Question

Update: I provided an answer of my own (reflecting the things I discovered since I asked the question). But there is still lot to be added. I'd love to hear about other people's opinions on the solutions and relations among them. In particular short, intuitive descriptions of the methods used. Come on, the bounty awaits ;-)

Now, this might look like a question into the history of Ising model but actually it's about physics. In particular I want to learn about all the approaches to Ising model that have in some way or other relation to Onsager's solution.

Also, I am asking three questions at once but they are all very related so I thought it's better to put them under one umbrella. If you think it would be better to split please let me know.

When reading articles and listening to lectures one often encounters so called Onsager's solution. This is obviously very famous, a first case of a complete solution of a microscopic system that exhibits phase transition. So it is pretty surprising that each time I hear about it, the derivation is (at least ostensibly) completely different.

To be more precise and give some examples:

The most common approach seems to be through computation of eigenvalues of some transfer matrix.
There are few approaches through Peierl's contour model. This can then be reformulated in terms of a model of cycles on edges and then one can either proceed by cluster expansion or again by some matrix calculations.

The solutions differ in what type of matrix they use and also whether or not they employ Fourier transform.

Now, my questions (or rather requests) are:

Try to give another example of an approach that might be called Onsager's solution (you can also include variations of the ones I already mentioned).
Are all of these approaches really that different? Argue why or why not some (or better yet, all) of them could happen to be equivalent.
What approach did Onsager actually take in his original paper. In other words, which of the numerous Onsager's solutions is actually the Onsager's solution.

For 3.: I looked at the paper for a bit and I am a little perplexed. On the one hand it looks like it might be related to transfer matrix but on the other hand it talks about quaternion algebras. Now that might be just a quirk of Onsager's approach to 4x4 matrices that pop basically in every other solution but I'll need some time to understand it; so any help is appreciated.

Answer 1

I wish I could do your question justice, but I will content myself with a remark on the connection between two of the solution methods mentioned in Barry McCoy's article, namely Baxter's commuting transfer matrix method, and Onsager's original algebraic approach.

In a certain sense, these methods have to be considered distinct since Baxter's method is applicable to a vast family of additional models, whereas Onsager's method applies only to Ising and closely related models. A related fact is that, while the free energy and order parameter have been computed for a great many two-dimensional models, only for Ising are the correlation functions completely understood. (They can be written in terms of simple determinants.) Among solvable two-dimensional models, Ising appears to be very special. It lies at the intersection of many infinite families of models. Although all solvable lattice models have lots of unexpected structure - in particular, they have infinitely many conserved quantities - Ising is even more special. Onsager's original method of solution exploited some of this special structure - in particular, the direct-product structure of the transfer matrices.

Since Baxter's commuting transfer matrix method does not exploit this special structure, it can be used to solve the many other models that don't have it. His method uses the Yang-Baxter relation to establish that the transfer matrices commute for different values of the spectral parameter (which, in the Ising model, parametrizes the difference between horizontal and vertical coupling strengths). Since the eigenvectors must therefore be independent of the spectral parameter, one can derive functional relations for the eigenvalues, which can then be solved.

Onsager's method was expanded upon by Dolan and Grady, who showed that a certain set of commutation relations implies the existence of an infinite set of conservation laws. In the 1980s, a solvable n-state generalization of the Ising model, known as the superintegrable chiral Potts model, was discovered that satisfies Dolan and Grady's conditions and, as a consequence, has transfer matrices with the same direct-product structure that Onsager exploited in 1944. Interestingly, the superintegrable chiral Potts model corresponds to a special point in a one-parameter family of solvable models, the integrable chiral Potts models. The latter are solvable by Baxter's method, but can be solved by Onsager's method only at the superintegrable point. There seems to be a lot of work going on currently on the correlation functions of the superintegrable chiral Potts model.

The other solution methods that Barry McCoy mentions in his Scholarpedia article - Kaufman's free fermions, the combinatorial method, Baxter and Enting's 399th solution - also seem to make use of the particular structure of the Ising model. In this sense, they are more akin to Onsager's original method than to Baxter's commuting-transfer-matrix method. As you have already suggested, there may be some equivalences among them, but I would have to give this more study before commenting further.