Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2

TL;DR

This paper derives integral formulas for polynomial solutions of the qKZ equations, demonstrating their role as ground states of the XXZ spin chain at Delta = -1/2 and establishing links with Temperley-Lieb loop models.

Contribution

It provides explicit integral representations for solutions of the qKZ equations at specific parameters and proves their connection to the ground state of the XXZ spin chain and loop models.

Findings

Integral formulas for polynomial solutions of qKZ equations.

Ground state eigenvector of XXZ spin chain at Delta = -1/2.

Relation between XXZ ground states and Temperley-Lieb loop models.

Abstract

Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new statement relating the ground state components of XXZ spin chains and Temperley-Lieb loop models is formulated and proved.