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

We show that a two-body Jastrow wave function is able to capture the ground-state properties of the S = 1 Heisenberg chain with nearest-neighbor superexchange J and single-ion anisotropy term D, in both the topological and large-D phases (with D/J >= 0). Here, the optimized Jastrow pseudopotential assumes a very simple form in Fourier space, i.e., v(q) approximate to 1/q(2), which is able to give rise to a finite string-order parameter in the topological regime. The results are analyzed 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 = +/- 1. Here, two phases are present at low temperatures: the first one is a diluted gas of dipoles (bound neutral pairs of particles), which are randomly oriented (describing the large-D 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.