Jordan cells of periodic loop models

TL;DR

This paper investigates Jordan cells in transfer matrices of periodic loop models, revealing their connection to logarithmic conformal field theories and providing criteria for their occurrence across sectors.

Contribution

It introduces criteria for the existence of Jordan cells in the transfer matrix of periodic loop models, linking algebraic structures to conformal field theory properties.

Findings

Jordan cells indicate logarithmic CFT behavior

Criteria for Jordan cells within and between sectors are established

Connection between algebraic structures and conformal field theories is clarified

Abstract

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, T_N, is an element of the periodic Temperley-Lieb algebra EPTL_N(\beta, \alpha), where N is the number of sites on a section of the cylinder, and \beta = -(q+1/q) = 2 \cos \lambda and \alpha the weights of contractible and non-contractible loops. The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6\lambda^2/(\pi(\lambda-\pi)). The abstract element T_N acts naturally on (a sum of) spaces V_N^d, similar to those upon which the standard modules of the (classical) Temperley-Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a {\em twist parameter} v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of T_N both between sectors with distinct defect numbers and within a given sector.