Ising’s Roots and the Transfer-Matrix Eigenvalues
Abstract
:1. Introduction
2. Ising’s Method and the Solution for a Two-State Model
2.1. Definition of States, Configurations, and Energy Places
2.2. Ising’s Solution of the Two-State Model: An Auxiliary Function
2.3. Reformulation of the Ising Problem with the Hamiltonian and Transfer Matrix for the Two-State Model
3. Ising’s Solution of the Three-State Model
4. Transfer-Matrix Formulation for Ising’s Three-State Model
5. Conclusions and Further Developments
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Schottky’s Idea and Transverse States
References
- Ising, E. Beitrag zur Theorie des Ferro- und Paramagnetismus; Dissertation zur Erlangung der Doktorwürde der Mathematisch- Naturwissenschaftlichen Fakultät der Hamburgischen Universität Vorgelegt von Ernst Ising aus Bochum. Hamburg 1924. An Excerpt of the Thesis “Contribution to the Theory of Ferromagnetism” Translated by Jane Ising and Tom Cummings Can Be Found on the Webpage of the Bibliotheca Augustina. Available online: http://www.icmp.lviv.ua/ising/books.html (accessed on 9 May 2024).
- Ising, E. Beitrag zur Theorie des Ferromagnetismus. Z. Physik 1925, 31, 253–258. [Google Scholar] [CrossRef]
- Pauli, W. Les théories quantiques du magnétisme: L’électron magnétique. In Rapports et Discussions du Sixième Conseil de Physique tenu à Bruxelles du 20 au 25 Octobre 1930; Gautirer-Villars: Paris, France, 1932; pp. 175–280. [Google Scholar]
- Bitter, F. Introduction to Ferromagnetism; McGraw-Hill: New York, NY, USA, 1937. [Google Scholar]
- Kramers, H.A.; Wannier, G.H. Statistics of the Two-Dimensional Ferromagnet. Part I. Phys. Rev. 1941, 60, 252. [Google Scholar]
- Kramers, H.A.; Wannier, G.H. Statistics of the Two-Dimensional Ferromagnet. Part II. Phys. Rev. 1941, 60, 263. [Google Scholar]
- Montroll, E. Statistical Mechanics of Nearest Neighbor Systems. J. Chem. Phys. 1941, 9, 706–721. [Google Scholar] [CrossRef]
- Potts, R.B. Some Generalized Order-Disorder Transformations. Proc. Camb. Philos. Soc. 1952, 48, 106. [Google Scholar] [CrossRef]
- Wu, F.Y. The Potts Model. Rev. Mod. Phys. 1982, 54, 235–268. [Google Scholar] [CrossRef]
- Brush, S.G. History of the Lenz-Ising Model. Rev. Mod. Phys. 1967, 39, 883–893. [Google Scholar] [CrossRef]
- Kobe, S. Ernst Ising (1900–1998). Braz. J. Phys. 2000, 30, 649–653. [Google Scholar] [CrossRef]
- Niss, M. History of the Lenz-Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena. Arch. Hist. Exact Sci. 2005, 59, 267–318. [Google Scholar] [CrossRef]
- Folk, R. The Survival of Ernst Ising and the Struggle to Solve His Model. In Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory; Holovatch, Y., Ed.; World Scientific: Singapore, 2022; Volume 7, pp. 1–77. [Google Scholar]
- Chang, T.S.; Ho, C.C. Arrangements with given number of neighbours. Proc. Royal Soc. Lond. Ser. A 1942, 180, 345–365. [Google Scholar]
- Stanley, H.E. Introduction to Phase Transitions and Critical Phenomena (International Series of Monographs on Physics); Oxford University Press: Oxford, UK, 1987; 336p. [Google Scholar]
- Heisenberg, W. Zur Theorie des Ferromagnetismus. Zeitschr. Phys. 1928, 49, 619–636. [Google Scholar] [CrossRef]
- McCoy, B.M.; Wu, T.T. The Two-Dimensional Ising Model, 2nd ed.; Dover Publications: New York, NY, USA, 2001. [Google Scholar]
- Domb, C. Configurational studies of Potts models. J. Phys. A 1974, 7, 1335–1348. [Google Scholar] [CrossRef]
- Yeomans, J.M. Statistical Mechanics of Phase Transitions; Clarendon Press: Oxford, UK, 1992. [Google Scholar]
- Blume, M. Theory of the first-order magnetic phase change in UO2. Phys. Rev. 1966, 141, 517–524. [Google Scholar] [CrossRef]
- Capel, H.W. On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 1966, 32, 966–988. [Google Scholar] [CrossRef]
- Blume, M.; Emery, V.J.; Griffiths, R.B. Ising model for the λ-transition and phase separation in He3–He4 mixtures. Phys. Rev. A 1971, 4, 1071–1077. [Google Scholar] [CrossRef]
- Moueddene, L.; Fytas, N.G.; Holovatch, Y.; Kenna, R.; Berche, B. Critical and tricritical singularities from small-scale Monte Carlo simulations: The Blume-Capel model in two dimensions. J. Stat. Mech. Theory Exp. 2024, 2024, 023206. [Google Scholar] [CrossRef]
- Fisher, M.E. Scaling, Universality and RenormalizationGroup Theory. In Critical Phenomena; Proceedings of the Summer School Held at the University of Stellenbosch, South Africa, 18–29 January 1982; Lecture Notes in Physics; Hahne, F.J.W., Ed.; Springer: Berlin/Heidelberg, Germany, 1983; Volume 186. [Google Scholar]
- Krasnytska, M.; Sarkanych, P.; Berche, B.; Holovatch, Y.; Kenna, R. Potts Model with Invisible States: A Review. Eur. J. Phys. Spec. Top. 2023, 232, 1681–1691. [Google Scholar] [CrossRef]
- Sarkanych, P.; Holovatch, Y.; Kenna, R. Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states. Phys. Lett. A 2017, 381, 3589–3593. [Google Scholar] [CrossRef]
- Sarkanych, P.; Holovatch, Y.; Kenna, R. Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields. J. Phys. A 2018, 51, 505001. [Google Scholar]
- Folk, R.; Holovatch, Y. Schottky’s forgotten step to the Ising model. Eur. Phys. J. H 2022, 47, 9. [Google Scholar] [CrossRef]
- Schottky, W. Über die Drehung der Atomachsen in festen Körpern. (Mit magnetischen, thermischen und chemischen Beziehungen). Physik. Zeitschr. 1922, 23, 448–455. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Folk, R.; Holovatch, Y. Ising’s Roots and the Transfer-Matrix Eigenvalues. Entropy 2024, 26, 459. https://doi.org/10.3390/e26060459
Folk R, Holovatch Y. Ising’s Roots and the Transfer-Matrix Eigenvalues. Entropy. 2024; 26(6):459. https://doi.org/10.3390/e26060459
Chicago/Turabian StyleFolk, Reinhard, and Yurij Holovatch. 2024. "Ising’s Roots and the Transfer-Matrix Eigenvalues" Entropy 26, no. 6: 459. https://doi.org/10.3390/e26060459