A new recursion formula is presented for the correlation functions of the integrable spin
1/2 XXX chain with inhomogeneity. It links the correlators involving
n consecutive lattice sites to those with
n−1 and
n−2 sites. In a series of papers by V. Korepin and two of the present authors, it was discovered that the correlators have a certain specific structure as functions of the inhomogeneity parameters. The formula mentioned above makes it possible to prove this structure directly, as well as to obtain an exact description of the rational functions that were left undetermined in the earlier work.
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.