Evolution of the critical points with the density of bosons under local three-body interactions

We study the quantum phase transition in the one-dimensional Bose–Hubbard model with three-body local interactions for densities ρ=2ρ=2, 3, 4 and 5 by implementing the Density Matrix Renormalization Group method. This system shows two types of phases. One phase is incompressible (corresponding to the Mott insulator state) and the other compressible (the superfluid state). The critical points found correspond to a value of the hopping parameter for which the gap energy is equal to zero. We found that the critical point increases slowly as the density of the system increases.