How are Clifford algebras and spinors used to study the Ising model?

Question

I've heard Clifford algebras and spinors are useful tools to study the Ising model, but I've never find any good discussion on this matter. Also, as far as I know, in the original solutions of the 2-D Ising model, Onsager made use of the theory of Lie algebras. I imagine that these are equivalent approaches but how are they related?

EDIT: I realized my post have two 'close' votes, so let me try to add a few more comments.

DISCLAIMER: I don't know much about Clifford algebras and I apologize if what follows is too basic or if the objects I'm about to mention are not easily related. But to better understand the relations between the following concepts is the main purpose of my question.

In his book The method of second quantization, F. Berezin defines a Clifford algebra (or Spinor algebra) to be an algebra $K_{n}$ with generators $k_{1},...,k_{n}$ such that: \begin{eqnarray} k_{i}k_{j}+k_{j}k_{i} = 2\delta_{ij} \tag{1}\label{1} \end{eqnarray} According to Berezin, to each Grassmann algebra $\mathcal{G}_{n}$ with $n$ generators there is a closely related Clifford algebra $K_{2n}$ with duplicated number of generators.

After I posted the question I realized there are some articles/texts discussing the issue of solving the Ising model (especially the 2-D) using Clifford algebras and all the references I found seems to treat the problem in a practical way, by using the same notion of Clifford algebra introduced in Berezin's book (see, e.g. Itzykson's article). In particular, for the 2-D Ising, the generators $k_{i}$ satisfying (\ref{1}) are representations of Pauli matrices in some space. However, these are expositions which address the problem under the physicist point of view, and the underlying spaces and rigorous definitions are not made clear. In particular, I'd like to understand the connection between these objects and the following definition, which is the definition I know of a Clifford algebra as it is defined in mathematics books.

Definition: Let $V$ be a $\mathbb{K}$-vector space, $\varphi: V \times V \to \mathbb{K}$ a symmetric bilinear map and $\Phi: V \to \mathbb{K}$ the quadratic form associated to $\varphi$. A Clifford algebra $\mathcal{Cl}(V, \Phi)$ associated to $V$ and $\Phi$ is a $\mathbb{K}$-associative algebra with unit together with a linear map $i_{\Phi}:V \to \mathcal{Cl}(V,\Phi)$ such that:

(a) $(i_{\Phi}(v))^{2}=\Phi(v)\cdot 1$, $\forall v \in V$,

(b) (Universal Property) For every $\mathbb{K}$-algebra $A$ and every linear map $f: V \to A$ such that $(f(v))^{2}=\Phi(v)\cdot 1_{A}$ ($\forall v \in V$), there exists a unique $\mathbb{K}$-homomorphism $\bar{f}: \mathcal{Cl}(V,\Phi)\to A$ such that $f = \bar{f}\circ i_{\Phi}$.

The above definition is rather general and abstract and I believe Berezin's definition of $K_{n}$ is just a particular case of it when $V$ is finite dimensional. For a treatment of spin systems, I believe $V$ should be taken as $\mathbb{C}$ or something like this.

Thus, my objective with this question is to understand how the Clifford algebra approach used by physicists to study the Ising model can be put in rigorous mathematical terminology.

As a final remark, I should mention that one of the first places where this kind of analysis was made seems to be Lieb, Mattis & Schultz's paper, which is more didatically discussed in Lieb's "Models in Statistical Physics", part of this book. There one can find a more careful exposition of the topic, but for unexperienced students like myself, is still very hard to connect the dots back to the Clifford algebra. The idea is to use the tranfer matrix to motivate the definition of spaces $H_{i}$ $(i=1,...,n$) which are defined to be generated by $\binom{1}{0}$ and $\binom{0}{1}$ and set $H := H_{1}\otimes \cdots \otimes H_{n}$. Then, the Pauli matrices mentioned before become operators on $H$ which act only at one entry of it. Again, it is not clear to me the connection of these objects and the Clifford algebra as introduced before.

Answer 1

Another solution of 2D Ising model (2009) [also at arXiv:0805.0225]

The partition function of the Ising model on a two-dimensional regular lattice is calculated by using the matrix representation of a Clifford algebra (the Dirac algebra), with number of generators equal to the number of lattice sites. It is shown that the partition function over all loops in a 2D lattice including self-intersecting ones is the trace of a polynomial in terms of Dirac matrices. The polynomial is an element of the rotation group in the spinor representation. Thus, the partition function is a function of a character on an orthogonal group of a high degree in the spinor representation.

Answer 2

The section 12.2 of Huang's treatise on Statistical Mechanics (1963) gives hints of possible connections between the Ising model and Clifford algebras, but of course $\textit{under the physicist point of view}$. Below I summarize the solution of the model without magnetic field, without proof and proper mathematical definitions. This is $\textit{not}$ an answer (Ceci n'est pas une pip), but maybe you can identify some element towards a rigorous and clear mathematical formulation.

The partition function of the Ising model on a squared lattice with $n^{2}$ spins, without external field, with isotropic exchange coupling $\epsilon$, is \begin{equation} Z(\beta\epsilon)=\text{Tr}\mathsf{\:{P}}^{n}, \end{equation}

with the $2^{n}\times2^{n}$ matrix $\mathsf{{P}}$ given by

\begin{equation} \mathsf{{P}}=\left(2\sinh\left(2\beta\epsilon\right)\right)^{n/2}\left[\frac{1}{2}\left(1+\mathsf{{U}}\right)\mathsf{{V}}^{+}+\frac{1}{2}\left(1-\mathsf{{U}}\right)\mathsf{{V}}^{-}\right], \end{equation}

being $\beta$ the inverse temperature, and \begin{align} \mathsf{{V}}^{\pm} & =e^{\pm i\phi\Gamma_{1}\Gamma_{2n}}\left[\prod_{\alpha=1}^{n-1}e^{-i\phi\Gamma_{2\alpha+1}\Gamma_{2\alpha}}\right]\left[\prod_{\lambda=1}^{n}e^{-i\theta\Gamma_{2\lambda}\Gamma_{2\lambda-1}}\right]\\ \mathsf{{U}} & =i^{n}\Gamma_{1}\Gamma_{2}\cdots\Gamma_{2n}, \end{align}

where $\phi=\beta\epsilon$, $\tanh\theta=e^{-2\phi}$, and the $2n$ matrices $\Gamma_{\mu}$ $\left(\mu=1,\cdots,2n\right)$ (-of size $2^{n}\times2^{n}$) defined by the anticommutation rule \begin{equation} \Gamma_{\mu}\Gamma_{\nu}+\Gamma_{\nu}\Gamma_{\mu}=2\delta_{\mu\nu}. \end{equation}