How many Onsager's solutions are there?

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 ;-)

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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.

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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:

1. The most common approach seems to be through computation of eigenvalues of some transfer matrix.
2. 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:

1. 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).
2. 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.
3. 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.

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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 am about halfway the most important part of Onsager's paper, so I'll try to summarize what I've understood so far, I'll edit later when I have more to say.

Onsager starts by using the 1D model to illustrate his methodology and fix some notations, so I'm gonna follow him but I'll use some more "modern" notations.

In the 1D Ising model, only neighbouring spins interact, therefore, the energy of interactions is represented by

$$E=-J\mu^{(k)}\mu^{(k-1)}$$

where $J$ is the interaction strength.

The partition function is

$$Z = \sum_{\mu^{(1)},\ldots,\mu^{(N)}=\pm 1} e^{-\sum_k J\mu^{(k)}\mu^{(k-1)}/kT}$$

Onsager notes that the exponential can be seen as a matrix component:

$$\langle \mu^{(k-1)}| V | \mu^{(k)} \rangle = e^{-J\mu^{(k)}\mu^{(k-1)}/kT}$$

The partition sum becomes the trace of a matrix product in this notation

$$Z = \sum_{\mu^{(1)},\mu^{(N)}=\pm 1} \langle \mu^{(1)}| V^{N-1} | \mu^{(N)} \rangle$$

So for large powers $N$ of $V$, the largest eigenvalue will dominate. In this case, $V$ is just a $2\times 2$ matrix and the largest eigenvalue is $2\cosh(J/kT)=2\cosh(H)$, introducing $H=J/kT$.

Now, to construct the 2D Ising model, Onsager proposes to build it by adding a 1D chain to another 1D chain, and then repeat the procedure to obtain the full 2D model.

First, he notes that the energy of the newly added chain $\mu$ will depend on the chain $\mu'$ to which it is added as follows:

$$E = -\sum_{j=1}^n J \mu_j \mu'_j $$

But if we exponentiate this to go to the partition formula, we get the $n$th power of the matrixwe defined previously, so using notation that Onsager introduced there

$$ V_1 = (2 \sinh(2H))^{n/2} \exp(H^{*}B)$$

with $H^{*}=\tanh^{-1}(e^{-2H})$ and $B=\sum_j C_j$ with $C_j$ the matrix operator that works on a chain as follows

$$C_j |\mu_1,\ldots,\mu_j,\ldots,\mu_n \rangle = |\mu_1,\ldots,-\mu_j,\ldots,\mu_n \rangle $$

Then, to account for the energy contribution from spins within a chain, he notes that the total energy is

$$E = -J' \sum_{j=1}^n \mu_j\mu_{j+1}$$

adding periodicity, that is the $n$th atom is a neighbor to the 1st. Also note that the interaction strength should not be equal to the interchain interaction strength. He introduces new matrix operators $s_j$ which act on a chain as

$$s_j|\mu_1,\ldots,\mu_j,\ldots,\mu_n \rangle = \mu_j |\mu_1,\ldots,\mu_j,\ldots,\mu_n \rangle $$

and in this way constructs a matrix

$$V_2 = \exp(H'A) = \exp(H'\sum_j s_j s_{j+1})$$

Now, the 2D model can be constructed by adding a chain through application of $V_1$ and then define the internal interactions by using $V_2$. So one gets the following chain of operations

$$\cdots V_2 V_1 V_2 V_1 V_2 V_1 V_2 V_1 V_2 V_1$$

It is thus clear that the matrix to be analyzed in our 2D model is $V=V_2 V_1$. This is our new eigenvalue problem:

$$\lambda | \mu_1,\ldots,\mu_n \rangle = \exp(H'\sum_j s_j s_{j+1}) \sum_{\mu'_1,\ldots,\mu'_n=\pm 1} \exp(H\sum_j \mu_j \mu'_{j})| \mu'_1,\ldots,\mu'_n \rangle$$

Now, the quaternions come into play. Onsager notes that the operators $s_j$ and $C_j$ he constructed form a quaternion algebra.

Basically, the basis elements $(1,s_j,C_j,s_jC_j)$ generate the quaternions and since for different $j$'s the operators commute, we have a tensor product of quaternions, thus a quaternion algebra.

-- To be continued --

Answer 2

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.

Answer 3

Seeing as no one is trying to give an answer, I'll take a stab at it myself.

Shortly after writing this question I learned (in this cute answer of Raskolnikov's) about Baxter's wonderful book on exact solutions in statistical mechanics. Slowly but surely I realized that Ising model has been solved so many times by some many different methods by virtually every famous physicist (I'll list some of the solutions later ) that it became clear that my question is hopefully inadequate and only reflects my huge ignorance.

To make up for that, I started reading papers. The Onsagar's paper itself came out in 1944. In 1949 there appeared Bruria Kaufman's paper where she notes that the transfer matrix can be interpreted as $2^n$-dimensional representation of $2n$-dimensional rotations. So she introduces spinor analysis (e.g. Pauli and Dirac matrices) and goes ahead to solve the problem. I must say I am in love with this approach (okay, you got me, I am a group person).

In 1952 Kac and Ward used a purely combinatorial method of some polygons (which I don't yet quite understand, but it probably has to do with Peierl's contours). Other papers note the duality with free fermionic field. Or note that Ising is just a special case of Random Cluster Model; or a dimer model. These papers carry names (in no particular order) such as Potts, Ward, Kac, Kasteleyn, Yang, Baxter, Fisher, Montroll and others. It's quite obvious that it will take me some time to understand (or indeed, even read) all those papers.

So I took a different road and used asked google. Querying all the names above at once returns precious gems:

1. Amazing article over at Scholarpedia. It contains historical treatment, main methods of solution, references to the papers I mentioned and much much more.
2. paper History of Lenz-Ising model
3. paper Spontaneous magnetization of the Ising model

Answer 4

Not to state the obvious, but it seems the information at the scholaropaedia article which @marek mentioned in his answer, is more comprehensive than any answer I or anyone else is likely to come up with.

To quote this article "there are five different methods which have been used to compute the free energy of the Ising model". For details best check out the link above. Anything more I add will just be repetition.

As for the bounty, it should go to Barry McCoy - the author of the scholaropaedia article ;)