OPEN PROBLEMS TREATED BUT NOT PUBLISHED IN ISING THESIS (DISCUSSION)

Reinhard Folk

Institute for Theoretical Physics, University Linz, Austria

In his thesis [1] Ernst Ising solved the problem of a chain of elementary magnets in an external magnetic field and showed that there is no phase transition when the magnetic field is set to zero. Goal of the thesis in fact was to proof the existence of the ferromagnetic phase. In order to do so he extended his model for the chain (1) by allowing more than two orientations of the elementary magnets at one place, (2) by considering two chains (the Ising ladder [2]) and (3) by including next-nearest neighbor interaction in addition to the nearest neighbor interaction [3]. It should be noted that in order to calculate the magnetization he had to include into his calculations of the free energy (partition function) an external magnetic field. It is tried to discuss the results he obtained without knowledge of spin, transfer matrix, duality etc. in comparison with present results.

References:

[1] E. Ising, Beitrag zur Theorie des Ferro- und Paramagnetismus. Dissertati- on zur Erlangung der Doktorwürde der Mathematisch-Naturwissenschaftlichen Fakultät der Hamburgischen (Universität vorgelegt von Ernst Ising aus Bo- chum, Hamburg, 1924), see: http://www.icmps.lviv.ua/ising/books.html. 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. A complete translation by B. Berche, Yu. Ho- lovatch, R. Kenna and the author is in preparation. The publication of E. Ising results is found in Beitrag zur Theorie des Ferromagnetismus Zeitschr. f. Phys. 31: 253 - 258 (1925)
[2] Weiguo Yin, Frustration-driven unconventional phase transitions at finite temperature in a one-dimensional ladder Ising model (2020) arXiv:2006.08921v2 Gao Xing-Ru, Ising Model on an Infinite Ladder Lattice Commun. Theor. Phys. 48, 553 (2007) R. Mejdani, et al. Ladder Ising spin configurations ii. magnetic properties. physica status solidi (b) 197, 153 - 164 (1996).
[3] Farid Taherkhani, et al. On the existence of an analytic solution to the 1-D Ising model with nearest and next-nearest neighbor interactions in the presence of a magnetic field, Phase Transitions, 84/1, 77 - 84 (2011)

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