The Q-operator for Root-of-Unity Symmetry in Six Vertex Model

TL;DR

This paper constructs an explicit Q-operator for the six-vertex model at roots of unity, revealing its symmetry properties and deriving related functional relations from the Bethe equation.

Contribution

It introduces a new explicit Q-operator incorporating $sl_2$-loop-algebra symmetry and derives comprehensive functional relations for the root-of-unity six-vertex model.

Findings

Q-operator explicitly constructed with $sl_2$-loop-algebra symmetry

Functional relations derived from Bethe equations

Illustrative calculations confirming the Q-operator's properties

Abstract

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the $Q$-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.