Reinhard Folk photo

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

Reinhard Folk (Personal webpage)

Institute for Theoretical Physics, University Linz, Austria

In his thesis [1] Ernst Ising solved the problem of a chain of elementarymagnets in an external magnetic field and showed that there is no phasetransition when the magnetic field is set to zero. Goal of the thesis in factwas to proof the existence of the ferromagnetic phase. In order to do so heextended his model for the chain (1) by allowing more than two orientationsof the elementary magnets at one place, (2) by considering two chains (theIsing ladder [2]) and (3) by including next-nearest neighbor interaction inaddition to the nearest neighbor interaction [3]. It should be noted that inorder to calculate the magnetization he had to include into his calculationsof the free energy (partition function) an external magnetic field. It is triedto 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 Bochum, 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. Holovatch, 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).