SIMULATIONS OF CRITICAL PROPERTIES OF CLASSICAL SPIN MODELS
Kozłowski P., Pawlicki P., Kamieniarz Grzegorz
Computational Physics Division Institute of Physics, A. Mickiewicz University,
ul. Umultowska 85, PL 61-624 Poznań, Poland
e-mail: gjk@pearl.amu.edu.pl
DOI: 10.12921/cmst.1997.03.01.39-54
OAI: oai:lib.psnc.pl:481
Abstract:
Two methods of simulations are described: an exact one – transfer matrix
technique and a statistical one – Monte Carlo method. Both of them are
applied to investigate critical properties of classical spin models. To do this
we also exploit finite size scaling and the critical point ratio of the square
of the second moment of the order parameter to its fourth moment. General
definition of a classical spin model as well as particular definitions of
models are presented. Results of both methods are in good agreement and,
moreover, they are consistent with numerical results provided by literature.
References:
[1] J. Ashkin and E. Teller, Phys. Rev. 64 (1943), 178.
[2] C. Fan, Phys. Lett. 3 9 A (1972), 136.
[3] P. Pawlicki, J. Rogiers Physica A 214 (1995), 277.
[4] M. Blum Phys. Rev. 141 (1966), 517.
[5] H. W. Capel, Physica (Utr.) 32 (1966), 966; 33 (1967), 295; 37 (1967), 423.
[6] H. E. Stanley Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford 1971.
[7] G. Kamieniarz, P. Kozłowski and R. Dekeyser, Phys.Rev. E 55 (1997), 3724.
[8] G. Kamieniarz and H. W. J. Blöte, J. Phys. A 26 (1993), 201.
[9] H. W. J. Blöte and G. Kamieniarz, Physica A 196 (1993), 455.
[10] G. Kamieniarz, P. Kozłowski and R. Dekeyser, Acta Magnetica, XI 1996/1997.
[11] K. Binder, Z. Phys. B 43 (1981), 119.
[12] A. Milchev, D. W. Heermann, and K. Binder, J. Stat. Phys. 44 (1986), 749.
[13] K. Binder and D. P. Landau, Phys. Rev. Lett. 52 (1984), 318.
[14] K. K. Kaski, K. Binder, and J. D. Gunton, Phys. Rev. B 29 (1984), 3996.
[15] M. N. Barber, R. B. Pearson, D. Toussaint and J. L. Richardson, Phys. Rev. B 32 (1985), 1720.
[16] D. P. Landau, J. Magn. Magn. Mater. 31 (1983), 11.
[17] H. W. J. Blöte, E. Luijten and J. R. Herringa, J. Phys. A 28 (1995), 6289.
[18] D. P. Landau and K. Binder, Phys. Rev. B 31 (1985), 5946.
[19] K. Binder, M. Nauenberg, V. Privman and A. P. Young, Phys. Rev. B 31 (1985), 1498.
[20] R. N. Bhatt and A. P. Young; Phys. Rev. Lett. 54 (1985), 924.
[21] G. Kamieniarz, P. Pawlicki, L. Dębski, and H.W.J. Blöte, Acta Magnetica XI 1996/1997 .
[22] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, J. Chem. Phys. 21 (1953), 1087.
[23] R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982).
[24] R. V. Ditzian, J. R. Banavar, G. S. Grest and L. P. Kadanoff, Phys. Rev. B 22 (1980), 2542.
[25] J. Chahine, J. R. Drugowich de Felicio and N. Caticha, J. Phys. A 22 (1989), 1639.
[26] G. Mazzeo, E. Carlon and H. van Beijeren, Phys. Rev. Lett. 74 (1995), 1391.
[27] P. D. Beale, Phys. Rev. B 33 (1986), 1717.
[28] T. W. Burkhardt, Phys. Rev. B 14 (1967), 1196;
T. W. Burkhardt and H. J. F. Knops, Phys.Rev. B 15 (1977), 1602.
[29] N. B. Wilding and P. Nielaba, Phys. Rev. E 53 (1996), 926.
