The Jordan structure of two-dimensional loop models
Abstract
We show how to use the link representation of the transfer matrix DN of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter \beta=2
\cos (\pi (1-a/b)), a,b\in {N} and, more specifically, partition functions of the corresponding Q-Potts spin models, with Q = β2. The braid limit of DN is shown to be a central element FN(β) of the Temperley-Lieb algebra TLN(β), its eigenvalues are determined and, for generic β, a basis of its eigenvectors is constructed using the Wenzl-Jones 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 FN and DN are shown to have Jordan blocks between sectors d and d' when d - d' < 2b and (d+d')/2 \equiv b-1 ~ {mod}\, 2b (d > d'). When a and b do not satisfy the previous constraint, DN is diagonalizable.- Publication:
-
Journal of Statistical Mechanics: Theory and Experiment
- Pub Date:
- April 2011
- DOI:
- 10.1088/1742-5468/2011/04/P04007
- arXiv:
- arXiv:1101.2885
- Bibcode:
- 2011JSMTE..04..007M
- Keywords:
-
- Mathematical Physics;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 55 pages