Quantum phase transitions in optical lattices beyond Bogoliubov   approximation
  We study the quantum phase transition in optical lattices using ordinary Bose Hubbard Hamiltonian within two loop approximation in variational perturbation theory. We have shown that this approximation can reproduce superfluid Mott insulator transition in contrast to the simple Bogoliubov or Hartee - Fock - Popov approximations.   The superfluid fraction and ground state energy per particle vs input parameters of the model are studied. 
H. Kleinert, Z. Narzikulov and Abdulla Rakhimov
cond-mat.quant-gas