Finite size lattice results for the two-boundary Temperley--Lieb loop model

TL;DR

This thesis analyzes the two-boundary Temperley--Lieb loop model on an infinite lattice, explicitly calculating eigenvectors, normalizations, and correlation functions, revealing connections to symplectic characters and conformal invariance in critical systems.

Contribution

It provides explicit solutions for the eigenvector and normalization of the TL(1) model with two boundaries, linking these to symplectic characters and deriving exact correlation functions.

Findings

Eigenvector solutions satisfy the q-deformed Knizhnik--Zamolodchikov equation.

Normalization is expressed as a product of four symplectic Weyl characters.

Exact correlation functions relate to symplectic characters and parafermionic observables.

Abstract

This thesis is concerned with aspects of the integrable Temperley--Lieb loop (TL($n$)) model on a vertically infinite lattice with two non-trivial boundaries. When $n=1$ the ground state eigenvector of the transfer matrix of this model can be interpreted as a probability distribution of the possible states of the system. Because of special properties the transfer matrix has at $n=1$, we can show that the eigenvector is a solution of the q-deformed Knizhnik--Zamolodchikov equation, and we use this fact to explicitly calculate some of the components of the eigenvector. In addition, recursive properties of the transfer matrix allow us to compute the normalisation of the eigenvector, and show that it is the product of four Weyl characters of the symplectic group. The boundary condition of this model lends itself to calculations relating to horizontal percolation. One of these calculations is a type of correlation function that can be interpreted as the density of percolation cluster crossings between the two boundaries of the lattice. It is an example of a class of parafermionic observables recently introduced in an attempt to rigorously prove conformal invariance of the scaling limit of critical two-dimensional lattice models. We derive an exact expression for this correlation function, and find that it can be expressed in terms of the same symplectic characters as the normalisation. In order to better understand these solutions, we use Sklyanin's scheme to perform separation of variables on the symplectic character, transforming the multivariate character into a product of single variable polynomials. Analysing the asymptotics of these polynomials will lead, via the inverse transformation, to the asymptotic limit of the symplectic character, and thus to the asymptotic limit of the ground state normalisation and correlation function of the loop model.