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

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... ase 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.