  We investigate the ground state properties of a newly discovered phase of one
dimensional lattice bosons with extended interactions (see E. G. Dalla Torre et
al., Phys. Rev. Lett. \textbf{97}, 260401 (2006)). The new phase, termed the
Haldane Insulator (HI) in analogy with the gapped phase of spin-1 chains, is
characterized by a non local order parameter, which can only be written as an
infinite string in terms of the bosonic densities. We show that the string
order can nevertheless be probed with physical fields that couple locally, via
the effect those fields have on the quantum phase transitions separating the
exotic phase from the conventional Mott and density wave phases. Using a field
theoretical analysis we show that a perturbation which breaks lattice inversion
symmetry gaps the critical point separating the Mott and Haldane phases and
eliminates the sharp distinction between them. This is remarkable given that
neither of these phases involves broken inversion symmetry. We also investigate
the evolution of the phase diagram with the tunable coupling between parallel
chains in an optical lattice setup. We find that inter-chain tunneling destroys
the direct phase transition between the Mott and Haldane insulators by
establishing an intermediate superfluid phase. On the other hand coupling the
chains only by weak repulsive interactions does not modify the structure of the
phase diagram. The theoretical predictions are confirmed with numerical
calculations using the Density Matrix Renormalization Group (DMRG).
