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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.

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  • $\begingroup$ Clifford algebras and spinors generally are certainly not equivalent to Lie algebras in any sense that I know, but any time that they occur should make you think of the $\mathsf D_n$ series of Lie algebras, especially $\mathsf D_4$, and maybe also the $\mathsf B_n$ series. I don't know whether these Lie algebras occur in any direct way in connection to the Ising model; Zhang - Clifford algebra approach of Ising model discusses Jordan algebras, which suggest a connection to exceptional groups (themselves 2 $\endgroup$
    – LSpice
    Jan 2, 2021 at 16:16
  • $\begingroup$ 1 connected to $\mathsf D_4$); see Octonions and the standard model (11th in the series, with links to previous) for a recent discussion of physics connections to Jordan algebras and the octonions, although not to the Ising model in this case. $\endgroup$
    – LSpice
    Jan 2, 2021 at 16:17
  • $\begingroup$ My comment just addresses the relationship between the mathematical definition of Clifford algebra that you quote and the physicist's definition as in Berezin's. Assuming the mathematical definition one has to construct one such algebra, which by the universal property is then unique (up to isomorphism, of course). Thus, for a vector space $V$ over any field (one may assume that the field is not of characteristic 2, to be able to use the polarization identiies) consider its tensor algebra $TV$ (the direct sum of all its tensor powers) and assume that $V$ has a symmetric bilinear form $b$ . . . $\endgroup$
    – F Zaldivar
    Jan 3, 2021 at 19:33
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    $\begingroup$ and consider the two-sided ideal ${\mathcal I}(V,b)\subseteq TV$ generated by the elements of the form $u\otimes v +v\otimes u-2b(u,v)$ for $u,v\in V$. The quotient $\text{Cl}(V,b)=TV/{\mathcal I}(V,b)$ is the Clifford algebra as one can easily verify. In the physicist's version, it is assumed that $V$ has an orthogonal basis with respect to the bilinear form $b$ and you get Berenzin's description of the corresponding Clifford algebra. $\endgroup$
    – F Zaldivar
    Jan 3, 2021 at 19:37
  • $\begingroup$ @FZaldivar thanks for the comment! What you said is really enlightening! I think we shall demand $V$ to be finite-dimensional too in order to have Berezin's description, right? Because there we have a finite number of generators. Also, according to Berezin, $k_{n}$ are generators because every element of the Clifford algebra $f \in K_{n}$ can be written as $f = \sum_{l=0}^{n}\alpha_{i_{1},...,i_{l}}(k_{i_{1}}\cdots k_{i_{l}})$, where $\alpha_{i_{1},...,i_{l}}$ are numbers. Does this representation follow from your construction? I think I'd need a bit more work there, right? $\endgroup$
    – IamWill
    Jan 5, 2021 at 18:11

2 Answers 2

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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.

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    $\begingroup$ Does this paper describe how the Clifford-algebraic/spinorial approach relates to the Lie-algebraic approach? (I don't know anything about either approach.) $\endgroup$
    – LSpice
    Jan 2, 2021 at 17:16
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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}

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