Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries

TL;DR

This paper constructs a three-parameter family of auxiliary matrices for the six-vertex model at roots of unity, revealing their role in solving the model, their symmetry-breaking effects, and their geometric interpretation via quantum group theory.

Contribution

It introduces a new family of auxiliary matrices based on quantum group intertwiners, providing a geometric understanding of symmetries and degeneracies at roots of unity.

Findings

Constructed a three-parameter family of auxiliary matrices.

Derived a functional relation enabling eigenvalue problem solutions.

Linked auxiliary matrix parameters to geometric structures and symmetries.

Abstract

The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$ a three parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged center of the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a three-dimensional complex hypersurface which remains invariant under the action of an infinite-dimensional group of analytic transformations, called the quantum coadjoint action.