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

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.

💡 Research Summary

The paper investigates a special class of polynomial solutions of the quantum Knizhnik‑Zamolodchikov (qKZ) equations associated with the six‑vertex R‑matrix and establishes a direct link between these solutions and the ground‑state eigenvector of the antiferromagnetic XXZ spin‑½ chain at anisotropy Δ = −½. The authors begin by recalling the definition of the six‑vertex R‑matrix R(u) depending on a deformation parameter q and a spectral parameter u, and they formulate the qKZ equations for a vector‑valued function Φ(z₁,…,z_N) of N inhomogeneity variables. While the generic solution of the qKZ system is a complicated meromorphic function, the authors show that when q is a primitive third root of unity, q = e^{±2πi/3}, the system admits a family of polynomial solutions.

These polynomial solutions are constructed explicitly as multiple contour integrals. For an odd number of vertical lines N = 2n + 1 they introduce a set of indices {a₁,…,a_n}⊂{1,…,N} and define
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