Quantum phase transitions in optical lattices beyond the Bogoliubov approximation

We study the quantum phase transition from a superfluid to a Mott insulator in optical lattices using a Bose-Hubbard Hamiltonian. For this purpose we develop a field theoretical approach in terms of path integral formalism to calculate the second-order quantum corrections to the energy density as well as to the superfluid fraction in cubic optical lattices. Using the present approach, the condensate fraction and ground-state energy are calculated as functions of the s-wave scattering length. In contrast to the Bogoliubov model, which is, technically speaking, a one-loop approximation, we carry the calculation up to two loops and improve the result further by variational perturbation theory. The result suggests that the quantum phase transition exists.