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We present first-principle numerical calculations for few particle solutionsof the attractive Bose-Hubbard model with periodic boundary conditions. We showthat the low-energy many-body states found by numerical diagonalization can bewritten as translational superposition states of compact composite systems ofparticles. These compact states break the translational symmetry of the problemand their center-of-mass and internal excitations offer simple explanations ofthe energy spectrum of the full model.

Relative and center-of-mass motion in the attractive Bose-Hubbard model