We study SU (3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU (3)-invariant Heisenberg chain.

We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian 𝒴(gl3) on the Bethe vectors are considered. These actions are relevant for the calculation of correlation functions and form factors of local operators of the underlying quantum models.

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed 'Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime −1 < q < 0, level one infinite dimensional representation (q−vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo et al. are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using q−bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for q generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the q−Onsager algebra (generic non-diagonal case) or the augmented q−Onsager algebra (generic diagonal case). MSC: 81R50; 81R10; 81U15.Keywords: XXZ open spin chain; q−Onsager algebra; q−vertex operators; Thermodynamic limit Potts models [PeAMT, Dav, vGR] are explicit counterexamples of this idea (see also [Ar, AS]).

We present in a unified and detailed way the nested Bethe ansatz for closed spin chains based on or (super)algebras, with arbitrary representations (i.e. ‘spins’) on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on or and for some particular super-spin chains. We give the Bethe equations and the form of the Bethe vectors. The case of gl(2|1), gl(2|2) and gl(4|4) superalgebras (that are related to AdS/CFT correspondence) is also detailed.

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.

The modified algebraic Bethe ansatz, introduced by Crampé and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-1 2 chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized 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. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-1 2 chain on the segment with two upper triangular boundaries. MSC: 82B23; 81R12

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.

We study SU (3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain determinant representations for form factors of diagonal entries of the monodromy matrix. This representation can be used for the calculation of form factors and correlation functions of the XXX SU (3)-invariant Heisenberg chain.