Phase transitions in three-dimensional bosonic systems in optical lattices

We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper er al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al, our finding indicates no superfluid state with zero condensate fraction.