Off-diagonal Bethe ansatz solutions of the anisotropic spin-12 chains with arbitrary boundary fields

The anisotropic spin-12 chains with arbitrary boundary fields are diagonalized with the off-diagonal Bethe ansatz method. Based on the properties of the R-matrix and the K-matrices, an operator product identity of the transfer matrix is constructed at some special points of the spectral parameter. Combining with the asymptotic behavior (for XXZ case) or the quasi-periodicity properties (for XYZ case) of the transfer matrix, the extended T–Q ansatzes and the corresponding Bethe ansatz equations are derived.