The Jordan structure of twodimensional loop models
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 FortuinKasteleyn model with various boundary conditions and parameter \beta=2
\cos (\pi (1a/b)), a,b\in {N} and, more specifically, partition functions of the corresponding QPotts spin models, with Q = β^{2}. The braid limit of D_{N} is shown to be a central element F_{N}(β) of the TemperleyLieb algebra TL_{N}(β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the WenzlJones projector. With any element of this basis is associated a number of defects d, 0 <= d <= N, and the basis vectors with the same d span a sector. Because components of these eigenvectors are singular when b \in
{Z}^* and a \in 2 {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 b1 ~ {mod}\, 2b (d > d'). When a and b do not satisfy the previous constraint, D_{N} is diagonalizable.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 April 2011
 DOI:
 10.1088/17425468/2011/04/P04007
 arXiv:
 arXiv:1101.2885
 Bibcode:
 2011JSMTE..04..007M
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics
 EPrint:
 55 pages