivardière, J.: Spin-1 lattice-gas model. I. Condensation and solidification of a simple fluid. Phys. Rev. A 11, 2079 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    Sivardière, J., Lajzerowicz, J.: Spin-1 lattice-gas model. II. Condensation and phase separation in a binary fluid. Phys. Rev. A 11, 2090 (1975)ADSCrossRefGoogle Scholar
  13. 13.
    Sivardière, J., Lajzerowicz, J.: Spin-1 lattice-gas model. III. Tricritical points in binary and ternary fluids. Phys. Rev. A 11, 2101 (1975)ADSCrossRefGoogle Scholar
  14. 14.
    Schick, M., Shih, W.H.: Spin-1 model of a microemulsion. Phys. Rev. B 34, 1797 (1986)ADSCrossRefGoogle Scholar
  15. 15.
    Newman, K.E., Dow, J.D.: Zinc-blende–diamond order-disorder transition in metastable crystalline (GaAs)1−xGe2 x alloys. Phys. Rev. B 27, 7495 (1983)ADSCrossRefGoogle Scholar
  16. 16.
    Tanaka, M., Kawabe, T.: Spin-one Ising model including biquadratic interaction with positive coupling constant. J. Phys. Soc. Jpn. 54, 2194 (1985)ADSCrossRefGoogle Scholar
  17. 17.
    Kivelson, S.A., Emery, V.J., Lin, H.Q.: Doped antiferromagnets in the weak-hopping limit. Phys. Rev. B 42, 6523 (1990)ADSCrossRefGoogle Scholar
  18. 18.
    Mathonière, C., Nuttal, C.J., Carling, S.G., Day, P.: Ferrimagnetic mixed-valency and mixed-metal tris (oxalato) iron (III) compounds: synthesis, structure, and magnetism. Inorg. Chem. 35, 1201 (1996)CrossRefGoogle Scholar
  19. 19.
    Plascak, J.A., Moreira, J.G., Sa Barreto, F.C.: Mean field solution of the general spin Blume–Capel model. Phys. Lett. A 173, 360 (1993)ADSCrossRefGoogle Scholar
  20. 20.
    Keskin, M., Canko, O., Ertaş, M.: Kinetics of the spin-2 Blume–Capel model under a time-dependent oscillating external field. J. Exper. Theo. Phys. 105, 1190 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Bahmad, L., Benyoussef, A., El Kenz, A.: Effects of a random crystal field on the spin-2 Blume–Capel model. Phys. Rev. B 76, 094412 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    Pena Lara, D., Plascak, J.A.: General spin Ising model with diluted and random crystal field in the pair approximation. Physica A 260, 443 (1998)ADSCrossRefGoogle Scholar
  23. 23.
    Canko, O., Albayrak, E.: Pair-approximation method for the quantum transverse spin-2 Ising model with a trimodal-random field. Phys. Lett. A 340, 18 (2005)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Canko, O., Albayrak, E., Keskin, M.: The quantum transverse spin-2 Ising model with a bimodal random-field in the pair approximation. J. Magn. Magn. Mater. 294, 63 (2005)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Jiang, W., Wei, G.Z., Xin, Z.H.: Phase diagrams and tricritical behavior in a spin-2 transverse Ising model with a crystal field on honeycomb lattice. J. Magn. Magn. Mater. 220, 96 (2000)ADSCrossRefGoogle Scholar
  26. 26.
    Jiang, W., Wei, G.Z., Xin, Z.H.: Phase diagrams and tricritical behavior of spin-2 Ising model with a transverse crystal field. Phys. Stat. Solid. B 221, 759 (2000)ADSCrossRefGoogle Scholar
  27. 27.
    Jiang, W., Wei, G.Z., Xin, Z.H.: Transverse Ising model with a crystal field for the spin-2. Phys. Stat. Solid. B 225, 215 (2001)ADSCrossRefGoogle Scholar
  28. 28.
    Liang, Y.Q., Wei, G.Z., Zhang, Q., Song, G.L.: Phase diagrams and tricritical behaviour of the spin-2 Ising model in a longitudinal random field. Chin. Phys. Lett. 21, 378 (2004)ADSCrossRefGoogle Scholar
  29. 29.
    Liang, Y.Q., Wei, G.Z., Song, L.L., Song, G.L., Zang, S.L.: Phase diagram and tricritical behavior of a spin-2 transverse Ising model in a random field. Commun. Theor. Phys. 42, 623 (2004)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Yigit, A., Albayrak, E.: Phase diagrams of the spin-2 Ising model in the presence of a quenched diluted crystal field distribution. Chin. Phys. B 21, 110503 (2012)CrossRefGoogle Scholar
  31. 31.
