Zoom seminar in the Auditorium
The critical behaviour of spin clusters and their interfaces in two dimensional local lattice models where spins take n values is well understood in the n=2, Ising case. Much less is known for n>2. Despite numerical evidence that these geometrical objects host a variety of critical exponents, there is no satisfactory approach to compute them exactly. In the Ising case, one is able to derive critical exponents through a mapping to O(N=1) loop models. Transfer matrices of such models of non intersecting loops are based on the (dilute) Temperley-Lieb algebra. The relation of this algebra to U_q(sl_2) makes it possible to reformulate the model as a vertex model which is known to exhibit integrable points. In this talk, we will present lattice models whose configuration are subgraphs of the lattice, called webs. Webs are higher rank analogues of loops and can be seen as invariants of quantum groups in tensor products of fundamental representations. We focus on the rank 2 case and describe how such web models can be mapped to spin models with n>2 such that webs are mapped to interfaces of spin clusters. We then show that the web models exhibit integrable points for some tuning of their parameters. The main difficulty in deriving that is that the local space of states is a direct sum of irreducible representations.
