Phase diagram of spin-1 bosons on one-dimensional lattices
  Spinor Bose condensates loaded in optical lattices have a rich phase diagram characterized by different magnetic order. Here we apply the Density Matrix Renormalization Group to accurately determine the phase diagram for spin-1 bosons loaded on a one-dimensional lattice. The Mott lobes present an even or odd asymmetry associated to the boson filling. We show that for odd fillings the insulating phase is always in a dimerized state. The results obtained in this work are also relevant for the determination of the ground state phase diagram of the S=1 Heisenberg model with biquadratic interaction. 
Matteo Rizzi, Davide Rossini, Gabriele De Chiara, Simone Montangero,   and Rosario Fazio