Physics of Scale Activities

1944
The Onsager solution to the Ising model
The canonical paper offering an exact solution to the two- dimensional Ising model of a ferromagnet. This is usually taken as a moment of crisis for classical mean field theory, since the specific heat for the Onsager solution was logarithmically infinite rather than merely discontinuously finite, and the partition function was non- analytic at the critical temperature. Though a virtuoso calculation, the Onsager solution was based on a very abstract model far from laboratory measurement. Extending it to a physical three dimensions seemed out of the question at the time, but for obvious reasons it became the gold standard for comparison with non-classical examples of critical behavior. Perturbation series expansions by Domb, Sykes, Fisher, and others subsequently became an especially fruitful domain of calculation in which reference to the Onsager solution was crucial to developing adequate models. One might also ask, what was the relation of the Onsager solution to the eventual development of non-perturbative approaches to critical phenomena? Where and when did the Onsager solution take on heuristic significance of the sort it could not claim when it was first published?
--Karl Hall
 
Onsager solution: sources
Primary:
L. Onsager, "Crystal statistics. I. A two-dimensional model with an order-disorder transition," Phys. Rev. 65 (1944): 117-149.


Fisher lists reviews on related mathematical techniques developed in the wake of the Onsager solution:


Newell and Montroll, Rev. Mod. Phys. 25 (1953): 352.
Domb, Advanc. Phys. 9 (1960), nos. 34, 35.
Dykhne and Rumer, Soviet Physics Uspekhi 4 (1962): 698; Uspekhi fiz. nauk 75 (1961): 101.
Montroll, Applied Combinatorial Mathematics (1964)
Green and Hurst, Order-Disorder Phenomena (1964)
Mattis, Theory of Magnetism (1965)


It would be of some interest to ascertain which physicists found which texts useful when first learning how to calculate quantities of interest to the theory of critical phenomena. Where did you learn your Onsager? Or better: how did you pick up the techniques necessary to master the content of the original paper?


Secondary:
Brush 1983, 244.
Domb 1996, 129.
Fisher 1967, 667.
Hoddeson et al. 1992, 530.

-- Karl Hall-February 25, 2002