One-dimensional extended Bose-Hubbard model with a confining potential:   a DMRG analysis
  The extended Bose-Hubbard model in a quadratic trap potential is studied using a finite-size density-matrix renormalization group method (DMRG). We compute the boson density profiles, the local compressibility and the hopping correlation functions. We observed the phase separation induced by the trap in all the quantities studied and conclude that the local density approximation is valid in the extended Bose-Hubbard model. From the plateaus obtained in the local compressibility it was possible to obtain the phase diagram of the homogeneous system which is in agreement with previous results. 
Laura Urba, Emil Lundh, Anders Rosengren