Exact solutions of the $C_n$ quantum spin chain

TL;DR

This paper derives exact solutions for the $C_n$ quantum spin chain using a generalized nested off-diagonal Bethe ansatz, applicable to both periodic and open boundary conditions, and demonstrates the method with the $C_3$ case.

Contribution

It introduces a generalized method for solving $C_n$ quantum spin chains with complex boundary conditions, extending the Bethe ansatz approach to high-rank Lie algebra models.

Findings

Eigenvalues of transfer matrices obtained

Homogeneous and inhomogeneous $T-Q$ relations derived

Method applicable to other Lie algebra-based models

Abstract

We study the exact solutions of quantum integrable model associated with the $C_n$ Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the $C_3$ as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous $T-Q$ relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the $C_n$ model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.