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

A Jastrow wave function for the spin-1 Heisenberg chain: the string order revealed by the mapping to the classical Coulomb gas