  The Haldane Insulator is a gapped phase characterized by an exotic non-local
order parameter. The parameter regimes at which it might exist, and how it
competes with alternate types of order, such as supersolid order, are still
incompletely understood. Using the Stochastic Green Function (SGF) quantum
Monte Carlo (QMC) and the Density Matrix Renormalization Group (DMRG), we study
numerically the ground state phase diagram of the one-dimensional bosonic
Hubbard model (BHM) with contact and near neighbor repulsive interactions. We
show that, depending on the ratio of the near neighbor to contact interactions,
this model exhibits charge density waves (CDW), superfluid (SF), supersolid
(SS) and the recently identified Haldane insulating (HI) phases. We show that
the HI exists only at the tip of the unit filling CDW lobe and that there is a
stable SS phase over a very wide range of parameters.
