The Jordan Structure of Two Dimensional Loop Models

Abstract

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 $\beta =2\cos (\pi (1-a/b)),a,b\in \mathbb{N}$ 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 $b\in {\mathbb{Z}}^{\ast }$ 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 $d^{\prime }$ when $d-d^{\prime }<2b$ and $(d+d^{\prime })/2\equiv b-1\text{mod}2b$ ($d>d^{\prime }$). When $a$ and $b$ do not satisfy the previous constraint, $D_N$ is diagonalizable.

55 pages

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