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Hard core bosons in a large class of one or two dimensional flat band systemshave an upper critical density, below which the ground states can be describedcompletely. At the critical density, the ground states are Wigner crystals. Ifone adds a particle to the system at the critical density, the ground state andthe low lying multi particle states of the system can be described as a Wignercrystal with an additional pair of particles. The energy band for the pair isseparated from the rest of the multi-particle spectrum. The proofs use aGerschgorin type of argument for block diagonally dominant matrices. In certainone-dimensional or tree-like structures one can show that the pair islocalised, for example in the chequerboard chain. For this one-dimensionalsystem with periodic boundary condition the energy band for the pair is flat,the pair is localised.

Pair formation of hard core bosons in flat band systems