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

Abstract

The anisotropic spin-1 2 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 ansatzs and the corresponding Bethe ansatz equations are derived.

Exact spectrum of the spin-s Heisenberg chain with generic non-diagonal boundaries

“…In section 5, we summarize our results and give some discussions. In appendix A, we prove that each solution of our functional equations can be parameterized in terms of a variety of inhomogeneous T − Q relations 1 It should be emphasized that the Bethe-type eigenstates of the spin-1 2 XXX chain with generic boundaries had challenged for many years and were conjectured in [67] and derived in [68] very recently after the discovery of the inhomogeneous T − Q relation in [56][57][58][59]. Only together with the very inhomogeneous T −Q relation, the SoV state [47][48][49][50] might be transformed into a Bethe state which possesses a well-defined homogeneous limit.…”

“…Recently, based on the fundamental properties of the R-matrix and the K-matrices for quantum integrable models, a systematic method for solving the eigenvalue problem of integrable models with generic boundary conditions, i.e., the off-diagonal Bethe ansatz (ODBA) method was proposed in [56][57][58][59] and several long-standing models [56][57][58][59][60][61] were then solved. Subsequently, the nested-version of ODBA for the models associated with su(n) algebra [62], the application to the integrable models beyond A-type [63] and the thermodynamic analysis based on the ODBA solutions [64] were developed.…”

“…Especially, the eigenstate problem for such kind of models with generic inhomogeneity was first approached via the SoV method [47][48][49][50]. A set of Bethe states for was then constructed in [67] and a method for retrieving the Bethe states based on the inhomogeneous T − Q relation [56][57][58][59] and the SoV basis [47][48][49][50] was developed in [68]. The latter method allows one to reach the homogeneous limit of the SoV eigenstates 1 and provides a clear connection among the SoV approach, the algebraic Bethe Ansatz and the ODBA.…”

“…Therefore the relations (3.12)-(3.17) are believed to completely characterize the spectrum of the fundamental spin-( 1 2 , s) transfer matrix t ( 1 2 ,s) (u). For the s = 1 2 case, the relations (3.12)-(3.17) are reduced to those used in [56][57][58][59] to determine the spectrum of the corresponding transfer matrix. The eigenstates associated with each solution of the resulting relations were constructed in [47][48][49][50] in the framework of the SoV method.…”

“…In this particular case one can choose 1 = −1, 2 3 = −1 and therefore c = 0. The corresponding T −Q ansatz (4.27), under the similar analysis as that in [56][57][58][59], is naturally reduced to the conventional one [55] obtained by the algebraic Bethe ansatz.…”