  Bosons interacting repulsively on a lattice with a flat lowest band energy
dispersion may, at sufficiently small filling factors, enter into a
Wigner-crystal-like phase. This phase is a consequence of the dispersionless
nature of the system, which in turn implies the occurrence of single-particle
localised eigenstates. We investigate one of these systems - the sawtooth
lattice - filled with strongly repulsive bosons at filling factors
infinitesimally above the critical point where the crystal phase is no longer
the ground state. We find, the the hard-core limit, that the crystal retains
its structure in all but one of its cells, where it is broken. The broken cell
corresponds to an exotic kind of repulsively bound state, which becomes
delocalised. We investigate the excitation spectrum of the system analytically
and find that the bound state behaves as a single particle hopping on an
effective lattice with reduced periodicity, and is therefore gapless. Thus, the
addition of a single particle to a flat-banded system at critical filling is
found to be enough to make kinetic behaviour manifest.
