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.