Influence of trapping potentials on the phase diagram of bosonic atoms in optical lattices

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We study the effect of external trapping potentials on the phase diagram of bosonic atoms in optical lattices. We introduce a generalized Bose-Hubbard Hamiltonian that includes the structure of the energy levels of the trapping potential, and show that these levels are in general populated both at finite and zero temperature. We characterize the properties of the superfluid transition for this situation and compare them with those of the standard Bose-Hubbard description. We briefly discuss similar behaviors for fermionic systems.