Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz

Abstract

Abstract. We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

Slavnov and Gaudin–Korepin formulas for models withoutU(1) symmetry: the XXX chain on the segment

“…In the case of the Heisenberg spin chain on the segment, this is a consequence of the breaking of the U (1) symmetry by off-diagonal boundaries. Many approaches have been developed to handle this problem, including generalizations of the Bethe ansatz to consider special non-diagonal boundaries, see for instance [9,26,4,29,1] and references therein, the SoV method [15,14,28,13,21], the functional method [16], the q-Onsager approach [8] and the non-polynomial solution from the homogeneous Baxter T-Q relation [25].Recently, the ABA has been generalized to include models with general boundary couplings [3,5,11,6,2]. The modified algebraic Bethe ansatz (MABA) has a distinct feature: the creation operator used to construct the eigenstates has an off-shell structure which leads to an inhomogeneous term in the eigenvalues and in the Bethe equations of the model.…”

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

“…For the spin-1 2 case, the Bethe states corresponding to the T − Q relation (4.16) were constructed in [68] (also conjectured in [67]) with the helps of the SoV basis proposed in [47][48][49][50]. It is interesting that the resulting Bethe states directly induces the homogeneous limits of the SoV states constructed in [47][48][49][50].…”

“…Since all the eigenvalues of the transfer matrix belong to the solution set of (3.12)-(3.17), we conclude that in the spin-1 2 case our functional relations characterize the spectrum completely. It is remarked that the corresponding Bethe states were given in [67,68]. These Bethe states have well-defined homogeneous limits and allows one to study the corresponding homogeneous open chain directly.…”

“…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.…”

“…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.…”