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.

We investigate the thermodynamic limit of the one-dimensional ferromagnetic XXZ model with twisted (or antiperiodic) boundary conditions. It is shown that the distribution of the Bethe roots of the inhomogeneous Bethe ansatz equations (BAEs) for the ground state as well as for the low-lying excited states satisfy the string hypothesis, although the inhomogeneous BAEs are not in the standard product form which has made the study of the corresponding thermodynamic limit nontrivial. We also obtain the twisted boundary energy induced by the nontrivial twisted boundary conditions in the thermodynamic limit.