The Jordan Structure of Two Dimensional Loop Models

TL;DR

This paper analyzes the algebraic structure of two-dimensional loop models at criticality, revealing how their transfer matrices relate to the Temperley-Lieb algebra and identifying conditions for diagonalizability and Jordan block formation.

Contribution

It introduces a link representation approach for the transfer matrix of loop models, determines eigenvalues, constructs eigenvectors using Wenzl-Jones projectors, and characterizes Jordan blocks based on boundary parameters.

Findings

Eigenvalues of the braid limit of the transfer matrix are determined.

A basis of eigenvectors is constructed using Wenzl-Jones projectors.

Jordan blocks occur under specific boundary conditions and parameter constraints.

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 $TL_N(\beta)$, 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 $a \in 2 \mathbb{Z} + 1$, 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.