  We show that a two-body Jastrow wave function is able to capture the
ground-state properties of the $S=1$ antiferromagnetic Heisenberg chain with
the single-ion anisotropy term, in both the topological and trivial phases.
Here, the optimized Jastrow pseudo potential assumes a very simple form in
Fourier space, i.e., $v_{q} \approx 1/q^2$, which is able to give rise to a
finite string-order parameter in the topological regime. The results are
analysed by using an exact mapping from the quantum expectation values over the
variational state to the classical partition function of the one-dimensional
Coulomb gas of particles with charge $q=\pm 1$. Here, two phases are present at
low temperatures: the first one is a diluted gas of dipoles (bound states of
particles with opposite charges), which are randomly oriented (describing the
trivial phase); the other one is a dense liquid of dipoles, which are aligned
thanks to the residual dipole-dipole interactions (describing the topological
phase, with the finite string order being related to the dipole alignment). Our
results provide an insightful interpretation of the ground-state nature of the
spin-1 antiferromagnetic Heisenberg model.
