Hidden order in bosonic gases confined in one dimensional optical lattices

We analyze the effective Hamiltonian arising from a suitable power series expansion of the overlap integrals of Wannier functions for confined bosonic atoms in a one-dimensional (1D) optical lattice. For certain constraints between the coupling constants, we construct an explicit relationship between such an effective bosonic Hamiltonian and the integrable spin-S anisotropic Heisenberg model. The former results are therefore integrable by construction. The field theory is governed by an anisotropic nonlinear -model with singlet and triplet massive excitations; this result holds also in the generic non-integrable cases. The criticality of the bosonic system is investigated. The schematic phase diagram is drawn. Our study sheds light on the hidden symmetry of the Haldane type for 1D bosons.