Topological defects in Temperley-Lieb lattice models

par Jonathan Belletete (CY Cergy Paris Université)

vendredi 18 mars 2022, 09:00 → 10:00 Europe/Paris

112 (Braconnier)

The Temperley-Lieb lattice models are a family of statistical models based on random knots: one constructs a family of knots on a given two dimensional surface, typically a torus or a cylinder, using a fixed number of strings each crossing other strings a fixed number of time, then assign to each crossing a statistical weight using a set of local rules. It has been shown that these models reproduce the statistical properties of various physically relevant systems, like the XXZ spin chain and the Fibonacci anyon model.

In an effort to describe physical models with different types of boundary conditions, physicists introduced  lattice defects: certain strings are "defective" and the rules for assigning statistical weight to their crossings is different from the others. These different rules were established using numerical studies and various ad-hoc arguments, on a per-model basis.

In this talk, I will present a formal algebraic formulation of these models and a way of understanding these defects as functors acting on categories of modules. I will show how to use these functors to obtain actual proofs of physical results which had been obtained numerically, and then extend them to certain cases which cannot be tackled using the physicists approach. 

Based on 1811.02551 and 2003.11293 .