  Using density matrix renormalization group numerical calculations, we study
the phase diagram of the half filled Bose-Hubbard system in the sawtooth
lattice with strong frustration in the kinetic energy term. We focus in
particular on values of the hopping terms which produce a flat band and show
that, in the presence of contact and near neighbor repulsion, three phases
exist: Mott insulator (MI), charge density wave (CDW), and the topological
Haldane insulating (HI) phase which displays edge states and particle imbalance
between the two ends of the system. We find that, even though the entanglement
spectrum in the Haldane phase is not doubly degenerate, it is in excellent
agreement with the entanglement spectrum of the Affleck-Kennedy-Lieb-Tasaki
(AKLT) state built in the Wannier basis associated with the flat band. This
emphasizes that the absence of degeneracy in the entanglement spectrum is not
necessarily a signature of a non-topological phase, but rather that the
(hidden) protecting symmetry involves non-local states. Finally, we also show
that the HI phase is stable against small departure from flatness of the band
but is destroyed for larger ones.
