I will start by introducing a scheme for quantum communication using an unmodulated and unmeasured spin chain. It presents an alternative to converting between static and flying qubits in order to connect up distinct quantum processors. I present some approaches to accomplish perfect quantum communication through a spin chain despite the dispersion of quantum information in the chain. I also discuss the accomplishment of gates between distant spins thorough a spin chain. Apart from transfer, a chain can also be used to simultaneously generate and distribute a maximally entangled state between distant sites, as I illustrate through a chain of coupled qutrits.

In recent years it has been realized that Hermiticity of the Hamiltonian is not a necessary condition for the reality of its eigenvalues. In many cases it is sufficient for the Hamiltonian to have an unbroken PT symmetry, but the natural Hilbert-space metric of such theories is not positive definite, which raises severe problems as to their physical interpretation. However, it turns out to be possible to construct another,dynamically-determined, metric eta=exp(iQ) that is indeed positive definite. For the most part Q can only be constructed perturbatively,but there are some cases where it is known exactly. With the aid of Q one can construct an equivalent Hermitian Hamiltonian, which, however, may be rather complicated and non-local.

We analyze the continuum limit of the integrable 3 3 times bar(3) times 3 times bar(3)... sl(2/1) superspin chain. We discover profoundly new features, including a continuous spectrum of conformal weights, whose numerical evidence is infinite degeneracies of the scaled gaps in the thermodynamic limit. This indicates that the corresponding conformal field theory has a non compact target space (even though our lattice model involves only finite dimensional representations). We argue that our results are compatible with this theory being the level k=3D1, SU(2/1) WZW model (whose precise definition requires some care). In doing so, we establish several new results for this model. With regard to potential applications to the spin quantum Hall effect, we conclude that the continuum limit of the 3 times bar (3) times 3 times bar(3)... sl(2,1) integrable superspin chain is not the same as (and is in fact very different from) the continuum limit of the corresponding chain with two-superspin interactions only, which is known to be a model for the spin quantum Hall effect. The study of possible RG flows between the two theories is left for further study.

In this talk I will present a short review on the algebraic Bethe ansatz technique and on the recently found solution of the so-called inverse problem. I will show how this solution provides a means for the explicit and exact computation of correlation functions in spin chains and summarize some of the many results obtained in this direction by members of the theory group at the ENS-Lyon in the last years. Finally I will present some work still in progress which intends the generalization of these techniques to the case of spin chains in the presence of impurities.