  We carry out an extensive study of the phase diagram of the extended Bose
Hubbard model, with a mean filling of one boson per site, in one dimension by
using the density matrix renormalization group and show that it can have
Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane-insulator
(HI) phases depending on the precise value of filling and how edge states are
handled. We show that the critical exponents and central charge for the HI-DW,
MI-HI and SF-MI transitions are consistent with those for models in the
two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT)
universality classes, respectively; and we suggest that the SF-HI transition
may be more exotic than a simple BKT transition.
