The Jordan Structure of Two Dimensional Loop Models

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 $\beta =2\cos (\pi (1-a\mathrm{/}b)),a,b\in \mathbb{N}$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q={\beta }^2$. The braid limit of $D_N$ is shown to be a central element $F_N(\beta )$ of the Temperley-Lieb algebra $T{L}_N(\beta )$, its eigenvalues are determined and, for generic $\beta$, 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$,thelinkrepresentationsof$F_N$and$D_N$areshowntohaveJordanblocksbetweensectors$d$and$d’$when$d-d’ < 2b$and$(d+d’)/2 \equiv b-1 \ $\text{mod}$ \ 2b$($d>d’$)\mathrm{.}When$a$and$b$donotsatisfythepreviousconstraint,$D_N$ is diagonalizable.