The Jordan structure of two-dimensional loop models

Abstract

We show how to use the link representation of the transfer matrixDN of loop models on the lattice to calculate partition functions, at criticality, of theFortuin–Kasteleyn model with various boundary conditions and parameter and, more specifically, partition functions of the correspondingQ-Potts spin models, withQ = β2. The braid limit ofDN is shown to be a centralelement FN(β) of theTemperley–Lieb algebra TLN(β), its eigenvalues are determined and, for genericβ, a basis of its eigenvectors is constructed using the Wenzl–Jones projector.With any element of this basis is associated a number of defectsd,0 ≤ d ≤ N, and the basis vectorswith the same d span a sector. Because components of these eigenvectors are singular when and , the link representations of FN and DN are shown to have Jordan blocks between sectorsd andd′ whend − d′ < 2b and (d > d′). Whena andb do not satisfy theprevious constraint, DN is diagonalizable.