Relative position between a pair of spin model subfactors, I

In this article, we investigate relative position between a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$ arising from two complex Hadamard matrices of order $2$ as well as order $4$. More precisely, we characterize when the two subfactors are equal, compute the Pimsner-Popa probabilistic constant and the Connes-St{\o}rmer relative entropy between them. To the best of our knowledge, this article is the first instance in the literature that the exact value of the Connes-St{\o}rmer relative entropy for a pair of (non-trivial) subfactors has been obtained. We construct en route a family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\in\mathbb{N}$. As a corollary, a rigidity of the angle between the two subfactors is established. Finally, as a pleasant application of the relative entropy, we characterize when the pair of spin model subfactors form a commuting square.