二维Loop模型的Jordan结构
The Jordan Structure of Two Dimensional Loop Models
摘要：
我们展示了如何使用环模型的转移矩阵$D_N$在晶格上的链接表示计算配分函数，在临界状态下，具有各种边界条件和参数$β= 2 \cos(π(1-a/b)), a,b\in \mathbb N$的Fortuin-Kasteleyn模型，更具体地说，配分函数的相应的$Q$ -Potts自旋模型，与$Q=β^2$。辫子极限的$D_N$被证明是一个中心元素$F_N(β)$的Temperley-Lieb代数$TL_N(β)$，它的特征值被确定，并为通用的$β$，其特征向量的基础上构造使用Wenzl-Jones投影。与该基的任何元素相关联的是多个缺陷$d$、$0\le d\le N$，并且具有相同$d$的基向量跨越扇区。因为当$b \in \mathbb{Z}^*$和$a \in 2 \mathbb{Z} + 1$时这些特征向量的分量是奇异的，所以当$d-d' < 2b$和$(d+d')/2 \equiv b-1 \ \textrm{mod} \ 2b$（$d>d'$）时，$F_N$和$D_N$的链路表示被示出为在扇区$d$和$d'$之间具有Jordan块。当$a$和$b$不满足前面的约束时，$D_N$是可对角化的。
英文摘要：
We show how to use the link representation of the transfer matrix $D_N$ of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter $β= 2 \cos(π(1-a/b)), a,b\in \mathbb N$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q=β^2$. The braid limit of $D_N$ is shown to be a central element $F_N(β)$ of the Temperley-Lieb algebra $TL_N(β)$, its eigenvalues are determined and, for generic $β$, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. To any element of this basis is associated a number of defects $d$, $0\le d\le N$, and the basis vectors with the same $d$ span a sector. Because components of these eigenvectors are singular when $b \in \mathbb{Z}^*$ and $a \in 2 \mathbb{Z} + 1$, the link representations of $F_N$ and $D_N$ are shown to have Jordan blocks between sectors $d$ and $d'$ when $d-d' < 2b$ and $(d+d')/2 \equiv b-1 \ \textrm{mod} \ 2b$ ($d>d'$). When $a$ and $b$ do not satisfy the previous constraint, $D_N$ is diagonalizable.
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