JAMP  Vol.6 No.5 , May 2018
On the Solution of One-Dimensional Ising Models
Abstract: In this paper it is shown that the thermodynamic limit of the partition function of the statistical models under consideration on a one-dimensional lattice with an arbitrary finite number of interacting neighbors is expressed in terms of the principal eigenvalue of a matrix of finite size. The high sparseness of these matrices for any number of interactions makes it possible to perform an effective numerical analysis of the macro characteristics of these models.
Cite this paper: N. Kharchenko, Y. (2018) On the Solution of One-Dimensional Ising Models. Journal of Applied Mathematics and Physics, 6, 960-967. doi: 10.4236/jamp.2018.65082.

[1]   Cambardella, P., Dallmeyer, A., Maiti, K., Malagoli, M., Eberhard, W., Karn, K. and Carbone, C. (2002) Ferromagnetism in One-Dimensional Monoatomic Metal Chains. Nature, 416, 301-304.

[2]   Andriushchenko, P.D. and Nefedev, K.V. (2013) Magnetic Phase Transition in the Lattice Ising Model. Advanced Material Research, 718, 166-171.

[3]   Baxter, R.J. (1982) Exactly Solved Models in Statistical Mechanics. Academic Press, New York and London.

[4]   Dmitriev, A.A., Katrakhov, V.V. and Kharchenko, Yu.N. (2004) Root Transfer Matrices in Ising Models. Nauka, Moscow. [In Russian]

[5]   Horn, R. and Johnson, C. (1986) Matrix Analysis. Cambridge University Press, London.

[6]   Katrakhov, V.V. and Kharchenko, Yu.N. (2006) Two-Dimensional Four-Line Models of the Ising Model Type. Theoretical and Mathematical Physics, 149, 1545-1558.

[7]   Arnalds, U.B., Cyico, J., Stopfel, Y., Kapaklis, V., Barenbold, O., Verschuren, M.A., Volff, U., Neu, V., Bergman, A. and Hjorvarsson, B. (2016) A New Look on the Two-Dimensional Ising Model: Thermal Artificial Spins. New Journal of Physics, 18, Article ID: 023008.

[8]   Fan, Y. (2011) One-Dimensional Ising Model with k-Spin Interactions. European Journal of Physics, 32, 1643.