On the critical behavior of the spin-s Ising model

Authors

  • Magdy Amin Faculty of Science - Minia University
  • M. Mubark Faculty of Science, Minia University
  • Y. Amin Faculty of Science, Minia University

DOI:

https://doi.org/10.31349/RevMexFis.69.021701

Keywords:

Ising model; exact solution; transfer matrix method; thermodynamic and magnetic properties

Abstract

The spin-s one dimensional Ising model is studied analytically within the framework of transfer matrix method. Exact results for some thermodynamical properties such as the internal energy, the entropy, the magnetization and the magnetic susceptibility are obtained for general spin-s in the absence (presence) of a magnetic field. The critical behavior of the thermodynamical properties are analysed for some values of spin-s (1/2, 1 and 3/2) at different temperature and field. The asymptotic behavior of these properties are investigated especially close to the critical temperature T → 0 and when T → ∞.

References

E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik 31 (1925) 253.

L. Onsager, Crystal statistics. I. A two-dimensional model with an orderdisorder transition, Phys. Rev. 65 (1944) 117.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 (1953) 1087.

M. E. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).

A. F. Sonsin, M. R. Cortes, D. R. Nunes, J. V. Gomes, and R. S. Costa, Analysis of 3D Ising model using Metropolis algorithms, J. Phys.: Conf. Ser. 630 (2015) 012057.

M. Blume, Theory of the first-order magnetic phase change inUO2, Phys. Rev. 141 (1966) 517.

H. W. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting, Physica 32 (1966) 966.

M. Blume, V. J. Emery and R. B. Griffiths, Ising model for the λ transition and phase separation in He3-He4 mixtures, Phys. Rev. A4 (1971) 1071.

T. Horiguchi, A spin-one Ising model on a honeycomb lattice, Phys. Lett. A 113 (1986) 425.

F. Y. Wu, On Horiguchi’s solution of the Blume-EmeryGriffiths model, Phys. Lett. A 116 (1986) 245.

T. Kaneyoshi, Contribution to the theory of spin-1 Ising models, J. Phys. Soc. Japan 56 (1987) 933.

T. Kaneyoshi, The phase transition of the spin-one Ising model with a random crystal field, J. Phys. C: Solid State Phys. 21 (1988) L679.

A. Rosengren and R. Haggkvist, Rigorous solution of a twodimensional Blume-Emery-Griffiths model, Phys. Rev. Lett. 63 (1989) 660.

M. Kolesik and L. Samaj, Solvable cases of the general spinone Ising model on the honeycomb lattice, Int. J. Mod. Phys. B 6 (1992) 1529.

A. Lipowski and M. Suzuki, On the exact solution of twodimensional spin S Ising models A, Physica A 193 (1993) 141.

T. Morita, Exactly Solved Two-Dimensional Ising Model with Spin S Greater than 1/2, J. Phys. Soc. Jpn 62 (1993) 4218.

N. Sh. Izmailian and N. S. Ananikian, General spin- 3 2 Ising model in a honeycomb lattice: Exactly solvable case, Phys. Rev. B 50 (1994) 6829.

T. Horiguchi and Y. Honda, Spin-3/2 Ising Model and AshkinTeller Model, Prog. Theor. Phys. 93 (1995) 981.

Q. Zhang, G.-Z. Wei, and Y.-Q. Liang, Phase diagrams and tricritical behavior in spin-1 Ising model with biaxial crystal-field on honeycomb lattice, J. Magn. Magn. Mat. 253 (2002) 45.

H. Ez-Zahraouy and A. Kassou-Ou-Ali, Phase diagrams of the spin-1 Blume-Capel film with an alternating crystal field, Phys. Rev. B 69 (2004) 064415.

R. Zivieri, Critical behavior of the classical spin-1 Ising model for magnetic systems, AIP Advances 12 (2022) 035326.

W. Heisenberg, Zur Theorie des Ferromagnetismus. SS. 619- 636, Zeitschrift fur Physik (in German) 49 (1928) 619.

I. Affleck, Field Theory Methods and Quantum Critical Phenomena (Les Houches lectures 1988), in Fields, strings and critical phenomena, eds. E. Brezin and J. Zinn-Justin (North-Holland, 1990).

N. D. Mermin and H. Wagner, Absence of ferromagnetism or anti-ferromagnetism in one- or two-dimensional Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133.

P. C. Hohenberg, Existence of long-range order in one- and two-dimensions, Phys. Rev. 158 (1976) 383.

B. E. Halperin, On the Hohenberg-Mermin-Wagner Theorem and Its Limitations, J. Stat. Phys. 175 (2018) 521.

R. Zivieri, Absence of Spontaneous Spin Symmetry Breaking in 1D and 2D Quantum Ferromagnetic Systems with Bilinear and Biquadratic Exchange Interactions, Symmetry 12 (2020) 2061.

J -H Chen, M E Fisher and B G Nickel, Unbiased Estimation of Corrections to Scaling by Partial Differential Approximants, Phys. Rev. Lett. 48 (1982) 630.

A M Ferrenberg and D P Landau, Critical behavior of the threedimensional Ising model: A high-resolution Monte Carlo study, Phys. Rev. B 44 (1991) 5081.

H W Blote, E Luijten and J R Heringa, Ising universality in three dimensions: a Monte Carlo study, J. Phys. A 28 (1995) 6289.

S Seth, Combinatorial approach to exactly solve the 1D Ising model, Eur. J. Phys. 38 (2017) 015104.

H A Kramers and G H Wannier, Statistics of the TwoDimensional Ferromagnet. Part I, Phys. Rev. 60 (1941) 252.

F A Kassan-Ogly, One-dimensional ising model with nextnearest-neighbour interaction in magnetic field, Phase Transition 74 (2001) 353.

O. Pujol and J. P. Perez, A synthetic approach to the transfer matrix method in classical and quantum physics, Eur. J. Phys. 28 (2007) 679.

T. Guidi et al., Direct observation of finite size effects in chains of antiferromagnetically coupled spins, Nat. Commun 6 (2015) 7061.

W. Wang, R. Diaz-Mendez and R. Capdevila, Solving the onedimensional Ising chain via mathematical induction: an intuitive approach to the transfer matrix, Eur J. Phys. 40 (2019) 065102.

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

T Antal, M Droz and Z Racz, Probability distribution of magnetization in the one-dimensional Ising model: effects of boundary conditions, J. Phys. A 37 (2004) 1465.

A. Proshkin, F. Kassan-Ogly, A. Zarubin, T. Ponomareva and I. Menshikh, Critical phenomena in 1D Ising model with arbitrary spin, EPJ Web of Conferences 185 (2018) 03004.

M. E. Amin, M. Mubarak and Y. Amin, Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions, Z. Naturforsch A 75 (2020) 175.

Downloads

Published

2023-03-01

How to Cite

[1]
M. Amin, M. Mubark, and Y. Amin, “On the critical behavior of the spin-s Ising model”, Rev. Mex. Fís., vol. 69, no. 2 Mar-Apr, pp. 021701 1–, Mar. 2023.

Issue

Section

17 Thermodynamics and Statistical Physics