[30] H. W. J. Blöte and M. P. M. de Nijs, Phys. Rev. B 37 (1988), 1766.
Two methods of simulations are described: an exact one – transfer matrix
technique and a statistical one – Monte Carlo method. Both of them are
applied to investigate critical properties of classical spin models. To do this
we also exploit finite size scaling and the critical point ratio of the square
of the second moment of the order parameter to its fourth moment. General
definition of a classical spin model as well as particular definitions of
models are presented. Results of both methods are in good agreement and,
moreover, they are consistent with numerical results provided by literature.
[1] J. Ashkin and E. Teller, Phys. Rev. 64 (1943), 178.
[2] C. Fan, Phys. Lett. 3 9 A (1972), 136.
[3] P. Pawlicki, J. Rogiers Physica A 214 (1995), 277.
[4] M. Blum Phys. Rev. 141 (1966), 517.
[5] H. W. Capel, Physica (Utr.) 32 (1966), 966; 33 (1967), 295; 37 (1967), 423.
[6] H. E. Stanley Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford 1971.
[7] G. Kamieniarz, P. Kozłowski and R. Dekeyser, Phys.Rev. E 55 (1997), 3724.
[8] G. Kamieniarz and H. W. J. Blöte, J. Phys. A 26 (1993), 201.
[9] H. W. J. Blöte and G. Kamieniarz, Physica A 196 (1993), 455.
[10] G. Kamieniarz, P. Kozłowski and R. Dekeyser, Acta Magnetica, XI 1996/1997.
[11] K. Binder, Z. Phys. B 43 (1981), 119.
[12] A. Milchev, D. W. Heermann, and K. Binder, J. Stat. Phys. 44 (1986), 749.
[13] K. Binder and D. P. Landau, Phys. Rev. Lett. 52 (1984), 318.
[14] K. K. Kaski, K. Binder, and J. D. Gunton, Phys. Rev. B 29 (1984), 3996.
[15] M. N. Barber, R. B. Pearson, D. Toussaint and J. L. Richardson, Phys. Rev. B 32 (1985), 1720.
[16] D. P. Landau, J. Magn. Magn. Mater. 31 (1983), 11.
[17] H. W. J. Blöte, E. Luijten and J. R. Herringa, J. Phys. A 28 (1995), 6289.
[18] D. P. Landau and K. Binder, Phys. Rev. B 31 (1985), 5946.
[19] K. Binder, M. Nauenberg, V. Privman and A. P. Young, Phys. Rev. B 31 (1985), 1498.
[20] R. N. Bhatt and A. P. Young; Phys. Rev. Lett. 54 (1985), 924.
[21] G. Kamieniarz, P. Pawlicki, L. Dębski, and H.W.J. Blöte, Acta Magnetica XI 1996/1997 .
[22] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, J. Chem. Phys. 21 (1953), 1087.
[23] R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982).
[24] R. V. Ditzian, J. R. Banavar, G. S. Grest and L. P. Kadanoff, Phys. Rev. B 22 (1980), 2542.
[25] J. Chahine, J. R. Drugowich de Felicio and N. Caticha, J. Phys. A 22 (1989), 1639.
[26] G. Mazzeo, E. Carlon and H. van Beijeren, Phys. Rev. Lett. 74 (1995), 1391.
[27] P. D. Beale, Phys. Rev. B 33 (1986), 1717.
[28] T. W. Burkhardt, Phys. Rev. B 14 (1967), 1196;
T. W. Burkhardt and H. J. F. Knops, Phys.Rev. B 15 (1977), 1602.
[29] N. B. Wilding and P. Nielaba, Phys. Rev. E 53 (1996), 926.
[30] H. W. J. Blöte and M. P. M. de Nijs, Phys. Rev. B 37 (1988), 1766.
