Quantum Phases of Long Range 1-D Bose-Hubbard Model: Field Theoretic and   DMRG Study at Different Densities
  We use Abelian Bosonization and density matrix renormalization group method to study the effect of density on quantum phases of long range 1-D Bose-Hubbard model. We predict the existence of supersolid phase and also other quantum phases for this system. We have analyzed the role of long range interaction parameter on solitonic phase near half filling. We discuss the effect of dimerization in nearest neighbor hopping and interaction terms on the plateau phase at the half filling. 
Manoranjan Kumar, Sujit Sarkar and S. Ramasesha