  We consider the problem of constructing the stationary state following a
quantum quench, using the exact overlaps for finite size integrable models. We
focus on the isotropic Heisenberg spin chain with initial state N\'eel or
Majumdar-Ghosh (dimer), although the proposed approach is valid for an
arbitrary integrable model. We consider only eigenstates which do not contain
zero-momentum strings because the latter are affected by fictitious
singularities that are very difficult to take into account. We show that the
fraction of eigenstates that do not contain zero-momentum strings is vanishing
in the thermodynamic limit. Consequently, restricting to this part of the
Hilbert space leads to vanishing expectation values of local observables.
However, it is possible to reconstruct the asymptotic values by properly
reweighting the expectations in the considered subspace, at the price of
introducing finite-size corrections. We also develop a Monte Carlo sampling of
the Hilbert space which allows us to study larger systems. We accurately
reconstruct the expectation values of the conserved charged and the root
distributions in the stationary state, which turn out to match the exact
thermodynamic results. The proposed method can be implemented even in cases in
which an analytic thermodynamic solution is not obtainable.