    Ertaş, M., Deviren, B., Keskin, M.: Dynamic phase transitions and dynamic phase diagrams of the spin-2 Blume–Capel model under an oscillating magnetic field within the effective-field theory. J. Magn. Magn. Mater. 324, 704 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Iwashita, T., Satou, R., Imada, T., Idogaki, T.: Magnetization and ground state spin structures of Ising spin system with biquadratic exchange interaction. Phys. B 284, 1203 (2000)ADSCrossRefGoogle Scholar
  33. 33.
    Saber, M., Tucker, J.W.: Theoretical study of the quenched diluted spin 2 Ising ferromagnet in a transverse field. Phys. A 217, 407 (1995)CrossRefGoogle Scholar
  34. 34.
    Iwashita, T., Uragami, K., Muraoka, Y., Kinoshita, T., Idogaki, T.: Monte Carlo simulations of the spin-2 Blume–Emery–Griffiths model, international conference on magnetism (ICM 2009). J. Phys. 200, 022020 (2010)Google Scholar
  35. 35.
    Jabar, A., Masrour, R., Jetto, K., Bahmad, L., Benyoussef, A., Hamedoun, M.: Monte Carlo simulations of the spin-2 Blume–Emery–Griffiths model with four-spin interactions. Superlatt. Microstruct. 100, 818 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    Erdinç, A., Canko, O., Albayrak, E.: The spin-2 antiferromagnet on the Bethe lattice. Eur. Phys. J. B 52, 521 (2006)ADSCrossRefGoogle Scholar
  37. 37.
    Hachem, N., Lafhal, A., Zahir, H., El Bouziani, M., Madani, M., Alrajhi, A.: The spin-2 Blume–Capel model by position space renormalization group. Superlatt. Microstruct. 111, 927 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    Mansuripur, M.: Magnetization reversal, coercivity, and the process of thermomagnetic recording in thin films of amorphous rare earth-transition metal alloys. J. Appl. Phys. 61, 1580 (1987)ADSCrossRefGoogle Scholar
  39. 39.
    Kahn, O.: In: Molecular Magnetism: From Molecular Assemblies to the Devices Coronado, E., Delhaès, P., Gatteschi, D., Miller, J. (eds.), Springer, Berlin (1996)Google Scholar
  40. 40.
    Weng, X.M., Li, Z.Y.: Transverse-random-field mixed Ising model with arbitrary spins. Phys. Rev. B 53, 12142 (1996)ADSCrossRefGoogle Scholar
  41. 41.
    Iwashita, T., Saton, R., Imada, T., Miyoshi, Y., Idogaki, T.: Mixed Ising spin system with higher-order spin interaction. J. Magn. Magn. Mater. 226–230, 577 (2001)CrossRefGoogle Scholar
  42. 42.
    Zhang, Q., Wei, G., Xin, Z., Liang, Y.: Effective-field theory and Monte Carlo study of a layered mixed spin-1 and spin-2 Ising system on honeycomb lattice. J. Magn. Magn. Mater. 280, 14 (2004)ADSCrossRefGoogle Scholar
  43. 43.
    Albayrak, E., Yigit, A.: The critical behavior of the mixed spin-1 and spin-2 Ising ferromagnetic system on the Bethe lattice. Phys. A 349, 471 (2005)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wie, G.Z., Gu, Y.W., Liu, J.: Mean-field and Monte Carlo studies of a mixed spin-1 and spin-2 Ising system with different anisotropies. Phys. Rev. B 74, 024422 (2006)ADSCrossRefGoogle Scholar
  45. 45.
    Čanová, L., Strečka, J., Jaščur, M.: Exact results of the mixed-spin Ising model on a decorated square lattice with two different decorating spins of integer magnitudes. Int. J. Mod. Phys. B 22, 2355 (2008)CrossRefGoogle Scholar
  46. 46.
    Deviren, B., Ertaş, M., Keskin, M.: The effective-field theory studies of critical phenomena in a mixed spin-1 and spin-2 Ising model on honeycomb and square lattices. Phys. A 389, 2036 (2010)CrossRefGoogle Scholar
  47. 47.
    Masrour, R., Jabar, A., Benyoussef, A., Hamedoun, M.: Spin-1 and -2 bilayer Bethe lattice: a Monte Carlo study. J. Magn. Magn. Mater. 401, 700 (2016)ADSCrossRefGoogle Scholar
  48. 48.
