Fusion Operators in the Generalized $\tau^{(2)}$-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin

TL;DR

This paper constructs fusion operators in the generalized $ au^{(2)}$-model, verifies their relations, and demonstrates the $sl_2$-loop-algebra symmetry in higher-spin XXZ spin chains at roots of unity, linking to the chiral Potts model.

Contribution

It introduces fusion operators in the generalized $ au^{(2)}$-model and establishes the $sl_2$-loop-algebra symmetry for higher-spin XXZ chains at roots of unity.

Findings

Fusion operators satisfy fusion relations with truncation identity.

$sl_2$-loop-algebra symmetry exists for root-of-unity higher-spin XXZ chains.

Explicit identification of evaluation parameters via Fabricius-McCoy current.

Abstract

We construct the fusion operators in the generalized $\tau^{(2)}$-model using the fused $L$-operators, and verify the fusion relations with the truncation identity. The algebraic Bethe ansatz discussion is conducted on two special classes of $\tau^{(2)}$ which include the superintegrable chiral Potts model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and demonstrate that the $sl_2$-loop-algebra symmetry exists for the root-of-unity XXZ spin chain with a higher spin, where the evaluation parameters for the symmetry algebra are identified by the explicit Fabricius-McCoy current for the Bethe states. Parallels are also drawn to the comparison with the superintegrable chiral Potts model.