Form factors for local spin operators of the XXZ Heisenberg spin-1 2 finite chain are computed. Representation theory of Drinfel'd twists for the quantum affine algebra U q (ŝl 2 ) in finite dimensional modules is used to calculate scalar products of Bethe states (leading to Gaudin formula) and to solve the quantum inverse problem for local spin operators in the finite chain. Hence, we obtain the representation of the n-spin correlation functions in terms of expectation values (in ferromagnetic reference state) of the operator entries of the quantum monodromy matrix satisfying Yang-Baxter algebra. This leads to the direct calculation of the form factors of the XXZ Heisenberg spin-1 2 finite chain as determinants of usual functions of the parameters of the model. A two-point correlation function for adjacent sites is also derived using similar techniques. * On leave of absence from the St Petersburg branch of the Steklov Mathematical Institute,

Using the algebraic Bethe ansatz method, and the solution of the quantum inverse scattering problem for local spins, we obtain multiple integral representations of the n-point correlation functions of the XXZ Heisenberg spin-1 2 chain in a constant magnetic field. For zero magnetic field, this result agrees, in both the massless and massive (anti-ferromagnetic) regimes, with the one obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q (ŝl 2 ) together with the corner transfer matrix approach (massive regime).

We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin- 1/2 chain (with magnetic field) in the disordered regime as well as to the density–density correlation function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.

We consider the XXZ spin chain with diagonal boundary conditions in the framework of algebraic Bethe Ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk inverse problem, we calculate the elementary building blocks for the correlation functions. In the limit of half-infinite chain, they are obtained as multiple integrals of usual functions, similar to the case of periodic boundary conditions.