The Z_6-symmetric model partition function on triangular lattice




Statistical mechanics, Z_Q-symmetric model, partition function, triangular lattice, phase transition


There is a study on a square lattice that can predict the existence of multiple phase transitions on a complex plane. We extend the study on the different types of ZQ-symmetric model and different lattices in order to provide more evidence to the existence of multiple phase transitions. We focus on the ZQ-symmetric model with the nearest neighbour interaction on the six spin directions between molecular dipole, i.e. Q = 6 on a triangular lattice. Mainly, the model is defined on the triangular lattice graph with the nearest neighbour interaction. By using the transfer matrix approach, the partition functions are computed for increasing lattice sizes. The roots of polynomial partition function are also computed and plotted in the complex Argand plane. The specific heat equation is used for further comparison. The result supports the existence of the multiple phase transitions by the emergence of the multiple line curves in the locus of zeros distribution for specific type of energy level.

Author Biographies

Nor Sakinah Mohd Manshur, International Islamic University Malaysia

Computational and Theoretical Sciences Department, Kulliyyah of Sciences

Siti Fatimah Zakaria, International Islamic University Malaysia

Assistant Professor, Computational and Theoretical Sciences Department, Kulliyyah of Sciences

Nasir Ganikhodjaev, International Islamic University Malaysia

Professor Dr, Computational and Theoretical Sciences Department, Kulliyyah of Sciences


P. Martin, Potts models and related problems in statistical mechanics, vol. 5, World Scientific Publishing Company, Teaneck N. J., 1991.

E. Ising, Beitrag zur theorie des ferromagnetismus, Zeitschrift für Phys., vol. 31, no. 1, pp. 253–258, 1925.

F. Zakaria and S. Manshur, The zeros distribution of Z_5-symmetric model on a triangular lattice, 2019. Submitted to Malaysian Journal of Fundamental and Applied Sciences.

B. Kaufman, Crystal statistics. II. Partition function evaluated by Spinor analysis, Phys. Rev., vol. 76, no. 8, pp. 1232–1243, 1949.

L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev., vol. 65, no. 3–4, p. 117, 1944.

R. J. Baxter, Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model: II, J. Stat. Phys., vol. 149, no. 6, pp. 1164–1167, 2012.

M. E. Fisher, The nature of critical points. University of Colorado Press, 1965.

F. Zakaria, Analytic properties of Potts and Ising model partition functions and the relationship between analytic properties and phase transitions in equilibrium statistical mechanics, Doctoral dissertation, University of Leeds, United Kingdom, 2016.

P. P. Martin and S. F. Zakaria, Zeros of the 3-state Potts model partition function for the square lattice revisited, J. Stat. Mech. Theory Exp. vol. 2019, no. 8, p. 84003, 2019.

S. Ono, Y. Karaki, M. Suzuki, and C. Kawabata, Statistical mechanics of three-dimensional finite Ising model, Phys. Lett. A, vol. 24, no. 12, pp. 703–704, 1967.

S. Katsura, Distribution of roots of the partition function in the complex temperature plane, Prog. Theor. Phys., vol. 38, no. 6, pp. 1415–1417, 1967.

R. Abe, Logarithmic singularity of specific heat near the transition point in the Ising model, Prog. Theor. Phys., vol. 37, no. 6, pp. 1070–1079, 1967.

R. Diestel, Graph Theory. 3rd Edition, Springer, 2006.

R. Diestel, Graph Theory, Electronic Edition 2000 (vol. 173), Springer-Verlag, New York, 2000.

R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press Limited, London, 1982.

K. Huang, Statistical mechanics, 2nd edition, John Wiley & Sons Ltd, Canada, p. 493, 1987.

A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes. McGraw-Hill, 2001.

J. M. Yeomans, Statistical mechanics of phase transitions. Clarendon Press, United Kingdom, 1992.

C.-N. Chen, C.-K. Hu, and F. Y. Wu, Partition function zeros of the square lattice Potts model, Phys. Rev. Lett., vol. 76, no. 2, p. 169, 1996.