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(β, α), where N is the number of sites on a section of the cylinder, and β= -(q+1/q) = 2 \cos λand αthe 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=1-6λ^2/(π(λ-π)). 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.

42 pages

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• Modular invariant partition function of critical dense polymers
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model
• Two-point boundary correlation functions of dense loop models
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Fusion of irreducible modules in the periodic Temperley--Lieb algebra
• Spin chains as modules over the affine Temperley-Lieb algebra
• Cyclic representations of the periodic Temperley Lieb algebra, complex Virasoro representations and stochastic processes