  We explain how to compute correlation functions at zero temperature within
the framework of the quantum version of the Separation of Variables (SoV) in
the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted
(quasi-periodic) boundary conditions. We first detail all steps of our method
in the case of anti-periodic boundary conditions. The model can be solved in
the SoV framework by introducing inhomogeneity parameters. The action of local
operators on the eigenstates are then naturally expressed in terms of multiple
sums over these inhomogeneity parameters. We explain how to transform these
sums over inhomogeneity parameters into multiple contour integrals. Evaluating
these multiple integrals by the residues of the poles outside the integration
contours, we rewrite this action as a sum involving the roots of the Baxter
polynomial plus a contribution of the poles at infinity. We show that the
contribution of the poles at infinity vanishes in the thermodynamic limit, and
that we recover in this limit for the zero-temperature correlation functions
the multiple integral representation that had been previously obtained through
the study of the periodic case by Bethe Ansatz or through the study of the
infinite volume model by the q-vertex operator approach. We finally show that
the method can easily be generalized to the case of a more general non-diagonal
twist: the corresponding weights of the different terms for the correlation
functions in finite volume are then modified, but we recover in the
thermodynamic limit the same multiple integral representation than in the
periodic or anti-periodic case, hence proving the independence of the
thermodynamic limit of the correlation functions with respect to the particular
form of the boundary twist.
