The spectral problem of the Heisenberg XXZ spin-1 2 chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to N, the length of the chain, and which satisfies a set of Bethe equations with an additional term.

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-1 2 chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix. MSC: 82B23; 81R12

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Abstract. We consider the XXX spin-1 2 Heisenberg chain on the circle with an arbitrary twist. We characterize its spectral problem using the modified algebraic Bethe anstaz and study the scalar product between the Bethe vector and its dual. We obtain modified Slavnov and Gaudin-Korepin formulas for the model. Thus we provide a first example of such formulas for quantum integrable models without U(1) symmetry characterized by an inhomogenous Baxter T-Q equation. The study of quantum integrable models with U(1) symmetry by the Bethe ansatz (BA) methods [4,11,34] provides exact solutions which found applications in a wide range of domains such as: statistical physics, condensed matter physics, high energy physics, mathematical physics, etc. One of the major accomplishments of the method has been the obtaining of form factors, for models related to gl 2 and gl 3 families of symmetry, in the compact form of determinants [8,9,21,22,24,26,31]. In particular, for models related to the gl 2 symmetry, the key results are the Slavnov [35] and the 20,25] formulas, which provide, respectively, the scalar product between an eigenstate and an arbitrary state and the norm of the eigenstates.In the case of models without U(1) symmetry, the usual BA techniques in general fail to provide a complete description of the spectrum 1 . Thus alternative methods have been developed, for instance, the separation of variables (SoV) [15,16,23,30,33], the commuting transfer matrices method [3], the functional method [18] or the q-Onsager approach [2]. Recently, key steps have been accomplished for the Bethe ansatz solution of such models. On the one hand, a new family of inhomogeneous Baxter T-Q equation to determine the eigenvalues has been proposed by the off-diagonal Bethe ansatz (ODBA) [12,37]. On the other hand, the construction of the off-shell Bethe vector has been done in the context of a modified algebraic Bethe ansatz (MABA) approach [1,5,6,10]. Let us remember that previous developments in the BA technique, in particular the obtainment of the eigenvectors of the XXX chain on the segment with upper-triangular boundaries [7], brought important insights to the MABA.1 In some cases, some gauge transformation can allow to apply the ABA, see for example the XYZ spin chain [4,36]. For the XXX case, that we consider here, the GL(2) symmetry allows one to restore the U(1) symmetry [14] and the usual ABA applies, provided that the twist is a non-singular matrix. Also, in the context of open XXZ spin chains, constraints on the parameters of the model allow one to apply the usual techniques, see [28].

We propose new families of solutions of the D (2) n+1 boundary Yang-Baxter equation.The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the D (2) n+1 Dynkin diagram, namely, U q (B n−p ) ⊗ U q (B p ), where p = 0, . . . , n. These transfer matrices also have a p ↔ n − p duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix.

We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K-matrices have an upper triangular form. We show that the main difficulty entailed by those form of the K-matrices is the construction of the excited states. However, it is possible to treat this problem with aid of an auxiliary transfer matrix and by means of a generalized creation operator.

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We consider the Temperley-Lieb (TL) open quantum spin chain with "free" boundary conditions associated with the spin-s representation of quantum-deformed sl(2). We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an offshell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.