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Recent experiments revealed the importance of higher-band effects for theMott insulator (MI) -- superfluid transition (SF) of ultracold bosonic atoms ormixtures of bosons and fermions in deep optical lattices [Best \emph{et al.},PRL \textbf{102}, 030408 (2009); Will \emph{et al.}, Nature \textbf{465}, 197(2010)]. In the present work, we derive an effective lowest-band Hamiltonian in3D that generalizes the standard Bose-Fermi Hubbard model taking these effectsas well as nonlinear corrections of the tunneling amplitudes mediated byinterspecies interactions into account. It is shown that a correct descriptionof the lattice states in terms of the bare-lattice Wannier functions ratherthan approximations such as harmonic oscillator states is essential. Incontrast to self-consistent approaches based on effective Wannier functions ourapproach provides a quantitative explanation of the observed reduction of thesuperfluid phase for repulsive interspecies interactions.

Multi-band and nonlinear hopping corrections to the 3D Bose-Fermi Hubbard model