Cold bosons in optical lattices: a tutorial for Exact Diagonalization
  Exact diagonalization techniques are a powerful method for studying many-body problems. Here, we apply this method to systems of few bosons in an optical lattice, and use it to demonstrate the emergence of interesting quantum phenomena like fragmentation and coherence. Starting with a standard Bose-Hubbard Hamiltonian, we first revise the characterization of the superfluid to Mott insulator transitions. We then consider an inhomogeneous lattice, where one potential minimum is made much deeper than the others. The Mott insulator phase due to repulsive on-site interactions then competes with the trapping of all atoms in the deep potential. Finally, we turn our attention to attractively interacting systems, and discuss the appearance of strongly correlated phases and the onset of localization for a slightly biased lattice. The article is intended to serve as a tutorial for exact diagonalization of Bose-Hubbard models. 
David Ravent\'os, Tobias Gra{\ss}, Maciej Lewenstein, Bruno   Juli\'a-D\'iaz
cond-mat.quant-gas