# Exact solution of the $sp(4)$ integrable spin chain with generic boundaries

## TL;DR

This paper extends the off-diagonal Bethe ansatz method to solve the $sp(4)$ integrable spin chain with generic boundaries, providing explicit eigenvalues and demonstrating the method's generalizability to higher rank models.

## Contribution

It develops a generalized off-diagonal Bethe ansatz approach for the $sp(4)$ model, deriving complete eigenvalues including for off-diagonal boundary conditions, and shows how to extend it to $sp(2n)$ models.

## Key findings

- Derived complete eigenvalues for $sp(4)$ spin chain with various boundary conditions.
- Obtained inhomogeneous $T-Q$ relations for off-diagonal boundary cases.
- Demonstrated the method's applicability to higher rank $sp(2n)$ models.

## Abstract

The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the $sp(4)$ (or $C_2$) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the same $T-Q$ relations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneous $T-Q$ relations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to generic $sp(2n)$ (i.e., $C_n$) integrable model.