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 --

This post imported from StackExchange Physics at 2014-04-01 16:36 (UCT), posted by SE-user Raskolnikov