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The Haldane Insulator is a gapped phase characterized by an exotic non-localorder parameter. The parameter regimes at which it might exist, and how itcompetes with alternate types of order, such as supersolid order, are stillincompletely understood. Using the Stochastic Green Function (SGF) quantumMonte Carlo (QMC) and the Density Matrix Renormalization Group (DMRG), we studynumerically the ground state phase diagram of the one-dimensional bosonicHubbard model (BHM) with contact and near neighbor repulsive interactions. Weshow 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 thatthe HI exists only at the tip of the unit filling CDW lobe and that there is astable SS phase over a very wide range of parameters.

Competing Supersolid and Haldane Insulator phases in the extended one-dimensional bosonic Hubbard model