# Towards the solution of an integrable $D_2^{(2)}$ spin chain

## TL;DR

This paper proposes a Bethe ansatz solution for a specific branch of the $D_2^{(2)}$ spin chain, addressing a longstanding gap in understanding its eigenvalues, especially for the branch that resisted previous solutions.

## Contribution

It introduces a Bethe ansatz approach for the epsilon=1 branch of the $D_2^{(2)}$ spin chain, including a proposal for the missing eigenvalues, advancing the understanding of this integrable model.

## Key findings

- Proposed a Bethe ansatz solution for the epsilon=1 branch.
- Identified limitations of the solution, as it only covers eigenvalues with odd degeneracy.
- Suggested a method to find the missing eigenvalues.

## Abstract

Two branches of integrable open quantum-group invariant $D_{n+1}^{(2)}$ quantum spin chains are known. For one branch (epsilon=0), a complete Bethe ansatz solution has been proposed. However, the other branch (epsilon=1) has so far resisted solution. In an effort to address this problem, we consider here the simplest case n=1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.