  In the one-dimensional Bose-Hubbard model with on-site and nearest neighbor
interactions, a gapped phase characterized by an exotic non-local order
parameter emerges, the Haldane insulator. Bose-Hubbard models with
cavity-mediated global range interactions display phase diagrams, which are
very similar to those with nearest neighbor repulsive interactions, but the
Haldane phase remains elusive there. Here we study the one-dimensional
Bose-Hubbard model with nearest-neighbor and cavity-mediated global-range
interactions and scrutinize the existence of a Haldane Insulator phase. With
the help of extensive quantum Monte-Carlo simulations we find that in the
Bose-Hubbard model with only cavity-mediated global-range interactions no
Haldane phase exists. For a combination of both interactions, the Haldane
Insulator phase shrinks rapidly with increasing strength of the cavity-mediated
global-range interactions. Thus, in spite of the otherwise very similar
behavior the mean-field like cavity-mediated interactions strongly suppress the
non-local order favored by nearest neighbor repulsion in some regions of the
phase diagram.
