# Off-diagonal Bethe Ansatz on the $so(5)$ spin chain

## TL;DR

This paper develops an off-diagonal Bethe Ansatz approach to solve the $so(5)$ quantum spin chain with various boundary conditions, deriving new spectral relations and extending the method to broader classes of models.

## Contribution

It introduces a novel inhomogeneous $T-Q$ relation for the $so(5)$ chain with non-diagonal boundaries and generalizes the approach to $so(2n+1)$ chains.

## Key findings

- Derived operator product identities for the $so(5)$ chain.
- Constructed a new inhomogeneous $T-Q$ relation for non-diagonal boundaries.
- Extended the method to $so(2n+1)$ quantum integrable spin chains.

## Abstract

The $so(5)$ (i.e., $B_2$) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in \cite{NYReshetikhin1}, while for the non-diagonal boundary case, a new inhomogeneous $T-Q$ relation is constructed. The present method can be directly generalized to deal with the $so(2n+1)$ (i.e., $B_n$) quantum integrable spin chains with general boundaries.