Mean-field approaches to the Bose-Hubbard model with three-body local   interaction
  The zero temperature properties of the generalized Bose-Hubbard model including three-body interactions are studied on a mean-field level. We obtain analytical results using the so-called perturbative mean-field method and more detailed numerical results using the Gutzwiller product state variational Ansatz. These two approaches yield equivalent results which compare well on a qualitative level with recent exact results obtained in the literature. 
Tomasz Sowi\'nski and Ravindra W. Chhajlany
cond-mat.quant-gas