Jordan cells of periodic loop models

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.