TITLE: What's the relation between spin model for subfactors theory and physics?
QUESTION [6 upvotes]: In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset &w\Delta w^*\end{matrix}$$
where $\Delta\subset M_n(\mathbb{C})$ is the algebra of diagonal matrices and
where $w\in M_n(\mathbb{C})$ is a complex Hadamard matrix, i.e. a unitary all
whose entries are of modulus $n^{-1/2}$ (Source: this paper of T. Banica. See also JS chapter 6).    
In the sense of physics, a spin model is  a mathematical model used to explain magnetism (see its wiki page).
Question: What's the relation between them?

REPLY [2 votes]: I am not familiar with spin models in the sense of subfactors, but my guess is the link between the two is conformal quantum field theory. The scaling limit of a critical spin model (in the sense of physics) usually gives a CQFT. On the other hand there is also a correspondence between subfactors and CQFTs. I think there is work by Wassermann for constructing CQFTs out of subfactors. V. Jones also has some conjectures about all subfactors arising from CQFTs in some way.