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

A. V. Razumov ¹, Yu. G. Stroganov ¹, Paul Zinn-Justin ²

Journal of Physics A Mathematical and Theoretical 40 (2007) 11827-11847

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.

• 1. Division of Theoretical Physics,
  Institut for High Energy Physics
• 2. Laboratoire de Physique Théorique et Hautes Energies (LPTHE),
  CNRS : UMR7589 – Université Paris VI – Pierre et Marie Curie – Université Paris VII – Paris Diderot