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
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