    Korkmaz, T., Temizer, Ü.: Dynamic compensation temperature in the mixed spin-1 and spin-2 Ising model in an oscillating field on alternate layers of a hexagonal lattice. J. Magn, Magn. Mater. 324, 3876 (2012)ADSCrossRefGoogle Scholar
  49. 49.
    Strečka, J., Čanová, L.: Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice. Condens. Matter Phys. 9, 179 (2006)CrossRefGoogle Scholar
  50. 50.
    Madani, M., Gaye, A., El Bouziani, M., Alrajhi, A.: Migdal–Kadanoff solution of the mixed spin-1 and spin-3/2 Blume–Capel model with different single-ion anisotropies. Phys. A 437, 396 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    El Bouziani, M., Madani, M., Gaye, A., Alrajhi, A.: Phase diagrams of the semi-infinite Blume–Capel model with mixed spins (SA = 1 and SB = 3/2) by Migdal Kadanoff renormalization group. W. J. Condens. Matt. Phys. 6, 109 (2016)ADSGoogle Scholar
  52. 52.
    Migdal, A.A.: Phase transitions in gauge and spin-lattice systems. Zh. Eksp. Teor. Fiz. 69, 1457, (1975) [Sov. Phys. JETP 42 (1975) 743]Google Scholar
  53. 53.
    Kadanoff, L.P.: Notes on Migdal’s recursion formulas. Ann. Phys. 100, 359 (1976)ADSCrossRefGoogle Scholar
  54. 54.
    Hasenbusch, M.: Finite size scaling study of lattice models in the three-dimensional Ising universality class. Phys. Rev. B 82, 174433 (2010)ADSCrossRefGoogle Scholar
  55. 55.
    Nienhuis, B., Nauenberg, M.: First-order phase transitions in renormalization-group theory. Phys. Rev. Lett. 35, 477 (1975)ADSCrossRefGoogle Scholar
  56. 56.
    Benayad, N.: Real-space renormalization group investigation of pure and disordered mixed spin Ising models on d-dimensional lattices. Z. Phys. B—Condensed Matter 81, 99 (1990)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Zaim, N., Zaim, A., Kerouad, M.: The phase diagrams of a spin 1/2 core and a spin 1 shell nanoparticle with a disordered interface. Superlatt. Microstruct. 100, 490 (2016)ADSCrossRefGoogle Scholar
  58. 58.
    El Antari, A., Zahir, H., Hasnaoui, A., Hachem, N., Alrajhi, A., Madani, M., El Bouziani, M.: Mixed spin-1/2 and spin-5/2 model by renormalization group theory: recursion equations and thermodynamic study. Int. J. Theor. Phys. 57, 2330 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zouhair, S., Monkade, M., Bourass, M., El Antari, A., El Bouziani, M., Madani, M., Alrajhi, A.: Random crystal field in a mixed spin S = 1/2 and S = 3/2 Ising model by renormalization group theory. J. Supercond. Nov. Magn. (2018). Google Scholar
  60. 60.
    Binder, K., Young, A.P.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801 (1986)ADSCrossRefGoogle Scholar
  61. 61.
    Kimura, T., Kumai, R., Tokura, Y., Li, J.Q., Matsui, Y.: Successive structural transitions coupled with magnetotransport properties in LaSr2Mn2O7. Phys. Rev. B 58, 11081 (1998)ADSCrossRefGoogle Scholar
  62. 62.
    Sata, T., Yamaguchi, T., Matsusaki, K.: Interaction between anionic polyelectrolytes and anion exchange membranes and change in membrane properties. J. Membr. Sci. 100, 229 (1995)CrossRefGoogle Scholar
  63. 63.
    Hui, K.: Reentrant behavior of an in-plane antiferromagnet in a magnetic field. Phys. Rev. B 38, 802 (1988)ADSCrossRefGoogle Scholar
  64. 64.
    Jaščur, M., Strečka, J.: Reentrant transitions of a mixed-spin Ising model on the diced lattice. Condens. Matter Phys. 8, 869 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Equipe de Physique Théorique Laboratoire L.P.M.C, Département de Physique, Faculté des SciencesUniversité Chouaib DoukkaliEl JadidaMorocco
  2. 2.Département de Physique-ChimieCRMEFMeknèsMorocco
  3. 3.Laboratoire LS3M, Faculté PolydisciplinaireUniversité Hassan IKhouribgaMorocco
  4. 4.Department of Information Technologies, Faculty of Education and Applied SciencesHodeidah UniversityRaymahYemen

Personalised recommendations