Jones proposed the study of two subfactors of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. The Pimsner-Popa probabilistic constant, Sano-Watatani angle, interior and exterior angle, and Connes-St{\o}rmer relative entropy (along with a slight variant of it) are a few key invariants for pair of subfactors that analyze their relative position. In practice, however, the explicit computation of these invariants is often difficult. In this article, we provide an in-depth analysis of a special class of two subfactors, namely a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$. We first characterize when two distinct $n\times n$ complex Hadamard matrices give rise to distinct spin model subfactors. Then, a detailed investigation has been carried out for pairs of (Hadamard equivalent) complex Hadamard matrices of order $2\times 2$ as well as Hadamard inequivalent complex Hadamard matrices of order $4\times 4$. To the best of our knowledge, this article is the first instance in the literature where the exact value of the Pimsner-Popa probabilistic constant and the noncommutative relative entropy for pairs of (non-trivial) subfactors have been obtained. Furthermore, we prove the factoriality of the intersection of the corresponding pair of subfactors using the `commuting square technique'. En route, we construct an infinite family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\geq 2$. As a consequence, the rigidity of the interior angle between the spin model subfactors is established. Last but not least, we explicitly compute the Sano-Watatani angle between the spin model subfactors.

翻译：Jones 提出将$II_1$因子中的两个子因子视为希尔伯特空间中两个闭子空间的一种量子化。Pimsner-Popa概率常数、Sano-Watatani角、内角与外角，以及Connes-Størmer相对熵（及其微小变体）是分析子因子对相对位置的几个关键不变量。然而在实践中，这些不变量的显式计算往往较为困难。本文对一类特殊的两个子因子——即超有限$II_1$因子$R$中的自旋模型子因子对——进行了深入分析。我们首先刻画了两个不同的$n\times n$复Hadamard矩阵何时会产生不同的自旋模型子因子。随后，针对$2\times 2$阶（Hadamard等价）复Hadamard矩阵对以及$4\times 4$阶Hadamard非等价复Hadamard矩阵对，开展了详尽研究。据我们所知，本文是文献中首次获得（非平凡）子因子对的Pimsner-Popa概率常数和非交换相对熵精确值的工作。此外，我们利用“交换平方技巧”证明了相应子因子对交的因子性。在此过程中，我们构造了$R$的一个无限族潜在新子因子。所有这些子因子都是不可约的，且Jones指标为$4n,n\geq 2$。由此，自旋模型子因子之间内角的刚性得以确立。最后但同样重要的是，我们显式计算了自旋模型子因子之间的Sano-Watatani角。